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A coupled ES-BEM and FM-BEM for structural acoustic problems
14
A coupled ES-BEM and FM-BEM for structural acoustic problems F. Wu a),b) , G.R. Liu b) , G.Y. Li c) , Y.J. Liu d) and Z.C. He e) (Received: 8 December 2013; Revised: 7 July 2014; Accepted: 9 July 2014) In this paper, a coupled numerical method of the edge-based smoothed nite element (ES-FEM) with the fast multipole BEM (FM-BEM) is proposed to ana- lyze structural acoustic problems. The vibrating structure is modeled using the so-called ES-FEM-DSG3 method, where the 3-node linear triangle plate ele- ments based on the ReissnerMindlin plate theory with the discrete shear gap (DSG) technique for overcoming the shear locking are applied. The edge-based gradient smoothing operations are applied to softenthe overly-stiff behavior in the standard FEM, which signicantly reduces the inherent numerical dis- persion error. The normal velocities on the surface of the structure are imposed as boundary conditions for the acoustic domain which is modeled using the FM-BEM for both the interior and exterior acoustic domains. Comparing with the conventional BEM, the matrix vector multiplication and the memory re- quirement in the FM-BEM are reduced dramatically. The coupled ES-FEM/ FM-BEM method takes the advantages of both ES-FEM and FM-BEM, which can avoid drawbacks of the overly-stiff behavior in FEM and computational inefciency in the conventional BEM. Two numerical examples are presented to verify and demonstrate the effectiveness of the combined method: one aca- demic problem for studying in detail the accuracy and efciency of the present method, and one application to a practical vehicle noise simulation. © 2014 Institute of Noise Control Engineering. Primary subject classication: 75.3; Secondary subject classication: 75.5 1 INTRODUCTION Noises generated by a vibrating thin structure can be commonly found in numerous engineering systems, such as aircrafts, sea vessels and the land vehicles. These acoustic problems are closely associated to uidstructure interactions and are of increasing con- cern when designing passenger transportation systems. Many researchers have done great work to solve this kind of problems. The widely used methods are the standard nite element method (FEM) and/or the boundary element method (BEM) 13 . It is, however, widely realized that substantial improvements are needed to address the accuracy, efciency and robust- ness of these methods for coupled structural acoustic problems. FEM is extensively preferred to model the structure part of structural acoustic problems, where lower-order ReissnerMindlin plate element is often chosen due to its efciency and simplicity 46 . However, this method is well-known to have two inherent drawbacks. Firstly, the ReissnerMindlin plate elements often suffer from the so-called shear lockingproblem in the thin plate case, which is resulted from the incorrect transverse shear strains under the pure bending condition 7 . The other one is the overly-stiff property of FEM in the numerical solution, which leads to a signicant loss of accuracy 8 . In order to eliminate shear lockingphenomenon, many numerical techniques and effective improvement of formulations have been proposed, such as the selective reduced integration scheme 911 , free formulation method 12 , and mixed formulation/ hybrid elements 13,14 . Unfortunately, all these methods have some drawbacks like instability due to rank de- ciency, inaccuracy and complex formulation. Then many new numerical techniques were developed to a) State KeyLab of Advanced Technology for Vehicle Body Design and Manufacture, Hunan University, Changsha, 410082, CHINA; email: [email protected]. b) School of Aerospace Systems, University of Cincinnati 2851 Woodside Dr, Cincinnati, OH 45221, USA. c) State KeyLab of Advanced Technology for Vehicle Body Design and Manufacture, Hunan University, Changsha, 410082, CHINA; email: [email protected] (Corresponding author). d) Mechanical Engineering, University of Cincinnati, 598 Rhodes, Cincinnati, OH 45221, USA. e) State KeyLab of Advanced Technology for Vehicle Body Design and Manufacture, Hunan University, Changsha, 410082, CHINA. 196 Noise Control Engr. J. 62 (4), July-August 2014 Published by INCE/USA in conjunction with KSNVE
Transcript
Page 1: A coupled ES-BEM and FM-BEM for structural acoustic problems

A coupled ES-BEM and FM-BEM for structural acoustic problemsF Wua)b) GR Liub) GY Lic) YJ Liud) and ZC Hee)

(Received 8 December 2013 Revised 7 July 2014 Accepted 9 July 2014)

In this paper a coupled numerical method of the edge-based smoothed finiteelement (ES-FEM) with the fast multipole BEM (FM-BEM) is proposed to ana-lyze structural acoustic problems The vibrating structure is modeled using theso-called ES-FEM-DSG3 method where the 3-node linear triangle plate ele-ments based on the ReissnerndashMindlin plate theory with the discrete shear gap(DSG) technique for overcoming the shear locking are applied The edge-basedgradient smoothing operations are applied to ldquosoftenrdquo the ldquooverly-stiffrdquo behaviorin the standard FEM which significantly reduces the inherent numerical dis-persion error The normal velocities on the surface of the structure are imposedas boundary conditions for the acoustic domain which is modeled using theFM-BEM for both the interior and exterior acoustic domains Comparing withthe conventional BEM the matrix vector multiplication and the memory re-quirement in the FM-BEM are reduced dramatically The coupled ES-FEMFM-BEM method takes the advantages of both ES-FEM and FM-BEM whichcan avoid drawbacks of the ldquooverly-stiffrdquo behavior in FEM and computationalinefficiency in the conventional BEM Two numerical examples are presentedto verify and demonstrate the effectiveness of the combined method one aca-demic problem for studying in detail the accuracy and efficiency of the presentmethod and one application to a practical vehicle noise simulation copy 2014Institute of Noise Control Engineering

Primary subject classification 753 Secondary subject classification 755

1 INTRODUCTION

Noises generated by a vibrating thin structure canbe commonly found in numerous engineering systemssuch as aircrafts sea vessels and the land vehiclesThese acoustic problems are closely associated tofluidndashstructure interactions and are of increasing con-cern when designing passenger transportation systemsMany researchers have done great work to solve thiskind of problems The widely used methods are thestandard finite element method (FEM) andor the

boundary element method (BEM)1ndash3 It is howeverwidely realized that substantial improvements areneeded to address the accuracy efficiency and robust-ness of these methods for coupled structural acousticproblems

FEM is extensively preferred to model the structurepart of structural acoustic problems where lower-orderReissnerndashMindlin plate element is often chosen due toits efficiency and simplicity4ndash6 However this methodis well-known to have two inherent drawbacks Firstlythe ReissnerndashMindlin plate elements often suffer fromthe so-called ldquoshear lockingrdquo problem in the thin platecase which is resulted from the incorrect transverseshear strains under the pure bending condition7 Theother one is the ldquooverly-stiff rdquo property of FEM inthe numerical solution which leads to a significantloss of accuracy8 In order to eliminate ldquoshear lockingrdquophenomenon many numerical techniques and effectiveimprovement of formulations have been proposedsuch as the selective reduced integration scheme9ndash11free formulation method12 and mixed formulationhybrid elements1314 Unfortunately all these methodshave some drawbacks like instability due to rank defi-ciency inaccuracy and complex formulation Thenmany new numerical techniques were developed to

a) State Key Lab of Advanced Technology for Vehicle BodyDesign and Manufacture Hunan University Changsha410082 CHINA email wufeifrankgmailcom

b) School of Aerospace Systems University of Cincinnati2851 Woodside Dr Cincinnati OH 45221 USA

c) State Key Lab of Advanced Technology for Vehicle BodyDesign and Manufacture Hunan University Changsha410082 CHINA email gylihnucn (Correspondingauthor)

d) Mechanical Engineering University of Cincinnati 598Rhodes Cincinnati OH 45221 USA

e) State Key Lab of Advanced Technology for Vehicle BodyDesign and Manufacture Hunan University Changsha410082 CHINA

196 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

further enhance the stability and accuracy of numericalsolution Such as enhanced assumed strain (EAS)methods1516 and assumed natural strain (ANS) meth-ods1718 Recently the discrete shear gap (DSG)method which works for elements of different ordersand shapes was developed for overcoming the ldquoshearlockingrdquo phenomenon In this work DSG is chosendue to its several superior properties19 In the otherfrontier of developing advanced FEM and overcomingthe ldquooverly-stiff rdquo drawback of FEM a family ofsmoothed finite element methods (S-FEM) was devel-oped by Liu et al based on the so-called weakenedweak formulation using the gradient smoothing tech-nique20ndash22 These novel S-FEM methods have beenfound and proven to always be softer than the FEMcounterpart and offer desirable ways to effectivelyldquosoftenrdquo the numerical model The cell-based smooth-ing finite element method (CS-FEM) was firstly devel-oped which works well for heavily distorted elementsand the general n-sided polygonal elements20 Thennode-based smoothed finite element method (NS-FEM)21 was developed for overly-soft behavior so asto produce upper bound solutions (for force drivenproblems) It is found instable temporally due to theoverly-soft feature Therefore stabilization techniquesare needed when NS-FEM is used to solve dynamicproblems22 A soft yet temporal stable model knownas the edge-based smoothed finite element (ES-FEM)that exhibits ldquoultrardquo accuracy and super convergenceproperties was also formulated In addition it is foundto be the best performer of all the linear models devel-oped so far for structural dynamic problems23ndash25 Forthe above reasons we choose ES-FEM together withDSG technique to model the structure part of ourstructural acoustic problems

The standard BEM is based on the boundary integralequation (BIE) formulations which discretizes only theboundary of the problem domain leading to a small setof discretized system equations In addition it is idealfor handling problems of infinite domains Due to thesetwo major features BEM has been found effective forwave propagation problems such as the acoustics andparticularly attractive for acoustics in infinite exteriordomains BEM has been applied in acoustic area formore than four decades26ndash28 However due to its com-putational inefficiency in establishing the discretizedsystem equations that is usually fully-populated andbadly conditioned BEM is limited to solving modelswith small or medium sizes Over the past decadesmany techniques are applied to improve the overallsolution efficiency of BEM Techniques including H-matrices29 the wavelet basis30 the fast Fourier trans-form31 and the fast multipole method3233 are adoptedto accelerate the matrixndashvector multiplication Efficient

iterative solvers such as the generalized minimum res-idue (GMRES) method34 and the conjugate gradientsquared (CGS) method35 are also chosen to solve thesystem of equations of a BEM model Among all theseadvances in BEM the FM-BEM stands out for its out-standing efficiency and hence is chosen for modelingthe acoustic fluid especially where there is an exterioracoustic media involved

The FM-BEM method has two key techniques to beimplemented One is the fast multipole method (FMM)which was first introduced by Rokhlin36 in the mid1980s The fundamental principle of the FMM is a mul-tipole expansion of the kernel by which the direct con-nection between the source point and the collocationpoint is separated Another is the use of iterative equa-tion solvers such as the GMERS34 With these twotechniques the FM-BEM37ndash42 can reduce the matrixvector multiplication dramatically In addition thememory requirement is also reduced The applicationof FM-BEM in acoustics was introduced in detail byNail et al40

This research aims to take advantages of both ES-FEM and FM-BEM techniques to model structuralacoustic systems Through such coupling we attemptto avoid drawbacks of the ldquooverly-stiff rdquo behavior inthe FEM for the vibrating structure and computationalinefficiency in the BEM for the interior acoustic cham-ber and exterior acoustic media Considering the hugedifference in density between the solid structure andair as the acoustic media the feedback of the acousticair onto the structure is neglected in this work Thismeans that only a ldquoweak couplingrdquo or one way couplingfrom the solid structure to the air will be taken into con-sideration The ES-FEM is applied to approximate thevibrations of the structure whose normal velocities so-lution are imposed as boundary conditions for theacoustic domain The FM-BEM is used to simulateboth the frequency responses in interior region andsound field distributions in the exterior region

The paper is organized as follows In Sec 2 we be-gin with a brief description of basic ES-FEM formula-tions implemented with DSG method for 2D structuredomain In Sec 3 Galerkin weak form and discretizedsystem equations for 3D acoustic problems are firstlypresented and then the edge-based gradient smoothingoperation for 3D acoustic problems is briefly intro-duced In Sec 4 conventional BEM formulation for3D acoustic problems is firstly reviewed and then themultipole expansion theory which is the fundamental ofthe FM-BEM is simply described In Sec 5 numericalexamples and application are presented to demonstratethe efficiency and validity of the coupled ES-FEMFM-BEM Finally a summary is given in Sec 6 to concludethis work

197Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

2 ES-FEM FORMULATIONS FOR PLATESTRUCTURE

Consider a vibrating plate subjected to external exci-tations The vibrating plate is modeled using 3-nodestriangle plate elements based on the low order ReissnerndashMindlin plate theory The ReissnerndashMindlin theory isintended for thick plates in which the normal to themid-surface remains straight but not necessarily perpen-dicular to the mid-surface Due to its simplicity and effi-ciency ReissnerndashMindlin theory is useful and practicalfor modeling plates that are not ldquothinrdquo where the classicplate theory is no longer valid However when it is ap-plied to thin plates these low-order plate elements oftensuffer from the so-called ldquoshear lockingrdquo This is due tothat the transverse shear strains cannot vanish under thepure bending condition based on the ReissnerndashMindlintheory In order to eliminate the shear locking discreteshear gap triangular element (DSG) method19 has beenutilized to overcome the shear locking problem togetherwith the edge-based smoothed techniques For easy ref-erence this technique is termed as ES-FEM-DSG3where ldquo3rdquo stands for the fact that we use only triangularelements Because we use only triangular elements thegeometry of the plate can be practically arbitrary

21 ES-FEM-DSG3 Formulations Basedon the ReissnerndashMindlin Plate

Based on the ReissnerndashMindlin plates theory the un-known vector of three independent field variables at anypoint in the problem domain of structure can be definedas u = θx θy w

T where θx and θy are the rotationangles of the line normal to the undeformed neutral sur-face in the xndashz and yndashz planes respectively and w isthe deflection The dynamic variation equation forReissnerndashMindlin plate elements without damping canbe described as follows43Z

Ωs

deTbDbebdΩthornZΩs

deTs DsesdΩ

thornZΩs

duTrto2udΩZΓs

duTtdΩ frac14 0 eth1THORN

where the bending stiffness constitutive coefficients Dband transverse shear stiffness constitutive coefficientsDs are defined as

Db frac14 Et3

12 1 n2eth THORN1 n 0n 1 00 0 1 n=2

24

35

Ds frac14 ktG1 00 1

eth2THORN

in which E represents Youngs modulus n is the Poissonratio G is the shear modulus and k is the shear correc-tion factor that is set as 56 in this work

The unknown field variable displacements can beapproximated by nodal displacements using shapefunction

u frac14 Νsue frac14Xnifrac141

Ni xeth THORNui

du frac14 Nsdue frac14Xnifrac141

Ni xeth THORNdui eth3THORN

where Ns is generalized shape functions and ue is thevector of generalized nodal displacements for eachplate element Ni and ui = θxi θyi wi

T are the shapefunction and nodal variable at node i respectivelyApplying Eqn (3) the discretized system equations ofEqn (1) can be written in following matrix form43

Meurou thornKu frac14 F eth4THORNwhere

K frac14ZΩBb

TDbBbdΩthornZΩBs

TDsBsdΩ

The stiffness matrices eth5THORN

M frac14ZΩrNs

Tdiag t3

12t3

12t

NsdΩ

The mass matrix eth6THORNF frac14

ZΓs

NsTtdΓ

The vector of nodal forces eth7THORNwhereBb frac14 Bb1 Bb2 ⋯ Bbnfrac12 is the strainndashdeflectionmatrix for bending Bs frac14 Bs1 Bs2 ⋯ Bsnfrac12 is thestrainndashdeflection matrix for shearing t is the thickness ofthe plate and t is the external load on the plate where

Bbi frac14

Ni

x0 0

0 Ni

y0

Ni

y Ni

x0

2666664

3777775

Bsi frac14Ni 0

Ni

x

0 NiNi

y

264

375 eth8THORN

Using the smoothed strainndashdeflection matrix Bb

and Bs computed based on the edges of elements toreplace Bb and Bs the smoothed stiffness can beexpressed as

K frac14ZΩ

BbTDb BbdΩthorn

BsTDs BsdΩ eth9THORN

198 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

The details of computing the smoothing strainndashdeflection matrices based on the edges can be foundin the following sub-section

Finally the ES-FEM-DSG3 formulation for struc-tural domain then can be written as

Meurouthorn Ku frac14 F eth10THORN

22 Edge-Based Smoothing Operationfor the Plate Structure

In this section edge-based smoothed finite elementmethod or ES-FEM for plates is introduced The platedomain is first discretized using a set of 3-node trian-gles just as in the standard FEM Because the ReissnerndashMindlin theory uses the derivatives of the deflection tocompute the strains only the stiffness matrix is smoothedin the ES-FEM The assembling of the stiffness matrixand the integration is based on the smoothing domainswhich is associated with the edges of the triangles Usingthe edges of these triangles we are able to construct Ns

smoothing domains For edge k the smoothing domainΩk is constructed by connecting the centroids of theneighbor triangles and the end-points of edge k Asshown in Fig 1 for interior edges the smoothing domainΩk is a quadrangle which is the assembly of the sub-domains of two neighboring elements while for edgeson the plate boundary the smoothing domain Ωk is onlya single (triangular) sub-domain The following are the

details of the calculation of the smoothed stiffness matrixfor the 2D structure problem

The smoothing operation is firstly applied to the bend-ing (in-plane) strain and the shear (off-plane) stain of theplate over each of the edge-based smoothing domains

laquob xketh THORN frac14 1Ak

ZΩk

laquob xeth THORNdΩ

laquos xketh THORN frac14 1Ak

ZΩk

laquos xeth THORNdΩ eth11THORN

where Ak is the area of the smoothing domain Ωkwhich can be calculated as follows

Ak frac14ZΩk

dΩ frac14 13

XNek

ifrac141

Aie eth12THORN

in which Nek is the number of the sub-domain of edge k

(that is either 2 for interior edges or 1 for edges on theplate boundary) and Ai

e is the area of ith sub-domain ina triangle element

Based on the assumption made in the ReissnerndashMindlin plate theory the bending strain can beexpressed as follows5

laquob xketh THORN frac14Xi2Mk

Bbi xketh THORNui eth13THORN

whereMk is the total number of vertex of the smoothingdomain which is either 4 for interior edges and 3 foredges on the plate boundary

When the thickness of plates becomes small theReissnerndashMindlin plates often suffer the shear lockingphenomenon In order to avoid such a locking theshear strain is calculated using discrete shear gap trian-gular element (DSG) method19 Combining with thesmoothing operation the smoothed shear strain matrixcan be calculated as follows

laquos xketh THORN frac14Xi2Mk

Bsi xketh THORNui eth14THORN

where the smoothed strain matrix Bbi xketh THORN and Bsi xketh THORNin Eqns (13) and (14) can be calculated by combiningwith the Eqns (11) and (12)

Bbi xketh THORN frac14 1Ak

XNek

ifrac141

13AieBbi xketh THORN

Bsi xketh THORN frac14 1Ak

XNek

ifrac141

13AieBsi xketh THORN eth15THORN

More details about Bbi(xk) and Bsi(xk) based onReissnerndashMindlin plate theory and the discrete shear

Field nodes Centroid of triangle

Edge of triangleBoundary of kΩ

Smoothing domain

Global boundary

Fig 1mdashTriangular mesh for a plate forES-FEM model and edge-basedsmoothing domains constructed byconnecting the centroids of theneighbor triangles and the twoend-points of edge k

199Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

gap triangular element (DSG) method can be found inRef 19

Using the smoothed strain matrix Bs and Bb the

smoothed strain stiffness matrix Kbketh THORN and smoothed

shear stiffness matrix Ksketh THORNfor the edge-based smoothed

domain k can be evaluated as

Kbketh THORN frac14

ZΩk

BbTDb BbdΩ

frac14XNs

kfrac141

Ak BbTDb Bb

Ksketh THORN frac14

ZΩk

BsTDs BsdΩ

frac14XNs

kfrac141

Ak BsTDs Bs eth16THORN

The global smoothed bending stiffness Kb and globalsmoothed shear stiffness Ks based on the edges can be as-sembled just as the same procedure as in the standard FEMThen the global smoothed stiffness can be evaluated as

