Application of the perfectly matched absorbing layer model

GEOPHYSICS, VOL. 66, NO. 1 (JANUARY-FEBRUARY 2001); P. 294–307, 13 FIGS.

Application of the perfectly matched absorbinglayer model to the linear elastodynamic problemin anisotropic heterogeneous media

Francis Collino and Chrysoula Tsogka∗

ABSTRACT

We present and analyze a perfectly matched, absorb-ing layer model for the velocity-stress formulation ofelastodynamics. The principal idea of this method con-sists of introducing an absorbing layer in which wedecompose each component of the unknown into twoauxiliary components: a component orthogonal to theboundary and a component parallel to it. A system ofequations governing these new unknowns then is con-structed. A damping term finally is introduced for thecomponent orthogonal to the boundary. This layer modelhas the property of generating no reflection at the inter-face between the free medium and the artificial absorb-ing medium. In practice, both the boundary conditionintroduced at the outer boundary of the layer and the dis-persion resulting from the numerical scheme produce asmall reflection which can be controlled even with verythin layers. As we will show with several experiments,this model gives very satisfactory results; namely, the re-flection coefficient, even in the case of heterogeneous,anisotropic media, is about 1% for a layer thickness offive space discretization steps.

INTRODUCTION

The simulation of waves by finite-difference or finite-element methods in unbounded domains requires a specialtreatment for the boundaries of the necessarily truncated com-putational domain. Two solutions have been proposed for thispurpose: absorbing boundary conditions (ABCs) and absorb-ing layers. ABCs were introduced by B. Engquist and A. Majda(1977) for the acoustic wave equation. They consist of introduc-ing some suitable local boundary conditions that simulate theoutgoing nature of the waves impinging on the borders. Thismethod works particularly well for absorbing waves nearly nor-mally incident to the artificial boundaries. For waves traveling

Manuscript received by the Editor August 26, 1998; revised manuscript received July 28, 1999.∗INRIA, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France. E-mail: [email protected]© 2001 Society of Exploration Geophysicists. All rights reserved.

obliquely to the boundary, higher-order ABCs must be used toachieve acceptable accuracy (Hagstrom, 1997).

For elastic waves, the situation is more complex. First, thetransparent condition, i.e., the exact condition relating normalstress and displacement on a line for outgoing waves, is nolonger a scalar but a matrix integro-differential relation. Itsapproximation by partial differential equations, which is theusual way to construct ABCs, leads to a very complex systemof equations, especially for higher-order methods. The stabilityof the coupled problem composed of the elastodynamic systemcompleted by these artificial conditions is then very difficultto analyze, and the situation is even more intricate when dis-cretization is considered (Chalindar, 1987). To overcome thesedifficulties, Higdon (1991, 1992) proposed combining severalfirst-order boundary conditions designed for the wave equa-tion, each of them being associated with either the pressure-wave velocity or the shear-wave velocity. These conditions aretheoretically stable (Higdon, 1990), relatively easy to imple-ment, and efficient for waves traveling in a direction close to thenormal of the artificial boundaries. They also can be adapted tothe case of surface waves (Sochacki et al., 1987). However, forother directions of propagation, important spurious reflectionsmay occur. Some authors, Peng and Toksoz (1995), for instance,have proposed optimizing the coefficients of Higdon’s methodto make these reflections decrease. However, numerical exper-iments still show relatively strong spurious reflections in somesituations and stability problems when higher-order numericalschemes are used (Simone and Hestholm, 1998).

Layer models are an alternative to ABCs. The idea is tosurround the domain of interest by some artificial absorbinglayers in which waves are trapped and attenuated. For elas-tic waves, several models have been proposed. For instance,Sochacki et al. (1987) suggest adding inside the layers someattenuation term, proportional to the first time derivative ofthe displacement to the elastodynamic equations. This tech-nique, inspired by physics, is quite delicate in practice. Themain difficulty is that when entering the layers, the wave “sees”the change in impedance of the medium and then is reflected

294

PML for Elastodynamics 295

artificially into the domain of interest. The use of smooth andnot too high attenuation profiles allows us to weaken the dif-ficulty but requires the use of thick layers (Israeli and Orszag,1981).

In this paper, we propose adapting a layer technique intro-duced by Berenger (1994) for Maxwell’s equations and that isnow the most widely used method for the simulation of elec-tromagnetic waves in nonbounded domains (Berenger, 1996,1997; Zhao and Cangellaris, 1996; Teixeira and Chew, 1998;Turkel and Yefet, 1998). This technique consists of designingan absorbing layer called a perfectly matched layer (PML) thatpossesses the property of generating no reflection at the in-terface between the free medium and the artificial absorbingmedium. This property allows us to use a very high damping pa-rameter inside the layer and consequently a small layer widthwhile achieving a quasi-perfect absorption of the waves. To ourknowledge, this method previously has been used only once forelastic waves simulation. In Hastings et al. (1996), the authorspropose the use of PML for the compressional and shear po-tentials formulation. In our paper, the PMLs are incorporatedinto the stress-velocity formulation.

The present paper is organized as follows. In section two, weconstruct the PML model in the general case of an evolutionproblem. It is based on an interpretation of the PML model ofBerenger (1994) as a change of variable in the complex plane[compare Rappaport (1995), Collino (1996)]. In section three,we apply the previous model to the velocity-stress formulationfor elastodynamics and we study the properties of the con-tinuous PML model via a plane-wave analysis. To show thegenerality of the PML model, two numerical schemes are pre-sented, in section four. The first is the classical Virieux finite-differences scheme (Virirux, 1986), and the second is a mixedfinite-element scheme (Becache et al., 1997) which allows usto consider anisotropic media as well. Finally, in section five,we present several numerical results which show the efficiencyof this model even in the case of heterogeneous, anisotropicelastic media and their superiority when compared with theclassical ABC method.

PML MODEL FOR A GENERAL EVOLUTION PROBLEM

We will present in this section the basic principles of thePML model in the general case of an evolution problem. Thiswill allow us to generalize it to other models of wave propa-gation. Consider a general evolution problem of the followingform

(a) ∂tv − A∂xv − B∂yv = 0,(1)

(b) v(t = 0) = v0,

where, in the general case, v is a m-vector, A is an m × m ma-trix, x ∈ R, and y ∈ R

n−1. To simplify the presentation, in whatfollows, we consider the scalar case with m = 1 and n = 2. More-over, we assume that the initial condition v0 is zero on the righthalf-space. We would like to replace problem (1) by an equiv-alent one posed in the left half-space. The basic principle ofthe PML model is to couple the equation in the left half-spacewith an equation in the right half-space such that there is noreflection at the interface, and that the wave decreases expo-nentially inside the right half-space. Thus, we first introducethe following system,

∂tv‖ − B∂yv = 0,

(2)∂tv

⊥ − A∂xv = 0,

with v = v‖ + v⊥, and where superscript ‖ means that we keeponly the derivative parallel to the interface, i.e., the y-derivative(superscript ⊥ means that only the x-derivative is considered).It is easy to see that system (2) implies (1)-(a). Second, we in-troduce a function d(x), zero on the left half-space and positiveon the right-half space. This function will play the role of thedamping factor. We define now a new wave, u, solution of thefollowing system,

∂t u⊥ + d(x)u⊥ − A∂x u = 0,

∂t u‖ − B∂yu = 0,

u(t = 0) = v0,

(3)

with u = u‖ + u⊥. It is easy to see that u satisfies the same systemof equations as v in the left half-space. If we look for time-harmonic solutions of system (1) with angular frequency ω, weget

iωv − A∂x v − B∂y v = 0.

