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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media Francis COLLINO, Chrysoula TSOGKA

N ˚ 3471 Aoˆut 1998 ` THEME 4

ISSN 0249-6399

apport de recherche

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media Francis COLLINO , Chrysoula TSOGKAy Theme 4 | Simulation et optimisation de systemes complexes Projet Ondes Rapport de recherche n3471 | Ao^ut 1998 | 28 pages

Abstract: We present and analyze a perfectly matched absorbing layer model for the velocity-stress formulation of elastodynamics. This layer has the astonishing property of generating no re ection at the interface between the free medium and the arti cial absorbing medium. This allows us to obtain very low spurious re ection even with very thin layers. Several experiments show the eciency and the generality of the model. Key-words: absorbing layers, absorbing conditions, mixed nite elements, anisotropic

(Resume : tsvp)

 y

INRIA-rocquencourt. [email protected] INRIA-rocquencourt. [email protected] Unit´e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T´el´ephone : 01 39 63 55 11 - International : +33 1 39 63 55 11 T´el´ecopie : (33) 01 39 63 53 30 - International : +33 1 39 63 53 30

Application du modele de couches absorbantes parfaitement adaptees (PML) au probleme de l'elastodynamique lineaire en milieu heterogene anisotrope

Resume : Nous presentons et analysons un modele de couches absorbantes parfaitement adaptees (PML) pour la formulation en vitesse-contraintes de l'elastodynamique. Ce modele a la propriete etonnante de ne generer aucune re ection parasite a l'interface entre le milieu elastique et la couche absorbante. Ceci nous permet d'obtenir des re ections tres faibles m^eme dans le cas de couches nes. Plusieurs experiences numeriques montrent l'ecacite et la generalite du modele. Mots-cle : couches absorbantes, conditions absorbantes, elastodynamique, elements nis mixtes, anisotropie

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media3

Table of Contents

1 Introduction 2 Pml model for a general evolution problem 3 Pml model for elastodynamics 3.1 The elastodynamic problem . . 3.2 Plane wave analysis . . . . . . 3.2.1 In nite absorbing layer 3.2.2 Finite absorbing layer .

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3 4 7

7 8 8 9

4 The discrete PML model

11

5 Dispersion Analysis 6 Numerical Results

15 18

4.1 The Virieux nite-dierences scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 The nite-element scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6.1 6.2 6.3 6.4 6.5

Homogeneous, isotropic elastic medium-Virieux scheme . . . . . . . Heterogeneous, isotropic elastic medium-Virieux scheme . . . . . . Heterogeneous, isotropic elastic medium. Finite-element scheme . . Homogeneous, anisotropic elastic medium. Finite-element scheme . Re ection Coecients . . . . . . . . . . . . . . . . . . . . . . . . .

7 Conclusions

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19 20 21 23 24

26

1 Introduction The simulation of waves by nite-dierences or nite-elements methods in unbounded domains requires a speci c treatment for the boundaries of the necessarily truncated computational domain. Two solutions have been proposed for this purpose : absorbing boundary conditions (ABC) and absorbing layers. ABC's have been introduced by B. Engquist and A. Majda for the acoustic wave equation [11]. They consist in adding to the wave equation some suitable local boundary conditions that simulate the outgoing nature of the waves impinging on the borders. This method works particularly well for absorbing waves nearly normally incident to the arti cial boundaries. For waves traveling obliquely, higher order ABC must be used to achieve acceptable accuracy, [12]. For elastic waves, the situation is more complex. First, the transparent condition, i.e. the exact condition relating normal stress and displacement on a line for outgoing waves, is no longer a scalar but a matrix integrodierential relation. Its approximation by partial dierential equations, which is the usual way to make ABC, leads to a very complex system of equations, especially for higher order methods. The stability of the coupled problem composed of the elastodynamic system completed with these arti cial conditions is then very dicult to analyze and the situation is even more intricate when discretization is considered, [8]. To overcome these diculties, Higdon [14], [16], proposed to combine several rst order boundary conditions designed for the wave equation, each of them being associated with either the pressure waves velocity or the shear waves velocity. These conditions are theoretically stable, [15], relatively easy to implement and ecient for the waves traveling in a direction close to the normal of the arti cial boundaries. They can be adapted to the case of surface waves too, [21]. However, for other directions of propagation, important spurious re ections may occur. Some authors, [18] for instance, have proposed to optimize the coecients of Higdon's method in order to make these re ections decrease. However, numerical experiments still show relatively strong spurious re ections in some situations and stability problems when higher order numerical schemes are used, [20]. Layers models are an alternative to ABC. The idea is to surround the domain of interest by some arti cial absorbing layers in which waves are trapped and attenuated. For elastic waves, several models have been proposed. For instance, Sochacki et al., [21] suggest to add inside the layers some attenuation term, proportional to the rst time derivative of the displacement to the elastodydamic equations. This technique is inspired by Physics and revealed to be quite delicate in practice. The main diculty is that, when entering the layers, the waves \sees" the change in impedance of the medium and then is re ected arti cially into the domain of RR n3471

4

F. Collino and C. Tsogka

interest. The use of smooth and not too high attenuation pro les allows us to weaken the diculty but require the use of thick layers, [17]. In this paper, we propose to adapt a layer technique introduced in [5] for the Maxwell's equations by Berenger and that is now the most widely used method for the simulation of electromagnetic waves in non bounded domains, [6], [7], [23] . This technique consists in designing an absorbing layer called perfectly matched layer (PML) that possesses the astonishing property of generating no re ection at the interface between the free medium and the arti cial absorbing medium. This property allow us to use a very high damping parameter inside the layer and consequently a small layer width while achieving a quasi-perfect absorption of the waves. To our knowledge, this method has been already used only once for elastic waves simulation. In [13], the authors propose the use of PML for the compressionnal and shear potentials formulation. In our paper, the PML are incorporated in the stress-velocity formulation. The present paper is organized as follows. In section 2 we construct the PML model in the general case of an evolution problem. It is based on an interpretation of the PML model of Berenger [5] as a change of variable in the complex plane c.f [9], [19]. In section 3, we apply the previous model to the velocity stress formulation for elastodynamics and we study the properties of the continuous PML model via a plane wave analysis. In order to show the generality of the PML model, two numerical schemes are presented (section 4). The rst one is the classical Virieux nite-dierences scheme [22] while the second one is a mixed nite-element scheme [3] which allow us to consider anisotropic media as well. We then study in section 5 the properties of the discrete model in the case of the nite element scheme in terms of a numerical dispersion analysis. Finally in section 6 we present several numerical results which show the eciency of this model even in the case of heterogeneous, anisotropic elastic media and its superiority when compared to the classical ABC's conditions.

2 Pml model for a general evolution problem We will present in this section the basic principles of the P.M.L model in the general case of an evolution problem. This will allow us to generalize it to other models of wave propagation. Consider a general evolution problem of the following form, posed initially in the space IRm (1)

(a) @t v ? A@x v ? B@y v = 0 (b) v(t = 0) = v0 ;

where v is a m-vector, A is a m  m matrix and B@y =

n X j =2

Bj @yj with Bj some m  m matrices. Moreover, we

assume that the initial condition v0 is supported in IRn? = f(x; y2 ; ::; yn ) 2 IRn ; x < 0g as shown in Figure 1. y PML support of initial data

0

x

Figure 1: Geometry of the problem. We would like to replace problem (1) by an equivalent one posed in the left half-space. Following the basic principle of the PML model, that is to couple the equation in the left half-space with an equation in the right half-space such that there is no re ection at the interface and that the wave decreases exponentially inside

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Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media5 the layer. We rst introduce the following system

v = vk + v? @t vk ? B@y v = 0 @t v? ? A@x v = 0 ;

(2)

where index k means that we keep only the derivative parallel to the interface, i.e. the y-derivative (while index ? means that only the x-derivative is considered). It is easy to see that system (2) is equivalent to(1)-(a).

