Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019

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Comput. Methods Appl. Mech. Engrg.

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Finite element computation of absorbing boundary conditionsfor time-harmonic wave problems

Denis Duhamel *, Tien-Minh NguyenUniversité Paris-Est, UR Navier, Ecole des Ponts ParisTech, 6 et 8 Avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne la Vallée, cedex 2, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 January 2008Received in revised form 6 May 2009Accepted 7 May 2009Available online 12 May 2009

Keywords:Absorbing boundary conditionsWaveguideFinite elementPeriodic medium

0045-7825/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cma.2009.05.002

* Corresponding author. Tel.: +33 1 64 15 37 28; faE-mail address: duhamel@lami.enpc.fr (D. Duham

This paper proposes a new method, in the frequency domain, to define absorbing boundary conditions forgeneral two-dimensional problems. The main feature of the method is that it can obtain boundary con-ditions from the discretized equations without much knowledge of the analytical behavior of the solu-tions and is thus very general. It is based on the computation of waves in periodic structures andneeds the dynamic stiffness matrix of only one period in the medium which can be obtained by standardfinite element software. Boundary conditions at various orders of accuracy can be obtained in a simpleway. This is then applied to study some examples for which analytical or numerical results are available.Good agreements between the present results and analytical solutions allow to check the efficiency andthe accuracy of the proposed method.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Wave problems in unbounded media can occur in many appli-cations in mechanics and engineering such as in acoustics, solidmechanics, electromagnetics, etc. It is well known that analyticalsolutions for such problems are available only for some specialcases. On the contrary, numerical methods can be applied to manycomplex problems. Physically, for problems in infinite domains,the energy is produced by sources in the region to be analyzedand must escape to infinity. For methods solving the problem ona bounded domain like the finite element method, it introducesthe difficulty of an artificial boundary to get a bounded domain.This boundary must be such that the energy crosses it withoutreflection and special conditions must be specified at the artificialboundary to reproduce this phenomena. Generally, these can beclassified into local or global boundary conditions. With a globalcondition all degrees of freedom (dofs) on the boundary are cou-pled while a local condition connects only neighboring dofs.

The first global method which has been used for solving suchproblems was the boundary element method. This method is welladapted for infinite domains and is described in numerous classicaltextbooks like [1–5]. It consists in solving an equation on theboundary of the domain only and the radiation conditions aretaken into account analytically. It also reduces the dimension ofthe problem to a surface in 3D and to a curve in 2D decreasing thusthe size of the linear problem to solve. However, the final probleminvolves full matrices which are also generally non symmetrical. It

ll rights reserved.

x: +33 1 64 15 37 41.el).

is also mainly limited to linear problems and to homogeneous do-mains or otherwise one has to introduce special and complex tech-niques to deal with non linear or non homogeneous situations.There are also singularities in the integrals which need specialattention for the numerical integrations. So this method is interest-ing and has been extensively used but it can lead to heavy compu-tations when the number of degrees of freedom increases. Moreinformation on such techniques can be found in the historicaland review papers [6,7].

In the other approaches, the computational domain is truncatedat some distance and boundary conditions are imposed at this arti-ficial boundary. These conditions at finite distance must simulateas closely as possible the exact radiation condition at infinity. Anapproach leading to a global boundary condition is the Dirichletto Neumann (DtN) mapping proposed by Keller and Givoli [8,9]and in an earlier version by Hunt et al. [10,11]. It consists in divid-ing the domain into a finite part containing the sources and an infi-nite domain of simple shape. The solution in the infinite domain issolved analytically, for example by series expansions, and an exactimpedance relation is obtained on the boundary between the finiteand infinite domains. This relation links the variable and its normalderivative on the whole boundary. The DtN mapping is thus non lo-cal and every node on the boundary is connected to all other nodes.This gives a full matrix for the nodes of the boundary which par-tially destroys the sparse matrix of the FEM and increases substan-tially the computing resources needed to get the solution. Thesolution has to be found in the exterior domain by analytical ornumerical methods. When the analytical solution can be found, itis generally under the form of a series expansion. The number ofterms in the expansion must be sufficient for an accurate solution

Γ

Γ

int

ext

Ω

A

A

+

−Ω int

Fig. 1. Computational domain.

D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019 3007

which can lead to heavy computations. Developments of the meth-od can be found in [12,13]. An application to the case of wave scat-tering in plates is also found in [14].

The other methods are local and the condition at a node in-volves only neighboring nodes which make them less demandingin computing resources and much easier to implement in a finiteelement code but also less accurate. A first possibility of such ap-proaches is the use of infinite elements as proposed by Bettesset al. [15–18] and Astley [19]. It consists in developing special ele-ments with a behavior at infinity reflecting that of analytical solu-tions obtained for the same type of problems. For wave problems,it involves complex-valued basis functions with outwarding prop-agation wave-like behavior in the radial direction. The elementswere further developed by Burnett [20] to considered other coordi-nate systems such as expansions in prolate coordinates. This meth-od is interesting but the inclusion of infinite elements requires thedevelopment of special elements and these elements can dependon decay parameters which have to be accurately chosen. A reviewof these methods has been proposed by Gerdes [21].

In the perfectly matched layer proposed by Bérenger [22,23],originally for electromagnetic waves, an exterior layer of finitethickness is introduced around the bounded domain. The absorp-tion in this domain is increasing as we move towards the exteriorsuch that outgoing waves are absorbed before reaching the exte-rior truncation boundary. The number of elements in the layer,its thickness, the variation of the absorption properties have tobe carefully chosen to optimize the efficiency of the method. Thisefficiency is better for a layer with a large thickness but this canlead to a significant increase in the number of elements in the finiteelement model. Various developments of the method can be foundin [24] and its optimization in [25]. Otherwise, various classes ofabsorbing boundary conditions were also developed by Engquistand Majda [26]. They consist in the numerical approximation ofdifferential operators on the boundary. For instance, examples ofthe application of Bayliss–Turkel conditions are presented byStrouboulis et al. [27]. However, the more accurate boundary con-ditions involve high order derivatives on the boundary which aredifficult to implement in the finite element method [28]. Finally,it was proved by Asvadurov et al. [25] and Guddati and Lim [29]that, in fact, the perfectly matched layer and the absorbing bound-ary conditions were closely connected. The Helmholtz equationwas also solved with these two boundary conditions by Heikkolaet al. [30] and these conditions were compared and optimized tominimize the reflection. Other boundary conditions involving onlysecond order derivatives have also been proposed. They introduceauxiliary variables and systems of equations on the boundarywhich lead to high order boundary conditions, see [31] for a reviewof such non-reflecting boundary (NRBC) methods. They weremainly developed for acoustic problems but in [32] a local bound-ary condition for elastic waves has been proposed. In [33] animpedance boundary condition in new coordinates was developedfor the convected Helmholtz equation. For fluid dynamic problems,[34] developed Lagrange multipliers for imposing various absorb-ing boundary conditions for cases where the type and the numberof boundary conditions can change, for instance as the flowchanges from subsonic to supersonic regimes and its direction var-ies with time. A general review of the methods described in theprecedent paragraphs for various dynamic, acoustic and wavepropagation problems can be found in [35–38]. Comparisons arealso made between the different methods.