K frac14 Kb thorn Ks eth17THORNIf the smoothing operation based on the edges is only

applied to bending stiffness Kb or shear stiffness Ksand the other one remains un-smoothed we can obtaintwo variant forms of ES-FEM where the global stiff-ness are computed using

K Beth THORN frac14 Kb thornKs eth18THORN

where K Beth THORN is global smoothed stiffness with only thebending stain smoothed It is denoted as ES(B)-FEMAlternatively

K Seth THORN frac14 Kb thorn Ks eth19THORN

where K Seth THORN is global smoothed stiffness with only theshear stain smoothed It is denoted as ES(S)-FEM

3 ES-FEM FORMULATIONS FOR 3DACOUSTIC PROBLEMS

Note that the ES-FEM is applicable also to acousticproblems and this section briefs the process

31 GS-Galerkin Weak Form and DiscretizedSystem Equations

In the acoustic domain we firstly define an enclosedcavity Ωf with Neumann boundary ΓN assuming thatthe fluid is homogeneous inviscid compressible and

only undergoes small translational movement Lettingp denote the acoustic pressure and k denote the wavenumber the governing equation for the sound pressurecan be expressed as

Δpthorn k2p frac14 0 in Ωf eth20THORN

where Δ is the Laplace operator the wave number canbe written as k = oc o is the angular frequency ofthe pressure oscillation and c is the speed of soundtraveling in the acoustic fluid

The Neumann boundary of the acoustic domain canbe defined as the following

rpn frac14 jrovn on ΓN eth21THORN

where j frac14 ffiffiffiffiffiffiffi1p

r is the density of medium vn denotesnormal velocity on the boundary The field variablepressure can be approximated using a shape functiondefined as

p frac14Xmifrac141

Nipi frac14 Np eth22THORN

where Pi denotes the unknown nodal pressure and Ni

are shape functions in node i N is the generated shapefunction and P is the vector of generated pressure foreach tetrahedron element

Applying the Eqn (22) using shape function as theweight function the standard Galerkin weak form foracoustic problem without acoustical damping can bewritten as25

ZΩrNrNP dΩthorn k2

ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth23THORN

Using the smoothed item rN based on the edges ofelements to replace the gradient component rN thegeneralized smoothed Garlerkin (GS-Galerkin) weakformulation for acoustic problem can be written as

rN rNPdΩthorn k2ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth24THORN

Finally the discretized system equations in Eqn (24)can be written in following matrix form

K frac14ZΩ

rN T rNdΩ eth25THORN

200 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

where

K frac14ZΩ

rN T rNdΩ

The smoothed acoustical stiffness matrix

eth26THORNΜ frac14

ZΩNTNdΩ

The acoustical mass matrix eth27THORN

F frac14ZΓN

NTvndΓ

The vector of nodal acoustic forces eth28THORN

Pf gT frac14 p1 p2⋯ pnf gNodal acoustic pressure in the domain

eth29THORN

32 Edge-Based Gradient SmoothingOperation for 3D Acoustic Domain

In this section the formulation of ES-FEM for 3Dacoustic fluid is presented The acoustic domain is di-vided exactly as that of standard FEM using four nodetetrahedral elements The edge-based gradient smooth-ing domains which are also serving as integrationdomains are then formed in association with these tet-rahedral elements As shown in Fig 2 the sub-smooth-ing domain of edge k in cell i is created by connectingthe centroid of cell i to the two end-nodes of the edgek and the related surface triangles

For acoustic problems the gradient smoothing oper-ation will be applied over each edge-based smoothingdomain on the velocity v The smoothed velocity whichis deduced by the gradient of acoustic pressure isdenoted as

v xketh THORN frac14 1Vk

ZΩk

v xeth THORNdΩ eth30THORN

where Vk frac14ZΩk

dΩ denotes the volume of smoothing

domain for edge kThe smoothed velocity can be expressed in terms of

acoustic pressure by applying the Greens theorem

v xketh THORN frac14 1jroVk

ZΩk

rpdΩ

frac14 1jroVk

ZΓk

pndΓ eth31THORN

Substituting the field variable (acoustic pressure) in-terpolation in form of Eqn (22) into Eqn (31) thesmoothed velocity for edge k can be denoted as the fol-lowing matrix form

v xketh THORN frac14 1jro

XI2Mk

Bi xketh THORNpi eth32THORN

where Mk represents the total number of nodes in thesmoothing domain of edge k Bi can be defined as

BTi xketh THORN frac14 bi1 bi2 bi3

eth33THORN

bip frac14 1Vk

ZΓk

Ni xeth THORNnp xeth THORNdΓ eth34THORN

Finally the smoothed stiffness matrix shown inEqn (25) can be assembled based on the smoothed Bas

Kketh THORN frac14

ZΩk

BT BdΩ frac14XNs

kfrac141

Vk BT B eth35THORN

Owning to the compact supports of the FEM shapefunctions the assembled smoothed stiffness matrixEqn (35) is banded and symmetric Therefore systemequations can be solved efficiently even though it dis-cretizes the entire domain as long as the domain isenclosed However when it is used for exterior media(that is infinite) some kind of non-reflecting boundarytechniques44 must be used In such cases the FM-BEMcan be a better choice because there is no need for artificialnon-reflecting boundary

n

n

n

n

n

n

n

n

Edge k

Fig 2mdash3D edge-based smoothing domainsconstructed by connecting the centroidof cell i to end-nodes of the edge kand the related surface triangles

201Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

4 FAST MULTIPOLE BEM FORACOUSTIC PROBLEMS

41 Conventional BEM Formulationsfor Acoustic Problems

In this section we first review the conventional BEMformulation for Helmholtz equations The fundamen-tal solution or the full-space Greens function foracoustic problems is well-known and can be denotedas follows45

G x yeth THORN frac14 ejk xyj j

4p x yj j eth36THORN

where j frac14 ffiffiffiffiffiffiffi1p

k is the wavenumber and |x y| isthe distance between the collocation point x and thesource point y

Combining the conventional boundary integral equa-tion (CBIE) and the hypersingular boundary integralequation (HBIE) a well-known integral equationnamed as CHBIE formulation for Helmholtz equationin Eqn (20) without the incident wave can be writtenas45

G x yeth THORNn yeth THORN p yeth THORNdΓ yeth THORN thorn C xeth THORNp xeth THORN

24

35

thorn aZΓ

2G x yeth THORNn yeth THORNn xeth THORN p yeth THORNdΓ yeth THORN

frac14ZΓ

G x yeth THORNq yeth THORNdΓ yeth THORN

thorn aZΓ

G x yeth THORNn xeth THORN q yeth THORNdΓ yeth THORN C xeth THORNq xeth THORN

24

35

8x 2 Γ eth37THORNwhere q is defined as q frac14 p

n The constant C(x) is set as12 for smooth surface around x and the coupling con-stant a is defined as jk

Dividing the boundary into N surface elements thediscretized form of the CHBIE formulation can beexpressed as45

XNjfrac141

fijpj frac14XNjfrac141

gijqj eth38THORN

where

fijpj frac14ZΔΓj

G x yeth THORNn yeth THORN pjdΓ yeth THORN thorn 1

2dijpj

thorn aZΔΓj

2G x yeth THORNn yeth THORNn xeth THORNpjdΓ yeth THORN

gijqj frac14ZΔΓj

G x yeth THORNqjdΓ yeth THORN

thorn aZΔΓj

G x yeth THORNn xeth THORN qjdΓ yeth THORN 1

2dijqj

264

375 eth39THORN

where dij is the Kronecker Delta and ΔΓj denoteselement j

The discretized form of the BurtonndashMiller formula-tion in Eqn (38) can be transformed to the followingsystem of equations by moving the known terms tothe right-hand side and the unknown terms to the left-hand side

a11 a12 ⋯ a1Na21 a22 ⋯ a2N⋮ ⋮ ⋱ ⋮aN1 aN2 ⋯ aNN

2664

3775

l1l2⋮lN

8gtgtltgtgt

9gtgt=gtgt

frac14b1b2⋮bN

8gtgtltgtgt

9gtgt=gtgtor Alfrac14b eth40THORN

where A l and b are the system matrix unknown vec-tor and known vector respectively

42 The Fast Multipole Method Implementedin BEM

There are two main techniques applied to improvethe efficiency of the conventional BEM Firstly the fastmultipole method (FMM) is employed to speed up thematrixndashvector multiplication in Al then an efficient it-erative solver such as the generalized minimum residuemethod (GMRES) will be applied to solve the systemof equations given by Eqn (40) With FMM the fastmultipole boundary element method can be con-structed The fundamental principle of the FMM is amultipole expansion of the kernel in which the directconnection between the source point and the colloca-tion point is separated The details of the derivationsof the FM-BEM formulations can be found in Refs 40and 45 With the fast multipole BEM acoustic BEMmodels with DOFs up to several millions have beensolved on laptop PCs with a RAM size of only 8 GB

5 NUMERICAL EXAMPLES

In this section two numerical applications of 3Dcases are presented in order to verify the effectivenessof the proposed combination of ES-FEM and FM-BEM formulations Because of the huge differenceexisting in terms of mass density of the structure and

202 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

air acoustic modes are not coincident with the struc-tural modes it is thus practical to neglect direct interac-tions between the structure and air46 meaning that thestructure dynamics is assumed not to be influenced bythe fluid For comparison the results obtained fromthe FEM with extremely fine mesh are also providedas the reference results The purpose of the first exam-ple of a simple elastic plate backed by a closed acousticcavity is to show the advantages of ES-FEM and FM-BEM The second example is an application of thepresent combined methods to a practical problem in ve-hicle engineering

51 Box with Flexible Plate on Top

In this subsection a weak coupling model of a flex-ible plate and air cavity is established The model isshown in Fig 3 The weakly coupled model is a combi-nation of the flexible plate on the top and a closedacoustic cavity attached The elastic plate is made ofaluminum (r = 2700 kgm3 n = 03 and E = 71 Gpa)The acoustic cavity is full of air (r = 121 kgm3 and

c = 343 ms) The plate which has a dimension of050 m 060 m and a thickness of t = 0003 m is sim-ply supported on all the four edges The closed acousticcavity has a dimension of 050 m 060 m 040 mThe remaining walls (except the coupled wall) of cavityare assumed to be rigid with the surface velocity fixedat v = 0

The top elastic plate is divided with ReissnerndashMind-lin triangle plate elements An evenly distributed timeharmonic load equal to 100 N is applied at the centerof plate (point A in Fig 3) First the forced frequencyresponses are computed at the center of the plate usingdifferent methods including FEM ES-FEM ES(B)-FEMand ES(S)-FEM with same model (155 nodes 264 ele-ments) The frequency ranges from 1 to 1000 Hz Thereference result is provided using FEM with much smal-ler elements (1265 nodes 2390 elements)

As shown in Fig 4 in the low frequency domain (0to 200 Hz) results obtained from FEM and ES-FEMshow excellent agreements with the reference resultdemonstrating that both FEM and ES-FEM can provideaccuracy results in low frequencies As the frequencyincreases the deviation between FEM result and thereference result becomes larger suggesting that the ac-curacy of the FEM result decreases with the increase ofthe frequency We also note that the eigen-frequenciesin FEM result (peaks in response curve) become higherand higher compared to the reference result This devi-ation mainly results from the inherent drawback ofldquoover-stiffnessrdquo in FEM based on the standard weakformulation The ES-FEM provides much more accu-rate result in higher frequency range compared to theFEM model using the same mesh From Fig 5 wecan see that ES(B)-FEM can also produce results simi-lar to that of ES-FEM The softening effect of ES(B)-FEM is almost equal to that of ES-FEM In additionas showed in Fig 6 the response curves obtained from

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFEMReference

Fig 4mdashFrequency responses computed at point A using ES-FEM and FEM for the plate alone

Point A-exciting point

Point B-a response point in the acoustic domain

Aluminum Plate ( s)

Acoustic domain ( f)

Fig 3mdashA flexible aluminum plate backed by abox of air

203Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

ES(S)-FEM have little difference from that of FEM (us-ing the same mesh) which means that the softening ef-fect by the edge-based smoothing on the off-plane shearstrain is minimum and can be neglected Therefore itcan be concluded that the total softening effects ofES-FEM are mainly due to smoothing the in-planebending strain

The sound pressure level (SPL) responses at point Bin acoustic domain (Fig 3) are also computed usingvarious combination of methods and the results areplotted in Fig 7 The normal velocity of the flexibleplate which provides the boundary condition of acous-tic domain is approximated using ES-FEM and FEMThe 3D acoustic domain is divided using tetrahedronelements (1045 nodes 6335 elements) for FEM andES-FEM If FM-BEM is chosen only the surface ofthe 3D acoustic domain is discretized with triangle ele-ments and hence the number of elements is much

smaller (634 nodes 1264 elements) The computationis performed for frequencies ranging from 1 to700 Hz For comparison the numerical result obtainedby the coupled FEMFEM with a very fine mesh(15864 nodes and 82858 elements) is presented asthe reference

As shown in Fig 7 the coupled FEMFEM gives theleast accurate results compared to all the other modelsThe over-stiffness phenomenon of FEM in 3D acousticproblems can also be observed and it becomes muchmore pronounced with the increase of the frequencyThe stiffness matrix in coupled ES-FEMES-FEM issofter and hence the results in high frequency rangeshow better agreements with reference results The cou-pled ES-FEMFM-BEM model has almost the samelevel accuracy as the coupled ES-FEMES-FEM modelIt is found that the FM-BEM can offer accurate resultsfor interior acoustic problems

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(B)-FEMES-FEMFEMReference

Fig 5mdashFrequency responses computed at point A using ES(B)-FEM ES-FEM and FEM for theplate alone

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(S)-FEMES-FEMFEMReference

Fig 6mdashFrequency response analysis in point A using ES(S)-FEM ES-FEM and FEM for theplate alone

204 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 2: A coupled ES-BEM and FM-BEM for structural acoustic problems

further enhance the stability and accuracy of numericalsolution Such as enhanced assumed strain (EAS)methods1516 and assumed natural strain (ANS) meth-ods1718 Recently the discrete shear gap (DSG)method which works for elements of different ordersand shapes was developed for overcoming the ldquoshearlockingrdquo phenomenon In this work DSG is chosendue to its several superior properties19 In the otherfrontier of developing advanced FEM and overcomingthe ldquooverly-stiff rdquo drawback of FEM a family ofsmoothed finite element methods (S-FEM) was devel-oped by Liu et al based on the so-called weakenedweak formulation using the gradient smoothing tech-nique20ndash22 These novel S-FEM methods have beenfound and proven to always be softer than the FEMcounterpart and offer desirable ways to effectivelyldquosoftenrdquo the numerical model The cell-based smooth-ing finite element method (CS-FEM) was firstly devel-oped which works well for heavily distorted elementsand the general n-sided polygonal elements20 Thennode-based smoothed finite element method (NS-FEM)21 was developed for overly-soft behavior so asto produce upper bound solutions (for force drivenproblems) It is found instable temporally due to theoverly-soft feature Therefore stabilization techniquesare needed when NS-FEM is used to solve dynamicproblems22 A soft yet temporal stable model knownas the edge-based smoothed finite element (ES-FEM)that exhibits ldquoultrardquo accuracy and super convergenceproperties was also formulated In addition it is foundto be the best performer of all the linear models devel-oped so far for structural dynamic problems23ndash25 Forthe above reasons we choose ES-FEM together withDSG technique to model the structure part of ourstructural acoustic problems

The standard BEM is based on the boundary integralequation (BIE) formulations which discretizes only theboundary of the problem domain leading to a small setof discretized system equations In addition it is idealfor handling problems of infinite domains Due to thesetwo major features BEM has been found effective forwave propagation problems such as the acoustics andparticularly attractive for acoustics in infinite exteriordomains BEM has been applied in acoustic area formore than four decades26ndash28 However due to its com-putational inefficiency in establishing the discretizedsystem equations that is usually fully-populated andbadly conditioned BEM is limited to solving modelswith small or medium sizes Over the past decadesmany techniques are applied to improve the overallsolution efficiency of BEM Techniques including H-matrices29 the wavelet basis30 the fast Fourier trans-form31 and the fast multipole method3233 are adoptedto accelerate the matrixndashvector multiplication Efficient

iterative solvers such as the generalized minimum res-idue (GMRES) method34 and the conjugate gradientsquared (CGS) method35 are also chosen to solve thesystem of equations of a BEM model Among all theseadvances in BEM the FM-BEM stands out for its out-standing efficiency and hence is chosen for modelingthe acoustic fluid especially where there is an exterioracoustic media involved

The FM-BEM method has two key techniques to beimplemented One is the fast multipole method (FMM)which was first introduced by Rokhlin36 in the mid1980s The fundamental principle of the FMM is a mul-tipole expansion of the kernel by which the direct con-nection between the source point and the collocationpoint is separated Another is the use of iterative equa-tion solvers such as the GMERS34 With these twotechniques the FM-BEM37ndash42 can reduce the matrixvector multiplication dramatically In addition thememory requirement is also reduced The applicationof FM-BEM in acoustics was introduced in detail byNail et al40

This research aims to take advantages of both ES-FEM and FM-BEM techniques to model structuralacoustic systems Through such coupling we attemptto avoid drawbacks of the ldquooverly-stiff rdquo behavior inthe FEM for the vibrating structure and computationalinefficiency in the BEM for the interior acoustic cham-ber and exterior acoustic media Considering the hugedifference in density between the solid structure andair as the acoustic media the feedback of the acousticair onto the structure is neglected in this work Thismeans that only a ldquoweak couplingrdquo or one way couplingfrom the solid structure to the air will be taken into con-sideration The ES-FEM is applied to approximate thevibrations of the structure whose normal velocities so-lution are imposed as boundary conditions for theacoustic domain The FM-BEM is used to simulateboth the frequency responses in interior region andsound field distributions in the exterior region

The paper is organized as follows In Sec 2 we be-gin with a brief description of basic ES-FEM formula-tions implemented with DSG method for 2D structuredomain In Sec 3 Galerkin weak form and discretizedsystem equations for 3D acoustic problems are firstlypresented and then the edge-based gradient smoothingoperation for 3D acoustic problems is briefly intro-duced In Sec 4 conventional BEM formulation for3D acoustic problems is firstly reviewed and then themultipole expansion theory which is the fundamental ofthe FM-BEM is simply described In Sec 5 numericalexamples and application are presented to demonstratethe efficiency and validity of the coupled ES-FEMFM-BEM Finally a summary is given in Sec 6 to concludethis work

197Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

2 ES-FEM FORMULATIONS FOR PLATESTRUCTURE

Consider a vibrating plate subjected to external exci-tations The vibrating plate is modeled using 3-nodestriangle plate elements based on the low order ReissnerndashMindlin plate theory The ReissnerndashMindlin theory isintended for thick plates in which the normal to themid-surface remains straight but not necessarily perpen-dicular to the mid-surface Due to its simplicity and effi-ciency ReissnerndashMindlin theory is useful and practicalfor modeling plates that are not ldquothinrdquo where the classicplate theory is no longer valid However when it is ap-plied to thin plates these low-order plate elements oftensuffer from the so-called ldquoshear lockingrdquo This is due tothat the transverse shear strains cannot vanish under thepure bending condition based on the ReissnerndashMindlintheory In order to eliminate the shear locking discreteshear gap triangular element (DSG) method19 has beenutilized to overcome the shear locking problem togetherwith the edge-based smoothed techniques For easy ref-erence this technique is termed as ES-FEM-DSG3where ldquo3rdquo stands for the fact that we use only triangularelements Because we use only triangular elements thegeometry of the plate can be practically arbitrary

21 ES-FEM-DSG3 Formulations Basedon the ReissnerndashMindlin Plate

Based on the ReissnerndashMindlin plates theory the un-known vector of three independent field variables at anypoint in the problem domain of structure can be definedas u = θx θy w