In the same way, the time-harmonic solutions of system (3)satisfy

(iω + d(x))u⊥ − A∂x u = 0,

iωu‖ − B∂y u = 0,

with u = u‖ + u⊥ or, equivalently,

iωu⊥ − iω

iω + d(x)A∂x u = 0,

iωu‖ − B∂y u = 0,

with u = u‖ + u⊥. We can remark now that the PML modelconsists of the simple substitution

∂x → ∂x = iω

iω + d(x)∂x ,

which implies the complex change of variables

x(x) = x − i

ω

∫ x

0d(s) ds.

We now can study the properties of this model using a plane-wave analysis. We seek particular solutions of system (1) in thefollowing form,

v = v0 exp[−i(kx x + ky y − ωt)],(4)

k = (kx , ky) ∈ R × R, ω ∈ R.

Formula (4) is a plane wave propagating in the direction k withthe phase velocity ω/|k|. To be a solution to problem (1), v hasto satisfy the following relation:

iv0ω − iAv0kx − iBv0ky = 0. (5)

In the same way, we seek particular solutions of system (3) inthe following form

u‖ = a‖ exp[−i(kx x(x) + ky y − ωt)],

u⊥ = a⊥ exp[−i(kx x(x) + ky y − ωt)],

u = u‖ + u⊥.

296 Collino and Tsogka

To be a solution to problem (3), u‖ and u⊥ have to satisfy thefollowing relations:

ia‖ω − iB(a⊥ + a‖)ky = 0,(6)

(iω + d(x))a⊥ − i

(1 − id(x)

ω

)A(a⊥ + a‖)kx = 0

or, equivalently,

ia‖ω − i B(a⊥ + a‖)ky = 0,(7)

ia⊥ω − i A(a⊥ + a‖)kx = 0.

Adding the two equalities of (7) gives

i(a⊥ + a‖)ω − iA(a⊥ + a‖)kx − iB(a⊥ + a‖)ky = 0. (8)

We can remark now that if we chose a⊥ and a‖ such that

a⊥ + a‖ = v0,

equation (8) becomes the same as (5). Moreover, from (7) weget

a‖ = Bv0ky

ω, a⊥ = Av0

kx

ω, (9)

which gives the plane-wave solution of (3). We then have thefollowing property: The plane wave solution of system (3) canbe written in the form

u = v0 exp[−i(kx x + ky y − ωt)] exp[−kx

ω

∫ x

0d(s) ds

],

and satisfies:

1) u ≡ v in the left half-space, x ≤ 0, which means that wehave no reflection at the interface; the layer model ismatched perfectly.

2) u is damped in the right half-space,3) the damping coefficient in the absorbing half-space is

αd = ‖u(x)‖‖v(x)‖ = exp

[−kx

ω

∫ x

0d(s) ds

]. (10)

Relation (10) implies that the waves decrease exponentiallyin the absorbing half-space. The damping coefficient (αd ) de-pends on the direction of propagation of the wave. More pre-cisely, αd decreases very quickly for a wave propagating nor-mally to the interface and decreases more and more slowlyas the direction of propagation approaches the parallel to theinterface.

PML MODEL FOR ELASTODYNAMICS

In this section, we present the PML model for the continu-ous elastodynamic problem. As we have seen in section two, weknow how to construct a PML model for an evolution problemof the form (1). Thus, to apply the technique described in theprevious section to elastodynamics, we need to write the elas-todynamic problem in the form of system (1). This can be doneeasily once we consider the mixed velocity-stress formulationfor elastodynamics.

The elastodynamic problem

We consider the 2-D elastodynamic problem written asa first-order hyperbolic system, the so called velocity-stresssystem

�∂tv − div σ = 0,(11)

A∂tσ − ε(v) = 0,

together with initial conditions. We suppose, as in the previ-ous section, that the initial condition is supported in the lefthalf-space. In equation (11), v = (vx , vy) denotes the velocity,σ the stress tensor, and � = �(x) the density. If u = (ux , uy) isthe displacement, then v = ∂t u. We denote ε(u) the linearizedstrain tensor, i.e.,

εi j (u) = 12

(∂ j ui + ∂i u j ).

The stress tensor is related to the strain tensor by the general-ized Hooke’s law

σ = σ (u)(x, t) = C(x)ε(u)(x, t),

where stiffness C(x) is a 4 × 4 positive tensor having the clas-sical properties of symmetry (Auld, 1973). In system (11), weused the compliance tensor

A = A(x) = C−1(x).

Finally, we identify the stress tensor σ with the following vector(still denoted by σ )

σ = [σ1, σ2, σ3]T ,

σ1 = σxx ; σ2 = σyy; σ3 = σxy .

We can write (11) in the following matrix form

�∂tv = D‖∂yσ + D⊥∂xσ in �,

A∂tσ = E‖∂yv + E⊥∂xv in � ,

with

D‖ =[

0 0 1

0 1 0

], D⊥ =

[1 0 0

0 0 1

],

E‖ =[

0 0 1

0 1 0

]T

, E⊥ =[

1 0 0

0 0 1

]T

.

We can apply now the same technique as in section two, and weget the following system in the perfectly matched layer(x > 0):

v = v‖ + v⊥; σ = σ ‖ + σ⊥,

�∂tv‖ = D‖∂yσ ; A∂tσ

‖ = E‖∂yv,

�∂tv⊥ + d(x)v⊥ = D⊥∂xσ ; Aσ⊥ = E⊥∂xv,

where d(x) denotes the damping factor. In a homogeneous,isotropic elastic medium, the matrix C(x) depends on the Lamecoefficients (λ, µ) of the medium. In this case, the PML model


can be written in the following form:

v = v‖ + v⊥; σ = σ ‖ + σ⊥,

�(∂t + d(x))v⊥x = ∂xσxx ; �∂tv

‖x = ∂yσxy,

�(∂t + d(x))v⊥y = ∂xσxy; �∂tv

‖y = ∂yσyy,

(∂t + d(x))σ⊥xx = (λ + 2µ)∂xvx ; ∂tσ

‖xx = λ∂yvy,

(∂t + d(x))σ⊥yy = λ∂xvx ; ∂tσ

‖yy = (λ + 2µ)∂yvy,

(∂t + d(x))σ⊥xy = µ∂xvy; ∂tσ

‖xy = µ∂yvx .