Secondly we introduce some positive scalar function d(x), which will play the role of the damping factor and we de ne a new wave, u, solution of (1)-(a) in the left half-space and satisfying a new system in the right half-space, involving a damping on the normal component, so that u satis es (3) and (4)

@t u ? A@x u ? B@y u = 0; x < 0 u(t = 0) = v0 ; x < 0 ; u = u k + u? @t u? + d(x)u? ? A@x u = 0; x > 0; @t uk ? B@y u = 0; x > 0; u(t = 0) = 0; x > 0 :

Another way to set the PML model is to assume that d(x) is extended by zero in IRn?

d(x) = 0; 8x  0 ; and that (uk ; u?) is sought in IRn solution of (5)

@t u? + d(x)u? ? A@x u = 0 @t uk ? B@y u = 0 u(t = 0) = v0 with u = uk + u? :

If we look for the stationary solutions of system (1) with frequency !, we get

i!v^ ? A@x v^ ? B@y v^ = 0 : In the same way, the stationary solutions of system (5), satisfy (i! + d(x))^u? ? A@x u^ = 0 i!u^k ? B@y u^ = 0 with u^ = u^k + u^? ; or, equivalently

i!u^? ? i! +i!d(x) A@x u^ = 0 i!u^k ? B@y u^ = 0 with u^ = u^k + u^? : We can remark now, that u^ and v^, satisfy the same equations in the area where d is zero, whereas inside the layer, the PML model consist in the simple substitution

@x ! @x~ = i! +i!d(x) @x ; RR n3471

6

F. Collino and C. Tsogka

which implies the complex change of variables

Z x x~(x) = x ? !i d(s)ds :

0 We can study now the properties of this model using a plane wave analysis. We seek particular solutions of system (1) in the following form v = v0 e?i(kx x+ky y?!t); vo 2 IRm (6) k = (kx ; ky ) 2 IR  IRn?1 ; ! 2 IR : Formula (6) is a plane wave propagating in the direction k with the phase velocity !=jkj. In order to be a solution of problem (1), V has to satisfy the following relations (7) iv0 ! ? iAv0 kx ? iBv0 ky = 0 : In the same way, we seek particular solutions of system (5) in the following form uk = ak e?i(kx x~(x)+ky y?!t) u? = a? e?i(kx x~(x)+ky y?!t) u = uk + u? :

In order to be a solution of problem (5), uk and u? have to satisfy the following relations iak ! ? iB (a? + ak )ky = 0   (8) (i! + d(x))a? ? i 1 ? id(x) A(a? + ak )k = 0 ; or, equivalently (9)

x

!

iak ! ? iB (a? + ak )ky = 0 ia? ! ? iA(a? + ak )kx = 0 :

Adding the two equalities of (9), gives (10) i(a? + ak )! ? iA(a? + ak )kx ? iB (a? + ak )ky = 0 : We can remark now that if we chose a? and ak such that a? + a k = v 0 ; equation (10) becomes the same as (7). Moreover from (9) we get (11)

ak = Bv0 k!y ;

a? = Av0 k!x ;

which gives the plane wave solution of (5). We then have the following property: The plane wave U solution of system (5) can be written in the form k R u = v0 e?i(kx x+ky y?!t) e? !x x d(s)ds ; and satis es :  u  v in the left half-space x  0, which means that we have no re ection at the interface : the layer model is perfectly mached.  u is damped in the right half-space,  the damping coecient in the absorbing layer is ku(x)k = e? k!x R x d(s)ds : kv(x)k Remark 1 Notice that the damping is exponentially decreasing and it depends on the direction of propagation of the wave. More precisely, it decreases very fast for a wave propagating normally to the interface and increases when the direction of propagation approaches the parallel to the interface. 0

0

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Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media7

3 Pml model for elastodynamics In this section we present the PML model for the continuous elastodynamic problem. As we have seen in section 2 we know how to construct a PML model for an evolution problem of the form (1). Thus, in order to apply the technique described in the previous section to elastodynamics we need to write the elastodynamic problem in the form of system (1). This can be easily done once we consider the mixed velocity-stress formulation for elastodynamics.

3.1 The elastodynamic problem

We consider the 2D elastodynamic problem written as a rst order hyperbolic system, the so called velocitystress system

% @v @t ? div A @ @t ? "(v)

(12)

= 0

+ initial condition.

= 0

We suppose as in the previous section that the initial condition is supported in IR2? . In (12) v = (vx ; vy ) denotes the velocity,  the stress tensor and % = %(x) the density. If u = (ux ; uy ) is the displacement then We denote "(u) the deformation tensor, i.e.,

v = @u @t : 



@ui + @uj : "ij (u) = 12 @x j @xi The stress tensor is related to the deformation tensor by Hooke's law

 = (u)(x; t) = C (x)"(u)(x; t) ; where C (x) is a 4  4 positive tensor having the classical properties of symmetry [1]. In system (12) we used A = A(x) = C ?1 (x) : Finally we identify the tensor  with the following vector (still denoted by )

 = [1 ; 2 ; 3 ]t ; 1 = xx ; 2 = yy ; 3 = xy : We can write (12) in the following matrix form k @ ? @ % @v @t = D @y + D @x in

k @v + E ? @v in ; A @ = E @t @y @x

with

Dk =

"

0 0 1 1 0 0 ; D? = ; 0 1 0 0 0 1

2

0 03 0 1 77 ; 1 05 2

6

E k = 64 RR n3471

#

"

2 6

E ? = 64

#

1 0 0

0 0 1 2

3 7 7 5

;

8

F. Collino and C. Tsogka

We can apply now the same technique as in section 2 and we get the following system in the Perfectly Matched Layer(x > 0) v = vk + v? k k @ % @v = D @t @y ? % @v@t + d(x)v? = D? @ @x ;

and

 = k + ? k k @v = E A @ @t? @y @ @v ; A @t + d(x)A? = E ? @x

where d(x) denotes the damping factor. In an homogeneous, isotropic elastic medium, the matrix C (x) depends on the Lame coecients (; ) of the medium. In this case system (12) can be written in the following form

@xx @vx @vy x @xx @xy % @v @t = @x + @y ; @t = ( + 2) @x +  @y @yy @vy @vx y @xy @yy % @v @t = @x + @y ; @t = ( + 2) @y +  @x @vx y ; @@txy =  @v @x +  @y ;

(13)

and the PML model becomes v = vk + v?