In the present study, another local method is proposed. Thismethod works on discrete systems directly, in contrast with manyexisting absorbing boundary conditions which are written on thecontinuous differential equations and discretized after. The princi-ple of the method is to compute wave propagations in groups ofelements near the boundary from the dynamic stiffness matrix of

these elements. Then, a boundary condition is obtained for cancel-ling the reflected waves. This condition is finally written as animpedance boundary condition relating the force and displace-ment degrees of freedom on the boundary. The approach is basedon the waveguide theory proposed by Mace et al. [39], Duhamelet al. [40], Duhamel [41] and is used to determine absorbingboundary conditions at the truncation boundary of 2D periodicmedia. Only information related to one period, obtained fromany standard FE software (the discrete stiffness and mass matricesand the nodal coordinates) are required to formulate the method.The advantage is that it can be applied to media with various com-plex behaviors.

This paper is outlined as follows. In Section 2, the methodologyfor determining absorbing boundary conditions for periodic mediais described. Then, a discussion for the application of the method togeneral media is proposed. In Section 3, a simple application is pro-posed to show the results of the method in a case where detailedcomputations can be done. In Section 4, two examples of finite ele-ment computations in acoustics and elastodynamics are presented.They allow to check the efficiency and accuracy of the proposedmethod for more complex cases. Finally, the paper is closed withsome conclusions.

2. Absorbing boundary conditions

We suppose that we want to solve a mechanical problem on aninfinite domain exterior to the bounded domain Xint (see Fig. 1).The infinite domain is approximated by the finite domain X whichis exterior to Xint and is limited by the exterior boundary Cext . Weare looking for a solution with radiating condition at infinity whichmeans that the solution should be outgoing near the boundary Cext .Near this exterior boundary the solution can be seen as composedof incident waves denoted Aþ and reflected waves A�. For a per-fectly absorbing boundary, one should have A� ¼ 0. In fact thiscondition is very difficult to implement in the numerical solutionsof such problems. Indeed, only the global solution is easily com-puted but the decomposition into incident and reflected waves isdifficult to obtain. The problem is thus to find an appropriateboundary condition to impose on the exterior boundary to finallyget A� � 0 on the solution. To be easily included in a finite elementmodel the searched boundary condition should be local and thecondition at a node of the boundary should involve only neighbor-ing nodes.

The approach proposed in this paper consists in studying thisproblem by first considering the case of periodic media. For thiscase, positive and negative waves and their amplitudes Aþ and

3008 D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019

A� can be computed by the method presented below. Then an ex-act boundary condition can be formulated for a half-plane bound-ary. It is further shown how this condition can be approximated bya local condition on the boundary. As homogeneous media are spe-cial cases of periodic media, the method presented here appliesalso to homogeneous media. Before considering the general case,a simple example for the Klein Gordon equation will be presented.

2.1. A simple example

Consider first the stationary Klein Gordon equation given by

d2u

dx2 þ ðk2 �m2Þu ¼ 0; ð1Þ

where u is the solution and k;m are real parameters. This equationis discretized with linear two nodes elements such that

uðnÞ ¼ u1N1ðnÞ þ u2N2ðnÞ; ð2Þ

where N1ðnÞ ¼ ð1� nÞ=2;N2ðnÞ ¼ ð1þ nÞ=2;u1 and u2 are the valuesof the function at the two nodes of the element. The discretizationof the first and second parts of relation (1) leads, for an element oflength l, to the matrices

k ¼ �1l

1 �1�1 1

� �m ¼ l

62 11 2

� �ð3Þ

and the dynamic stiffness matrix of one element is given by

d ¼ �1l

1 �1�1 1

� �þ l

6ðk2 �m2Þ

2 11 2

� �: ð4Þ

Waves of propagating constant el are such that

u2 ¼ elu1; ð5Þf2 þ elf1 ¼ 0 ð6Þ

leading in an element to

e2ld12 þ ðd11 þ d22Þel þ d21 ¼ 0; ð7Þ

where dij are the components of the matrix d. Taking into accountthe symmetries in the matrix d, this yields

e2l þ 2d11

d12el þ 1 ¼ 0; ð8Þ

whose solutions are

el� ¼ � d11

d12�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid11

d12

� �2

� 1

sð9Þ

with

d11

d12¼ �1� l2ðk2 �m2Þ=3

1þ l2ðk2 �m2Þ=6: ð10Þ

For l2ðk2 �m2Þ � 1, one gets, at first order,

el� � 1� ilffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

qð11Þ

meaning

l� � �ilffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

q: ð12Þ

There are thus two waves in an element, such that

uþfþ

� �¼

1d11 þ elþd12

� �and

u�f�

� �¼

1d11 þ el�d12

� �ð13Þ

and the general solution is given by

u

f

� �¼ aþ

uþfþ

� �þ a�

u�f�

� �: ð14Þ

The condition for only outgoing waves is thus a� ¼ 0 on the rightboundary and aþ ¼ 0 on the left boundary leading respectively tothe conditions

f=u ¼ fþ=uþ on the right;f=u ¼ f�=u� on the left:

ð15Þ

The condition in the first case is

f=u ¼ d11 þ elþd12

¼ d11 þ � d11

d12þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid11

d12

� �2

� 1

s0@ 1Ad12

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

11 � d212

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ðk2 �m2Þ þ 1

12ððk2 �m2ÞlÞ2

r

¼ iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

12ðk2 �m2Þl2

rð16Þ

� iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

q: ð17Þ

In the second case, one gets

f=u ¼ d11 þ el�d12 � �iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

q: ð18Þ

We recognize approximations of the classical absorbing boundaryconditions which have been obtained here directly from the discret-ized equations. Compared to the classical boundary condition onthe right (and exact in this case) f=u ¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

p, the relative error

isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

12 ðk2 �m2Þl2

qwhich depends mainly on the size of the ele-

ment relatively to the wavelength. The present boundary conditionhas been obtained entirely from the discrete matrices without anyknowledge of the analytical solution of the problem.

To estimate the reflection coefficient created by such a bound-

ary condition consider an incident wave Aeiffiffiffiffiffiffiffiffiffiffiffik2�m2p

x on the bound-

ary. A reflected wave RAe�iffiffiffiffiffiffiffiffiffiffiffik2�m2p

x is created. The total solutionand its associated force are given by

uðxÞ ¼ Aeiffiffiffiffiffiffiffiffiffiffiffik2�m2p

x þ RAe�iffiffiffiffiffiffiffiffiffiffiffik2�m2p

x;

f ðxÞ ¼ Aiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

qeiffiffiffiffiffiffiffiffiffiffiffik2�m2p

x � RAiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

qe�i

ffiffiffiffiffiffiffiffiffiffiffik2�m2p

x: ð19Þ

Writing the boundary condition (16), for instance by taking theboundary at x ¼ 0, yields

f ð0Þiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

p� �uð0Þ

¼ 1� R1þ R

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

12ðk2 �m2Þl2

r: ð20Þ

So the reflection coefficient is finally given by

R ¼1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

12 ðk2 �m2Þl2

q1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

12 ðk2 �m2Þl2

q � 148ðk2 �m2Þl2

: ð21Þ

This coefficient is low and of second order whenffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �m2

pl� 1.