(12)

Plane-wave analysis

Infinite absorbing layer.—In the case of a homogeneous,isotropic elastic medium, following the same technique as insection two, we can show that the displacement plane waves,U j , j = p, s, corresponding to solutions of system (11), can bewritten in the following form,

Up = Apdp exp[

iωVp

(t − cos(θ)x + sin(θ)y

Vp

)],

(13)

Us = Asds exp[

iωVs

(t − − sin(θ)x + cos(θ)y

Vs

)],

where Vp =√(λ + 2µ)/� is the pressure-wave velocity, Vs =√µ/� is the shear-wave velocity, θ gives the direction of wave

propagation, d j , j = p, s defines the direction of particle mo-tion, and A j , j = p, s is the amplitude of the wave. We can re-mark then that the displacement plane waves, U j (for j = p, s),solutions of system (12), can be written as

U p = Apdp exp[iωVp

(t − cos(θ)x p + sin(θ)y

Vp

)],

(14)

U s = Asds exp[iωVs

(t − − sin(θ)xs + cos(θ)y

Vs

)],

where we simply substituted x j , j = p, s for x in (13), withx j , j = p, s defined in the same way as in section two (ω re-placed by ωVj ),

x j (x) = x − i

ωVj

∫ x

0d(s) ds, j = p, s.

Moreover, it can be shown that U j , j = p, s, satisfies

1) U j ≡ U j , for j = p, s in the left half-space x ≤ 0 (no re-flection).

2) U j , j = p, s, is damped in the right half-space.3) the damping coefficient in the absorbing layer is

‖U p(x)‖‖Up(x)‖ = exp

[−cos θ

Vp

∫ x

0d(s) ds

],

‖U s(x)‖‖Us(x)‖ = exp

[−cos θ

Vs

∫ x

0d(s) ds

].

Finite absorbing layer.—In practice, we take a finite absorb-ing layer by introducing a boundary at x = δ, with a Dirichletcondition. This new boundary produces a reflection, but be-cause the wave decreases exponentially in the layer, the reflec-

tion coefficient quickly becomes very small. This coefficientdepends on the choice of d(x) and on the size δ of the layer. Tostudy the properties of the model in this case, we recall someclassical results for the elastodynamic problem.

Consider the elastodynamic problem with a homogeneousDirichlet condition(v = 0) on the boundary x = 0, as shownin Figure 1. Take, for example, the case of an incident planewave P :

U incp = Aincdp exp

[iωVp

(t − dp · x

Vp

)], dp = (cos θ, sin θ).

We know then (compare Achenbach [1984], Vol. 1, §5.6,p. 173) that the incident wave is reflected into a pressure waveUr

p and a shear wave Urs , given by

Urp = Ar

pdrp exp

[iωVp

(t − dr

p · x

Vp

)],

drp = (−cos θ1, sin θ1),

Urs = Ar

sdrs exp

[iωVp

(t − ps · x

Vp

)],

drs = (−sin θ2, cos θ2),

ps = (cos θ2, sin θ2),

and we have the reflection coefficients

Rpp =∥∥Ur

p

∥∥∥∥U incp

∥∥ = cos(θ − θ2)cos(θ + θ2)

,

Rps =∥∥Ur

s

∥∥∥∥U incp

∥∥ = sin θ2

cos(θ + θ2),

with sin θ2 = Vs sin θ/Vp . In the same way, in the case of anincident plane wave S, we have

Rss =∥∥Ur

s

∥∥∥∥U incs

∥∥ = sin(θ + θ2)sin(θ − θ2)

,

Rsp =∥∥Ur

p

∥∥∥∥U incs

∥∥ = sin θ2

sin(θ − θ2),

with cos θ2 = Vp cos θ/Vs . We can consider now the case ofthe finite layer. Given that the plane waves U j , j = p, s,corresponding to solutions of system (11), are given by equa-tions (13), we can compute the reflection coefficients inducedby the layer of width δ.

FIG. 1. Schematic view of the reflection of an incident P-wave(left) and of an incident S-wave (right) in the case of a homo-geneous Dirichlet boundary condition.


The case of a plane wave P

As we have shown previously, there is no reflection at theinterface x = 0 when the plane wave U p penetrates the lossymedium. After traveling a length δ, we can compute U p byusing formula (14), obtaining

U p = Up exp

[−cos θ

Vp

∫ δ

0d(s) ds

].

Then plane wave U p gives at the boundary x = δ, two reflectedwaves U r

p and U rs , which are damped until penetrating again

the elastic medium at the interface x = 0. It is easy to see thatthe reflection coefficient is given in this case by

Rδpp =

∥∥U rp(x)

∥∥‖Up(x)‖ = Rppexp

[−2

cos θ

Vp

∫ δ

0d(s) ds

], x < 0,

Rδps =

∥∥U rs(x)

∥∥‖Up(x)‖ = Rpsexp

[−2

cos θ

Vp

∫ δ

0d(s) ds

], x < 0.

(15)

The case of a plane wave S

In the case of an incident plane wave S, we obtain similarly

Rδss =

∥∥U rs(x)

∥∥‖Us(x)‖ = Rssexp

[−2

cos θ

Vs

∫ δ

0d(s) ds

], x < 0,

Rδsp =

∥∥U rp(x)

∥∥‖Us(x)‖ = Rsp exp

[−2

cos θ

Vs

∫ δ

0d(s) ds

], x < 0.

(16)

Relations (15) and (16) imply that the reflection can be madeas weak as desired by choosing the damping factor d(x) largeenough. However, this is no longer true when we consider thediscrete PML model. As we will see in Appendix A, the dis-crete absorbing layer model is not matched perfectly. That is aconsequence of the numerical dispersion, which introduces areflection at the interface.

The discrete PML model

In this section, we present the discrete PML model for elasto-dynamics. To do so, we introduce two numerical schemes forthe discretization of system (11). The first is a finite-differencescheme, introduced in Virieux (1986), and the second is ob-tained using a new mixed finite element (Becache et al., 1997)in a regular mesh. The use of this finite element has the advan-tage of leading to an explicit scheme (mass lumping), even inthe case of an anisotropic elastic medium.

The Virieux finite-difference scheme

This scheme uses a staggered grid formulation, so that if vx

is computed at the points (i, j) (xi = ih, y j = jh) of a referencegrid, then vy is computed at the points (i + 1

2 , j + 12 ), σxx and σyy

at (i + 12 , j), and σxy at (i, j + 1

2 ). The discrete PML model as-sociated to the Virieux finite-difference scheme can be writtenas follows:

