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Review Article An Overview of Recent Advances in the Iterative Analysis of Coupled Models for Wave Propagation D. Soares Jr. 1 and L. Godinho 2 1 Structural Engineering Department, Federal University of Juiz de Fora, Cidade Universit´ aria, 36036-330 Juiz de Fora, MG, Brazil 2 CICC, Department of Civil Engineering, University of Coimbra, 3030-788 Coimbra, Portugal Correspondence should be addressed to D. Soares Jr.; delfi[email protected] Received 18 September 2013; Accepted 25 November 2013; Published 14 January 2014 Academic Editor: Daniel Dias-da-Costa Copyright © 2014 D. Soares Jr. and L. Godinho. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Wave propagation problems can be solved using a variety of methods. However, in many cases, the joint use of different numerical procedures to model different parts of the problem may be advisable and strategies to perform the coupling between them must be developed. Many works have been published on this subject, addressing the case of electromagnetic, acoustic, or elastic waves and making use of different strategies to perform this coupling. Both direct and iterative approaches can be used, and they may exhibit specific advantages and disadvantages. is work focuses on the use of iterative coupling schemes for the analysis of wave propagation problems, presenting an overview of the application of iterative procedures to perform the coupling between different methods. Both frequency- and time-domain analyses are addressed, and problems involving acoustic, mechanical, and electromagnetic wave propagation problems are illustrated. 1. Introduction e analysis of wave propagation, either involving electro- magnetic, acoustic, or elastic waves, has been widely studied by researchers using different strategies and methodologies, as can be seen, for example, in [110], among many others. In many cases, the interaction between different types of media, such as fluid-solid or soil-structure interaction problems, poses significant challenges that can hardly be tackled by means of a single numerical method, requiring the joint use of different procedures to model different parts of the problem. Indeed, taking into consideration the specificities and partic- ular features of distinct numerical methods, their combined use, as coupled or hybrid models, has been proposed by many authors, in order to explore the individual advantages of each technique. In acoustic and elastodynamic problems, coupled models, including, for example, the joint use of the boundary element method (BEM) and the method of fundamental solutions (MFS) [11] or of the BEM and the meshless Kansa’s method [12], have been successfully applied. Similarly, when mod- elling dynamic fluid-structure and soil-structure interac- tions, wave propagation in elastic media with heterogeneities, or the transmission of ground-borne vibration, coupled models using the finite element method (FEM) and the BEM have been extensively documented in the literature [1319], mostly using the FEM to model the structure and the BEM to model the hosting infinite or semiinfinite medium. Although these approaches can be quite useful in addressing many engineering problems, they mostly corre- spond to standard direct coupling methodologies and thus exhibit well-known limitations. Indeed, directly coupling distinct methods involves assembling a single system matrix, accounting for the contributions of each method and for the required coupling interface conditions, which frequently becomes poorly conditioned due to the different nature of the methods. Since this system is formed from the contributions of distinct methods, it is also usually not possible to make use of their individual advantages in terms of optimized solvers or memory storage (e.g., in BEM-FEM the final system will no longer be banded and symmetric, etc.). In addition to this limitation, by forming a single system of equations, a very large problem usually arises, leading to increased computational efforts and thus to a loss of performance. All these limitations have justified the appearance of itera- tive algorithms to obtain accurate solutions in a more efficient Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 426283, 21 pages http://dx.doi.org/10.1155/2014/426283
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Page 1: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Review ArticleAn Overview of Recent Advances in the Iterative Analysis ofCoupled Models for Wave Propagation

D Soares Jr1 and L Godinho2

1 Structural Engineering Department Federal University of Juiz de Fora Cidade Universitaria 36036-330 Juiz de Fora MG Brazil2 CICC Department of Civil Engineering University of Coimbra 3030-788 Coimbra Portugal

Correspondence should be addressed to D Soares Jr delfimsoaresufjfedubr

Received 18 September 2013 Accepted 25 November 2013 Published 14 January 2014

Academic Editor Daniel Dias-da-Costa

Copyright copy 2014 D Soares Jr and L GodinhoThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Wave propagation problems can be solved using a variety of methods However in many cases the joint use of different numericalprocedures to model different parts of the problem may be advisable and strategies to perform the coupling between them mustbe developed Many works have been published on this subject addressing the case of electromagnetic acoustic or elastic wavesand making use of different strategies to perform this coupling Both direct and iterative approaches can be used and they mayexhibit specific advantages and disadvantages This work focuses on the use of iterative coupling schemes for the analysis ofwave propagation problems presenting an overview of the application of iterative procedures to perform the coupling betweendifferent methods Both frequency- and time-domain analyses are addressed and problems involving acoustic mechanical andelectromagnetic wave propagation problems are illustrated

1 Introduction

The analysis of wave propagation either involving electro-magnetic acoustic or elastic waves has been widely studiedby researchers using different strategies and methodologiesas can be seen for example in [1ndash10] among many others Inmany cases the interaction between different types of mediasuch as fluid-solid or soil-structure interaction problemsposes significant challenges that can hardly be tackled bymeans of a single numericalmethod requiring the joint use ofdifferent procedures to model different parts of the problemIndeed taking into consideration the specificities and partic-ular features of distinct numerical methods their combineduse as coupled or hybridmodels has been proposed bymanyauthors in order to explore the individual advantages of eachtechnique

In acoustic and elastodynamic problems coupledmodelsincluding for example the joint use of the boundary elementmethod (BEM) and the method of fundamental solutions(MFS) [11] or of the BEM and the meshless Kansarsquos method[12] have been successfully applied Similarly when mod-elling dynamic fluid-structure and soil-structure interac-tions wave propagation in elasticmedia with heterogeneities

or the transmission of ground-borne vibration coupledmodels using the finite element method (FEM) and theBEM have been extensively documented in the literature[13ndash19] mostly using the FEM to model the structure andthe BEM to model the hosting infinite or semiinfinitemedium Although these approaches can be quite useful inaddressing many engineering problems they mostly corre-spond to standard direct coupling methodologies and thusexhibit well-known limitations Indeed directly couplingdistinct methods involves assembling a single system matrixaccounting for the contributions of each method and forthe required coupling interface conditions which frequentlybecomes poorly conditioned due to the different nature of themethods Since this system is formed from the contributionsof distinct methods it is also usually not possible to make useof their individual advantages in terms of optimized solversor memory storage (eg in BEM-FEM the final system willno longer be banded and symmetric etc) In addition tothis limitation by forming a single system of equationsa very large problem usually arises leading to increasedcomputational efforts and thus to a loss of performance

All these limitations have justified the appearance of itera-tive algorithms to obtain accurate solutions in amore efficient

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 426283 21 pageshttpdxdoiorg1011552014426283

2 Journal of Applied Mathematics

manner Perhaps one of the first iterative techniques to bedeveloped for general problems is the well-known Schwarzalternating strategy [20 21] in which the domain of analysisis partitioned in overlapping subdomains and the solution isfound by successively iterating along these subdomains untilconvergence is reached This classical and simple to imple-ment algorithm has been applied to many problem typesincluding potential problems [22] or electromagnetic wavepropagation problems [23] However for the case of acousticproblems or elastic wave propagation problems formulatedin the frequency domain the special oscillatory structure ofthe solution leads to severe convergence problemswhenusingsuch classic approaches and more sophisticated and difficultto implement strategies must be defined

In recent years more elaborate iterative domain decom-position techniques have been proposed and discussed inorder to analyze a wide range of problems providing goodresults especially in terms of flexibility and efficiency Mostlythese techniques have been applied to nontransient applica-tions and they usually consider the analysis of coupled mod-els taking into account the interaction of different discretiza-tion methods physical phenomena and so forth In fact forcomplex models iterative domain decomposition techniquesare recommended usually providing a better approach forthe analysis Indeed a proper numerical simulation is hardlyachieved by a single numerical technique in those casesmostly because complex and quite different phenomenainteract requiring particularized advanced expertise andorlarge scale problems are involved demanding high computa-tional efforts

Nowadays several works are available discussing iterativenonoverlapping partitioned analysis Taking into accountelliptic problems Rice et al [24] presented a quite completediscussion considering several interface relaxation proce-dures and comparing formulations and performances Asa matter of fact most of the publications on the topic arefocused on elliptic models few being devoted to hyperbolicproblems Taking into account computationalmechanics oneof the first publications on the topicwas presented by Lin et al[25] which discussed a relaxed iterative procedure to couplethe FEM and the BEM considering linear static analysesSimilar approaches have been presented later on consideringpotential andmechanical static linear analyses [26 27] In theworks of Elleithy et al [28 29] concerning mechanical staticand potential problems the authors propose that the domainof the original problem is subdivided into subdomains eachof them modeled by the finite element or boundary elementmethods the coupling between the different subdomains isperformed using smoothing operators on the interdomainboundaries Their strategy allows separate computations forthe BEM and FEM subdomains with successive update ofthe boundary conditions at the interfaces being performeduntil convergence is achieved In [30ndash32] similar approachesfor the analysis of different linear problems using domaindecomposition techniques were also presented Furtherdevelopments of these strategies to nonlinear analysis in solidmechanics can also be found in the works of Elleithy et al[33] using an interface relaxation finite element-boundary

element coupling method (FEM-BEM coupling) for elasto-plastic analysis or Jahromi et al [34] who established acoupling procedure based on a sequential iterative Dirichlet-Neumann coupling algorithm for nonlinear soil-structureinteraction It must be noted that the described works referto nontransient problems either linear or nonlinear and noapplication to wave propagation analysis is focused on inthese works

Taking into account time-domain wave propagationmodels the first work on the topic seems to have beenpresented by Soares et al [35] who described a relaxedFEM-BEM iterative coupling procedure to analyze dynamicnonlinear problems considering different time discretiza-tions within each sub-domain of the model Later on thistechnique has been further developed to analyze other wavepropagation models including acoustic elastic and electro-magnetic wave propagation or solid-fluid interaction takinginto account several different numerical procedures using theFEM and the BEM [36ndash45] or the meshless local Petrov-Galerkin method [46] Most of these works are focused onthe iterative coupling of different numerical discretizationtechniques and a review considering the iterative couplingof the FEM and the BEM taking into account some wavepropagation models in computational mechanics has beenpresented in [47] The coupling of acoustic and mechanicwave propagation models on the other hand has beenreviewed in [48] taking into account different domaindecomposition techniques and considering several numericaldiscretization techniques

In the analysis of wave propagation using frequency-domain formulations iterative coupling procedures can befound in the literature mostly considering acoustic-acousticand acoustic-elastodynamic coupling [49ndash54] As it hasbeen reported frequency-domain wave propagation analysesusually give rise to ill-posed problems and in these cases theconvergence of the iterative coupling algorithm can be eithertoo slow or unachievableThis is the case in acoustic-acousticacoustic-elastodynamic and elastodynamic-elastodynamicinteracting models and as discussed in this work conver-gence can be hardly achieved if no special procedure isconsidered especially if higher frequencies are focused on Asreferred in the literature in order to deal with this ill-posedproblem and ensure convergence of the iterative couplingalgorithm special techniques such as the adoption of optimalrelaxation parameters must be considered

In this work time- and frequency-domain analyses ofwave propagation models are reviewed taking into accountrelaxed iterative coupling procedures In this context severalwave propagation models (such as electromagnetic acousticmechanic) are considered and several numerical procedures(such as the finite element method the boundary elementmethod and meshless methods) are employed to discretizethe model In the iterative coupling approach each sub-domain of the global model is analyzed independently (as anuncoupled model) and a successive renewal of the variablesat the common interfaces is performed until convergenceis achieved These iterative methodologies exhibit severaladvantages when compared to standard coupling schemesfor instance

Journal of Applied Mathematics 3

(i) different subdomains can be analysed separatelyleading to smaller and better-conditioned systems ofequations (different solvers suitable for each sub-domain may be employed)

(ii) only interface routines are required when one wishesto use existing codes to build coupling algorithms(thus coupled systems may be solved by separateprogrammodules taking full advantage of specializedfeatures and disciplinary expertise)

(iii) matching nodes at common interfaces are notrequired greatly improving the flexibility and versa-tility of the coupled analyses especiallywhendifferentdiscretization methods are considered

(iv) matching time steps at common interfaces are notrequired (in time-domain analysis) allowing optimaltemporal discretizations within each sub-domainimproving accuracy and stability aspects

(v) nonlinear analyses (as well as other iterative-basedanalyses)may be carried out in the same iterative loopof the iterative coupling not introducing a relevantextra computational effort for the model

(vi) more efficient analyses can be obtained once theglobal model can be reduced to several subdomainswith reduced size matrices

As a matter of fact Gauzellino et al [55] compared theiterative domain decomposition and global solution takinginto account three-dimensional Helmholtz problems Theirnumerical results show that iterative domain decompositionmethods perform far better than globalmethods In additionthey observed that iterative domain decomposition methodsinvolving small subdomains work better than those withsubdomains involving a large number of elements Similarresults have been obtained by Soares et al [51] taking intoaccount two-dimensional Helmholtz problems

To give a detailed overview of the recent developmentsfound in many of the referred works the remainder ofthis paper will address a number of application examplesconcerning different phenomena and methods First thegoverning equations related to wave propagation models aregenerically and briefly presented In the sequence an effi-cient iterative coupling technique is described including themathematical derivation of the optimized relaxationmethod-ology Some numerical applications are finally presentedillustrating the accuracy performance and potentialities ofthe discussed procedures taking into account different wavepropagation models and discretization techniques

2 Governing Equations

Wave propagation phenomena may be generically describedby the following timefrequency-domain governing equa-tions

1198880(119909 119905) + 119888

1(119909) (119909 119905) + 119888

2(119909) (119909 119905)

+ 1198883(119909) 120597119891 (119906 (119909 119905)) = 0

(1a)

1198880 (119909 120596) minus 120596

21198881 (119909) 119906 (119909 120596) + 1198941205961198882 (

119909) 119906 (119909 120596)

+ 1198883(119909) 120597119891 (119906 (119909 120596)) = 0

(1b)

which can be further generalized in order to consider morecomplex behavior such as time varying coefficients (119888

119897(119909 119905)

119897 = 1 2 3) nonlinearities (119888119897(119909 119906(119909 119905)) 119897 = 0 1 2 3 etc)

Equation (1a) stands for the time domain governing equationwhereas (1b) stands for its frequency-domain counterpart(overbars indicate frequency-domain values) In these equa-tions 119906 represents the incognita field which can be scalarvectorial and so forth according to the physical model infocus 119888

119894stands for a general coefficient representation which

can as well be a scalar a tensor and so forth Overdots standfor time derivatives whereas 120597119891 indicates a spatial derivativeoperator The complex number is denoted by 119894 and the timefrequency and space domains are represented by 119905 120596 and119909 respectively (in this case 119909 isin Ω where Ω is the spatialdomain of the model)

The boundary conditions (119909 isin Γ where Γ is theboundary of the model) may be generically described as (forsimplicity from this point onwards overbars are no longerused to indicate frequency-domain values and 120589 stands for 119905or 120596 according to the case of analysis)

119891 (119906 (119909 120589) V (119909 120589)) = 119888 (119909 120589) (2)

where once again 119888 stands for known terms In (2) 119891stands for a generic function representing the combinationof its arguments The variable V which may be consideredprescribed at the boundary of the model is a functionof 119906 and it is usually expressed considering some normalprojection (normal to the boundary) of the spatial derivativesof 119906 (ie V = 120597119891

119899(119906))

To completely define the model initial conditions (whichare usually adopted null in frequency-domain analyses) mustalso be defined In this case a generic representation can begiven by 119891(119906(119909 119905 = 0) (119909 119905 = 0)) = 119888(119909) where notationanalogous to that of (2) is considered

Taking into account coupled models in which differentdomains interact by a common interface interface conditionsmust be stated indicating how the domains interactThis canbe generically expressed as

1198911(119906 (119909minus 120589) V (119909minus 120589)) = 119891

2(119906 (119909+ 120589) V (119909+ 120589)) (3)

where 119909 isin Γ119868 119909minus isin Γ

119868cup Ω1 119909+ isin Γ

119868cup Ω2 and Γ

119868is

the common interface between domains Ω1and Ω

2 In (3)

functions 1198911and 119891

2describe how the interaction between

the coupled domains takes place by relating their boundaryvalues on the common interface

3 Iterative Coupling Analysis

In order to enable the coupling between sectioned domainsof a global model an iterative procedure is employed herewhich performs a successive renewal of the relevant vari-ables at the common interfaces This approach is based onthe imposition of prescribed boundary conditions properlyevaluated at the interfaces of the sectioned domains allowingeach domain of the global model to be analyzed separately

4 Journal of Applied Mathematics

Since the sectioned domains are analyzed separately the rele-vant systems of equations are formed independently beforethe iterative process starts (in the case of linear analyses)and are kept constant along the iterative process renderinga very efficient procedure The separate treatment of thesectioned domains allows independent discretizations to beconsidered on each domain without any special requirementof matching nodes along the common interfaces Moreoverin the case of time-domain analysis different time-stepsmay also be considered for each domain Thus the couplingalgorithm can be presented for a generic case in whichthe interface nodes may not match and the interface timeinstants are disconnected allowing exploiting the benefits ofthe iterative coupling formulation

To ensure andor to speed up convergence a relaxationparameter 120582 is introduced in the iterative coupling algorithmThe effectiveness of the iterative process is strongly related tothe selection of this relaxation parameter since an inappro-priate selection for 120582 can significantly increase the number ofiterations in the analysis or even worse make convergenceunfeasible As it has been reported [49 51] frequency-domain analyses usually give rise to ill-posed problems andin these cases the convergence of simple iterative couplingalgorithms can either be too slow or unachievable In order todeal with ill-posed problems and ensure convergence of theiterative coupling algorithm an optimal iterative procedureis adopted here with optimal relaxation parameters beingcomputed at each iterative step As it is illustrated in thenext section the introduction of these optimal relaxationparameters allows the iterative coupling technique to bevery effective especially in the frequency domain ensuringconvergence at a low number of iterative steps

31 Iterative Algorithm Initially in the kth iterative step ofthe coupled analysis of domains 1 and 2 the so-called domain1 is analyzed and the variables 119906 or V at the common interfacesof the domain are computed taking into account prescribedvalues of V or 119906 at these common interfacesThese prescribedvalues of V or 119906 are provided from the previous iterative step(in the first iterative step null or previous time-step valuesmay be considered) Once the variables 119906 or V are computedthey are applied to evaluate the boundary conditions thatare prescribed at the common interfaces of domain 2 asdescribed by (3) Taking into account these prescribed 119906 or Vboundary conditions the so-called domain 2 is analyzed andthe variables V or 119906 at the common interfaces of the domainare computed Then the computed V or 119906 values are appliedto evaluate the boundary conditions that are prescribed atthe common interfaces of domain 1 reinitiating the iterativecycle A sketch of this cycle is depicted in Figure 1

As previously discussed relaxation parameters must beconsidered in order to ensure andor to speed up theconvergence of the iterative process Thus the values thatare computed after the analysis of the sectioned domain maybe combined with its previous iterative step counterpartrelaxing the computation of the actual iterative step valueMathematically this can be represented as follows

119910(119896+1)

= (120582) 119910(119896+120582)

+ (1 minus 120582) 119910(119896) (4)

where 120582 is the adopted relaxation parameter and 119910 stands for119906 or V according to the case of analysis one should note that119910(119896+120582) is the value computed at the end of the iterative step

before the application of the relaxation parameterA proper selection for 120582 at each iterative step is extremely

important for the effectiveness of the iterative couplingprocedure In order to obtain an easy to implement efficientand effective expression for the relaxation parameter compu-tation optimal 120582 values are deduced in Section 32

32 Optimal Relaxation Parameter In order to evaluate anoptimal relaxation parameter the following square errorfunctional is minimized here

120576 (120582) =

10038171003817100381710038171003817Y(119896+1) (120582) minus Y(119896) (120582)1003817100381710038171003817

1003817

2

(5)

where Y stands for a vector whose entries are 119906 or V valuescomputed at the common interfaces

Taking into account the relaxation of the field values forthe (119896 + 1) and (119896) iterations (6a) and (6b) may be writtenbased on the definition in (4)

Y(119896+1) = (120582)Y(119896+120582) + (1 minus 120582)Y(119896) (6a)

Y(119896) = (120582)Y(119896+120582minus1) + (1 minus 120582)Y(119896minus1) (6b)Substituting (6a) and (6b) into (5) yields

120576 (120582) =

10038171003817100381710038171003817(120582)W(119896+120582) + (1 minus 120582)W(119896)1003817100381710038171003817

1003817

2

= (1205822)

10038171003817100381710038171003817W(119896+120582)1003817100381710038171003817

1003817

2

+ 2120582 (1 minus 120582) (W(119896+120582)W(119896))

+ (1 minus 120582)210038171003817100381710038171003817W(119896)1003817100381710038171003817

1003817

2

(7)

where the inner product definition is employed (eg(WW) = W

2) and new variables as defined in thefollowing are considered

W (119896+120582) = Y(119896+120582) minus Y(119896+120582minus1) (8)To find the optimal 120582 that minimizes the functional 120576(120582)

(7) is differentiated with respect to 120582 and the result is set tozero described as follows

(120582)

10038171003817100381710038171003817W(119896+120582)1003817100381710038171003817

1003817

2

+ (1 minus 2120582) (W(119896+120582)W(119896))

+ (120582 minus 1)

10038171003817100381710038171003817W(119896)1003817100381710038171003817

1003817

2

= 0

(9)

Rearranging the terms in (9) yields

120582 =

(W(119896)W(119896) minusW(119896+120582))1003817100381710038171003817W(119896) minusW(119896+120582)100381710038171003817

1003817

2(10)

which is an easy to implement expression that provides anoptimal value for the relaxation parameter 120582 at each iterativestepThis expression requires a low computational cost whencompared to other alternatives that can be found in theliterature (see eg [28 29]) and it provides very good resultsas it has been reported taking into account different physicalmodels and domain analyses [43 44 51ndash54] The iterativeprocess is relatively insensitive to the value of the relaxationparameter adopted for the first iterative step and 120582 = 05 canbe considered in this case for instance

