A mixed finite element approach for viscoelastic wave - CiteSeer

Computational Geosciences 8: 255–299, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands.

A mixed finite element approach for viscoelastic wavepropagation

Eliane Bécache, Abdelaâziz Ezziani and Patrick JolyINRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France

E-mail: [email protected]

Received 5 November 2003; accepted 10 September 2004

In this paper, we are interested in the modeling of wave propagation in viscoelastic media.We present a family of models which generalize the Zener’s model. We achieve its mathemat-ical analysis: existence and uniqueness of solutions, energy decay and propagation with finitespeed. For the numerical resolution, we extend a mixed finite element method proposed in [8].This method combines mass lumping with a centered explicit scheme for time discretization.For the resulting scheme, we prove a discrete energy decay result and provide a sufficient sta-bility condition. For the numerical simulation in open domains we adapt the perfectly matchedlayers techniques to viscoelastic waves [23]. Various numerical results are presented.

Keywords: energy dissipation, finite velocity propagation, mixed finite element, stabilityanalysis, viscoelastic waves, Zener’s model

1. Introduction

For the numerical simulation of wave propagation in solids, particularly for theapplications in geophysics, it is now commonly admitted that it is important to takeinto account the attenuation effects due to the visco-elastic nature of the medium[12,16]. One of the first problem that one meets is the choice of the appropriate visco-elasticity model, from the wide variety of models provided by the physical literature[10,21,30,42]. The generalized Zener’s models, that we shall consider in this paper,has the advantage to be presented in a unified framework (that we shall describe in sec-tion 2) and to offer a sufficiently large flexibility to permit to take into account most ofthe interesting phenomena from the geophysical point of view.

The numerical treatment of such media is already a rather old subject since it be-gan about 15 years ago (see, for instance, the works of Carcione et al. [15,17,18]). Mostof the methods developed up to now are based on rather simple finite difference meth-ods [11,20,38]. In this spirit, one reference work is due to Robertson et al. [41] whoachieved in particular a stability and accuracy analysis of their method. Note howeverthat their analysis, based on the Fourier method, is restricted to 1D homogeneous me-dia. It is known that a robust way to treat heterogeneous media is to use finite element

256 E. Bécache et al. / Mixed finite elements for viscoelastic wave propagation

methods. In the context of visco-elastic waves, it seems that there are very few existingworks, in particular, of mathematical nature. Let us mention however the work doneby Janovský et al. [36] in which the authors propose a finite element method in spaceand a quadrature rules in time to solve an isotropic viscoelastic problem based on a for-mulation in displacement and an integral representation of the viscoelastic model (seealso [43] for quasi-static problems), or more recently (of much less mathematical na-ture) by Isedman et al. [35] about space–time finite elements. Let us finally emphasizethe work of Ha et al. [32] where they are interested in the nonconforming finite elementapproximation of a viscoelastic complex model in the frequency domain, which makespossible to allow the coefficients of the model to depend on ω. Recently, in [8], wedevelop a new mixed finite element method for elastodynamics equations. This methodis especially designed for regular meshes but presents the interest to allow the couplingwith the fictitious domain method to treat complicated geometries of the propagationdomain [7,22]. Our objective in this paper is essentially to extend (and analyze) thismethod to (a class of) visco-elastic media. More precisely, there are two main parts inthis article:

• Provide a rather complete mathematical theory for the generalized Zener’s model.This is the object of section 3. The main interest of this section is to be preparatory tothe numerical analysis of section 4 and we do not pretend that neither the results, northe method we use to prove them, are very original. In fact, the mathematical analysisof wave propagation in viscoelastic media has been initiated about forty years ago.The basic general theory was developed in particular by Gurtin and Sternberg [31]and first existence and uniqueness results, which however can not be applied to themodels that we consider here, are due to Duvaut and Lions [26]. Concerning Zener’smodel, in dimension 1, we must cite the Ph.D. thesis of Canadas [14]. Since then,a lot of progress has been accomplished and, rather recently, a complete monographhas been written by Fabrizio and Morro [28] on the mathematical problems of lin-ear visco-elasticity. Their results, very general, are essentially based on the use ofLaplace transform methods and are applicable to a large class of models that includegeneralized Zener’s models. However, it is not clear that our results of section 3are contained in [28] (for instance, we use a different approach based on semi-grouptheory). In particular, the results concerning energy decay (theorem 3.3) and finitevelocity propagation (theorem 3.5) seem to be new.

• Construct a numerical method that generalizes the one of [8] and guarantees theexplicit nature of the numerical scheme as well as its stability even in the case ofheterogeneous media (section 4). This can be achieved by working with the stress-displacement formulation of the propagation equations (we used the velocity-stressformulation in [8]), by using mass lumping as in [8] (this is one of the main interestof the new finite element) and – this is the main point – by a pplying a specific timestepping of the constitutive law. The stability analysis is based on a discrete energydecay result (theorem 4.1) that mimics the continuous result of theorem 3.3. Weobtain a sufficient stability condition (theorem 4.2) that coincides in the 1D homoge-

E. Bécache et al. / Mixed finite elements for viscoelastic wave propagation 257

neous case with the stability condition that can be obtained by the Fourier method. Inthis sense, our analysis generalizes the one by Robertson et al. [41].

Sections 3 and 4 are the two main sections of this article. They are preceded, insection 2, by a brief recap of visco-elasticity theory with a more detailed presentation ofgeneralized Zener’s models that are the object of our study, In section 5, we shall presentvarious numerical results obtained with our method. This section includes in particular adescription of the way we generalized the Perfectly Matched Layers method (a methodto treat transparent boundaries [9,24]) to viscoelastic waves (section 5.1) and a numericalsimulation in a “realistic” model for which we have designed specifically the parametersof our Zener’s model in order to realize a quasi Q-constant model (section 5.2.3).

2. Viscoelastic models

2.1. The general models

The linear viscoelastic models [30] take into account the waves absorption phe-nomenon, in such models if we consider the strain tensor at time t :

εij (u) = 1

2

(∂ui

∂xj

+ ∂uj

∂xi

)(1)

associated to a displacement field u(x, t) (x ∈ � ⊂ Rn, n = 1, 2, 3), the linear vis-

coelastic material to be one fore which the stress tensor σ is related to ε by a convolutionintegral [21,30,42] as follows:

σij (x, t) =∫ t

−∞Gijkl(t − τ)

∂εkl(τ )

∂τdτ, (2)

where G is a tensor of order 4, symmetric:

Gijkl = Gjikl = Gijlk, (3)

which is called the tensorial relaxation function.Formula (2) expresses that the state of the stress σ at time t depends on the history

of the strain, this is why we call such models: “memory models”. In the literature, wefind several equivalent ways to present the law (2). For example, a simple integration byparts leads to:

σij (x, t) = Gijkl(0)εkl(t) +∫ t

0εkl(t − τ)

∂Gijkl(τ )

∂τdτ. (4)

We suppose thereafter that the functions are causal, i.e. equal to 0 for t < 0. Theexpression (2) can be expressed like a convolution product:

σ = G ∗ ∂ε

∂t= R ∗ ε, (5)


where R = ∂G/∂t is the derivative (in the sense of distributions) of G, with

(R ∗ ε)ij =d∑

k,l=1

Rijkl ∗ εkl. (6)

Note that relation (2) generalizes the Hooke’s law, and the elastic case is obtained withthe following choice:

G = C(x)H(t),

where H is the Heaviside’s function and C the standard elastic tensor.An alternative form of the stress–strain relation can be obtained by inverting rela-

tion (2):

εij (t) =∫ t

−∞Jijkl(t − τ)

∂σkl(τ )

∂τdτ, (7)

where J is the tensorial creep function:

Jijkl = Jjikl = Jijlk. (8)

2.2. The differential models

The type of “differential–integral” equations presented previously is not easy to besolved in a numerical simulation. A more practical way to present the viscoelastic con-stitutive law is to write it in a differential form, based on mechanical analogies which useassemblies of spring and dashpot in the monodimensional case (rheological elementarymodel) [10,21,31]:

P(D)σ = Q(D)ε. (9)

In (9) D is the differential operator:

Di : f → ∂if

∂ti∀i ∈ N,

P and Q are differential polynomials

P(D) =N∑

m=0

amDm, Q(D) =M∑

m=0

bmDm. (10)

Gurtin and Sternberg [31] showed equivalence between the integral and differential rep-resentation in the case M = N under the assumptions:

• aN, bN �= 0.

• The initial conditions (Compatibility condition):

N∑m=n

am

∂m−n

∂tm−nσ (0) =

N∑m=n

bm

∂m−n

∂tm−nε(0) ∀n = 1, . . . , N. (11)


If these assumptions are verified, there exists a relaxation function G related through therelaxation integral law (2). Moreover G is unique, and is the solution of the initial-valueproblem governed by the differential equation:

P(D)G = b0 on ]0,+∞[, (12)

subject to the initial conditions:

G(0) = bN

aN

,∂n

∂tnG(0) = 1

aN

[bN−n −

n−1∑m=0

aN−n+m

∂m

∂tmG(0)

]∀n = 1, . . . , N.

(13)

2.3. Zener’s model

Differential form. In the monodimensional case, the commonly differential viscoelas-tic models are the Maxwell, Kelvin–Voigt and Zener’s one [12,14]. In this study andbelow, we focus on an extension of the Zener’s model to higher dimensions, for whichthe differential polynomials P and Q are of degree one. The constitutive law is writtenin differential form:

σ + τ0∂σ

∂t= Cε(u) + τ0Dε

(∂u

∂t

), (14)

where τ0 > 0 is a relaxation time, C and D are two tensors of order 4 symmetric,positive definite. We will reconsider further the properties of these tensors.

Integral form. The relaxation time τ0 is positive, the relation (14) can be rewritten inthe form:

1

τ0

(σ − Dε(u)

) + ∂t

(σ − Dε(u)

) = − 1

τ0(D − C)ε(u) (15)

by multiplying this equation with e(1/τ0)τ , after integrating between 0 and t , we obtain(by choosing the initial condition σ0 = Dε(u0)):

σ (t) = Dε(u(t)

) +∫ t

0− 1

τ0(D − C)e−(t−τ )/τ0ε

(u(τ)

)dτ.

This gives the integral form of the Zener’s law:

σ (t) = G(0)ε(t) +∫ t

0

∂G

∂s(τ)ε(t − τ) dτ,

with

G(t) = C + (D − C)e−t/τ0 . (16)

The relaxation function G is written as a superposition of the elasticity tensor and anexponential function which is zero at infinity. We will see later that the tensor D − Cmust be positive in order to obtain a dissipative model (see section 3.3).