2.2. General impedance boundary condition

In this section we present the general outline of the method be-fore starting a more rigorous development in the following section.So, to extend the precedent example to more general cases, con-sider a vector function uðx; yÞ and a force vector fðx; yÞ acting ona line parallel to the y axis as in Fig. 2. They can be decomposedby a Fourier transform as

0

−1

1

Γext

x

x

1

2

m

−m

2

2

u(x,y)

f(x,y)

Fig. 2. Periodic medium near the exterior boundary.

1l

1

2

2l F

boundary

outgoing wavesincoming waves

b1

2b

Fig. 3. Periodic medium.

D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019 3009

uðx; yÞ ¼ 1

ð2pÞ2Z þ1

�1

Z þ1

�1uðkx; kyÞeiðkxxþkyyÞ dkx dky;

fðx; yÞ ¼ 1

ð2pÞ2Z þ1

�1

Z þ1

�1fðkx; kyÞeiðkxxþkyyÞ dkx dky:

ð22Þ

Let us suppose that u is solution of a linear operator

LðuÞ ¼Xn¼N

n¼0

Xa1þa2¼n

aa1a2

@nu@xa1@ya2

¼ 0: ð23Þ

In the Fourier domain, this relation yieldsXn¼N

n¼0

Xa1þa2¼n

aa1a2 ðikxÞa1 ðikyÞa2

!uðkx; kyÞ ¼ 0: ð24Þ

For a given value of ky, the precedent relation has non zero solutionsfor kx such that the determinantXn¼N

n¼0

Xa1þa2¼n

aa1a2 ðikxÞa1 ðikyÞa2

¼ 0: ð25Þ

Let us denote by kþj the nþ positive solutions such that Reðkþj Þ < 0 orReðkþj Þ ¼ 0 and the energy flux is directed towards positive values ofx. We denote by k�j the other solutions. We have the decomposition

uðx; kyÞ ¼Xj¼nþ

j¼1

aþj uþj eikþj x þXj¼n�

j¼1

a�j u�j eik�j x: ð26Þ

In the same way, for the force components

fðx; kyÞ ¼Xj¼nþ

j¼1

aþj fþj eikþj x þXj¼n�

j¼1

a�j f�j eik�j x; ð27Þ

where fþj and f�j are the force components respectively associated touþj and u�j . If the boundary is such that only positive waves exists atproximity, one has

uð0; kyÞ ¼Xj¼nþ

j¼1

aþj uþj ¼ Uþaþ;

fð0; kyÞ ¼Xj¼nþ

j¼1

aþj fþj ¼ Fþaþ; ð28Þ

where Uþ and Fþ are the matrices whose columns are respectivelyuþj and fþj . Eliminating the aþ coefficients, one gets

fð0; kyÞ ¼ FþðUþÞ�1uð0; kyÞ;fð0; yÞ ¼ ðG � uÞð0; yÞ

ð29Þ

with

GðkyÞ ¼ FþðUþÞ�1ðkyÞ ð30Þ

and * means the convolution.In the following this boundary condition will be computed di-

rectly from the discrete equations for general linear media.

2.3. Solution in a periodic medium

Consider an infinite two-dimensional periodic medium, asshown in Fig. 3. The elementary period is limited by the domainðx1; x2Þ 2 ½0; b1� � ½0; b2�. A function Uðx1; x2Þ defined on the two-dimensional periodic medium can be decomposed as an integralof pseudo periodic functions

Uðx1; x2Þ ¼Z p

b2

� pb2

eikx2 bUðx1; k; x2Þdk; ð31Þ

where bUðx1; k; x2Þ is a periodic function in x2 with period b2. Fromthe Fourier transform bUðx1; kÞ of Uðx1; x2Þ, one has

bUðx1; k; x2Þ ¼1

2pXþ1

m2¼�1

bUðx1; kþ 2pm2

b2Þei2pm2

b2x2

¼ 12p

Xþ1m2¼�1

ei2pm2b2

x2

Z þ1

�1e�iðkþ2pm2

b2ÞxUðx1; xÞdx

¼ 12p

Z þ1

�1e�ikxUðx1; xÞ

Xþ1m2¼�1

ei2pm2b2ðx2�xÞdx

¼ b2

2p

Z þ1

�1

Xþ1m2¼�1

dðx2 � x�m2b2Þe�ikxUðx1; xÞdx

¼ b2

2pXþ1

m2¼�1e�ikðx2þm2b2ÞUðx1; x2 þm2b2Þ: ð32Þ

This gives the relation inverse of (31). From relation (31), one seesthat the behavior in x2 of the general solution can be obtained fromfunctions as eikx2 bUðx1; k; x2Þ with bUðx1; k; x2Þ periodic in x2. Alongdirection 1, we use a decomposition in Bloch waves as it is usualin periodic media. Finally, the general solution can be obtained fromfunctions uðx1; k; x2Þ ¼ eikx2 bUðx1; k; x2Þ such that

uðx1; k; x2 þm2b2Þ ¼ eikm2b2 uðx1; k; x2Þ; ð33Þuðx1 þm1b1; k; x2Þ ¼ eim1luðx1; k; x2Þ; ð34Þ

where m1 and m2 are integers, k 2 R \ � pb2; p

b2

h iand l 2 C.

3010 D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019

The discrete dynamic equation of a cell (an elementary period)obtained from a FE model at a frequency x and for the time depen-dence e�ixt is given by

ðK� ixC�x2MÞq ¼ f; ð35Þ

where K, M and C are the stiffness, mass and damping matrices,respectively, f is the loading vector and q the vector of the degreesof freedom (dofs). Introducing the dynamic stiffness matrixeD ¼ K� ixC�x2M, decomposing the dofs into boundary ðBÞ andinterior ðIÞ dofs, and assuming that there are no external forces onthe interior nodes, result in the following equation:eDBB

eDBIeDIBeDII

" #qB

qI

� �¼

fB

0

� �: ð36Þ

The interior dofs can be eliminated using the second row of Eq. (36),which results in

qI ¼ �eD�1IIeDIBqB: ð37Þ

The first row of Eq. (36) becomes

fB ¼ eDBB � eDBIeD�1

IIeDIB

� �qB; ð38Þ

which can be written as

f ¼ Dq: ð39Þ

It should be noted that only boundary dofs are considered in thefollowing.