T where θx and θy are the rotationangles of the line normal to the undeformed neutral sur-face in the xndashz and yndashz planes respectively and w isthe deflection The dynamic variation equation forReissnerndashMindlin plate elements without damping canbe described as follows43Z

Ωs

deTbDbebdΩthornZΩs

deTs DsesdΩ

thornZΩs

duTrto2udΩZΓs

duTtdΩ frac14 0 eth1THORN

where the bending stiffness constitutive coefficients Dband transverse shear stiffness constitutive coefficientsDs are defined as

Db frac14 Et3

12 1 n2eth THORN1 n 0n 1 00 0 1 n=2

24

35

Ds frac14 ktG1 00 1

eth2THORN

in which E represents Youngs modulus n is the Poissonratio G is the shear modulus and k is the shear correc-tion factor that is set as 56 in this work

The unknown field variable displacements can beapproximated by nodal displacements using shapefunction

u frac14 Νsue frac14Xnifrac141

Ni xeth THORNui

du frac14 Nsdue frac14Xnifrac141

Ni xeth THORNdui eth3THORN

where Ns is generalized shape functions and ue is thevector of generalized nodal displacements for eachplate element Ni and ui = θxi θyi wi

T are the shapefunction and nodal variable at node i respectivelyApplying Eqn (3) the discretized system equations ofEqn (1) can be written in following matrix form43

Meurou thornKu frac14 F eth4THORNwhere

K frac14ZΩBb

TDbBbdΩthornZΩBs

TDsBsdΩ

The stiffness matrices eth5THORN

M frac14ZΩrNs

Tdiag t3

12t3

12t

NsdΩ

The mass matrix eth6THORNF frac14

ZΓs

NsTtdΓ

The vector of nodal forces eth7THORNwhereBb frac14 Bb1 Bb2 ⋯ Bbnfrac12 is the strainndashdeflectionmatrix for bending Bs frac14 Bs1 Bs2 ⋯ Bsnfrac12 is thestrainndashdeflection matrix for shearing t is the thickness ofthe plate and t is the external load on the plate where

Bbi frac14

Ni

x0 0

0 Ni

y0

Ni

y Ni

x0

2666664

3777775

Bsi frac14Ni 0

Ni

x

0 NiNi

y

264

375 eth8THORN

Using the smoothed strainndashdeflection matrix Bb

and Bs computed based on the edges of elements toreplace Bb and Bs the smoothed stiffness can beexpressed as

K frac14ZΩ

BbTDb BbdΩthorn

BsTDs BsdΩ eth9THORN

198 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

The details of computing the smoothing strainndashdeflection matrices based on the edges can be foundin the following sub-section

Finally the ES-FEM-DSG3 formulation for struc-tural domain then can be written as

Meurouthorn Ku frac14 F eth10THORN

22 Edge-Based Smoothing Operationfor the Plate Structure

In this section edge-based smoothed finite elementmethod or ES-FEM for plates is introduced The platedomain is first discretized using a set of 3-node trian-gles just as in the standard FEM Because the ReissnerndashMindlin theory uses the derivatives of the deflection tocompute the strains only the stiffness matrix is smoothedin the ES-FEM The assembling of the stiffness matrixand the integration is based on the smoothing domainswhich is associated with the edges of the triangles Usingthe edges of these triangles we are able to construct Ns

smoothing domains For edge k the smoothing domainΩk is constructed by connecting the centroids of theneighbor triangles and the end-points of edge k Asshown in Fig 1 for interior edges the smoothing domainΩk is a quadrangle which is the assembly of the sub-domains of two neighboring elements while for edgeson the plate boundary the smoothing domain Ωk is onlya single (triangular) sub-domain The following are the

details of the calculation of the smoothed stiffness matrixfor the 2D structure problem

The smoothing operation is firstly applied to the bend-ing (in-plane) strain and the shear (off-plane) stain of theplate over each of the edge-based smoothing domains

laquob xketh THORN frac14 1Ak

ZΩk

laquob xeth THORNdΩ

laquos xketh THORN frac14 1Ak

ZΩk

laquos xeth THORNdΩ eth11THORN

where Ak is the area of the smoothing domain Ωkwhich can be calculated as follows

Ak frac14ZΩk

dΩ frac14 13

XNek

ifrac141

Aie eth12THORN

in which Nek is the number of the sub-domain of edge k

(that is either 2 for interior edges or 1 for edges on theplate boundary) and Ai

e is the area of ith sub-domain ina triangle element

Based on the assumption made in the ReissnerndashMindlin plate theory the bending strain can beexpressed as follows5

laquob xketh THORN frac14Xi2Mk

Bbi xketh THORNui eth13THORN

whereMk is the total number of vertex of the smoothingdomain which is either 4 for interior edges and 3 foredges on the plate boundary

When the thickness of plates becomes small theReissnerndashMindlin plates often suffer the shear lockingphenomenon In order to avoid such a locking theshear strain is calculated using discrete shear gap trian-gular element (DSG) method19 Combining with thesmoothing operation the smoothed shear strain matrixcan be calculated as follows

laquos xketh THORN frac14Xi2Mk

Bsi xketh THORNui eth14THORN

where the smoothed strain matrix Bbi xketh THORN and Bsi xketh THORNin Eqns (13) and (14) can be calculated by combiningwith the Eqns (11) and (12)

Bbi xketh THORN frac14 1Ak

XNek

ifrac141

13AieBbi xketh THORN

Bsi xketh THORN frac14 1Ak

XNek

ifrac141

13AieBsi xketh THORN eth15THORN

More details about Bbi(xk) and Bsi(xk) based onReissnerndashMindlin plate theory and the discrete shear

Field nodes Centroid of triangle

Edge of triangleBoundary of kΩ

Smoothing domain

Global boundary

Fig 1mdashTriangular mesh for a plate forES-FEM model and edge-basedsmoothing domains constructed byconnecting the centroids of theneighbor triangles and the twoend-points of edge k

199Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

gap triangular element (DSG) method can be found inRef 19

Using the smoothed strain matrix Bs and Bb the

smoothed strain stiffness matrix Kbketh THORN and smoothed

shear stiffness matrix Ksketh THORNfor the edge-based smoothed

domain k can be evaluated as

Kbketh THORN frac14

ZΩk

BbTDb BbdΩ

frac14XNs

kfrac141

Ak BbTDb Bb

Ksketh THORN frac14

ZΩk

BsTDs BsdΩ

frac14XNs

kfrac141

Ak BsTDs Bs eth16THORN

The global smoothed bending stiffness Kb and globalsmoothed shear stiffness Ks based on the edges can be as-sembled just as the same procedure as in the standard FEMThen the global smoothed stiffness can be evaluated as

K frac14 Kb thorn Ks eth17THORNIf the smoothing operation based on the edges is only

applied to bending stiffness Kb or shear stiffness Ksand the other one remains un-smoothed we can obtaintwo variant forms of ES-FEM where the global stiff-ness are computed using

K Beth THORN frac14 Kb thornKs eth18THORN

where K Beth THORN is global smoothed stiffness with only thebending stain smoothed It is denoted as ES(B)-FEMAlternatively

K Seth THORN frac14 Kb thorn Ks eth19THORN

where K Seth THORN is global smoothed stiffness with only theshear stain smoothed It is denoted as ES(S)-FEM

3 ES-FEM FORMULATIONS FOR 3DACOUSTIC PROBLEMS

Note that the ES-FEM is applicable also to acousticproblems and this section briefs the process

31 GS-Galerkin Weak Form and DiscretizedSystem Equations

In the acoustic domain we firstly define an enclosedcavity Ωf with Neumann boundary ΓN assuming thatthe fluid is homogeneous inviscid compressible and

only undergoes small translational movement Lettingp denote the acoustic pressure and k denote the wavenumber the governing equation for the sound pressurecan be expressed as

Δpthorn k2p frac14 0 in Ωf eth20THORN

where Δ is the Laplace operator the wave number canbe written as k = oc o is the angular frequency ofthe pressure oscillation and c is the speed of soundtraveling in the acoustic fluid

The Neumann boundary of the acoustic domain canbe defined as the following

rpn frac14 jrovn on ΓN eth21THORN

where j frac14 ffiffiffiffiffiffiffi1p

r is the density of medium vn denotesnormal velocity on the boundary The field variablepressure can be approximated using a shape functiondefined as

p frac14Xmifrac141

Nipi frac14 Np eth22THORN

where Pi denotes the unknown nodal pressure and Ni

are shape functions in node i N is the generated shapefunction and P is the vector of generated pressure foreach tetrahedron element

Applying the Eqn (22) using shape function as theweight function the standard Galerkin weak form foracoustic problem without acoustical damping can bewritten as25

ZΩrNrNP dΩthorn k2

ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth23THORN

Using the smoothed item rN based on the edges ofelements to replace the gradient component rN thegeneralized smoothed Garlerkin (GS-Galerkin) weakformulation for acoustic problem can be written as

rN rNPdΩthorn k2ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth24THORN

Finally the discretized system equations in Eqn (24)can be written in following matrix form

K frac14ZΩ

rN T rNdΩ eth25THORN

200 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

where

K frac14ZΩ

rN T rNdΩ

The smoothed acoustical stiffness matrix

eth26THORNΜ frac14

ZΩNTNdΩ

The acoustical mass matrix eth27THORN

F frac14ZΓN

NTvndΓ

The vector of nodal acoustic forces eth28THORN

Pf gT frac14 p1 p2⋯ pnf gNodal acoustic pressure in the domain

eth29THORN

32 Edge-Based Gradient SmoothingOperation for 3D Acoustic Domain

In this section the formulation of ES-FEM for 3Dacoustic fluid is presented The acoustic domain is di-vided exactly as that of standard FEM using four nodetetrahedral elements The edge-based gradient smooth-ing domains which are also serving as integrationdomains are then formed in association with these tet-rahedral elements As shown in Fig 2 the sub-smooth-ing domain of edge k in cell i is created by connectingthe centroid of cell i to the two end-nodes of the edgek and the related surface triangles

For acoustic problems the gradient smoothing oper-ation will be applied over each edge-based smoothingdomain on the velocity v The smoothed velocity whichis deduced by the gradient of acoustic pressure isdenoted as

v xketh THORN frac14 1Vk

ZΩk

v xeth THORNdΩ eth30THORN

where Vk frac14ZΩk

dΩ denotes the volume of smoothing

domain for edge kThe smoothed velocity can be expressed in terms of

acoustic pressure by applying the Greens theorem

v xketh THORN frac14 1jroVk

ZΩk

rpdΩ

frac14 1jroVk

ZΓk

pndΓ eth31THORN

Substituting the field variable (acoustic pressure) in-terpolation in form of Eqn (22) into Eqn (31) thesmoothed velocity for edge k can be denoted as the fol-lowing matrix form

v xketh THORN frac14 1jro

XI2Mk

Bi xketh THORNpi eth32THORN

where Mk represents the total number of nodes in thesmoothing domain of edge k Bi can be defined as

BTi xketh THORN frac14 bi1 bi2 bi3

eth33THORN

bip frac14 1Vk

ZΓk

Ni xeth THORNnp xeth THORNdΓ eth34THORN

Finally the smoothed stiffness matrix shown inEqn (25) can be assembled based on the smoothed Bas

Kketh THORN frac14

ZΩk

BT BdΩ frac14XNs

kfrac141

Vk BT B eth35THORN

Owning to the compact supports of the FEM shapefunctions the assembled smoothed stiffness matrixEqn (35) is banded and symmetric Therefore systemequations can be solved efficiently even though it dis-cretizes the entire domain as long as the domain isenclosed However when it is used for exterior media(that is infinite) some kind of non-reflecting boundarytechniques44 must be used In such cases the FM-BEMcan be a better choice because there is no need for artificialnon-reflecting boundary

n

n

n

n

n

n

n

n

Edge k

Fig 2mdash3D edge-based smoothing domainsconstructed by connecting the centroidof cell i to end-nodes of the edge kand the related surface triangles

201Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

4 FAST MULTIPOLE BEM FORACOUSTIC PROBLEMS

41 Conventional BEM Formulationsfor Acoustic Problems

In this section we first review the conventional BEMformulation for Helmholtz equations The fundamen-tal solution or the full-space Greens function foracoustic problems is well-known and can be denotedas follows45

G x yeth THORN frac14 ejk xyj j

4p x yj j eth36THORN

where j frac14 ffiffiffiffiffiffiffi1p

k is the wavenumber and |x y| isthe distance between the collocation point x and thesource point y

Combining the conventional boundary integral equa-tion (CBIE) and the hypersingular boundary integralequation (HBIE) a well-known integral equationnamed as CHBIE formulation for Helmholtz equationin Eqn (20) without the incident wave can be writtenas45

G x yeth THORNn yeth THORN p yeth THORNdΓ yeth THORN thorn C xeth THORNp xeth THORN

24

35

thorn aZΓ

2G x yeth THORNn yeth THORNn xeth THORN p yeth THORNdΓ yeth THORN

frac14ZΓ

G x yeth THORNq yeth THORNdΓ yeth THORN

thorn aZΓ

G x yeth THORNn xeth THORN q yeth THORNdΓ yeth THORN C xeth THORNq xeth THORN

24

35

8x 2 Γ eth37THORNwhere q is defined as q frac14 p

n The constant C(x) is set as12 for smooth surface around x and the coupling con-stant a is defined as jk

Dividing the boundary into N surface elements thediscretized form of the CHBIE formulation can beexpressed as45

XNjfrac141

fijpj frac14XNjfrac141

gijqj eth38THORN

where

fijpj frac14ZΔΓj

G x yeth THORNn yeth THORN pjdΓ yeth THORN thorn 1

2dijpj

thorn aZΔΓj

2G x yeth THORNn yeth THORNn xeth THORNpjdΓ yeth THORN

gijqj frac14ZΔΓj

G x yeth THORNqjdΓ yeth THORN

thorn aZΔΓj

G x yeth THORNn xeth THORN qjdΓ yeth THORN 1

2dijqj

264

375 eth39THORN

where dij is the Kronecker Delta and ΔΓj denoteselement j

The discretized form of the BurtonndashMiller formula-tion in Eqn (38) can be transformed to the followingsystem of equations by moving the known terms tothe right-hand side and the unknown terms to the left-hand side

a11 a12 ⋯ a1Na21 a22 ⋯ a2N⋮ ⋮ ⋱ ⋮aN1 aN2 ⋯ aNN

2664

3775

l1l2⋮lN

8gtgtltgtgt

9gtgt=gtgt

frac14b1b2⋮bN

8gtgtltgtgt

9gtgt=gtgtor Alfrac14b eth40THORN

where A l and b are the system matrix unknown vec-tor and known vector respectively

42 The Fast Multipole Method Implementedin BEM

There are two main techniques applied to improvethe efficiency of the conventional BEM Firstly the fastmultipole method (FMM) is employed to speed up thematrixndashvector multiplication in Al then an efficient it-erative solver such as the generalized minimum residuemethod (GMRES) will be applied to solve the systemof equations given by Eqn (40) With FMM the fastmultipole boundary element method can be con-structed The fundamental principle of the FMM is amultipole expansion of the kernel in which the directconnection between the source point and the colloca-tion point is separated The details of the derivationsof the FM-BEM formulations can be found in Refs 40and 45 With the fast multipole BEM acoustic BEMmodels with DOFs up to several millions have beensolved on laptop PCs with a RAM size of only 8 GB

5 NUMERICAL EXAMPLES

In this section two numerical applications of 3Dcases are presented in order to verify the effectivenessof the proposed combination of ES-FEM and FM-BEM formulations Because of the huge differenceexisting in terms of mass density of the structure and

202 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

air acoustic modes are not coincident with the struc-tural modes it is thus practical to neglect direct interac-tions between the structure and air46 meaning that thestructure dynamics is assumed not to be influenced bythe fluid For comparison the results obtained fromthe FEM with extremely fine mesh are also providedas the reference results The purpose of the first exam-ple of a simple elastic plate backed by a closed acousticcavity is to show the advantages of ES-FEM and FM-BEM The second example is an application of thepresent combined methods to a practical problem in ve-hicle engineering

51 Box with Flexible Plate on Top

In this subsection a weak coupling model of a flex-ible plate and air cavity is established The model isshown in Fig 3 The weakly coupled model is a combi-nation of the flexible plate on the top and a closedacoustic cavity attached The elastic plate is made ofaluminum (r = 2700 kgm3 n = 03 and E = 71 Gpa)The acoustic cavity is full of air (r = 121 kgm3 and

c = 343 ms) The plate which has a dimension of050 m 060 m and a thickness of t = 0003 m is sim-ply supported on all the four edges The closed acousticcavity has a dimension of 050 m 060 m 040 mThe remaining walls (except the coupled wall) of cavityare assumed to be rigid with the surface velocity fixedat v = 0

The top elastic plate is divided with ReissnerndashMind-lin triangle plate elements An evenly distributed timeharmonic load equal to 100 N is applied at the centerof plate (point A in Fig 3) First the forced frequencyresponses are computed at the center of the plate usingdifferent methods including FEM ES-FEM ES(B)-FEMand ES(S)-FEM with same model (155 nodes 264 ele-ments) The frequency ranges from 1 to 1000 Hz Thereference result is provided using FEM with much smal-ler elements (1265 nodes 2390 elements)

As shown in Fig 4 in the low frequency domain (0to 200 Hz) results obtained from FEM and ES-FEMshow excellent agreements with the reference resultdemonstrating that both FEM and ES-FEM can provideaccuracy results in low frequencies As the frequencyincreases the deviation between FEM result and thereference result becomes larger suggesting that the ac-curacy of the FEM result decreases with the increase ofthe frequency We also note that the eigen-frequenciesin FEM result (peaks in response curve) become higherand higher compared to the reference result This devi-ation mainly results from the inherent drawback ofldquoover-stiffnessrdquo in FEM based on the standard weakformulation The ES-FEM provides much more accu-rate result in higher frequency range compared to theFEM model using the same mesh From Fig 5 wecan see that ES(B)-FEM can also produce results simi-lar to that of ES-FEM The softening effect of ES(B)-FEM is almost equal to that of ES-FEM In additionas showed in Fig 6 the response curves obtained from

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFEMReference

Fig 4mdashFrequency responses computed at point A using ES-FEM and FEM for the plate alone

Point A-exciting point

Point B-a response point in the acoustic domain

Aluminum Plate ( s)

Acoustic domain ( f)

Fig 3mdashA flexible aluminum plate backed by abox of air

203Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

ES(S)-FEM have little difference from that of FEM (us-ing the same mesh) which means that the softening ef-fect by the edge-based smoothing on the off-plane shearstrain is minimum and can be neglected Therefore itcan be concluded that the total softening effects ofES-FEM are mainly due to smoothing the in-planebending strain

The sound pressure level (SPL) responses at point Bin acoustic domain (Fig 3) are also computed usingvarious combination of methods and the results areplotted in Fig 7 The normal velocity of the flexibleplate which provides the boundary condition of acous-tic domain is approximated using ES-FEM and FEMThe 3D acoustic domain is divided using tetrahedronelements (1045 nodes 6335 elements) for FEM andES-FEM If FM-BEM is chosen only the surface ofthe 3D acoustic domain is discretized with triangle ele-ments and hence the number of elements is much

smaller (634 nodes 1264 elements) The computationis performed for frequencies ranging from 1 to700 Hz For comparison the numerical result obtainedby the coupled FEMFEM with a very fine mesh(15864 nodes and 82858 elements) is presented asthe reference

As shown in Fig 7 the coupled FEMFEM gives theleast accurate results compared to all the other modelsThe over-stiffness phenomenon of FEM in 3D acousticproblems can also be observed and it becomes muchmore pronounced with the increase of the frequencyThe stiffness matrix in coupled ES-FEMES-FEM issofter and hence the results in high frequency rangeshow better agreements with reference results The cou-pled ES-FEMFM-BEM model has almost the samelevel accuracy as the coupled ES-FEMES-FEM modelIt is found that the FM-BEM can offer accurate resultsfor interior acoustic problems