@ + d(x))v? = @xx % ( @t x @x @ @ % ( @t + d(x))vy? = @xxy @ + d(x))? = ( + 2) @vx ( @t xx @x @v @ ? = x ( @t + d(x))yy @x @ @v ? = y ( @t + d(x))xy @x

(14)

;  = k + ? k

x = @xy ; % @v @t @y k @ @v y ; % @t = @yyy

k y ; @@txx =  @v @y k y ; @@tyy = ( + 2) @v @y

k x ; @@txy =  @v @y :

3.2 Plane wave analysis

3.2.1 In nite absorbing layer

In the case of a homogeneous, isotropic elastic medium, following the same technique as in section 2, we can show that the plane waves, Uj ; j = p; s, solution of system (13) can be written in the following form  x  y Up = Ap d~p ei!Vp (t? Vp ) Pressure wave (15) ? x  y ) Shear wave ; Vs Us = As d~s ei!Vs (t? cos( ) +sin( )

sin( ) +cos( )

p

p

where Vp = ( + 2)=% is the Pressure waves velocity, Vs = =% is the Shear waves velocity,  gives the direction of wave propagation, d~j ; j = p; s de nes the direction of particle motion and Aj ; j = p; s is the amplitude of the waves. We can remark then that the plane waves, Uej ; j = p; s, solutions of system (14) can be written as  xp  y ) Pressure wave Vp Uep = Ap d~p ei!Vp (t? (16) ?  xs  y ) Shear wave ; Vs Ues = As d~s ei!Vs (t? cos( )~ +sin( )

sin( )~ +cos( )

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Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media9

where we simply substituted x by x~j ; j = p; s in (15) and where x~j ; j = p; s is de ned in the same way as in section 2 (! replaced by !Vj ) Z x i x~j (x) = x ? !V d(s)ds; j = p; s : j 0 Moreover we can show that Uej ; j = p; s satisfy :  Uej  Uj ; for j = p; s in the left half-space x  0 (no re ection),  Uej ; j = p; s are damped in the right half-space,  the damping coecient in the absorbing layer is kUep (x)k = e? Vp R x d(s)ds ; kUp (x)k kUes (x)k = e? Vs R x d(s)ds : kUs (x)k cos

cos

0

0

3.2.2 Finite absorbing layer

In practice, we take a nite absorbing layer by introducing a boundary at x = , with a Dirichlet condition as we can see in Figure 2. y PML v = 0

support of initial data

δ

0

x

Figure 2: A nite PML layer. This new boundary produces a re ection, but since the wave decreases exponentially in the layer, the re ection coecient becomes quickly very small. This coecient depends on the choice of d(x) and on the size  of the layer. In order to study the PML layer properties in this case, we recall some classical results for the elastodynamic problem. Consider the elastodynamic problem with a homogeneous Dirichlet condition(~v = 0) on the boundary x = 0 as shown in Figure 3. Take for example the case of an incident plane wave P d~p
~ x U inc = Ainc d~p ei!Vp (t? Vp ) ; d~p = (cos ; sin ) : p

y r

Us r

Up

θ

2

θ

1

x

θ inc

Up

RR n3471

Figure 3: Re ection of an incident P wave.

10 by

F. Collino and C. Tsogka

We know then (c.f [1]) that the incident wave is re ected into a pressure wave Upr and a shear wave Usr given


d~r ~ x

p Upr = Arp d~rp ei!Vp (t? Vp ) ; p ~s
~ x Usr = Ars d~rs ei!Vp (t? Vp ) ;

and we have the re ection coecients

d~rp = (? cos 1 ; sin 1 ) d~rs = (? sin 2 ; cos 2 ); p~s = (cos 2 ; sin 2 ) ;

kUpr k = cos( ? 2 ) Rpp = kU inc p k cos( + 2 ) r k U s k = sin 2 Rps = kU inc p k cos( + 2 ) with sin 2 = Vs sin =Vp :

In the same way in the case of an incident plane wave S we have y r

Up r

Us

θ

θ

2

x

1

θ inc

Us

Figure 4: Re ection of an incident S wave. r s k = sin( + 2 ) Rss = kkUUinc s k sin( ? 2 ) k Upr k 2 Rsp = kU inc k = sin(sin  ? 2 ) s with cos 2 = Vp cos =Vs :

We can consider now the case of the nite PML layer. Given that the plane waves, Uj ; j = p; s, solution of system (13) are given by equations (15) we can compute the re ection coecients induced by the PML layer of length .

The case of a plane wave P

As we have shown previously there is no re ection at the interface x = 0 when the plane wave Uep penetrates y PML ~r ~r Up Us v=0 ~ Up =Up

0

~ Up

δ

x

Figure 5: A plane wave P the lossy medium. After traveling a length  we can compute Uep using the formula (16), obtaining  R Uep = Up e? Vp d(s)ds : cos

0

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Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media11

Then the plane wave Uep , gives at the boundary x = , two re ected waves Uepr and Uesr , which are damped till penetrating again the elastic medium at the interface x = 0. It is easy to see that the re ection coecient is given in this case by R er  = kUp (x)k = Rpp e?2 Vp  d(s)ds ; x < 0 Rpp kUp (x)k R r k  = Ues (x)k = Rps e?2 Vp  d(s)ds ; x < 0 : Rps kUp (x)k cos

(17)

cos

The case of a plane wave S

0

0

In the case of an incident plane wave S, we obtain similarly r  = kUes (x)k = Rss e?2 Vs R  d(s)ds ; x < 0 Rss kUs(x)k er R k  = Up (x)k = Rsp e?2 Vs  d(s)ds ; x < 0 : Rsp kUs (x)k cos

(18)

cos

0

0

Remark 2 Relations (17) and (18) imply that the re ection can be made as weak as desired by choosing the damping factor d(x) large enough. However this is no longer true when we consider the discrete PML model. As we will see in the next section, the discrete absorbing layer model is not perfectly matched. That is a consequence of the numerical dispersion, which introduces a re ection at the interface.

4 The discrete PML model In this section we present the discrete PML model for elastodynamics. To do so we introduce two numerical schemes for the discretization of system (12). The rst one is a nite-dierences scheme, introduced by Virieux in [22] and the second one is obtained using a new mixed nite element [3] in a regular mesh. The use of this nite element has the main advantage of leading to an explicit scheme (mass lumping), even in the case of an anisotropic elastic medium.

4.1 The Virieux nite-dierences scheme

This scheme uses a staggered grid formulation, so that, if vx is computed at the points (i; j ) (xi = ih; yj = jh) of a reference grid, then vy is computed at the points (i + 21 ; j + 12 ), xx and yy at (i + 21 ; j ) and xy at (i; j + 12 ). The disrcete PML model associated to the Virieux nite-dierences scheme can be written as follows (vx )ni;j = (vxx )ni;j + (vxy )ni;j (xx )in+;j ? (xx )ni?+ ;j (vxx )ni;j+1 ? (vxx )ni;j x (vxx )ni;j+1 + (vxx )ni;j + di =
t 2 %h n + n+ (vxy )ni;j+1 ? (vxy )ni;j y (vxy )ni;j+1 + (vxy )ni;j (xy )i;j ? (xy )i;j? + dj = ;
t 2 %h 1 2

1 2 1 2

1 2

1 2 1 2

(vy )ni ;j = (vyx )ni ;j + (vyy )ni ;j (vyx )ni +1;j (vyx )ni +1;j ? (vyx )ni ;j x + di
t (vyy )ni +1;j ? (vyy )ni ;j (vyy )ni +1;j y + dj
t 1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

RR n3471

1 2

1 2

1 2

1 2

1 2

1 2 1 2

n+ (xy )ni +;j ? (xy )i;j + (vyx )ni ;j = 2 %h n + y n + (vy )i ;j (yy )i ;j ? (xy )ni +;j = ; 2 %h 1 2