The periodic cell is assumed to be meshed with an equal num-ber of nodes on their opposite sides. The boundary dofs are decom-posed into left ðLÞ, right ðRÞ, bottom ðBÞ, top ðTÞ dofs and associatedcorners ðLBÞ, ðRBÞ, ðLTÞ and ðRTÞ as shown in Fig. 4. The longitudinaldofs vector is defined as ql ¼ t ½tqL

tqRtqLB

ttqRBtqRT

tqLT �.Thus, Eq. (39) is rewritten as

Dll DlB DlT

DBl DBB DBT

DTl DTB DTT

264375 ql

qB

qT

264375 ¼ f l

fB

fT

264375: ð40Þ

Using the pseudo periodic condition (34) and the effort equilibriumat the bottom side of the cell, relations between the transverse dofsare given by

1

2

RB

T TRTL

R

Longitudinal DOFs

Transverse DOFs

LB

L

B

Transverse DOFs

IInternal DOFs

1

b2

b

Fig. 4. A cell in the periodic medium.

qT ¼ eikb2 qB;

fB þ e�ikb2 fT ¼ 0:ð41Þ

Multiplying the third row of Eq. (40) with e�ikb2 , taking the sum ofthe second and third rows of Eq. (40), using conditions (41), lead to

ðDBl þ e�ikb2 DTlÞql þ ðDBB þ DTT þ e�ikb2 DTB þ eikb2 DBTÞqB ¼ 0 ð42Þ

so

qB ¼ �ðDBB þ DTT þ e�ikb2 DTB þ eikb2 DBTÞ�1ðDBl þ e�ikb2 DTlÞql: ð43Þ

Using (41) and (43), the first row of Eq. (40) becomes

f l ¼ ½Dll � ðDlB þ eikb2 DlTÞðDBB þ DTT þ e�ikb2 DTB þ eikb2 DBTÞ�1

� ðDBl þ e�ikb2 DTlÞ�ql; ð44Þ

which can be written as

f l ¼ Dlql: ð45Þ

Using the pseudo periodic conditions (34) also lead to the followingrelations between longitudinal dofs

qR ¼ eilqL;

qRB ¼ eilqLB;

qRT ¼ eiðlþkb2ÞqLB;

qLT ¼ eikb2 qLB:

ð46Þ

From the pseudo periodic conditions (46), it can be seen that allcomponents of the vector ql depend on the set of dofs defined byqr ¼ t½tqL

tqLB�. This can be expressed as

ql ¼ ðW0 þ eilW1Þqr ; ð47Þ

where the matrices W0 and W1 depend on the wavenumber k andare given by

W0 ¼

I OO OO IO OO OO eikb2 I

2666666664

3777777775W1 ¼

O OI OO OO IO eikb2 IO O

2666666664

3777777775: ð48Þ

The equilibrium conditions between adjacent cells are given by

eilfL þ fR ¼ 0

eilfLB þ fRB þ eiðl�kb2ÞfLT þ e�ikb2 fRT ¼ 0ð49Þ

that can be written as

ðeilW�0 þW�

1Þf l ¼ 0; ð50Þ

where ðÞ� denotes the operator of complex conjugate andtranspose.

Combining (45), (47) and (50), lead to

ðeilW�0 þW�

1ÞDlðW0 þ eilW1Þqr ¼ 0 ð51Þ

that can be written as

ðA0 þ eilðA1 þ A2Þ þ e2ilA3Þqr ¼ 0; ð52Þ

where

A0 ¼W�1DlW0;

A1 ¼W�0DlW0;

A2 ¼W�1DlW1;

A3 ¼W�0DlW1:

ð53Þ

The eigenvalue eil and the eigenvector qr are thus solutions of aquadratic eigenvalue problem. It is convenient to transform theproblem (52) into another linear eigenvalue problem as

D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019 3011

eil A3 OO A3

� �qreqr

� �¼

O A3

�A0 �ðA1 þ A2Þ

� �qreqr

� �ð54Þ

with ~qr ¼ eilqr .From Eqs. (44) and (45), one can notice that

tDlðkÞ ¼ Dlð�kÞ: ð55Þ

Moreover, from (48), we have

W�j ðkÞ ¼ tWjðkÞ ¼ tWjð�kÞ for j ¼ 0;1 ð56Þ

and from (53)

tA0ðkÞ ¼ A3ð�kÞ;tA1ðkÞ ¼ A1ð�kÞ;tA2ðkÞ ¼ A2ð�kÞ;tA3ðkÞ ¼ A0ð�kÞ:

ð57Þ

It can be easily shown by taking the determinant of the matrix inrelation (52) that if eilj is an eigenvalue for the wavenumberk; e�ilj is also an eigenvalue for the wavenumber �k. These repre-sent a pair of positive and negative-going waves, respectively. The2n eigensolutions of Eq. (54) can be split into two sets of nþ andn� eigensolutions with 2n ¼ nþ þ n�, which are denoted byðeilþ

j ;qþj Þ and ðeil�j ;q�j Þ respectively, with the first set such that

jeilþj j 6 1. In the case jeilþ

j j ¼ 1, the first set of positive-going wavesmust contain waves propagating in the positive direction such thatRefixqH

j frjg > 0 where fr

j is the reduced set of boundary force dofs ofleft cells on right cells and is given by

frj ¼

fL

fLB þ e�ikb2 fLT

� �¼W�

0f lj ¼W�

0DlðW0 þ eilj W1Þqj: ð58Þ

In the second set of negative-going waves, the eigenvalues eil�j are

associated with waves such that RefixqHj fr

jg < 0.With the eigenvector qj and the force component of relation

(58), we introduce the state vector

xjðkÞ ¼qjðkÞfr

j ðkÞ

" #¼

qjðkÞðA1ðkÞ þ eiljðkÞA3ðkÞÞqjðkÞ

" #: ð59Þ

In this relation qjðkÞ is the eigenvector associated to eiljðkÞ. One canalso introduce

yjð�kÞ ¼ tpjð�kÞðA2ðkÞ þ eiljðkÞA3ðkÞÞ tpjð�kÞh i

: ð60Þ

In this relation pjð�kÞ is the eigenvector associated to e�iljðkÞ sincewe have seen that e�iljðkÞ is also an eigenvalue of (52) for the wave-number �k. From relation (52) written for the eigenvector qjðkÞ,multiplying this relation by e�iljðkÞ and then on the left by tpið�kÞ,one gets

tpið�kÞðe�iljðkÞA0ðkÞ þ ðA1ðkÞ þ A2ðkÞÞ þ eiljðkÞA3ðkÞÞqjðkÞ ¼ 0:

ð61Þ

In the same way, writing relation (52) for the eigenvector pið�kÞ,taking the transpose of the relation, using relations (57) and multi-plying on the right by qjðkÞ, leads, after a global multiplication byeiliðkÞ, to the following relation:

tpið�kÞðeiliðkÞA3ðkÞ þ ðA1ðkÞ þ A2ðkÞÞ þ e�iliðkÞA0ðkÞÞqjðkÞ ¼ 0:

ð62Þ

The difference between the two precedent relations yields

ðeiliðkÞ � eiljðkÞÞtpið�kÞðA3ðkÞ � e�iliðkÞe�iljðkÞA0ðkÞÞqjðkÞ ¼ 0: ð63Þ

In the case eiliðkÞ – eiljðkÞ, we get

tpið�kÞðeiliðkÞA3ðkÞ � e�iljðkÞA0ðkÞÞqjðkÞ ¼ 0: ð64Þ

Now it is possible to compute the product yið�kÞ xjðkÞ by

yið�kÞ xjðkÞ ¼ tpið�kÞðA2ðkÞ þ eiliðkÞA3ðkÞÞqjðkÞ

þ tpið�kÞðA1ðkÞ þ eiljðkÞA3ðkÞÞqjðkÞ

¼ tpið�kÞðA2ðkÞ þ eiliðkÞA3ðkÞÞqjðkÞ � tpið�kÞðA2ðkÞ

þ e�iljðkÞA0ðkÞÞqjðkÞ ¼ didij: ð65Þ

The result of relation (64) has been used in the caseeiliðkÞ – eiljðkÞ and di is a factor depending on the eigenvector i. Thisgives orthogonality relations on the statevectors associated to theeigenvalues.