Isotropic model. By analogy with the purely elastic case, we consider that the mediumcharacterized by the law (14) is isotropic if there are coefficients λ(x), µ(x), γλ(x),γµ(x) such that:

(Cσ )ij = λδij σkk + 2µσij , (Dσ )ij = λγλδij σkk + 2µγµσij ∀i, j = 1, n,

and we define the coefficients:

vp =√

λ + 2µ

ρ, vs =

√µ

ρ, τp = τ0

λγλ + 2µγµ

λ + 2µ, τs = τ0γµ, (17)

with vp (resp. vs) is the velocity and τp (resp. τs) is the relaxation time associated withthe pressure (resp. shear) wave.

Monodimensional Zener’s model. We obtain the 1D Zener’s model, if we take C = µ

and D = τ−10 µτ1 with the condition τ1 > τ0 (equivalent to D − C > 0 in general

dimension). In this case, the constitutive law becomes:

σ + τ0∂tσ = µ∂xu + µτ1∂2xtu. (18)

2.4. Generalized Zener’s model

To characterize the attenuation of waves in a viscoelastic material, we introduce thequality factor or dissipation factor Q. If we make the Fourier transform of equation (5)we will have the relation between stress σ and strain ε in the frequency domain

σ (ω) = M(ω)ε(ω), (19)

where M (resp. σ , ε) is the Fourier transform of R (resp. σ, ε). Here M is the complexmodulus. The quality factor is defined as [12,16]:

Q(ω) = �e M(ω)

m M(ω), (20)

where �e M(ω) is the real part of M and m M(ω) is the imaginary one. For 1D Zener’smodel (18) the quality factor is:

Q(ω) = 1 + τ1τ0ω2

ω(τ1 − τ0). (21)

In geophysics [44], an important class of materials is characterized by the fact that thequality Q almost constant independent of the frequency over the seismic frequencyrange, we speak of constant-Q materials. A single Zener’s model is clearly not suf-ficient (see (21)) and that is why it is interesting to consider a so called generalizedZener’s model.

Starting from the simple Zener’s model, it is easy to construct a more complexmodel, composed by a number of Zener’s elements in parallel [16], as illustrated in the1D case in figure 1. In this figure, the parameters Ei, Ei , i = 1, . . . , k, are the springs


Figure 1. Generalized Zener’s model.

elasticity moduli and ηi , i = 1, . . . , k, the dashpots viscosity parameters. In higherdimension the generalized Zener’s model consists in representing the stress tensor as asuperposition of elementary stresses

σ =k∑

i=1

σi, (22)

where σi satisfies the following elementary relation:

σi + τ 0i ∂tσi = Ciε(u) + τ 0

i Diε(∂tu), i = 1, . . . , k. (23)

3. Mathematical analysis

3.1. Model problem

We are interested in the wave propagation in a medium modeled by a Zener’slaw (14) in dimension n (n = 1, 2, 3). Our goal is to determine the displacement field u


and the stress tensor σ :

ρ∂2u

∂t2− div σ = f in R

n×]0, T ], (i)

σ + τ0∂σ

∂t= Cε(u) + τ0Dε

(∂u

∂t

)in R

n×]0, T ], (ii)

u(x, 0) = u0,∂u

∂t(x, 0) = u1 in R

n, (iii)

σ (x, 0) = σ0 in Rn, (iv)

(24)

with

(div σ )i =n∑

j=1

∂σij

∂xj

∀i = 1, n,

where ρ is the mass density, τ0 a relaxation time, f the force density. C and D are twotensors 4 × 4 satisfying the properties:

Cijkl = Cjikl = Cklij , Dijkl = Djikl = Dklij , (25)

and there exist M−, M+, two positive constants, such that:

0 < M−|σ |2 � Cσ : σ � M+|σ |2, ∀σ ∈ Lsym(R

n)

a.e. x ∈ Rn,

0 < M−|σ |2 � Dσ : σ � M+|σ |2, ∀σ ∈ Lsym(R

n)

a.e. x ∈ Rn,

(26)

where

Lsym(R

n) = {

σ ∈ L(R

n) | σij = σji, ∀i, j = 1, . . . , n

},

L(R

n) = {

σ : Rn → R

n | σ is linear},

σ : σ = σij σij , ∀(σ, σ ) ∈ [L

(R

n)]2

(scalar product in L(R

n)),

|σ | = (σ : σ )1/2 =(∑

i,j

σ 2ij

)1/2

(norm in L(R

n)),

Gσ = Gijklσkl, ∀(G, σ ) ∈ L(L

(R

n),L

(R

n)) × L

(R

n).

Remark 3.1. For any tensor G, the symmetry property (25) implies:

Gσ : σ = Gσ : σ ∀(σ, σ ) ∈ [L

(R

n)]2

.

Moreover we consider the following assumptions:

• ρ, τ0,C and D are measurable.

0 < ρ− � ρ(x) � ρ+ < +∞ a.e. x ∈ Rn. (27)

• 0 < τ− � τ0(x) � τ+ < +∞ a.e. x ∈ Rn.

• The absorption condition: Z = D − C is positive definite:

0 < M−|σ |2 � Zσ : σ � M+|σ |2, ∀σ ∈ Lsym(R

n)

a.e. x ∈ Rn. (28)


3.2. Existence and uniqueness result

We consider the functional spaces:

L2(R

n,Lsym(R

n)) =

{σ : R

n → Lsym(R

n) ∣∣∣ ∫

Rn

|σ |2 dx < ∞},

Xsym(R

n) = {

σ ∈ L2(R

n,Lsym(R

n)) | div σ ∈ [

L2(R

n)]n}

,

(29)

and for any symmetric positive definite tensor C, we denote by:

• (. : .)C the scalar product in Lsym(Rn) and | · |C its associated norm:

Lsym(R

n) × Lsym

(R

n) −→ R,

(σ, ε) −→ Cσ : ε ≡ (σ : ε)C .

• 〈. , . 〉C the scalar product in L2(Rn,Lsym(Rn)) and ‖ · ‖C its associated norm:

L2(R

n,Lsym(R

n)) × L2

(R

n,Lsym(R

n)) −→ R,

(σ, ε) −→∫

Rn

Cσ : ε dx ≡ 〈σ, ε〉C .

We write the problem (24) as a first-order evolution system by introducing the variablesv = ∂u/∂t and s = σ −Cε(u) (difference between the viscoelastic stress and the purelyelastic stress τ0 = 0). We will see in the next section, that this new variable s alsoplays a role in the displacement-stress mixed formulation. With these new variables, theproblem (24) is rewritten as:

∂u

∂t− v = 0,

∂v

∂t− 1

ρdiv

(Cε(u)

) − 1

ρdiv s = f

ρ,

∂s

∂t− Zε(v) + 1

τ0s = 0,

u(x, 0) = u0, v(x, 0) = u1, s(x, 0) = s0 = σ0 − Cε(u0),

(30)

or, setting W = (u, v, s)T: { dW

dt+ �W = F,

W(0) = W0,

(31)

where

�W =

−v

− 1

ρdiv

(Cε(u)

) − 1

ρdiv s

−Zε(v) + 1

τ0s

, W0 =(

u0

v0

s0

), F =

0f

ρ

0

. (32)


We introduce the Hilbert space:

H = [H 1

(R

n)]n × [

L2(R

n)]n × L2

(R

n,Lsym(R

n))

, (33)

with the scalar product:

(W1,W2)H = (u1, u2)ρ + ⟨ε(u1), ε(u2)

⟩C

+ (v1, v2)ρ + 〈s1, s2〉Z−1, (34)

where W1 = (u1, v1, s1)T and W2 = (u2, v2, s2)

T and we set:

(u1, u2)ρ =∫

Rn

ρ u1 · u2 dx,

where u · v denotes the Euclidien scalar product in Rn.

We consider the operator � : D(�) ⊂ H → H defined by (32), where

D(�) = {(u, v, s) ∈ H | s + Cε(u) ∈ Xsym(

Rn), v ∈ [

H 1(R

n)]n}

. (35)

Remark 3.2. The scalar product (34) is well defined thanks to conditions (26)–(28).

The proof of the existence and uniqueness of the solution relies on the use of Hille–Yosida’s theorem [13], which requires the following lemma:

Lemma 3.1. The operator � + λI is maximal monotone for all λ > 1/2.

Proof. Monotony. Let W = (u, v, s)t ∈ D(�), we have

(�W,W)H = −∫

Rn

ρu · v dx −∫

Rn

Cε(u) : ε(v) dx −∫

Rn

div(Cε(u) + s

) · v dx

−∫

Rn

[ε(v) − Z−1

τ s] : s dx,

= −∫

Rn

ρu · v dx +∫

Rn

Z−1τ s : s dx,

with Zτ = τ0Z. In addition:

|W |2H =∫

Rn

ρ|u|2 dx +∫

Rn

Cε(u) : ε(u) dx +∫

Rn

ρ|v|2 dx +∫

Rn

Z−1s : s dx.

We deduce that

(�W,W)H + λ|W |2H �∫

Rn

ρ(λ|u|2 − uv + λ|v|2)

dx,

�(

λ − 1

2

) ∫Rn

ρ(|u|2 + |v|2)

dx.

In the last inequality we see that � + λI is monotone for all λ > 1/2.


Surjectivity. Let us show that � + νI is surjective for all ν > 0. This is equivalentto: ∀F = (f, g, h)T ∈ H , ∃W = (u, v, s)T ∈ D(�) solution of the system

νu − v = f, (a)

− 1

ρdiv

(Cε(u)

) + νv − 1

ρdiv s = g, (b)

−Z(x)ε(v) + 1 + ντ0

τ0s = h. (c)

(36)

If (36) has a solution, v and s can be easily eliminated and u must be solution of theequation

− div(Zε(u)

) + ρν2u = ρg + ρνf + div(

τ0

1 + ντ0h

)− div

(τ0

1 + ντ0Zε(f )

), (37)

where

Z = C + ντ0

1 + ντ0Z. (38)

The variational formulation of (37) can be written as follows:{find u ∈ [

H 1(R

n)]n

such that:a(u, u) = l(u), ∀u ∈ [

H 1(R

n)]n

,(39)

with

a(u, u) = ⟨Zε(u), ε(u)

⟩ + ν2(ρ u, u),

l(u) = (ρg, u) + ν(ρf, u) −⟨

τ0

1 + ντ0h, ε(u)

⟩+

⟨τ0

1 + ντ0Z ε(f ), ε(u)

⟩,

(40)

where (. , . ) and 〈. , . 〉 indicate the scalar product respectively in [L2(Rn)]n andL2(Rn,Lsym(Rn)).