1 2

1 2

1 2

1 2 1 2

1

1 2

1 2

1

1 2 1 2

1 2

1 2

12

F. Collino and C. Tsogka x )n+ + ( y )n+ (xx )in+;j = (xx xx i ;j i ;j x )n+ + ( x )n? x )n+ ? ( x )n? (xx (xx xx i ;j xx i ;j (vx )ni ;j ? (vx )ni;j i ;j i ;j x + d = (  + 2  ) i
t 2 h n + n ? n ? n + y ) y ) y ) (xx ? (xx (y ) + (xx (v )n ? (vy )ni ;j? i ;j i ;j + dy xx i ;j i ;j =  y i ;j ; j
t 2 h 1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1

1 2

1 2

1 2

1 2

1 2

x )n+ + ( y )n+ (yy )in+;j = (yy yy i ;j i ;j x )n+ ? ( x )n? x )n+ + ( x )n? (yy (yy yy i ;j yy i ;j (vx )ni ;j ? (vx )ni;j i ;j i ;j x + d =  i
t 2 h n + n ? n ? n + y y y y (yy )i ;j ? (yy )i ;j (yy )i ;j + (yy )i ;j (vy )ni ;j ? (vy )ni ;j? y + dj = ( + 2) ;
t 2 h 1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1

1 2

1 2

1 2

1 2

1 2

n+ = ( x )n+ + ( y )n+ (xy )i;j xy i;j xy i;j x )n+ ? ( x )n? x )n+ + ( x )n? (xy (xy (vy )ni ;j ? (vy )ni? ;j xy i;j xy i;j i;j i;j x + di =
t 2 h n + n + n ? n ? y y y y (xy )i;j ? (xy )i;j ( ) + (xy )i;j (vx )ni;j ? (vx )ni;j y xy i;j + d =  : j
t 2 h where we used the notation ik = i + k and j k = j + k. The previous system of equations corresponds to the PML model in the corners, where we need a damping in both directions. For the computation of the solution inside the PML layers in the x direction we use the same system of equations with dy = 0, while for PML layers in the y direction we take dx = 0. We present in Figure 6 the values of dx and dy for the dierent PML layers. 1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2

1 2 1 2

1 2

1 2

1 2

1

1 2

y

111 000 0000000000000 1111111111111 000 111 000 111 000 111 00 11 PML-x: dy = 0 000 111 000 111 00 11 000 111 0000000000000 1111111111111 000 111 000 111 000 111 000 111 000 111 000 111 PML-y: dx = 0 000 111 000 111 000 111 000 111 000 111 000 111 00000 111 11PML-x,y 111 000 000 111 000 111 000 111 000 111 000 111 000 111

support of

initial datas

111 000 000 111 000 111 000 111 000 111 000 111 000 111

00 11 00000000000000 11111111111111 11 00 00 0000000000000011 11111111111111 00 11 00 0000000000000011 0011111111111111 11

x

Figure 6: PML layers.

4.2 The nite-element scheme

For this scheme, the velocity is approximated by piecewise constant functions and the stress tensor  by Q1 functions with some particular continuity properties :  xy is continuous in both directions : it is approximated by Q1 continuous functions.  xx is continuous only in the x direction : it is approximated by Q1 functions continuous in x and discontinuous in y direction. INRIA

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media13

 yy is continuous only in the y direction : it is approximated by Q1 functions continuous in y and

discontinuous in x direction. We present in Figure 7 the mixed nite element used.

11 00 11 00

11 00 00 11

h σ xx b σ xx

(vx, vy) 11 00 00 11 00 11

d

σyy g

σyy 11 00 00 11

1 0 1 0

11 00 00 11

σxy

Figure 7: The nite element On a regular grid, the velocity v = (vx ; vy ) is computed at the points (i ; j ) and the stress tensor at the points (i; j ). We consider the general case of an anisotropic elastic material described by the elasticity matrix C , and we denote by A the inverse of C 1 2

2

a11 a12 a13 A = C ?1 = 4 a12 a22 a23 a13 a23 a33

1 2

3 5

The numerical scheme can then be written in the following way (see [4] for details). For the rst equation of system (12) we have (vx )ni +1;j ? (vx )ni ;j 1 (h )n+ ? (h )n+ + (b )n+ ? (b )n+ = xx i;j xx i ;j xx i;j
t 2%h xx i ;j  n + n + n + +( ) ? ( ) + ( ) ? ( )n+ 1 2

1 2

1 2

1 2

1 2

1

1

1 2

xy i;j

xy i;j 12

1 2 1

1

1

1 2

1

1 2

1

xy i1 ;j21

xy i1 ;j2

(vy )ni +1;j ? (vy )ni ;j 1 ( )n+ ? ( )n+ + ( )n+ ? ( )n+ = xy i;j xy i ;j xy i;j
t 2%h xy i ;j  d )n+ ? ( d )n+ + ( g )n+ ? ( g )n+ +(yy yy i ;j yy i;j yy i ;j i;j 1 2

1 2

1 2

1 2

1 2

1

1

1 2

1 2

1 2

1 2 1

1

1

1 2 1

1

1

1 2

1 2

The second equation of system (12) results in the following 5  5 local system, coupling the ve degrees of freedom associated to  at each vertex (see Figure 7). n? ni;j+ ? i;j = A?h 1 Bhx + A?h 1 Bhy
t 1 2

where

RR n3471

 h  = xx ; 2 2a11 6 6 0 6 6 1 Ah = 2 666 a12 6 6 a12 4 2a13

1 2

b ;  d ;  g ; xy t xx yy yy 0 a12 a12 2a13 2a11 a12 a12 2a13 a12 2a22 0 2a23 a12 0 2a22 2a23 2a13 2a23 2a23 4a33

3 7 7 7 7 7 7 7 7 7 5

14

F. Collino and C. Tsogka

and

Bhx = h1 [B1x ; B2x ; 0; 0; B5x ]t

Bhy = h1 [0; 0; B3y ; B4y ; B5y ]t

with

B1x = (vx )ni ;j ? (vx )ni? ;j B2x = (vx )ni ;j? ? (vx )ni? ;j? B3y = (vy )ni ;j ? (vy )ni ;j? B4y = (vy )ni? ;j ? (vy )ni? ;j? B5x = (vy )ni ;j ? (vy )ni? ;j + (vy )ni ;j? ? (vy )ni? ;j? B5y = (vx )ni ;j ? (vx )ni ;j? + (vx )ni? ;j ? (vx )ni? ;j? In the case of an isotropic elastic material the matrix A?h 1 is given by 1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

2

a 6 6 c 6 6 A?h 1 = 666 b 6 6 b 4

c a b b

b b a c

b b c a

0 0 0 0

0 0 0 0 d

with

1 2

1 2

1 2

1 2

1 2

1 2

3 7 7 7 7 7 7 7 7 7 5

2 2 2 a = 2(2(+2+)2?)  ; b = 2 ; c = 2( + 2) ; d = 2 :

(19)

We can write now the PML model associated to this scheme, using the same technique as for the Virieux scheme (vx )ni ;j = (vxx )ni ;j + (vxy )ni ;j (vxx )ni +1;j (vxx )ni +1;j ? (vxx )ni ;j x + di
t (vxy )ni +1;j ? (vxy )ni ;j (vxy )ni +1;j y + di
t 1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

