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Wave propagation in anisotropic elastic materials and curvilinear coordinates using a summation-by-parts finite difference method N. Anders Petersson and Bjorn Sjogreen

Submitted to Journal of Computational Physics

October 25, 2014; Revised March 4, 2015

LLNL-JRNL-663238

October 1, 2007

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Wave propagation in anisotropic elastic materials and

curvilinear coordinates using a summation-by-parts finite

di↵erence method

N. Anders Petersson1, Bjorn Sjogreen

Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, POBox 808, Livermore CA 94551.

Abstract

We develop a fourth order accurate finite di↵erence method for solving thethree-dimensional elastic wave equation in general heterogeneous anisotropicmaterials on curvilinear grids. The proposed method is an extension of themethod for isotropic materials, previously described in the paper by Sjogreenand Petersson [J. Sci. Comput. 52 (2012)]. The proposed method discretizesthe anisotropic elastic wave equation in second order formulation, using anode centered finite di↵erence method that satisfies the principle of summa-tion by parts. The summation by parts technique results in a provably stablenumerical method that is energy conserving. We also generalize and eval-uate the super-grid far-field technique for truncating unbounded domains.Unlike the commonly used perfectly matched layers (PML), the super-gridtechnique is stable for general anisotropic material, because it is based ona coordinate stretching combined with an artificial dissipation. As a result,the discretization satisfies an energy estimate, proving that the numericalapproximation is stable. We demonstrate by numerical experiments that, ifthe super-grid layers are su�ciently wide, the errors due to truncating thedomain are of the same order as, or smaller than, the propagation errorsfrom the interior of the domain. Applications of the proposed method aredemonstrated by three-dimensional simulations of anisotropic wave propa-gation in crystals.

Keywords: Anisotropy, elastic wave equation, curvilinear coordinates,far-field closure, summation-by-parts

Email addresses: petersson1@llnl.gov (N. Anders Petersson),sjogreen2@llnl.gov (Bjorn Sjogreen)

1Corresponding author

Preprint submitted to Journal of Computational Physics March 4, 2015

1. Introduction

This paper describes a fourth order accurate numerical method for cal-culating wave propagation in general anisotropic elastic materials, i.e., ma-terials in which waves propagate with di↵erent speeds in di↵erent direc-tions. Such materials occur in several applications. One class of anisotropicmaterials are crystals. Here the directionally dependent wave propagationproperties follow from the symmetries and structure of the atomic bondsin the crystal. In seismic applications, isotropic layered materials behaveanisotropically when they are subjected to waves where the wavelength ismuch longer than the thickness of the layers [1]. Furthermore, fractures inan isotropic material can lead to directionally dependent wave propagationproperties [2], i.e., anisotropic behavior. More generally, spatial homoge-nization of a fine grained heterogeneous isotropic elastic material is knownto result in a coarser grained elastic model with anisotropic properties [3, 4].

Many wave propagation codes for isotropic elastic materials are basedon finite di↵erence methods on staggered grids [5, 6, 7]. These methodsapproximate the elastic wave equation in first order velocity-stress formula-tion. Unfortunately, the staggered grid approach is non-trivial to general-ize to general anisotropic materials. The fundamental di�culty is to placethe dependent variables on the staggered grid, such that all terms in theanisotropic Hooke’s law can be approximated accurately and, at the sametime, making the numerical method stable. Since an isotropic material hasanisotropic properties when the equations are transformed to curvilinear co-ordinates, similar di�culties occur for staggered grid methods on curvilinearmeshes. Node centered methods, which discretize the elastic wave equationin second order displacement formulation, do not have this di�culty. Forexample, the spectral element method, described in [8], is naturally formu-lated for general linear stress-strain relationships, and has successfully beenused for modeling general anisotropy [9] as well as realistic topography usingcurvilinear (unstructured hexahedral) meshes [10].