According to (28) and Korn’s inequality [26,39] in [H 1(Rn)]n, the bilinear forma(. , . ) is continuous coercive in [H 1(Rn)]n for all ν �= 0. The Lax–Milgram’stheorem implies that the problem (40) has a unique solution u in [H 1(Rn)]n. Thedata (f, h) belonging to [H 1(Rn)]n ×L2(Rn,Lsym(Rn)), we then obtain the existence ofs ∈ L2(Rn,Lsym(Rn)) by setting:

s = ντ0

1 + ντ0Zε(u) + τ0

1 + ντ0h − τ0

1 + ντ0Zε(f ). (41)

The existence of v ∈ [H 1(Rn)]n directly follows from (36)-(a). Finally if we use theequation (36)-(b), we easily check that:

div(Cε(u) + s

) ∈ [L2(

Rn)]n

.

We have shown that � + νI is surjective, for all ν > 0 and thus, in particular, forν = λ + 1 which concludes the proof. �

Now, we can state the existence and uniqueness theorem:


Theorem 3.1. If the initial data (u0, u1, σ0) ∈ ([H 1(Rn)]n)2 × Xsym(Rn) then for anyf ∈ C1(0, T ; [L2(Rn)]n), the model problem (24) has a unique solution (u, σ ) whichsatisfies: {

u ∈ C1(0, T ; [

H 1(R

n)]n) ∩ C2

(0, T ; [

L2(R

n)]n)

,

σ ∈ C0(0, T ;Xsym

(R

n)) ∩ C1

(0, T ;L2

(R

n,Lsym(R

n)))

.

Proof. Under the assumption W0 ∈ D(�) and thanks to the Hille–Yosida’s theo-rem [13] and the lemma 3.1, we deduce that the problem (30) has a unique solutionW ∈ C0(0, T ;D(�)) ∩ C1(0, T ;H). Then we have:

• u ∈ C1(0, T ; [H 1(Rn)]n),• v = (∂u/∂t) ∈ C0(0, T ; [H 1(Rn)]n) ∩ C1(0, T ; [L2(Rn)]n),• σ = s + Cε(u) ∈ C0(0, T ;Xsym(Rn))) and s ∈ C1(0, T ;L2(Rn,Lsym(Rn)))),

what involves:{u ∈ C1

(0, T ; [

H 1(R

n)]n) ∩ C2

(0, T ; [

L2(R

n)]n)

,

σ ∈ C0(0, T ;Xsym

(R

n)) ∩ C1

(0, T ;L2

(R

n,Lsym(R

n)))

. �

It is not difficult to obtain a similar result in the case of the more general model (23).In this case, we look for the displacement field u and the “elementary” stress tensors σi

which compose σ (see (22)) solutions of problem:ρ∂2

t tu −k∑

i=1

div σi = f,

σi + τ 0i ∂tσi = Ciε(u) + τ 0

i Diε(∂tu), i = 1, . . . , k,

u(x, 0) = u0, ∂tu(x, 0) = u1,

σi(x, 0) = σ 0i , i = 1, . . . , k.

(42)

And the following theorem can be easily proved:

Theorem 3.2. If (u0, u1, σ01 , . . . , σ 0

k ) ∈ ([H 1(Rn)]n)2 × (Xsym(Rn))k. For any f ∈C1(0, T ; [L2(Rn)]n). The problem (42) has a unique solution (u, σ1, . . . , σk) satisfying:{

u ∈ C1(0, T ; [

H 1(R

n)]n) ∩ C2

(0, T ; [

L2(R

n)]n)

,

σi ∈ C0(0, T ;Xsym

(R

n)) ∩ C1

(0, T ;L2

(R

n,Lsym(R

n)))

, ∀i = 1, . . . , k.

3.3. Energy decay

In order to show that our model describes an absorbing medium, we point out adecreasing quantity that will be called energy of the model.


Definition 3.1. Considering (u, σ ) the strong solution of system (24), we define theenergy of (u, σ ) at time t :

E(u, σ, t) = 1

2

∥∥∥∥∂u

∂t

∥∥∥∥2

ρ

+ 1

2

∥∥ε(u)∥∥2

C+ 1

2‖s‖2

Z−1 , s = σ − Cε(u). (43)

Remark 3.3.

• The energy quantity is decomposed into two parts:

– The quantity 12‖∂tu‖2

ρ dx + 12‖ε(u)‖2

C corresponds to the standard energy in thepurely elastic case (τ0 = 0).

– The quantity 12‖s‖2

Z−1 , is the norm of the difference between viscoelastic stress andelastic stress which is due to the viscoelasticity.

• Taking the regularity of (u, σ ) into account, the energy E verifies:

E ∈ C1(0, T ); E(t) < +∞ ∀t � 0.

We obtain the following result:

Theorem 3.3. The energy E(t) satisfies the identity:

dE

dt= −‖s‖2

Z−1τ

+ (f, ∂tu). (44)

In particular, in absence of source term (f = 0) the energy E decreases in time.

Proof. We can rewrite equations (i) and (ii) of problem (24) with respect to u and s:

ρ∂2u

∂t2− div s − div

(Cε(u)

) = f , (45a)

Z−1∂ts + Z−1τ s = ε(∂tu). (45b)

Multiplying the first equation (45a) by ∂tu and the second one (45b) by s, we get afterintegration on R

n:

1

2

d

dt‖∂tu‖2

ρ − (div s, ∂tu) − (div

(Cε(u)

), ∂tu

) − (f, ∂tu) = 0, (46a)

1

2

d

dt‖s‖2

Z−1 + ‖s‖2Zτ

− ⟨ε(∂tu), s

⟩ = 0. (46b)

If we integrate by parts the second and the third term of the first equation, we have:

1

2

d

dt‖∂tu‖2

ρ + 1

2

d

dt

∥∥ε(u)∥∥2

C+ ⟨

s, ε(∂tu)⟩ − (f, ∂tu) = 0, (47)

which gives us (44) by adding (46b) and (47).


We deduce the energy decay for f = 0 since assumption (28) shows that Z ispositive definite and consequently −‖s‖2

Z−1τ

is a dissipation term. �

Remark 3.4. Note the important role of the absorption condition Z > 0 (positive def-inite), as well as in the existence and uniqueness result than in the energy dissipationresult.

Again, we can easily extend this result to the general model (42).

Theorem 3.4. The energy quantity

E(t) = 1

2‖∂tu‖2

ρ + 1

2

n∑i=1

[∥∥ε(u)∥∥2

Ci+ ‖si‖2

Z−1i

], (48)

associated to the solution (u, σ1, . . . , σk) of the problem (42), verifies:

dE

dt= −

n∑i=1

‖si‖2(Zτ

i )−1 + (f, ∂tu) � (f, ∂tu), (49)

where

si = σi − Ciε(u), Zi = Di − Ci , Zτi = τ 0

i Zi , ∀i = 1, . . . , k.

3.4. Finite velocity propagation

In this paragraph we are interested to the problem of finite velocity propagation forthe viscoelastic waves. We will show that for data with compact support the solutionremains with compact support at any time. Furthermore, we will determine an optimumsolution support with a geometrical study. This study can be considered as an extensionto higher dimensions of Canadas’s work [14], done for 1D problem (18).

Let G be a symmetric definite positive tensor of order four and ν a non zero vector.We introduce the Chrystoffel’s tensor [1]:

(G, ν) = ij = Gikj lνkνl.

This is a symmetric positive definite tensor.

Remark 3.5. The Chrystoffel’s tensor appears when one performs a plane wave analysisfor the homogeneous purely elastic problem (τ0 = 0, f = 0), looking for particularsolutions of the form (i2 = −1):{

u(x, t) = ei(ωt−k·x) d, d ∈ Rn, ω ∈ R, k ∈ R

n,

σ (x, t) = ei(ωt−k·x) D, D ∈ Ls(R

n).

(50)

This leads to the dispersion relation:

(C, k) d = ρω2d ⇐⇒ (C, ν) d = ρV 2d


with ν = k/|k| the unit propagation direction and V = ω/|k| the phase velocity (and V 2

appears as one eigenvalues of (1/ρ) (C, ν)).

Let ν be a vector of the unit sphere Sn−1, we define the largest eigenvalue of(1/ρ) (D, ν):

Vp(x, ν) =(

supv �=0

(D, ν)v · v

ρ|v|2)1/2

, (51)

and

V +p (ν) = sup

x∈Rn

{Vp(x, ν)

}. (52)

We have chosen the notation (51) by analogy to the velocity of P waves in isotropicelastic medium. In this case Vp(x, ν) = vp(x) for all directions ν. We consider themobile half-spaces with velocity V > 0:

Eν(V , t) = {x ∈ R

n | x · ν � V t}. (53)

We have the theorem.

Theorem 3.5. If the data (u0, u1, σ0, f ) have a compact support and satisfy:

supp u0 ∪ supp u1 ∪ supp σ0 ∪ supp f (. , t) ⊂ K,

where K is a compact of Rn, then

supp u(. , t) ⊂ K + E(t) ∀0 � t � T

with

E(t) =⋂

ν∈Sn−1

Eν

(V +

p (ν), t).

Proof. We will make the proof when the data are regular. We assume that K = B(0, R)

is a ball of center 0 and radius R. We reduce to the general case while proceeding theregularization and by using the fact that any compact is limiting of a finished union ofclosed balls. We consider ν a vector of the unit sphere of R

n, the mobile half-space:

�tν = (

B(0, R) + Eν(V , t))c = {

x ∈ Rn | x.ν > R + V t

},

�tν is the mobile half-space which propagates in the direction ν with a velocity V ,

tν = {x ∈ R | x.ν = R + V t} the boundary of �t

ν (see figure 2) and dσ the sur-face measurement on t

ν .Note that, by construction the initial data verify:{

u0(x) = u1(x) = 0, σ0(x) = 0 ∀x ∈ �0ν,

f (x, t) = 0 ∀t > 0, ∀x ∈ �tν .

(54)


Figure 2. Mobile half-space.

In particular, at time t = 0, the solution (u, σ ) vanishes in �tν . The idea of the demon-

stration is to find V (large enough) so that the solution vanishes in �tν .