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(B)-FEMES-FEMFEMReference

Fig 5mdashFrequency responses computed at point A using ES(B)-FEM ES-FEM and FEM for theplate alone

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(S)-FEMES-FEMFEMReference

Fig 6mdashFrequency response analysis in point A using ES(S)-FEM ES-FEM and FEM for theplate alone

204 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 3: A coupled ES-BEM and FM-BEM for structural acoustic problems

2 ES-FEM FORMULATIONS FOR PLATESTRUCTURE

Consider a vibrating plate subjected to external exci-tations The vibrating plate is modeled using 3-nodestriangle plate elements based on the low order ReissnerndashMindlin plate theory The ReissnerndashMindlin theory isintended for thick plates in which the normal to themid-surface remains straight but not necessarily perpen-dicular to the mid-surface Due to its simplicity and effi-ciency ReissnerndashMindlin theory is useful and practicalfor modeling plates that are not ldquothinrdquo where the classicplate theory is no longer valid However when it is ap-plied to thin plates these low-order plate elements oftensuffer from the so-called ldquoshear lockingrdquo This is due tothat the transverse shear strains cannot vanish under thepure bending condition based on the ReissnerndashMindlintheory In order to eliminate the shear locking discreteshear gap triangular element (DSG) method19 has beenutilized to overcome the shear locking problem togetherwith the edge-based smoothed techniques For easy ref-erence this technique is termed as ES-FEM-DSG3where ldquo3rdquo stands for the fact that we use only triangularelements Because we use only triangular elements thegeometry of the plate can be practically arbitrary

21 ES-FEM-DSG3 Formulations Basedon the ReissnerndashMindlin Plate

Based on the ReissnerndashMindlin plates theory the un-known vector of three independent field variables at anypoint in the problem domain of structure can be definedas u = θx θy w

T where θx and θy are the rotationangles of the line normal to the undeformed neutral sur-face in the xndashz and yndashz planes respectively and w isthe deflection The dynamic variation equation forReissnerndashMindlin plate elements without damping canbe described as follows43Z

Ωs

deTbDbebdΩthornZΩs

deTs DsesdΩ

thornZΩs

duTrto2udΩZΓs

duTtdΩ frac14 0 eth1THORN

where the bending stiffness constitutive coefficients Dband transverse shear stiffness constitutive coefficientsDs are defined as

Db frac14 Et3

12 1 n2eth THORN1 n 0n 1 00 0 1 n=2

24

35

Ds frac14 ktG1 00 1

eth2THORN

in which E represents Youngs modulus n is the Poissonratio G is the shear modulus and k is the shear correc-tion factor that is set as 56 in this work

The unknown field variable displacements can beapproximated by nodal displacements using shapefunction

u frac14 Νsue frac14Xnifrac141

Ni xeth THORNui

du frac14 Nsdue frac14Xnifrac141

Ni xeth THORNdui eth3THORN

where Ns is generalized shape functions and ue is thevector of generalized nodal displacements for eachplate element Ni and ui = θxi θyi wi

T are the shapefunction and nodal variable at node i respectivelyApplying Eqn (3) the discretized system equations ofEqn (1) can be written in following matrix form43

Meurou thornKu frac14 F eth4THORNwhere

K frac14ZΩBb

TDbBbdΩthornZΩBs

TDsBsdΩ

The stiffness matrices eth5THORN

M frac14ZΩrNs

Tdiag t3

12t3

12t

NsdΩ

The mass matrix eth6THORNF frac14

ZΓs

NsTtdΓ

The vector of nodal forces eth7THORNwhereBb frac14 Bb1 Bb2 ⋯ Bbnfrac12 is the strainndashdeflectionmatrix for bending Bs frac14 Bs1 Bs2 ⋯ Bsnfrac12 is thestrainndashdeflection matrix for shearing t is the thickness ofthe plate and t is the external load on the plate where

Bbi frac14

Ni

x0 0

0 Ni

y0

Ni

y Ni

x0

2666664

3777775

Bsi frac14Ni 0

Ni

x

0 NiNi

y

264

375 eth8THORN

Using the smoothed strainndashdeflection matrix Bb

and Bs computed based on the edges of elements toreplace Bb and Bs the smoothed stiffness can beexpressed as

K frac14ZΩ

BbTDb BbdΩthorn

BsTDs BsdΩ eth9THORN

198 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

The details of computing the smoothing strainndashdeflection matrices based on the edges can be foundin the following sub-section

Finally the ES-FEM-DSG3 formulation for struc-tural domain then can be written as

Meurouthorn Ku frac14 F eth10THORN

22 Edge-Based Smoothing Operationfor the Plate Structure

In this section edge-based smoothed finite elementmethod or ES-FEM for plates is introduced The platedomain is first discretized using a set of 3-node trian-gles just as in the standard FEM Because the ReissnerndashMindlin theory uses the derivatives of the deflection tocompute the strains only the stiffness matrix is smoothedin the ES-FEM The assembling of the stiffness matrixand the integration is based on the smoothing domainswhich is associated with the edges of the triangles Usingthe edges of these triangles we are able to construct Ns

smoothing domains For edge k the smoothing domainΩk is constructed by connecting the centroids of theneighbor triangles and the end-points of edge k Asshown in Fig 1 for interior edges the smoothing domainΩk is a quadrangle which is the assembly of the sub-domains of two neighboring elements while for edgeson the plate boundary the smoothing domain Ωk is onlya single (triangular) sub-domain The following are the

details of the calculation of the smoothed stiffness matrixfor the 2D structure problem

The smoothing operation is firstly applied to the bend-ing (in-plane) strain and the shear (off-plane) stain of theplate over each of the edge-based smoothing domains

laquob xketh THORN frac14 1Ak

ZΩk

laquob xeth THORNdΩ

laquos xketh THORN frac14 1Ak

ZΩk

laquos xeth THORNdΩ eth11THORN

where Ak is the area of the smoothing domain Ωkwhich can be calculated as follows

Ak frac14ZΩk

dΩ frac14 13

XNek

ifrac141

Aie eth12THORN

in which Nek is the number of the sub-domain of edge k

(that is either 2 for interior edges or 1 for edges on theplate boundary) and Ai

e is the area of ith sub-domain ina triangle element

Based on the assumption made in the ReissnerndashMindlin plate theory the bending strain can beexpressed as follows5

laquob xketh THORN frac14Xi2Mk

Bbi xketh THORNui eth13THORN

whereMk is the total number of vertex of the smoothingdomain which is either 4 for interior edges and 3 foredges on the plate boundary

When the thickness of plates becomes small theReissnerndashMindlin plates often suffer the shear lockingphenomenon In order to avoid such a locking theshear strain is calculated using discrete shear gap trian-gular element (DSG) method19 Combining with thesmoothing operation the smoothed shear strain matrixcan be calculated as follows

laquos xketh THORN frac14Xi2Mk

Bsi xketh THORNui eth14THORN

where the smoothed strain matrix Bbi xketh THORN and Bsi xketh THORNin Eqns (13) and (14) can be calculated by combiningwith the Eqns (11) and (12)

Bbi xketh THORN frac14 1Ak

XNek

ifrac141

13AieBbi xketh THORN

Bsi xketh THORN frac14 1Ak

XNek

ifrac141

13AieBsi xketh THORN eth15THORN

More details about Bbi(xk) and Bsi(xk) based onReissnerndashMindlin plate theory and the discrete shear

Field nodes Centroid of triangle

Edge of triangleBoundary of kΩ

Smoothing domain

Global boundary

Fig 1mdashTriangular mesh for a plate forES-FEM model and edge-basedsmoothing domains constructed byconnecting the centroids of theneighbor triangles and the twoend-points of edge k

199Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

gap triangular element (DSG) method can be found inRef 19

Using the smoothed strain matrix Bs and Bb the

smoothed strain stiffness matrix Kbketh THORN and smoothed

shear stiffness matrix Ksketh THORNfor the edge-based smoothed

domain k can be evaluated as

Kbketh THORN frac14

ZΩk

BbTDb BbdΩ

frac14XNs

kfrac141

Ak BbTDb Bb

Ksketh THORN frac14

ZΩk

BsTDs BsdΩ

frac14XNs

kfrac141

Ak BsTDs Bs eth16THORN

The global smoothed bending stiffness Kb and globalsmoothed shear stiffness Ks based on the edges can be as-sembled just as the same procedure as in the standard FEMThen the global smoothed stiffness can be evaluated as

K frac14 Kb thorn Ks eth17THORNIf the smoothing operation based on the edges is only

applied to bending stiffness Kb or shear stiffness Ksand the other one remains un-smoothed we can obtaintwo variant forms of ES-FEM where the global stiff-ness are computed using

K Beth THORN frac14 Kb thornKs eth18THORN

where K Beth THORN is global smoothed stiffness with only thebending stain smoothed It is denoted as ES(B)-FEMAlternatively

K Seth THORN frac14 Kb thorn Ks eth19THORN

where K Seth THORN is global smoothed stiffness with only theshear stain smoothed It is denoted as ES(S)-FEM

3 ES-FEM FORMULATIONS FOR 3DACOUSTIC PROBLEMS

Note that the ES-FEM is applicable also to acousticproblems and this section briefs the process

31 GS-Galerkin Weak Form and DiscretizedSystem Equations

In the acoustic domain we firstly define an enclosedcavity Ωf with Neumann boundary ΓN assuming thatthe fluid is homogeneous inviscid compressible and

only undergoes small translational movement Lettingp denote the acoustic pressure and k denote the wavenumber the governing equation for the sound pressurecan be expressed as

Δpthorn k2p frac14 0 in Ωf eth20THORN

where Δ is the Laplace operator the wave number canbe written as k = oc o is the angular frequency ofthe pressure oscillation and c is the speed of soundtraveling in the acoustic fluid

The Neumann boundary of the acoustic domain canbe defined as the following

rpn frac14 jrovn on ΓN eth21THORN

where j frac14 ffiffiffiffiffiffiffi1p

r is the density of medium vn denotesnormal velocity on the boundary The field variablepressure can be approximated using a shape functiondefined as

p frac14Xmifrac141

Nipi frac14 Np eth22THORN

where Pi denotes the unknown nodal pressure and Ni

are shape functions in node i N is the generated shapefunction and P is the vector of generated pressure foreach tetrahedron element

Applying the Eqn (22) using shape function as theweight function the standard Galerkin weak form foracoustic problem without acoustical damping can bewritten as25

ZΩrNrNP dΩthorn k2

ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth23THORN

Using the smoothed item rN based on the edges ofelements to replace the gradient component rN thegeneralized smoothed Garlerkin (GS-Galerkin) weakformulation for acoustic problem can be written as

rN rNPdΩthorn k2ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth24THORN

Finally the discretized system equations in Eqn (24)can be written in following matrix form

K frac14ZΩ

rN T rNdΩ eth25THORN

200 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

where

K frac14ZΩ

rN T rNdΩ

The smoothed acoustical stiffness matrix

eth26THORNΜ frac14

ZΩNTNdΩ

The acoustical mass matrix eth27THORN

F frac14ZΓN

NTvndΓ

The vector of nodal acoustic forces eth28THORN

Pf gT frac14 p1 p2⋯ pnf gNodal acoustic pressure in the domain

eth29THORN

32 Edge-Based Gradient SmoothingOperation for 3D Acoustic Domain

In this section the formulation of ES-FEM for 3Dacoustic fluid is presented The acoustic domain is di-vided exactly as that of standard FEM using four nodetetrahedral elements The edge-based gradient smooth-ing domains which are also serving as integrationdomains are then formed in association with these tet-rahedral elements As shown in Fig 2 the sub-smooth-ing domain of edge k in cell i is created by connectingthe centroid of cell i to the two end-nodes of the edgek and the related surface triangles

For acoustic problems the gradient smoothing oper-ation will be applied over each edge-based smoothingdomain on the velocity v The smoothed velocity whichis deduced by the gradient of acoustic pressure isdenoted as

v xketh THORN frac14 1Vk

ZΩk

v xeth THORNdΩ eth30THORN

where Vk frac14ZΩk

dΩ denotes the volume of smoothing

domain for edge kThe smoothed velocity can be expressed in terms of

acoustic pressure by applying the Greens theorem

v xketh THORN frac14 1jroVk

ZΩk

rpdΩ

frac14 1jroVk

ZΓk

pndΓ eth31THORN

Substituting the field variable (acoustic pressure) in-terpolation in form of Eqn (22) into Eqn (31) thesmoothed velocity for edge k can be denoted as the fol-lowing matrix form

v xketh THORN frac14 1jro

XI2Mk

Bi xketh THORNpi eth32THORN

where Mk represents the total number of nodes in thesmoothing domain of edge k Bi can be defined as

BTi xketh THORN frac14 bi1 bi2 bi3

eth33THORN

bip frac14 1Vk

ZΓk

Ni xeth THORNnp xeth THORNdΓ eth34THORN

Finally the smoothed stiffness matrix shown inEqn (25) can be assembled based on the smoothed Bas

Kketh THORN frac14

ZΩk

BT BdΩ frac14XNs

kfrac141

Vk BT B eth35THORN

Owning to the compact supports of the FEM shapefunctions the assembled smoothed stiffness matrixEqn (35) is banded and symmetric Therefore systemequations can be solved efficiently even though it dis-cretizes the entire domain as long as the domain isenclosed However when it is used for exterior media(that is infinite) some kind of non-reflecting boundarytechniques44 must be used In such cases the FM-BEMcan be a better choice because there is no need for artificialnon-reflecting boundary

n

n

n

n

n

n

n

n

Edge k

Fig 2mdash3D edge-based smoothing domainsconstructed by connecting the centroidof cell i to end-nodes of the edge kand the related surface triangles

201Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

4 FAST MULTIPOLE BEM FORACOUSTIC PROBLEMS

41 Conventional BEM Formulationsfor Acoustic Problems

In this section we first review the conventional BEMformulation for Helmholtz equations The fundamen-tal solution or the full-space Greens function foracoustic problems is well-known and can be denotedas follows45

G x yeth THORN frac14 ejk xyj j

4p x yj j eth36THORN

where j frac14 ffiffiffiffiffiffiffi1p

k is the wavenumber and |x y| isthe distance between the collocation point x and thesource point y

Combining the conventional boundary integral equa-tion (CBIE) and the hypersingular boundary integralequation (HBIE) a well-known integral equationnamed as CHBIE formulation for Helmholtz equationin Eqn (20) without the incident wave can be writtenas45

G x yeth THORNn yeth THORN p yeth THORNdΓ yeth THORN thorn C xeth THORNp xeth THORN

24

35

thorn aZΓ

2G x yeth THORNn yeth THORNn xeth THORN p yeth THORNdΓ yeth THORN

frac14ZΓ

G x yeth THORNq yeth THORNdΓ yeth THORN

thorn aZΓ

G x yeth THORNn xeth THORN q yeth THORNdΓ yeth THORN C xeth THORNq xeth THORN

24

35

8x 2 Γ eth37THORNwhere q is defined as q frac14 p

n The constant C(x) is set as12 for smooth surface around x and the coupling con-stant a is defined as jk

Dividing the boundary into N surface elements thediscretized form of the CHBIE formulation can beexpressed as45

XNjfrac141

fijpj frac14XNjfrac141

gijqj eth38THORN

where

fijpj frac14ZΔΓj

G x yeth THORNn yeth THORN pjdΓ yeth THORN thorn 1

2dijpj

thorn aZΔΓj

2G x yeth THORNn yeth THORNn xeth THORNpjdΓ yeth THORN

gijqj frac14ZΔΓj

G x yeth THORNqjdΓ yeth THORN

thorn aZΔΓj

G x yeth THORNn xeth THORN qjdΓ yeth THORN 1

2dijqj

264

375 eth39THORN

where dij is the Kronecker Delta and ΔΓj denoteselement j

The discretized form of the BurtonndashMiller formula-tion in Eqn (38) can be transformed to the followingsystem of equations by moving the known terms tothe right-hand side and the unknown terms to the left-hand side

a11 a12 ⋯ a1Na21 a22 ⋯ a2N⋮ ⋮ ⋱ ⋮aN1 aN2 ⋯ aNN

2664

3775

l1l2⋮lN

8gtgtltgtgt

9gtgt=gtgt

frac14b1b2⋮bN

8gtgtltgtgt

9gtgt=gtgtor Alfrac14b eth40THORN

where A l and b are the system matrix unknown vec-tor and known vector respectively

42 The Fast Multipole Method Implementedin BEM

There are two main techniques applied to improvethe efficiency of the conventional BEM Firstly the fastmultipole method (FMM) is employed to speed up thematrixndashvector multiplication in Al then an efficient it-erative solver such as the generalized minimum residuemethod (GMRES) will be applied to solve the systemof equations given by Eqn (40) With FMM the fastmultipole boundary element method can be con-structed The fundamental principle of the FMM is amultipole expansion of the kernel in which the directconnection between the source point and the colloca-tion point is separated The details of the derivationsof the FM-BEM formulations can be found in Refs 40and 45 With the fast multipole BEM acoustic BEMmodels with DOFs up to several millions have beensolved on laptop PCs with a RAM size of only 8 GB

5 NUMERICAL EXAMPLES

In this section two numerical applications of 3Dcases are presented in order to verify the effectivenessof the proposed combination of ES-FEM and FM-BEM formulations Because of the huge differenceexisting in terms of mass density of the structure and

202 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

air acoustic modes are not coincident with the struc-tural modes it is thus practical to neglect direct interac-tions between the structure and air46 meaning that thestructure dynamics is assumed not to be influenced bythe fluid For comparison the results obtained fromthe FEM with extremely fine mesh are also providedas the reference results The purpose of the first exam-ple of a simple elastic plate backed by a closed acousticcavity is to show the advantages of ES-FEM and FM-BEM The second example is an application of thepresent combined methods to a practical problem in ve-hicle engineering

51 Box with Flexible Plate on Top

In this subsection a weak coupling model of a flex-ible plate and air cavity is established The model isshown in Fig 3 The weakly coupled model is a combi-nation of the flexible plate on the top and a closedacoustic cavity attached The elastic plate is made ofaluminum (r = 2700 kgm3 n = 03 and E = 71 Gpa)The acoustic cavity is full of air (r = 121 kgm3 and

c = 343 ms) The plate which has a dimension of050 m 060 m and a thickness of t = 0003 m is sim-ply supported on all the four edges The closed acousticcavity has a dimension of 050 m 060 m 040 mThe remaining walls (except the coupled wall) of cavityare assumed to be rigid with the surface velocity fixedat v = 0

The top elastic plate is divided with ReissnerndashMind-lin triangle plate elements An evenly distributed timeharmonic load equal to 100 N is applied at the centerof plate (point A in Fig 3) First the forced frequencyresponses are computed at the center of the plate usingdifferent methods including FEM ES-FEM ES(B)-FEMand ES(S)-FEM with same model (155 nodes 264 ele-ments) The frequency ranges from 1 to 1000 Hz Thereference result is provided using FEM with much smal-ler elements (1265 nodes 2390 elements)

As shown in Fig 4 in the low frequency domain (0to 200 Hz) results obtained from FEM and ES-FEMshow excellent agreements with the reference resultdemonstrating that both FEM and ES-FEM can provideaccuracy results in low frequencies As the frequencyincreases the deviation between FEM result and thereference result becomes larger suggesting that the ac-curacy of the FEM result decreases with the increase ofthe frequency We also note that the eigen-frequenciesin FEM result (peaks in response curve) become higherand higher compared to the reference result This devi-ation mainly results from the inherent drawback ofldquoover-stiffnessrdquo in FEM based on the standard weakformulation The ES-FEM provides much more accu-rate result in higher frequency range compared to theFEM model using the same mesh From Fig 5 wecan see that ES(B)-FEM can also produce results simi-lar to that of ES-FEM The softening effect of ES(B)-FEM is almost equal to that of ES-FEM In additionas showed in Fig 6 the response curves obtained from

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFEMReference

Fig 4mdashFrequency responses computed at point A using ES-FEM and FEM for the plate alone