+ (vxx )ni ;j 1 (h )n+ ? (h )n+ + (b )n+ ? (b )n+ = xx i;j xx i ;j xx i;j 2 2%h xx i ;j + (vxy )ni ;j 1 ( )n+ ? ( )n+ + ( )n+ ? ( )n+ = xy i ;j xy i;j xy i ;j 2 2%h xy i;j 1 2

1 2

1 2

1 2

1 2

1 2

1

1 2

1 2

1 2

1 2

1

1

1 2 1

1

1 2 1

1

1

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2





1 2

(vy )ni ;j = (vyx )ni ;j + (vyy )ni ;j (vyx )ni +1;j ? (vyx )ni ;j (vyx )ni +1;j + (vyx )ni ;j 1 ( )n+ ? ( )n+ + ( )n+ ? ( )n+ x + d = xy i;j xy i ;j xy i;j i
t 2 2%h xy i ;j (vyy )ni +1;j + (vyy )ni ;j (vyy )ni +1;j ? (vyy )ni ;j 1 (d )n+ ? (d )n+ + (g )n+ ? (g )n+ y + d = yy i ;j yy i;j yy i ;j i
t 2 2%h yy i;j and n+ + (y )n+ ni;j+ = (x )i;j i;j x n+ x n? (x )ni;j+ ? (x )ni;j? x ( )i;j + ( )i;j = M ?1 B x + d i h h
t 2 n? y n? y n+ (y )ni;j+ ? (y )i;j y ( )i;j + ( )i;j = M ?1 B y + d i h h
t 2 1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1

1 2

1 2

1 2

1 2

1 2

1

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1

1 2 1

1

1 2 1

1

1

1 2

1 2





1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

INRIA

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media15

5 Dispersion Analysis In this section we will study the properties of the discrete PML model for elastodynamics in terms of a numerical dispersion analysis, that is a plane wave analysis. We present this analysis only for the nite-element scheme. We consider the case of an homogeneous isotropic elastic medium characterized by the Lame coecients ;  and the density %. The PML layer is in the x ? direction (Figure 1) and we consider the case of a in nite layer. We look for plane wave solutions of the form (vl )n+ = (^vl ) e?iky (j )h+i!(n+ )
t ; l =?; k 1 2

1

1

2 x i 12 ;j2 21 x i 12 1 1 (vyl )ni 21+;j2 12 = (^vyl )i 21 e?iky (j 2 )h+i!(n+ 21 )
t ; h;l )n = (^ h;l )i e?iky jh+i!n
t ; (xx i;j xx b;l n b;l (xx )i;j = (^xx )i e?iky jh+i!n
t ; d;l )n = (^ d;l )i e?iky jh+i!n
t ; (yy i;j yy g;l n g;l )i e?iky jh+i!n
t ; (yy )i;j = (^yy l )n = (^ l )i e?iky jh+i!n
t ; (xy i;j xy

and we set

l =?; l =?; l =?; l =?; l =?; l =?;

k k k k k k

(^vx )i = (^vxk )i + (^vx? )i ; (^vy )i = (^vyk )i + (^vy? )i h )i = (^ h;k )i + (^ h;? )i ; (^ b )i = (^ b;k )i + (^ b;? )i (^xx xx xx xx xx xx d d; k d; ? g g; k g;? )i (^yy )i = (^yy )i + (^yy )i ; (^yy )i = (^yy )i + (^yy k )i + (^? )i : (^xy )i = (^xy xy 1 2

1 2

1 2

1 2

1 2

1 2

Plugging these expression into the discrete scheme, gives + (^xy )i At k = ?Ay (^xy )i 2%h
t (^vx )i h h Ci ? iky h=2 (^xx )i ? (^xx )i = e (^ v )
t x i 2%h b )i ? (^ b )i (^  xx xx ? ik h= 2 y +e 2%h g )i + (^ d )i (^yy At (^vk ) yy = ? Ay y
t i 2 %h  Ci ? ky h  (^xy )i ? (^xy )i ; (^ v ) = cos
t x i 2 %h 1

1 2

1 2

1

1 2

1

1

1 2

1 2

1

1 2

At (^k ) = ?Ay (^vx )i + (^vx )i?
t xy i 2h   (^ Ci (^? ) =  cos ky h vy )i ? (^vy )i? ;
t xy i 2 h 1 2

1 2

1 2

1 2

At (^h;k ) = ?bAy (^vy )i + (^vy )i?
t xx i h Ci (^h;? ) = ae?iky h=2 + ceiky h=2  (^vx )i
t xx i At (^b;k ) = ?bAy (^vy )i + (^vy )i?
t xx i h Ci (^b;? ) = aeiky h=2 + ce?iky h=2  (^vx )i
t xx i 1 2

1 2

RR n3471

1 2

1 2

1 2

1 2

? (^vx )i? h

? (^vx )i? h

1 2

1 2

;

16 and

F. Collino and C. Tsogka

At (^d;k ) = ?Ay a(^vy )i + c(^vy )i?
t yy i h   Ci (^d;? ) = 2b cos ky h (^vx )i ? (^vx )i?
t yy i 2 h + a (^ v ) c (^ v ) y y ? At g;k i i
t (^yy )i = ?Ay  h Ci (^g;? ) = 2b cos ky h (^vx )i ? (^vx )i? ;
t yy i 2 h 1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

where a; b; c are given by (19) and At ; Ay and Cl are de ned by 







t ; A = 2i sin ky h At = 2i sin !
y 2    2 !
t t Cl = 2i sin 2 + dl
t cos !
2 ; l = i; i : 1 2

After some tedious calculations, we can rewrite the previous system of equations with v^x ; v^y as the only unknowns. We obtain A2t (^v ) =  A2 (^vx )i + 2(^vx )i + (^vx )i?
t2 x i % y 4h2 ! (^vx )i ? (^vx )i? (^vx )i ? (^vx )i a + c cos( k h ) y + %D ? h2 Di h2 Di i   (^vy )i ? (^vy )i (^vy )i ? (^vy )i? ?Ay % cos ky2h + 2 2h Di 2h2 Di ! (^vy )i ? (^vy )i? + h 2 Di (20) A2t (^v ) = 1 A2 c(^vy )i + 2a(^vy )i + c(^vy )i?
t2 y i % y 2h2 !   (^vy )i ? (^vy )i (^vy )i ? (^vy )i?  k h y 2 + % cos 2 h 2 Di Di ? h 2 Di Di   (^vx )i ? (^vx )i (^vx )i ? (^vx )i? b k h y + ?Ay % cos 2 h2 Di h2 Di ! (^vx)i ? (^vx )i? + 2h2 D ; i 3 2

1 2

1 2

3 2

1 2

1 2

1 2

1 2

1 2

1

3 2

1 2

1 2

1 2

1

3 2

1 2

1 2

1 2

3 2

1 2

1 2

1 2

3 2

1

1 2

1 2

1 2

1 2

1 2

3 2

1 2

1 2

1

1 2

3 2

1 2

with

Dl = ACl ; l = i; i : 1 2

t

Let us consider now the case of a medium with a unique in nite layer for which

di =



if i  0 d1 if i > 0 ; di =

0

1 2



if i < 0 d1 if i > 0 ;

0

INRIA

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media17

d remaining undetermined yet, and let us look for particular solution of the form 1 2