Journal of Applied Mathematics 5

Begi

nnin

g of

iter

ativ

e ana

lysis

Analysis of domain 1

Interface condition

considering space(time) compatibility

Interface condition

considering space(time) compatibility

Analysis of domain 2

computed computed

computedcomputed

Introduction of relaxation parameters

End

of it

erat

ive a

naly

sis

f1minus21 (uminus minus) = f1minus2

2 (u+ +)

f2minus11 (u+ +) = f2minus1

2 (uminus minus)

uminus or minus is u+ or + is

+ or u+ isminus or uminus is

Figure 1 Sketch of the iterative coupling algorithm

Interface of domain 1 Interface of

domain 2

y+1

y+2

y+3

y+4

y+5

y+6y+7

y+8

y+9

yminus1

yminus2

yminus3

yminus4

yminus5

(a)

yminus(tminus)

y+(t+)

yminus(tminus minus Δtminus)

y+(t+ minus Δt+)

tminust+tminus minus Δtminust+ minus Δt+

(b)

Figure 2 (a) Sketch for a spatial interpolation of nodal values on the interface 119910+1= 119868(119910

minus

1 119910minus

2) 119910+2= 119868(119910

minus

1 119910minus

2) 119910+3= 119868(119910

minus

2 119910minus

3) and so forth

(b) sketch for a temporal interpolation of time-step values on the interface 119910+(119905+) = 119868(119910minus(119905minus) 119910minus(119905minus minus Δ119905minus)) and so forth where 119868 stands fora linear interpolation function

33 Interface Compatibility As previously discussed inde-pendent spatial (and temporal in time-domain analysis)discretizations may be considered for each domain of themodel not requiring matching nodes (or equal time steps)at the common interfaces Thus special procedures must beemployed to ensure the interface spatial (and temporal) com-patibility In order to do so interpolation and extrapolationprocedures are considered here These procedures can begenerically described by

119910 (119909119894 120589) =

119869

sum

119895=1

120572119895119910 (119909119895 120589) (11a)

119910 (119909 119905119899) = 1205730119910 (119909 119905

119898) +

119869

sum

119895=1

120573119895119910 (119909 (119905 minus 119895Δ119905)

119898119899) (11b)

where (11a) stands for spatial interpolations and (11b) standsfor time interpolationsextrapolations (120572

119895and 120573

119895stand

for spatial interpolation coefficients and time interpola-tionextrapolation coefficients respectively where Δ119905 rep-resents the time step) In Figure 2 simple sketches for thespatial and temporal interpolation procedures are depictedtaking into account linear interpolations

Although time interpolations usually can be carried outwithout further difficulties time extrapolations may give riseto instabilities if not properly elaboratedThus extrapolationsshould be performed in consonancewith the field approxima-tions being adopted within each time step and with the timediscretization procedures being considered in the analysis inorder to formulate a consistent procedure Once a consistentmethodology is elaborated time interpolationextrapolation

6 Journal of Applied Mathematics

procedures can be employed with confidence as referred inthe literature [47 48] and illustrated in the next sectionOne should notice that usually different optimal (optimalin terms of accuracy stability and efficiency) time stepsare required when taking into account different numericalmethods spatial discretizations material properties physicalphenomena and so forth Thus in some cases consideringdifferent time steps within each domain of a coupled modelis of maximal importance to allow the effectiveness of theanalysis

Using space(time) interpolation(extrapolation) proce-dures optimal modeling of each sectioned domain may beachieved which is very important inwhat concerns flexibilityefficiency accuracy and stability aspects

4 Numerical Applications

In this section the general procedures previously discussedare particularized and briefly detailed taking into accountdifferent physicalmodels anddiscretization techniquesThusthe discussed iterative coupling methodology is appliedconsidering a wide range of wave propagation models andnumerical methods richly illustrating its performance andpotentialities

In this context time- and frequency-domain analysesare carried out here and electromagnetic acoustic andmechanical wave propagation phenomena (as well as theirinteractions) are discussed in the applications that followMoreover different numerical techniques (such as the finiteelement method the boundary element method and mesh-less methods) are applied to discretize the different domainsof the model illustrating the versatility and generality of thediscussed iterative method

41 Electromagnetic Waves In electromagnetic models vec-torial wave equations describe the electric and the magneticfield evolution [56 57] In this case (1a) can be rewritten as(in this subsection time-domain analyses are focused on)

nabla times (120583(119909)minus1nabla times E (119909 119905)) + 120576 (119909) E (119909 119905) = minus J (119909 119905) (12a)

nabla times (120576(119909)minus1nabla timesH (119909 119905)) + 120583 (119909) H (119909 119905)

= nabla times (120576(119909)minus1J (119909 119905))

(12b)

and (3) can be rewritten as

n (119909) times (E (119909+ 119905) minus E (119909minus 119905)) = 0 (13a)

(D (119909+ 119905) minusD (119909

minus 119905)) sdot n (119909) = 120588 (119909 119905) (13b)

(B (119909+ 119905) minus B (119909minus 119905)) sdot n (119909) = 0 (13c)

n (119909) times (H (119909+ 119905) minusH (119909

minus 119905)) = J (119909 119905) (13d)

where E and H are the electric and magnetic field intensityvectors respectively D and B represent the electric andmagnetic flux densities respectively and J and 120588 stand for theelectric current and electric charge density respectively Theparameters 120576 and 120583 denote respectively the permittivity and

permeability of themediumand itswave propagation velocityis specified as 119888 = (120576120583)

minus12 n is the normal vector fromdomain 1 to domain 2 Equations (13a) and (13b) state that thetangential component of E is continuous across the interfaceand that the normal component of D has a step of surfacecharge on the interface surface respectively Equations (13c)and (13d) state that the normal component ofB is continuousacross the interface and that the tangential component ofH iscontinuous across the interface if there is no surface currentpresent respectively

In the present application the electromagnetic fieldssurrounding infinitely long wires are studied [41] Two casesof analysis are focused here namely (a) case 1 where onewireis considered (b) case 2 where two wires are employed Forboth cases the wires are carrying time-dependent currents(ie 119868(119905) = 119905 or 119868(119905) = 119905

2) and they are located along theadopted 119911-axis A sketch of the model is depicted in Figure 3

The spatial and temporal evolution of the electric fieldintensity vector is analyzed here taking into account a finiteelement method (FEM)mdashboundary element method (BEM)coupled formulation In this context the FEM is appliedto model the region close to the wires whereas the BEMsimulates the remaining infinity domain As it is well knownthe BEM employs fundamental solutions which fulfill theradiation conditionThus this formulation is very suitable toperform infinite domain analysis once reflected waves frominfinity are avoided [58]

The adopted spatial discretization is also described inFigure 3 In this case 2344 linear triangular finite elementsand 80 linear boundary elements are employed in the analyses(see references [57 58] for more details regarding the FEMand the BEMapplied to electromagnetic analyses)The radiusof the FEM-BEM interface is defined by 119877 = 1m andmatching nodes are considered at the interface For temporaldiscretization the selected time step is given byΔ119905 = 5sdot10minus11sfor both domainsThephysical properties of themedium (air)are 120583 = 12566 sdot 10minus6Hm and 120576 = 88544 sdot 10minus12 Fm

Figure 4 shows the modulus of the electric field intensityobtained at points A and B (see Figure 3) considering theiterative couplingmethodology Analytical time histories [58]are also depicted in Figure 4 highlighting the good accuracyof the numerical results In Figure 5 charts are displayedindicating the percentage of occurrence of different relax-ation parameter values (evaluated according to expression(10)) in each analysis As can be observed for all consideredcases optimal relaxation parameters aremostly in the interval07 le 120582 le 08 In fact an optimal relaxation parameterselection is extremely case dependent It is function of thephysical properties of the model geometric aspects adoptedspatial and temporal discretizations and so forth Equation(10) provides a simple expression to evaluate this complexparameter

In order to illustrate the effectiveness of the methodologywhen considering different time discretizations for differentdomains Figure 6 depicts results that are computed consider-ing Δ119905 = 25 sdot 10minus11 s for the FEM and Δ119905 = 20 sdot 10minus10 s for theBEM (ie a difference of 8 times between the time steps) Forsimplicity results are presented considering just the first case

Journal of Applied Mathematics 7

R

Wire AB

FEM-BEMinterface

x

y

z

(a)

R

Wire A

B

interface

Wire

FEM-BEM

x

y

z

(b)

Figure 3 Sketch of the electromagnetic models and adopted FEMBEM spatial discretizations (a) case 1 one wire (b) case 2 two wires

000 025 050 075

0

1

2

3

4

5

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(a)

000 025 050 075

0

2

4

6

8

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(b)

AnalyticalFEM-BEM

000 025 050 075

0

1

2

3

4

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(c)

AnalyticalFEM-BEM

000 025 050 075

0

2

4

6

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(d)

Figure 4 Time history results for the electric field intensity at points A and B considering 119868(119905) = 119905 and (a) case 1 and (b) case 2 119868(119905) = 1199052 and(c) case 1 and (d) case 2

8 Journal of Applied Mathematics

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(a)

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(b)

00 02 04 06 08 100

5

10

15

20

25

30

Occ

urre

nce (

)

Relaxation parameter

(c)

00 02 04 06 08 100

5

10

15

20

25

30O

ccur

renc

e (

)

Relaxation parameter

(d)

Figure 5 Percentage of occurrence of different relaxation parameter values during the analysis considering 119868(119905) = 119905 and (a) case 1 and (b)case 2 119868(119905) = 1199052 and (c) case 1 and (d) case 2

of analysis that is case 1 and 119868(119905) = 119905 As one can observein Figure 6(a) good results are still obtained taking intoaccount the iterative formulation in spite of the existing timedisconnections at the interface In Figure 6(b) the evolutionof the relaxation parameter is depicted taking into accountthis last configuration As one can observe in this caseoptimal relaxation parameter values are between 07 and 10and mostly concentrate on the interval (09 10) In fact itis expected that these values get closer to 10 when smallertime steps are considered In the present analysis an averagenumber of 492 iterations per time step is obtained (takinginto account 800 FEM time steps) which is a relatively lownumber illustrating the good performance of the technique(it must be remarked that a tight tolerance criterion wasadopted for the convergence of the iterative analysis)

42 Acoustic Waves In acoustic models a scalar wave equa-tion describes the acoustic pressure field evolution [1] In thiscase (1b) can be rewritten as (in this subsection frequency-domain analyses are focused)

nabla sdot (120581 (119909) nabla119901 (119909 120596)) + 1205962120588 (119909) 119901 (119909 120596) = 120574 (119909 120596) (14)

and (3) can be rewritten as

(119901 (119909+ 120596) minus 119901 (119909

minus 120596)) = 0 (15a)

(119902 (119909+ 120596) minus 119902 (119909

minus 120596)) = 119892 (119909 120596) (15b)

where 119901 is the hydrodynamic pressure and 120574 and 119892 standfor domain and surface sources respectively The parameters120588 and 120581 denote respectively the mass density and com-pressibility of the medium and its wave propagation velocity

Journal of Applied Mathematics 9

000 025 050 075

0

1

2

3

4

5

Point B

Point A

AnalyticalFEM-BEM

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

(a)

0 1000 2000 3000 4000

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

(b)Figure 6 Results considering different time steps for each domain (a) electric field intensity at points A and B (b) optimal relaxationparameters for each iterative step

50

50

x

y

R=10

120592 = 30000ms

120592 = 30000ms

120592 = 15000ms

S (minus50 00)

(a)

x (m)

y(m

)65

6

55

5

45

4

35135 14 145 15 155 16 165

(b)

Figure 7 (a) Sketch for the heterogeneous medium with multiple subregions (b) boundary and domain point distribution considering thespatial discretization of an inclusion and adjacent fluid

is specified as 120592 = (120581120588)12 The hydrodynamic fluxes on

the interfaces are represented by 119902 and they are defined by119902 = 120581 nabla119901 sdot n where n is the normal vector from domain1 to domain 2 Equation (15a) states that the pressure iscontinuous across the interface whereas (15b) states that theflux is continuous across the interface if there is no surfacesource

The advantages of using iterative coupling procedures arerevealed when more complex configurations are analyzedIn this subsection the case of a heterogeneous domaincomposed of a homogeneous fluid incorporating multiplecircular inclusions with different properties is analyzed

For this purpose consider the host medium to allow thepropagation of sound with a velocity of 1500ms and thismedium is excited by a line source located at 119909

119904= minus50m

and 119910119904= 00m Within this fluid consider the presence of

8 circular inclusions all of them are with unit radius andfilled with a different fluid allowing sound waves to travel at3000ms as depicted in Figure 7

The above-described system has been analyzed takinginto account the proposed iterative coupling proceduremaking use of the Kansarsquos method (KM) to model all theinclusions and of the method of fundamental solutions(MFS) to model the host fluid (see references [12 59ndash61]

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

2 Journal of Applied Mathematics

manner Perhaps one of the first iterative techniques to bedeveloped for general problems is the well-known Schwarzalternating strategy [20 21] in which the domain of analysisis partitioned in overlapping subdomains and the solution isfound by successively iterating along these subdomains untilconvergence is reached This classical and simple to imple-ment algorithm has been applied to many problem typesincluding potential problems [22] or electromagnetic wavepropagation problems [23] However for the case of acousticproblems or elastic wave propagation problems formulatedin the frequency domain the special oscillatory structure ofthe solution leads to severe convergence problemswhenusingsuch classic approaches and more sophisticated and difficultto implement strategies must be defined

In recent years more elaborate iterative domain decom-position techniques have been proposed and discussed inorder to analyze a wide range of problems providing goodresults especially in terms of flexibility and efficiency Mostlythese techniques have been applied to nontransient applica-tions and they usually consider the analysis of coupled mod-els taking into account the interaction of different discretiza-tion methods physical phenomena and so forth In fact forcomplex models iterative domain decomposition techniquesare recommended usually providing a better approach forthe analysis Indeed a proper numerical simulation is hardlyachieved by a single numerical technique in those casesmostly because complex and quite different phenomenainteract requiring particularized advanced expertise andorlarge scale problems are involved demanding high computa-tional efforts

Nowadays several works are available discussing iterativenonoverlapping partitioned analysis Taking into accountelliptic problems Rice et al [24] presented a quite completediscussion considering several interface relaxation proce-dures and comparing formulations and performances Asa matter of fact most of the publications on the topic arefocused on elliptic models few being devoted to hyperbolicproblems Taking into account computationalmechanics oneof the first publications on the topicwas presented by Lin et al[25] which discussed a relaxed iterative procedure to couplethe FEM and the BEM considering linear static analysesSimilar approaches have been presented later on consideringpotential andmechanical static linear analyses [26 27] In theworks of Elleithy et al [28 29] concerning mechanical staticand potential problems the authors propose that the domainof the original problem is subdivided into subdomains eachof them modeled by the finite element or boundary elementmethods the coupling between the different subdomains isperformed using smoothing operators on the interdomainboundaries Their strategy allows separate computations forthe BEM and FEM subdomains with successive update ofthe boundary conditions at the interfaces being performeduntil convergence is achieved In [30ndash32] similar approachesfor the analysis of different linear problems using domaindecomposition techniques were also presented Furtherdevelopments of these strategies to nonlinear analysis in solidmechanics can also be found in the works of Elleithy et al[33] using an interface relaxation finite element-boundary

element coupling method (FEM-BEM coupling) for elasto-plastic analysis or Jahromi et al [34] who established acoupling procedure based on a sequential iterative Dirichlet-Neumann coupling algorithm for nonlinear soil-structureinteraction It must be noted that the described works referto nontransient problems either linear or nonlinear and noapplication to wave propagation analysis is focused on inthese works

Taking into account time-domain wave propagationmodels the first work on the topic seems to have beenpresented by Soares et al [35] who described a relaxedFEM-BEM iterative coupling procedure to analyze dynamicnonlinear problems considering different time discretiza-tions within each sub-domain of the model Later on thistechnique has been further developed to analyze other wavepropagation models including acoustic elastic and electro-magnetic wave propagation or solid-fluid interaction takinginto account several different numerical procedures using theFEM and the BEM [36ndash45] or the meshless local Petrov-Galerkin method [46] Most of these works are focused onthe iterative coupling of different numerical discretizationtechniques and a review considering the iterative couplingof the FEM and the BEM taking into account some wavepropagation models in computational mechanics has beenpresented in [47] The coupling of acoustic and mechanicwave propagation models on the other hand has beenreviewed in [48] taking into account different domaindecomposition techniques and considering several numericaldiscretization techniques

In the analysis of wave propagation using frequency-domain formulations iterative coupling procedures can befound in the literature mostly considering acoustic-acousticand acoustic-elastodynamic coupling [49ndash54] As it hasbeen reported frequency-domain wave propagation analysesusually give rise to ill-posed problems and in these cases theconvergence of the iterative coupling algorithm can be eithertoo slow or unachievableThis is the case in acoustic-acousticacoustic-elastodynamic and elastodynamic-elastodynamicinteracting models and as discussed in this work conver-gence can be hardly achieved if no special procedure isconsidered especially if higher frequencies are focused on Asreferred in the literature in order to deal with this ill-posedproblem and ensure convergence of the iterative couplingalgorithm special techniques such as the adoption of optimalrelaxation parameters must be considered

In this work time- and frequency-domain analyses ofwave propagation models are reviewed taking into accountrelaxed iterative coupling procedures In this context severalwave propagation models (such as electromagnetic acousticmechanic) are considered and several numerical procedures(such as the finite element method the boundary elementmethod and meshless methods) are employed to discretizethe model In the iterative coupling approach each sub-domain of the global model is analyzed independently (as anuncoupled model) and a successive renewal of the variablesat the common interfaces is performed until convergenceis achieved These iterative methodologies exhibit severaladvantages when compared to standard coupling schemesfor instance

Journal of Applied Mathematics 3

(i) different subdomains can be analysed separatelyleading to smaller and better-conditioned systems ofequations (different solvers suitable for each sub-domain may be employed)

(ii) only interface routines are required when one wishesto use existing codes to build coupling algorithms(thus coupled systems may be solved by separateprogrammodules taking full advantage of specializedfeatures and disciplinary expertise)

(iii) matching nodes at common interfaces are notrequired greatly improving the flexibility and versa-tility of the coupled analyses especiallywhendifferentdiscretization methods are considered

(iv) matching time steps at common interfaces are notrequired (in time-domain analysis) allowing optimaltemporal discretizations within each sub-domainimproving accuracy and stability aspects

(v) nonlinear analyses (as well as other iterative-basedanalyses)may be carried out in the same iterative loopof the iterative coupling not introducing a relevantextra computational effort for the model

(vi) more efficient analyses can be obtained once theglobal model can be reduced to several subdomainswith reduced size matrices

As a matter of fact Gauzellino et al [55] compared theiterative domain decomposition and global solution takinginto account three-dimensional Helmholtz problems Theirnumerical results show that iterative domain decompositionmethods perform far better than globalmethods In additionthey observed that iterative domain decomposition methodsinvolving small subdomains work better than those withsubdomains involving a large number of elements Similarresults have been obtained by Soares et al [51] taking intoaccount two-dimensional Helmholtz problems

To give a detailed overview of the recent developmentsfound in many of the referred works the remainder ofthis paper will address a number of application examplesconcerning different phenomena and methods First thegoverning equations related to wave propagation models aregenerically and briefly presented In the sequence an effi-cient iterative coupling technique is described including themathematical derivation of the optimized relaxationmethod-ology Some numerical applications are finally presentedillustrating the accuracy performance and potentialities ofthe discussed procedures taking into account different wavepropagation models and discretization techniques

2 Governing Equations

Wave propagation phenomena may be generically describedby the following timefrequency-domain governing equa-tions

1198880(119909 119905) + 119888

1(119909) (119909 119905) + 119888

2(119909) (119909 119905)

+ 1198883(119909) 120597119891 (119906 (119909 119905)) = 0

(1a)

1198880 (119909 120596) minus 120596

21198881 (119909) 119906 (119909 120596) + 1198941205961198882 (

119909) 119906 (119909 120596)

+ 1198883(119909) 120597119891 (119906 (119909 120596)) = 0

(1b)

which can be further generalized in order to consider morecomplex behavior such as time varying coefficients (119888

119897(119909 119905)

119897 = 1 2 3) nonlinearities (119888119897(119909 119906(119909 119905)) 119897 = 0 1 2 3 etc)

Equation (1a) stands for the time domain governing equationwhereas (1b) stands for its frequency-domain counterpart(overbars indicate frequency-domain values) In these equa-tions 119906 represents the incognita field which can be scalarvectorial and so forth according to the physical model infocus 119888

119894stands for a general coefficient representation which

can as well be a scalar a tensor and so forth Overdots standfor time derivatives whereas 120597119891 indicates a spatial derivativeoperator The complex number is denoted by 119894 and the timefrequency and space domains are represented by 119905 120596 and119909 respectively (in this case 119909 isin Ω where Ω is the spatialdomain of the model)

The boundary conditions (119909 isin Γ where Γ is theboundary of the model) may be generically described as (forsimplicity from this point onwards overbars are no longerused to indicate frequency-domain values and 120589 stands for 119905or 120596 according to the case of analysis)

119891 (119906 (119909 120589) V (119909 120589)) = 119888 (119909 120589) (2)

where once again 119888 stands for known terms In (2) 119891stands for a generic function representing the combinationof its arguments The variable V which may be consideredprescribed at the boundary of the model is a functionof 119906 and it is usually expressed considering some normalprojection (normal to the boundary) of the spatial derivativesof 119906 (ie V = 120597119891

119899(119906))

To completely define the model initial conditions (whichare usually adopted null in frequency-domain analyses) mustalso be defined In this case a generic representation can begiven by 119891(119906(119909 119905 = 0) (119909 119905 = 0)) = 119888(119909) where notationanalogous to that of (2) is considered