(vx )ni, j = (vx

x

)ni, j + (vy

x

)ni, j ,(

vxx

)n+1i, j − (vx

x

)ni, j

�t+ dx

i

(vx

x

)n+1i, j + (vx

x

)ni, j

2

=(σxx )

n+ 12

i12 , j

− (σxx )n+ 1

2

i− 1

2 , j

�h,

(v

yx)n+1

i, j − (vyx)n

i, j

�t+ d y

j

(v

yx)n+1

i, j + (vyx)n

i, j

2

=(σxy)

n+ 12

i, j12

− (σxy)n+ 1

2

i, j− 1

2

�h,

(vy)n

i12 , j

12

= (vxy

)ni

12 , j

12

+ (vyy

)ni

12 , j

12,

(vx

y

)n+1

i12 , j

12

− (vxy

)ni

12 , j

12

�t+ dx

i12

(vx

y

)n+1

i12 , j

12

+ (vxy

)ni

12 , j

12

2

=(σxy)

n+ 12

i1, j12

− (σxy)n+ 1

2

i, j12

�h,

(v

yy)n+1

i12 , j

12

− (vyy)n

i12 , j

12

�t+ d y

j12

(vyy )n+1

i12 , j

12

+ (vyy)n

i12 , j

12

2

=(σyy)

n+ 12

i12 , j1

− (σxy)n+ 1

2

i12 , j

�h,

(σxx )n+ 1

2

i12 , j

= (σ xxx

)n+ 12

i12 , j

+ (σ yxx

)n+ 12

i12 , j

,

(σ x

xx

)n+ 12

i12 , j

− (σ xxx

)n− 12

i12 , j

�t+ dx

i12

(σ x

xx

)n+ 12

i12 , j

+ (σ xxx

)n− 12

i12 , j

2

= (λ + 2µ)(vx )n

i1, j− (vx )n

i, j

h,

(σ

yxx)n+ 1

2

i12 , j

− (σ yxx)n− 1

2

i12 , j

�t+ d y

j

(σ

yxx)n+ 1

2

i12 , j

+ (σ yxx)n− 1

2

i12 , j

2

= λ

(vy)n

i12 , j

12

− (vy)n

i12 , j

− 12

h,

(σyy)n+ 1

2

i12 , j

= (σ xyy

)n+ 12

i12 , j

+ (σ yyy

)n+ 12

i12 , j

,

(σ x

yy

)n+ 12

i12 , j

− (σ xyy

)n− 12

i12 , j

�t+ dx

i12

(σ x

yy

)n+ 12

i12 , j

+ (σ xyy

)n− 12

i12 , j

2


= λ(vx )n

i1, j− (vx )n

i, j

h,

(σ

yyy)n+ 1

2

i12 , j

− (σ yyy)n− 1

2

i12 , j

�t+ d y

j

(σ

yyy)n+ 1

2

i12 , j

+ (σ yyy)n− 1

2

i12 , j

2

= (λ + 2µ)

(vy)n

i12 , j

12

− (vy)n

i12 , j

− 12

h,

(σxy)n+ 1

2

i, j12

= (σ xxy

)n+ 12

i, j12

+ (σ yxy

)n+ 12

i, j12,

(σ x

xy

)n+ 12

i, j12

− (σ xxy

)n− 12

i, j12

�t+ dx

i

(σ x

xy

)n+ 12

i, j12

+ (σ xxy

)n− 12

i, j12

2

= µ

(vy)n

i12 , j

12

− (vy)n

i− 1

2 , j12

h,

(σ

yxy)n+ 1

2

i, j12

− (σ yxy)n− 1

2

i, j12

�t+ d y

j12

(σ

yxy)n+ 1

2

i, j12

+ (σ yxy)n− 1

2

i, j12

2

= µ(vx )n

i, j1 − (vx )ni, j

h,

where we used the notation i k = i + k and j k = j + k.The previous system of equations corresponds to the PML

model in the corners, where we need damping in both direc-tions. For the computation of the solution inside the PML layersin the x direction, we use the same system of equations withd y = 0, and for PML layers in the y direction, we take dx = 0.We present, in Figure 2, the values of dx and d y for the differentPML layers.

THE FINITE-ELEMENT SCHEME

For this scheme, the velocity is approximated by piecewiseconstant functions and the stress tensor σ by piecewise lin-ear functions with some particular continuity properties. More

FIG. 2. The different PML layers: In the corners, both dx andd y are positive, and d y = 0 (resp. dx = 0) inside the PML layersin the x (resp. y) direction.

precisely, σxy is continuous in both directions and σxx (resp.σyy) is continuous only in the x (resp. y) direction. We present,in Figure 3, the mixed finite element used. On a regular grid,

the velocity v = (vx , vy) is computed at the points (i12 , j

12 ) and

the stress tensor at the points (i, j). We consider the generalcase of an anisotropic elastic material described by the elastic-ity matrix C , and we denote by A the inverse of C

A = C−1 =

a11 a12 a13

a12 a22 a23

a13 a23 a33

.

The numerical scheme then can be written in the following way[see Becache et al. (1998) for details]. For the first equation ofsystem (11), we have

(vx )n+1

i12 , j

12

− (vx )n

i12 , j

12

�t= 1

2�h

((σ h

xx

)n+ 12

i1, j− (σ h

xx

)n+ 12

i, j

+ (σ bxx

)n+ 12

i1, j1 − (σ bxx

)n+ 12

i, j1 + (σxy)n+ 1

2i, j1 − (σxy)

n+ 12

i, j

+ (σxy)n+ 1

2i1, j1 − (σxy)

n+ 12

i1, j

),

(vy)n+1

i12 , j

12

− (vy)n

i12 , j

12

�t= 1

2�h

((σxy)

n+ 12

i1, j− (σxy)

n+ 12

i, j

+ (σxy)n+ 1

2i1, j1 − (σxy)

n+ 12

i, j1 + (σ dyy

)n+ 12

i, j1 − (σ dyy

)n+ 12

i, j

+ (σ gyy

)n+ 12

i1, j1 − (σ gyy

)n+ 12

i1, j

).

The second equation of system (11) results in the following5 × 5 local system, coupling the five degrees of freedom asso-ciated to σ at each vertex (see Figure 3).

�n+ 1

2i, j − �

n− 12

i, j

�t= A−1

h Bxh + A−1

h B yh ,

FIG. 3. The finite element: for the stress tensor, five degreesof freedom are associated at each summit of the element, andfor the velocity, two degrees of freedom are associated at thecenter of the element.


where

� = [σ hxx , σ

bxx , σ

dyy, σ

gyy, σxy

]t,

Ah = 12

2a11 0 a12 a12 2a13

0 2a11 a12 a12 2a13

a12 a12 2a22 0 2a23

a12 a12 0 2a22 2a23

2a13 2a13 2a23 2a23 4a33

,

and

Bxh = 1

h

[Bx

1 , Bx2 , 0, 0, Bx

5

]t,

B yh = 1

h

[0, 0, B y

3 , B y4 , B y

5

]t,

with

Bx1 = (vx )n

i12 , j

12

− (vx )n

i− 1

2 , j12,

Bx2 = (vx )n

i12 , j

− 12

− (vx )n

i− 1

2 , j− 1

2,

B y3 = (vy)n

i12 , j

12

− (vy)n

i12 , j

− 12,

B y4 = (vy)n

i− 1

2 , j12

− (vy)n

i− 1

2 , j− 1

2,

Bx5 = (vy)n

i12 , j

12

− (vy)n

i− 1

2 , j12

+ (vy)n

i12 , j

− 12

− (vy)n

i− 1

2 , j− 1

2,

B y5 = (vx )n

i12 , j

12

− (vx )n

i12 , j

− 12

+ (vx )n

i− 1

2 , j12

− (vx )n

i− 1

2 , j− 1

2.