Point A-exciting point

Point B-a response point in the acoustic domain

Aluminum Plate ( s)

Acoustic domain ( f)

Fig 3mdashA flexible aluminum plate backed by abox of air

203Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

ES(S)-FEM have little difference from that of FEM (us-ing the same mesh) which means that the softening ef-fect by the edge-based smoothing on the off-plane shearstrain is minimum and can be neglected Therefore itcan be concluded that the total softening effects ofES-FEM are mainly due to smoothing the in-planebending strain

The sound pressure level (SPL) responses at point Bin acoustic domain (Fig 3) are also computed usingvarious combination of methods and the results areplotted in Fig 7 The normal velocity of the flexibleplate which provides the boundary condition of acous-tic domain is approximated using ES-FEM and FEMThe 3D acoustic domain is divided using tetrahedronelements (1045 nodes 6335 elements) for FEM andES-FEM If FM-BEM is chosen only the surface ofthe 3D acoustic domain is discretized with triangle ele-ments and hence the number of elements is much

smaller (634 nodes 1264 elements) The computationis performed for frequencies ranging from 1 to700 Hz For comparison the numerical result obtainedby the coupled FEMFEM with a very fine mesh(15864 nodes and 82858 elements) is presented asthe reference

As shown in Fig 7 the coupled FEMFEM gives theleast accurate results compared to all the other modelsThe over-stiffness phenomenon of FEM in 3D acousticproblems can also be observed and it becomes muchmore pronounced with the increase of the frequencyThe stiffness matrix in coupled ES-FEMES-FEM issofter and hence the results in high frequency rangeshow better agreements with reference results The cou-pled ES-FEMFM-BEM model has almost the samelevel accuracy as the coupled ES-FEMES-FEM modelIt is found that the FM-BEM can offer accurate resultsfor interior acoustic problems

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(B)-FEMES-FEMFEMReference

Fig 5mdashFrequency responses computed at point A using ES(B)-FEM ES-FEM and FEM for theplate alone

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(S)-FEMES-FEMFEMReference

Fig 6mdashFrequency response analysis in point A using ES(S)-FEM ES-FEM and FEM for theplate alone

204 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 4: A coupled ES-BEM and FM-BEM for structural acoustic problems

The details of computing the smoothing strainndashdeflection matrices based on the edges can be foundin the following sub-section

Finally the ES-FEM-DSG3 formulation for struc-tural domain then can be written as

Meurouthorn Ku frac14 F eth10THORN

22 Edge-Based Smoothing Operationfor the Plate Structure

In this section edge-based smoothed finite elementmethod or ES-FEM for plates is introduced The platedomain is first discretized using a set of 3-node trian-gles just as in the standard FEM Because the ReissnerndashMindlin theory uses the derivatives of the deflection tocompute the strains only the stiffness matrix is smoothedin the ES-FEM The assembling of the stiffness matrixand the integration is based on the smoothing domainswhich is associated with the edges of the triangles Usingthe edges of these triangles we are able to construct Ns

smoothing domains For edge k the smoothing domainΩk is constructed by connecting the centroids of theneighbor triangles and the end-points of edge k Asshown in Fig 1 for interior edges the smoothing domainΩk is a quadrangle which is the assembly of the sub-domains of two neighboring elements while for edgeson the plate boundary the smoothing domain Ωk is onlya single (triangular) sub-domain The following are the

details of the calculation of the smoothed stiffness matrixfor the 2D structure problem

The smoothing operation is firstly applied to the bend-ing (in-plane) strain and the shear (off-plane) stain of theplate over each of the edge-based smoothing domains

laquob xketh THORN frac14 1Ak

ZΩk

laquob xeth THORNdΩ

laquos xketh THORN frac14 1Ak

ZΩk

laquos xeth THORNdΩ eth11THORN

where Ak is the area of the smoothing domain Ωkwhich can be calculated as follows

Ak frac14ZΩk

dΩ frac14 13

XNek

ifrac141

Aie eth12THORN

in which Nek is the number of the sub-domain of edge k

(that is either 2 for interior edges or 1 for edges on theplate boundary) and Ai

e is the area of ith sub-domain ina triangle element

Based on the assumption made in the ReissnerndashMindlin plate theory the bending strain can beexpressed as follows5

laquob xketh THORN frac14Xi2Mk

Bbi xketh THORNui eth13THORN

whereMk is the total number of vertex of the smoothingdomain which is either 4 for interior edges and 3 foredges on the plate boundary

When the thickness of plates becomes small theReissnerndashMindlin plates often suffer the shear lockingphenomenon In order to avoid such a locking theshear strain is calculated using discrete shear gap trian-gular element (DSG) method19 Combining with thesmoothing operation the smoothed shear strain matrixcan be calculated as follows

laquos xketh THORN frac14Xi2Mk

Bsi xketh THORNui eth14THORN

where the smoothed strain matrix Bbi xketh THORN and Bsi xketh THORNin Eqns (13) and (14) can be calculated by combiningwith the Eqns (11) and (12)

Bbi xketh THORN frac14 1Ak

XNek

ifrac141

13AieBbi xketh THORN

Bsi xketh THORN frac14 1Ak

XNek

ifrac141

13AieBsi xketh THORN eth15THORN

More details about Bbi(xk) and Bsi(xk) based onReissnerndashMindlin plate theory and the discrete shear

Field nodes Centroid of triangle

Edge of triangleBoundary of kΩ

Smoothing domain

Global boundary

Fig 1mdashTriangular mesh for a plate forES-FEM model and edge-basedsmoothing domains constructed byconnecting the centroids of theneighbor triangles and the twoend-points of edge k

199Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

gap triangular element (DSG) method can be found inRef 19

Using the smoothed strain matrix Bs and Bb the

smoothed strain stiffness matrix Kbketh THORN and smoothed

shear stiffness matrix Ksketh THORNfor the edge-based smoothed

domain k can be evaluated as

Kbketh THORN frac14

ZΩk

BbTDb BbdΩ

frac14XNs

kfrac141

Ak BbTDb Bb

Ksketh THORN frac14

ZΩk

BsTDs BsdΩ

frac14XNs

kfrac141

Ak BsTDs Bs eth16THORN

The global smoothed bending stiffness Kb and globalsmoothed shear stiffness Ks based on the edges can be as-sembled just as the same procedure as in the standard FEMThen the global smoothed stiffness can be evaluated as

K frac14 Kb thorn Ks eth17THORNIf the smoothing operation based on the edges is only

applied to bending stiffness Kb or shear stiffness Ksand the other one remains un-smoothed we can obtaintwo variant forms of ES-FEM where the global stiff-ness are computed using

K Beth THORN frac14 Kb thornKs eth18THORN

where K Beth THORN is global smoothed stiffness with only thebending stain smoothed It is denoted as ES(B)-FEMAlternatively

K Seth THORN frac14 Kb thorn Ks eth19THORN

where K Seth THORN is global smoothed stiffness with only theshear stain smoothed It is denoted as ES(S)-FEM

3 ES-FEM FORMULATIONS FOR 3DACOUSTIC PROBLEMS

Note that the ES-FEM is applicable also to acousticproblems and this section briefs the process

31 GS-Galerkin Weak Form and DiscretizedSystem Equations

In the acoustic domain we firstly define an enclosedcavity Ωf with Neumann boundary ΓN assuming thatthe fluid is homogeneous inviscid compressible and

only undergoes small translational movement Lettingp denote the acoustic pressure and k denote the wavenumber the governing equation for the sound pressurecan be expressed as

Δpthorn k2p frac14 0 in Ωf eth20THORN

where Δ is the Laplace operator the wave number canbe written as k = oc o is the angular frequency ofthe pressure oscillation and c is the speed of soundtraveling in the acoustic fluid

The Neumann boundary of the acoustic domain canbe defined as the following

rpn frac14 jrovn on ΓN eth21THORN

where j frac14 ffiffiffiffiffiffiffi1p

r is the density of medium vn denotesnormal velocity on the boundary The field variablepressure can be approximated using a shape functiondefined as

p frac14Xmifrac141

Nipi frac14 Np eth22THORN

where Pi denotes the unknown nodal pressure and Ni

are shape functions in node i N is the generated shapefunction and P is the vector of generated pressure foreach tetrahedron element

Applying the Eqn (22) using shape function as theweight function the standard Galerkin weak form foracoustic problem without acoustical damping can bewritten as25

ZΩrNrNP dΩthorn k2

ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth23THORN

Using the smoothed item rN based on the edges ofelements to replace the gradient component rN thegeneralized smoothed Garlerkin (GS-Galerkin) weakformulation for acoustic problem can be written as

rN rNPdΩthorn k2ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth24THORN

Finally the discretized system equations in Eqn (24)can be written in following matrix form

K frac14ZΩ

rN T rNdΩ eth25THORN

200 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

where

K frac14ZΩ

rN T rNdΩ

The smoothed acoustical stiffness matrix

eth26THORNΜ frac14

ZΩNTNdΩ

The acoustical mass matrix eth27THORN

F frac14ZΓN

NTvndΓ

The vector of nodal acoustic forces eth28THORN

Pf gT frac14 p1 p2⋯ pnf gNodal acoustic pressure in the domain

eth29THORN

32 Edge-Based Gradient SmoothingOperation for 3D Acoustic Domain

In this section the formulation of ES-FEM for 3Dacoustic fluid is presented The acoustic domain is di-vided exactly as that of standard FEM using four nodetetrahedral elements The edge-based gradient smooth-ing domains which are also serving as integrationdomains are then formed in association with these tet-rahedral elements As shown in Fig 2 the sub-smooth-ing domain of edge k in cell i is created by connectingthe centroid of cell i to the two end-nodes of the edgek and the related surface triangles

For acoustic problems the gradient smoothing oper-ation will be applied over each edge-based smoothingdomain on the velocity v The smoothed velocity whichis deduced by the gradient of acoustic pressure isdenoted as

v xketh THORN frac14 1Vk

ZΩk

v xeth THORNdΩ eth30THORN

where Vk frac14ZΩk

dΩ denotes the volume of smoothing

domain for edge kThe smoothed velocity can be expressed in terms of

acoustic pressure by applying the Greens theorem

v xketh THORN frac14 1jroVk

ZΩk

rpdΩ

frac14 1jroVk

ZΓk

pndΓ eth31THORN

Substituting the field variable (acoustic pressure) in-terpolation in form of Eqn (22) into Eqn (31) thesmoothed velocity for edge k can be denoted as the fol-lowing matrix form

v xketh THORN frac14 1jro

XI2Mk

Bi xketh THORNpi eth32THORN

where Mk represents the total number of nodes in thesmoothing domain of edge k Bi can be defined as

BTi xketh THORN frac14 bi1 bi2 bi3

eth33THORN

bip frac14 1Vk

ZΓk

Ni xeth THORNnp xeth THORNdΓ eth34THORN

Finally the smoothed stiffness matrix shown inEqn (25) can be assembled based on the smoothed Bas

Kketh THORN frac14

ZΩk

BT BdΩ frac14XNs

kfrac141

Vk BT B eth35THORN

Owning to the compact supports of the FEM shapefunctions the assembled smoothed stiffness matrixEqn (35) is banded and symmetric Therefore systemequations can be solved efficiently even though it dis-cretizes the entire domain as long as the domain isenclosed However when it is used for exterior media(that is infinite) some kind of non-reflecting boundarytechniques44 must be used In such cases the FM-BEMcan be a better choice because there is no need for artificialnon-reflecting boundary

n

n

n

n

n

n

n

n

Edge k

Fig 2mdash3D edge-based smoothing domainsconstructed by connecting the centroidof cell i to end-nodes of the edge kand the related surface triangles

201Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

4 FAST MULTIPOLE BEM FORACOUSTIC PROBLEMS

41 Conventional BEM Formulationsfor Acoustic Problems

In this section we first review the conventional BEMformulation for Helmholtz equations The fundamen-tal solution or the full-space Greens function foracoustic problems is well-known and can be denotedas follows45

G x yeth THORN frac14 ejk xyj j

4p x yj j eth36THORN

where j frac14 ffiffiffiffiffiffiffi1p

k is the wavenumber and |x y| isthe distance between the collocation point x and thesource point y

Combining the conventional boundary integral equa-tion (CBIE) and the hypersingular boundary integralequation (HBIE) a well-known integral equationnamed as CHBIE formulation for Helmholtz equationin Eqn (20) without the incident wave can be writtenas45

G x yeth THORNn yeth THORN p yeth THORNdΓ yeth THORN thorn C xeth THORNp xeth THORN

24

35

thorn aZΓ

2G x yeth THORNn yeth THORNn xeth THORN p yeth THORNdΓ yeth THORN

frac14ZΓ

G x yeth THORNq yeth THORNdΓ yeth THORN

thorn aZΓ

G x yeth THORNn xeth THORN q yeth THORNdΓ yeth THORN C xeth THORNq xeth THORN

24

35

8x 2 Γ eth37THORNwhere q is defined as q frac14 p

n The constant C(x) is set as12 for smooth surface around x and the coupling con-stant a is defined as jk

Dividing the boundary into N surface elements thediscretized form of the CHBIE formulation can beexpressed as45

XNjfrac141

fijpj frac14XNjfrac141

gijqj eth38THORN

where

fijpj frac14ZΔΓj

G x yeth THORNn yeth THORN pjdΓ yeth THORN thorn 1

2dijpj

thorn aZΔΓj

2G x yeth THORNn yeth THORNn xeth THORNpjdΓ yeth THORN

gijqj frac14ZΔΓj

G x yeth THORNqjdΓ yeth THORN

thorn aZΔΓj

G x yeth THORNn xeth THORN qjdΓ yeth THORN 1

2dijqj

264

375 eth39THORN

where dij is the Kronecker Delta and ΔΓj denoteselement j

The discretized form of the BurtonndashMiller formula-tion in Eqn (38) can be transformed to the followingsystem of equations by moving the known terms tothe right-hand side and the unknown terms to the left-hand side

a11 a12 ⋯ a1Na21 a22 ⋯ a2N⋮ ⋮ ⋱ ⋮aN1 aN2 ⋯ aNN

2664

3775

l1l2⋮lN

8gtgtltgtgt

9gtgt=gtgt

frac14b1b2⋮bN

8gtgtltgtgt

9gtgt=gtgtor Alfrac14b eth40THORN

where A l and b are the system matrix unknown vec-tor and known vector respectively

42 The Fast Multipole Method Implementedin BEM

There are two main techniques applied to improvethe efficiency of the conventional BEM Firstly the fastmultipole method (FMM) is employed to speed up thematrixndashvector multiplication in Al then an efficient it-erative solver such as the generalized minimum residuemethod (GMRES) will be applied to solve the systemof equations given by Eqn (40) With FMM the fastmultipole boundary element method can be con-structed The fundamental principle of the FMM is amultipole expansion of the kernel in which the directconnection between the source point and the colloca-tion point is separated The details of the derivationsof the FM-BEM formulations can be found in Refs 40and 45 With the fast multipole BEM acoustic BEMmodels with DOFs up to several millions have beensolved on laptop PCs with a RAM size of only 8 GB

5 NUMERICAL EXAMPLES

In this section two numerical applications of 3Dcases are presented in order to verify the effectivenessof the proposed combination of ES-FEM and FM-BEM formulations Because of the huge differenceexisting in terms of mass density of the structure and

202 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

air acoustic modes are not coincident with the struc-tural modes it is thus practical to neglect direct interac-tions between the structure and air46 meaning that thestructure dynamics is assumed not to be influenced bythe fluid For comparison the results obtained fromthe FEM with extremely fine mesh are also providedas the reference results The purpose of the first exam-ple of a simple elastic plate backed by a closed acousticcavity is to show the advantages of ES-FEM and FM-BEM The second example is an application of thepresent combined methods to a practical problem in ve-hicle engineering

51 Box with Flexible Plate on Top

In this subsection a weak coupling model of a flex-ible plate and air cavity is established The model isshown in Fig 3 The weakly coupled model is a combi-nation of the flexible plate on the top and a closedacoustic cavity attached The elastic plate is made ofaluminum (r = 2700 kgm3 n = 03 and E = 71 Gpa)The acoustic cavity is full of air (r = 121 kgm3 and

c = 343 ms) The plate which has a dimension of050 m 060 m and a thickness of t = 0003 m is sim-ply supported on all the four edges The closed acousticcavity has a dimension of 050 m 060 m 040 mThe remaining walls (except the coupled wall) of cavityare assumed to be rigid with the surface velocity fixedat v = 0

The top elastic plate is divided with ReissnerndashMind-lin triangle plate elements An evenly distributed timeharmonic load equal to 100 N is applied at the centerof plate (point A in Fig 3) First the forced frequencyresponses are computed at the center of the plate usingdifferent methods including FEM ES-FEM ES(B)-FEMand ES(S)-FEM with same model (155 nodes 264 ele-ments) The frequency ranges from 1 to 1000 Hz Thereference result is provided using FEM with much smal-ler elements (1265 nodes 2390 elements)

As shown in Fig 4 in the low frequency domain (0to 200 Hz) results obtained from FEM and ES-FEMshow excellent agreements with the reference resultdemonstrating that both FEM and ES-FEM can provideaccuracy results in low frequencies As the frequencyincreases the deviation between FEM result and thereference result becomes larger suggesting that the ac-curacy of the FEM result decreases with the increase ofthe frequency We also note that the eigen-frequenciesin FEM result (peaks in response curve) become higherand higher compared to the reference result This devi-ation mainly results from the inherent drawback ofldquoover-stiffnessrdquo in FEM based on the standard weakformulation The ES-FEM provides much more accu-rate result in higher frequency range compared to theFEM model using the same mesh From Fig 5 wecan see that ES(B)-FEM can also produce results simi-lar to that of ES-FEM The softening effect of ES(B)-FEM is almost equal to that of ES-FEM In additionas showed in Fig 6 the response curves obtained from

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFEMReference

Fig 4mdashFrequency responses computed at point A using ES-FEM and FEM for the plate alone

Point A-exciting point

Point B-a response point in the acoustic domain

Aluminum Plate ( s)

Acoustic domain ( f)

Fig 3mdashA flexible aluminum plate backed by abox of air

203Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

ES(S)-FEM have little difference from that of FEM (us-ing the same mesh) which means that the softening ef-fect by the edge-based smoothing on the off-plane shearstrain is minimum and can be neglected Therefore itcan be concluded that the total softening effects ofES-FEM are mainly due to smoothing the in-planebending strain

The sound pressure level (SPL) responses at point Bin acoustic domain (Fig 3) are also computed usingvarious combination of methods and the results areplotted in Fig 7 The normal velocity of the flexibleplate which provides the boundary condition of acous-tic domain is approximated using ES-FEM and FEMThe 3D acoustic domain is divided using tetrahedronelements (1045 nodes 6335 elements) for FEM andES-FEM If FM-BEM is chosen only the surface ofthe 3D acoustic domain is discretized with triangle ele-ments and hence the number of elements is much

smaller (634 nodes 1264 elements) The computationis performed for frequencies ranging from 1 to700 Hz For comparison the numerical result obtainedby the coupled FEMFEM with a very fine mesh(15864 nodes and 82858 elements) is presented asthe reference

As shown in Fig 7 the coupled FEMFEM gives theleast accurate results compared to all the other modelsThe over-stiffness phenomenon of FEM in 3D acousticproblems can also be observed and it becomes muchmore pronounced with the increase of the frequencyThe stiffness matrix in coupled ES-FEMES-FEM issofter and hence the results in high frequency rangeshow better agreements with reference results The cou-pled ES-FEMFM-BEM model has almost the samelevel accuracy as the coupled ES-FEMES-FEM modelIt is found that the FM-BEM can offer accurate resultsfor interior acoustic problems

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(B)-FEMES-FEMFEMReference

Fig 5mdashFrequency responses computed at point A using ES(B)-FEM ES-FEM and FEM for theplate alone

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(S)-FEMES-FEMFEMReference

Fig 6mdashFrequency response analysis in point A using ES(S)-FEM ES-FEM and FEM for theplate alone