2.4. Absorbing boundary conditions

Fig. 2 presents the periodic medium near the exterior boundary.In this domain the solution is described by relation (31), yielding,respectively for the displacement and force components,

qðx1; x2Þ ¼Z p

b2

� pb2

q̂ðx1; k; x2Þeikx2 dk;

fðx1; x2Þ ¼Z p

b2

� pb2

f̂ðx1; k; x2Þeikx2 dk ð66Þ

with the force components given by relation (58). Introducing thestate vector x ¼ tðtq; tfÞ and decomposing this solution into the dif-ferent waves, we get

xðx1; x2Þ ¼Z p

b2

� pb2

x̂ðx1; k; x2Þeikx2 dk �Z p

b2

� pb2

Xj¼2n

j¼1

ajðx1; kÞxjðkÞeikx2 dk:

ð67Þ

The last relation is the approximation obtained by the finite ele-ment computation of wave solutions presented before. The condi-tion of outgoing waves means that there is no incoming wave, sothe amplitudes ajðx1; kÞ associated with incoming waves must equalzero. This condition is obtained by

y�l ð�kÞ Xj¼2n

j¼1

ajðx1; kÞxjðkÞ ¼ 0 for 1 6 l 6 n�: ð68Þ

In this relation y�l ð�kÞ are the vectors associated to the negative-going waves, given by relation (60). Using relation (65), one getsa�j ðx1; kÞ ¼ 0 for 1 6 j 6 n� for the amplitudes of the negative-goingwaves. Introducing the matrix Y with lines given by y�l leads to

Yð�kÞ x̂ðx1; k; x2Þ ¼ 0: ð69Þ

Decomposing now x̂ into its displacement and force components,doing the same thing for Yð�kÞ ¼ ½Q ð�kÞ Fð�kÞ� leads to

Q ð�kÞ q̂ðx1; k; x2Þ þ Fð�kÞ f̂ðx1; k; x2Þ ¼ 0: ð70Þ

The relation on the boundary is

f̂ðx1; k; x2Þ ¼ �F�1ð�kÞQ ð�kÞq̂ðx1; k; x2Þ ð71Þ

and then from relation (66)

fðx1; x2Þ ¼ �Z p

b2

� pb2

F�1ð�kÞQ ð�kÞq̂ðx1; k; x2Þeikx2 dk: ð72Þ

From the inverse relation (32), one also has

q̂ðx1; k; x2Þ ¼b2

2pXþ1

m2¼�1e�ikðx2þm2b2Þqðx1; x2 þm2b2Þ; ð73Þ

x

x

x

x

x

x

x

D

h

Source

x

x

x

x

x

x

θ

case a)

case b)

Fig. 5. Points used to estimate the boundary condition with sound pressurescreated by a plane wave (case a) and a point source (case b).

3012 D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019

which leads to

fðx1; x2Þ ¼ �b2

2p

Z pb2

� pb2

F�1ð�kÞQ ð�kÞ

�Xþ1

m2¼�1e�ikðx2þm2b2Þqðx1; x2 þm2b2Þdk: ð74Þ

Introducing the function

Gðx2Þ ¼ �b2

2p

Z pb2

� pb2

F�1ð�kÞQ ð�kÞe�ikx2 dk: ð75Þ

The final relation is

fðx1; x2Þ ¼Xþ1

m2¼�1Gðx2 þm2b2Þqðx1; x2 þm2b2Þ: ð76Þ

This is the impedance relation on the boundary obtained with theassumption that there is no negative-going wave. This relation isthe absorbing boundary condition we were looking for. It can becomputed from the wave vectors and the force components associ-ated with them. Relation (76) involves an infinite number of termson the boundary. This relation can also be written as

fðx1; x2Þ ¼Xþ1

m2¼�1Gðx2 þm2b2Þ

!qðx1; x2Þ

þXþ1

m2¼�1Gðx2 þm2b2Þðqðx1; x2 þm2b2Þ � qðx1; x2ÞÞ

¼Xþ1

m2¼�1Gðx2 þm2b2Þ

!qðx1; x2Þ

þ 12b2

Xþ1m2¼�1

m2b2Gðx2 þm2b2Þ !

ðqðx1; x2 þ b2Þ

� qðx1; x2 � b2ÞÞ þXþ1

m2¼�1Gðx2 þm2b2Þ½qðx1; x2 þm2b2Þ

� qðx1; x2Þ �12

m2ðqðx1; x2 þ b2Þ � qðx1; x2 � b2ÞÞ�: ð77Þ

If q is slowly varying the last term should be small and for practicalpurposes we will use the approximate relations at various ordersgiven by

fðx1; x2Þ � G0qðx1; x2Þ þG1

2b2ðqðx1; x2 þ b2Þ � qðx1; x2 � b2ÞÞ

þ G2

2b22

ðqðx1; x2 þ b2Þ þ qðx1; x2 � b2Þ � 2qðx1; x2ÞÞ þ

ð78Þ

with

G0 ¼Xþ1

m2¼�1Gðx2 þm2b2Þ ¼ �ðF�1Q Þð0Þ;

G1 ¼Xþ1

m2¼�1m2b2Gðx2 þm2b2Þ ¼ iðF�1Q Þ0ð0Þ;

G2 ¼Xþ1

m2¼�1ðm2b2Þ2Gðx2 þm2b2Þ ¼ ðF�1Q Þ00ð0Þ:

ð79Þ

Relation (78) involves a finite number of nodes around the pointwhere the relation is written. It depends on the number of nodeschosen to approximate the boundary condition. This number canbe 1 for a crude approximation involving only one node or can belarger. For a very large number of nodes, the condition tends to-wards the true absorbing condition for a half-plane in the periodicmedia given by (76). Up to now everything has been written forperiodic media but it is clear that homogeneous media are also

periodic media and so all that has been said applies also to homo-geneous media. The condition (78) can be seen as a generalizationof the Taylor approximation boundary condition proposed byEngquist and Majda [26]. But, while the boundary conditions in[26] were obtained by approximation of the exact continuous rela-tions for specific problems, they are obtained here directly and withgeneral applicability from the discretized equations.