For this, we use an energy technique. We consider the system of equations (45):

ρ∂2u

∂t2− div s − div

(Cε(u)

) = f, (55a)

Z−1τ s + Z−1∂ts = ε(∂tu). (55b)

Let us define the energy density:

e(x, t) = 1

2

(ρ

∣∣∣∣∂u

∂t

∣∣∣∣2

+ Cε : ε + Z−1s : s

), (56)

taking the scalar product of (55a) with ∂tu and of (55b) with s adding the two equationsand integrating on �t

ν , we obtain the following identity:∫�T

r

∂e

∂tdx +

∫�T

r

Z−1τ s : s dx +

∫ t

r

(s + Cε)ν · ∂tu dσ = 0, (57)

where ν is the normal vector to tν and σν = σij νj the matrix–vector product in R

n.Using the relation (valid for all e(x, t) ∈ C1(Rn × R

+)):

d

dt

∫�T

r

e(x, t) dx =∫

�Tr

∂e

∂t(x, t) dx − V

∫ t

r

e(x, t) dσ, (58)

we deducedEν

dt(t) +

∫�T

r

Z−1τ s : s dx + 1

2

∫ t

r

φν(x, t) dσ = 0,

where Eν(t) = ∫�T

re(x, t) dx is the energy contained in �t

r and

φν(x, t) = Vρ|∂tu|2 + V |ε|2C + V |s|2Z−1 + 2(s + Cε)ν.∂tu, (59)


with

|ε|2C = Cε : ε and |s|2Z−1 = Z−1s : s.

We see that the energy decreases if the quadratic form is positive. We will use thefollowing lemma:

Lemma 3.2. For all v ∈ Rn, we have:

(Cε)ν · v � |ε|C( (C, ν)v · v

)1/2, ∀v ∈ R

n, (i)

sν · v � |s|Z( (Z, ν)v · v)1/2

, ∀v ∈ Rn. (ii)

(60)

Proof. Let v ∈ Rn

(Cε)ν · v = Cijklεklνj vi

= Cijklεkl(ν ⊗ v)ij = Cε : ν ⊗ v = (ε : ν ⊗ v)C

with (ν ⊗ v)ij = 12 (νivj + νjvi). As (. : .)C defines a scalar product in Ls(Rn), we can

use the Cauchy–Schwarz inequality and we obtain

(ε : ν ⊗ v)C � |ε|C|ν ⊗ v|C .

Then to prove (i), it is sufficient to show that

|ν ⊗ v|2C = (C, ν)v · v.

Indeed

|ν ⊗ v|2C = Cν ⊗ v : ν ⊗ v = Cijkl(ν ⊗ v)kl(ν ⊗ v)ij

= Cijklνkvlνivj

= (Cijklνiνk)vlvj = (Clkjiνiνk)vjvl

= lj (C, ν)vj vl = (C, ν)v · v.

In the same way, for (ii), we have

sν · v = s : ν ⊗ v

= s : Z−1Zν ⊗ v

= (Z−1s : ν ⊗ v

)Z

�∣∣Z−1s

∣∣Z|ν ⊗ v|Z = |s|Z−1

( (Z, ν)v · v

)1/2. �

Using this lemma and according to (59) we obtain:

φν(x, t) � Vρ|∂tu|2 + V |ε|2C + V |s|2Z−1 − 2|ε|C

( (C, ν)∂tu · ∂tu

)1/2

− 2|s|Z−1

( (Z, ν)∂tu · ∂tu

)1/2


� Vρ|∂tu|2 + V(|ε|2C + |s|2

Z−1

)− 2

(|ε|2C + |s|2Z−1

)1/2( (C, ν)∂tu · ∂tu + (Z, ν)∂tu · ∂tu

)1/2

� Vρ|∂tu|2 + V(|ε|2C + |s|2

Z−1

) − 2(|ε|2C + |s|2

Z−1

)1/2( (C + Z, ν)∂tu · ∂tu

)1/2

� Vρ|∂tu|2 + V(|ε|2C + |s|2

Z−1

) − 2(|ε|2C + |s|2

Z−1

)1/2( (D, ν)∂tu · ∂tu

)1/2.

(61)

Using definition (51) of Vp, inequality (61) becomes

φν(x, t) � Vρ |∂tu|2 + V(|ε|2C + |s|2Zτ

) − 2Vp√

ρ(|ε|2C + |s|2Zτ

)1/2|∂tu|. (62)

The second term of this inequality is positive if

ρ(V 2

p − V 2)

� 0 ⇐⇒ V � Vp.

When choosing V = V +p (ν) given by (52), we will have φν � 0. Therefore Eν is

decreasing and we have:

Eν(t) � Eν(0) = 0 (by construction).

This implies that u(x, t) = 0 in �tν and consequently supp u(., t) ⊂ {�t

ν}c = B(0, R) +Eν(V

+p (ν), t). The demonstration being valid for all ν ∈ Sn−1 unit sphere of R

n, then wehave:

supp u(., t) ⊂⋂

ν∈Sn−1

{�t

ν

}c = B(0, R) + E(t). �

Remark 3.6. In the isotropic case, Vp does not depend on ν:

Vp(x, ν) =(

λ(x)γλ(x) + 2µ(x)γµ(x)

ρ(x)

)1/2

=(

λ(x)γλ(x) + 2µ(x)γµ(x)

λ(x) + 2µ(x)

λ(x) + 2µ(x)

ρ

)1/2

=√

τp(x)

τ0(x)vp(x).

This shows that

V +(ν) = V + = supx∈Rn

√τp(x)

τ0(x)vp(x)

which implies supp u(., t) ⊂ B(0, R + V +t) (see figure 3, n = 2).


Figure 3. Support of u in isotropic medium.

Figure 4. Support of u in anisotropic medium.

4. Numerical analysis

Our objective is to develop an efficient numerical method for solving the modelproblem (24). We extend the mixed finite element with mass lumping presented in [8,45]for elastodynamic equations to viscoelastic equations. The basis of this method is amixed formulation of the problem in which the stress tensor σ is searched in the spaceof symmetric H(div) tensors, namely H sym(div), and the velocity (or displacement) issearched in L2 [4].

4.1. Reformulation of the model problem

To apply this method in the viscoelasticity case, we reformulate the problem, i.e.instead of considering a constitutive law stress–strain, we will rewrite it in the form ofa strain–stress form. This is possible by introducing the unknown s = σ − Cε, as thedifference between the viscoelastic and the purely elastic stress.


We consider the model problem in a bounded open domain � ⊂ Rn{

ρ∂2t tu − div σ = f, (i)

σ + τ0∂tσ = Cε(u) + τ0Dε(∂tu), (ii)(63)

with the following initial conditions:

u(t = 0) = u0, σ (t = 0) = σ0, ∂tu(t = 0) = u1, (64)

and the boundary condition

u = 0 in ∂�. (65)

We introduce the new variable s, (63)-(ii) is rewritten in the form (45b), then thesystem (63) becomes ρ∂2

t tu − div σ = f,

Mτ s + M∂t s = ε(∂tu),

Aσ − As = ε(u),

(66)

where M = Z−1, Mτ = Z−1τ , A = C−1. What follows, we will take the problem

(64)–(66) as model problem.

Remark 4.1. We introduce the intermediate variable s, to obtain the mixed formulationand the space discretization. We will see later that we can eliminate σ or s after timediscretization.

4.2. Mixed formulation (displacement-stress)

We consider the functional spaces:

M = [L2

(�; R

n)]n

, H = L2(�;L(

Rn)) =

{σ : � → L

(R

n) ∣∣∣ ∫

�

|σ |2 dx < ∞},

(67)

we define the tensor space

X = {σ ∈ H | div σ ∈ M} (68)

and also the symmetric tensor space

Xsym = {σ ∈ X | σ ∈ Ls

(R

n)}

. (69)

We obtain the mixed formulation, by multiplying the first equation of (66) by u ∈ M ,the second by s ∈ Xsym, the third by σ ∈ Xsym and then integrating on �. After an


integration by part of the right-hand side of both last equations (the terms in which ε

appears) we will have the following mixed formulation:d2

dt2ρ

(u(t), u

) − b(u, σ (t)

) = (f, u) ∀u ∈ M ,

mτ

(s(t), s

) + d

dtm

(s(t), s

) + d

dtb

(u(t), s

) = 0 ∀s ∈ Xsym,

a(σ (t), σ

) − a(s(t), σ

) + b(u(t), σ

) = 0 ∀σ ∈ Xsym,

(70)

where

ρ(u, u) =∫

�

ρ u · u dx ∀(u, u) ∈ M × M,

m(s, s) =∫

�

Ms : s dx ∀(s, s) ∈ H × H,

mτ (s, s) =∫

�

Mτ s : s dx ∀(s, s) ∈ H × H,

a(σ, σ ) =∫

�

A σ : σ dx ∀(σ, σ ) ∈ H × H,

(71)

and

b(u, σ ) =∫

�

div σ · u dx ∀(u, σ ) ∈ M × X. (72)

4.3. Space discretization

We suppose that � is a union of parallelepiped and we consider a regular grid (Th)

of � composed of cubes K with edges h > 0. We introduce some finite-dimensionalapproximation spaces:

Mh ⊂ M, Xh ⊂ X, Xhsym = Xh ∩ Xsym.

The approximation problem of the formulation (70) consists in finding (uh(t), sh(t),

σh(t)) ∈ Mh × Xhsym × Xh

sym such that:d2

dt2ρ(uh, uh) − b(uh, σh) = (f, uh) ∀uh ∈ Mh,

mτ (sh, sh) + d

dtm(sh, sh) + d

dtb(uh, sh) = 0 ∀sh ∈ Xh

sym,

a(σh, σh) − a(sh, σh) + b(uh, σh) = 0 ∀σh ∈ Xhsym.

(73)

Since there are no continuity conditions required in M, it will be approximated withdiscontinuous functions. Therefore the mass matrix associated to ρ(uh, uh) is diagonalby construction. In order to achieve mass lumping on the mass matrices associated to


the bilinear forms m(sh, sh), mτ (sh, sh) and a(σh, σh), we use the approximation spaceXh

sym described in [8]. This result in the following choice:

Mh = {uh ∈ M | ∀K ∈ Th, (uh)i | K ∈ Q0

},

Xh = {σh ∈ X | ∀K ∈ Th, (σh)ij | K ∈ Q1

},

Xhsym = {

σh ∈ Xh | σh ∈ Lsym(R

n)}

,

(74)

where Q0 and Q1 are two polynomial spaces defined for k = 0 or 1 by

Qk ={

p | p(x1, . . . , xn) =∑

l∈{0,...,k}nal

n∏i=1

xlii

}.