Taking into account coupled models in which differentdomains interact by a common interface interface conditionsmust be stated indicating how the domains interactThis canbe generically expressed as

1198911(119906 (119909minus 120589) V (119909minus 120589)) = 119891

2(119906 (119909+ 120589) V (119909+ 120589)) (3)

where 119909 isin Γ119868 119909minus isin Γ

119868cup Ω1 119909+ isin Γ

119868cup Ω2 and Γ

119868is

the common interface between domains Ω1and Ω

2 In (3)

functions 1198911and 119891

2describe how the interaction between

the coupled domains takes place by relating their boundaryvalues on the common interface

3 Iterative Coupling Analysis

In order to enable the coupling between sectioned domainsof a global model an iterative procedure is employed herewhich performs a successive renewal of the relevant vari-ables at the common interfaces This approach is based onthe imposition of prescribed boundary conditions properlyevaluated at the interfaces of the sectioned domains allowingeach domain of the global model to be analyzed separately

4 Journal of Applied Mathematics

Since the sectioned domains are analyzed separately the rele-vant systems of equations are formed independently beforethe iterative process starts (in the case of linear analyses)and are kept constant along the iterative process renderinga very efficient procedure The separate treatment of thesectioned domains allows independent discretizations to beconsidered on each domain without any special requirementof matching nodes along the common interfaces Moreoverin the case of time-domain analysis different time-stepsmay also be considered for each domain Thus the couplingalgorithm can be presented for a generic case in whichthe interface nodes may not match and the interface timeinstants are disconnected allowing exploiting the benefits ofthe iterative coupling formulation

To ensure andor to speed up convergence a relaxationparameter 120582 is introduced in the iterative coupling algorithmThe effectiveness of the iterative process is strongly related tothe selection of this relaxation parameter since an inappro-priate selection for 120582 can significantly increase the number ofiterations in the analysis or even worse make convergenceunfeasible As it has been reported [49 51] frequency-domain analyses usually give rise to ill-posed problems andin these cases the convergence of simple iterative couplingalgorithms can either be too slow or unachievable In order todeal with ill-posed problems and ensure convergence of theiterative coupling algorithm an optimal iterative procedureis adopted here with optimal relaxation parameters beingcomputed at each iterative step As it is illustrated in thenext section the introduction of these optimal relaxationparameters allows the iterative coupling technique to bevery effective especially in the frequency domain ensuringconvergence at a low number of iterative steps

31 Iterative Algorithm Initially in the kth iterative step ofthe coupled analysis of domains 1 and 2 the so-called domain1 is analyzed and the variables 119906 or V at the common interfacesof the domain are computed taking into account prescribedvalues of V or 119906 at these common interfacesThese prescribedvalues of V or 119906 are provided from the previous iterative step(in the first iterative step null or previous time-step valuesmay be considered) Once the variables 119906 or V are computedthey are applied to evaluate the boundary conditions thatare prescribed at the common interfaces of domain 2 asdescribed by (3) Taking into account these prescribed 119906 or Vboundary conditions the so-called domain 2 is analyzed andthe variables V or 119906 at the common interfaces of the domainare computed Then the computed V or 119906 values are appliedto evaluate the boundary conditions that are prescribed atthe common interfaces of domain 1 reinitiating the iterativecycle A sketch of this cycle is depicted in Figure 1

As previously discussed relaxation parameters must beconsidered in order to ensure andor to speed up theconvergence of the iterative process Thus the values thatare computed after the analysis of the sectioned domain maybe combined with its previous iterative step counterpartrelaxing the computation of the actual iterative step valueMathematically this can be represented as follows

119910(119896+1)

= (120582) 119910(119896+120582)

+ (1 minus 120582) 119910(119896) (4)

where 120582 is the adopted relaxation parameter and 119910 stands for119906 or V according to the case of analysis one should note that119910(119896+120582) is the value computed at the end of the iterative step

before the application of the relaxation parameterA proper selection for 120582 at each iterative step is extremely

important for the effectiveness of the iterative couplingprocedure In order to obtain an easy to implement efficientand effective expression for the relaxation parameter compu-tation optimal 120582 values are deduced in Section 32

32 Optimal Relaxation Parameter In order to evaluate anoptimal relaxation parameter the following square errorfunctional is minimized here

120576 (120582) =

10038171003817100381710038171003817Y(119896+1) (120582) minus Y(119896) (120582)1003817100381710038171003817

1003817

2

(5)

where Y stands for a vector whose entries are 119906 or V valuescomputed at the common interfaces

Taking into account the relaxation of the field values forthe (119896 + 1) and (119896) iterations (6a) and (6b) may be writtenbased on the definition in (4)

Y(119896+1) = (120582)Y(119896+120582) + (1 minus 120582)Y(119896) (6a)

Y(119896) = (120582)Y(119896+120582minus1) + (1 minus 120582)Y(119896minus1) (6b)Substituting (6a) and (6b) into (5) yields

120576 (120582) =

10038171003817100381710038171003817(120582)W(119896+120582) + (1 minus 120582)W(119896)1003817100381710038171003817

1003817

2

= (1205822)

10038171003817100381710038171003817W(119896+120582)1003817100381710038171003817

1003817

2

+ 2120582 (1 minus 120582) (W(119896+120582)W(119896))

+ (1 minus 120582)210038171003817100381710038171003817W(119896)1003817100381710038171003817

1003817

2

(7)

where the inner product definition is employed (eg(WW) = W

2) and new variables as defined in thefollowing are considered

W (119896+120582) = Y(119896+120582) minus Y(119896+120582minus1) (8)To find the optimal 120582 that minimizes the functional 120576(120582)

(7) is differentiated with respect to 120582 and the result is set tozero described as follows

(120582)

10038171003817100381710038171003817W(119896+120582)1003817100381710038171003817

1003817

2

+ (1 minus 2120582) (W(119896+120582)W(119896))

+ (120582 minus 1)

10038171003817100381710038171003817W(119896)1003817100381710038171003817

1003817

2

= 0

(9)

Rearranging the terms in (9) yields

120582 =

(W(119896)W(119896) minusW(119896+120582))1003817100381710038171003817W(119896) minusW(119896+120582)100381710038171003817

1003817

2(10)

which is an easy to implement expression that provides anoptimal value for the relaxation parameter 120582 at each iterativestepThis expression requires a low computational cost whencompared to other alternatives that can be found in theliterature (see eg [28 29]) and it provides very good resultsas it has been reported taking into account different physicalmodels and domain analyses [43 44 51ndash54] The iterativeprocess is relatively insensitive to the value of the relaxationparameter adopted for the first iterative step and 120582 = 05 canbe considered in this case for instance

Journal of Applied Mathematics 5

Begi

nnin

g of

iter

ativ

e ana

lysis

Analysis of domain 1

Interface condition

considering space(time) compatibility

Interface condition

considering space(time) compatibility

Analysis of domain 2

computed computed

computedcomputed

Introduction of relaxation parameters

End

of it

erat

ive a

naly

sis

f1minus21 (uminus minus) = f1minus2

2 (u+ +)

f2minus11 (u+ +) = f2minus1

2 (uminus minus)

uminus or minus is u+ or + is

+ or u+ isminus or uminus is

Figure 1 Sketch of the iterative coupling algorithm

Interface of domain 1 Interface of

domain 2

y+1

y+2

y+3

y+4

y+5

y+6y+7

y+8

y+9

yminus1

yminus2

yminus3

yminus4

yminus5

(a)

yminus(tminus)

y+(t+)

yminus(tminus minus Δtminus)

y+(t+ minus Δt+)

tminust+tminus minus Δtminust+ minus Δt+

(b)

Figure 2 (a) Sketch for a spatial interpolation of nodal values on the interface 119910+1= 119868(119910

minus

1 119910minus

2) 119910+2= 119868(119910

minus

1 119910minus

2) 119910+3= 119868(119910

minus

2 119910minus

3) and so forth

(b) sketch for a temporal interpolation of time-step values on the interface 119910+(119905+) = 119868(119910minus(119905minus) 119910minus(119905minus minus Δ119905minus)) and so forth where 119868 stands fora linear interpolation function

33 Interface Compatibility As previously discussed inde-pendent spatial (and temporal in time-domain analysis)discretizations may be considered for each domain of themodel not requiring matching nodes (or equal time steps)at the common interfaces Thus special procedures must beemployed to ensure the interface spatial (and temporal) com-patibility In order to do so interpolation and extrapolationprocedures are considered here These procedures can begenerically described by

119910 (119909119894 120589) =

119869

sum

119895=1

120572119895119910 (119909119895 120589) (11a)

119910 (119909 119905119899) = 1205730119910 (119909 119905

119898) +

119869

sum

119895=1

120573119895119910 (119909 (119905 minus 119895Δ119905)

119898119899) (11b)

where (11a) stands for spatial interpolations and (11b) standsfor time interpolationsextrapolations (120572

119895and 120573

119895stand

for spatial interpolation coefficients and time interpola-tionextrapolation coefficients respectively where Δ119905 rep-resents the time step) In Figure 2 simple sketches for thespatial and temporal interpolation procedures are depictedtaking into account linear interpolations

Although time interpolations usually can be carried outwithout further difficulties time extrapolations may give riseto instabilities if not properly elaboratedThus extrapolationsshould be performed in consonancewith the field approxima-tions being adopted within each time step and with the timediscretization procedures being considered in the analysis inorder to formulate a consistent procedure Once a consistentmethodology is elaborated time interpolationextrapolation

6 Journal of Applied Mathematics

procedures can be employed with confidence as referred inthe literature [47 48] and illustrated in the next sectionOne should notice that usually different optimal (optimalin terms of accuracy stability and efficiency) time stepsare required when taking into account different numericalmethods spatial discretizations material properties physicalphenomena and so forth Thus in some cases consideringdifferent time steps within each domain of a coupled modelis of maximal importance to allow the effectiveness of theanalysis

Using space(time) interpolation(extrapolation) proce-dures optimal modeling of each sectioned domain may beachieved which is very important inwhat concerns flexibilityefficiency accuracy and stability aspects

4 Numerical Applications

In this section the general procedures previously discussedare particularized and briefly detailed taking into accountdifferent physicalmodels anddiscretization techniquesThusthe discussed iterative coupling methodology is appliedconsidering a wide range of wave propagation models andnumerical methods richly illustrating its performance andpotentialities

In this context time- and frequency-domain analysesare carried out here and electromagnetic acoustic andmechanical wave propagation phenomena (as well as theirinteractions) are discussed in the applications that followMoreover different numerical techniques (such as the finiteelement method the boundary element method and mesh-less methods) are applied to discretize the different domainsof the model illustrating the versatility and generality of thediscussed iterative method

41 Electromagnetic Waves In electromagnetic models vec-torial wave equations describe the electric and the magneticfield evolution [56 57] In this case (1a) can be rewritten as(in this subsection time-domain analyses are focused on)

nabla times (120583(119909)minus1nabla times E (119909 119905)) + 120576 (119909) E (119909 119905) = minus J (119909 119905) (12a)

nabla times (120576(119909)minus1nabla timesH (119909 119905)) + 120583 (119909) H (119909 119905)

= nabla times (120576(119909)minus1J (119909 119905))

(12b)

and (3) can be rewritten as

n (119909) times (E (119909+ 119905) minus E (119909minus 119905)) = 0 (13a)

(D (119909+ 119905) minusD (119909

minus 119905)) sdot n (119909) = 120588 (119909 119905) (13b)

(B (119909+ 119905) minus B (119909minus 119905)) sdot n (119909) = 0 (13c)

n (119909) times (H (119909+ 119905) minusH (119909

minus 119905)) = J (119909 119905) (13d)

where E and H are the electric and magnetic field intensityvectors respectively D and B represent the electric andmagnetic flux densities respectively and J and 120588 stand for theelectric current and electric charge density respectively Theparameters 120576 and 120583 denote respectively the permittivity and

permeability of themediumand itswave propagation velocityis specified as 119888 = (120576120583)

minus12 n is the normal vector fromdomain 1 to domain 2 Equations (13a) and (13b) state that thetangential component of E is continuous across the interfaceand that the normal component of D has a step of surfacecharge on the interface surface respectively Equations (13c)and (13d) state that the normal component ofB is continuousacross the interface and that the tangential component ofH iscontinuous across the interface if there is no surface currentpresent respectively

In the present application the electromagnetic fieldssurrounding infinitely long wires are studied [41] Two casesof analysis are focused here namely (a) case 1 where onewireis considered (b) case 2 where two wires are employed Forboth cases the wires are carrying time-dependent currents(ie 119868(119905) = 119905 or 119868(119905) = 119905

2) and they are located along theadopted 119911-axis A sketch of the model is depicted in Figure 3

The spatial and temporal evolution of the electric fieldintensity vector is analyzed here taking into account a finiteelement method (FEM)mdashboundary element method (BEM)coupled formulation In this context the FEM is appliedto model the region close to the wires whereas the BEMsimulates the remaining infinity domain As it is well knownthe BEM employs fundamental solutions which fulfill theradiation conditionThus this formulation is very suitable toperform infinite domain analysis once reflected waves frominfinity are avoided [58]

The adopted spatial discretization is also described inFigure 3 In this case 2344 linear triangular finite elementsand 80 linear boundary elements are employed in the analyses(see references [57 58] for more details regarding the FEMand the BEMapplied to electromagnetic analyses)The radiusof the FEM-BEM interface is defined by 119877 = 1m andmatching nodes are considered at the interface For temporaldiscretization the selected time step is given byΔ119905 = 5sdot10minus11sfor both domainsThephysical properties of themedium (air)are 120583 = 12566 sdot 10minus6Hm and 120576 = 88544 sdot 10minus12 Fm

Figure 4 shows the modulus of the electric field intensityobtained at points A and B (see Figure 3) considering theiterative couplingmethodology Analytical time histories [58]are also depicted in Figure 4 highlighting the good accuracyof the numerical results In Figure 5 charts are displayedindicating the percentage of occurrence of different relax-ation parameter values (evaluated according to expression(10)) in each analysis As can be observed for all consideredcases optimal relaxation parameters aremostly in the interval07 le 120582 le 08 In fact an optimal relaxation parameterselection is extremely case dependent It is function of thephysical properties of the model geometric aspects adoptedspatial and temporal discretizations and so forth Equation(10) provides a simple expression to evaluate this complexparameter

In order to illustrate the effectiveness of the methodologywhen considering different time discretizations for differentdomains Figure 6 depicts results that are computed consider-ing Δ119905 = 25 sdot 10minus11 s for the FEM and Δ119905 = 20 sdot 10minus10 s for theBEM (ie a difference of 8 times between the time steps) Forsimplicity results are presented considering just the first case

Journal of Applied Mathematics 7

R

Wire AB

FEM-BEMinterface

x

y

z

(a)

R

Wire A

B

interface

Wire

FEM-BEM

x

y

z

(b)

Figure 3 Sketch of the electromagnetic models and adopted FEMBEM spatial discretizations (a) case 1 one wire (b) case 2 two wires

000 025 050 075

0

1

2

3

4

5

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(a)

000 025 050 075

0

2

4

6

8

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(b)

AnalyticalFEM-BEM

000 025 050 075

0

1

2

3

4

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(c)

AnalyticalFEM-BEM

000 025 050 075

0

2

4

6

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(d)

Figure 4 Time history results for the electric field intensity at points A and B considering 119868(119905) = 119905 and (a) case 1 and (b) case 2 119868(119905) = 1199052 and(c) case 1 and (d) case 2

8 Journal of Applied Mathematics

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(a)

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(b)

00 02 04 06 08 100

5

10

15

20

25

30

Occ

urre

nce (

)

Relaxation parameter

(c)

00 02 04 06 08 100

5

10

15

20

25

30O

ccur

renc

e (

)

Relaxation parameter

(d)

Figure 5 Percentage of occurrence of different relaxation parameter values during the analysis considering 119868(119905) = 119905 and (a) case 1 and (b)case 2 119868(119905) = 1199052 and (c) case 1 and (d) case 2

of analysis that is case 1 and 119868(119905) = 119905 As one can observein Figure 6(a) good results are still obtained taking intoaccount the iterative formulation in spite of the existing timedisconnections at the interface In Figure 6(b) the evolutionof the relaxation parameter is depicted taking into accountthis last configuration As one can observe in this caseoptimal relaxation parameter values are between 07 and 10and mostly concentrate on the interval (09 10) In fact itis expected that these values get closer to 10 when smallertime steps are considered In the present analysis an averagenumber of 492 iterations per time step is obtained (takinginto account 800 FEM time steps) which is a relatively lownumber illustrating the good performance of the technique(it must be remarked that a tight tolerance criterion wasadopted for the convergence of the iterative analysis)

42 Acoustic Waves In acoustic models a scalar wave equa-tion describes the acoustic pressure field evolution [1] In thiscase (1b) can be rewritten as (in this subsection frequency-domain analyses are focused)

nabla sdot (120581 (119909) nabla119901 (119909 120596)) + 1205962120588 (119909) 119901 (119909 120596) = 120574 (119909 120596) (14)

and (3) can be rewritten as

(119901 (119909+ 120596) minus 119901 (119909

minus 120596)) = 0 (15a)

(119902 (119909+ 120596) minus 119902 (119909

minus 120596)) = 119892 (119909 120596) (15b)

where 119901 is the hydrodynamic pressure and 120574 and 119892 standfor domain and surface sources respectively The parameters120588 and 120581 denote respectively the mass density and com-pressibility of the medium and its wave propagation velocity

Journal of Applied Mathematics 9

000 025 050 075

0

1

2

3

4

5

Point B

Point A

AnalyticalFEM-BEM

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

(a)

0 1000 2000 3000 4000

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

(b)Figure 6 Results considering different time steps for each domain (a) electric field intensity at points A and B (b) optimal relaxationparameters for each iterative step

50

50

x

y

R=10

120592 = 30000ms

120592 = 30000ms

120592 = 15000ms

S (minus50 00)

(a)

x (m)

y(m

)65

6

55

5

45

4

35135 14 145 15 155 16 165

(b)

Figure 7 (a) Sketch for the heterogeneous medium with multiple subregions (b) boundary and domain point distribution considering thespatial discretization of an inclusion and adjacent fluid

is specified as 120592 = (120581120588)12 The hydrodynamic fluxes on

the interfaces are represented by 119902 and they are defined by119902 = 120581 nabla119901 sdot n where n is the normal vector from domain1 to domain 2 Equation (15a) states that the pressure iscontinuous across the interface whereas (15b) states that theflux is continuous across the interface if there is no surfacesource

The advantages of using iterative coupling procedures arerevealed when more complex configurations are analyzedIn this subsection the case of a heterogeneous domaincomposed of a homogeneous fluid incorporating multiplecircular inclusions with different properties is analyzed

For this purpose consider the host medium to allow thepropagation of sound with a velocity of 1500ms and thismedium is excited by a line source located at 119909

119904= minus50m

and 119910119904= 00m Within this fluid consider the presence of

8 circular inclusions all of them are with unit radius andfilled with a different fluid allowing sound waves to travel at3000ms as depicted in Figure 7

The above-described system has been analyzed takinginto account the proposed iterative coupling proceduremaking use of the Kansarsquos method (KM) to model all theinclusions and of the method of fundamental solutions(MFS) to model the host fluid (see references [12 59ndash61]

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Page 3: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 3

(i) different subdomains can be analysed separatelyleading to smaller and better-conditioned systems ofequations (different solvers suitable for each sub-domain may be employed)

(ii) only interface routines are required when one wishesto use existing codes to build coupling algorithms(thus coupled systems may be solved by separateprogrammodules taking full advantage of specializedfeatures and disciplinary expertise)

(iii) matching nodes at common interfaces are notrequired greatly improving the flexibility and versa-tility of the coupled analyses especiallywhendifferentdiscretization methods are considered

(iv) matching time steps at common interfaces are notrequired (in time-domain analysis) allowing optimaltemporal discretizations within each sub-domainimproving accuracy and stability aspects

(v) nonlinear analyses (as well as other iterative-basedanalyses)may be carried out in the same iterative loopof the iterative coupling not introducing a relevantextra computational effort for the model

(vi) more efficient analyses can be obtained once theglobal model can be reduced to several subdomainswith reduced size matrices

As a matter of fact Gauzellino et al [55] compared theiterative domain decomposition and global solution takinginto account three-dimensional Helmholtz problems Theirnumerical results show that iterative domain decompositionmethods perform far better than globalmethods In additionthey observed that iterative domain decomposition methodsinvolving small subdomains work better than those withsubdomains involving a large number of elements Similarresults have been obtained by Soares et al [51] taking intoaccount two-dimensional Helmholtz problems

To give a detailed overview of the recent developmentsfound in many of the referred works the remainder ofthis paper will address a number of application examplesconcerning different phenomena and methods First thegoverning equations related to wave propagation models aregenerically and briefly presented In the sequence an effi-cient iterative coupling technique is described including themathematical derivation of the optimized relaxationmethod-ology Some numerical applications are finally presentedillustrating the accuracy performance and potentialities ofthe discussed procedures taking into account different wavepropagation models and discretization techniques

2 Governing Equations

Wave propagation phenomena may be generically describedby the following timefrequency-domain governing equa-tions

1198880(119909 119905) + 119888

1(119909) (119909 119905) + 119888

2(119909) (119909 119905)

+ 1198883(119909) 120597119891 (119906 (119909 119905)) = 0

(1a)

1198880 (119909 120596) minus 120596

21198881 (119909) 119906 (119909 120596) + 1198941205961198882 (

119909) 119906 (119909 120596)

+ 1198883(119909) 120597119891 (119906 (119909 120596)) = 0

(1b)

which can be further generalized in order to consider morecomplex behavior such as time varying coefficients (119888

119897(119909 119905)

119897 = 1 2 3) nonlinearities (119888119897(119909 119906(119909 119905)) 119897 = 0 1 2 3 etc)