3. Simple examples

3.1. Estimation of the accuracy

In this section we try to estimate the quality of the proposedboundary condition compared with known relations for the simplecase of the two-dimensional acoustics. Consider first a plane waveincident on the plane y ¼ 0 at an angle h with the normal to theplane. Let us define points at a horizontal distance D from the ori-gin and with a vertical spacing h, see Fig. 5. The sound pressure atpoint ðD; lhÞ is given by

pla ¼ eiKðcos hDþsin hlhÞ; ð80Þ

where K ¼ x=c is the wavenumber and c is the sound velocity. Theanalytical force at the same points is the normal derivative in direc-tion 1 given by

f la ¼ iK cos heiKðcos hDþsin hlhÞ: ð81Þ

For a point source at origin, the pressure is solution of

Dpþ K2p ¼ �dðrÞ; ð82Þ

where r is the distance from the origin and dðÞ is the Dirac function.The solution of this equation for the time dependence e�ixt is givenby

GðrÞ ¼ i4

H0ðKrÞ; ð83Þ

where H0 is the Hankel function of zero order and first type. Theanalytical solution at each point l is

pla ¼

i4

H0 KffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 þ ðlhÞ2

q� �ð84Þ

and the analytical force at the same points is the normal derivative(in direction 1) given by

D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019 3013

f la ¼ �

iKD

4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 þ ðlhÞ2

q H1 KffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 þ ðlhÞ2

q� �: ð85Þ

The absorbing boundary condition described in the precedent sec-tion will allow to compute numerical forces f l

n at a node from theknowledge of pl

a. If the boundary condition was perfect one wouldhave f l

n ¼ f la but the proposed condition is approximate and one only

has f ln � f l

a. The error can be estimated by

e ¼ jfln � f l

ajjf l

aj: ð86Þ

The next step is to compute f ln from the method proposed in this pa-

per and the error by relation (86) to estimate the quality of theabsorbing condition.

3.2. Acoustic element

Consider the rectangular four nodes acoustic element of sizeb1 � b2. The elementary stiffness and mass matrices are given by

K ¼ 16b1b2

ð2b22 þ 2b2

1Þ ð�2b22 þ b2

1Þ ð�b22 � b2

1Þ ðb22 � 2b2

1Þð�2b2

2 þ b21Þ ð2b2

2 þ 2b21Þ ðb2

2 � 2b21Þ ð�b2

2 � b21Þ

ð�b22 � b2

1Þ ðb22 � 2b2

1Þ ð2b22 þ 2b2

1Þ ð�2b22 þ b2

1Þðb2

2 � 2b21Þ ð�b2

2 � b21Þ ð�2b2

2 þ b21Þ ð2b2

2 þ 2b21Þ

2666664

3777775;ð87Þ

M ¼ b1b2

36c2

4 2 1 22 4 2 11 2 4 22 1 2 4

2666437775 ð88Þ

and the dynamic stiffness matrix can then be determined byD ¼ K�x2M.

It can be noted that the reduced set of displacement dofs qr con-tains only qLB. Then, the matrices W0 and W1 have the followingforms:

W0 ¼ t½1 0 0 eikb2 �; W1 ¼ t½0 1 eikb2 0�: ð89Þ

The terms Aj in Eq. (53) are given by

A0ðkÞ ¼W�1ðkÞDlðkÞW0ðkÞ

¼ � 118

12b2

b1� 6

b1

b2þ 2K2b1b2

�þ 6

b2

b1þ 6

b1

b2þ K2b1b2

� �cosðkb2Þ

�;

A1ðkÞ ¼W�0ðkÞDlðkÞW0ðkÞ

¼ �19�6

b2

b1� 6

b1

b2þ 2K2b1b2

�þ �3

b2

b1þ 6

b1

b2þ K2b1b2

� �cosðkb2Þ

�;

A2ðkÞ ¼W�1ðkÞDlðkÞW1ðkÞ

¼ �19�6

b2

b1� 6

b1

b2þ 2K2b1b2

�þ �3

b2

b1þ 6

b1

b2þ K2b1b2

� �cosðkb2Þ

�¼ A1ðkÞ;

A3ðkÞ ¼W�0ðkÞDlðkÞW1ðkÞ

¼ � 118

12b2

b1� 6

b1

b2þ 2K2b1b2

�þ 6

b2

b1þ 6

b1

b2þ K2b1b2

� �cosðkb2Þ

�¼ A0ðkÞ:

ð90Þ

The eigensolutions of the spectral problem (52) are then deter-mined and are given by

eil ¼ 12A3

�ðA1 þ A2Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA1 þ A2Þ2 � 4A2

0

q� �; ð91Þ

e�il ¼ 12A3

�ðA1 þ A2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA1 þ A2Þ2 � 4A2

0

q� �: ð92Þ

The signs are selected as in Section 2.3. As there is only one dof inthis case, one has nþ ¼ 1 and after normalization one can chooseq1ðkÞ ¼ 1. From relation (60), taking also pjð�kÞ ¼ 1, yields

yð�kÞ ¼ A2 þ eilA3 1 �

¼ A32A3

�ðA1 þ A2Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA1 þ A2Þ2 � 4A2

0

q� �þ A2 1

� �¼ � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA1 þ A2Þ2 � 4A2

0

q1

h ithis gives, with the notation of relation (70),

Qð�kÞ ¼ �12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA1 þ A2Þ2 � 4A2

0

q;

Fð�kÞ ¼ 1:ð93Þ

Near k ¼ 0 (this means in fact near the normal incidence) one hasthe development

A0ðkÞ ¼� b2b1� 1

6 K2b1b2þ 136 6 b2

b1þ6 b1

b2þK2b1b2

� �ðkb2Þ2þOððkb2Þ4Þ;

A1ðkÞ ¼ b2b1� 1

3 K2b1b2þ 118 �3 b2

b1þ6 b1

b2þK2b1b2

� �ðkb2Þ2þOððkb2Þ4Þ;

A2ðkÞ ¼ A1ðkÞ;

A3ðkÞ ¼ A0ðkÞ

8>>>>>>>><>>>>>>>>:ð94Þ

and this leads to

Qð�kÞ ¼ �iKb2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

12ðKb1Þ2

r� 1� 1

21þ ðKb2Þ2

3 � ðKb1Þ26 � ðK

2b1b2Þ236

1� ðKb1Þ212

k2

K2

!þ Oððkb2Þ4Þ:

ð95Þ

From relation (58) one also has

f rðkÞ ¼W�0DlðW0 þ eilW1Þ ¼ A1 þ eilA3 ¼ Qð�kÞ: ð96Þ

Following the rule that the positive waves are such thatRefixqH

j frjg > 0, one has to choose the minus sign in relation (93).

For the case Kb1 � 1 and k� K , one has the approximation:

f ð0Þ ’ �iKb2: ð97Þ

The power across the boundary is thus

P ¼ 12

Reðfleft!rightv�Þ ¼12

Reðf rð0Þð�ixÞ�Þ ¼ 12

Kb2x > 0: ð98Þ

For the second order approximation one has

G0 ¼ �ðF�1QÞð0Þ ¼ iKb2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1

12ðKb1Þ2

r� iKb2;

G1 ¼ iðF�1QÞ0ð0Þ ¼ 0;

G2 ¼ ðF�1QÞ00ð0Þ ¼ ib2

K1þ ðKb2Þ2

3 � ðKb1Þ26 � ðK

2b1b2Þ236ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 112 ðKb1Þ2

q � ib2

K:

ð99Þ

The relation between forces and displacements dofs on the bound-ary of the element is thus given by using (78)

2

3014 D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019

f ðx1; x2Þ � iKb2qðx1; x2Þ þi

2Kb2ðqðx1; x2 þ b2Þ

þ qðx1; x2 � b2Þ � 2qðx1; x2ÞÞ: ð100Þ

At order 0 one finds the classical approximation of the radiatingboundary condition. The factor b2 is present because the force is cal-culated over an edge of an element of length b2.