204 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 5: A coupled ES-BEM and FM-BEM for structural acoustic problems

gap triangular element (DSG) method can be found inRef 19

Using the smoothed strain matrix Bs and Bb the

smoothed strain stiffness matrix Kbketh THORN and smoothed

shear stiffness matrix Ksketh THORNfor the edge-based smoothed

domain k can be evaluated as

Kbketh THORN frac14

ZΩk

BbTDb BbdΩ

frac14XNs

kfrac141

Ak BbTDb Bb

Ksketh THORN frac14

ZΩk

BsTDs BsdΩ

frac14XNs

kfrac141

Ak BsTDs Bs eth16THORN

The global smoothed bending stiffness Kb and globalsmoothed shear stiffness Ks based on the edges can be as-sembled just as the same procedure as in the standard FEMThen the global smoothed stiffness can be evaluated as

K frac14 Kb thorn Ks eth17THORNIf the smoothing operation based on the edges is only

applied to bending stiffness Kb or shear stiffness Ksand the other one remains un-smoothed we can obtaintwo variant forms of ES-FEM where the global stiff-ness are computed using

K Beth THORN frac14 Kb thornKs eth18THORN

where K Beth THORN is global smoothed stiffness with only thebending stain smoothed It is denoted as ES(B)-FEMAlternatively

K Seth THORN frac14 Kb thorn Ks eth19THORN

where K Seth THORN is global smoothed stiffness with only theshear stain smoothed It is denoted as ES(S)-FEM

3 ES-FEM FORMULATIONS FOR 3DACOUSTIC PROBLEMS

Note that the ES-FEM is applicable also to acousticproblems and this section briefs the process

31 GS-Galerkin Weak Form and DiscretizedSystem Equations

In the acoustic domain we firstly define an enclosedcavity Ωf with Neumann boundary ΓN assuming thatthe fluid is homogeneous inviscid compressible and

only undergoes small translational movement Lettingp denote the acoustic pressure and k denote the wavenumber the governing equation for the sound pressurecan be expressed as

Δpthorn k2p frac14 0 in Ωf eth20THORN

where Δ is the Laplace operator the wave number canbe written as k = oc o is the angular frequency ofthe pressure oscillation and c is the speed of soundtraveling in the acoustic fluid

The Neumann boundary of the acoustic domain canbe defined as the following

rpn frac14 jrovn on ΓN eth21THORN

where j frac14 ffiffiffiffiffiffiffi1p

r is the density of medium vn denotesnormal velocity on the boundary The field variablepressure can be approximated using a shape functiondefined as

p frac14Xmifrac141

Nipi frac14 Np eth22THORN

where Pi denotes the unknown nodal pressure and Ni

are shape functions in node i N is the generated shapefunction and P is the vector of generated pressure foreach tetrahedron element

Applying the Eqn (22) using shape function as theweight function the standard Galerkin weak form foracoustic problem without acoustical damping can bewritten as25

ZΩrNrNP dΩthorn k2

ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth23THORN

Using the smoothed item rN based on the edges ofelements to replace the gradient component rN thegeneralized smoothed Garlerkin (GS-Galerkin) weakformulation for acoustic problem can be written as

rN rNPdΩthorn k2ZΩNNPdΩ

jroZΓN

NvndΓfrac14 0 eth24THORN

Finally the discretized system equations in Eqn (24)can be written in following matrix form

K frac14ZΩ

rN T rNdΩ eth25THORN

200 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

where

K frac14ZΩ

rN T rNdΩ

The smoothed acoustical stiffness matrix

eth26THORNΜ frac14

ZΩNTNdΩ

The acoustical mass matrix eth27THORN

F frac14ZΓN

NTvndΓ

The vector of nodal acoustic forces eth28THORN

Pf gT frac14 p1 p2⋯ pnf gNodal acoustic pressure in the domain

eth29THORN

32 Edge-Based Gradient SmoothingOperation for 3D Acoustic Domain

In this section the formulation of ES-FEM for 3Dacoustic fluid is presented The acoustic domain is di-vided exactly as that of standard FEM using four nodetetrahedral elements The edge-based gradient smooth-ing domains which are also serving as integrationdomains are then formed in association with these tet-rahedral elements As shown in Fig 2 the sub-smooth-ing domain of edge k in cell i is created by connectingthe centroid of cell i to the two end-nodes of the edgek and the related surface triangles

For acoustic problems the gradient smoothing oper-ation will be applied over each edge-based smoothingdomain on the velocity v The smoothed velocity whichis deduced by the gradient of acoustic pressure isdenoted as

v xketh THORN frac14 1Vk

ZΩk

v xeth THORNdΩ eth30THORN

where Vk frac14ZΩk

dΩ denotes the volume of smoothing

domain for edge kThe smoothed velocity can be expressed in terms of

acoustic pressure by applying the Greens theorem

v xketh THORN frac14 1jroVk

ZΩk

rpdΩ

frac14 1jroVk

ZΓk

pndΓ eth31THORN

Substituting the field variable (acoustic pressure) in-terpolation in form of Eqn (22) into Eqn (31) thesmoothed velocity for edge k can be denoted as the fol-lowing matrix form

v xketh THORN frac14 1jro

XI2Mk

Bi xketh THORNpi eth32THORN

where Mk represents the total number of nodes in thesmoothing domain of edge k Bi can be defined as

BTi xketh THORN frac14 bi1 bi2 bi3

eth33THORN

bip frac14 1Vk

ZΓk

Ni xeth THORNnp xeth THORNdΓ eth34THORN

Finally the smoothed stiffness matrix shown inEqn (25) can be assembled based on the smoothed Bas

Kketh THORN frac14

ZΩk

BT BdΩ frac14XNs

kfrac141

Vk BT B eth35THORN

Owning to the compact supports of the FEM shapefunctions the assembled smoothed stiffness matrixEqn (35) is banded and symmetric Therefore systemequations can be solved efficiently even though it dis-cretizes the entire domain as long as the domain isenclosed However when it is used for exterior media(that is infinite) some kind of non-reflecting boundarytechniques44 must be used In such cases the FM-BEMcan be a better choice because there is no need for artificialnon-reflecting boundary

n

n

n

n

n

n

n

n

Edge k

Fig 2mdash3D edge-based smoothing domainsconstructed by connecting the centroidof cell i to end-nodes of the edge kand the related surface triangles

201Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

4 FAST MULTIPOLE BEM FORACOUSTIC PROBLEMS

41 Conventional BEM Formulationsfor Acoustic Problems

In this section we first review the conventional BEMformulation for Helmholtz equations The fundamen-tal solution or the full-space Greens function foracoustic problems is well-known and can be denotedas follows45

G x yeth THORN frac14 ejk xyj j

4p x yj j eth36THORN

where j frac14 ffiffiffiffiffiffiffi1p

k is the wavenumber and |x y| isthe distance between the collocation point x and thesource point y

Combining the conventional boundary integral equa-tion (CBIE) and the hypersingular boundary integralequation (HBIE) a well-known integral equationnamed as CHBIE formulation for Helmholtz equationin Eqn (20) without the incident wave can be writtenas45

G x yeth THORNn yeth THORN p yeth THORNdΓ yeth THORN thorn C xeth THORNp xeth THORN

24

35

thorn aZΓ

2G x yeth THORNn yeth THORNn xeth THORN p yeth THORNdΓ yeth THORN

frac14ZΓ

G x yeth THORNq yeth THORNdΓ yeth THORN

thorn aZΓ

G x yeth THORNn xeth THORN q yeth THORNdΓ yeth THORN C xeth THORNq xeth THORN

24

35

8x 2 Γ eth37THORNwhere q is defined as q frac14 p

n The constant C(x) is set as12 for smooth surface around x and the coupling con-stant a is defined as jk

Dividing the boundary into N surface elements thediscretized form of the CHBIE formulation can beexpressed as45

XNjfrac141

fijpj frac14XNjfrac141

gijqj eth38THORN

where

fijpj frac14ZΔΓj

G x yeth THORNn yeth THORN pjdΓ yeth THORN thorn 1

2dijpj

thorn aZΔΓj

2G x yeth THORNn yeth THORNn xeth THORNpjdΓ yeth THORN

gijqj frac14ZΔΓj

G x yeth THORNqjdΓ yeth THORN

thorn aZΔΓj

G x yeth THORNn xeth THORN qjdΓ yeth THORN 1

2dijqj

264

375 eth39THORN

where dij is the Kronecker Delta and ΔΓj denoteselement j

The discretized form of the BurtonndashMiller formula-tion in Eqn (38) can be transformed to the followingsystem of equations by moving the known terms tothe right-hand side and the unknown terms to the left-hand side

a11 a12 ⋯ a1Na21 a22 ⋯ a2N⋮ ⋮ ⋱ ⋮aN1 aN2 ⋯ aNN

2664

3775

l1l2⋮lN

8gtgtltgtgt

9gtgt=gtgt

frac14b1b2⋮bN

8gtgtltgtgt

9gtgt=gtgtor Alfrac14b eth40THORN

where A l and b are the system matrix unknown vec-tor and known vector respectively

42 The Fast Multipole Method Implementedin BEM

There are two main techniques applied to improvethe efficiency of the conventional BEM Firstly the fastmultipole method (FMM) is employed to speed up thematrixndashvector multiplication in Al then an efficient it-erative solver such as the generalized minimum residuemethod (GMRES) will be applied to solve the systemof equations given by Eqn (40) With FMM the fastmultipole boundary element method can be con-structed The fundamental principle of the FMM is amultipole expansion of the kernel in which the directconnection between the source point and the colloca-tion point is separated The details of the derivationsof the FM-BEM formulations can be found in Refs 40and 45 With the fast multipole BEM acoustic BEMmodels with DOFs up to several millions have beensolved on laptop PCs with a RAM size of only 8 GB

5 NUMERICAL EXAMPLES

In this section two numerical applications of 3Dcases are presented in order to verify the effectivenessof the proposed combination of ES-FEM and FM-BEM formulations Because of the huge differenceexisting in terms of mass density of the structure and

202 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

air acoustic modes are not coincident with the struc-tural modes it is thus practical to neglect direct interac-tions between the structure and air46 meaning that thestructure dynamics is assumed not to be influenced bythe fluid For comparison the results obtained fromthe FEM with extremely fine mesh are also providedas the reference results The purpose of the first exam-ple of a simple elastic plate backed by a closed acousticcavity is to show the advantages of ES-FEM and FM-BEM The second example is an application of thepresent combined methods to a practical problem in ve-hicle engineering

51 Box with Flexible Plate on Top

In this subsection a weak coupling model of a flex-ible plate and air cavity is established The model isshown in Fig 3 The weakly coupled model is a combi-nation of the flexible plate on the top and a closedacoustic cavity attached The elastic plate is made ofaluminum (r = 2700 kgm3 n = 03 and E = 71 Gpa)The acoustic cavity is full of air (r = 121 kgm3 and

c = 343 ms) The plate which has a dimension of050 m 060 m and a thickness of t = 0003 m is sim-ply supported on all the four edges The closed acousticcavity has a dimension of 050 m 060 m 040 mThe remaining walls (except the coupled wall) of cavityare assumed to be rigid with the surface velocity fixedat v = 0

The top elastic plate is divided with ReissnerndashMind-lin triangle plate elements An evenly distributed timeharmonic load equal to 100 N is applied at the centerof plate (point A in Fig 3) First the forced frequencyresponses are computed at the center of the plate usingdifferent methods including FEM ES-FEM ES(B)-FEMand ES(S)-FEM with same model (155 nodes 264 ele-ments) The frequency ranges from 1 to 1000 Hz Thereference result is provided using FEM with much smal-ler elements (1265 nodes 2390 elements)

As shown in Fig 4 in the low frequency domain (0to 200 Hz) results obtained from FEM and ES-FEMshow excellent agreements with the reference resultdemonstrating that both FEM and ES-FEM can provideaccuracy results in low frequencies As the frequencyincreases the deviation between FEM result and thereference result becomes larger suggesting that the ac-curacy of the FEM result decreases with the increase ofthe frequency We also note that the eigen-frequenciesin FEM result (peaks in response curve) become higherand higher compared to the reference result This devi-ation mainly results from the inherent drawback ofldquoover-stiffnessrdquo in FEM based on the standard weakformulation The ES-FEM provides much more accu-rate result in higher frequency range compared to theFEM model using the same mesh From Fig 5 wecan see that ES(B)-FEM can also produce results simi-lar to that of ES-FEM The softening effect of ES(B)-FEM is almost equal to that of ES-FEM In additionas showed in Fig 6 the response curves obtained from

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFEMReference

Fig 4mdashFrequency responses computed at point A using ES-FEM and FEM for the plate alone

Point A-exciting point

Point B-a response point in the acoustic domain

Aluminum Plate ( s)

Acoustic domain ( f)

Fig 3mdashA flexible aluminum plate backed by abox of air

203Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

ES(S)-FEM have little difference from that of FEM (us-ing the same mesh) which means that the softening ef-fect by the edge-based smoothing on the off-plane shearstrain is minimum and can be neglected Therefore itcan be concluded that the total softening effects ofES-FEM are mainly due to smoothing the in-planebending strain

The sound pressure level (SPL) responses at point Bin acoustic domain (Fig 3) are also computed usingvarious combination of methods and the results areplotted in Fig 7 The normal velocity of the flexibleplate which provides the boundary condition of acous-tic domain is approximated using ES-FEM and FEMThe 3D acoustic domain is divided using tetrahedronelements (1045 nodes 6335 elements) for FEM andES-FEM If FM-BEM is chosen only the surface ofthe 3D acoustic domain is discretized with triangle ele-ments and hence the number of elements is much

smaller (634 nodes 1264 elements) The computationis performed for frequencies ranging from 1 to700 Hz For comparison the numerical result obtainedby the coupled FEMFEM with a very fine mesh(15864 nodes and 82858 elements) is presented asthe reference

As shown in Fig 7 the coupled FEMFEM gives theleast accurate results compared to all the other modelsThe over-stiffness phenomenon of FEM in 3D acousticproblems can also be observed and it becomes muchmore pronounced with the increase of the frequencyThe stiffness matrix in coupled ES-FEMES-FEM issofter and hence the results in high frequency rangeshow better agreements with reference results The cou-pled ES-FEMFM-BEM model has almost the samelevel accuracy as the coupled ES-FEMES-FEM modelIt is found that the FM-BEM can offer accurate resultsfor interior acoustic problems

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(B)-FEMES-FEMFEMReference

Fig 5mdashFrequency responses computed at point A using ES(B)-FEM ES-FEM and FEM for theplate alone

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(S)-FEMES-FEMFEMReference

Fig 6mdashFrequency response analysis in point A using ES(S)-FEM ES-FEM and FEM for theplate alone

204 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 6: A coupled ES-BEM and FM-BEM for structural acoustic problems

where

K frac14ZΩ

rN T rNdΩ

The smoothed acoustical stiffness matrix

eth26THORNΜ frac14

ZΩNTNdΩ

The acoustical mass matrix eth27THORN

F frac14ZΓN

NTvndΓ

The vector of nodal acoustic forces eth28THORN

Pf gT frac14 p1 p2⋯ pnf gNodal acoustic pressure in the domain

eth29THORN

32 Edge-Based Gradient SmoothingOperation for 3D Acoustic Domain

In this section the formulation of ES-FEM for 3Dacoustic fluid is presented The acoustic domain is di-vided exactly as that of standard FEM using four nodetetrahedral elements The edge-based gradient smooth-ing domains which are also serving as integrationdomains are then formed in association with these tet-rahedral elements As shown in Fig 2 the sub-smooth-ing domain of edge k in cell i is created by connectingthe centroid of cell i to the two end-nodes of the edgek and the related surface triangles

For acoustic problems the gradient smoothing oper-ation will be applied over each edge-based smoothingdomain on the velocity v The smoothed velocity whichis deduced by the gradient of acoustic pressure isdenoted as

v xketh THORN frac14 1Vk

ZΩk

v xeth THORNdΩ eth30THORN

where Vk frac14ZΩk

dΩ denotes the volume of smoothing

domain for edge kThe smoothed velocity can be expressed in terms of

acoustic pressure by applying the Greens theorem

v xketh THORN frac14 1jroVk

ZΩk

rpdΩ

frac14 1jroVk

ZΓk

pndΓ eth31THORN

Substituting the field variable (acoustic pressure) in-terpolation in form of Eqn (22) into Eqn (31) thesmoothed velocity for edge k can be denoted as the fol-lowing matrix form

v xketh THORN frac14 1jro

XI2Mk

Bi xketh THORNpi eth32THORN

where Mk represents the total number of nodes in thesmoothing domain of edge k Bi can be defined as

BTi xketh THORN frac14 bi1 bi2 bi3

eth33THORN

bip frac14 1Vk

ZΓk

Ni xeth THORNnp xeth THORNdΓ eth34THORN

Finally the smoothed stiffness matrix shown inEqn (25) can be assembled based on the smoothed Bas

Kketh THORN frac14

ZΩk

BT BdΩ frac14XNs

kfrac141

Vk BT B eth35THORN

Owning to the compact supports of the FEM shapefunctions the assembled smoothed stiffness matrixEqn (35) is banded and symmetric Therefore systemequations can be solved efficiently even though it dis-cretizes the entire domain as long as the domain isenclosed However when it is used for exterior media(that is infinite) some kind of non-reflecting boundarytechniques44 must be used In such cases the FM-BEMcan be a better choice because there is no need for artificialnon-reflecting boundary

n

n

n

n

n

n

n

n

Edge k

Fig 2mdash3D edge-based smoothing domainsconstructed by connecting the centroidof cell i to end-nodes of the edge kand the related surface triangles

201Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

4 FAST MULTIPOLE BEM FORACOUSTIC PROBLEMS

41 Conventional BEM Formulationsfor Acoustic Problems

In this section we first review the conventional BEMformulation for Helmholtz equations The fundamen-tal solution or the full-space Greens function foracoustic problems is well-known and can be denotedas follows45

G x yeth THORN frac14 ejk xyj j

4p x yj j eth36THORN

where j frac14 ffiffiffiffiffiffiffi1p

k is the wavenumber and |x y| isthe distance between the collocation point x and thesource point y

Combining the conventional boundary integral equa-tion (CBIE) and the hypersingular boundary integralequation (HBIE) a well-known integral equationnamed as CHBIE formulation for Helmholtz equationin Eqn (20) without the incident wave can be writtenas45

G x yeth THORNn yeth THORN p yeth THORNdΓ yeth THORN thorn C xeth THORNp xeth THORN

24

35

thorn aZΓ

2G x yeth THORNn yeth THORNn xeth THORN p yeth THORNdΓ yeth THORN

frac14ZΓ

G x yeth THORNq yeth THORNdΓ yeth THORN

thorn aZΓ

G x yeth THORNn xeth THORN q yeth THORNdΓ yeth THORN C xeth THORNq xeth THORN

24

35

8x 2 Γ eth37THORNwhere q is defined as q frac14 p

n The constant C(x) is set as12 for smooth surface around x and the coupling con-stant a is defined as jk

Dividing the boundary into N surface elements thediscretized form of the CHBIE formulation can beexpressed as45

XNjfrac141

fijpj frac14XNjfrac141

gijqj eth38THORN

where

fijpj frac14ZΔΓj

G x yeth THORNn yeth THORN pjdΓ yeth THORN thorn 1

2dijpj

thorn aZΔΓj

2G x yeth THORNn yeth THORNn xeth THORNpjdΓ yeth THORN

gijqj frac14ZΔΓj

G x yeth THORNqjdΓ yeth THORN

thorn aZΔΓj

G x yeth THORNn xeth THORN qjdΓ yeth THORN 1

2dijqj

264

375 eth39THORN

where dij is the Kronecker Delta and ΔΓj denoteselement j

The discretized form of the BurtonndashMiller formula-tion in Eqn (38) can be transformed to the followingsystem of equations by moving the known terms tothe right-hand side and the unknown terms to the left-hand side

a11 a12 ⋯ a1Na21 a22 ⋯ a2N⋮ ⋮ ⋱ ⋮aN1 aN2 ⋯ aNN

2664

3775

l1l2⋮lN

8gtgtltgtgt

9gtgt=gtgt

frac14b1b2⋮bN

8gtgtltgtgt

9gtgt=gtgtor Alfrac14b eth40THORN

where A l and b are the system matrix unknown vec-tor and known vector respectively