The present paper has two objectives. First, we describe a fourth or-der accurate node centered finite di↵erence scheme for wave propagation ingeneral anisotropic elastic materials. Our scheme satisfies the principle ofsummation by parts (SBP) and is a generalization of the method describedin [11, 12], which is implemented in the elastic wave propagation open sourcecode SW4, version 1.0 [13]. The finite di↵erence scheme is fourth order ac-curate, stable, and energy conserving. We here generalize the method to a

2

fully anisotropic material in curvilinear coordinates, allowing for accuratemodeling of realistic topography. Our main motivation for using the sum-mation by parts method is to obtain a spatial discretization that satisfies anenergy estimate, which guarantees stability of the numerical approximationfor heterogeneous materials on curvilinear grids with free surface or Dirichletboundary conditions. We remark that our SBP method uses ghost pointsjust outside the boundaries to enforce the boundary conditions in a strong(point-wise) sense. There is a related SBP method that does not use ghostpoints and instead enforces the boundary conditions in a weak sense usingpenalty techniques, see e.g. [14, 15, 16].

The second objective is to analyze and numerically evaluate the super-grid far-field truncation technique for anisotropic elastic materials. Super-grid far-field conditions truncate very large or unbounded domains to finiteextent by adding sponge layers outside the domain of interest. Inside thelayers, the wave equation is modified by a combination of grid stretchingand high order artificial dissipation. Compared to perfectly matched lay-ers (PML) [17], the greatest strength of the super-grid technique is thatthe overall numerical method is provably stable, as long as the underlyingnumerical method is stable on a curvilinear grid. Note that the PML equa-tions can have unstable solutions (growing exponentially in time) for someanisotropic materials [18] that violate the so-called geometric stability con-dition [19]. We have previously proven that the isotropic discretized elasticwave equation with super-grid layers satisfies an energy estimate [12], pre-cluding exponential growth of the numerical solution. In the present paper,we extend that analysis to general anisotropic elastic materials on curvilineargrids. An additional strength of the super-grid technique is its simplicityand low computational cost. In contrast to the PML method, super-griddoes not rely on augmenting the wave equation with additional di↵erentialequations that govern additional dependent variables. A potential weaknessof the super-grid technique is that it does not achieve the ’perfect’ non-reflecting property of PML. However, if the super-grid layers are su�cientlywide, numerical experiments indicate that artificial reflections from the far-field truncation can be made to be of the same order, or smaller, than thepropagation errors from the interior of the domain.

This paper is organized as follows. In Section 2, we review the equationsof anisotropic elastic wave propagation in Cartesian coordinates. Section 3generalizes the results of Section 2 to curvilinear coordinates. The finitedi↵erence discretization is presented in Section 4, where we also present ane�cient way of estimating the stability limit for the time step. Section 5describes the super-grid technique and numerical experiments are presented

3

in Section 6. Here we verify the accuracy of the proposed finite di↵erencescheme, demonstrate the accuracy of the super-grid far-field truncation tech-nique, and verify energy conservation of the numerical solution. Conclusionsare given in Section 7.

2. The anisotropic elastic wave equation

To make the presentation self-contained and to define a consistent no-tation, we start by introducing the governing equations in a form that isamenable for constructing the SBP discretization. For further backgroundinformation, we recommend an advanced text book on solid mechanics,e.g. [20].

We consider the time-dependent elastic wave equation in a three-dimensionaldomain x 2 ⌦, where x = (x(1), x(2), x(3))T are the Cartesian coordinatesand u = (u(1), u(2), u(3))T are the Cartesian components of the three-dimensionaldisplacement vector. In displacement formulation, the elastic wave equationtakes the form

⇢

@

2u

@t

2= r · T + F, x 2 ⌦, t � 0, (1)

r · T = G

T

s

CG

s

u =: Lu, (2)

subject to appropriate initial and boundary conditions. Here, ⇢ is the den-sity, T is the stress tensor, and F is the external forcing per unit volume.The spatial operator L is called the 3⇥ 3 symmetric Kelvin-Christo↵el dif-ferential operator matrix [20]. Let T

ij

and E

ij

be the Cartesian componentsof the symmetric stress and strain tensors, respectively. We adopt Voigt’svector notation,