We compare four solutions in the following

(1) The zero order solution with the numerical computation ofQð0Þ leading to the relation f l

n ¼ G0pla.

(2) The zero order solution with the simplified computation ofQð0Þ leading to the relation f l

n ¼ iKb2pla.

(3) The second order solution with the numerical computation ofG2 ¼ ðF�1QÞ00ð0Þ � ððF�1QÞðdÞ þ ðF�1QÞð�dÞ � 2ðF�1QÞð0ÞÞ= d2

leading to the relation f ln ¼ G0pl

a þG2

2b22ðplþ1

a þ pl�1a � 2pl

aÞ.(4) The second order solution with the simplify computation of

Qð0Þ and ðF�1QÞ00ð0Þ leading to the relation f ln ¼ iKb2pl

aþi

2Kb2ðplþ1

a þ pl�1a � 2pl

aÞ.

3.3. Example

Here we compute the error of relation (86) for different cases asshown in Fig. 5. Case (a) is for a sound pressure created by a plane

0 100 200 300 400 500 600 700 800 900 1000−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

frequency (Hz)

log1

0(er

ror)

zero ordersimplified zero ordersecond ordersimplified second order

0 100 200 300 400 500 600 700 800 900 1000−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

frequency (Hz)

log1

0(er

ror)

zero ordersimplified zero ordersecond ordersimplified second order

a

b

Fig. 6. Error versus the frequency for a plane wave at 10� for (a) an element size0:01 m� 0:01 m and (b) an element size 0:05 m� 0:05 m.

wave while in case (b) the sound pressure is created by a pointsource. The acoustic element used for the computation of theboundary condition can be of size b1 � b2 ¼ 0:01 m� 0:01 m orb1 � b2 ¼ 0:05 m� 0:05 m. The sound velocity is c ¼ 340 m=s andthe distance between the points is h ¼ b2.

The first example is for a pressure created by a plane wave atthe incidence angle h ¼ 10�. The error for the four cases listed inthe precedent section are plotted in Fig. 6. The error is the samefor any point on the vertical axis. It can be observed that the secondorder relations are much better than the first ones as expected. Thecomparison of the two sizes for the acoustic element shows thatthe size 0:05 m� 0:05 m can reduce the accuracy of the solutionfor high frequencies and the second order condition. In these casesit is better to use elements with small sizes.

In Fig. 7 the error is plotted versus the angle of incidence of theplane wave. An acoustic element of size b1 � b2 ¼ 0:01 m� 0:01 mhas been used. The solution is accurate (error less than 1%) for an-gles up to 10� for a zero order condition and up to 30� for a secondorder condition. This clearly shows that second order conditionsare much better for waves at oblique incidence.

Finally the error is plotted versus the distance along the y axis inFig. 8 for a pressure created by a point source at distance D ¼ 1 m

0 10 20 30 40 50 60 70 80 90−7

−6

−5

−4

−3

−2

−1

0

1

angle

log1

0(er

ror)

zero ordersimplified zero ordersecond ordersimplified second order

Fig. 7. Error versus the angle of incidence.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−3

−2.5

−2

−1.5

−1

−0.5

Distance along y (m)

log1

0(er

ror)

zero ordersimplified zero ordersecond ordersimplified second order

Fig. 8. Error versus the distance for a point source.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

X (m)

Y (m

)

Configuration of the 2D domain

Fig. 9. Example of finite element domain with a point source in its center.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

Gre

en’s

func

tion

at (0

.3,0

), R

eal p

art

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Frequency (Hz)

Gre

en’s

func

tion

at (0

.3,0

), Im

agin

ary

part

a

b

Fig. 10. Comparison of analytical — and numerical Green’s functions, with thepresent method at order 0 –– and at order 2 for the 2D acoustics at point (0.3,0)with L ¼ 1 m and b ¼ 0:025 m: (a) real part, (b) imaginary part.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)

Gre

en’s

func

tion

at (0

.3,0

.3),

Rea

l par

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

Gre

en’s

func

tion

at (0

.3,0

.3),

Imag

inar

y pa

rt

a

b

Fig. 11. Comparison of analytical — and numerical Green’s functions, with thepresent method at order 0 –– and at order 2 for the 2D acoustics at point ð0:3; 0:3Þwith L ¼ 1 m and b ¼ 0:025 m: (a) real part, (b) imaginary part.

D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019 3015

on the x axis and at the frequency 1000 Hz. The reduction in accu-racy can be observed as we move along the y axis leading to greaterincidence angles in agreement with the precedent observation on

the plane waves. All these points confirm the accuracy of the meth-od proposed here.

4. Finite element examples

4.1. Acoustics

In this section we use the precedent boundary condition tosolve some finite element problems for different frequencies andmesh densities. We consider first a finite element acoustic problemon a square domain with a point source excitation in its center. Atwo-dimensional domain of size 1 m� 1 m is generated by Ansys.The domain and an example of mesh are presented in Fig. 9. Thesize of the acoustic element is b ¼ 0:025 m leading to 40� 40elements for the whole domain. In each case, only square elementsare used for the mesh. The sound velocity is c ¼ 340 m=s.Only mesh information, stiffness and mass matrices are pickedout and are then introduced into Matlab to get the results pre-sented below. The procedure is done over the frequency band½0; 2000 Hz�.

Numerical Green’s functions are calculated for zero and secondorder boundary conditions. The excitation is at point (0,0) and the

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

rela

tive

erro

r in

Gre

en’s

func

tion

at (0

.3,0

.3)

Fig. 12. Comparison of relative errors for 0 order — and second order –– boundaryconditions for the 2D acoustics at point ð0:3; 0:3Þ with L ¼ 1 m and b ¼ 0:025 m.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

Gre

en’s

func

tion

at (0

.3,0

.3),

Rea

l par

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

Gre

en’s

func

tion

at (0

.3,0

.3),

Imag

inar

y pa

rt

a

b

Fig. 13. Comparison of analytical — and numerical Green’s functions, with thepresent method at order 0 –– and at order 2 for the 2D acoustics at point (0.3,0.3)with L ¼ 1 m and b ¼ 0:0125 m: (a) real part, (b) imaginary part.

3016 D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019

analytical solution in infinite space is given by formula (84). TheGreen’s functions are presented in Fig. 10 for a point at (0.3,0) onthe horizontal axis and in Fig. 11 for a point at (0.3,0.3) alongthe diagonal. Good agreements between the two types of absorb-ing boundary conditions and the analytical solution can be ob-served. Both boundary conditions fail at low frequencies becausethe size of the domain is too small compared to the wavelength.Similarly the error for high frequencies are of same orders for bothboundary conditions. For intermediate frequencies the error islower for the second order boundary condition. This is more clearlyseen in Fig. 12 where the relative error for the point (0.3,0.3) isplotted versus the frequency.