With this choice, the displacement uh is constant per element and the stress tensor σh

(resp. sh) is a bilinear function in H(div), which imposes the continuity of its normalcomponents: this means that σii (resp. sii) is continuous through xi (i = 1, . . . , n) andthe symmetry of the tensor implies the continuity of σij (resp. sij ) in the two directions xi

and xj . In conclusion, there are five degrees of freedom for σh (resp. sh) per vertex andone degree of freedom for uh per element (see figure 5 for n = 2). The mass matricesare then evaluated thanks to Gauss–Lobatto’s formulas:

m(σh, σh) ≈ mh(σh, σh) =∑K∈Th

∮K

Mσh : σh dx,

mτ (σh, σh) ≈ mτ h(σh, σh) =∑K∈Th

∮K

Mτ σh : σh dx, (75)

a(σh, σh) ≈ ah(σh, σh) =∑K∈Th

∮K

Aσh : σh dx,

where∮K

f dx = h2

4

∑x∈Kv

f (x) ∀f ∈ C0(K) and Kv = {x ∈ R

n | x is a vertex of K}.

Figure 5. The degrees of freedom for Xhsym and Mh.


We introduce the basis functions BN1 = {ωi | i = 1, . . . N1} (resp. BN2 = {φi | i =1, . . . N2}) of Mh (resp. Xh

sym), where N1 = dim Mh (resp. N2 = dim Xhsym). If we

consider Uh = (U1, . . . , UN1), Sh = (S1, . . . , SN2) and �h = (�1, . . . , �N2) the coordi-nates of functions uh, sh and σh on these bases, then we can rewrite the problem (73) inthe following matrix form:

Mu

d2Uh

dt2− B�h = F,

MτSh + MsdSh

dt+ B� dUh

dt= 0,

A�h − ASh + B�Uh = 0.

(76)

The matrix Mu is diagonal, Mτ , Ms and A are block diagonals, the dimension of eachblock is the number degrees of freedom per vertex (5 for n = 2).

4.4. Time discretization

For the time discretization we construct a totally explicit centered finite differencescheme, by approximating the first equation at tn = n�t with the leap-frog scheme, thesecond equation at tn+1/2 = (n + 1

2)�t :Mu

Un+1h − 2Un

h + Un−1h

�t2− B�n

h = Fn,

Mτ

Sn+1h + Sn

h

2+ Ms

Sn+1h − Sn

h

�t+ B� Un+1

h − Unh

�t= 0,

A�n+1h − ASn+1

h + B�Un+1h = 0.

(77)

Note that we can eliminate �nh , which leads to

Mu

Un+1h − 2Un

h + Un−1h

�t2− BSn

h + BA−1B�Unh = 0,

Mτ

Sn+1h + Sn

h

2+ Ms

Sn+1h − Sn

h

�t+ B� Un+1

h − Unh

�t= 0.

(78)

4.5. Discrete energy and stability analysis

To prove the stability of the scheme (77) we will use an energy technique.

Definition 4.1. We introduce the discrete energy as follows:

En+1/2 = 1

2

∥∥∥∥un+1h − un

h

�t

∥∥∥∥2

ρh

+ 12ah

(σ n+1

h − sn+1h , σ n

h − snh

)+ 1

4

[∥∥sn+1h

∥∥2mh

+ ∥∥snh

∥∥2mh

]+ �t2

4bh

(sn+1h − sn

h

�t,un+1

h − unh

�t

),

(79)


where

‖uh‖2ρh

= ρh(uh, uh) ∀uh ∈ Mh,

‖sh‖mh2 = mh(sh, sh) ∀sh ∈ Xh

sym.

Remark 4.2. We can decompose the discrete energy as the sum of three terms:

• The first one,

1

2

∥∥∥∥un+1h − un

h

�t

∥∥∥∥2

ρh

+ 1

2ah

(σ n+1

h − sn+1h , σ n

h − snh

)corresponds to the classical discrete energy in the purely elastic case. It approximatesthe first part of the continuous energy (43): 1/2[‖∂tu‖2

ρ + ‖ε(u)‖2C].

• The second one, 14 [‖sn+1

h ‖2mh

+ ‖snh‖2

mh] due to the viscoelasticity, approximates

12‖s‖2

Z−1 .

• The last one,

�t2

4bh

(sn+1h − sn

h

�t,un+1

h − unh

�t

)is a small term of order O(�t2) which is due to the finite difference approximation.

We now prove an energy decay result (f = 0):

Theorem 4.1. The discrete energy verifies:

En+1/2 − En−1/2

�t= −1

8

[∥∥sn+1h + sn

h

∥∥2mτ h

+ ∥∥snh + sn−1

h

∥∥2mτ h

], (80)

where ‖sh‖2mτ h

= mτ h(sh, sh) ∀sh ∈ Xhsym.

Proof. We use the varitional formulation associated to the scheme (77):

ρh

(un+1

h − 2unh + un−1

h

�t2, uh

)− bh

(uh, σ

nh

) = 0 ∀uh ∈ Mh,

mτ h

(sn+1h + sn

h

2, sh

)+ mh

(sn+1h − sn

h

�t, sh

)+ bh

(un+1

h −unh

�t, sh

)= 0 ∀sh ∈ Xh

sym,

ah

(σ n+1

h , σh

) − ah

(sn+1h , σh

) + bh

(un+1

h , σh

) = 0 ∀σh ∈ Xhsym.

(81)


If we take uh = (un+1h − un−1

h )/(2�t) (the centered approximation of ∂tuh at tn) andsh = sn+1

h + snh , then we get:

1

2�t

[∥∥∥∥un+1h − un

h

�t

∥∥∥∥2

ρh

−∥∥∥∥un

h − un−1h

�t

∥∥∥∥2

ρh

]− bh

(un+1

h − un−1h

2�t, σ n

h

)= 0, (82)

1

2

∥∥sn+1h + sn

h

∥∥2mτ h

+ 1

�t

[∥∥sn+1h

∥∥2mh

− ∥∥snh

∥∥2mh

] + bh

(un+1

h − unh

�t, sn+1

h + snh

)= 0. (83)

Using the third equation of (81), we obtain:

bh

(un+1

h − un−1h

2�t, σ n

h

)= bh

(un+1

h − un−1h

2�t, σ n

h − snh

)+ bh

(h

un+1h − un−1

h

2�t, sn

h

)= − 1

2�t

[ah

(σ n+1

h − sn+1h , σ n

h − snh

)− ah

(σ n

h − snh, σ n−1

h − sn−1h

)] + bh

(un+1

h − un−1h

2�t, sn

h

), (84)

which allows us to rewrite (82) in the form:

1

2�t

[∥∥∥∥un+1h − un

h

�t

∥∥∥∥2

ρh

−∥∥∥∥un

h − un−1h

�t

∥∥∥∥2

ρh

]− bh

(un+1

h − un−1h

2�t, sn

h

)+ 1

2�t

[ah

(σ n+1

h − sn+1h , σ n

h − snh

) − ah

(σ n

h − snh , σ n−1

h − sn−1h

)] = 0. (85)

If we take the average of equation (83) at time tn+1/2 and tn−1/2, we get:

1

4

∥∥sn+1h + sn

h

∥∥2mτ h

+ 1

4

∥∥snh + sn−1

h

∥∥2mτ h

+ 1

2�t

[∥∥sn+1h

∥∥2mh

− ∥∥sn−1h

∥∥2mh

]+ bh

(un+1

h − unh

2�t, sn+1

h + snh

)+ bh

(un

h − un−1h

2�t, sn

h + sn−1h

)= 0. (86)

We decompose the two last terms of this formula in the form:bh

(un+1

h − unh

2�t, sn+1

h + snh

)= bh

(un+1

h − unh

2�t, sn+1

h − snh

)+ 2bh

(un+1

h − unh

2�t, sn

h

),

bh

(un

h − un−1h

2�t, sn

h + sn−1h

)= −bh

(un

h − un−1h

2�t, sn

h − sn−1h

)+ 2bh

(un

h − un−1h

2�t, sn

h

).

(87)Plugging (87) into (86), we get

−bh

(un+1

h − un−1h

2�t, sn

h

)= 1

2�t

(T

n+1/21 −T

n−1/21

)+ 1

4�t

(T

n+1/22 −T

n−1/22

)+T n, (88)


where

Tn+1/2

1 = 1

2

[∥∥sn+1h

∥∥2mh

+ ∥∥snh

∥∥2mh

], T

n+1/22 = bh

(un+1

h − unh, s

n+1h − sn

h

)and

T n = 1

2

[∥∥∥∥sn+1h + sn

h

2

∥∥∥∥2

mτ h

+∥∥∥∥sn

h + sn−1h

2

∥∥∥∥2

mτ h

]� 0.

After substitution of the quantity (88) in (85), we will find (80)

En+ 12 − En−1/2

�t= −T n,

which is the desired identity. �

In order to establish a sufficient stability condition, thanks to theorem 4.1, it sufficesto show that the energy En+1/2 is a positive quadratic form (the rest of the proof beingthen classical, see [37], for instance). This is where the CFL stability condition willoccur: the time step �t must be small enough in order to ensure the positivity of En+1/2.

To establish this stability condition, we shall have to come back to the matrix for-mulation (78) of our scheme and to introduce a new matrix:

K = (A−1 + M−1

s

)−1.

We shall also use the following notation for vector norms in RN :

• ‖.‖ will denote the usual Euclidien norm of RN and (. , .) the corresponding inner

product.

• If P denotes a positive definite symmetric matrix, we shall set:

‖U‖P = supU �=0

(PU, U)1/2

‖U‖ .

We can now state the main result of this section.

Theorem 4.2. A sufficient stability condition in L2 of the numerical scheme (77) isgiven by

�t

2‖B‖ � 1, (89)

where, by definition,

‖B‖ = supUh �=0,�h �=0

(B�h,Uh)

‖Uh‖Mu‖�h‖K

. (90)


Proof. We look for a condition under which the discrete energy is positive. To do this,we use the equivalent matrix form of En+1/2:

En+1/2 = 1

2

∥∥∥∥Un+1h − Un

h

�t

∥∥∥∥2

Mu

+ 1

2

(A−1B�Un+1

h , B�Unh

) + 1

4

(∣∣Sn+1h

∣∣2Ms

+ ∣∣Snh

∣∣2Ms

)+ �t2

4

(B� Un+1

h − Unh

�t,Sn+1

h − Snh

�t

).

As the following quantities verify:∥∥Sn+1

h

∥∥2Ms

+ ∥∥Snh

∥∥2Ms

= 1

2

(∥∥Sn+1h + Sn

h

∥∥2Ms

+ ∥∥Sn+1h − Sn

h

∥∥2Ms

),(

A−1B�Un+1h , B�Un

h

) = 1

4

∥∥B�(Un+1

h + Unh

)∥∥2A−1 − 1

4

∥∥B�(Un+1

h − Unh

)∥∥2A−1,

we can rewrite En+1/2 in the form: En+1/2 = En+1/21 + E

n+1/22 , where

En+1/21 = 1

2

∥∥∥∥B� Un+1h + Un

h

2

∥∥∥∥2

A−1

+ 1

2

∥∥∥∥Sn+1h + Sn

h

2

∥∥∥∥2

Ms

� 0,

En+1/22 = 1

2

∥∥∥∥Un+1h − Un

h

�t

∥∥∥∥2

Mu

− �t2

8

∥∥∥∥B� Un+1h − Un

h

�t

∥∥∥∥2

A−1

+ �t2

8

∥∥∥∥Sn+1h − Sn

h

�t

∥∥∥∥2

Ms

+ �t2

4

(B� Un+1

h − Unh

�t,Sn+1

h − Snh

�t

).