42 The Fast Multipole Method Implementedin BEM

There are two main techniques applied to improvethe efficiency of the conventional BEM Firstly the fastmultipole method (FMM) is employed to speed up thematrixndashvector multiplication in Al then an efficient it-erative solver such as the generalized minimum residuemethod (GMRES) will be applied to solve the systemof equations given by Eqn (40) With FMM the fastmultipole boundary element method can be con-structed The fundamental principle of the FMM is amultipole expansion of the kernel in which the directconnection between the source point and the colloca-tion point is separated The details of the derivationsof the FM-BEM formulations can be found in Refs 40and 45 With the fast multipole BEM acoustic BEMmodels with DOFs up to several millions have beensolved on laptop PCs with a RAM size of only 8 GB

5 NUMERICAL EXAMPLES

In this section two numerical applications of 3Dcases are presented in order to verify the effectivenessof the proposed combination of ES-FEM and FM-BEM formulations Because of the huge differenceexisting in terms of mass density of the structure and

202 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

air acoustic modes are not coincident with the struc-tural modes it is thus practical to neglect direct interac-tions between the structure and air46 meaning that thestructure dynamics is assumed not to be influenced bythe fluid For comparison the results obtained fromthe FEM with extremely fine mesh are also providedas the reference results The purpose of the first exam-ple of a simple elastic plate backed by a closed acousticcavity is to show the advantages of ES-FEM and FM-BEM The second example is an application of thepresent combined methods to a practical problem in ve-hicle engineering

51 Box with Flexible Plate on Top

In this subsection a weak coupling model of a flex-ible plate and air cavity is established The model isshown in Fig 3 The weakly coupled model is a combi-nation of the flexible plate on the top and a closedacoustic cavity attached The elastic plate is made ofaluminum (r = 2700 kgm3 n = 03 and E = 71 Gpa)The acoustic cavity is full of air (r = 121 kgm3 and

c = 343 ms) The plate which has a dimension of050 m 060 m and a thickness of t = 0003 m is sim-ply supported on all the four edges The closed acousticcavity has a dimension of 050 m 060 m 040 mThe remaining walls (except the coupled wall) of cavityare assumed to be rigid with the surface velocity fixedat v = 0

The top elastic plate is divided with ReissnerndashMind-lin triangle plate elements An evenly distributed timeharmonic load equal to 100 N is applied at the centerof plate (point A in Fig 3) First the forced frequencyresponses are computed at the center of the plate usingdifferent methods including FEM ES-FEM ES(B)-FEMand ES(S)-FEM with same model (155 nodes 264 ele-ments) The frequency ranges from 1 to 1000 Hz Thereference result is provided using FEM with much smal-ler elements (1265 nodes 2390 elements)

As shown in Fig 4 in the low frequency domain (0to 200 Hz) results obtained from FEM and ES-FEMshow excellent agreements with the reference resultdemonstrating that both FEM and ES-FEM can provideaccuracy results in low frequencies As the frequencyincreases the deviation between FEM result and thereference result becomes larger suggesting that the ac-curacy of the FEM result decreases with the increase ofthe frequency We also note that the eigen-frequenciesin FEM result (peaks in response curve) become higherand higher compared to the reference result This devi-ation mainly results from the inherent drawback ofldquoover-stiffnessrdquo in FEM based on the standard weakformulation The ES-FEM provides much more accu-rate result in higher frequency range compared to theFEM model using the same mesh From Fig 5 wecan see that ES(B)-FEM can also produce results simi-lar to that of ES-FEM The softening effect of ES(B)-FEM is almost equal to that of ES-FEM In additionas showed in Fig 6 the response curves obtained from

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFEMReference

Fig 4mdashFrequency responses computed at point A using ES-FEM and FEM for the plate alone

Point A-exciting point

Point B-a response point in the acoustic domain

Aluminum Plate ( s)

Acoustic domain ( f)

Fig 3mdashA flexible aluminum plate backed by abox of air

203Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

ES(S)-FEM have little difference from that of FEM (us-ing the same mesh) which means that the softening ef-fect by the edge-based smoothing on the off-plane shearstrain is minimum and can be neglected Therefore itcan be concluded that the total softening effects ofES-FEM are mainly due to smoothing the in-planebending strain

The sound pressure level (SPL) responses at point Bin acoustic domain (Fig 3) are also computed usingvarious combination of methods and the results areplotted in Fig 7 The normal velocity of the flexibleplate which provides the boundary condition of acous-tic domain is approximated using ES-FEM and FEMThe 3D acoustic domain is divided using tetrahedronelements (1045 nodes 6335 elements) for FEM andES-FEM If FM-BEM is chosen only the surface ofthe 3D acoustic domain is discretized with triangle ele-ments and hence the number of elements is much

smaller (634 nodes 1264 elements) The computationis performed for frequencies ranging from 1 to700 Hz For comparison the numerical result obtainedby the coupled FEMFEM with a very fine mesh(15864 nodes and 82858 elements) is presented asthe reference

As shown in Fig 7 the coupled FEMFEM gives theleast accurate results compared to all the other modelsThe over-stiffness phenomenon of FEM in 3D acousticproblems can also be observed and it becomes muchmore pronounced with the increase of the frequencyThe stiffness matrix in coupled ES-FEMES-FEM issofter and hence the results in high frequency rangeshow better agreements with reference results The cou-pled ES-FEMFM-BEM model has almost the samelevel accuracy as the coupled ES-FEMES-FEM modelIt is found that the FM-BEM can offer accurate resultsfor interior acoustic problems

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(B)-FEMES-FEMFEMReference

Fig 5mdashFrequency responses computed at point A using ES(B)-FEM ES-FEM and FEM for theplate alone

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(S)-FEMES-FEMFEMReference

Fig 6mdashFrequency response analysis in point A using ES(S)-FEM ES-FEM and FEM for theplate alone

204 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 7: A coupled ES-BEM and FM-BEM for structural acoustic problems

4 FAST MULTIPOLE BEM FORACOUSTIC PROBLEMS

41 Conventional BEM Formulationsfor Acoustic Problems

In this section we first review the conventional BEMformulation for Helmholtz equations The fundamen-tal solution or the full-space Greens function foracoustic problems is well-known and can be denotedas follows45

G x yeth THORN frac14 ejk xyj j

4p x yj j eth36THORN

where j frac14 ffiffiffiffiffiffiffi1p

k is the wavenumber and |x y| isthe distance between the collocation point x and thesource point y

Combining the conventional boundary integral equa-tion (CBIE) and the hypersingular boundary integralequation (HBIE) a well-known integral equationnamed as CHBIE formulation for Helmholtz equationin Eqn (20) without the incident wave can be writtenas45

G x yeth THORNn yeth THORN p yeth THORNdΓ yeth THORN thorn C xeth THORNp xeth THORN

24

35

thorn aZΓ

2G x yeth THORNn yeth THORNn xeth THORN p yeth THORNdΓ yeth THORN

frac14ZΓ

G x yeth THORNq yeth THORNdΓ yeth THORN

thorn aZΓ

G x yeth THORNn xeth THORN q yeth THORNdΓ yeth THORN C xeth THORNq xeth THORN

24

35

8x 2 Γ eth37THORNwhere q is defined as q frac14 p

n The constant C(x) is set as12 for smooth surface around x and the coupling con-stant a is defined as jk

Dividing the boundary into N surface elements thediscretized form of the CHBIE formulation can beexpressed as45

XNjfrac141

fijpj frac14XNjfrac141

gijqj eth38THORN

where

fijpj frac14ZΔΓj

G x yeth THORNn yeth THORN pjdΓ yeth THORN thorn 1

2dijpj

thorn aZΔΓj

2G x yeth THORNn yeth THORNn xeth THORNpjdΓ yeth THORN

gijqj frac14ZΔΓj

G x yeth THORNqjdΓ yeth THORN

thorn aZΔΓj

G x yeth THORNn xeth THORN qjdΓ yeth THORN 1

2dijqj

264

375 eth39THORN

where dij is the Kronecker Delta and ΔΓj denoteselement j

The discretized form of the BurtonndashMiller formula-tion in Eqn (38) can be transformed to the followingsystem of equations by moving the known terms tothe right-hand side and the unknown terms to the left-hand side

a11 a12 ⋯ a1Na21 a22 ⋯ a2N⋮ ⋮ ⋱ ⋮aN1 aN2 ⋯ aNN

2664

3775

l1l2⋮lN

8gtgtltgtgt

9gtgt=gtgt

frac14b1b2⋮bN

8gtgtltgtgt

9gtgt=gtgtor Alfrac14b eth40THORN

where A l and b are the system matrix unknown vec-tor and known vector respectively

42 The Fast Multipole Method Implementedin BEM

There are two main techniques applied to improvethe efficiency of the conventional BEM Firstly the fastmultipole method (FMM) is employed to speed up thematrixndashvector multiplication in Al then an efficient it-erative solver such as the generalized minimum residuemethod (GMRES) will be applied to solve the systemof equations given by Eqn (40) With FMM the fastmultipole boundary element method can be con-structed The fundamental principle of the FMM is amultipole expansion of the kernel in which the directconnection between the source point and the colloca-tion point is separated The details of the derivationsof the FM-BEM formulations can be found in Refs 40and 45 With the fast multipole BEM acoustic BEMmodels with DOFs up to several millions have beensolved on laptop PCs with a RAM size of only 8 GB

5 NUMERICAL EXAMPLES

In this section two numerical applications of 3Dcases are presented in order to verify the effectivenessof the proposed combination of ES-FEM and FM-BEM formulations Because of the huge differenceexisting in terms of mass density of the structure and

202 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

air acoustic modes are not coincident with the struc-tural modes it is thus practical to neglect direct interac-tions between the structure and air46 meaning that thestructure dynamics is assumed not to be influenced bythe fluid For comparison the results obtained fromthe FEM with extremely fine mesh are also providedas the reference results The purpose of the first exam-ple of a simple elastic plate backed by a closed acousticcavity is to show the advantages of ES-FEM and FM-BEM The second example is an application of thepresent combined methods to a practical problem in ve-hicle engineering

51 Box with Flexible Plate on Top

In this subsection a weak coupling model of a flex-ible plate and air cavity is established The model isshown in Fig 3 The weakly coupled model is a combi-nation of the flexible plate on the top and a closedacoustic cavity attached The elastic plate is made ofaluminum (r = 2700 kgm3 n = 03 and E = 71 Gpa)The acoustic cavity is full of air (r = 121 kgm3 and

c = 343 ms) The plate which has a dimension of050 m 060 m and a thickness of t = 0003 m is sim-ply supported on all the four edges The closed acousticcavity has a dimension of 050 m 060 m 040 mThe remaining walls (except the coupled wall) of cavityare assumed to be rigid with the surface velocity fixedat v = 0

The top elastic plate is divided with ReissnerndashMind-lin triangle plate elements An evenly distributed timeharmonic load equal to 100 N is applied at the centerof plate (point A in Fig 3) First the forced frequencyresponses are computed at the center of the plate usingdifferent methods including FEM ES-FEM ES(B)-FEMand ES(S)-FEM with same model (155 nodes 264 ele-ments) The frequency ranges from 1 to 1000 Hz Thereference result is provided using FEM with much smal-ler elements (1265 nodes 2390 elements)

As shown in Fig 4 in the low frequency domain (0to 200 Hz) results obtained from FEM and ES-FEMshow excellent agreements with the reference resultdemonstrating that both FEM and ES-FEM can provideaccuracy results in low frequencies As the frequencyincreases the deviation between FEM result and thereference result becomes larger suggesting that the ac-curacy of the FEM result decreases with the increase ofthe frequency We also note that the eigen-frequenciesin FEM result (peaks in response curve) become higherand higher compared to the reference result This devi-ation mainly results from the inherent drawback ofldquoover-stiffnessrdquo in FEM based on the standard weakformulation The ES-FEM provides much more accu-rate result in higher frequency range compared to theFEM model using the same mesh From Fig 5 wecan see that ES(B)-FEM can also produce results simi-lar to that of ES-FEM The softening effect of ES(B)-FEM is almost equal to that of ES-FEM In additionas showed in Fig 6 the response curves obtained from

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFEMReference

Fig 4mdashFrequency responses computed at point A using ES-FEM and FEM for the plate alone

Point A-exciting point

Point B-a response point in the acoustic domain

Aluminum Plate ( s)

Acoustic domain ( f)

Fig 3mdashA flexible aluminum plate backed by abox of air

203Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

ES(S)-FEM have little difference from that of FEM (us-ing the same mesh) which means that the softening ef-fect by the edge-based smoothing on the off-plane shearstrain is minimum and can be neglected Therefore itcan be concluded that the total softening effects ofES-FEM are mainly due to smoothing the in-planebending strain

The sound pressure level (SPL) responses at point Bin acoustic domain (Fig 3) are also computed usingvarious combination of methods and the results areplotted in Fig 7 The normal velocity of the flexibleplate which provides the boundary condition of acous-tic domain is approximated using ES-FEM and FEMThe 3D acoustic domain is divided using tetrahedronelements (1045 nodes 6335 elements) for FEM andES-FEM If FM-BEM is chosen only the surface ofthe 3D acoustic domain is discretized with triangle ele-ments and hence the number of elements is much

smaller (634 nodes 1264 elements) The computationis performed for frequencies ranging from 1 to700 Hz For comparison the numerical result obtainedby the coupled FEMFEM with a very fine mesh(15864 nodes and 82858 elements) is presented asthe reference

As shown in Fig 7 the coupled FEMFEM gives theleast accurate results compared to all the other modelsThe over-stiffness phenomenon of FEM in 3D acousticproblems can also be observed and it becomes muchmore pronounced with the increase of the frequencyThe stiffness matrix in coupled ES-FEMES-FEM issofter and hence the results in high frequency rangeshow better agreements with reference results The cou-pled ES-FEMFM-BEM model has almost the samelevel accuracy as the coupled ES-FEMES-FEM modelIt is found that the FM-BEM can offer accurate resultsfor interior acoustic problems

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(B)-FEMES-FEMFEMReference

Fig 5mdashFrequency responses computed at point A using ES(B)-FEM ES-FEM and FEM for theplate alone

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(S)-FEMES-FEMFEMReference

Fig 6mdashFrequency response analysis in point A using ES(S)-FEM ES-FEM and FEM for theplate alone

204 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 8: A coupled ES-BEM and FM-BEM for structural acoustic problems

air acoustic modes are not coincident with the struc-tural modes it is thus practical to neglect direct interac-tions between the structure and air46 meaning that thestructure dynamics is assumed not to be influenced bythe fluid For comparison the results obtained fromthe FEM with extremely fine mesh are also providedas the reference results The purpose of the first exam-ple of a simple elastic plate backed by a closed acousticcavity is to show the advantages of ES-FEM and FM-BEM The second example is an application of thepresent combined methods to a practical problem in ve-hicle engineering

51 Box with Flexible Plate on Top

In this subsection a weak coupling model of a flex-ible plate and air cavity is established The model isshown in Fig 3 The weakly coupled model is a combi-nation of the flexible plate on the top and a closedacoustic cavity attached The elastic plate is made ofaluminum (r = 2700 kgm3 n = 03 and E = 71 Gpa)The acoustic cavity is full of air (r = 121 kgm3 and

c = 343 ms) The plate which has a dimension of050 m 060 m and a thickness of t = 0003 m is sim-ply supported on all the four edges The closed acousticcavity has a dimension of 050 m 060 m 040 mThe remaining walls (except the coupled wall) of cavityare assumed to be rigid with the surface velocity fixedat v = 0

The top elastic plate is divided with ReissnerndashMind-lin triangle plate elements An evenly distributed timeharmonic load equal to 100 N is applied at the centerof plate (point A in Fig 3) First the forced frequencyresponses are computed at the center of the plate usingdifferent methods including FEM ES-FEM ES(B)-FEMand ES(S)-FEM with same model (155 nodes 264 ele-ments) The frequency ranges from 1 to 1000 Hz Thereference result is provided using FEM with much smal-ler elements (1265 nodes 2390 elements)

As shown in Fig 4 in the low frequency domain (0to 200 Hz) results obtained from FEM and ES-FEMshow excellent agreements with the reference resultdemonstrating that both FEM and ES-FEM can provideaccuracy results in low frequencies As the frequencyincreases the deviation between FEM result and thereference result becomes larger suggesting that the ac-curacy of the FEM result decreases with the increase ofthe frequency We also note that the eigen-frequenciesin FEM result (peaks in response curve) become higherand higher compared to the reference result This devi-ation mainly results from the inherent drawback ofldquoover-stiffnessrdquo in FEM based on the standard weakformulation The ES-FEM provides much more accu-rate result in higher frequency range compared to theFEM model using the same mesh From Fig 5 wecan see that ES(B)-FEM can also produce results simi-lar to that of ES-FEM The softening effect of ES(B)-FEM is almost equal to that of ES-FEM In additionas showed in Fig 6 the response curves obtained from

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFEMReference

Fig 4mdashFrequency responses computed at point A using ES-FEM and FEM for the plate alone

Point A-exciting point

Point B-a response point in the acoustic domain

Aluminum Plate ( s)

Acoustic domain ( f)

Fig 3mdashA flexible aluminum plate backed by abox of air

203Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

ES(S)-FEM have little difference from that of FEM (us-ing the same mesh) which means that the softening ef-fect by the edge-based smoothing on the off-plane shearstrain is minimum and can be neglected Therefore itcan be concluded that the total softening effects ofES-FEM are mainly due to smoothing the in-planebending strain

The sound pressure level (SPL) responses at point Bin acoustic domain (Fig 3) are also computed usingvarious combination of methods and the results areplotted in Fig 7 The normal velocity of the flexibleplate which provides the boundary condition of acous-tic domain is approximated using ES-FEM and FEMThe 3D acoustic domain is divided using tetrahedronelements (1045 nodes 6335 elements) for FEM andES-FEM If FM-BEM is chosen only the surface ofthe 3D acoustic domain is discretized with triangle ele-ments and hence the number of elements is much

smaller (634 nodes 1264 elements) The computationis performed for frequencies ranging from 1 to700 Hz For comparison the numerical result obtainedby the coupled FEMFEM with a very fine mesh(15864 nodes and 82858 elements) is presented asthe reference

As shown in Fig 7 the coupled FEMFEM gives theleast accurate results compared to all the other modelsThe over-stiffness phenomenon of FEM in 3D acousticproblems can also be observed and it becomes muchmore pronounced with the increase of the frequencyThe stiffness matrix in coupled ES-FEMES-FEM issofter and hence the results in high frequency rangeshow better agreements with reference results The cou-pled ES-FEMFM-BEM model has almost the samelevel accuracy as the coupled ES-FEMES-FEM modelIt is found that the FM-BEM can offer accurate resultsfor interior acoustic problems

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(B)-FEMES-FEMFEMReference

Fig 5mdashFrequency responses computed at point A using ES(B)-FEM ES-FEM and FEM for theplate alone

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(S)-FEMES-FEMFEMReference

Fig 6mdashFrequency response analysis in point A using ES(S)-FEM ES-FEM and FEM for theplate alone

204 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 9: A coupled ES-BEM and FM-BEM for structural acoustic problems

ES(S)-FEM have little difference from that of FEM (us-ing the same mesh) which means that the softening ef-fect by the edge-based smoothing on the off-plane shearstrain is minimum and can be neglected Therefore itcan be concluded that the total softening effects ofES-FEM are mainly due to smoothing the in-planebending strain