Equation (1a) stands for the time domain governing equationwhereas (1b) stands for its frequency-domain counterpart(overbars indicate frequency-domain values) In these equa-tions 119906 represents the incognita field which can be scalarvectorial and so forth according to the physical model infocus 119888

119894stands for a general coefficient representation which

can as well be a scalar a tensor and so forth Overdots standfor time derivatives whereas 120597119891 indicates a spatial derivativeoperator The complex number is denoted by 119894 and the timefrequency and space domains are represented by 119905 120596 and119909 respectively (in this case 119909 isin Ω where Ω is the spatialdomain of the model)

The boundary conditions (119909 isin Γ where Γ is theboundary of the model) may be generically described as (forsimplicity from this point onwards overbars are no longerused to indicate frequency-domain values and 120589 stands for 119905or 120596 according to the case of analysis)

119891 (119906 (119909 120589) V (119909 120589)) = 119888 (119909 120589) (2)

where once again 119888 stands for known terms In (2) 119891stands for a generic function representing the combinationof its arguments The variable V which may be consideredprescribed at the boundary of the model is a functionof 119906 and it is usually expressed considering some normalprojection (normal to the boundary) of the spatial derivativesof 119906 (ie V = 120597119891

119899(119906))

To completely define the model initial conditions (whichare usually adopted null in frequency-domain analyses) mustalso be defined In this case a generic representation can begiven by 119891(119906(119909 119905 = 0) (119909 119905 = 0)) = 119888(119909) where notationanalogous to that of (2) is considered

Taking into account coupled models in which differentdomains interact by a common interface interface conditionsmust be stated indicating how the domains interactThis canbe generically expressed as

1198911(119906 (119909minus 120589) V (119909minus 120589)) = 119891

2(119906 (119909+ 120589) V (119909+ 120589)) (3)

where 119909 isin Γ119868 119909minus isin Γ

119868cup Ω1 119909+ isin Γ

119868cup Ω2 and Γ

119868is

the common interface between domains Ω1and Ω

2 In (3)

functions 1198911and 119891

2describe how the interaction between

the coupled domains takes place by relating their boundaryvalues on the common interface

3 Iterative Coupling Analysis

In order to enable the coupling between sectioned domainsof a global model an iterative procedure is employed herewhich performs a successive renewal of the relevant vari-ables at the common interfaces This approach is based onthe imposition of prescribed boundary conditions properlyevaluated at the interfaces of the sectioned domains allowingeach domain of the global model to be analyzed separately

4 Journal of Applied Mathematics

Since the sectioned domains are analyzed separately the rele-vant systems of equations are formed independently beforethe iterative process starts (in the case of linear analyses)and are kept constant along the iterative process renderinga very efficient procedure The separate treatment of thesectioned domains allows independent discretizations to beconsidered on each domain without any special requirementof matching nodes along the common interfaces Moreoverin the case of time-domain analysis different time-stepsmay also be considered for each domain Thus the couplingalgorithm can be presented for a generic case in whichthe interface nodes may not match and the interface timeinstants are disconnected allowing exploiting the benefits ofthe iterative coupling formulation

To ensure andor to speed up convergence a relaxationparameter 120582 is introduced in the iterative coupling algorithmThe effectiveness of the iterative process is strongly related tothe selection of this relaxation parameter since an inappro-priate selection for 120582 can significantly increase the number ofiterations in the analysis or even worse make convergenceunfeasible As it has been reported [49 51] frequency-domain analyses usually give rise to ill-posed problems andin these cases the convergence of simple iterative couplingalgorithms can either be too slow or unachievable In order todeal with ill-posed problems and ensure convergence of theiterative coupling algorithm an optimal iterative procedureis adopted here with optimal relaxation parameters beingcomputed at each iterative step As it is illustrated in thenext section the introduction of these optimal relaxationparameters allows the iterative coupling technique to bevery effective especially in the frequency domain ensuringconvergence at a low number of iterative steps

31 Iterative Algorithm Initially in the kth iterative step ofthe coupled analysis of domains 1 and 2 the so-called domain1 is analyzed and the variables 119906 or V at the common interfacesof the domain are computed taking into account prescribedvalues of V or 119906 at these common interfacesThese prescribedvalues of V or 119906 are provided from the previous iterative step(in the first iterative step null or previous time-step valuesmay be considered) Once the variables 119906 or V are computedthey are applied to evaluate the boundary conditions thatare prescribed at the common interfaces of domain 2 asdescribed by (3) Taking into account these prescribed 119906 or Vboundary conditions the so-called domain 2 is analyzed andthe variables V or 119906 at the common interfaces of the domainare computed Then the computed V or 119906 values are appliedto evaluate the boundary conditions that are prescribed atthe common interfaces of domain 1 reinitiating the iterativecycle A sketch of this cycle is depicted in Figure 1

As previously discussed relaxation parameters must beconsidered in order to ensure andor to speed up theconvergence of the iterative process Thus the values thatare computed after the analysis of the sectioned domain maybe combined with its previous iterative step counterpartrelaxing the computation of the actual iterative step valueMathematically this can be represented as follows

119910(119896+1)

= (120582) 119910(119896+120582)

+ (1 minus 120582) 119910(119896) (4)

where 120582 is the adopted relaxation parameter and 119910 stands for119906 or V according to the case of analysis one should note that119910(119896+120582) is the value computed at the end of the iterative step

before the application of the relaxation parameterA proper selection for 120582 at each iterative step is extremely

important for the effectiveness of the iterative couplingprocedure In order to obtain an easy to implement efficientand effective expression for the relaxation parameter compu-tation optimal 120582 values are deduced in Section 32

32 Optimal Relaxation Parameter In order to evaluate anoptimal relaxation parameter the following square errorfunctional is minimized here

120576 (120582) =

10038171003817100381710038171003817Y(119896+1) (120582) minus Y(119896) (120582)1003817100381710038171003817

1003817

2

(5)

where Y stands for a vector whose entries are 119906 or V valuescomputed at the common interfaces

Taking into account the relaxation of the field values forthe (119896 + 1) and (119896) iterations (6a) and (6b) may be writtenbased on the definition in (4)

Y(119896+1) = (120582)Y(119896+120582) + (1 minus 120582)Y(119896) (6a)

Y(119896) = (120582)Y(119896+120582minus1) + (1 minus 120582)Y(119896minus1) (6b)Substituting (6a) and (6b) into (5) yields

120576 (120582) =

10038171003817100381710038171003817(120582)W(119896+120582) + (1 minus 120582)W(119896)1003817100381710038171003817

1003817

2

= (1205822)

10038171003817100381710038171003817W(119896+120582)1003817100381710038171003817

1003817

2

+ 2120582 (1 minus 120582) (W(119896+120582)W(119896))

+ (1 minus 120582)210038171003817100381710038171003817W(119896)1003817100381710038171003817

1003817

2

(7)

where the inner product definition is employed (eg(WW) = W

2) and new variables as defined in thefollowing are considered

W (119896+120582) = Y(119896+120582) minus Y(119896+120582minus1) (8)To find the optimal 120582 that minimizes the functional 120576(120582)

(7) is differentiated with respect to 120582 and the result is set tozero described as follows

(120582)

10038171003817100381710038171003817W(119896+120582)1003817100381710038171003817

1003817

2

+ (1 minus 2120582) (W(119896+120582)W(119896))

+ (120582 minus 1)

10038171003817100381710038171003817W(119896)1003817100381710038171003817

1003817

2

= 0

(9)

Rearranging the terms in (9) yields

120582 =

(W(119896)W(119896) minusW(119896+120582))1003817100381710038171003817W(119896) minusW(119896+120582)100381710038171003817

1003817

2(10)

which is an easy to implement expression that provides anoptimal value for the relaxation parameter 120582 at each iterativestepThis expression requires a low computational cost whencompared to other alternatives that can be found in theliterature (see eg [28 29]) and it provides very good resultsas it has been reported taking into account different physicalmodels and domain analyses [43 44 51ndash54] The iterativeprocess is relatively insensitive to the value of the relaxationparameter adopted for the first iterative step and 120582 = 05 canbe considered in this case for instance

Journal of Applied Mathematics 5

Begi

nnin

g of

iter

ativ

e ana

lysis

Analysis of domain 1

Interface condition

considering space(time) compatibility

Interface condition

considering space(time) compatibility

Analysis of domain 2

computed computed

computedcomputed

Introduction of relaxation parameters

End

of it

erat

ive a

naly

sis

f1minus21 (uminus minus) = f1minus2

2 (u+ +)

f2minus11 (u+ +) = f2minus1

2 (uminus minus)

uminus or minus is u+ or + is

+ or u+ isminus or uminus is

Figure 1 Sketch of the iterative coupling algorithm

Interface of domain 1 Interface of

domain 2

y+1

y+2

y+3

y+4

y+5

y+6y+7

y+8

y+9

yminus1

yminus2

yminus3

yminus4

yminus5

(a)

yminus(tminus)

y+(t+)

yminus(tminus minus Δtminus)

y+(t+ minus Δt+)

tminust+tminus minus Δtminust+ minus Δt+

(b)

Figure 2 (a) Sketch for a spatial interpolation of nodal values on the interface 119910+1= 119868(119910

minus

1 119910minus

2) 119910+2= 119868(119910

minus

1 119910minus

2) 119910+3= 119868(119910

minus

2 119910minus

3) and so forth

(b) sketch for a temporal interpolation of time-step values on the interface 119910+(119905+) = 119868(119910minus(119905minus) 119910minus(119905minus minus Δ119905minus)) and so forth where 119868 stands fora linear interpolation function

33 Interface Compatibility As previously discussed inde-pendent spatial (and temporal in time-domain analysis)discretizations may be considered for each domain of themodel not requiring matching nodes (or equal time steps)at the common interfaces Thus special procedures must beemployed to ensure the interface spatial (and temporal) com-patibility In order to do so interpolation and extrapolationprocedures are considered here These procedures can begenerically described by

119910 (119909119894 120589) =

119869

sum

119895=1

120572119895119910 (119909119895 120589) (11a)

119910 (119909 119905119899) = 1205730119910 (119909 119905

119898) +

119869

sum

119895=1

120573119895119910 (119909 (119905 minus 119895Δ119905)

119898119899) (11b)

where (11a) stands for spatial interpolations and (11b) standsfor time interpolationsextrapolations (120572

119895and 120573

119895stand

for spatial interpolation coefficients and time interpola-tionextrapolation coefficients respectively where Δ119905 rep-resents the time step) In Figure 2 simple sketches for thespatial and temporal interpolation procedures are depictedtaking into account linear interpolations

Although time interpolations usually can be carried outwithout further difficulties time extrapolations may give riseto instabilities if not properly elaboratedThus extrapolationsshould be performed in consonancewith the field approxima-tions being adopted within each time step and with the timediscretization procedures being considered in the analysis inorder to formulate a consistent procedure Once a consistentmethodology is elaborated time interpolationextrapolation

6 Journal of Applied Mathematics

procedures can be employed with confidence as referred inthe literature [47 48] and illustrated in the next sectionOne should notice that usually different optimal (optimalin terms of accuracy stability and efficiency) time stepsare required when taking into account different numericalmethods spatial discretizations material properties physicalphenomena and so forth Thus in some cases consideringdifferent time steps within each domain of a coupled modelis of maximal importance to allow the effectiveness of theanalysis

Using space(time) interpolation(extrapolation) proce-dures optimal modeling of each sectioned domain may beachieved which is very important inwhat concerns flexibilityefficiency accuracy and stability aspects

4 Numerical Applications

In this section the general procedures previously discussedare particularized and briefly detailed taking into accountdifferent physicalmodels anddiscretization techniquesThusthe discussed iterative coupling methodology is appliedconsidering a wide range of wave propagation models andnumerical methods richly illustrating its performance andpotentialities

In this context time- and frequency-domain analysesare carried out here and electromagnetic acoustic andmechanical wave propagation phenomena (as well as theirinteractions) are discussed in the applications that followMoreover different numerical techniques (such as the finiteelement method the boundary element method and mesh-less methods) are applied to discretize the different domainsof the model illustrating the versatility and generality of thediscussed iterative method

41 Electromagnetic Waves In electromagnetic models vec-torial wave equations describe the electric and the magneticfield evolution [56 57] In this case (1a) can be rewritten as(in this subsection time-domain analyses are focused on)

nabla times (120583(119909)minus1nabla times E (119909 119905)) + 120576 (119909) E (119909 119905) = minus J (119909 119905) (12a)

nabla times (120576(119909)minus1nabla timesH (119909 119905)) + 120583 (119909) H (119909 119905)

= nabla times (120576(119909)minus1J (119909 119905))

(12b)

and (3) can be rewritten as

n (119909) times (E (119909+ 119905) minus E (119909minus 119905)) = 0 (13a)

(D (119909+ 119905) minusD (119909

minus 119905)) sdot n (119909) = 120588 (119909 119905) (13b)

(B (119909+ 119905) minus B (119909minus 119905)) sdot n (119909) = 0 (13c)

n (119909) times (H (119909+ 119905) minusH (119909

minus 119905)) = J (119909 119905) (13d)

where E and H are the electric and magnetic field intensityvectors respectively D and B represent the electric andmagnetic flux densities respectively and J and 120588 stand for theelectric current and electric charge density respectively Theparameters 120576 and 120583 denote respectively the permittivity and

permeability of themediumand itswave propagation velocityis specified as 119888 = (120576120583)

minus12 n is the normal vector fromdomain 1 to domain 2 Equations (13a) and (13b) state that thetangential component of E is continuous across the interfaceand that the normal component of D has a step of surfacecharge on the interface surface respectively Equations (13c)and (13d) state that the normal component ofB is continuousacross the interface and that the tangential component ofH iscontinuous across the interface if there is no surface currentpresent respectively

In the present application the electromagnetic fieldssurrounding infinitely long wires are studied [41] Two casesof analysis are focused here namely (a) case 1 where onewireis considered (b) case 2 where two wires are employed Forboth cases the wires are carrying time-dependent currents(ie 119868(119905) = 119905 or 119868(119905) = 119905

2) and they are located along theadopted 119911-axis A sketch of the model is depicted in Figure 3

The spatial and temporal evolution of the electric fieldintensity vector is analyzed here taking into account a finiteelement method (FEM)mdashboundary element method (BEM)coupled formulation In this context the FEM is appliedto model the region close to the wires whereas the BEMsimulates the remaining infinity domain As it is well knownthe BEM employs fundamental solutions which fulfill theradiation conditionThus this formulation is very suitable toperform infinite domain analysis once reflected waves frominfinity are avoided [58]

The adopted spatial discretization is also described inFigure 3 In this case 2344 linear triangular finite elementsand 80 linear boundary elements are employed in the analyses(see references [57 58] for more details regarding the FEMand the BEMapplied to electromagnetic analyses)The radiusof the FEM-BEM interface is defined by 119877 = 1m andmatching nodes are considered at the interface For temporaldiscretization the selected time step is given byΔ119905 = 5sdot10minus11sfor both domainsThephysical properties of themedium (air)are 120583 = 12566 sdot 10minus6Hm and 120576 = 88544 sdot 10minus12 Fm

Figure 4 shows the modulus of the electric field intensityobtained at points A and B (see Figure 3) considering theiterative couplingmethodology Analytical time histories [58]are also depicted in Figure 4 highlighting the good accuracyof the numerical results In Figure 5 charts are displayedindicating the percentage of occurrence of different relax-ation parameter values (evaluated according to expression(10)) in each analysis As can be observed for all consideredcases optimal relaxation parameters aremostly in the interval07 le 120582 le 08 In fact an optimal relaxation parameterselection is extremely case dependent It is function of thephysical properties of the model geometric aspects adoptedspatial and temporal discretizations and so forth Equation(10) provides a simple expression to evaluate this complexparameter

In order to illustrate the effectiveness of the methodologywhen considering different time discretizations for differentdomains Figure 6 depicts results that are computed consider-ing Δ119905 = 25 sdot 10minus11 s for the FEM and Δ119905 = 20 sdot 10minus10 s for theBEM (ie a difference of 8 times between the time steps) Forsimplicity results are presented considering just the first case

Journal of Applied Mathematics 7

R

Wire AB

FEM-BEMinterface

x

y

z

(a)

R

Wire A

B

interface

Wire

FEM-BEM

x

y

z

(b)

Figure 3 Sketch of the electromagnetic models and adopted FEMBEM spatial discretizations (a) case 1 one wire (b) case 2 two wires

000 025 050 075

0

1

2

3

4

5

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(a)

000 025 050 075

0

2

4

6

8

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(b)

AnalyticalFEM-BEM

000 025 050 075

0

1

2

3

4

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(c)

AnalyticalFEM-BEM

000 025 050 075

0

2

4

6

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(d)

Figure 4 Time history results for the electric field intensity at points A and B considering 119868(119905) = 119905 and (a) case 1 and (b) case 2 119868(119905) = 1199052 and(c) case 1 and (d) case 2

8 Journal of Applied Mathematics

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(a)

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(b)

00 02 04 06 08 100

5

10

15

20

25

30

Occ

urre

nce (

)

Relaxation parameter

(c)

00 02 04 06 08 100

5

10

15

20

25

30O

ccur

renc

e (

)

Relaxation parameter

(d)

Figure 5 Percentage of occurrence of different relaxation parameter values during the analysis considering 119868(119905) = 119905 and (a) case 1 and (b)case 2 119868(119905) = 1199052 and (c) case 1 and (d) case 2

of analysis that is case 1 and 119868(119905) = 119905 As one can observein Figure 6(a) good results are still obtained taking intoaccount the iterative formulation in spite of the existing timedisconnections at the interface In Figure 6(b) the evolutionof the relaxation parameter is depicted taking into accountthis last configuration As one can observe in this caseoptimal relaxation parameter values are between 07 and 10and mostly concentrate on the interval (09 10) In fact itis expected that these values get closer to 10 when smallertime steps are considered In the present analysis an averagenumber of 492 iterations per time step is obtained (takinginto account 800 FEM time steps) which is a relatively lownumber illustrating the good performance of the technique(it must be remarked that a tight tolerance criterion wasadopted for the convergence of the iterative analysis)

42 Acoustic Waves In acoustic models a scalar wave equa-tion describes the acoustic pressure field evolution [1] In thiscase (1b) can be rewritten as (in this subsection frequency-domain analyses are focused)

nabla sdot (120581 (119909) nabla119901 (119909 120596)) + 1205962120588 (119909) 119901 (119909 120596) = 120574 (119909 120596) (14)

and (3) can be rewritten as

(119901 (119909+ 120596) minus 119901 (119909

minus 120596)) = 0 (15a)

(119902 (119909+ 120596) minus 119902 (119909

minus 120596)) = 119892 (119909 120596) (15b)

where 119901 is the hydrodynamic pressure and 120574 and 119892 standfor domain and surface sources respectively The parameters120588 and 120581 denote respectively the mass density and com-pressibility of the medium and its wave propagation velocity

Journal of Applied Mathematics 9

000 025 050 075

0

1

2

3

4

5

Point B

Point A

AnalyticalFEM-BEM

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

(a)

0 1000 2000 3000 4000

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

(b)Figure 6 Results considering different time steps for each domain (a) electric field intensity at points A and B (b) optimal relaxationparameters for each iterative step

50

50

x

y

R=10

120592 = 30000ms

120592 = 30000ms

120592 = 15000ms

S (minus50 00)

(a)

x (m)

y(m

)65

6

55

5

45

4

35135 14 145 15 155 16 165

(b)

Figure 7 (a) Sketch for the heterogeneous medium with multiple subregions (b) boundary and domain point distribution considering thespatial discretization of an inclusion and adjacent fluid

is specified as 120592 = (120581120588)12 The hydrodynamic fluxes on

the interfaces are represented by 119902 and they are defined by119902 = 120581 nabla119901 sdot n where n is the normal vector from domain1 to domain 2 Equation (15a) states that the pressure iscontinuous across the interface whereas (15b) states that theflux is continuous across the interface if there is no surfacesource

The advantages of using iterative coupling procedures arerevealed when more complex configurations are analyzedIn this subsection the case of a heterogeneous domaincomposed of a homogeneous fluid incorporating multiplecircular inclusions with different properties is analyzed

For this purpose consider the host medium to allow thepropagation of sound with a velocity of 1500ms and thismedium is excited by a line source located at 119909

119904= minus50m

and 119910119904= 00m Within this fluid consider the presence of

8 circular inclusions all of them are with unit radius andfilled with a different fluid allowing sound waves to travel at3000ms as depicted in Figure 7

The above-described system has been analyzed takinginto account the proposed iterative coupling proceduremaking use of the Kansarsquos method (KM) to model all theinclusions and of the method of fundamental solutions(MFS) to model the host fluid (see references [12 59ndash61]

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

4 Journal of Applied Mathematics

Since the sectioned domains are analyzed separately the rele-vant systems of equations are formed independently beforethe iterative process starts (in the case of linear analyses)and are kept constant along the iterative process renderinga very efficient procedure The separate treatment of thesectioned domains allows independent discretizations to beconsidered on each domain without any special requirementof matching nodes along the common interfaces Moreoverin the case of time-domain analysis different time-stepsmay also be considered for each domain Thus the couplingalgorithm can be presented for a generic case in whichthe interface nodes may not match and the interface timeinstants are disconnected allowing exploiting the benefits ofthe iterative coupling formulation

To ensure andor to speed up convergence a relaxationparameter 120582 is introduced in the iterative coupling algorithmThe effectiveness of the iterative process is strongly related tothe selection of this relaxation parameter since an inappro-priate selection for 120582 can significantly increase the number ofiterations in the analysis or even worse make convergenceunfeasible As it has been reported [49 51] frequency-domain analyses usually give rise to ill-posed problems andin these cases the convergence of simple iterative couplingalgorithms can either be too slow or unachievable In order todeal with ill-posed problems and ensure convergence of theiterative coupling algorithm an optimal iterative procedureis adopted here with optimal relaxation parameters beingcomputed at each iterative step As it is illustrated in thenext section the introduction of these optimal relaxationparameters allows the iterative coupling technique to bevery effective especially in the frequency domain ensuringconvergence at a low number of iterative steps