In the case of an isotropic elastic material, the matrix A−1h is

given by

A−1h =

a c b b 0

c a b b 0

b b a c 0

b b c a 0

0 0 0 0 d

,

with

a = 2(λ + 2µ)2 − λ2

2(λ + 2µ), b = λ

2,

c = λ2

2(λ + 2µ), d = µ

2. (17)

We now can write the PML model associated to this scheme,using the same technique as for the Virieux scheme,

(vx )n

i12 , j

12

= (vxx

)ni

12 , j

12

+ (vyx

)ni

12 , j

12,

(vx

x

)n+1

i12 , j

12

− (vxx

)ni

12 , j

12

�t+ dx

i12

(vx

x

)n+1

i12 , j

12

+ (vxx

)ni

12 , j

12

2

= 12�h

((σ h

xx

)n+ 12

i1, j− (σ h

xx

)n+ 12

i, j + (σ bxx

)n+ 12

i1, j1 − (σ bxx

)n+ 12

i, j1

),

(v

yx)n+1

i12 , j

12

− (vyx)n

i12 , j

12

�t+ d y

i12

(v

yx)n+1

i12 , j

12

+ (vyx)n

i12 , j

12

2

= 12�h

((σxy)

n+ 12

i, j1 − (σxy)n+ 1

2i, j + (σxy)

n+ 12

i1, j1 − (σxy)n+ 1

2i1, j

),

(vy)n

i12 , j

12

= (vxy

)ni

12 , j

12

+ (vyy

)ni

12 , j

12,

(vx

y

)n+1

i12 , j

12

− (vxy

)ni

12 , j

12

�t+ dx

i12

(vx

y

)n+1

i12 , j

12

+ (vxy )n

i12 , j

12

2

= 12�h

((σxy)

n+ 12

i1, j− (σxy)

n+ 12

i, j + (σxy)n+ 1

2i1, j1 − (σxy)

n+ 12

i, j1

),

(v

yy)n+1

i12 , j

12

− (vyy)n

i12 , j

12

�t+ d y

i12

(v

yy)n+1

i12 , j

12

+ (vyy)n

i12 , j

12

2

= 12�h

((σ d

yy

)n+ 12

i, j1 − (σ dyy

)n+ 12

i, j + (σ gyy

)n+ 12

i1, j1 − (σ gyy

)n+ 12

i1, j

),

and

�n+ 1

2i, j = (�x )

n+ 12

i, j + (� y)n+ 1

2i, j ,

(�x )n+ 1

2i, j − (�x )

n− 12

i, j

�t+ dx

i

(�x )n+ 1

2i, j + (�x )

n− 12

i, j

2= A−1

h Bxh ,

(� y)n+ 1

2i, j − (� y)

n− 12

i, j

�t+ d y

i

(� y)n+ 1

2i, j + (� y)

n− 12

i, j

2= A−1

h B yh .

The properties of this model are studied in terms of a numericaldispersion analysis presented in Appendix A. This analysis per-mits us to compute the reflection coefficients in the case of a ho-mogeneous, isotropic elastic solid. In this case, we have foundthat the reflection coefficients are approximately proportionalto h2. That shows the second-order accuracy of the discretePML model, which thus is no longer perfectly matched. Thismeans in particular that the damping factor cannot be chosento be too large with respect to the inverse of the discretizationstep. Of course, it must be large enough to allow the dampingof the wave inside the layer, so there is a trade-off (see the endof the Appendix A for more details).

The theoretical results presented in the previous sectionsare based on a plane-wave analysis and thus can be appliedonly to wave-propagation problems in homogeneous media.However, as we will see in the next sections, the discrete PMLmodel also can be used to compute numerically the solutionof the elastodynamic problem in heterogeneous, anisotropicelastic media, although we have no theoretical justification forthis extension.

NUMERICAL RESULTS

In the following examples, we simulate elastic wave propa-gation in a 2-D unbounded medium. We will present severalresults, including the case of heterogeneous, anisotropic elasticmedia. To show the generality of the method, we give some nu-merical results for both schemes: the Virieux finite-differencescheme and the finite-element scheme. A more complete anal-ysis is presented for the finite-element scheme; namely, the


discrete reflection coefficients are computed as a function ofthe incident angle. Let us first set some notation and give thecharacteristics of the numerical examples that will follow. Weconsider an unbounded domain � in 2-D, occupied by an elas-tic material, and we suppose that the initial condition (or thesource) is supported in a part of �. The material will be charac-terized by its wave velocities Vp (pressure) and Vs (shear) in theisotropic case and by its elasticity matrix C in the anisotropiccase. To solve numerically the elastodynamic problem in �, wedefine a bounded domain D of a simple geometry (here it willbe a square), containing the support of the initial data, with ab-sorbing layers (PML) on all four boundaries. For the discretiza-tion of the problem, we take a regular grid on D composed ofN × N square elements of edge h = 1/N . The time step then iscomputed following the CFL condition for each scheme. Moreprecisely, we take: �t = 0.9h/(

√2Vp) for the Virieux scheme

and �t = h/Vp for the finite-element scheme.For our numerical experiments, an explosive source located

at point S = (xs, ys) is used, that is,

f (x, t) = F(t)g(r).

The function F(t) is defined by

F(t) =−2π2 f 2

0 (t − t0) e−π2 f 20 (t−t0)2

if t ≤ 2t0

0 if t > 2t0, (18)

where t0 = 1/ f0, f0 = V s/(hNL ) is the central frequency, and NL

is the number of points per S wavelength. The function g(r) isa radial function given by

g(r) =(

1 − r2

a2

)3

1Ba

(x − xs

r,

y − ys

r

),

(19)r =

√(x − xs)2 + (y − ys)2, a = 5h,

where 1Ba is one on Ba , the disk of center S and radius a, andzero elsewhere. In the absorbing layers, we use the followingmodel for the damping parameter d(x),

d(x) = d0

(x

δ

)2

, (20)

where δ is the width of the layer and d0 is a function of thetheoretical reflection coefficient (R = Rδ

pp) [see relation (15)for θ = 0],

d0 = log(

1R

)3Vp

2δ. (21)

We first present some results in a homogeneous isotropicmedium.

Homogeneous, isotropic elastic medium, Virieux scheme

We consider here an elastic medium with Vp = 2000 m/s,Vs = 1400 m/s. The characteristics of the discretization are

N = 200, h = 0.15 m, S = (7.5 m, 7.5 m),

NL = 20, δ = 10 h and R = 0.001.

We present some snapshots of the solution in Figure 4. Thesnapshots show no reflection when the results are presentedon the normal scale. Indeed, for the PML model used in this

example, the reflection coefficient is about 0.1%, as can be seenby magnifying the results by a factor of 100 (see Figure 5). InFigure 5, we also present the results obtained using the first-order absorbing boundary condition introduced by Reynolds(1978). The reflection coefficient for this ABC is about 10%;the first-order ABC is not accurate enough for waves travelingobliquely to the boundary.

Heterogeneous, isotropic elastic medium, Virieux scheme

We consider a heterogeneous elastic medium; the interfaceis parallel to the x axis and located in the middle of the frame(see Figure 6). The wave velocities in the two media are V 1

p =2000 m/s, V 1

s = 1400 m/s (the upper medium), V 2p = 1000 m/s,

V 2s = 700 m/s (the lower medium). All the other parameters

are the same as in section A, where Vs = (V 1s + V 2

s )/2 has beenused for the computation of f0 (equation 18) and Vp = V 1

p tocompute �t . The source now is located in the lower mediumnear the interface (at point [15m, 16.2m]). We present somesnapshots of the solution in Figure 6. The reflection coefficientfor this example is about 0.1%, as we can see in Figure 6.