� = (T11, T22, T33, T23, T13, T12)T

, e = (E11, E22, E33, 2E23, 2E13, 2E12)T

.

which allows Hooke’s law to be expressed in terms of the 6 ⇥ 6 sti↵nessmatrix C, which is symmetric and positive definite [20]. Because C is sym-metric, it has 21 unique elements, corresponding to the 21 parameters of ageneral anisotropic material. With this notation, the strain vector and thedivergence of the stress tensor can be expressed in terms of the symmetricpart of the gradient operator and its transpose, respectively,

Lu = G

T

s

�, � = Ce, e = G

s

u, G

T

s

=

0

BB@

@1 0 0 0 @3 @2

0 @2 0 @3 0 @1

0 0 @3 @2 @1 0

1

CCA , (3)

4

where @k

= @/@x

(k).For the purpose of constructing a SBP discretization, we introduce the

notationG

T

s

� = P

T

1 @1� + P

T

2 @2� + P

T

3 @3�, (4)

where the matrices are defined by

P

T

1 =

0

BB@

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

1

CCA , P

T

2 =

0

BB@

0 0 0 0 0 1

0 1 0 0 0 0

0 0 0 1 0 0

1

CCA ,

and

P

T

3 =

0

BB@

0 0 0 0 1 0

0 0 0 1 0 0

0 0 1 0 0 0

1

CCA .

By using (4) in (2), we obtain

Lu =3X

j=1

P

T

j

@

j

0

@C

3X

j=k

P

k

@

k

u

1

A =3X

j=1

3X

k=1

@

j

�P

T

j

CP

k

@

k

u

�

=:3X

j=1

3X

k=1

@

j

⇣M

jk

@

k

u

⌘, (5)

where the 3⇥ 3 matrices on the right hand side are defined by

M

jk = P

T

j

CP

k

. (6)

Hence, each matrix M

jk contains a subset of the elements of C, as de-termined by the matrices P

j

. We refer to Appendix Appendix A for theexact expressions of M jk. From the positive definiteness of C it follows thatM

11,M

22, and M

33 are also positive definite. Moreover, the definition (6)shows that Mkj = (M jk)T . With this notation we can write

Lu = @1 (A1ru) + @2 (A2ru) + @3 (A3ru) , (7)

where

A1ru := M

11@1u+M

12@2u+M

13@3u, (8)

A2ru := M

21@1u+M

22@2u+M

23@3u, (9)

A3ru := M

31@1u+M

32@2u+M

33@3u. (10)

5

Because Lu is equal to the divergence of the stress tensor, we also have

A1ru =

0

BB@

T11

T12

T13

1

CCA , A2ru =

0

BB@

T12

T22

T23

1

CCA , A3ru =

0

BB@

T13

T23

T33

1

CCA . (11)

From (11) it follows that a boundary with unit normal n = (n(1), n

(2), n

(3))T

has boundary traction

n · T (u) = n

(1)A1ru+ n

(2)A2ru+ n

(3)A3ru. (12)

A free surface boundary condition corresponds to n · T (u) = 0.

2.1. Energy estimate

For a box-shaped domain, ⌦ = {0 x

(1) a

(1), 0 x

(2) a

(2), 0

x

(3) a

(3)}, we define the L2 scalar product of two real vector-valued func-

tions u(x) 2 <

3! <

q and v(x) 2 <

3! <

q by

(u,v)2 =

Z

⌦

qX

l=1

u

(l)v

(l)

!dx

(1)dx

(2)dx

(3). (13)

An energy estimate for the solution of the elastic wave equation can bederived by analyzing the scalar product between u

t

and (1),

(ut

, ⇢u

tt

)2 = (ut

,Lu)2 + (ut

,F)2 . (14)

From (7),

(v,Lu)2 = (v, @1(A1ru) + @2(A2ru) + @3(A3ru))2=: �S(v,u) +B(v,u). (15)