31 Iterative Algorithm Initially in the kth iterative step ofthe coupled analysis of domains 1 and 2 the so-called domain1 is analyzed and the variables 119906 or V at the common interfacesof the domain are computed taking into account prescribedvalues of V or 119906 at these common interfacesThese prescribedvalues of V or 119906 are provided from the previous iterative step(in the first iterative step null or previous time-step valuesmay be considered) Once the variables 119906 or V are computedthey are applied to evaluate the boundary conditions thatare prescribed at the common interfaces of domain 2 asdescribed by (3) Taking into account these prescribed 119906 or Vboundary conditions the so-called domain 2 is analyzed andthe variables V or 119906 at the common interfaces of the domainare computed Then the computed V or 119906 values are appliedto evaluate the boundary conditions that are prescribed atthe common interfaces of domain 1 reinitiating the iterativecycle A sketch of this cycle is depicted in Figure 1

As previously discussed relaxation parameters must beconsidered in order to ensure andor to speed up theconvergence of the iterative process Thus the values thatare computed after the analysis of the sectioned domain maybe combined with its previous iterative step counterpartrelaxing the computation of the actual iterative step valueMathematically this can be represented as follows

119910(119896+1)

= (120582) 119910(119896+120582)

+ (1 minus 120582) 119910(119896) (4)

where 120582 is the adopted relaxation parameter and 119910 stands for119906 or V according to the case of analysis one should note that119910(119896+120582) is the value computed at the end of the iterative step

before the application of the relaxation parameterA proper selection for 120582 at each iterative step is extremely

important for the effectiveness of the iterative couplingprocedure In order to obtain an easy to implement efficientand effective expression for the relaxation parameter compu-tation optimal 120582 values are deduced in Section 32

32 Optimal Relaxation Parameter In order to evaluate anoptimal relaxation parameter the following square errorfunctional is minimized here

120576 (120582) =

10038171003817100381710038171003817Y(119896+1) (120582) minus Y(119896) (120582)1003817100381710038171003817

1003817

2

(5)

where Y stands for a vector whose entries are 119906 or V valuescomputed at the common interfaces

Taking into account the relaxation of the field values forthe (119896 + 1) and (119896) iterations (6a) and (6b) may be writtenbased on the definition in (4)

Y(119896+1) = (120582)Y(119896+120582) + (1 minus 120582)Y(119896) (6a)

Y(119896) = (120582)Y(119896+120582minus1) + (1 minus 120582)Y(119896minus1) (6b)Substituting (6a) and (6b) into (5) yields

120576 (120582) =

10038171003817100381710038171003817(120582)W(119896+120582) + (1 minus 120582)W(119896)1003817100381710038171003817

1003817

2

= (1205822)

10038171003817100381710038171003817W(119896+120582)1003817100381710038171003817

1003817

2

+ 2120582 (1 minus 120582) (W(119896+120582)W(119896))

+ (1 minus 120582)210038171003817100381710038171003817W(119896)1003817100381710038171003817

1003817

2

(7)

where the inner product definition is employed (eg(WW) = W

2) and new variables as defined in thefollowing are considered

W (119896+120582) = Y(119896+120582) minus Y(119896+120582minus1) (8)To find the optimal 120582 that minimizes the functional 120576(120582)

(7) is differentiated with respect to 120582 and the result is set tozero described as follows

(120582)

10038171003817100381710038171003817W(119896+120582)1003817100381710038171003817

1003817

2

+ (1 minus 2120582) (W(119896+120582)W(119896))

+ (120582 minus 1)

10038171003817100381710038171003817W(119896)1003817100381710038171003817

1003817

2

= 0

(9)

Rearranging the terms in (9) yields

120582 =

(W(119896)W(119896) minusW(119896+120582))1003817100381710038171003817W(119896) minusW(119896+120582)100381710038171003817

1003817

2(10)

which is an easy to implement expression that provides anoptimal value for the relaxation parameter 120582 at each iterativestepThis expression requires a low computational cost whencompared to other alternatives that can be found in theliterature (see eg [28 29]) and it provides very good resultsas it has been reported taking into account different physicalmodels and domain analyses [43 44 51ndash54] The iterativeprocess is relatively insensitive to the value of the relaxationparameter adopted for the first iterative step and 120582 = 05 canbe considered in this case for instance

Journal of Applied Mathematics 5

Begi

nnin

g of

iter

ativ

e ana

lysis

Analysis of domain 1

Interface condition

considering space(time) compatibility

Interface condition

considering space(time) compatibility

Analysis of domain 2

computed computed

computedcomputed

Introduction of relaxation parameters

End

of it

erat

ive a

naly

sis

f1minus21 (uminus minus) = f1minus2

2 (u+ +)

f2minus11 (u+ +) = f2minus1

2 (uminus minus)

uminus or minus is u+ or + is

+ or u+ isminus or uminus is

Figure 1 Sketch of the iterative coupling algorithm

Interface of domain 1 Interface of

domain 2

y+1

y+2

y+3

y+4

y+5

y+6y+7

y+8

y+9

yminus1

yminus2

yminus3

yminus4

yminus5

(a)

yminus(tminus)

y+(t+)

yminus(tminus minus Δtminus)

y+(t+ minus Δt+)

tminust+tminus minus Δtminust+ minus Δt+

(b)

Figure 2 (a) Sketch for a spatial interpolation of nodal values on the interface 119910+1= 119868(119910

minus

1 119910minus

2) 119910+2= 119868(119910

minus

1 119910minus

2) 119910+3= 119868(119910

minus

2 119910minus

3) and so forth

(b) sketch for a temporal interpolation of time-step values on the interface 119910+(119905+) = 119868(119910minus(119905minus) 119910minus(119905minus minus Δ119905minus)) and so forth where 119868 stands fora linear interpolation function

33 Interface Compatibility As previously discussed inde-pendent spatial (and temporal in time-domain analysis)discretizations may be considered for each domain of themodel not requiring matching nodes (or equal time steps)at the common interfaces Thus special procedures must beemployed to ensure the interface spatial (and temporal) com-patibility In order to do so interpolation and extrapolationprocedures are considered here These procedures can begenerically described by

119910 (119909119894 120589) =

119869

sum

119895=1

120572119895119910 (119909119895 120589) (11a)

119910 (119909 119905119899) = 1205730119910 (119909 119905

119898) +

119869

sum

119895=1

120573119895119910 (119909 (119905 minus 119895Δ119905)

119898119899) (11b)

where (11a) stands for spatial interpolations and (11b) standsfor time interpolationsextrapolations (120572

119895and 120573

119895stand

for spatial interpolation coefficients and time interpola-tionextrapolation coefficients respectively where Δ119905 rep-resents the time step) In Figure 2 simple sketches for thespatial and temporal interpolation procedures are depictedtaking into account linear interpolations

Although time interpolations usually can be carried outwithout further difficulties time extrapolations may give riseto instabilities if not properly elaboratedThus extrapolationsshould be performed in consonancewith the field approxima-tions being adopted within each time step and with the timediscretization procedures being considered in the analysis inorder to formulate a consistent procedure Once a consistentmethodology is elaborated time interpolationextrapolation

6 Journal of Applied Mathematics

procedures can be employed with confidence as referred inthe literature [47 48] and illustrated in the next sectionOne should notice that usually different optimal (optimalin terms of accuracy stability and efficiency) time stepsare required when taking into account different numericalmethods spatial discretizations material properties physicalphenomena and so forth Thus in some cases consideringdifferent time steps within each domain of a coupled modelis of maximal importance to allow the effectiveness of theanalysis

Using space(time) interpolation(extrapolation) proce-dures optimal modeling of each sectioned domain may beachieved which is very important inwhat concerns flexibilityefficiency accuracy and stability aspects

4 Numerical Applications

In this section the general procedures previously discussedare particularized and briefly detailed taking into accountdifferent physicalmodels anddiscretization techniquesThusthe discussed iterative coupling methodology is appliedconsidering a wide range of wave propagation models andnumerical methods richly illustrating its performance andpotentialities

In this context time- and frequency-domain analysesare carried out here and electromagnetic acoustic andmechanical wave propagation phenomena (as well as theirinteractions) are discussed in the applications that followMoreover different numerical techniques (such as the finiteelement method the boundary element method and mesh-less methods) are applied to discretize the different domainsof the model illustrating the versatility and generality of thediscussed iterative method

41 Electromagnetic Waves In electromagnetic models vec-torial wave equations describe the electric and the magneticfield evolution [56 57] In this case (1a) can be rewritten as(in this subsection time-domain analyses are focused on)

nabla times (120583(119909)minus1nabla times E (119909 119905)) + 120576 (119909) E (119909 119905) = minus J (119909 119905) (12a)

nabla times (120576(119909)minus1nabla timesH (119909 119905)) + 120583 (119909) H (119909 119905)

= nabla times (120576(119909)minus1J (119909 119905))

(12b)

and (3) can be rewritten as

n (119909) times (E (119909+ 119905) minus E (119909minus 119905)) = 0 (13a)

(D (119909+ 119905) minusD (119909

minus 119905)) sdot n (119909) = 120588 (119909 119905) (13b)

(B (119909+ 119905) minus B (119909minus 119905)) sdot n (119909) = 0 (13c)

n (119909) times (H (119909+ 119905) minusH (119909

minus 119905)) = J (119909 119905) (13d)

where E and H are the electric and magnetic field intensityvectors respectively D and B represent the electric andmagnetic flux densities respectively and J and 120588 stand for theelectric current and electric charge density respectively Theparameters 120576 and 120583 denote respectively the permittivity and

permeability of themediumand itswave propagation velocityis specified as 119888 = (120576120583)

minus12 n is the normal vector fromdomain 1 to domain 2 Equations (13a) and (13b) state that thetangential component of E is continuous across the interfaceand that the normal component of D has a step of surfacecharge on the interface surface respectively Equations (13c)and (13d) state that the normal component ofB is continuousacross the interface and that the tangential component ofH iscontinuous across the interface if there is no surface currentpresent respectively

In the present application the electromagnetic fieldssurrounding infinitely long wires are studied [41] Two casesof analysis are focused here namely (a) case 1 where onewireis considered (b) case 2 where two wires are employed Forboth cases the wires are carrying time-dependent currents(ie 119868(119905) = 119905 or 119868(119905) = 119905

2) and they are located along theadopted 119911-axis A sketch of the model is depicted in Figure 3

The spatial and temporal evolution of the electric fieldintensity vector is analyzed here taking into account a finiteelement method (FEM)mdashboundary element method (BEM)coupled formulation In this context the FEM is appliedto model the region close to the wires whereas the BEMsimulates the remaining infinity domain As it is well knownthe BEM employs fundamental solutions which fulfill theradiation conditionThus this formulation is very suitable toperform infinite domain analysis once reflected waves frominfinity are avoided [58]

The adopted spatial discretization is also described inFigure 3 In this case 2344 linear triangular finite elementsand 80 linear boundary elements are employed in the analyses(see references [57 58] for more details regarding the FEMand the BEMapplied to electromagnetic analyses)The radiusof the FEM-BEM interface is defined by 119877 = 1m andmatching nodes are considered at the interface For temporaldiscretization the selected time step is given byΔ119905 = 5sdot10minus11sfor both domainsThephysical properties of themedium (air)are 120583 = 12566 sdot 10minus6Hm and 120576 = 88544 sdot 10minus12 Fm

Figure 4 shows the modulus of the electric field intensityobtained at points A and B (see Figure 3) considering theiterative couplingmethodology Analytical time histories [58]are also depicted in Figure 4 highlighting the good accuracyof the numerical results In Figure 5 charts are displayedindicating the percentage of occurrence of different relax-ation parameter values (evaluated according to expression(10)) in each analysis As can be observed for all consideredcases optimal relaxation parameters aremostly in the interval07 le 120582 le 08 In fact an optimal relaxation parameterselection is extremely case dependent It is function of thephysical properties of the model geometric aspects adoptedspatial and temporal discretizations and so forth Equation(10) provides a simple expression to evaluate this complexparameter

In order to illustrate the effectiveness of the methodologywhen considering different time discretizations for differentdomains Figure 6 depicts results that are computed consider-ing Δ119905 = 25 sdot 10minus11 s for the FEM and Δ119905 = 20 sdot 10minus10 s for theBEM (ie a difference of 8 times between the time steps) Forsimplicity results are presented considering just the first case

Journal of Applied Mathematics 7

R

Wire AB

FEM-BEMinterface

x

y

z

(a)

R

Wire A

B

interface

Wire

FEM-BEM

x

y

z

(b)

Figure 3 Sketch of the electromagnetic models and adopted FEMBEM spatial discretizations (a) case 1 one wire (b) case 2 two wires

000 025 050 075

0

1

2

3

4

5

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(a)

000 025 050 075

0

2

4

6

8

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(b)

AnalyticalFEM-BEM

000 025 050 075

0

1

2

3

4

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(c)

AnalyticalFEM-BEM

000 025 050 075

0

2

4

6

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(d)

Figure 4 Time history results for the electric field intensity at points A and B considering 119868(119905) = 119905 and (a) case 1 and (b) case 2 119868(119905) = 1199052 and(c) case 1 and (d) case 2

8 Journal of Applied Mathematics

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(a)

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(b)

00 02 04 06 08 100

5

10

15

20

25

30

Occ

urre

nce (

)

Relaxation parameter

(c)

00 02 04 06 08 100

5

10

15

20

25

30O

ccur

renc

e (

)

Relaxation parameter

(d)

Figure 5 Percentage of occurrence of different relaxation parameter values during the analysis considering 119868(119905) = 119905 and (a) case 1 and (b)case 2 119868(119905) = 1199052 and (c) case 1 and (d) case 2

of analysis that is case 1 and 119868(119905) = 119905 As one can observein Figure 6(a) good results are still obtained taking intoaccount the iterative formulation in spite of the existing timedisconnections at the interface In Figure 6(b) the evolutionof the relaxation parameter is depicted taking into accountthis last configuration As one can observe in this caseoptimal relaxation parameter values are between 07 and 10and mostly concentrate on the interval (09 10) In fact itis expected that these values get closer to 10 when smallertime steps are considered In the present analysis an averagenumber of 492 iterations per time step is obtained (takinginto account 800 FEM time steps) which is a relatively lownumber illustrating the good performance of the technique(it must be remarked that a tight tolerance criterion wasadopted for the convergence of the iterative analysis)

42 Acoustic Waves In acoustic models a scalar wave equa-tion describes the acoustic pressure field evolution [1] In thiscase (1b) can be rewritten as (in this subsection frequency-domain analyses are focused)

nabla sdot (120581 (119909) nabla119901 (119909 120596)) + 1205962120588 (119909) 119901 (119909 120596) = 120574 (119909 120596) (14)

and (3) can be rewritten as

(119901 (119909+ 120596) minus 119901 (119909

minus 120596)) = 0 (15a)

(119902 (119909+ 120596) minus 119902 (119909

minus 120596)) = 119892 (119909 120596) (15b)

where 119901 is the hydrodynamic pressure and 120574 and 119892 standfor domain and surface sources respectively The parameters120588 and 120581 denote respectively the mass density and com-pressibility of the medium and its wave propagation velocity

Journal of Applied Mathematics 9

000 025 050 075

0

1

2

3

4

5

Point B

Point A

AnalyticalFEM-BEM

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

(a)

0 1000 2000 3000 4000

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

(b)Figure 6 Results considering different time steps for each domain (a) electric field intensity at points A and B (b) optimal relaxationparameters for each iterative step

50

50

x

y

R=10

120592 = 30000ms

120592 = 30000ms

120592 = 15000ms

S (minus50 00)

(a)

x (m)

y(m

)65

6

55

5

45

4

35135 14 145 15 155 16 165

(b)

Figure 7 (a) Sketch for the heterogeneous medium with multiple subregions (b) boundary and domain point distribution considering thespatial discretization of an inclusion and adjacent fluid

is specified as 120592 = (120581120588)12 The hydrodynamic fluxes on

the interfaces are represented by 119902 and they are defined by119902 = 120581 nabla119901 sdot n where n is the normal vector from domain1 to domain 2 Equation (15a) states that the pressure iscontinuous across the interface whereas (15b) states that theflux is continuous across the interface if there is no surfacesource

The advantages of using iterative coupling procedures arerevealed when more complex configurations are analyzedIn this subsection the case of a heterogeneous domaincomposed of a homogeneous fluid incorporating multiplecircular inclusions with different properties is analyzed

For this purpose consider the host medium to allow thepropagation of sound with a velocity of 1500ms and thismedium is excited by a line source located at 119909

119904= minus50m

and 119910119904= 00m Within this fluid consider the presence of

8 circular inclusions all of them are with unit radius andfilled with a different fluid allowing sound waves to travel at3000ms as depicted in Figure 7

The above-described system has been analyzed takinginto account the proposed iterative coupling proceduremaking use of the Kansarsquos method (KM) to model all theinclusions and of the method of fundamental solutions(MFS) to model the host fluid (see references [12 59ndash61]

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 5: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 5

Begi

nnin

g of

iter

ativ

e ana

lysis

Analysis of domain 1

Interface condition

considering space(time) compatibility

Interface condition

considering space(time) compatibility

Analysis of domain 2

computed computed

computedcomputed

Introduction of relaxation parameters

End

of it

erat

ive a

naly

sis

f1minus21 (uminus minus) = f1minus2

2 (u+ +)

f2minus11 (u+ +) = f2minus1

2 (uminus minus)

uminus or minus is u+ or + is

+ or u+ isminus or uminus is

Figure 1 Sketch of the iterative coupling algorithm

Interface of domain 1 Interface of

domain 2

y+1

y+2

y+3

y+4

y+5

y+6y+7

y+8

y+9

yminus1

yminus2

yminus3

yminus4

yminus5

(a)

yminus(tminus)

y+(t+)

yminus(tminus minus Δtminus)

y+(t+ minus Δt+)

tminust+tminus minus Δtminust+ minus Δt+

(b)

Figure 2 (a) Sketch for a spatial interpolation of nodal values on the interface 119910+1= 119868(119910

minus

1 119910minus

2) 119910+2= 119868(119910

minus

1 119910minus

2) 119910+3= 119868(119910

minus

2 119910minus

3) and so forth

(b) sketch for a temporal interpolation of time-step values on the interface 119910+(119905+) = 119868(119910minus(119905minus) 119910minus(119905minus minus Δ119905minus)) and so forth where 119868 stands fora linear interpolation function

33 Interface Compatibility As previously discussed inde-pendent spatial (and temporal in time-domain analysis)discretizations may be considered for each domain of themodel not requiring matching nodes (or equal time steps)at the common interfaces Thus special procedures must beemployed to ensure the interface spatial (and temporal) com-patibility In order to do so interpolation and extrapolationprocedures are considered here These procedures can begenerically described by

119910 (119909119894 120589) =

119869

sum

119895=1

120572119895119910 (119909119895 120589) (11a)

119910 (119909 119905119899) = 1205730119910 (119909 119905

119898) +

119869

sum

119895=1

120573119895119910 (119909 (119905 minus 119895Δ119905)

119898119899) (11b)

where (11a) stands for spatial interpolations and (11b) standsfor time interpolationsextrapolations (120572

119895and 120573

119895stand

for spatial interpolation coefficients and time interpola-tionextrapolation coefficients respectively where Δ119905 rep-resents the time step) In Figure 2 simple sketches for thespatial and temporal interpolation procedures are depictedtaking into account linear interpolations

Although time interpolations usually can be carried outwithout further difficulties time extrapolations may give riseto instabilities if not properly elaboratedThus extrapolationsshould be performed in consonancewith the field approxima-tions being adopted within each time step and with the timediscretization procedures being considered in the analysis inorder to formulate a consistent procedure Once a consistentmethodology is elaborated time interpolationextrapolation

6 Journal of Applied Mathematics

procedures can be employed with confidence as referred inthe literature [47 48] and illustrated in the next sectionOne should notice that usually different optimal (optimalin terms of accuracy stability and efficiency) time stepsare required when taking into account different numericalmethods spatial discretizations material properties physicalphenomena and so forth Thus in some cases consideringdifferent time steps within each domain of a coupled modelis of maximal importance to allow the effectiveness of theanalysis

Using space(time) interpolation(extrapolation) proce-dures optimal modeling of each sectioned domain may beachieved which is very important inwhat concerns flexibilityefficiency accuracy and stability aspects

4 Numerical Applications

In this section the general procedures previously discussedare particularized and briefly detailed taking into accountdifferent physicalmodels anddiscretization techniquesThusthe discussed iterative coupling methodology is appliedconsidering a wide range of wave propagation models andnumerical methods richly illustrating its performance andpotentialities

In this context time- and frequency-domain analysesare carried out here and electromagnetic acoustic andmechanical wave propagation phenomena (as well as theirinteractions) are discussed in the applications that followMoreover different numerical techniques (such as the finiteelement method the boundary element method and mesh-less methods) are applied to discretize the different domainsof the model illustrating the versatility and generality of thediscussed iterative method

41 Electromagnetic Waves In electromagnetic models vec-torial wave equations describe the electric and the magneticfield evolution [56 57] In this case (1a) can be rewritten as(in this subsection time-domain analyses are focused on)

nabla times (120583(119909)minus1nabla times E (119909 119905)) + 120576 (119909) E (119909 119905) = minus J (119909 119905) (12a)

nabla times (120576(119909)minus1nabla timesH (119909 119905)) + 120583 (119909) H (119909 119905)

= nabla times (120576(119909)minus1J (119909 119905))

(12b)

and (3) can be rewritten as

n (119909) times (E (119909+ 119905) minus E (119909minus 119905)) = 0 (13a)

(D (119909+ 119905) minusD (119909

minus 119905)) sdot n (119909) = 120588 (119909 119905) (13b)