Heterogeneous, isotropic elastic medium,finite-element scheme

The heterogeneous elastic medium considered here is char-acterized by the velocity model presented in Figure 7; we have

FIG. 4. Snapshots of 2-D elastic finite-difference simulations(Virieux scheme), in a homogeneous medium, using a PMLlayer with width 10h. The snapshots are the norm of the velocity‖v‖ =

√v2

1 + v22 .


max Vp/min Vp = 2.1 and Vp = 1.6Vs . For the discretization, Vp

and Vs are piecewise constants (one value per element). Thesize of the grid is 200 × 200, h = 0.15 m, NL = 10, and the sourceis located at point (15 m, 3 m). We present, in Figure 8, the normof the velocity magnified by a factor of 20 for three experi-ments corresponding to different absorbing layers. More pre-cisely, relations (20) and (21) are used with δ = 5h − R = 0.01,δ = 10h − R = 0.001, and δ = 20h − R = 0.0001. Let us recallthat these values for R correspond to the theoretical (Rpp)reflection coefficients. The reflection coefficients obtained nu-merically are close to the theoretical ones. This is no longer trueif the number of meshes inside the layer is too small; the smallerR, the larger layer width required. This phenomenon resultsfrom the numerical dispersion (Collino and Monk, 1998).

Homogeneous, anisotropic elastic medium,finite-element scheme

In this example, we consider an anisotropic elastic solid, theapatite. We want to point out that in the case of anisotropic elas-tic materials, only lower-order absorbing boundary conditionsare known, and they are quite difficult to implement (Becache,1991). On the contrary, the generalization of the PML model tothe anisotropic case is straightforward, and there are no supple-mentary difficulties for the implementation. Moreover, the re-sults are very satisfactory, as we can see in the following figures.

The characteristics of the problem are N = 200, h = 0.15 m,and NL = 10, and the source is located at the center of the frameS = (15 m, 15 m). We present in Figure 9 three experiments,corresponding to δ = 5h − R = 0.01, δ = 10h − R = 0.001, andδ = 20h − R = 0.0001. We can see in Figure 9 that there is noreflection when the results are presented on the normal scale, aswas already the case for a homogeneous medium. However, wecan see some reflection by magnifying these results, as shownin Figure 10. The reflection coefficients obtained numericallyare close to the theoretical ones.

Reflection coefficients

In this section, we show reflection coefficients computedby using numerical simulations. We consider a homogeneousisotropic elastic solid (Vp = 5.710 m/s, Vs = 2.93 m/s), and weuse the finite-element scheme for the space discretization. Tocompute the reflection coefficients we first consider the solu-tion of the elastodynamic problem at points Mi located near the

FIG. 5. The the norm of the velocity at t = 13.8 ms: In (1), we present the results obtained with the PML model (δ = 10h) magnifiedby a factor of 100. In (2), we present the results obtained with a first-order ABC on the normal scale.

FIG. 6. Snapshots of 2-D elastic finite-differences simulations(Virieux scheme), in a heterogeneous medium, using a PMLlayer with length 10h. We present the norm of the velocity

‖v‖ =√

v21 + v2

2 , in the normal scale (1) and (2). To see somereflections, (3) is magnified by a factor of 1000.


FIG. 7. The velocity model for the heterogeneous medium;max Vp/min Vp = 2.1 and Vp = 1.6Vs .

FIG. 8. Snapshots of 2-D elastic finite-elements simulations, in aheterogeneous, medium, using the PML absorbing layer modelwith δ = 5h, δ = 10h, and δ = 20h. The snapshots are the normof the velocity at t = 14.2 ms magnified by a factor of 20.

upper boundary of domain D1 (D1 is a rectangular domainof size 200 m × 100 m, and a PML layer model is used on allboundaries of the domain). We denote by (V1)i the velocityvector at point Mi . Then to obtain a reference solution at pointsMi , we move the upper boundary of the domain 50 m above.That is, we solve the elastodynamic problem in the domain D2

(a rectangle of size 200 m × 150 m). We also use the PML layermodel on all boundaries of the domain, and we denote by (V2)i

the velocity vector at point Mi . The geometry of the problemwith the two computational domains is shown in Figure 11.In the case of a pure P-wave source, we define the reflectioncoefficients by

(Rpp)(θi ) = div(U2)i − div(U1)i

div(U2)i,

(22)

(Rps)(θi ) = curl(U2)i − curl(U1)i

curl(U2)i.

In the same way, we define

(Rss)(θi ) = curl(U2) − curl(U1)curl(U2)

,

(Rsp)(θi ) = div(U2) − div(U1)div(U2)

,

in the case of a pure S-wave source. For our examples, we usethe P-wave source function given by (18, 19) with NL = 16. Forthis source function, the curl of the velocity is smaller than the

FIG. 9. Snapshots of 2-D elastic finite-elements simulations, in ahomogeneous, anisotropic medium, using the PML absorbinglayer model with δ = 5h. The snapshots are the norm of thevelocity ‖v‖ =

√v2

1 + v22 .


divergence by a factor 10−10. The reflection coefficient Rps isneglected.

Relation (22) is used to compute two reflection coefficientsat each time step. The results presented in Figure 12 correspondto their mean value over [0, T ], for T = 25 s.

FIG. 10. Snapshots of 2-D elastic finite-elements simulations, ina homogeneous, anisotropic medium, using the PML absorbinglayer model with δ = 5h, δ = 10h, and δ = 20h. The snapshots arethe norm of the velocity at t = 13.2 ms magnified by a factorof 20.

FIG. 11. Geometry for computing reflection coefficients. Twocomputational domains are defined: D1 200 m × 100 m and D2200 m × 150 m.

The grid size is 400 × 200 for D1, 400 × 300 for D2, andh = 0.5 m for both domains. The source is located at point (200,80) of the grid. In the PML layers,

d(x) = d0

(x

δ

)4

, (23)

where δ is the length of the layer and d0 is given by

d0 = log(

1R

)4Vp

2δ.

We present three experiments with δ = 5h − R = 10−4, δ =10h − R = 10−6, and δ = 20h − R = 10−8. To examine the perfor-mance of the PML model, comparisons with the first-, second-,and third-order Higdon’s absorbing boundary conditions areperformed.

The reflection coefficients for the absorbing boundary con-ditions proposed by Higdon are the theoretical ones. We obtainthem by applying an operator of the form

m∏i=1

(βi

∂

∂t+ Vp

∂

∂n

),

with m = 1; β1 = 1, or m = 2; β1 = 1, β2 = Vp/Vs(= 1.95), orm = 3; β1 = 1, β2 = 1.5, β3 = Vp/Vs(= 1.95).

As we can see in Figure 12, for almost all angles of incidence,the PML absorbing layer model gives better results then theabsorbing boundary conditions of first and second order andthis even for the 5h length absorbing layer. Compared with thethird-order absorbing condition, the PML layers of 10h and 20hare substantially better except near the 60-degree angle. Wewant to point out that in this example we compare the reflectioncoefficients provided by the continuous Higdon model to theone given by the discrete PML model. It is known that theHigdon reflection coefficients increase when discretization isused.