The sound pressure level (SPL) responses at point Bin acoustic domain (Fig 3) are also computed usingvarious combination of methods and the results areplotted in Fig 7 The normal velocity of the flexibleplate which provides the boundary condition of acous-tic domain is approximated using ES-FEM and FEMThe 3D acoustic domain is divided using tetrahedronelements (1045 nodes 6335 elements) for FEM andES-FEM If FM-BEM is chosen only the surface ofthe 3D acoustic domain is discretized with triangle ele-ments and hence the number of elements is much

smaller (634 nodes 1264 elements) The computationis performed for frequencies ranging from 1 to700 Hz For comparison the numerical result obtainedby the coupled FEMFEM with a very fine mesh(15864 nodes and 82858 elements) is presented asthe reference

As shown in Fig 7 the coupled FEMFEM gives theleast accurate results compared to all the other modelsThe over-stiffness phenomenon of FEM in 3D acousticproblems can also be observed and it becomes muchmore pronounced with the increase of the frequencyThe stiffness matrix in coupled ES-FEMES-FEM issofter and hence the results in high frequency rangeshow better agreements with reference results The cou-pled ES-FEMFM-BEM model has almost the samelevel accuracy as the coupled ES-FEMES-FEM modelIt is found that the FM-BEM can offer accurate resultsfor interior acoustic problems

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(B)-FEMES-FEMFEMReference

Fig 5mdashFrequency responses computed at point A using ES(B)-FEM ES-FEM and FEM for theplate alone

0 100 200 300 400 500 600 700 800 900 100080

100

120

140

160

180

200

220

240

260

Frequency(Hz)

Res

pons

e(dB

)

ES(S)-FEMES-FEMFEMReference

Fig 6mdashFrequency response analysis in point A using ES(S)-FEM ES-FEM and FEM for theplate alone

204 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
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Page 10: A coupled ES-BEM and FM-BEM for structural acoustic problems

52 Automobile Passenger Compartmentwith a Flexible Roof

In this example the application of the present com-bined method (ES-FEM and FM-BEM) to a practicalproblem of vehicle engineering is examined The vehi-cle body is made of panels and is usually welded withnumerous thin steel plates among which the automo-bile coping is one of the largest structures in the vehi-cle The roof can be easily excited and undergoes lowamplitude vibration generating noises which contri-butes strongly to both the interior sound pressure level(SPL) in the automobile passenger compartment andthe exterior noise pressure distribution

In this study a weak coupling model between theflexible roof and the passenger compartment is estab-lished as shown in Fig 8 The boundary edges of theroof is totally fixed with w = 0 θx = 0 and θy = 0 Itis discretized using 422 ReissnerndashMindlin triangle plateelements with 241 nodes The elastic plate is made ofsteel (r = 7900 kgm3 n = 03 and E = 210 GPa) witha thickness of 0001 m The automobile passenger com-partment is divided using 139945 tetrahedron elementswith 26498 nodes for the FEM and ES-FEM Whenthe acoustic domain is calculated using FM-BEM onlythe surface of the 3D acoustic domain is meshed withconstant triangle elements that are much less in numb-ers (11550 elements and 5777 nodes) An evenly dis-tributed time harmonic load (100 N) is applied in themiddle of the coping (exciting point in Fig 8) Boththe interior the sound pressure level (SPL) and the exte-rior of sound pressure distribution are computed andexamined

The sound pressure level (SPL) responses calculatedat drivers ear point obtained using the coupled ES-FEMFM-BEM and coupled ES-FEMES-FEM areplotted in Fig 9 The results are compared against the

reference result that is calculated using coupled FEMFEM with 630441 elements and 114174 nodes

As shown in Fig 9 the results for this complicatedexample reinforces the finding from the previous sim-ple example The response results from the ES-FEMFM-BEM agree well with that from ES-FEMES-FEM Both results are much more accurate than theFEMFEM results using the same mesh In the low fre-quency range (0 to 40 Hz) all the coupled methods canproduce very accurate solutions which is in a goodagreement with the reference result As the frequencyincreases the result obtained from the coupled FEMFEM becomes inaccurate Both ES-FEMFM-BEMand ES-FEMES-FEM results have similar level of ac-curacy much more accurate than the FEM counterpartand the eigen-frequencies (peak in response curve) aremuch closer to that of the reference result

In order to examine the performance of the ES-FEMFM-BEM comparing with the conventional ES-FEM

100 200 300 400 500 600 70080

100

120

140

160

180

200

220

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 7mdashThe sound pressure level (SPL) responses computed at point B using ES-FEMFM-BEMES-FEMES-FEM and FEMFEM

Exciting point

Response point at drivers

left ear

Automobile coping ( sΩ )

Acoustic domain ( fΩ )

Fig 8mdashA weak coupling model combined bythe flexible coping and the passengercompartment

205Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
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  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 11: A coupled ES-BEM and FM-BEM for structural acoustic problems

BEM the forced frequency response at drivers ear pointare computed and plotted in Fig 10 It is observed thatthe results obtained from ES-FEMFM-BEM coincide withthe one from ES-FEMBEM This indicates that the FMMoperation does not lead to any loss of accuracy if the FMMparameters are chosen reasonably However the compu-tational efficiency is improved significantly via the FMMoperations The efficiency of ES-FEMFM-BEM is fur-ther evident in the following numerical example

Solving sound radiation problems is one of the mostimportant and useful application of the boundary inte-gral methods In this subsection we further explorethe boundary integral approaches using a larger scaleproblem The radiation of acoustic waves from vibrat-ing portions of the vehicle body is studied The vehiclebody model which is used in the previous case has anoverall dimensions of 27 m 14 m 13 m in the x yand z direction respectively and is meshed with 11550constant triangular elements (Fig 8) For data collec-tion for the velocity potential distribution a total of

1170 field points are placed on a semi-cylindrical sur-face with radius of 25 m shown in Fig 11 The har-monic vibrations of the roof along the z direction arecomputed by ES-FEM-DSG3 subjected to a harmonicload of 100 N with a frequency of 8213 Hz at the cen-ter of the coping (exciting point in Fig 8) The soundpressure distribution on the surface of the semi-columncylinder is computed using the FM-BEM and BEM andshown in Fig 12 It is found that sound pressure level(SPL) distribution obtained using the ES-FEMFM-BEM and ES-FEMBEM is almost the same whichdemonstrates that FM-BEM can solve the radiationproblem as the BEM without the loss of accuracy Toexamine the efficiency of various combination ofmodels the vehicle body is discretized using differentsize elements The CPU time used by the ES-FEMFM-BEM and the ES-FEMBEM codes is recordedand the comparison is shown in Fig 13 It is clearlyshown that the ES-FEMFM-BEM is much less time-consuming than ES-FEMBEM in solving all different

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMFM-BEMES-FEMES-FEMFEMFEMReference

Fig 9mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMES-FEM ES-FEMFM-BEM and FEMFEM

20 40 60 80 100 120 140 160 180 20040

60

80

100

120

140

160

180

200

Frequency(Hz)

Res

pons

e(dB

)

ES-FEMBEMES-FEMFM-BEM

Fig 10mdashThe sound pressure level (SPL) responses at vehicle drivers ear point obtained usingES-FEMBEM and ES-FEMFM-BEM

206 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 12: A coupled ES-BEM and FM-BEM for structural acoustic problems

sizes of models (DOFs from 4000 to 12000) The CPUtime for the ES-FEMFM-BEM code scales almost lin-early with the increase of the DOFs The conventionalBEM however scales about as a cubic function with

the DOFs and it can only solve models with up to17300 DOFs on the same PC

6 CONCLUSIONS AND DISCUSSIONS

In this paper a coupled ES-FEMFM-BEM methodis proposed for analyzing structural acoustic problemsOur combined approach takes the best advantages ofboth ES-FEM and FM-BEM and the inherent draw-backs of the ldquooverly-stiff rdquo in FEM and computationalinefficiency in BEM are overcome Numerical exam-ples of structural acoustic problems have demonstratedthe following features of the present method

1 For the ReissnerndashMindlin plates the total soften-ing effect of ES-FEM is mainly resulted bysmoothing of the in-plane bending strains whilesmoothing the shear strain has little effects

2 The coupled ES-FEMFM-BEM can producemuch more accurate results than that of theFEMFEM in middle frequency range for interioracoustic problems

3 The coupled ES-FEMFM-BEM produces almostthe same level of accuracy as the coupled ES-FEMES-FEM which means that the FMM operation inES-FEMFM-BEM does not lead to significant lossof accuracy

4 Owning to the FMM technique and the iterativeequation solver (GMERS) applied in FM-BEMcoupled ES-FEMFM-BEM is much more effi-cient than ES-FEMBEM for exterior noise radia-tion problems without losing accuracy It isfound that ES-FEMFM-BEM can be severaltimes faster than ES-FEMBEM which is espe-cially crucial for large-scale numerical acousticproblems

Vehicle model

Sound pressure on a semi-cylindrical surface

Fig 11mdashSemi-cylindrical surface forexamining the sound pressure excitedby a vibrating coping of vehicle

Computed sound-pressure distribution using ES-FEMBEM

b

a

Computed sound-pressure distribution using ES-FEMFM-BEM

SPL (dB)270265260255250245240235

SPL (dB)270265260255250245240235

Y

Z

X

Y

Z

X

Fig 12mdashComputed sound-pressuredistribution on a semi-cylindricalsurface for the vehicle body model(at 8213 Hz) using differentcombined methods

04 06 08 1 12 14 16 18

x 104

0

20

40

60

80

100

120

DOFs

CP

U ti

me(

sec

)

ES-FEMBEM

ES-FEMFM-BEM

Fig 13mdashCPU time used by the ES-FEMFM-BEM code compared with thatof the ES-FEMBEM code

207Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
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  • B31
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Page 13: A coupled ES-BEM and FM-BEM for structural acoustic problems

7 ACKNOWLEDGMENTS

The authors wish to thank the support NFSC(61232014) and also the Chinafunded PostgraduatesStudying Aboard Program for Building Top Universityand the National Natural Science Foundation of ChinaThis work is partially supported by University of Cin-cinnati The work by the 2nd author is partially sup-ported by the United States NSF Grant under theaward no 1214188 and partially by the United StatesARO contract no W911NF-12-1-0147

8 REFERENCES

1 GC Everstine ldquoFinite element formulations of structuralacoustics problemrdquo Computers amp Structures 65(3) 307ndash321(1997)

2 D Soares Jr ldquoAcoustic modeling by BEMndashFEM coupling pro-cedures taking into account explicit and implicit multi-domaindecomposition techniquesrdquo International Journal for Numeri-cal Methods in Engineering 78 1076ndash1093 (2009)

3 M Fischer and L Gaul ldquoFast BEM-FEM mortar couplingfor acoustic-structure interactionrdquo International Journal forNumerical Methods in Engineering 62(12) 1677ndash1690(2005)

4 ZO Cecil and RL Taylor The Finite Element Method SolidMechanics Vol 2 Butterworth-Heinemann (2000)

5 N Reddy Theory and Analysis of Elastic Plates and ShellsCRC Press Taylor and Francis Group New York (2006)

6 F Gruttmann and W Wagner ldquoA stabilized one-point inte-grated quadrilateral ReissnerndashMindlin plate elementrdquo Interna-tional Journal for Numerical Methods in Engineering 612273ndash2295 (2004)

7 J Mackerle ldquoFinite element linear and nonlinear static and dy-namic analysis of structural elements a bibliographyrdquo Engi-neering Computations 19(5) 520ndash594 (2002)

8 OC Zienkiewicz and RL Taylor The Finite Element MethodFifth Ed Butterworth-Heinemann Oxford (2000)

9 OC Zienkiewicz RL Taylor and JM Too ldquoReduced integra-tion techniques in general of plates and shellsrdquo InternationalJournal for Numerical Methods in Engineering 3 275ndash290(1971)

10 S Cen YQ Long and ZH Yao ldquoA new hybrid-enhanced dis-placement-based element for the analysis of laminated compos-ite platesrdquo Computers and Structures 80(9ndash10) 819ndash833(2002)

11 S Cen AK Soh YQ Long and ZH Yao ldquoA new 4-nodequadrilateral FE model with variable electrical degrees of free-dom for the analysis of piezoelectric laminated compositeplatesrdquo Composite Structures 58(4) 583ndash599 (2002)

12 PG Bergan and X Wang ldquoQuadrilateral plate bending ele-ments with shear deformationsrdquo Computers and Structures19(1ndash2) 25ndash34 (1984)

13 SW Lee and THH Pian ldquoFinite elements based upon Mind-lin plate theory with particular reference to the four-node iso-parametric elementrdquo AIAA Journal 16 29ndash34 (1978)

14 SW Lee and C Wong ldquoMixed formulation finite elements forMindlin theory plate bendingrdquo International Journal for Nu-merical Methods in Engineering 18 1297ndash1311 (1982)

15 JMA Ceacutesar de Saacute RM Natal Jorge RA Fontes Valente andPMA Areias ldquoDevelopment of shear locking-free shell ele-ments using an enhanced assumed strain formulationrdquo Int JNumer Methods Engrg 53 1721ndash1750 (2002)

16 RPR Cardoso JW Yoon M Mahardika S Choudhry RJAlves de Sousa and RA Fontes Valente ldquoEnhanced assumedstrain (EAS) and assumed natural strain (ANS) methods for

one-point quadrature solid-shell elementsrdquo Int J Numer Meth-ods Engrg 156ndash187 (2008)

17 JL Batoz and I Katili ldquoOn a simple triangular ReissnerMind-lin plate element based on incompatible modes and discreteconstraintsrdquo Int J Numer Methods Engrg 1603ndash1632(1992)

18 OC Zienkiewicz RL Taylor P Papadopoulos and E OnateldquoPlate bending elements with discrete constraints new triangu-lar elementsrdquo Comput Struct 505ndash522 (1990)

19 KU Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shell fi-nite elementsrdquo Comput Struct 75 321ndash34 (2000)

20 GR Liu TT Nguyen KY Dai and KY Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering 71902ndash30 (2007)

21 GR Liu TT Nguyen XH Nguyen and KY Lam ldquoA node-based smoothed finite element method for upper bound solu-tion to solid problems (NS-FEM)rdquo Comput Struct 87 14ndash26(2009a)

22 ZQ Zhan and GR Liu ldquoTemporal stabilization of the node-based smoothed finite element method and solution bound oflinear elastostatics and vibration problemsrdquo ComputationalMechanics 46 229ndash246 (2010)

23 ZC He GR Liu ZH Zhong GY Zhang and AG ChengldquoCoupled analysis of 3D structuralndashacoustic problems usingthe edge-based smoothed finite element methodfinite elementmethodrdquo Finite Elements in Analysis and Design 46 1114ndash1121(2010)

24 GR Liu TT Nguyen and KY Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static and dy-namic problems of solid mechanicsrdquo J Sound Vibr 3201100ndash1130 (2009)

25 ZC He GR Liu ZH Zhong SC Wu GY Zhang and AGCheng ldquoAn edge-based smoothed finite element method (ES-FEM) for analyzing three-dimensional acoustic problemsrdquoComputer Methods in Applied Mechanics and Engineering199(1ndash4) 20ndash33 (2009)

26 HA Schenck ldquoImproved integral formulation for acoustic ra-diation problemsrdquo J Acoust Soc Am 44(1) 41ndash58 (1968)

27 AJ Burton and GF Miller ldquoThe application of the integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Societyof London Series A Mathematical Physical amp EngineeringSciences 323(1553)201ndash210 (1971)

28 O Estorff Boundary Elements in Acoustics Advances andApplications WIT Press Southampton (2000)

29 W Hackbusch ldquoA sparse matrix arithmetic based on H-matricesPart I Introduction to H-matricesrdquo Computing 62(2) 89ndash108(1999)

30 G Beylkin A Coifman and V Rokhlin ldquoFast wavelet trans-forms and numerical algorithms Irdquo Communications on Pureand Applied Mathematics 141ndash183 (1991)

31 G Golub and CV Loan Matrix Computations 3rd Ed TheJohns Hopkins University Press Baltimore (1996)

32 L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics 73 325ndash348(1987)

33 V Rokhlin ldquoA fast algorithm for the discrete Laplace transfor-mationrdquo Journal of Complex 4(1)12ndash32 (1988)

34 Y Saad and MH Schultz ldquoGMRES a generalized minimal re-sidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing 7856ndash869 (1986)

35 P Sonneveld ldquoCGS A fast Lanczos-type solver for nonsym-metric linear systems SIAMrdquo Journal on Scientific and Statis-tical Computing 10 36ndash52 (1986)

36 V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics 60(2)187ndash207 (1985)

208 Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
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  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
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  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46
Page 14: A coupled ES-BEM and FM-BEM for structural acoustic problems

37 S Amini and ATJ Profit ldquoAnalysis of a diagonal form of thefast multipole algorithm for scattering theoryrdquo BIT NumericalMathematics 585ndash602 (1999)

38 Tetsuya Sakuma and Yosuke Yasuda ldquoFast multipole boundaryelement method for large-scale steady-state sound field analy-sis Part I setup and validationrdquo Acta Acustica united withAcustica 88(4) 513ndash525 (2002)

39 S Schneider ldquoApplication of fast methods for acoustic scatter-ing and radiation problemsrdquo Journal of Computational Acous-tics 11(3) 387ndash401 (2003)

40 A Nail D Gumerov and D Ramani Fast Multipole Methodsfor the Helmholtz Equation in Three Dimensions ElsevierScience (2005)

41 ZS Chen H Waubke and W Kreuzer ldquoA formulation of thefast multipole boundary element method (FMBEM) for acous-tic radiation and scattering from three-dimensional structuresrdquoJournal of Computational Acoustics 303ndash320 (2008)

42 HJ Wu YJ Liu and WK Jiang ldquoA fast multipole boundaryelement method for 3D multi-domain acoustic scattering pro-blems based on the BurtonndashMiller formulationrdquo EngineeringAnalysis with Boundary Elements 36(5) 779ndash788 (2012)

43 GR Liu Meshfree Methods Moving beyond the Finite Ele-ment Method 2nd Ed CRC Press (2009)

44 GR Liu and JD Achenbach ldquoA strip element method forstress-analysis of anisotropic linearly elastic solidsrdquo Journalof Applied Mechanics Transactions of the ASME 61(2) 270ndash277(1994)

45 YJ Li Fast Multipole Boundary Element Method mdash Theoryand Applications in Engineering Cambridge University Press(2009)

46 R Citarella L Federico and A Cicatiello ldquoModal acoustictransfer vector approach in a FEMndashBEM vibro-acoustic analy-sisrdquo Engineering Analysis with Boundary Elements 31 248ndash258(2007)

209Noise Control Engr J 62 (4) July-August 2014 Published by INCEUSA in conjunction with KSNVE

  • s1
  • aff1
  • aff2
  • aff3
  • aff4
  • aff5
  • s2
  • s2A
  • E1
  • E2
  • E3
  • E4
  • E5
  • E6
  • E7
  • E8
  • E9
  • E10
  • s2B
  • E11
  • E12
  • E13
  • E14
  • E15
  • F1
  • E16
  • E17
  • E18
  • E19
  • s3
  • s3A
  • E20
  • E21
  • E22
  • E23
  • E24
  • E25
  • E26
  • E27
  • E28
  • E29
  • s3B
  • E30
  • E31
  • E32
  • E33
  • E34
  • E35
  • F2
  • s4
  • s4A
  • E36
  • E37
  • E38
  • E39
  • E40
  • s4B
  • s5
  • s5A
  • F4
  • F3
  • F5
  • F6
  • s5B
  • F7
  • F8
  • F9
  • F10
  • s6
  • F11
  • F12
  • F13
  • B1
  • B2
  • B3
  • B4
  • B5
  • B6
  • B7
  • B8
  • B9
  • B10
  • B11
  • B12
  • B13
  • B14
  • B15
  • B16
  • B17
  • B18
  • B19
  • B20
  • B21
  • B22
  • B23
  • B24
  • B25
  • B26
  • B27
  • B28
  • B29
  • B30
  • B31
  • B32
  • B33
  • B34
  • B35
  • B36
  • B37
  • B38
  • B39
  • B40
  • B41
  • B42
  • B43
  • B44
  • B45
  • B46

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