(B (119909+ 119905) minus B (119909minus 119905)) sdot n (119909) = 0 (13c)

n (119909) times (H (119909+ 119905) minusH (119909

minus 119905)) = J (119909 119905) (13d)

where E and H are the electric and magnetic field intensityvectors respectively D and B represent the electric andmagnetic flux densities respectively and J and 120588 stand for theelectric current and electric charge density respectively Theparameters 120576 and 120583 denote respectively the permittivity and

permeability of themediumand itswave propagation velocityis specified as 119888 = (120576120583)

minus12 n is the normal vector fromdomain 1 to domain 2 Equations (13a) and (13b) state that thetangential component of E is continuous across the interfaceand that the normal component of D has a step of surfacecharge on the interface surface respectively Equations (13c)and (13d) state that the normal component ofB is continuousacross the interface and that the tangential component ofH iscontinuous across the interface if there is no surface currentpresent respectively

In the present application the electromagnetic fieldssurrounding infinitely long wires are studied [41] Two casesof analysis are focused here namely (a) case 1 where onewireis considered (b) case 2 where two wires are employed Forboth cases the wires are carrying time-dependent currents(ie 119868(119905) = 119905 or 119868(119905) = 119905

2) and they are located along theadopted 119911-axis A sketch of the model is depicted in Figure 3

The spatial and temporal evolution of the electric fieldintensity vector is analyzed here taking into account a finiteelement method (FEM)mdashboundary element method (BEM)coupled formulation In this context the FEM is appliedto model the region close to the wires whereas the BEMsimulates the remaining infinity domain As it is well knownthe BEM employs fundamental solutions which fulfill theradiation conditionThus this formulation is very suitable toperform infinite domain analysis once reflected waves frominfinity are avoided [58]

The adopted spatial discretization is also described inFigure 3 In this case 2344 linear triangular finite elementsand 80 linear boundary elements are employed in the analyses(see references [57 58] for more details regarding the FEMand the BEMapplied to electromagnetic analyses)The radiusof the FEM-BEM interface is defined by 119877 = 1m andmatching nodes are considered at the interface For temporaldiscretization the selected time step is given byΔ119905 = 5sdot10minus11sfor both domainsThephysical properties of themedium (air)are 120583 = 12566 sdot 10minus6Hm and 120576 = 88544 sdot 10minus12 Fm

Figure 4 shows the modulus of the electric field intensityobtained at points A and B (see Figure 3) considering theiterative couplingmethodology Analytical time histories [58]are also depicted in Figure 4 highlighting the good accuracyof the numerical results In Figure 5 charts are displayedindicating the percentage of occurrence of different relax-ation parameter values (evaluated according to expression(10)) in each analysis As can be observed for all consideredcases optimal relaxation parameters aremostly in the interval07 le 120582 le 08 In fact an optimal relaxation parameterselection is extremely case dependent It is function of thephysical properties of the model geometric aspects adoptedspatial and temporal discretizations and so forth Equation(10) provides a simple expression to evaluate this complexparameter

In order to illustrate the effectiveness of the methodologywhen considering different time discretizations for differentdomains Figure 6 depicts results that are computed consider-ing Δ119905 = 25 sdot 10minus11 s for the FEM and Δ119905 = 20 sdot 10minus10 s for theBEM (ie a difference of 8 times between the time steps) Forsimplicity results are presented considering just the first case

Journal of Applied Mathematics 7

R

Wire AB

FEM-BEMinterface

x

y

z

(a)

R

Wire A

B

interface

Wire

FEM-BEM

x

y

z

(b)

Figure 3 Sketch of the electromagnetic models and adopted FEMBEM spatial discretizations (a) case 1 one wire (b) case 2 two wires

000 025 050 075

0

1

2

3

4

5

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(a)

000 025 050 075

0

2

4

6

8

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(b)

AnalyticalFEM-BEM

000 025 050 075

0

1

2

3

4

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(c)

AnalyticalFEM-BEM

000 025 050 075

0

2

4

6

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(d)

Figure 4 Time history results for the electric field intensity at points A and B considering 119868(119905) = 119905 and (a) case 1 and (b) case 2 119868(119905) = 1199052 and(c) case 1 and (d) case 2

8 Journal of Applied Mathematics

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(a)

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(b)

00 02 04 06 08 100

5

10

15

20

25

30

Occ

urre

nce (

)

Relaxation parameter

(c)

00 02 04 06 08 100

5

10

15

20

25

30O

ccur

renc

e (

)

Relaxation parameter

(d)

Figure 5 Percentage of occurrence of different relaxation parameter values during the analysis considering 119868(119905) = 119905 and (a) case 1 and (b)case 2 119868(119905) = 1199052 and (c) case 1 and (d) case 2

of analysis that is case 1 and 119868(119905) = 119905 As one can observein Figure 6(a) good results are still obtained taking intoaccount the iterative formulation in spite of the existing timedisconnections at the interface In Figure 6(b) the evolutionof the relaxation parameter is depicted taking into accountthis last configuration As one can observe in this caseoptimal relaxation parameter values are between 07 and 10and mostly concentrate on the interval (09 10) In fact itis expected that these values get closer to 10 when smallertime steps are considered In the present analysis an averagenumber of 492 iterations per time step is obtained (takinginto account 800 FEM time steps) which is a relatively lownumber illustrating the good performance of the technique(it must be remarked that a tight tolerance criterion wasadopted for the convergence of the iterative analysis)

42 Acoustic Waves In acoustic models a scalar wave equa-tion describes the acoustic pressure field evolution [1] In thiscase (1b) can be rewritten as (in this subsection frequency-domain analyses are focused)

nabla sdot (120581 (119909) nabla119901 (119909 120596)) + 1205962120588 (119909) 119901 (119909 120596) = 120574 (119909 120596) (14)

and (3) can be rewritten as

(119901 (119909+ 120596) minus 119901 (119909

minus 120596)) = 0 (15a)

(119902 (119909+ 120596) minus 119902 (119909

minus 120596)) = 119892 (119909 120596) (15b)

where 119901 is the hydrodynamic pressure and 120574 and 119892 standfor domain and surface sources respectively The parameters120588 and 120581 denote respectively the mass density and com-pressibility of the medium and its wave propagation velocity

Journal of Applied Mathematics 9

000 025 050 075

0

1

2

3

4

5

Point B

Point A

AnalyticalFEM-BEM

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

(a)

0 1000 2000 3000 4000

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

(b)Figure 6 Results considering different time steps for each domain (a) electric field intensity at points A and B (b) optimal relaxationparameters for each iterative step

50

50

x

y

R=10

120592 = 30000ms

120592 = 30000ms

120592 = 15000ms

S (minus50 00)

(a)

x (m)

y(m

)65

6

55

5

45

4

35135 14 145 15 155 16 165

(b)

Figure 7 (a) Sketch for the heterogeneous medium with multiple subregions (b) boundary and domain point distribution considering thespatial discretization of an inclusion and adjacent fluid

is specified as 120592 = (120581120588)12 The hydrodynamic fluxes on

the interfaces are represented by 119902 and they are defined by119902 = 120581 nabla119901 sdot n where n is the normal vector from domain1 to domain 2 Equation (15a) states that the pressure iscontinuous across the interface whereas (15b) states that theflux is continuous across the interface if there is no surfacesource

The advantages of using iterative coupling procedures arerevealed when more complex configurations are analyzedIn this subsection the case of a heterogeneous domaincomposed of a homogeneous fluid incorporating multiplecircular inclusions with different properties is analyzed

For this purpose consider the host medium to allow thepropagation of sound with a velocity of 1500ms and thismedium is excited by a line source located at 119909

119904= minus50m

and 119910119904= 00m Within this fluid consider the presence of

8 circular inclusions all of them are with unit radius andfilled with a different fluid allowing sound waves to travel at3000ms as depicted in Figure 7

The above-described system has been analyzed takinginto account the proposed iterative coupling proceduremaking use of the Kansarsquos method (KM) to model all theinclusions and of the method of fundamental solutions(MFS) to model the host fluid (see references [12 59ndash61]

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

6 Journal of Applied Mathematics

procedures can be employed with confidence as referred inthe literature [47 48] and illustrated in the next sectionOne should notice that usually different optimal (optimalin terms of accuracy stability and efficiency) time stepsare required when taking into account different numericalmethods spatial discretizations material properties physicalphenomena and so forth Thus in some cases consideringdifferent time steps within each domain of a coupled modelis of maximal importance to allow the effectiveness of theanalysis

Using space(time) interpolation(extrapolation) proce-dures optimal modeling of each sectioned domain may beachieved which is very important inwhat concerns flexibilityefficiency accuracy and stability aspects

4 Numerical Applications

In this section the general procedures previously discussedare particularized and briefly detailed taking into accountdifferent physicalmodels anddiscretization techniquesThusthe discussed iterative coupling methodology is appliedconsidering a wide range of wave propagation models andnumerical methods richly illustrating its performance andpotentialities

In this context time- and frequency-domain analysesare carried out here and electromagnetic acoustic andmechanical wave propagation phenomena (as well as theirinteractions) are discussed in the applications that followMoreover different numerical techniques (such as the finiteelement method the boundary element method and mesh-less methods) are applied to discretize the different domainsof the model illustrating the versatility and generality of thediscussed iterative method

41 Electromagnetic Waves In electromagnetic models vec-torial wave equations describe the electric and the magneticfield evolution [56 57] In this case (1a) can be rewritten as(in this subsection time-domain analyses are focused on)

nabla times (120583(119909)minus1nabla times E (119909 119905)) + 120576 (119909) E (119909 119905) = minus J (119909 119905) (12a)

nabla times (120576(119909)minus1nabla timesH (119909 119905)) + 120583 (119909) H (119909 119905)

= nabla times (120576(119909)minus1J (119909 119905))

(12b)

and (3) can be rewritten as

n (119909) times (E (119909+ 119905) minus E (119909minus 119905)) = 0 (13a)

(D (119909+ 119905) minusD (119909

minus 119905)) sdot n (119909) = 120588 (119909 119905) (13b)

(B (119909+ 119905) minus B (119909minus 119905)) sdot n (119909) = 0 (13c)

n (119909) times (H (119909+ 119905) minusH (119909

minus 119905)) = J (119909 119905) (13d)

where E and H are the electric and magnetic field intensityvectors respectively D and B represent the electric andmagnetic flux densities respectively and J and 120588 stand for theelectric current and electric charge density respectively Theparameters 120576 and 120583 denote respectively the permittivity and

permeability of themediumand itswave propagation velocityis specified as 119888 = (120576120583)

minus12 n is the normal vector fromdomain 1 to domain 2 Equations (13a) and (13b) state that thetangential component of E is continuous across the interfaceand that the normal component of D has a step of surfacecharge on the interface surface respectively Equations (13c)and (13d) state that the normal component ofB is continuousacross the interface and that the tangential component ofH iscontinuous across the interface if there is no surface currentpresent respectively

In the present application the electromagnetic fieldssurrounding infinitely long wires are studied [41] Two casesof analysis are focused here namely (a) case 1 where onewireis considered (b) case 2 where two wires are employed Forboth cases the wires are carrying time-dependent currents(ie 119868(119905) = 119905 or 119868(119905) = 119905

2) and they are located along theadopted 119911-axis A sketch of the model is depicted in Figure 3

The spatial and temporal evolution of the electric fieldintensity vector is analyzed here taking into account a finiteelement method (FEM)mdashboundary element method (BEM)coupled formulation In this context the FEM is appliedto model the region close to the wires whereas the BEMsimulates the remaining infinity domain As it is well knownthe BEM employs fundamental solutions which fulfill theradiation conditionThus this formulation is very suitable toperform infinite domain analysis once reflected waves frominfinity are avoided [58]

The adopted spatial discretization is also described inFigure 3 In this case 2344 linear triangular finite elementsand 80 linear boundary elements are employed in the analyses(see references [57 58] for more details regarding the FEMand the BEMapplied to electromagnetic analyses)The radiusof the FEM-BEM interface is defined by 119877 = 1m andmatching nodes are considered at the interface For temporaldiscretization the selected time step is given byΔ119905 = 5sdot10minus11sfor both domainsThephysical properties of themedium (air)are 120583 = 12566 sdot 10minus6Hm and 120576 = 88544 sdot 10minus12 Fm

Figure 4 shows the modulus of the electric field intensityobtained at points A and B (see Figure 3) considering theiterative couplingmethodology Analytical time histories [58]are also depicted in Figure 4 highlighting the good accuracyof the numerical results In Figure 5 charts are displayedindicating the percentage of occurrence of different relax-ation parameter values (evaluated according to expression(10)) in each analysis As can be observed for all consideredcases optimal relaxation parameters aremostly in the interval07 le 120582 le 08 In fact an optimal relaxation parameterselection is extremely case dependent It is function of thephysical properties of the model geometric aspects adoptedspatial and temporal discretizations and so forth Equation(10) provides a simple expression to evaluate this complexparameter

In order to illustrate the effectiveness of the methodologywhen considering different time discretizations for differentdomains Figure 6 depicts results that are computed consider-ing Δ119905 = 25 sdot 10minus11 s for the FEM and Δ119905 = 20 sdot 10minus10 s for theBEM (ie a difference of 8 times between the time steps) Forsimplicity results are presented considering just the first case

Journal of Applied Mathematics 7

R

Wire AB

FEM-BEMinterface

x

y

z

(a)

R

Wire A

B

interface

Wire

FEM-BEM

x

y

z

(b)

Figure 3 Sketch of the electromagnetic models and adopted FEMBEM spatial discretizations (a) case 1 one wire (b) case 2 two wires

000 025 050 075

0

1

2

3

4

5

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(a)

000 025 050 075

0

2

4

6

8

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(b)

AnalyticalFEM-BEM

000 025 050 075

0

1

2

3

4

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(c)

AnalyticalFEM-BEM

000 025 050 075

0

2

4

6

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(d)

Figure 4 Time history results for the electric field intensity at points A and B considering 119868(119905) = 119905 and (a) case 1 and (b) case 2 119868(119905) = 1199052 and(c) case 1 and (d) case 2

8 Journal of Applied Mathematics

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(a)

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(b)

00 02 04 06 08 100

5

10

15

20

25

30

Occ

urre

nce (

)

Relaxation parameter

(c)

00 02 04 06 08 100

5

10

15

20

25

30O

ccur

renc

e (

)

Relaxation parameter

(d)

Figure 5 Percentage of occurrence of different relaxation parameter values during the analysis considering 119868(119905) = 119905 and (a) case 1 and (b)case 2 119868(119905) = 1199052 and (c) case 1 and (d) case 2

of analysis that is case 1 and 119868(119905) = 119905 As one can observein Figure 6(a) good results are still obtained taking intoaccount the iterative formulation in spite of the existing timedisconnections at the interface In Figure 6(b) the evolutionof the relaxation parameter is depicted taking into accountthis last configuration As one can observe in this caseoptimal relaxation parameter values are between 07 and 10and mostly concentrate on the interval (09 10) In fact itis expected that these values get closer to 10 when smallertime steps are considered In the present analysis an averagenumber of 492 iterations per time step is obtained (takinginto account 800 FEM time steps) which is a relatively lownumber illustrating the good performance of the technique(it must be remarked that a tight tolerance criterion wasadopted for the convergence of the iterative analysis)

42 Acoustic Waves In acoustic models a scalar wave equa-tion describes the acoustic pressure field evolution [1] In thiscase (1b) can be rewritten as (in this subsection frequency-domain analyses are focused)

nabla sdot (120581 (119909) nabla119901 (119909 120596)) + 1205962120588 (119909) 119901 (119909 120596) = 120574 (119909 120596) (14)

and (3) can be rewritten as

(119901 (119909+ 120596) minus 119901 (119909

minus 120596)) = 0 (15a)

(119902 (119909+ 120596) minus 119902 (119909

minus 120596)) = 119892 (119909 120596) (15b)

where 119901 is the hydrodynamic pressure and 120574 and 119892 standfor domain and surface sources respectively The parameters120588 and 120581 denote respectively the mass density and com-pressibility of the medium and its wave propagation velocity

Journal of Applied Mathematics 9

000 025 050 075

0

1

2

3

4

5

Point B

Point A

AnalyticalFEM-BEM

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

(a)

0 1000 2000 3000 4000

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

(b)Figure 6 Results considering different time steps for each domain (a) electric field intensity at points A and B (b) optimal relaxationparameters for each iterative step

50

50

x

y

R=10

120592 = 30000ms

120592 = 30000ms

120592 = 15000ms

S (minus50 00)

(a)

x (m)

y(m

)65

6

55

5

45

4

35135 14 145 15 155 16 165

(b)

Figure 7 (a) Sketch for the heterogeneous medium with multiple subregions (b) boundary and domain point distribution considering thespatial discretization of an inclusion and adjacent fluid

is specified as 120592 = (120581120588)12 The hydrodynamic fluxes on

the interfaces are represented by 119902 and they are defined by119902 = 120581 nabla119901 sdot n where n is the normal vector from domain1 to domain 2 Equation (15a) states that the pressure iscontinuous across the interface whereas (15b) states that theflux is continuous across the interface if there is no surfacesource

The advantages of using iterative coupling procedures arerevealed when more complex configurations are analyzedIn this subsection the case of a heterogeneous domaincomposed of a homogeneous fluid incorporating multiplecircular inclusions with different properties is analyzed

For this purpose consider the host medium to allow thepropagation of sound with a velocity of 1500ms and thismedium is excited by a line source located at 119909

119904= minus50m

and 119910119904= 00m Within this fluid consider the presence of

8 circular inclusions all of them are with unit radius andfilled with a different fluid allowing sound waves to travel at3000ms as depicted in Figure 7

The above-described system has been analyzed takinginto account the proposed iterative coupling proceduremaking use of the Kansarsquos method (KM) to model all theinclusions and of the method of fundamental solutions(MFS) to model the host fluid (see references [12 59ndash61]

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 7

R

Wire AB

FEM-BEMinterface

x

y

z

(a)

R

Wire A

B

interface

Wire

FEM-BEM

x

y

z

(b)

Figure 3 Sketch of the electromagnetic models and adopted FEMBEM spatial discretizations (a) case 1 one wire (b) case 2 two wires

000 025 050 075

0

1

2

3

4

5

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(a)

000 025 050 075

0

2

4

6

8

Point B

Point A

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

Time (10minus8 s)

(b)

AnalyticalFEM-BEM

000 025 050 075

0

1

2

3

4

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(c)

AnalyticalFEM-BEM

000 025 050 075

0

2

4

6

Point B

Point A

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus15

Vm

)

(d)

Figure 4 Time history results for the electric field intensity at points A and B considering 119868(119905) = 119905 and (a) case 1 and (b) case 2 119868(119905) = 1199052 and(c) case 1 and (d) case 2

8 Journal of Applied Mathematics

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(a)

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(b)

00 02 04 06 08 100

5

10

15

20

25

30

Occ

urre

nce (

)

Relaxation parameter

(c)

00 02 04 06 08 100

5

10

15

20

25

30O

ccur

renc

e (

)

Relaxation parameter

(d)

Figure 5 Percentage of occurrence of different relaxation parameter values during the analysis considering 119868(119905) = 119905 and (a) case 1 and (b)case 2 119868(119905) = 1199052 and (c) case 1 and (d) case 2

of analysis that is case 1 and 119868(119905) = 119905 As one can observein Figure 6(a) good results are still obtained taking intoaccount the iterative formulation in spite of the existing timedisconnections at the interface In Figure 6(b) the evolutionof the relaxation parameter is depicted taking into accountthis last configuration As one can observe in this caseoptimal relaxation parameter values are between 07 and 10and mostly concentrate on the interval (09 10) In fact itis expected that these values get closer to 10 when smallertime steps are considered In the present analysis an averagenumber of 492 iterations per time step is obtained (takinginto account 800 FEM time steps) which is a relatively lownumber illustrating the good performance of the technique(it must be remarked that a tight tolerance criterion wasadopted for the convergence of the iterative analysis)

42 Acoustic Waves In acoustic models a scalar wave equa-tion describes the acoustic pressure field evolution [1] In thiscase (1b) can be rewritten as (in this subsection frequency-domain analyses are focused)

nabla sdot (120581 (119909) nabla119901 (119909 120596)) + 1205962120588 (119909) 119901 (119909 120596) = 120574 (119909 120596) (14)

and (3) can be rewritten as

(119901 (119909+ 120596) minus 119901 (119909

minus 120596)) = 0 (15a)

(119902 (119909+ 120596) minus 119902 (119909

minus 120596)) = 119892 (119909 120596) (15b)

where 119901 is the hydrodynamic pressure and 120574 and 119892 standfor domain and surface sources respectively The parameters120588 and 120581 denote respectively the mass density and com-pressibility of the medium and its wave propagation velocity

Journal of Applied Mathematics 9

000 025 050 075

0

1

2

3

4

5

Point B

Point A

AnalyticalFEM-BEM

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

(a)

0 1000 2000 3000 4000

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

(b)Figure 6 Results considering different time steps for each domain (a) electric field intensity at points A and B (b) optimal relaxationparameters for each iterative step

50

50

x

y

R=10

120592 = 30000ms

120592 = 30000ms

120592 = 15000ms

S (minus50 00)

(a)

x (m)

y(m

)65

6

55

5

45

4

35135 14 145 15 155 16 165

(b)

Figure 7 (a) Sketch for the heterogeneous medium with multiple subregions (b) boundary and domain point distribution considering thespatial discretization of an inclusion and adjacent fluid

is specified as 120592 = (120581120588)12 The hydrodynamic fluxes on

the interfaces are represented by 119902 and they are defined by119902 = 120581 nabla119901 sdot n where n is the normal vector from domain1 to domain 2 Equation (15a) states that the pressure iscontinuous across the interface whereas (15b) states that theflux is continuous across the interface if there is no surfacesource