Here, S and B represent interior and boundary terms, respectively. Afterintegration by parts, the interior term can be written

S(v,u) = (@1v,M11@1u+M

12@2u+M

13@3u)2

+ (@2v,M21@1u+M

22@2u+M

23@3u)2

+ (@3v,M31@1u+M

32@2u+M

33@3u)2. (16)

6

The definition of M

jk in (6) gives, @j

v

T

M

jk

@

k

u = @

j

v

T

P

T

j

CP

k

@

k

u =

(Pj

@

j

v)TCP

k

@

k

u, so that

S(v,u) =3X

j=1

3X

k=1

(Pj

@

j

v, CP

k

@

k

u)2 =⇣ 3X

j=1

P

j

@

j

v, C

3X

k=1

P

k

@

k

u

⌘

2

= (Gs

v, CG

s

u)2.

It follows from the positive definiteness of C that S is symmetric and positivesemi-definite,

S(v,u) = S(u,v), S(u,u) = (Gs

u, CG

s

u)2 � 0. (17)

Because C is positive definite, the null-space of S consists of functions u

such that G

s

u = 0. It is straightforward to show that there are six lin-early independent functions that satisfy G

s

u = 0, corresponding to solidbody translations in the three coordinate directions and solid body rota-tions around the three coordinate axes.

The boundary term of (15) satisfies

B(v,u) =

Za

(2)

x

(2)=0

Za

(3)

x

(3)=0

⇥v

T

A1ru

⇤a

(1)

x

(1)=0dx

(2)dx

(3)

+

Za

(1)

x

(1)=0

Za

(3)

x

(3)=0

⇥v

T

A2ru

⇤a

(2)

x

(2)=0dx

(1)dx

(3)

+

Za

(1)

x

(1)=0

Za

(2)

x

(2)=0

⇥v

T

A3ru

⇤a

(3)

x

(3)=0dx

(1)dx

(2). (18)

Obviously, B(v,u) = 0 if v satisfies homogeneous Dirichlet conditions. Thefirst term on the right hand side of (18) is evaluated along the boundariesx

(1) = 0 and x

(1) = a

(1), respectively. Here the normal is n = (⌥1, 0, 0)T ,and A1ru equals the boundary traction. Hence, if a free surface conditionis imposed along x

(1) = 0 or x

(1) = a

(1), we have A1ru = 0. The sameargument applies to the second and third terms. Therefore, B(v,u) = 0 ifu satisfies free surface conditions on all boundaries. In summary,

B(v,u) = 0, if v = 0, or n · T (u) = 0, for x 2 @⌦. (19)

From (14) and (15) it follows that

1

2

d

dt

�k

p

⇢u

t

k

22 + S(u,u)

�= B(u

t

,u) + (ut

,F)2. (20)

7

The terms on the left hand side, kp

⇢u

t

k

22 and S(u,u), represent the kine-

matic and potential energies, respectively. The boundary term B(ut

,u) = 0if u either satisfies homogeneous Dirichlet or free surface conditions, becauseif u = 0 on the boundary then also u

t

= 0.By integrating (20) in time,

E(T ) = E(0) +

ZT

t=0(u

t

,F)2 dt, E(t) :=1

2

�k

p

⇢u

t

k

22 + (G

s

u, C(Gs

u))2�,

if B(ut

,u) = 0. This shows that the solution of the elastic wave equationsubject to homogeneous Dirichlet or free surface boundary conditions is awell-posed problem. In the absence of external forcing (F = 0), we getE(t) = E(0) for all t > 0, i.e., the total energy of the solution is conserved.

3. Generalization to curvilinear coordinates

In this section we consider non-rectangular domains. Our presentationis essentially a generalization of the technique developed in [21].