Moreover, the quantity En+1/22 satisfies

En+1/22 � 1

2

([Mu − �t2

4BA−1B�

]Un+1

h − Unh

�t,Un+1

h − Unh

�t

)+ �t2

8

∥∥∥∥Sn+1h − Sn

h

�t

∥∥∥∥2

Ms

− �t2

8

(BM−1

s B� Un+1h − Un

h

�t,Un+1

h − Unh

�t

)− �t2

8

∥∥∥∥Sn+1h − Sn

h

�t

∥∥∥∥2

Ms

= 1

2

(Mu

Un+1h − Un

h

�t,Un+1

h − Unh

�t

)− �t2

8

(B

(A−1 + M−1

s

)B� Un+1

h − Unh

�t,Un+1

h − Unh

�t

).

The positivity of En+1/2 is ensured under the inequality

�t2

4

(B

(A−1 + M−1

s

)B�Uh,Uh

)� (MuUh,Uh) ∀Uh ∈ Xh

sym,

which is equivalent to: (�t2/4)‖L‖ � 1, where

L = B(A−1 + M−1

s

)B� and ‖L‖ = sup

U �=0

(LU,U)

(MuU,U). (91)


This implies the stability sufficient condition (89) thanks to the following lemma. �

Lemma 4.1. If ‖B‖ and ‖L‖ are resp. defined by (90) and (91) we have the identity‖B‖2 = ‖L‖.

Proof. We show the two inequalities ‖B‖2 � ‖L‖ and ‖B‖2 � ‖L‖:

1. ‖B‖2 � ‖L‖. Let (�,U) ∈ Xhsym × Mh:

(B�,U) = (�,B�U) � ‖�‖K‖B�U‖K−1

= ‖�‖K

(BK

−1B�U,U)1/2

= ‖�‖K(LU,U)1/2

� ‖L‖1/2‖�‖K‖U‖Mu.

2. ‖B‖2 � ‖LL‖. Let U ∈ Mh, if we introduce � = K−1B�U , we have

(LU,U) = (BK

−1B�U,U) = (B�,U) � ‖B‖ ‖U‖Mu

‖�‖K. (92)

The norm ‖�‖2K

is written as

‖�‖2K

= (K�,�) = (B�U,�) = (B�,U) � ‖B‖ ‖U‖Mu‖�‖K, (93)

which gives us ‖L‖ � ‖B‖2. �

Remark 4.3. We did not succeed to express the norm ‖B‖ (see (90)) in terms of thebilinear forms b(. , .), ρ(. , .), a(. , .) and m(. , .). However, when one tends to the purelyelastic case, i.e. when the matrix Z = D − C tends to zero, then the matrix Ms tends to0 and K tends to A. Therefore, in the limit Z → 0, we get

‖B‖ = supUh �=0,Sh �=0

(B�h,Uh)

‖Uh‖Mu‖�h‖K

= supbh(σ, u)

‖u‖ρh‖σ‖ah

, (94)

and one recovers the stability condition for the elastodynamic equations [45].

The particular case of a homogeneous isotropic medium. In this case, the tensor C andD are given by

(Cσ )ij = λσkkδij + 2µσij , (Dσ )ij = λγλσkkδij + 2µγµσij ∀i, j = 1, n.

We find ‖B‖ = vp2√

τp/h√

τ0, then the numerical scheme is stable, if the followingsufficient stability condition is satisfied:

�t � h

cp,∞, cp,∞ = vp

√τp

τ0, τp = τ0

λγλ + 2µγµ

λ + 2µ. (95)

cp,∞ is the P waves velocity in high frequencies [3,5].


Remark 4.4. The condition (95) is also necessary, as it can be proven by a Fourier analy-sis in a homogeneous 1D medium (see [6]) by using a study in only one direction.

4.6. The discrete generalized Zener model

In the same way, we can extend the preview studies to the generalized Zenermodel (42). If we introduce si = σi − Ciε(u), we get the reformulate problem:

ρ∂2t u −

k∑i=1

div σi = f,

Mi,τ si + Mi∂t si = ε(∂tu) ∀i = 1, . . . , k,

Aiσi − Aisi = ε(u) ∀i = 1, . . . , k,

(96)

where Mi = Z−1i , Mi,τ = Z−1

i,τ , Ai = C−1i .

The discrete schemeMu

Un+1h − 2Un

h + Un−1h

�t2−

k∑i=1

BSni,h +

k∑i=1

BA−1i B�Un

h = Fn,

Mi,τ

Sn+1i,h + Sn

i,h

2+ Mi,s

Sn+1i,h − Sn

i,h

�t+ B� Un+1

h − Unh

�t= 0 ∀i = 1, . . . , k.

(97)

It is also easy to obtain a discrete energy decay result and a stability condition. Thediscrete energy

En+1/2g = 1

2

∥∥∥∥un+1h − un

h

�t

∥∥∥∥2

ρ

+k∑

i=1

[1

4

(∥∥sn+1i,h

∥∥2mi

h

+ ∥∥sni,h

∥∥2mi

h

)+ �t2

4b

(un+1

h − unh

�t,un+1

i,h − uni,h

�t

)+ 1

2ai

h

(σ n+1

i,h − sn+1i,h , σ n

i,h − sni,h

)],

satisfies the dissipation result:

En+1/2g − E

n−1/2g

�t= −1

8

k∑i=1

[∥∥sn+1i,h + sn

i,h

∥∥2mi

τh

+ ∥∥sni,h + sn−1

i,h

∥∥2mi

τh

].

Consequently the sufficient stability condition is given by the following inequality:

�t2

4

∥∥∥∥∥k∑

i=1

Li,h

∥∥∥∥∥ � 1, (98)

where Li,h = B(A−1i + M−1

i,s )B� ∀i = 1, . . . , k.


In practice, instead of working with the condition (98), we use the following suffi-cient condition:

�t2

4

k∑i=1

‖Li,h‖ � 1, (99)

which gives us in the homogeneous isotropic case the stability condition:

�t � h

(k∑

i=1

c2i

)−1/2

, ci = vp,i

(τp,i

τ0,i

)1/2

, (100)

where vp,i (resp. τ0,i and τp,i) is the velocity (resp. the two relaxation times) of P wavescorrespond to the the ith elementary law in the generalized model (42).

5. Numerical results

5.1. Treatment of unbounded domain

We treat the viscoelastic wave propagation in an unbounded domain by applyingthe PML technique. The general principle of the method is to surround the domain of in-terest with absorbing layers. The particularity of the absorbing perfectly matched layersis that they are perfectly matched, which means that there is no reflection at the inter-face between the viscoelastic medium and the absorbing layer. Therefore the restrictionof the solution to the viscoelastic medium coincides with the exact solution. Further-more, the transmitted wave decreases exponentially inside the layer. The PML modelwas first introduced by Bérenger [9] for solving the Maxwell’s equations in unboundeddomains and has been since then extended to other models, as elastodynamics [24,29]and acoustics [25,33,34]. We have extended it here to the viscoelastodynamic equations.

5.1.1. Adaptation of the PML technique to viscoelastic mediaWe present the PML model, by adapting the method used for the elastody-

namic [24] to the viscoelastic problem. In [24] the authors present the basic principles ofthe method for a first-order hyperbolic system, and apply it for the elastodynamic equa-tions with a mixed formulation velocity-stress, which consists essentially in two steps:first a splitting of the differential operators as a sum of derivatives with respect to x1 andto x2, second adding a damping term on the unknowns associated to the derivatives withrespect to x1 (for a vertical layers). Here we present the method for the second orderdisplacement–stress formulation of the visco-elastodynamic equations. We consider ourproblem in the reformulated form:

ρ∂2t u − div σ = 0,

Mτ s + M∂ts − ε(∂tu) = 0,

Aσ − As − ε(u) = 0,

(101)


with the initial conditions (64), and assuming that the initial data have a compact supportin the propagation space.

In order to split the differential operators, we first rewrite the system (101) asρ∂2

t u = D2∂x2σ + D1∂x1σ,

Mτ s + M∂ts = E2∂2x2t

u + E1∂2x1t

u,

Aσ − As = E2∂x2u + E1∂x1u,

(102)

where

σ = (σ11, σ22, σ12)T, D2 =

[0 0 10 1 0

], D1 =

[1 0 00 0 1

],

E2 = (D2)T and E1 = (D1)

T.

We now introduce the splitting:

u = u2 + u1, σ = σ2 + σ1, s = s2 + s1, (103a) ρ∂2t u2 = D2∂x2σ,

Mτ s2 + M∂ts2 = E2∂2x2t

u,

Aσ2 − As2 = E2∂x2u.

(103b)

ρ∂2t u1 = D1∂x1σ,

Mτ s1 + M∂ts1 = E1∂2x1t

u,

Aσ1 − As1 = E1∂x1u.

(103c)

The damping terms will concern only equation (103c) containing derivatives with respectto x1. In order to introduce these damping terms, we recall the interpretation of PMLsin the frequency domain (see [19,23,24,40]) which consists in the following complexchange of variables:

x1 −→ x1 + 1

iω

∫ x

0d1(s) ds. (104)

This is also equivalent to the substitution:

∂x1 −→ η1∂x1, η1 = iω

iω + d1(x1), (105)

where d1(x1) is a damping function equal to 0 in the propagation medium and positivein the absorbing medium (figure 6).

To apply this principle, we first rewrite (103c) in the frequency domain −ρω2u1 = D1∂x1σ,

Mτ s1 + iωMs1 = iωE1∂x1u,

Aσ1 − As1 = E1∂x1u.

(106)


Figure 6. Support of initial data.

If we use the change of variables (105), we get −ρω2u1 = η1D1∂x1σ,

Mτ s1 + iωMs1 = iωη1E1∂x1u,

Aσ1 − As1 = η1E1∂x1u.