The advantages of using iterative coupling procedures arerevealed when more complex configurations are analyzedIn this subsection the case of a heterogeneous domaincomposed of a homogeneous fluid incorporating multiplecircular inclusions with different properties is analyzed

For this purpose consider the host medium to allow thepropagation of sound with a velocity of 1500ms and thismedium is excited by a line source located at 119909

119904= minus50m

and 119910119904= 00m Within this fluid consider the presence of

8 circular inclusions all of them are with unit radius andfilled with a different fluid allowing sound waves to travel at3000ms as depicted in Figure 7

The above-described system has been analyzed takinginto account the proposed iterative coupling proceduremaking use of the Kansarsquos method (KM) to model all theinclusions and of the method of fundamental solutions(MFS) to model the host fluid (see references [12 59ndash61]

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

8 Journal of Applied Mathematics

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(a)

00 02 04 06 08 100

3

6

9

12

15

Occ

urre

nce (

)

Relaxation parameter

(b)

00 02 04 06 08 100

5

10

15

20

25

30

Occ

urre

nce (

)

Relaxation parameter

(c)

00 02 04 06 08 100

5

10

15

20

25

30O

ccur

renc

e (

)

Relaxation parameter

(d)

Figure 5 Percentage of occurrence of different relaxation parameter values during the analysis considering 119868(119905) = 119905 and (a) case 1 and (b)case 2 119868(119905) = 1199052 and (c) case 1 and (d) case 2

of analysis that is case 1 and 119868(119905) = 119905 As one can observein Figure 6(a) good results are still obtained taking intoaccount the iterative formulation in spite of the existing timedisconnections at the interface In Figure 6(b) the evolutionof the relaxation parameter is depicted taking into accountthis last configuration As one can observe in this caseoptimal relaxation parameter values are between 07 and 10and mostly concentrate on the interval (09 10) In fact itis expected that these values get closer to 10 when smallertime steps are considered In the present analysis an averagenumber of 492 iterations per time step is obtained (takinginto account 800 FEM time steps) which is a relatively lownumber illustrating the good performance of the technique(it must be remarked that a tight tolerance criterion wasadopted for the convergence of the iterative analysis)

42 Acoustic Waves In acoustic models a scalar wave equa-tion describes the acoustic pressure field evolution [1] In thiscase (1b) can be rewritten as (in this subsection frequency-domain analyses are focused)

nabla sdot (120581 (119909) nabla119901 (119909 120596)) + 1205962120588 (119909) 119901 (119909 120596) = 120574 (119909 120596) (14)

and (3) can be rewritten as

(119901 (119909+ 120596) minus 119901 (119909

minus 120596)) = 0 (15a)

(119902 (119909+ 120596) minus 119902 (119909

minus 120596)) = 119892 (119909 120596) (15b)

where 119901 is the hydrodynamic pressure and 120574 and 119892 standfor domain and surface sources respectively The parameters120588 and 120581 denote respectively the mass density and com-pressibility of the medium and its wave propagation velocity

Journal of Applied Mathematics 9

000 025 050 075

0

1

2

3

4

5

Point B

Point A

AnalyticalFEM-BEM

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

(a)

0 1000 2000 3000 4000

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

(b)Figure 6 Results considering different time steps for each domain (a) electric field intensity at points A and B (b) optimal relaxationparameters for each iterative step

50

50

x

y

R=10

120592 = 30000ms

120592 = 30000ms

120592 = 15000ms

S (minus50 00)

(a)

x (m)

y(m

)65

6

55

5

45

4

35135 14 145 15 155 16 165

(b)

Figure 7 (a) Sketch for the heterogeneous medium with multiple subregions (b) boundary and domain point distribution considering thespatial discretization of an inclusion and adjacent fluid

is specified as 120592 = (120581120588)12 The hydrodynamic fluxes on

the interfaces are represented by 119902 and they are defined by119902 = 120581 nabla119901 sdot n where n is the normal vector from domain1 to domain 2 Equation (15a) states that the pressure iscontinuous across the interface whereas (15b) states that theflux is continuous across the interface if there is no surfacesource

The advantages of using iterative coupling procedures arerevealed when more complex configurations are analyzedIn this subsection the case of a heterogeneous domaincomposed of a homogeneous fluid incorporating multiplecircular inclusions with different properties is analyzed

For this purpose consider the host medium to allow thepropagation of sound with a velocity of 1500ms and thismedium is excited by a line source located at 119909

119904= minus50m

and 119910119904= 00m Within this fluid consider the presence of

8 circular inclusions all of them are with unit radius andfilled with a different fluid allowing sound waves to travel at3000ms as depicted in Figure 7

The above-described system has been analyzed takinginto account the proposed iterative coupling proceduremaking use of the Kansarsquos method (KM) to model all theinclusions and of the method of fundamental solutions(MFS) to model the host fluid (see references [12 59ndash61]

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 9: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 9

000 025 050 075

0

1

2

3

4

5

Point B

Point A

AnalyticalFEM-BEM

Time (10minus8 s)

Elec

tric

al fi

eld

inte

nsity

Ez

(10minus7

Vm

)

(a)

0 1000 2000 3000 4000

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

(b)Figure 6 Results considering different time steps for each domain (a) electric field intensity at points A and B (b) optimal relaxationparameters for each iterative step

50

50

x

y

R=10

120592 = 30000ms

120592 = 30000ms

120592 = 15000ms

S (minus50 00)

(a)

x (m)

y(m

)65

6

55

5

45

4

35135 14 145 15 155 16 165

(b)

Figure 7 (a) Sketch for the heterogeneous medium with multiple subregions (b) boundary and domain point distribution considering thespatial discretization of an inclusion and adjacent fluid

is specified as 120592 = (120581120588)12 The hydrodynamic fluxes on

the interfaces are represented by 119902 and they are defined by119902 = 120581 nabla119901 sdot n where n is the normal vector from domain1 to domain 2 Equation (15a) states that the pressure iscontinuous across the interface whereas (15b) states that theflux is continuous across the interface if there is no surfacesource

The advantages of using iterative coupling procedures arerevealed when more complex configurations are analyzedIn this subsection the case of a heterogeneous domaincomposed of a homogeneous fluid incorporating multiplecircular inclusions with different properties is analyzed

For this purpose consider the host medium to allow thepropagation of sound with a velocity of 1500ms and thismedium is excited by a line source located at 119909

119904= minus50m

and 119910119904= 00m Within this fluid consider the presence of

8 circular inclusions all of them are with unit radius andfilled with a different fluid allowing sound waves to travel at3000ms as depicted in Figure 7

The above-described system has been analyzed takinginto account the proposed iterative coupling proceduremaking use of the Kansarsquos method (KM) to model all theinclusions and of the method of fundamental solutions(MFS) to model the host fluid (see references [12 59ndash61]

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

0

001

002

003

004

Angle (rad)

Am

plitu

de (P

a)

minus001

minus002

minus003

minus004

minus005

minus006

(a)

0 1 2 3 4 5 6Angle (rad)

0

002

004

006

008

Am

plitu

de (P

a)

minus002

minus004

minus006

minus008

(b)Figure 8Hydrodynamic pressures along the common interface of the 8th inclusion for (a)120596 = 400Hz and (b)120596 = 1000Hz (mdash real-iterative- - - - Imag-iterative I Real-direct ◻ Imag-direct)

for more details regarding the KM and the MFS appliedto acoustic analyses) One should note that since the realsource is positioned at the outer region the iterative processis initialized with the analysis of the MFS model consideringprescribed Neumann boundary conditions at the commoninterfaces Once the boundary pressures for the outer regionare computed these values are transferred to the closedregions by imposing Dirichlet boundary conditions incor-porating information about the influence of each inclusionon the remaining heterogeneities Then each KM subregionis analysed independently and the internal boundary values(normal fluxes) are evaluated autonomously for each inclu-sion The iterative procedure then goes further includingthe calculation of the relaxation parameter at each iterativestep as well as the correction of boundary variables untilconvergence is achieved

To model the system each MFS boundary is discretizedby 55 points 331 KM domain points are equally distributedwithin each inclusion and 66 KM boundary points (around31 points per wavelength) are used (see Figure 7(b)) Thecomplexity of the model hinders the definition of a closedform solution thus the results are checked against a numer-ical model which performs the direct (ie noniterative)coupling between both methods In that model 66 boundarypoints are used in the MFS to define the boundary ofeach inclusion and 66 and 331 KM boundary and domainpoints respectively are adopted for the discretization of eachinclusion (analogously to the iterative coupling procedure)Figure 8 compares the responses computed by the iterativeand the direct coupling methodologies Results are depictedalong the boundary of the 8th inclusion for excitation fre-quencies of 400Hz (Figure 8(a)) and 1000Hz (Figure 8(b))As can be observed in the figure there is a perfect matchbetween both approaches with the iterative procedure clearlyconverging to the correct solution

It is important at this point to highlight the differencesin the computational times of the direct and of the iterative

coupling approaches For the present model configurationthe direct coupling approach had to deal simultaneously with528 boundary points and a total of 2648 internal points (ieconsidering a coupled matrix of dimension 3704) which isimplied in 37389 s of CPU time in a Matlab implementation(being this CPU time independent of the frequency in focus)For the iterative coupling approach using 55 boundary pointsfor the MFS and 66 boundary points for the KM it waspossible to obtain analogous results considering 1206 s ofCPU time for the frequency of 200Hz and 3232 s for thefrequency of 1000Hz (ie 323 and 864 of the com-putational cost of the direct coupling methodology resp)Even if the same number of boundary points is used in theiterative coupling approach for the MFS (ie 66 points) thefinal CPU time would just increase up to 3571 s (955 ofthe computational cost of the direct coupling methodology)These results are summarized in Table 1 where the numberof iterations and the CPU time are presented for the firstscenarios (ie 55 boundary points for the MFS) and forfrequencies between 50Hz and 1000HzThe values describedin Table 1 further confirm that the difference in calculationtimes between the iterative and the direct coupling approachis striking and reveal an excellent gain in performancefavouring the iterative coupling technique It is importantto understand that this gain is strongly related to the pos-sibility of dealing with smaller-sized matrices when usingthe iterative coupling procedure Moreover it is possible toinvert (or triangularize etc) the relevant matrices only atthe first iterative step and then proceed with the calculationsusing the inverted matrices (or forwardback substitutingetc) As a consequence after the first iteration only matrixvector multiplication operations are required and very highsavings in what concerns computational time are achieved Infact considering that the number of operations required formatrix inversion can be assumed to be of the order of1198733 (119873being thematrix size) a simple calculation allows concludingthat for the current model the relative cost of inverting the

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 11

Table 1 Total number of iterations and relative CPU time (itera-tivedirect coupling) for the acoustic model

Frequency (Hz) Iterations Relative CPU time ()50 14 209100 18 243150 29 319200 31 323250 26 278300 40 386350 31 313400 32 320450 30 302500 49 449550 48 440600 45 418650 68 594700 34 328750 39 366800 110 920850 98 828900 78 675950 104 8831000 100 864

eight KM matrices (each one being a square matrix with397 times 397 entries) and the MFS matrix (with 440 times 440entries) is less than 2 of the cost of inverting a larger 3704times 3704 matrix as required for the direct coupling strategySimilar conclusions can be obtained considering other solverprocedures such as matrix triangularizations demonstratingthat a considerably less expensive methodology is obtainedif the different subdomains are analysed separately (evenconsidering an eventual high number of iterative steps in theiterative analysis)

Analyzing the difference in computational times betweenthe two analyzed frequencies (ie 200Hz and 1000Hz) alsoreveals a significant difference between themThis differenceis related to the number of iterations required for conver-gence which was higher when the excitation frequency of1000Hz was considered The plot in Figure 9 indicates thenumber of iterations required for convergence along a rangeof frequencies between 10Hz and 1000Hz using a constantnumber of boundary (55 for theMFS and 66 for the KM) andinternal points (331 for the KM) As expected the number ofiterations increases with the frequency It is interesting to notethat the maximum necessary number of iterations occurredfor a frequency of 990Hz requiring 170 iterations and a CPUtime of 5480 s to converge which is less than 15 of the CPUtime required by the direct coupling for the same frequency

In Figure 10 the wavefield produced within and aroundthe inclusions is illustrated for excitation frequencies of600Hz and 1000Hz As expected as the frequency increasesthe multiple inclusions generate progressively more complexwave fields with the interaction between them becomingvery significant for the higher frequency Observation of

250

200

150

100

50

00 200 400 600 800 1000

Num

ber o

f ite

ratio

ns

Frequency (Hz)

Figure 9 Total number of iterations considering different frequen-cies and 55 collocation points for the MFS and 66 boundary pointsfor the KM at each circular inclusion

these results also reveals a strong shadow effect produced bythe inclusions with much lower amplitudes being registeredin the region behind the inclusions placed further awayfrom the source This effect is even more pronounced forthe higher frequency Interestingly for both frequenciesthe space between the two lines of inclusions works as aguiding path along which the sound energy travels with lessattenuation

43 Mechanical Waves In dynamic models a vectorial waveequation describes the displacement field evolution [1] In thiscase considering linear behaviour (1a) can be rewritten as (inthis subsection time-domain analyses are focused)

nabla times (120583 (119909) nabla times u (119909 119905))

minus nabla ((120578 (119909) + 2120583 (119909)) nabla sdot u (119909 119905)) + 120588 (119909) u (119909 119905)

= f (119909 119905)

(16)

and (3) can be rewritten as

(u (119909+ 119905) minus u (119909minus 119905)) = 0 (17a)

(120590 (119909+ 119905) minus 120590 (119909

minus 119905))n (119909) = 120591 (119909 119905) (17b)

where u is the displacement vector and f and 120591 stand fordomain and surface forces respectively The terms 120578 and 120583denote the so-called Lame parameters and 120588 is the massdensity of the medium In this case the wave propagationvelocities are specified as 119888

119904= (120583120588)

12 (shear wave) and119888119889= ((120578 + 2120583)120588)

12 (dilatational wave) The stress tensoris denoted by 120590 and n is the normal vector from domain 1to domain 2 Equation (17a) states that the displacements arecontinuous across the interface whereas (17b) states that thetractions are continuous across the interface if there are nosurface forces on it

One main advantage of the discussed coupling algorithmis that other iterative processes can be carried out in the same

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

12 Journal of Applied Mathematics

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(a)

6

4

2

0

minus2

y(m

)

x (m)

015

01

005

0 5 10 15

(b)

Figure 10 3D plots of the sound field for frequencies of (a) 600Hz and (b) 1000Hz

y

x

R

Af(t)

(a)

FEM

TD-BEM

(b)

D-BEM

TD-BEM

(c)

Figure 11 (a) Sketch of the circular cavity (b) FEM-BEM discretization (c) BEM-BEM discretization

iterative loop needed for the couplingThus consideration ofcoupled nonlinear models as for example may not demanda superior amount of computational effort which is verybeneficial

In the present application a nonlinear model is consid-ered and elastoplastic analyses are carried out (for detailsabout elastoplastic analyses one is referred to [33ndash35 62ndash64]) Moreover two discretization approaches are employedhere one taking into account FEM-BEM coupling proce-dures and another considering BEM-BEM coupled tech-niques (for more details about these coupled models oneis referred to [37 43]) In this context a nonlinear infinitydomain is analyzed here in which a circular cavity is loadedThe region expected to develop plastic strains is discretized bythe finite element method in the case of the FEM-BEM cou-pled analysis or by the domain boundary element method(D-BEM) in the case of the BEM-BEM coupled analysisTheremainder of the infinity domain is discretized by the time-domain boundary element method (TD-BEM) A sketch of

Elastic Elastoplastic

0 2 4 6 8 10 12 14 16 18 20 22 24 26

000

001

002

003

004

005

006

007

TD-BEMFEM TD-BEMD-BEM

Disp

lace

men

t (m

)

Time (s)

Figure 12 Radial displacement considering coupled TD-BEMD-BEM and coupled TD-BEMFEM analyses linear and nonlinearresults at point A

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 13

0 100 200 300 400 500 600

3

4

5

6

7

ElasticElastoplastic

Num

ber o

f ite

ratio

ns

Time step

(a)

ElasticElastoplastic

0 500 1000 1500 2000 2500 3000 3500

05

06

07

08

09

10

Rela

xatio

n pa

ram

eter

Iteration

1500 1600 1700 1800 1900 2000

075080085090095100

Zoom

(b)

Figure 13 TD-BEMFEM analyses (a) number of iterations per time step considering optimal relaxation parameters (b) optimal relaxationparameters for each iterative step

the model is depicted in Figure 11 as well as the adopteddiscretizations The FEM-BEM discretization is depictedin Figure 11(b) In this case 1944 linear triangular finite ele-ments and 80 linear boundary elements are employed in thecoupled analysis The BEM-BEM discretization is depictedin Figure 11(c) In this case 46 linear boundary elementsare employed in the BEM-BEM coupled analysis (20 linearboundary elements for the TD-BEM and 26 linear boundaryelements for the D-BEM) as well as 270 linear triangular cells(D-BEM formulation) In the BEM-BEM coupled analysisthe double symmetry of the problem is taken into accountAn interesting feature of the boundary element formulationis that symmetric bodies under symmetric loads can beanalysed without discretization of the symmetry axes Thiscan be accomplished by an automatic condensation processwhich integrates over reflected elements and performs theassemblage of the finalmatrices in reduced size [64]The timediscretization adopted is given byΔ119905 = 004 s for the FEMandΔ119905 = 020 s for the D-BEM and the TD-BEM

The physical properties of the model are 120583 = 2652 sdot

108Nm2 120578 = 2274 sdot 10

8Nm2 and 120588 = 1804 sdot 103 kgm3

A perfectly plastic material obeying theMohr-Coulomb yieldcriterion is assumed where 119888

119900= 48263sdot10

6Nm2 (cohesion)and 120601 = 30

∘ (internal friction angle) The geometry of theproblem is defined by 119877 = 3048m (the radius of the TD-BEM circular mesh is given by 5119877)

In Figure 12 the displacement time history at point A isdepicted considering linear and nonlinear analyses As onecan notice good agreement is observed between the FEM-BEM and BEM-BEM results It is important to highlight thatfor the FEM-BEM analyses a difference of 5 times betweenthe FEM and BEM time steps is considered illustrating theeffectiveness of the time interpolationextrapolation proce-dures adopted in the analyses

The number of iterations per time step and the optimalrelaxation parameters evaluated at each iterative step aredepicted in Figure 13 taking into account the FEM-BEMcoupled analyses As one may observe basically the same

Table 2 Total number of iterations (considering all time steps) forthe dynamic model

Relaxation parameter Elastic analysis Elastoplastic analysis100 3730 3740090 3392 3443080 3973 3993070 4772 4777Optimal 3287 3346

computational effort (ie number of iterative steps) is nec-essary for both linear and nonlinear analyses highlightingthe efficiency of the proposed methodology for complexphenomena modeling It is also important to remark thelow number of iterative steps necessary for convergencewith a maximum of 7 iterations being necessary within atime step taking into account the entire linear and non-linear analyses For the focused configurations the optimalrelaxation parameters are intricately distributed within theinterval (075 100) as depicted in Figure 13(b)

In Table 2 the total number of iterations is presentedconsidering analyses with optimal relaxation parameters andwith some constant preselected 120582 values As onemay observean inappropriate selection for the relaxation parameter canconsiderably increase the associated computational effortThus the optimization technique is extremely important inorder to provide a robust and efficient iterative couplingformulation In Figure 14 the computed 120590

119909119910stresses are

depicted considering the BEM-BEM elastoplastic analysisAn advantage of the D-BEM is that it employs nodal stressequations [37] allowing computing continuous stress fieldsin counterpart to the FEM which computes stresses basedon displacement derivatives obtaining discontinuous stressfields at element interfaces

44 Coupled Acoustic-Mechanical Waves In this case differ-ent wave equations as indicated in Sections 42 and 43 (see

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

14 Journal of Applied Mathematics

0

(a)

(b)

(d)

(c)

minus0083334

minus016667

minus025

minus033334

minus041667

minus05

minus058334

minus066667

minus075

Figure 14 Spatial and temporal evolution of 120590119909119910considering elastoplastic analysis (scale factor 68947 sdot 106Nm2) (a) 119905 = 4 s (b) 119905 = 8 s (c)

119905 = 12 s (d) 119905 = 16 s

0 50

2

4

6

8

10

Concrete wallWater

Source

Receiver

x (m)

y(m

)

minus5minus10

(a)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(b)

0 50

2

4

6

8

10

x (m)

y(m

)

minus5

(c)

0 50

2

4

6

8

10

minus5

x (m)

y(m

)

(d)Figure 15 (a) Sketch of the coupled acoustic-dynamic model (b) MFS collocation points (o) and virtual sources (x) (c) FEMmesh when 20nodes are used along the solid-fluid interface (d) node distribution for the meshless methods when 20 nodes are used along the solid-fluidinterface

(14) and (16)) describe different domains of the globalmodelThe interface conditions for the acoustic-dynamic coupling(3) can then be written as (in this subsection frequency-domain analyses are focused)

(n (119909) sdot 120590 (119909+ 120596)n (119909) minus 119901 (119909minus 120596)) = 0 (18a)

(minus1205962n (119909) sdot u (119909+ 120596) minus 120592(119909minus)2119902 (119909minus 120596)) = 0 (18b)

where u is the displacement vector and 120590 is the stress tensorof the dynamic model (domain 1) 119901 is the hydrodynamicpressure and 119902 is the hydrodynamic flux of the acousticmodel (domain 2) n is the normal vector from domain 1 todomain 2 The acoustic wave propagation velocity is denotedby 120592 Equation (18a) states that the normal components ofthe dynamic tractions are equal to the acoustic pressures

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 15: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 15

0

4

0 50 100 150

Reference-realReference-imag

Frequency (Hz)

Am

plitu

deminus4

(a)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(b)

0 50 100 150Frequency (Hz)