Assume that there is a one-to-one mapping x = x(r) : [0, 1]3 ! ⌦ ⇢ <

3,

x(r) =⇣x

(1)(r), x(2)(r), x(3)(r)⌘T

, r = (r(1), r(2), r(3))T , 0 r

(k) 1,

for k = 1, 2, 3, from the unit cube in parameter space to the domain ⌦ inphysical space. Denote partial di↵erentiation with respect to the parametercoordinates by e@

k

= @/@r

(k). The relation between @

i

and e@j

can be ex-pressed in terms of the forward mapping function x = x(r), or its inverse,

r = r(x) : ⌦ ! [0, 1]3, where r(x) =�r

(1)(x), r(2)(x), r(3)(x)�T

. By thechain rule,

e@

q

=3X

p=1

@x

(p)

@r

(q)@

p

, @

i

=3X

j=1

⇠

ij

e@

j

, ⇠

ij

=@r

(j)

@x

(i). (21)

for q = 1, 2, 3 and i = 1, 2, 3, respectively. The derivatives of the forwardand inverse mapping functions define the covariant and contravariant basevectors,

a

k

:= e@

k

x =

0

BB@

@x

(1)/@r

(k)

@x

(2)/@r

(k)

@x

(3)/@r

(k)

1

CCA , a

k := rr

(k) =

0

BB@

@r

(k)/@x

(1)

@r

(k)/@x

(2)

@r

(k)/@x

(3)

1

CCA =

0

BB@

⇠1k

⇠2k

⇠3k

1

CCA ,

(22)

8

(2)

1

Γ3r (1)

Γ4

Γ3

Γ2Γ1

Γ2

Γ4

r(2)

x(2)

x(1)

r(1)

r

Γ

Figure 1: The mapping between physical (Cartesian) space (left) and parameter space(right) in the two-dimensional case. Here, boundary segments �1, �2, �3, and �4 aremapped to to r

(1) = 0, r(1) = 1, r(2) = 0, and r

(2) = 1, respectively.

for k = 1, 2, 3, respectively. It is well-known that the contravariant basevectors can be expressed in terms of the covariant base vectors (see [22] fordetails),

a

i =1

J

(aj

⇥ a

k

) , (i, j, k) cyclic, J = det |a1 a2 a3| . (23)

Here, J is the Jacobian of the forward mapping function. The above relationprovides a convenient way of calculating the metric coe�cients ⇠

ij

, whichare needed in the curvilinear formulation of the elastic wave equation. Inthe following we assume that the mapping is non-singular, with 0 < J < 1.

In Cartesian coordinates (left side of Figure 1), the elastic wave equationtakes the form (1)-(2). In curvilinear coordinates, it is natural to formulatethe elastic wave equation as

⇢J

@

2u

@t

2= JLu+ JF, r 2 [0, 1]3, t � 0. (24)

We introduce the curvilinear mapping into (4) to obtain

G

s

u =3X

j=1

P

j

@

j

u =3X

j=1

P

j

3X

k=1

⇠

jk

e@

k

u =3X

k=1

eP

k

e@

k

u,

9

where

eP

k

=3X

j=1

⇠

jk

P

j

.

This definition gives the divergence of the stress tensor in curvilinear coor-dinates,

Lu = G

T

s

CG

s

u =3X

j=1

eP

T

j

e@

j

⇣C

3X

k=1

eP

k

e@

k

u

⌘

=1

J

3X

j=1

3X

k=1

e@

j

⇣J

eP

T

j

C

eP

k

e@

k

u

⌘. (25)

Here we used the metric identities

@1(J⇠k1) + @2(J⇠

k2) + @3(J⇠k3) = 0, k = 1, 2, 3, (26)

which follow from (23) (also see [22]). This identity allows the term J

eP

T

j

to be moved inside the di↵erentiation e@j

on the right hand side of (25). Wedefine the matrices

N

jk = J

eP

T

j

C

eP

k

, (27)

and rewrite (25) on the same form as (7)–(10),

JLu = @1

⇣eA1eru

⌘+ @2

⇣eA2eru

⌘+ @3

⇣eA3eru

⌘, (28)

where

eA1eru := N

11@1u+N

12@2u+N

13@3u, (29)

eA2eru := N

21@1u+N

22@2u+N

23@3u, (30)

eA3eru := N

31@1u+N

32@2u+N

33@3u. (31)

The definition (27) makes it straightforward to verify that the matrices N jk

have the same properties as the matrices M

jk, i.e., N11, N

22, and N

33 arepositive definite and N

kj = (N jk)T .