(107)

In order to come back in the time domain, it is necessary to introduce some auxiliaryunknowns (U1, S1 and �1), and it is easy to show that we finally obtain the followingsystem for the PML in the x1 direction:

ρ∂2t U1 = D1∂x1σ,

Mτ S1 + M∂tS1 = E1∂2x1t

u,

A�1 − AS1 = E1∂x1u,

∂tU1 = ∂tu1 + d1(x1)u1,

∂t�1 = ∂tσ1 + d1(x1)σ1,

∂tS1 = ∂ts1 + d1(x1)s1.

(108)

We replace ∂2t U1 by ∂2

t u1 +d(x1)∂tu1 and as s does not appear in the first three equationswe eliminate the last equation and we replace S1 by s1. The system (108) is written thenin the form:

ρ(∂2t u1 + d1(x1)∂tu1

) = D1∂x1σ,

Mτ s1 + M∂ts1 = E1∂2x1t

u,

A�1 − As1 = E1∂x1u,

∂t�1 = ∂tσ1 + d1(x1)σ1.

(109)


Finally, the PML model (in the direction x1) associated to the problem (101) is given bythe following system:

u = u2 + u1, σ = σ2 + σ1,

ρ∂2t u2 = D2∂x2σ, ρ

(∂2t u1 + d1(x1)∂tu1

) = D1∂x1σ,

Mτ s2 + Mpts2 = E2∂2x2t

u, Mτ s1 + M∂t s1 = E1∂2x1t

u,

Aσ2 − A s2 = E2∂x2u, A�1 − As1 = E1∂x1u,

∂t�1 = ∂tσ1 + d1(x1)σ1.

(110)

We treat in the same way the PML layer in the direction x2, and, finally, the general PMLmodel is written:

u = u2 + u1, σ = σ2 + σ1,

ρ(∂2t u2 + d2(x2)∂tu2

) = D2∂x2σ, ρ(∂2t u1 + d1(x1)∂tu1

) = D1∂x1σ,

Mτ s2 + M∂ts2 = E2∂2x2t

u, Mτ s1 + M∂ts1 = E1∂2x1t

u,

A�2 − As2 = E2∂x2u, A�1 − As1 = E1∂x1u,

∂t�2 = ∂tσ2 + d2(x2)σ2, ∂t�1 = ∂tσ1 + d1(x1)σ1.

(111)

5.1.2. The discrete PML modelTo approximate the PML model, we use the mixed finite element method with mass

lumping presented in section 4.3 for space discretization and a second order centeredfinite difference scheme for time discretization, which gives us the following system:

uh = uh,1 + uh,2, σh = σh,1 + σh,2, sh = sh,1 + sh,2,

Mu

[un+1

h,m − 2unh,m + un−1

h,m

�t+ Du,m

un+1h,m − un

h,m

�t

]− Biσ

nh,m = f n,

Mτ

sn+1h,m + sn

h,m

2+ Ms

sn+1h,m − sn

h,m

�t+ B�

i

un+1h,m − un

h,m

�t= 0,

A�n+1h,m − Asn+1

h,m + B�i u

n+1h,m = 0,

�n+1h,m − �n

h,m

�t− σ n+1

h,m − σ nh,m

�t− Dσ,m

σ n+1h,m + σ n

h,m

2= 0,

for m = 1, 2,

(112)

where Mu, Ms, Mτ and A are the same mass matrices as in section 4 and

(Bm)i,j = bm(ω, φj ) =∫

�

Dm∂xmφjωi dx, 1 � i � N1, 1 � j � N2,

(Du,m)i,j =∫

�

dm(xm)ωiωj dx, 1 � i � N1, 1 � j � N1,

(Dσ,m)i,j =∫

�

dm(xm)φi : φj dx, 1 � i � N2, 1 � j � N2,

for m = 1, 2.

(113)


5.2. Numerical results

5.2.1. Monodimensional caseFor the 1D problem, it is easy to validate the numerical scheme suggested in the

previous section, since we have an exact solution for a particular choice of initial con-ditions. It is supposed here that � is the unit segment ]0, 1[ and the following data areconsidered:

f (x, t) = u(0, t) = u(1, t) = 0,

u0(x) = sin(πx), u1(x) = 0,

σ0(x) = µτ1

τ0π cos(πx),

and coefficients of the homogeneous viscoelastic medium

ρ = 1, µ = 1, τ0 = 1, τ1 = 1.2.

The exact solution (u, σ ) of the problem (24) in 1D is calculated by the separation ofvariables method, i.e. we look for a solution in the form

u(x, t) = U(t) sin(πx),

σ (x, t) = �(t)π cos(πx),

U(0) = 1, U (0) = 0, �(0) = µτ1

τ0.

(114)

Using (114) in (24), we find easily{U + π2� = 0,

� + τ0� = U + τ1U ,(115)

which implies that U is a solution of the following third order equation:τ0

d3U

dt3+ d2U

dt2+ π2τ1

dU

dt+ π2U = 0,

U(0) = 1, U (0) = 0, U(0) = −π 2µτ1

τ0.

(116)

Using the Fourier transform, we show that the solution of this system can be written inthe form

U(t) = CeS∗t + eηt [A cos ω∗t + B sin ω∗t],where S∗ and η ± iω∗ are the solutions of the following equation

τ0X3 + X2 + τ1π

2X + π2 = 0.

A,B, and C are real constants which we can determine from the initial conditions.For the numerical solution, we choose as discretization parameters �t = 0.01

(satisfies the stability condition) and h = 1/80. We denote by uexa the displacement forthe exact solution and by unum displacement for the solution obtained by our numericalscheme. We present in figure 7 the variation of the error |uexa − unum| with respect


Figure 7. Error variation.

Figure 8. Numerical and exact displacement at point x = 0.5.

to time and space and in figure 8 the displacement field, at point x = 0.5 (where thedisplacement is maximum) with respect to time for both the exact and the approximatesolution. These two figures show that there is a very good agreement between the twosolution (less than 0.022% of error). Then, in order to study the convergence of ourscheme, we change the discretization parameters h and �t , while keeping the ratio �t/h

fixed equal to 0.9 � CFL = 0.91287. In figure 9 we present the L∞-norm of thecorresponding error with respect to the space step h (on loglog scale). Figure 9 showsthat there is convergence and that the error is of order 2.

5.2.2. Numerical simulation in 2D caseIn order to show the properties of wave propagation in viscoelastic media, we

present in the following simulations a comparison between the wave propagation in anelastic and a viscoelastic isotropic homogeneous medium. In these experiments we con-


h �t ‖uexa − unum‖∞10−1 9 × 10−2 6.11162 × 10−03

5 × 10−2 4.5 × 10−2 1.46935 × 10−03

2.5 × 10−2 2.25 × 10−2 3.61645 × 10−04

1.25 × 10−2 1.125 × 10−2 8.97108 × 10−05

6.25 × 10−3 5.625 × 10−3 2.23409 × 10−05

3.125 × 10−3 2.8125 × 10−3 5.57442 × 10−06

Figure 9. Convergence.

sider a domain � = [0, 10] × [0, 10] and we take a regular grid composed of squares ofedges h = 0.1, a time step �t satisfying the stability condition: �t = CFLh, whereCFL = (vp

√τp/τ0)

−1 (resp. CFL = v−1p ) in the viscoelastic medium (resp. elastic

medium). We take the initial data u0, u1, σ0 equal to 0 and we consider a compactsupported source f located around the point source S(xs, zs) and verifies:

f (x, t) = F(t)�g(r), r = ((x − xs)

2 + (z − zs)2)1/2

, (117)

with

F(t) ={

−2π2f 20 (t − t0)e

−π2f 20 (t−t0)

2if t � 2t0,

0 if not.

t0 = 1

f0, f0 = vs

hNL

central frequency, (118)

�g(r) =(

1 − r2

a2

)1Ba

�e, �e =(

x − xs

r,z − zs

r

),

where NL is the number of points per S wave-length and 1Bathe indicatrice function

associated to the disk Ba of center S and radius a = 5h.The isotropic elastic medium is characterized by the wave velocities vp, vs and the

mass density ρ:

ρ = 1, vp = 2.74, vs = 1.43 (119)

and the viscoelastic medium is characterized by the same values vp, vs, ρ and the relax-ation times τ0, τp and τs

τ0 = 0, 7, τp = 1.0133, τs = 1.01470. (120)

Energy dissipation. We make a first experiment in the domain � with free bound-ary conditions at the outer boundaries and the point source located in the center of thedomain. We compare in figure 10 the evolution of discrete energy for elastic and the


Figure 10. Discrete energy.

Figure 11. Computational domain.

viscoelastic one. Figure 10 shows that after extinction of the source f the elastic en-ergy (dashed line) is constant while the viscoelastic energy (solid line) is exponentiallydecreasing, which confirms our predictions.

Wave propagation in homogeneous, isotropic media. Now, we compare the wave prop-agation in the elastic medium defined in (119) with the one in the viscoelastic mediumdefined in (119)–(120). The domain � is surrounded with PML absorbing layers oflength δ = 10h (see figure 11). In figure 12, we represent at the top (resp. at the bot-tom) the restriction of the elastic (resp. viscoelastic) displacement to the upper domain[−δ, 10 + δ] × [5, 10 + δ] (resp. lower domain [−δ, 10 + δ] × [−δ, 5]). We observethat the viscoelastic wave propagates faster and is more damped than the elastic wave.Furthermore, the PML layers absorbs very well the waves.

Efficiency of PML model. In order to show that the absorbing layers are well adaptedto the viscoelastic problem, we consider three simulations in:

1. A bounded domain with Dirichlet boundary condition.

2. A big domain to simulate an unbounded domain.


Figure 12. Elastic waves at the top, viscoelastic waves at the bottom.

Figure 13. Seismograms for different domains.

3. A bounded domain with layers PML length δ = 10h,

for which we compare the displacement variation in time at the observation point xo =9.9 (see figure 11) located near the interface between the domain � and the artificiallayers.

In figure 13 we present the three seismograms: in a domain with Dirichlet bound-ary condition (dash-dotted), in a domain with PML layers (solid) and in an unboundeddomain (dotted). We remark that for the first domain, in spite of the damping of vis-coelastic waves, the reflected waves are clearly visible and for the two last domains theboth graphs coincide very well. In order to see the reflection due to the use of PMLlayers, we present the difference between these two displacements, in figure 14. Thisshows that the reflection does not exceed 8 ·10−4, so 0.017% of maximum displacement.


Figure 14. Reflected waves.