101

10minus1

10minus3

10minus5

Abso

lute

erro

r

MFS-MLPGMFS-CMMFS-FEM

(c)Figure 16 (a) Reference pressure result absolute error considering (b) 20 and (c) 40 boundary nodes in the solid along the solid-fluidinterface

and (18b) relates the normal components of the dynamicaccelerations to the acoustic fluxes

In the present application a model in which a concretewall of 100 m high is coupled to a fluid waveguide filledwith water is analyzed A sketch of the model is depicted inFigure 15(a) For this case a pressure source is positioned inthe waveguide at (minus100 05) illuminating the system Theconcrete structure corresponds to a wall with variable cross-section exhibiting thicknesses of 40m at its basis and of20m at its top

To simulate this coupled system several approachesare employed For the fluid medium the MFS is used inall cases allowing the use of the Greenrsquos function for awaveguide (see [61] for details concerning this function)ThisGreenrsquos function is written as a summation of modes and itsconvergence is very difficult when the source and the receiver

are positioned along the same vertical line thus posing severedifficulties for its use together with a BEM formulation Thestructure is modelled using three different methods namelythe FEM a local collocation method (CM) (see [65] fordetails about this procedure) and a meshless local Petrov-Galerking technique (MLPG) (see [65 66] for details aboutthis procedure) Representations of the node distribution foreach one of the methods can be found in Figures 15(b)ndash15(d)

Since no analytical solution can be found in the literaturefor the present case a numerical solutionmaking use of a fullBEM model ensuring the correct coupling between the solidand the fluid is used For this case the rigid bottom of thewaveguide is accounted for using an image-source Greenrsquosfunction while the free surface is fully discretized up to adistance of 600m from the concrete wall after this an ane-choic termination is considered imposing adequate Robin

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 16: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

16 Journal of Applied Mathematics

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(a)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(b)

0

20

40

60

80

100

0 50 100 150

Optimized

Frequency (Hz)

Num

ber o

f ite

ratio

ns

120582 = 05

(c)

Figure 17 Number of iterations required for convergence (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

boundary conditions For the solid full-space Greenrsquosfunctions are adopted and 60 nodal points are used alongthe solid-fluid interface ensuring that accurate results can beobtained A total of 796 boundary elements are used to buildthis model Details on the mathematical formulation of thistechnique can be found in the works of Tadeu and Godinho[7]

Figure 16(a) illustrates the reference response in termsof real and imaginary components of the acoustic pressureat a receiver located at (minus10 30) for frequencies between1Hz and 150Hz This position is chosen so that the effectof the vibration of the concrete wall and thus of the solid-fluid coupling can be evident in the responses As can beseen in the figure two peaks with significant amplitudecan be observed corresponding to vibration modes of thewall coupled to the fluid after these peaks the responseexhibits a smoother form Figures 16(b) and 16(c) illustrate

the absolute difference to the reference solution calculatedfor the three different approaches namely theMFS-FEM theMFS-CM and the MFS-MLPG For the fluid 10 nodes arepositioned along the interface while for the solid results arepresented for 20 (Figure 16(b)) and 40 nodes (Figure 16(c))When 20 nodes are used the responses provided by the threeapproaches are very similar with the two meshless methodsexhibiting a lower error level at the lower frequencies andwith a worse behaviour of the CM being observable in thehigher frequencies Observing the figure it is apparent thatthe MLPG is providing a more accurate response throughoutthe analysed frequency range exhibiting a lower error thanthe FEM even at high frequencies When more nodes areused (Figure 16(c)) the error levels provided by all methodsimprove although the MLPG still exhibits a better overallbehaviour than the remaining methods

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 17: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 17

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(a)

1

2

3

4

5

6

7

8

9

10

11

12

y(m

)

0 2 4 6 8x (m)

minus4 minus2

(b)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(c)

0 2 4 6 8

1

2

3

4

5

6

7

8

9

10

11

12

x (m)

y(m

)

minus4 minus2

(d)

Figure 18 Real ((a) and (b)) and imaginary ((c) and (d)) parts of the deformation of the solid structure (amplified) when the excitationfrequency is 125Hz Results are shown for 20 ((a) and (c)) and 40 ((b) and (d)) boundary nodes in the solid along the fluid-solid interfacewhen using the FEM (times) CM (∘) and MLPG (∙)

It is important to notice that although different errorlevels are observed for each method the iterative couplingalgorithm always quickly converges even for frequencies inthe vicinity of the response peaks referred to before Figure 17illustrates the number of iterations required for convergencefor the three approaches considering 40 nodes along theinterface to model the solid Clearly all three approachesexhibit very similar curves requiring similar numbers ofiterations for the iterative process to reach convergence ateach frequency It is also very clear that for this case the

number of iterations is always small slightly exceeding 20iterations only at a few specific frequencies For the remainingfrequencies only about 10 to 15 iterations are necessaryto attain convergence In the same figure the number ofiterations required when a fixed relaxation parameter is used(ie 120582 = 05) is also depicted Comparison between thecurves calculated with optimal and fixed parameters revealsa striking difference with the optimal parameter always lead-ing to significantly less iterations and ensuring convergencefor all frequencies When the relaxation parameter is fixed

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 18: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

18 Journal of Applied Mathematics

0

05

10

0 25 50 75 100

AbsoluteImaginaryReal

Iteration

minus05

minus10

120582

(a)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(b)

0

05

10

0 25 50 75 100

Iteration

minus05

minus10

120582

AbsoluteImaginaryReal

(c)Figure 19 Variation of the complex relaxation parameter throughout the iterative process (a) MFS-FEM (b) MFS-CM (c) MFS-MLPG

convergence cannot be reached at two sets of frequenciesassociated with specific dynamic behaviours of the systemThese results once again illustrate the importance of usinga well-chosen relaxation parameter to ensure that effectiveanalyses are obtained

The set of plots shown in Figure 18 illustrates the defor-mation of the structure at frequency 125Hz In the plottedresults a grey patch is used to identify the reference responsewhile marks are used to depict the (amplified) deformedshape of the structure when analysed by the three iterativelycoupled approaches The left column reveals the response for20 nodes positioned in the solid along the interface whereasthe right column shows the equivalent result computed for40 nodes It can be observed that for all approaches theresponse improves significantly when more nodes are usedindicating that the convergent behaviour of the methods canbe observed The computed responses reveal very similarshape and displacement amplitudes when compared to thereference solution with the response provided by the MFS-CM approach being somewhat worse than the remainingtwo In fact the MFS-MLPG and the MFS-FEM exhibit verysimilar behaviours with lower discrepancies being registeredfor the meshless method Finally Figure 19 illustrates thevariation of the complex relaxation parameter throughout theiterative process showing its real and imaginary componentstogether with its absolute value Those plots reveal a verysimilar evolution of the parameter for all combinations ofmethods again this indicates that the discussed iterativeprocedure is quite independent of the discretizationmethodsinvolved in the analyses

5 Conclusions

This paper presents an overview of the application of itera-tive coupling strategies to the analysis of wave propagationproblems Different methods were considered including

mesh-based and meshless methods ranging from the moreclassic BEM and FEM to the less usual MLPG collocationmethods or MFS Several examples of the iterative cou-pling technique were presented in Section 4 including theapplication of the scheme to electromagnetic acoustic elas-ticelastoplastic and acoustic-elastic interaction problemsThe generality and flexibility of the iterative scheme allowedan efficient analysis of these problems either using time orfrequency domainmodelsTheuse of an optimized relaxationparameter (which is the basis of this scheme) proved tobe quite important clearly accelerating (or in some casesensuring) convergence for all tested cases this parameterwasshown to unpredictably vary throughout the iterative processand thus its appropriate recalculation at each iterative stepbecomes importantThe illustrated analyses clearly indicatedthat the strategy can be effectively used for different methodsand that the performance of the iterative technique is quiteinsensitive to the discretization methods employed in theanalyses

It should be highlighted that the coupling techniquepresented here is based on previous experience and works ofthe authors and although the paper is focused in wave prop-agation problems the iterative strategy presently discussedcan be regarded as a quite generic framework to performthe coupling between different methods in many types ofapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by CNPq (Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico) and FAPEMIG

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 19: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 19

(Fundacao de Amparo a Pesquisa do Estado deMinas Gerais)is greatly acknowledged

References

[1] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak amp CompanyNew York NY USA 1973

[2] J D Achenbach W Lin and L M Keer ldquoMathematicalmodelling of ultrasonic wave scattering by sub-surface cracksrdquoUltrasonics vol 24 no 4 pp 207ndash215 1986

[3] R A Stephen ldquoA review of finite differencemethods for seismo-acoustics problems at the seafloorrdquo Reviews of Geophysics vol26 no 3 pp 445ndash458 1988

[4] J Dominguez Boundary Elements in Dynamics SouthamptonComputational Mechanics Publications 1993

[5] A Karlsson and K Kreider ldquoTransient electromagnetic wavepropagation in transverse periodic mediardquo Wave Motion vol23 no 3 pp 259ndash277 1996

[6] D Clouteau G Degrande and G Lombaert ldquoNumericalmodelling of traffic induced vibrationsrdquoMeccanica vol 36 no4 pp 401ndash420 2001

[7] A Tadeu and L Godinho ldquoScattering of acoustic waves bymovable lightweight elastic screensrdquo Engineering Analysis withBoundary Elements vol 27 no 3 pp 215ndash226 2003

[8] J L Wegner M M Yao and X Zhang ldquoDynamic wave-soil-structure interaction analysis in the time domainrdquo Computersamp Structures vol 83 no 27 pp 2206ndash2214 2005

[9] Y B Yang and L C Hsu ldquoA review of researches on ground-borne vibrations due to moving trains via underground tun-nelsrdquo Advances in Structural Engineering vol 9 no 3 pp 377ndash392 2006

[10] Y B Yang andH H HungWave Propagation for Train-InducedVibrations A FiniteInfinite Element ApproachWorld Scientific2009

[11] I Castro and A Tadeu ldquoCoupling of the BEMwith theMFS forthe numerical simulation of frequency domain 2-D elastic wavepropagation in the presence of elastic inclusions and cracksrdquoEngineering Analysis with Boundary Elements vol 36 no 2 pp169ndash180 2012

[12] L Godinho and A Tadeu ldquoAcoustic analysis of heterogeneousdomains coupling the BEM with Kansas methodrdquo EngineeringAnalysis with Boundary Elements vol 36 no 6 pp 1014ndash10262012

[13] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[14] O von Estorff and M J Prabucki ldquoDynamic response inthe time domain by coupled boundary and finite elementsrdquoComputational Mechanics vol 6 no 1 pp 35ndash46 1990

[15] G Kergourlay E Balmes and D Clouteau ldquoModel reductionfor efficient FEMBEM couplingrdquo in Proceedings of the 25thInternational Conference on Noise and Vibration Engineering(ISMA rsquo00) vol 25 pp 1189ndash1196 September 2000

[16] E Savin and D Clouteau ldquoElastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with abounded medium Application to seismic soil-structure inter-action (SSSI)rdquo International Journal for Numerical Methods inEngineering vol 54 no 4 pp 607ndash630 2002

[17] C C Spyrakos and C Xu ldquoDynamic analysis of flexible massivestrip-foundations embedded in layered soils by hybrid BEM-FEMrdquoComputersamp Structures vol 82 no 29-30 pp 2541ndash25502004

[18] M Adam and O von Estorff ldquoReduction of train-inducedbuilding vibrations by using open and filled trenchesrdquo Comput-ers amp Structures vol 83 no 1 pp 11ndash24 2005

[19] L Andersen and C J C Jones ldquoCoupled boundary andfinite element analysis of vibration from railway tunnels-acomparison of two- and three-dimensional modelsrdquo Journal ofSound and Vibration vol 293 no 3 pp 611ndash625 2006

[20] H A Schwarz ldquoUeber einige Abbildungsaufgabenrdquo Journal furfie Reine und Angewandte Mathematik vol 70 pp 105ndash1201869

[21] M J Gander ldquoSchwarz methods over the course of timerdquoElectronic Transactions on Numerical Analysis vol 31 pp 228ndash255 2008

[22] L Ling and E J Kansa ldquoPreconditioning for radial basisfunctions with domain decompositionmethodsrdquoMathematicaland Computer Modelling vol 40 no 13 pp 1413ndash1427 2004

[23] V Dolean M El Bouajaji M J Gander S Lanteri andR Perrussel ldquoDomain decomposition methods for electro-magnetic wave propagation problems in heterogeneous mediaand complex domainsrdquo in Domain Decomposition Methods inScience and Engineering pp 15ndash26 Springer Berlin Germany2011

[24] J R Rice P Tsompanopoulou and E Vavalis ldquoInterfacerelaxation methods for elliptic differential equationsrdquo AppliedNumerical Mathematics vol 32 no 2 pp 219ndash245 2000

[25] C-C Lin E C Lawton J A Caliendo and L R Anderson ldquoAniterative finite element-boundary element algorithmrdquo Comput-ers amp Structures vol 59 no 5 pp 899ndash909 1996

[26] QDeng ldquoAn analysis for a nonoverlapping domain decomposi-tion iterative procedurerdquo SIAM Journal on Scientific Computingvol 18 no 5 pp 1517ndash1525 1997

[27] D Yang ldquoA parallel iterative nonoverlapping domain decom-position procedure for elliptic problemsrdquo IMA Journal ofNumerical Analysis vol 16 no 1 pp 75ndash91 1996

[28] W M Elleithy H J Al-Gahtani and M El-Gebeily ldquoIterativecoupling of BE and FE methods in elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 25 no 8 pp 685ndash6952001

[29] W M Elleithy and M Tanaka ldquoInterface relaxation algorithmsfor BEM-BEM coupling and FEM-BEM couplingrdquo ComputerMethods in Applied Mechanics and Engineering vol 192 no 26-27 pp 2977ndash2992 2003

[30] B Yan J Du N Hu and H Sekine ldquoDomain decompositionalgorithm with finite element-boundary element couplingrdquoApplied Mathematics and Mechanics vol 27 no 4 pp 519ndash5252006

[31] Y Boubendir A Bendali and M B Fares ldquoCoupling of anon-overlapping domain decomposition method for a nodalfinite element method with a boundary element methodrdquoInternational Journal for Numerical Methods in Engineering vol73 no 11 pp 1624ndash1650 2008

[32] G H Miller and E G Puckett ldquoA Neumann-Neumann pre-conditioned iterative substructuring approach for computingsolutions to Poissonrsquos equation with prescribed jumps on anembedded boundaryrdquo Journal of Computational Physics vol235 pp 683ndash700 2013

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 20: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

20 Journal of Applied Mathematics

[33] W M Elleithy M Tanaka and A Guzik ldquoInterface relaxationFEM-BEM coupling method for elasto-plastic analysisrdquo Engi-neering Analysis with Boundary Elements vol 28 no 7 pp 849ndash857 2004

[34] H Z Jahromi B A Izzuddin and L Zdravkovic ldquoA domaindecomposition approach for coupled modelling of nonlin-ear soil-structure interactionrdquo Computer Methods in AppliedMechanics andEngineering vol 198 no 33 pp 2738ndash2749 2009

[35] D Soares Jr O von Estorff and W J Mansur ldquoIterativecoupling of BEM and FEM for nonlinear dynamic analysesrdquoComputational Mechanics vol 34 no 1 pp 67ndash73 2004

[36] J Soares O von Estorff and W J Mansur ldquoEfficient non-linear solid-fluid interaction analysis by an iterative BEMFEMcouplingrdquo International Journal for Numerical Methods in Engi-neering vol 64 no 11 pp 1416ndash1431 2005

[37] D Soares Jr J A M Carrer and W J Mansur ldquoNon-linearelastodynamic analysis by the BEM an approach based on theiterative coupling of the D-BEM and TD-BEM formulationsrdquoEngineering Analysis with Boundary Elements vol 29 no 8 pp761ndash774 2005

[38] O von Estorff and C Hagen ldquoIterative coupling of FEM andBEM in 3D transient elastodynamicsrdquoEngineering Analysis withBoundary Elements vol 29 no 8 pp 775ndash787 2005

[39] D Soares Jr and W J Mansur ldquoDynamic analysis of fluid-soil-structure interaction problems by the boundary elementmethodrdquo Journal of Computational Physics vol 219 no 2 pp498ndash512 2006

[40] D Soares Jr ldquoNumerical modelling of acoustic-elastodynamiccoupled problems by stabilized boundary element techniquesrdquoComputational Mechanics vol 42 no 6 pp 787ndash802 2008

[41] D Soares Jr ldquoA time-domain FEM-BEM iterative couplingalgorithm to numerically model the propagation of electromag-netic wavesrdquo Computer Modeling in Engineering and Sciencesvol 32 no 2 pp 57ndash68 2008

[42] A Warszawski D Soares Jr and W J Mansur ldquoA FEM-BEMcoupling procedure to model the propagation of interactingacoustic-acousticacoustic-elastic waves through axisymmetricmediardquo Computer Methods in Applied Mechanics and Engineer-ing vol 197 no 45 pp 3828ndash3835 2008

[43] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[44] D Soares Jr ldquoFluid-structure interaction analysis by optimisedboundary element-finite element coupling proceduresrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 184ndash195 2009

[45] D Soares Jr ldquoAcoustic modelling by BEM-FEM couplingprocedures taking into account explicit and implicit multi-domain decomposition techniquesrdquo International Journal forNumerical Methods in Engineering vol 78 no 9 pp 1076ndash10932009

[46] D Soares Jr ldquoAn iterative time-domain algorithm for acoustic-elastodynamic coupled analysis considering meshless localPetrov-Galerkin formulationsrdquoComputerModeling in Engineer-ing and Sciences vol 54 no 2 pp 201ndash221 2009

[47] D Soares ldquoFEM-BEM iterative coupling procedures to analyzeinteracting wave propagation models fluid-fluid solid-solidand fluid-solid analysesrdquo Coupled Systems Mechanics vol 1 pp19ndash37 2012

[48] D Soares Jr ldquoCoupled numerical methods to analyze interact-ing acoustic-dynamic models by multidomain decompositiontechniquesrdquo Mathematical Problems in Engineering vol 2011Article ID 245170 28 pages 2011

[49] J-D Benamou and B Despres ldquoA domain decompositionmethod for the Helmholtz equation and related optimal controlproblemsrdquo Journal of Computational Physics vol 136 no 1 pp68ndash82 1997

[50] A Bendali Y Boubendir and M Fares ldquoA FETI-like domaindecompositionmethod for coupling finite elements and bound-ary elements in large-size problems of acoustic scatteringrdquoComputers amp Structures vol 85 no 9 pp 526ndash535 2007

[51] D Soares Jr L Godinho A Pereira and C Dors ldquoFrequencydomain analysis of acoustic wave propagation in heterogeneousmedia considering iterative coupling procedures between themethod of fundamental solutions and Kansarsquos methodrdquo Inter-national Journal for Numerical Methods in Engineering vol 89no 7 pp 914ndash938 2012

[52] D Soares and L Godinho ldquoAn optimized BEM-FEM itera-tive coupling algorithm for acoustic-elastodynamic interactionanalyses in the frequency domainrdquoComputers amp Structures vol106-107 pp 68ndash80 2012

[53] L Godinho and D Soares ldquoFrequency domain analysis offluid-solid interaction problems bymeans of iteratively coupledmeshless approachesrdquo Computer Modeling in Engineering ampSciences vol 87 no 4 pp 327ndash354 2012

[54] L Godinho and D Soares ldquoFrequency domain analysis ofinteracting acoustic-elastodynamic models taking into accountoptimized iterative coupling of different numerical methodsrdquoEngineering Analysis with Boundary Elements vol 37 no 7 pp1074ndash1088 2013

[55] PMGauzellino F I Zyserman and J E Santos ldquoNonconform-ing finite elementmethods for the three-dimensional helmholtzequation terative domain decomposition or global solutionrdquoJournal of Computational Acoustics vol 17 no 2 pp 159ndash1732009

[56] J P A Bastos and N Ida Electromagnetics and Calculation ofFields Springer New York NY USA 1997

[57] J M Jin J Jin and J M Jin The Finite Element Method inElectromagnetics Wiley New York NY USA 2002

[58] D Soares Jr and M P Vinagre ldquoNumerical computation ofelectromagnetic fields by the time-domain boundary elementmethod and the complex variable methodrdquo Computer Modelingin Engineering and Sciences vol 25 no 1 pp 1ndash8 2008

[59] L Godinho A Tadeu and P Amado Mendes ldquoWave prop-agation around thin structures using the MFSrdquo ComputersMaterials and Continua vol 5 no 2 pp 117ndash127 2007

[60] L Godinho E Costa A Pereira and J Santiago ldquoSomeobservations on the behavior of the method of fundamentalsolutions in 3d acoustic problemsrdquo International Journal ofComputational Methods vol 9 no 4 Article ID 1250049 2012

[61] E G A Costa L Godinho J A F Santiago A Pereira and CDors ldquoEfficient numerical models for the prediction of acousticwave propagation in the vicinity of a wedge coastal regionrdquoEngineering Analysis with Boundary Elements vol 35 no 6 pp855ndash867 2011

[62] W F Chen and D J Han Plasticity for Structural EngineersSpring New York NY USA 1988

[63] A S Khan and S Huang ContinuumTheory of Plasticity JohnWiley amp Sons New York NY USA 1995

[64] J C F TellesThe Boundary Element Method Applied to InelasticProblems Spring Berlin Germany 1983

[65] S N Atluri The Meshless Method (MLPG) for Domain amp BIEDiscretizations vol 677 Tech Science Press Forsyth Ga USA2004

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 21: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Journal of Applied Mathematics 21

[66] J R Xiao and M A McCarthy ldquoA local heaviside weightedmeshless method for two-dimensional solids using radial basisfunctionsrdquo Computational Mechanics vol 31 no 3-4 pp 301ndash315 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 22: Review Article An Overview of Recent Advances in the ...downloads.hindawi.com/journals/jam/2014/426283.pdf · they observed that iterative domain decomposition methods involving small

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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