3.1. Boundary conditions

To transform a free surface boundary condition to curvilinear coordi-nates, we first note that the boundary normal can be represented by the

10

metric derivatives. For example, along r

(1) = 0 or r

(1) = 1, the outwardlydirected unit normal satisfies

n :=

0

BB@

n

(1)

n

(2)

n

(3)

1

CCA = ⌥

rr

(1)

|rr

(1)|

=⌥1q

(⇠11)2 + (⇠21)

2 + (⇠31)2

0

BB@

⇠11

⇠21

⇠31

1

CCA ,

for r

(1) = 0, or r

(1) = 1, where the minus sign applies to r

(1) = 0. Thecomponents of the stress tensor are given by (11). Using (21), the gradientof u in (8)-(10) can be expressed in terms of derivatives with respect to theparameter coordinates. After some straightforward algebra,

n · T =⌥1

J |rr

(1)|

⇣N

11@1u+N

12@2u+N

13@3u

⌘, r

(1) = 0, 1. (32)

The definition of eA1eru in (29) finally gives

n · T =⌥1

J |rr

(1)|

eA1eru, r

(1) = 0, 1. (33)

In a similar way,

n · T =⌥1

J |rr

(2)|

eA2eru, r

(2) = 0, 1, (34)

n · T =⌥1

J |rr

(3)|

eA3eru, r

(3) = 0, 1. (35)

3.2. Energy estimate

By using (24)-(28), the elastic wave equation in curvilinear coordinatescan be written

⇢

@

2u

@t

2=

1

J

h@1

⇣eA1eru

⌘+ @2

⇣eA2eru

⌘+ @3

⇣eA3eru

⌘i+ F, (36)

for r 2 [0, 1]3 and t � 0. In curvilinear coordinates, the volume element inan integral is scaled by J , and the L2 scalar product (13) becomes

(u,v)2 =

Z

r2[0,1]3

qX

l=1

u

(l)v

(l)

!J dr

(1)dr

(2)dr

(3). (37)

An energy estimate can be derived in the same way as in the Cartesiancase, because the factor J in the scalar product cancels the 1/J on the right

11

hand side of (36). Partial integration gives a spatial decomposition of theform (15). The only di↵erence is that the matrices M

jk, which describethe material properties in the Cartesian case, are replaced by the matri-ces N jk, which describe the corresponding material properties in parameterspace r 2 [0, 1]3. Since this is the only di↵erence, the interior term S(v,u)is symmetric and positive semi-definite also in curvilinear coordinates. Incurvilinear coordinates, free surface conditions take the form (33)-(35), andDirichlet boundary conditions are transformed trivially. Hence, the bound-ary term B(u

t

,u) = 0 if u either satisfies free surface or homogeneous Dirich-let conditions. Under such boundary conditions, the elastic wave equationin curvilinear coordinates is therefore a well-posed problem.

3.3. Isotropic elastic material in curvilinear coordinates

In the special case of an isotropic elastic material, we have

M

11iso

=

0

BB@

2µ+ � 0 0

0 µ 0

0 0 µ

1

CCA , M

12iso

=

0

BB@

0 � 0

µ 0 0

0 0 0

1

CCA , M

13iso

=

0

BB@

0 0 �

0 0 0

µ 0 0

1

CCA ,

M

21iso

= (M12iso

)T , M

22iso

=

0

BB@

µ 0 0

0 2µ+ � 0

0 0 µ

1

CCA , M

23iso

=

0

BB@

0 0 0

0 0 �

0 µ 0

1

CCA ,

M

31iso

= (M13iso

)T , M

32iso

= (M23iso

)T , M

33iso

=

0

BB@

µ 0 0

0 µ 0

0 0 2µ+ �

1

CCA .