P wave (V^ )i

(21)

(V^ )i

1 2

(V^ )i

1 2

1 2

1 2

1 2

1 2

1 2

S wave (V^ )i

= d~p e?ikx (i )h + Rpp d~rp eikx (i )h +Rps d~rs eikx (i )h for i  0 = Tpp dep e?iekx (i )h + Tps des e?iekx (i )h for i  0

1 2

1 2

1 2

= d~s e?ikx (i )h + Rss d~rs eikx (i )h +Rsp d~rp eikx (i )h for i + 12  0 = Tss des e?iekx (i )h + Tsp dep e?iekx (i )h for i  0 ; 1 2

1 2

1 2

1 2

1 2

where

1 2

"

(^v ) (V^ )i = (^vx)i yi 1 2

#

1 2

:

1 2 1 2

dp

Tpp

d

8

d 1/2

Tps

Rpp Rps 1 0

1 0

1 0

1 0

1 0

1 0

-3/2 -1/2

1 0

1/2

3/2

5/2

1 0

1 0

i

Figure 8: Schematic view on the plane waves solution in a single layer discrete model The equations (20) are satis ed for i (free medium) and (23) (lossy medium). 2 X t (22)
t2 (23) with

1 2

6= 0 if kx and ekx are solutions of the two relations of dispersion (22)

v^x = 1 K^ 11 K^ 12 v^x h2 K^ 12 K^ 22







"

#

Xt2 v^x = 1 Ke 11 Ke 12
t2 v^x h2 Ke 12 Ke 22 



v^x v^x



v^x v^x



; ;

K^ 11 = Vp2 Xx(1 ? 4Xy ) + Vs2 Xy (1 ? Xx ) K^ 22 = Vp2 Xy (1 ? 4Xx) + Vs2 Xx (1 ? Xy ) q K^ 12 = (Vp2 ? Vs2 ) XxXy (1 ? Xx )(1 ? Xy )     X = sin2 !
t ; X = sin2 kx h t

2

x

2

(V 2 ? 2V 2 )2 Xy = sin2 ky2h ;  = p 4V 4 s ; = 41 



p

RR n3471

18

F. Collino and C. Tsogka

and

Ke 11 = Vp2 Xe x2 (1 ? 4Xy ) + Vs2 Xy (1 ? Xex ) D e 2 e K22 = Vp Xy (1 ? 4Xex ) + Vs2 Xe x2 (1 ? Xy ) D (Vp2 ? Vs2 ) q e e K12 = X X (1 ? Xe )(1 ? X ) De ! x y ? x
y !
t + d1
t cos ? !
t
e 2 i sin k h x 2 2 2 :
? Xex = sin 2 ; De = 2i sin !
2 t Now, the system of equations at i = 0 gives the value of the re ection coecients Rpp (resp. Rss ) and Rps (resp. Rsp ). We have used the software MAPLE to compute its Taylor expansion with respect to the discretization e

1 2

step and we obtain

d1 ? 2d 4! d1 ? 2d Rps = 4! d1 ? 2d Rss = 4! d1 ? 2d Rsp = 4! Rpp =

(24)

with

1 2

1 2 1 2

1 2

? ?


1 ? 2r12 (Vp2 + Vs2 ) Vr0 h + O(h2 )

p


3 ? 2r12 (Vp2 + Vs2 ) ky h + O(h2 ) ?
1 ? 2r22 r12 (Vp2 + Vs2 ) r0 h + O(h2 )

Vp

?


2r22 r12 (Vp2 + Vs2 ) ? 1 ? 2r22 ky h + O(h2 )

q 2 k2 r0 = !2 ? Vp2 ky2 ; r1 = !y2 ; r2 = VVs2 ; r3 = r1 :

2 From relations (24), we can easily remark that the best choice for d consists in taking p

1 2

d = d21 : 1 2

In that case, the rst order terms in (24) disappear and we obtain that the re ection coecients are roughly proportional to d1 (d1 + i!)h2. Two consequences can be deduced from this result. First the numerical scheme is consistent with the continuous PML model : the spurious re ection for a given value of d1 is only due to the dispersion of the numerical scheme. Let us remark that the second order accuracy is recovered in the h2 dependency of the re ection coecients. Then, from a more practical point of view, this dispersion analysis implies that we can not use a value of d1 too large for a given discretization step. In the case of a nite length layer with a given number of nodes, the layer is characterized by (

(d ; d ; :::; dnl ? ); (d1 ; d2 ; :::; dnl ); 1 2

3 2

1 2

we are then led to nd a trade o between choosing the di+ 's too weak (which would imply a strong re ection due to the Dirichlet boundary condition) or too large (which would imply spurious re ections due to the dispersion). A partial answer to this problem is to use a smooth pro le for d(x), (di+ = d(h(i + 21 )), as is done in the classical layers models, [17]. An alternative is to determine the best coecients for d(x) by minimizing the numerical re ection coecients as is done in [10] for the Helmholtz equation. 1 2

1 2

6 Numerical Results In the following examples, we simulate elastic wave propagation in a 2-D unbounded medium. We will present several results including the case of heterogeneous, anisotropic elastic media. In order to show the generality of the method we give some numerical results for both schemes : the Virieux nite-dierences scheme and the nite-elements scheme. A more complete analysis is presented for the nite-element scheme, namely the discrete re ection coecients are computed as a function of the incident angle. Let us rst set some notation and give the characteristics of the numerical examples that will follow. We consider an unbounded domain in 2D, INRIA

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media19

occupied by an elastic material and we suppose that the initial condition (or the source) is supported in a part of . The material will be characterized by its wave velocities Vp (pressure) and Vs (share) in the isotropic case and by its elasticity matrix C in the anisotropic case. In order to solve numerically the elastodynamic problem in we de ne a bounded domain D of a simple geometry (here it will be a square), containing the support of the initial data, with absorbing layers (PML) on all four boundaries. For the discretisation of the problem we take a regular grid of D composed by N  N square elements of edge h = 1=N . The time step is then computed following the CFL condition for each scheme, more precisely we take : 
t = p0:9h for the Virieux scheme, 2Vp


t = Vh for the nite-elements scheme. p

For our numerical experiments we use an explosive source located at the point S = (xs1 ; xs2 ), that is

f (x; t) = F (t)~g(r) ; with (25)

(

?22 f02 (t ? t0 )e? f (t?t )

if t  2t0 ; 0 if t > 2t0 t0 = f1 ; f0 = Vhs N1 is the central frequency, 0 L NL is the number of points per S wave length;

F (t) =

2 2 0

0

2

and ~g(r) is a radial function : s s 2 3 ~g(r) = 1 ? ar 2 1Ba x1 ?r x1 ; x2 ?r x2 q r = (x1 ? xs1 )2 + (x2 ? xs2 )2 ; a = 5h ; 

(26)







with 1Ba the characteristic function of Ba the disk of center S and and radius a. In the absorbing layers we use the following model for the damping parameter d(x)  2 d(x) = d0 x

 ) (see where  is the length of the layer and d0 is a function of the theoretical re ection coecient (R = Rpp relation (17) for  = 0)   d0 = log R1 32Vp :

We rst present some results in an homogeneous isotropic medium.