The numerical results presented in the sections above showthe generality of the PML model: These models can be im-plemented easily in different numerical schemes. They givesatisfactory results even in the case of anisotropic, heteroge-neous elastic media. The reflection coefficients are close to thetheoretical ones for all the presented experiments. They areabout 1% for a 5h PML layer width, 0.1% for a 10h PML layerwidth, and 0.01% for a 20h PML layer width in all experi-ments. In practice, one should choose the length of the layer

FIG. 12. Comparison of Rpp for the Higdon ABCs and PML.


as a function of the scale of the problem and the requestedreflection coefficient. In the numerical examples we have pre-sented, the scale of the problems was about 15 wavelengthsof P-wave (λp) in each direction. In this case, a PML layerwith δ = 5h (that is, about 0.5 λp) is sufficient and gives betterresults than the second-order absorbing condition. However,for larger-scale computations, the numerical dispersion will bemore important, and thus a PML layer with δ = 5h probablywould not give the expected results. Thus, for large-scale prob-lems, one should consider a PML layer with δ ≥ 10h. We wantto point out that even in this case, the additional cost remainsmodest, given that the layer represents only a small portion ofthe total grid. Let us finally remark that the PML model canbe extended easily to the case of Rayleigh waves. (Experimen-tal results not shown in this paper have demonstrated a goodabsorption of Rayleigh waves).

CONCLUSIONS

We have presented in this paper a generalization of the PMLabsorbing layer model for the elastodynamic problem in thecase of heterogeneous, anisotropic media. The implementa-tion of this model was presented in the 2-D case but it canbe extended straightforwardly to the 3-D case. The superiorityof this model compared with the absorbing boundary condi-tions proposed by Higdon was shown by numerical results inthe case of an isotropic, homogeneous medium. Moreover, wehave shown by several numerical examples the efficiency of thismodel even in the case of heterogeneous, anisotropic media.To conclude, let us point out the ease of the implementationof this model in the case of an anisotropic, elastic medium, incomparison with the complexity of the respective absorbingboundary conditions.

REFERENCES

Achenbach, J. D., 1984, Wave Propagation in Elastic Solids, Vols. I andII: Elsevier Science Publ. B.V.

Auld, B. A., 1973, Acoustic fields and elastic waves in solids: John Wiley& Sons, Inc.

Becache, E., 1991, Resolution par une methode d’equations integralesd’un probleme de diffraction d’ondes elastiques transitoires par unefissure.: Ph.D. Thesis, Univ. of Paris 6.

Becache, E., Joly, P., and Tsogka, C., 1997, Elements finis mixtes et con-densation de masse en elastodynamique lineaire: (i) construction:C. R. Acad. Sci. Paris Ser. I Math., 325, 545–550.

——— 1998, Fictitious domain method applied to the scattering by a

crack of transient elastic waves in anisotropic media: A new familyof mixed finite elements leading to explicit schemes: Mathematicaland numerical aspects of wave propagation: SIAM, 322–326.

Berenger, J., 1994, A perfectly matched layer for the absorption ofelectromagnetic waves: J. Comput. Phys., 114 185–200.

——— 1996, Three-dimensional perfectly matched layer for the ab-sorption of electromagnetic waves: J. Comput. Phys., 127, 363–379.

——— 1997, Improved PML for the FDTD Solution of Wave-Structure Interaction Problems: IEEE Trans. Antennas and Propa-gation, 45, 466–473.

Chalindar, B., 1987, Conditions aux limites artificielles pour lesequations de l’elastodynamique: Ph.D. thesis, Univ. Saint-Etienne.

Collino, F., 1996, Perfectly matched absorbing layers for the paraxialequations: J. Comput. Phys., 131, 164–180.

Collino, F., and Monk, P., 1998, Optimizing the perfectly matched lay-ers: Comput. Methods Appl. Mech. Engrg., 164, No. 1–2, 157–171.

Engquist, B., and Majda, A., 1997, Absorbing boundary conditions forthe numerical simulation of Waves: Math. Comp., 31, 629–651.

Hagstrom, T., 1997, On High-order radiation boundary condition, inEngquist, B., and Kriegsmann, G. A., Eds, Computational WavePropagation: Springer-Verlag New York, Inc., 86, 1–21.

Hastings, F., Schneider, J. B., and Broschat, S. L., 1996, Applicationof the perfectly matched layer (PML) absorbing boundary condi-tion to elastic wave propagation: J. Acoust. Soc. Am., 100, 3061–3069.

Higdon, R., 1990, Radiation boundary conditions for elastic wave prop-agation: SIAM J. Numer. Anal., 27, 831–870.

——— 1991, Absorbing boundary conditions for elastic waves: Geo-physics, 56, 231–241.

——— 1992, Absorbing boundary conditions for acoustic and elasticwaves in stratified media: J. Comput. Phys., 101, 386–418.

Israeli, M., and Orszag, S., 1981, Approximation of radiation boundaryconditions: J. Comput. Phys., 41, 115–135.

Peng, C., and Toksoz, M., 1995, An optimal absorbing boundary con-dition for elastic wave modeling: Geophysics, 60, 296–301.

Rappaport, C., 1995, Perfectly matched absorbing conditions based onanisotropic lossy mapping of space, IEEE Microwave and GuidedWave Letters, 5, No. 3, 90–92.

Reynolds, A., 1978, Boundary conditions for the numerical solution ofwave propagation problems: Geophysics, 43 1099–1110.

Simone, A., and Hestholm, S., 1998, Instability in applying absorb-ing boundary conditions to high-order seismic modeling algorithms:Geophysics, 63, 1017–1023.

Sochacki, J., Kubichek, R., George, J., Fletcher, W., and Smithson, S.,1987, Absorbing boundary conditions and surface waves: Geo-physics: 52, 60–71.

Teixeira, F. L., and Chew, W. C., 1998, Analytical derivation of a confor-mal perfectly matched absorber for electromagnetic waves: Micro.Opt. Tech. Lett., 17, 231–236.

Turkel, E., and Yefet, A., 1998, Absorbing pml boundary layers forwave-like equations. Absorbing boundary conditions: Appl. Numer.Math., 27, No. 4, 553–557.

Virieux, J., 1986, P-sv wave propagation in heterogeneous media:Velocity-stress finite difference method: Geophysics, 51, 889–901.

Zhao, L., and Cangellaris, A. C., 1996, A general approach for develop-ing unsplit-field time-domain implementations of perfectly matchedlayers for FDTD grid truncation: IEEE Trans. Microwave TheoryTech., 44, 2555–2563.