5.2.3. Realistic simulationsNow, one considers a realistic experiment modeling an oil reservoir located in a

domain � = [0, 270] × [800, 960] (Mc Elroy in Texas) with a strong productive zonebetween 880 m and 900 m. The heterogeneous medium is characterized by the physicaldata:

depth (m) vp (m/s) vs (m/s) Qp Qs ρ (kg/m3)

800.00 5924.884 2928.865 67.06922 20.00 2280.533900810.00 5773.175 2738.594 35.43660 20.00 2242.557601820.00 5313.743 3004.746 21.44790 20.00 2295.163535830.00 5311.784 2564.933 17.41970 20.00 2206.127967840.00 5986.338 2993.962 31.28279 20.00 2293.101428850.00 6176.533 3050.588 40.08961 20.00 2303.867950860.00 5619.798 3065.524 34.76888 20.00 2306.682782870.00 5163.070 2940.252 53.11094 20.00 2282.747271880.00 5191.213 2473.413 36.02421 20.00 2186.179681890.00 5613.954 2583.599 24.61245 20.00 2210.130769900.00 5579.241 2644.431 31.45738 20.00 2223.027083910.00 5712.608 2905.288 57.79010 20.00 2275.930476920.00 6219.111 2876.129 47.12633 20.00 2270.198248930.00 6386.336 3111.822 27.59576 20.00 2315.343230940.00 6489.850 2866.232 133.56708 20.00 2268.242737950.00 6517.794 2866.232 24.26944 20.00 2268.242737960.00 6111.120 2866.232 63.14770 20.00 2268.242737

Remark 5.1. ρ is calculated from the values of vs by using the Gardner’s law [2]: ρ =av

1/4s with a = 0.31 when ρ is in g/cm3 and v in m/s.

To approach the quality factors, we have used the Q+ optimization algorithm pre-sented in [27] in a range frequencies [fa, fb] = [20, 200] and the generalized Zener’s


Figure 15. P wave velocity.

Figure 16. S wave velocity.

Figure 17. Approximation of Qs = 20 and Qp = 57.7901.


Table 1Optimized relaxation times for Qs = 20 and Qp = 57.7901.

Elementary model τ0 (ms) τs (ms) τp (ms)

1 7.9577 9.9145 8.62382 2.5165 2.5664 2.53363 0.79577 1.0160 0.8652

Figure 18. xs = 5 m, zs = 880 m.


Figure 18. (Continued.)

Figure 19. xs = 135 m, zs = 805 m.

model with 3 elementary laws (see figure 17 for Qs = 20 and Qp = 57.7901).We consider a regular grid composed of squares of edges h = 1 m, the time step�t = 8.272 · 10−5 satisfies the stability condition. The source f is defined by (117)–(118) with f0 = 100 Hz. We present in figures 18 and 19 the snapshots of uz componentwith two different point sources. One of the interest of these experiments is that the


Figure 19. (Continued.)

two localizations of the source give rise to different phenomenons. In the first case, onecan observe guided waves while in the second case one observes transmission-reflexionphenomenons.

Aknowledgements

The authors would like to thank W. Mulder and R.E. Plessix from Shell companyfor providing us with the realistic data.

References

[1] B.A. Auld, Acoustic Fields and Elastic Waves in Solids, Vols. I, II (Wiley, New York, 1973).[2] K.D. Ayon and R.S. Robert, Predicting density using vs and gardner’s relationship, Technical Report,

CREWES (1997).[3] E. Bécache, A. Ezziani and P. Joly, Modeling of wave propagation in linear viscoelastic media, in:

Mathematical Modelling of Wave Phenomena, Mathematical Modelling in Physics, Engineering andCognitive Sciences, Vol. 7 (2002).


[4] E. Bécache, A. Ezziani and P. Joly, Mathematical and numerical modeling of wave propagation inlinear viscoelastic media, in: Mathematical and Numerical Aspects of Wave Propagation (Springer,Berlin, 2003) pp. 916–921.

[5] E. Bécache, A. Ezziani and P. Joly, Modélisation de la propagation d’ondes dans les milieux viscoée-lastiques linéaires I, Analyse mathématique, Technical Report 4785, INRIA (2003).

[6] E. Bécache, A. Ezziani and P. Joly, Modélisation de la propagation d’ondes dans les milieux viscoée-lastiques linéaires II, Analyse numérique, Technical Report 5159, INRIA (2004).

[7] E. Bécache, P. Joly and C. Tsogka, Fictitious domains, mixed finite elements and perfectly matchedlayers for 2-D elastic wave propagation, J. Comput. Acoust. 9(3) (2001) 1175–1201.

[8] E. Bécache, P. Joly and C. Tsogka, A new family of mixed finite elements for the linear elastodynamicproblem, SIAM J. Numer. Anal. 39(6) (2002) 2109–2132.

[9] J.P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput.Phys. (1994).

[10] D.R. Bland, The Theory of Linear Viscoelasticity (Pergamon, New York, 1960).[11] T. Bohlen, Parallel 3-D viscoelastic finite-difference seismic modelling, Comput. Geosci. 28(8) (2002)

887–899.[12] T. Bourbié, O. Coussy and B. Zinszner, Acoustique des Milieux Poreux (l’Institut Français du Pétrole,

1986).[13] H. Brezis, Analyse Fonctionnelle – Théorie et Applications (Masson, Paris, 1983).[14] G. Canadas, Etude mathématique et numérique d’une équation hyperbolique du troisième ordre inter-

venant en seismique, Ph.D. thesis, Université de Pau (1979).[15] J.M. Carcione, Seismic modeling in viscoelastic media, Geophys. 58 (1995) 110–120.[16] J.M. Carcione, Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic and Porous

Media (Pergamon, New York, 2001).[17] J.M. Carcione, D. Kosloff and R. Kosloff, Wave propagation simulation in a linear viscoelastic

medium, Geophys. J. Roy. Astr. Soc. 93 (1988) 393–407.[18] J.M. Carcione, D. Kosloff and R. Kosloff, Wave propagation simulation in a visco-elastic medium,

Geophys. J. Roy. Astr. Soc. 95 (1988) 597–611.[19] W.C. Chew and W.H. Weedon, A 3D perfectly matched medium from modified Maxwell’s equations

with stretched coordinates, IEEE Microwave Opt. Technol. Lett. 7 (1994) 599–604.[20] T. Chichinina, G.R. Jarillo and V. Sabinin, Numerical model of seismic wave propagation in vis-

coelastic media, in: Mathematical and Numerical Aspects of Wave Propagation (Springer, Berlin,2003) pp. 922–927.

[21] R.M. Christensen, Theory of Viscoelasticity – An Introduction (Academic Press, New York, 1982).[22] F. Collino, P. Joly and F. Millot, Fictitious domain method for unsteady problems: Application to

electromagnetic scattering, J. Comput. Phys. 138(2) (1997) 907–938.[23] F. Collino and P.B. Monk, Optimizing the perfectly matched layer. Exterior problems of wave propa-

gation, Comput. Methods Appl. Mech. Engrg. (1998).[24] F. Collino and C. Tsogka, Application of the PML absorbing layer model to the linear elastodynamic

problem in anisotropic heteregeneous media, Geophys. 66 (2001) 294–307.[25] J. Diaz and P. Joly, Stabilized perfectly matched layer for advective acoustics, in: Mathematical and

Numerical Aspects of Wave Propagation (Springer, Berlin, 2003) pp. 115–119.[26] G. Duvaut and J.L. Lions, Inéquations en Mécanique et en Physique (Dunod, Paris, 1972).[27] H. Emmerich and M. Korn, Incorporation of attenuation into time–domain computations, Geophys.

52(9) (1987) 1252–1264.[28] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity (SIAM, Philadelphia,

PA, 1992).[29] S. Fauqueux, Eléments finis mixtes spectraux et couches absorbantes parfaitement adaptées pour la

propagation d’ ondes élastiques en régime transitoire, Ph.D. thesis, Université Paris 9 (2003).[30] Y.C. Fung, Foundations of Solid Mechanics (Prentice-Hall, Englewood, Cliffs, NJ, 1965).


[31] M.E. Gurtin and E. Sternberg, On the linear theory of viscoelasticity, Arch. Rational Mech. Anal. 11(1962) 291–356.

[32] T. Ha, J.E. Santos and D. Sheen, Nonconforming finite element methods for the simulation of wavesin viscoelastic solids, Comput. Methods Appl. Mech. Engrg. 191 (2002) 5647–5670.

[33] T. Hagstrom and I. Nazarov, Absorbing layers and radiation boundary conditions for jet flow simu-lations, in: Proc. of the 8th AIAA/CEAS Aeroacoustics Conf., Breckenridge, CO, USA (17–19 June2002) AIAA paper, pp. 2002–2606.

[34] F.Q. Hu, A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables,J. Comput. Phys. 173 (2001) 455–480.

[35] A. Idesman, R. Niekamp and E. Stein, Finite elements in space and time for generalized viscoelasticMaxwell model, Comput. Mech. 27 (2001) 49–60.

[36] V. Janovský, S. Shaw, M.K. Warby and J.R Whiteman, Numerical methods for treating problems ofviscoelastic isotropic solid deformation, J. Comput. Appl. Math. 63 (1995) 91–107.

[37] P. Joly, Variational methods for time-dependent wave propagation problems, in: Topics in Computa-tional Wave Propagation Direct and Inverse Problems, Lecture Notes in Computational Science andEngineering, Vol. 31 (Springer, New York, 2003) pp. 201–264.

[38] G.A. McMechan and T. Xu, Efficient 3-D viscoelastic modeling with application to near-surface landseismic data, Geophys. 69(2) (1998) 601–612.

[39] J.A. Nitsche, On Korn’s second inequality, RAIRO Numer. Anal. 15(3) (1981) 237–248.[40] C.M. Rappaport, Perfectly matched absorbing conditions based on anisotropic lossy mapping of

space, IEEE Microwave Guided Wave Lett. 3 (1995) 90–92.[41] J.O.A. Robertsson, J.O. Blanch and W.W. Symes, Viscoelastic finite difference modeling, Geophys.

59(9) (1994) 1444–1456.[42] J. Salencon, Viscoélasticité (Presse de l’École Nationale des Ponts et Chaussées, Paris, 1983).[43] S. Shaw, M.K. Warby, J.R Whiteman, C. Dawson and M.F Wheeler, Numerical techniques for the

treatment of quasistatic viscoelastic stress problems in linear isotropic solids, Comput. Methods Appl.Mech. Engrg. 118 (1994) 211–237.

[44] T.A. Spencer, J.R. Sonnad and T.M. Butler, Seismic Q stratigraphy or dissipation, Geophys. 47 (1982)16–24.

[45] C. Tsogka, Modélisation mathématique et numérique de la propagation des ondes élastiques tridimen-sionnelles dans les milieux fissurés, Ph.D. thesis, Université Paris 9 (1999).

A mixed finite element approach for viscoelastic wave - CiteSeer

Documents

Transcript of A mixed finite element approach for viscoelastic wave - CiteSeer