6.1 Homogeneous, isotropic elastic medium-Virieux scheme

We consider here an elastic medium with Vp = 2000m=s; Vs = 1400m=s. The characteristics of the discretization are N = 200; h = 0:15m; S = (7:5m; 7:5m); NL = 20;  = 10h and R = 0:001: We rst present some snapshots of the solution on the normal scale in Figure 9. We can remark in the snapshots presented on Figure 9 that we can see no re ection when the results are presented on the normal scale. If we want to see some re ection we have to magnify the results. In this example the re ection coecient is about 0:1% as we remark by magnifying the results by a factor 200 (see Figure 10).

RR n3471

20

F. Collino and C. Tsogka

t = 2:76ms

t = 5:53ms

t = 8:35ms

t = 11:1ms

t = 13:8ms

t = 16:7ms

Figure 9: Snapshots of 2-D elastic nite-dierences simulations, using the PML absorbing layer model. The p snapshots are the norm of the velocity kvk = v12 + v12

Figure 10: The snapshot at t = 13:8ms magni ed here by a factor 200.

6.2 Heterogeneous, isotropic elastic medium-Virieux scheme

We consider an heterogeneous elastic medium, the interface is parallel to the x1 axis and located in the middle of the frame (see Figure 11). The wave velocities in the two media are Vp1 = 2000m=s; Vs1 = 1400m=s (the upper medium), Vp2 = 1000m=s; Vs2 = 700m=s (the lower medium). All the other parameters are the same as in section 6.1 where we take Vs = (Vs1 + Vs2 )=2 for the computation of f0 (equation 25) and Vp = Vp1 to compute
t. The source is now located in the lower medium near the interface (at point(15m; 16:2m)). We present some snapshots of the solution on the normal scale on Figure 11. The re ection coecient for this example is about 0:1% as we can see on Figure 12.

INRIA

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media21

t = 8:35ms

t = 13:8ms

t = 16:7ms

Figure 11: Snapshots of 2-D elastic nite-dierences simulations, using the PML absorbing layer model. The p snapshots are the norm of the velocity kvk = v12 + v12 .

(1)

(2)

Figure 12: Snapshot of 2-D elastic nite-dierences simulations, using the PML absorbing layer model. We represent here the norm of the velocity at time t = 16:7ms, (1) is magni ed by a factor 100 and (2) by a factor 1000.

6.3 Heterogeneous, isotropic elastic medium. Finite-element scheme

The heterogeneous elastic medium considered here is characterized by the velocity model presented on Figure Vp = 2:1 and V = 1:6V . For the discretization we take V and V piecewise constants (one 13, we have max p s p s min Vp value per element).

RR n3471

22

F. Collino and C. Tsogka The Vp velocity model

Vp (m/s) above 1365 1300 - 1365 1235 - 1300 1170 - 1235 1105 - 1170 1040 - 1105 975 - 1040 910 - 975 845 - 910 780 - 845 715 - 780 below 715

Vp Figure 13: The velocity model for the heterogeneous medium, max min Vp = 2:1 and Vp = 1:6Vs . The size of the grid is 200  200, h = 0:15m, NL = 10 and the source is located at the point (15m; 3m). We present, on Figure 14, three experiments with dierent sizes of the absorbing layer, namely, in the rst we take  = 5h ? R = 0:01, in the second  = 10h ? R = 0:001 and nally  = 20h ? R = 0:0001.

t = 3:55ms

t = 7:1ms

t = 10:65ms

Figure 14: Snapshots of 2-D elastic nite-elements simulations, using the PML absorbing layer model with p  = 5h. The snapshots are the norm of the velocity kvk = v12 + v12

INRIA

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media23

5h

10h

20h

t = 14:2ms

t = 17:75ms Figure 15: Snapshots of 2-D elastic nite-elements simulations, using the PML absorbing layer model with  = 5h,  = 10h and  = 20h. All snapshots are magni ed by a factor of 10. on Figure 15 we present the snapshots of the solution magni ed by a factor of 10. We remind that the re ection coecients depend on the length of the layer, more precisely we have :  A re ection of about 1% for  = 5h,  a re ection of about 0:1% for  = 10h,  and a re ection of about 0:01% for  = 20h. These re ection coecients which are theoretical are con rmed by the numerical examples.

6.4 Homogeneous, anisotropic elastic medium. Finite-element scheme

In this example we consider an anisotropic elastic solid, the apatite. We want to point out that in the case of anisotropic elastic materials only lower order absorbing boundary conditions are known and they are quite dicult to implement [2]. On the contrary the generalization of the PML model to the anisotropic case is straightforward and there are no supplementary diculties for the implementation. Moreover the results are very satisfactory as we can see in the following Figures. The characteristics of the problem are : N = 200, h = 0:15m, NL = 10 and the source is located at the center of the frame S = (15m; 15m). We present on Figure 16 three experiments, corresponding to :  = 5h ? R = 0:01,  = 10h ? R = 0:001 and  = 20h ? R = 0:0001. We can see on Figure 16 that there is no re ection when the results are presented on the normal scale as it was already the case for an homogeneous medium. However, we can see some re ection by magnifying these results as shown on Figure 17. The re ection coecients obtained numerically are close to the theoretical one, more precisely we obtain :  A re ection of about 1% for  = 5h,  a re ection of about 0:1% for  = 10h, RR n3471

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t = 6:6ms

t = 13:2ms

t = 16:5ms

Figure 16: Snapshots of 2-D elastic nite-elements simulations, using the PML absorbing layer model with p  = 5h. The snapshots are the norm of the velocity kvk = v12 + v12 5h

10h

20h

t = 13:2ms

t = 16:5ms Figure 17: Snapshots of 2-D elastic nite-elements simulations, using the PML absorbing layer model with p  = 5h,  = 10h and  = 20h. The snapshots are the norm of the velocity kvk = v12 + v12 magni ed by a factor 10.

 and a re ection of about 0:02% for  = 20h.

6.5 Re ection Coecients

In this section we present re ection coecients computed using numerical simulations. We consider an homogeneous isotropic elastic solid (Vp = 5:710m=s, Vs = 2:93m=s) and we use the nite-element scheme for the space discretization. INRIA

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media25 PML D1 PML PML

1S 0

D1

θ

PML

PML

Figure 18: Geometry for computing re ection coecients. Figure 18 shows the geometry of the problem consisting of two computational domains D1 and D2 . To compute the re ection coecient we consider the solution at points near the upper boundary of D1 as shown on Figure 18. Denoting by (U1 )i (resp. (U2 )i ) the velocity at point Mi in D1 (resp. D2 ) we take 2 )i ? div(U1 )i (Rpp )(i ) = div(Udiv( U2 )i (27) curl( U ) ? curl(U1 )i : 2 i (Rps )(i ) = curl(U2 )i That is the expression for the re ection coecients in the case of a pure P-wave source. In the same way we can de ne 2 ) ? curl(U1 ) (Rss )(i ) = curl(Ucurl( U2 ) 2 ) ? div(U1 ) ; (Rsp )(i ) = div(Udiv( U2 ) in the case of a pure S wave source. For our examples we use a P-wave source function described by (25,26) with NL = 16. For this source function the rotational of the velocity is smaller then the divergence by a factor 10?10 so that the re ection coecient Rps is negligible.