APPENDIX A

DISPERSION ANALYSIS

In this section, we will study the properties of the discretePML model for elastodynamics in terms of a numerical dis-persion analysis, that is, a plane-wave analysis. We present thisanalysis only for the finite-element scheme. We consider thecase of a homogeneous, isotropic elastic medium character-ized by the Lame coefficients λ, µ and the density �. We use aninfinite PML layer in the x-direction. We look for plane-wavesolutions of the form

(V �)n+ 1

2

i12 , j

12

= (V �)i

12

exp[−iky

(j

12)

h + iω

(n + 1

2

)�t

],

(��)ni, j = (��)i exp[−iky jh + iωn�t],

V � = [v�x , v

�y

]T, �� = [σ h,�

xx , σ b,�xx , σ d,�

yy , σ g,�yy , σ �

xy

]T,

for � = ⊥, ‖, and we set

(V )i

12

= (V ‖)i

12

+ (V ⊥)i

12; (�)i = (�‖)i + (�⊥)i

Plugging these expressions into the discrete scheme and elim-inating the unknowns associated to σ lead to

A2t

�t2(vx )

i12

= µ

�A2

y

(vx )i

32

+ 2(vx )i

12

+ (vx )i− 1

2

4h2

+ a + c cos(kyh)�D

i12

(vx )

i32

− (vx )i

12

h2 Di1

−(vx )

i12

− (vx )i− 1

2

h2 Di

− Ay

µ

�cos(

kyh

2

)

× (vy)

i32

− (vy)i

12

2h2 Di1+

(vy)i

12

− (vy)i− 1

2

2h2 Di

+(vy)

i32

− (vy)i− 1

2

h2 Di

12

, (A-1)

A2t

�t2(vy)

i12

= 1�

A2y

c(vy)i

32

+ 2a(vy)i

12

+ c(vy)i− 1

2

2h2

+ µ

�cos2

(kyh

2

) (vy)i

32

− (vy)i

12

h2 Di1 Di

12

−(vy)

i12

− (vy)i− 1

2

h2 Di Di

12

− Ay

b

�cos(

kyh

2

)

× (vx )

i32

− (vx )i

12

h2 Di1+

(vx )i

12

− (vx )i− 1

2

h2 Di

+(vx )

i32

− (vx )i− 1

2

2h2 Di

12

, (A-2)

where a, b, c are given by (17) and At , Ay , C�, and D� are definedby

At = 2i sin(

ω�t

2

), Ay = 2i sin

(kyh

2

),

C� = 2i sin(

ω�t

2

)+ d��t cos

(ω�t

2

), � = i, i

12 ,

D� = C�

At, � = i, i

12 .

Let us consider now the case of a medium with a unique infinitelayer for which

di ={

0 if i ≤ 0

d∞ if i > 0, d

i12

={

0 if i < 0

d∞ if i > 0,

d 12

remaining undetermined yet, and let us look for particular

solution of the form (see Figure A-1)

P-wave

(V )i

12

= dp e−ikx(

i12)

h + Rppdrpeikx

(i

12)

h + Rpsdrseikx

(i

12)

h

for i12 ≤ 0,

(V )i

12

= Tppd pe−i kx(

i12)

h + Tps dse−i kx(

i12)

h for i12 ≥ 0.

S-wave

(V )i

12

= dse−ikx(i

12)

h + Rssdrseikx

(i

12)

h + Rspdrpeikx

(i

12)

h

for i12 ≤ 0,

(V )i

12

= Tss dse−i kx(

i12)

h + Tspdpe−i kx(

i12)

h for i12 ≥ 0.

FIG. A-1. Schematic view of the plane-wave solution in a single-layer discrete model.


Equations (A-1) and (A-2) are satisfied for i12 �= 0 if kx and

kx are solutions of the two relations of dispersion (A-3) (freemedium) and (A-4) (lossy medium).

X2t

�t2

[vx

vy

]= 1

h2

[K 11 K 12

K 12 K 22

][vx

vy

], (A-3)

X2t

�t2

[vx

vy

]= 1

h2

[K11 K12

K12 K22

][vx

vy

], (A-4)

with

K 11 = V 2p Xx (1 − 4αX y) + V 2

s X y(1 − β Xx ),

K 22 = V 2p X y(1 − 4αXx ) + V 2

s Xx (1 − β X y),

K 12 = (V 2p − V 2

s

)√Xx X y(1 − Xx )(1 − X y),

Xt = sin2(

ω�t

2

), Xx = sin2

(kx h

2

),

X y = sin2(

kyh

2

), α =

(V 2

p − 2V 2s

)24V 4

p

, β = 14,

and

K11 = V 2p

X x

D2(1 − 4αX y) + V 2

s X y(1 − β X x ),

K22 = V 2p X y(1 − 4α X x ) + V 2

s

X x

D2(1 − β X y),

K12 =(V 2

p − V 2s

)D

√X x X y(1 − X x )(1 − X y),

X x = sin2(

kx h

2

),

D =2i sin

(ω�t

2

)+ d∞�t cos

(ω�t

2

)

2i sin(

ω�t

2

) .

Now the system of equations (A-3) and (A-4) at i12 = 0 gives the

value of the reflection coefficients Rpp(resp. Rss) and Rps(resp.Rsp). We have used the software MAPLE to compute theirTaylor expansions with respect to the discretization step h, andwe obtained

Rpp =d∞ − 2d 1

2

4ω

(1 − 2r2

1

(V 2

p + V 2s

)) r0

Vph + O(h2),

Rps =d∞ − 2d 1

2

4ω

(3 − 2r2

1

(V 2

p + V 2s

))kyh + O(h2),

(A-5)

Rss =d∞ − 2d 1

2

4ω

(1 − 2r2

2 r21

(V 2

p + V 2s

)) r0

Vph + O(h2),

Rsp =d∞ − 2d 1

2

4ω

(2r2

2 r21

(V 2

p + V 2s

)−1−2r22

)kyh + O(h2),

with

r0 =√

ω2 − V 2p k2

y, r1 = k2y

ω2, r2 = V 2

s

V 2p

, r3 = 1r2

.

From relations (A-5), we can remark easily that the best choicefor d 1

2consists of taking

d 12

= d∞2

.

In that case, the first-order terms in (A-5) disappear and weobtain that the reflection coefficients are approximately pro-portional to d∞(d∞ + iω)h2. Two consequences can be deducedfrom this result. First, the numerical scheme is consistent withthe continuous PML model: The spurious reflection for a givenvalue of d∞ results only from the dispersion of the numericalscheme. Let us remark that the second-order accuracy is recov-ered in the h2 dependency of the reflection coefficients. Then,from a more practical point of view, this dispersion analysisimplies that we cannot use a value of d∞ too large for a givendiscretization step.

In the case of a finite length layer with a given number ofnodes, the layer is characterized by

(

d 12, d 3

2, . . . , d

n�− 12

),(

d1, d2, . . . , dn�

),

We then are led to find a trade-off between choosing the di+ 1

2s

too small (which would imply a strong reflection caused bythe Dirichlet boundary condition) or too large (which wouldimply spurious reflections caused by the dispersion). A par-tial answer to this problem is to use a smooth profile for d(x),(d

i+ 12

= d(h(i + 12 )), as is done in the classical layers models

(Israeli and Orszag, 1981). An alternative is to determine thebest coefficients for d(x) by minimizing the numerical reflec-tion coefficients, as is done in Collino and Monk (1998) for theHelmholtz equation.

Documents

Application of the perfectly matched absorbing layer model