The same results are presented in Fig. 13 for the point ð0:3; 0:3Þand a mesh density of 80� 80 elements. The solution is clearlymuch better at high frequencies meaning that the errors seen inFigs. 10 and 11 can be explained by elements too large for thesefrequencies and not by the quality of the boundary conditions. InFig. 14 the domain is now 2 m� 2 m with 80� 80 elements, sowith elements of the same size as for Figs. 10 and 11. Results areplotted for the point (0.3,0.3). Now the improvement is clearlyseen for low frequencies while there is no difference for highfrequencies.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

Gre

en’s

func

tion

at (0

.3,0

.3),

Rea

l par

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

Gre

en’s

func

tion

at (0

.3,0

.3),

Imag

inar

y pa

rt

a

b

Fig. 14. Comparison of analytical — and numerical Green’s functions, with thepresent method at order 0 –– and at order 2 for the 2D acoustics at pointð0:3; 0:3Þ with L ¼ 2 m and b ¼ 0:025 m: (a) real part, (b) imaginary part.

0 0.5 1 1.5 2x 104

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10−12

Frequency (Hz)

Gre

en’s

func

tion

at (0

.5,0

), R

eal p

art Analytic

Numeric: L=1m, b=0.025mNumeric: L=1m, b=0.0125mNumeric: L=1m, b=0.00625m

0 0.5 1 1.5 2x 104

−1

−0.5

0

0.5

1

1.5

2

2.5x 10−12

Frequency (Hz)

Gre

en’s

func

tion

at (0

.5,0

), Im

agin

ary

part

AnalyticNumeric: L=1m, b=0.025mNumeric: L=1m, b=0.0125mNumeric: L=1m, b=0.00625m

0 0.5 1 1.5 2x 104

−1

−0.5

0

0.5

1

1.5x 10−12

Frequency (Hz)

Gre

en’s

func

tion

at (0

,0.5

), R

eal p

art Analytic

Numeric: L=1m, b=0.025mNumeric: L=1m, b=0.0125mNumeric: L=1m, b=0.00625m

0 0.5 1 1.5 2x 104

−5

0

5

10

15

20

25x 10−13

Frequency (Hz)

Gre

en’s

func

tion

at (0

,0.5

), Im

agin

ary

part

AnalyticNumeric: L=1m, b=0.025mNumeric: L=1m, b=0.0125mNumeric: L=1m, b=0.00625m

a

c d

b

Fig. 15. Comparison of analytical and numerical Green’s functions for the 2D elasticity with different sizes of elements: (a) real part at (0,0.5), (b) imaginary part at (0,0.5), (c)real part at (0.5,0), (d) imaginary part at (0.5,0).

0 0.5 1 1.5 2x 104

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10−12

Frequency (Hz)

Gre

en’s

func

tion

at (0

.5,0

), R

eal p

art Analytic

Numeric: L=1m, b=0.025mNumeric: L=2m, b=0.025mNumeric: L=4m, b=0.025m

0 0.5 1 1.5 2x 104

−1

−0.5

0

0.5

1

1.5

2

x 10−12

Frequency (Hz)

Gre

en’s

func

tion

at (0

.5,0

), Im

agin

ary

part

AnalyticNumeric: L=1m, b=0.025mNumeric: L=2m, b=0.025mNumeric: L=4m, b=0.025m

0 0.5 1 1.5 2x 104

−1

−0.5

0

0.5

1

1.5

2

x 10−12

Frequency (Hz)

Gre

en’s

func

tion

at (0

,0.5

), R

eal p

art Analytic

Numeric: L=1m, b=0.025mNumeric: L=2m, b=0.025mNumeric: L=4m, b=0.025m

0 0.5 1 1.5 2x 104

−5

0

5

10

15

20

x 10−13

Frequency (Hz)

Gre

en’s

func

tion

at (0

,0.5

), Im

agin

ary

part

AnalyticNumeric: L=1m, b=0.025mNumeric: L=2m, b=0.025mNumeric: L=4m, b=0.025m

a

c d

b

Fig. 16. Comparison of analytical and numerical Green’s functions for the 2D elasticity with different sizes of the domain: (a) real part at (0,0.5), (b) imaginary part at (0,0.5),(c) real part at (0.5,0), (d) imaginary part at (0.5,0).

D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019 3017

3018 D. Duhamel, T.-M. Nguyen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3006–3019

4.2. Two-dimensional elastodynamics

The analytical solution in direction en of the two-dimensionalelastodynamics case when submitted to a unit force at origin indirection em, is given by

GnmðrÞ ¼i

4lAdnm þ B

xnxm

r2

h ið101Þ

with

A ¼ H0ðKT rÞ � 1KT r

H1ðKT rÞ � bH1ðKLrÞ½ �;

B ¼ �2Aþ ½H0ðKT rÞ þ b2H0ðKLrÞ�;

where H0 and H1 are the Hankel functions of first type, of orderszero and one respectively. The wavenumbers are KL ¼ x=cL andKT ¼ x=cT for the longitudinal and transverse waves respectively.The velocities cL; cT and the ratio between them b are given by,

b ¼ cTcL

c2L ¼

kþ2lq

c2T ¼

lq

8>><>>: with k ¼ Emð1þ mÞð1� 2mÞ and l ¼ E

2ð1þ mÞ :

ð102Þ

In this example, the same global meshes as for the acoustic case areused. The boundary condition is computed from the square fournodes elements. The material is steel with E ¼ 2:1011 Pa;m ¼ 0:3and q ¼ 7800 kg=m3 and plane strain conditions are used in thecomputation. The sizes of the periodic cell can beb ¼ 0:025 m;0:0125 m or 0:00625 m.

In Fig. 15, numerical solutions are compared with analyticalsolutions for the point (0.5,0) and (0,0.5) for different sizes ofthe cell. The curves represent the real and imaginary parts of thefirst component of the displacement for an excitation at origin indirection 1. The same remarks as the previous examples can bemade: 1Þ a denser mesh leads to lower errors over the high fre-quency band ½8:103 Hz;20:103 Hz�; 2 for the low frequency band½0; 8:103 Hz�, numerical results are different from the analyticalsolutions due to the finite size of the domain ðL ¼ 1 mÞ.

In Fig. 16, the results for these two points are presented whenthe size of the domain is increased successively withL ¼ 1 m;2 m and 4 m. In this case, when the size of the cell is fixedto b ¼ 0:025 m, a larger domain leads to lower errors over the lowfrequency band ½0; 8:103 Hz�. The results are the same as

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 104

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency (Hz)

rela

tive

erro

r in

Gre

en’s

func

tion

at (0

.5,0

)

Fig. 17. Comparison of relative errors for 0 order — and second order –– boundaryconditions for the 2D elastodynamics at point (0.5,0) with L ¼ 1 m and b ¼ 0:025 m.

previously in the high frequency band because in this domainthe precision depends on the size of the elements and not on thesize of the global domain. Some improvements are however seenfor intermediate frequencies.

In Fig. 17 the error for boundary conditions of zero and secondorders are plotted versus the frequency. The second order condi-tion is much more accurate for intermediate frequencies as forthe acoustic case.

5. Conclusion

In this paper, a method to determine absorbing boundary con-ditions for two-dimensional periodic media has been presented.It works directly on the discretized equations. The boundary condi-tion is first obtained as a global impedance relation and is thenlocalized into boundary conditions of various orders. In two exam-ples, good agreements were observed when compared with analyt-ical solutions.

In any case, the proposed method is efficient because it requiresonly the discrete dynamic matrices which can be obtained by anystandard finite element software. This method could be used formedia with more complex behaviors than those presented in theprecedent examples.

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