Remark 3 Relation 27 is used to compute re ection coecients at each time step, to obtain results presented

on Figure 19 we integrate over time. The grid size is 400  200 for D1 , 400  300 for D2 and h = 0:5m for both domains. The source is located at point (200; 80) of the grid. As we can see on Figure 18 the domain is surrounded by absorbing layers, in which we take  4 d(x) = d0 x (28)

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





d0 = log R1 42Vp : We present three experiments with  = 5h ? R = 10?4,  = 10h ? R = 10?6 and  = 20h ? R = 10?8 . To

examine the performance of the PML model, comparisons with the 1st, 2nd and 3rd order Higdons absorbing boundary conditions are presented.

RR n3471

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F. Collino and C. Tsogka

Reflection Coefficients 0 -20 -40

Rpp (dB)

-60 -80 -100 Higdon 1 Higdon 2

-120

Higdon 3 PML 5

-140

PML 10 PML 20

-160 -180 0

10

20

30 40 50 Incidence Angle (degree)

60

70

80

Figure 19: Comparison of Rpp for the Higdon ABC's and PML.

Remark 4 The re ection coecients for the absorbing boundary conditions proposed by Higdon are the theoretical ones. We obtain them by applying an operator of the form m Y i=1





@ +V @ : i @t p @n

For the example we have taken  For m = 1 : 1 = 1,  For m = 2 : 1 = 1, 2 = Vp =Vs (= 1:95),  For m = 2 : 1 = 1, 2 = 1:5 and 3 = Vp =Vs (= 1:95).

As we can see on Figure 6.5 for almost all angles of incidence the PML absorbing layer model gives better results then the absorbing boundary conditions of rst and second order and this even for the 5h length absorbing layer. Compared to the third order absorbing condition the PML layers of 10h and 20h are substantially better except near the 60 degree angle. We want to point out that in this example we compare the re ection coecients provided by the continuous Higdon model to the one given by the discrete PML model. It is known that the Higdon re ection coecients increase when discretization is used. The numerical results presented in sections 6.1, 6.2, 6.3, 6.4 and 6.5 show the generality of the PML model which can be easily applied in several numerical schemes and gives satisfactory results even in the case of anisotropic heterogeneous elastic media. The re ection coecients are near the theoretical ones for all the presented experiments, so we can resume the following : For the considered models of the damping factor the re ection coecients are about 1% for a 5h PML layer length, below 0:2% for a 10h PML layer length and below 0:1% for a 20h PML layer length for all experiments. In practice one should choose the length of the layer as a function of the scale of the problem and the requested re ection coecient. In the numerical examples that we have presented the scale of the problems was about 15 wavelengths of P-wave (p ) in each direction, in this case a PML layer with  = 5h (that is about 0.5 p ) is sucient and gives better results than the second order absorbing condition. However, for larger scale computations the numerical dispersion will be more important and thus a PML layer with  = 5h would probably not give the expected results. Thus for large scale problems one should consider a PML layer with   10h. We want to point out, that even in this case the additional cost remains modest, given that the layer only represents a small portion of the total grid. Let us nally remark that the PML model can be easily extended to the case of Rayleigh waves. (Experimental results not shown in this paper have demonstrated a good absorbtion of Rayleigh waves).

7 Conclusions We have presented in this paper a generalization of the PML absorbing layer model for the elastodynamic problem in the case of heterogeneous, anisotropic media. The implementation of this model was presented in the 2D case but it can be straightforwardly extended to the 3D case. The superiority of this model compared INRIA

Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media27

to the absorbing boundary conditions proposed by Higdon was shown by numerical results in the case of an isotropic homogeneous medium. Moreover we have shown by several numerical examples the eciency of this model : remarkable results even in the case of heterogeneous, anisotropic media. Finaly we want to point out the ease of the generalization and the implementation of this model in the case of an anisotropic elastic medium in comparison with the complexity of the respective absorbing boundary conditions.

References [1] B. A. Auld. Acoustic Fields and Elastic Waves in Solids, volume I et II. Wiley, 1973. [2] E. Becache. Resolution par une methode d'equations integrales d'un probleme de diraction d'ondes elastiques transitoires par une ssure. PhD thesis, PARIS 6, 1991. [3] E. Becache, P. Joly, and C. Tsogka. Elements nis mixtes et condensation de masse en elastodynamique lineaire. (i) construction. C.R. Acad. Sci. Paris, t. 325, Serie I:545{550, 1997. [4] E. Becache, P. Joly, and C. Tsogka. Fictitious domain method applied to the scattering by a crack of transient elastic waves in anisotropic media : a new family of mixed nite elements leading to explicit schemes. pages 322{326. SIAM, 1998. [5] J.P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. Journal of Comp. Physics., 114:185{200, 1994. [6] J.P. Berenger. Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. Journal of Comp. Physics., 114(2):363{379, 1996. [7] J.P. Berenger. Improved PML for the FDTD Solution of Wave-Structure Interaction Problems . IEEE transactions on Antennas and Propagation., 45(3):363{379, march 1997. [8] B. Chalindar. Conditions aux limites arti cielles pour les equations de l'elastodynamique. PhD thesis, Saint Etienne (France), 1987. [9] F. Collino. Perfectly matched absorbing layers for the paraxial equations. J. of Comp. Phy., (131):164{180, 1996. [10] F. Collino and P.B. Monk. Optimizing the perfectly matched layers. Comput. Methods Apl. Mech. Engrg., pages {, to appear 1998. [11] B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31(139):629{651, Juillet 1977. [12] T. Hagstrom. On high-order radiation boundary condition. In B. Engquist and G. A. Kriegsmann, editors, Computational Wave Propagation, volume 86 of The IMA Volumes in Math. and Applications. Springer, 1997. [13] F. Hastings, J.B. Schneider, and S. L. Broschat. Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation . J. Acoust. Soc. Am., 100(5):3061{ 3069, November 1996. [14] R.C. Higdon. Absorbing boundary conditions for elastic waves . Geophysics., 56( 2):231{241, February 1991. [15] R.L. Higdon. Radiation boundary conditions for elastic wave propagation. SIAM J. Numer. Anal., 27(4):831{870, April 1990. [16] R.L. Higdon. Absorbing boundary conditions for acoustic and elastic waves in strati ed media. J. Comput. Phys., 101(2):386{418, 1992. [17] M. Israeli and S.A. Orszag. Approximation of radiation boundary conditions. Journal of Comp. Physics., 41:115{135, 1981. [18] C. Peng and M.N. Toksoz. An optimal absorbing boundary condition for elastic wave modeling . Geophysics., 60 (1):296{301, January-February 1995. RR n3471

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[19] C. M. Rappaport. Perfectly matched absorbing conditions based on anisotropic lossy maping of space. IEEE Micr. Guid. Wave Lett., 5(90), 1995. [20] A. Simone and S. Hestholm. Instability in applying absorbing boundary conditions to high-order seismic modeling algorithms . Geophysics., 63( 3):1017{1023, May-June 1998. [21] J. Sochacki, R. Kubichek, J. George, W.R. Fletcher, and S. Smithson. Absorbing boundary conditions and surface waves . Geophysics., 52 (1):60{71, January 1987. [22] J. Virieux. P-sv wave propagation in heterogeneous media :velocity-stress nite dierence method. Geophysics, 51(4):889{901, 1986. [23] L. Zhao and A.C. Cangellaris. A general approach to for developping unsplit- eld time-domain implementations of perfectly matched layers for FDTD grid truncation. IEEE trans. Microwave Theory Tech.., 44:2555{2563, 1996.

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