Here, � and µ are the first and second Lame parameters, respectively.In curvilinear coordinates, the corresponding material properties are de-

scribed by the matrices N ij , defined in (27). For example, N11 satisfies

N

11iso

= Jµ

�⇠

211 + ⇠

221 + ⇠

231

�

0

BB@

1 0 0

0 1 0

0 0 1

1

CCA+J(�+µ)

0

BB@

⇠

211 ⇠11⇠21 ⇠11⇠31

⇠11⇠21 ⇠

221 ⇠21⇠31

⇠11⇠31 ⇠21⇠31 ⇠

231

1

CCA .

The remaining N

ij are of a similar form. For a general curvilinear mapping,note that the transformed matrices do not have any zero elements. Hence,because of the coordinate mapping, the isotropic material has anisotropicproperties in curvilinear parameter space.

12

4. Discretization of the elastic wave equation

To conserve space we only describe the discretization in curvilinear co-ordinates. The Cartesian case follows by using the semi-trivial mappingfunction x

(k)(r(k)) = a

(k)r

(k), k = 1, 2, 3, where a

(k) are constants.We consider the elastic wave equation in curvilinear coordinates (24)

where JLu is given by (28). We re-order the terms of the spatial operatorsuch that

Lu =1

J

h@1(N

11@1u) + @2(N

22@2u) + @3(N

33@3u) + @1(N

12@2u) + @1(N

13@3u)

+@2(N21@1u) + @2(N

23@3u) + @3(N

31@1u) + @3(N

32@2u)

i. (38)

A uniform grid, r(1)i

= (i � 1)h1, i = 0, . . . , n1 + 1, r(2)j

= (j � 1)h2, j =

0, . . . , n2+1, and r

(3)k

= (k�1)h3, k = 0, . . . , n3+1 discretizes the domain inparameter space. Here, the grid spacings are h1 = 1/(n1�1), h2 = 1/(n2�1),and h3 = 1/(n3�1). The points outside the domain are ghost points, whichare used to impose the boundary conditions.

Before presenting our spatial discretization of (38), we first review somewell-known properties of summation-by-parts (SBP) finite di↵erence opera-tors in a one-dimensional setting.

4.1. SBP finite di↵erence operators

Assume that a one-dimensional domain is discretized by the uniform gridx

i

= (i� 1)h for i = 0, . . . , n+1, where the domain boundaries are at i = 1and i = n. Let u

i

be a real-valued function defined on the grid. We saythat the di↵erence operator D, approximating d/dx, satisfies the propertyof SBP if

(u,Dv)h1 = �(Du, v)

h1 � u1v1 + u

n

v

n

, (39)

in a scalar product,

(u, v)h1 = h

nX

i=1

!

i

u

i

v

i

, 0 < !

i

< 1, (40)

where !i

are the weights in the discrete scalar product. The grid functionDu

i

is defined at all points i = 1, . . . , n. Away from the boundaries, Du

i

equals a centered di↵erence operator. In order to satisfy (39), the coe�cientsin D are modified at a few points near each boundary. When using a scalarproduct of the form (40), it is well known that (39) can only be satisfied

13

if the order of the truncation error in Du is reduced by a factor of two ata few points near each boundary. It is possible to improve the truncationerror near the boundary by using so-called full norm SBP operators [23].However, these operators can lead to instabilities with variable coe�cientsand will not be used here.

In the following our presentation assumes a scalar product of the form(40).

SBP operators of order p away from the boundaries and order p/2 nearthe boundaries, for p = 2, 4, 6, 8, are well-documented in the literature, seee.g. [23]. It is theoretically possible to derive even higher order accurate SBPoperators, but the stencils become very wide and the coe�cients depend ona number of parameters which can be di�cult to determine.

Second derivative terms of the type (a(x)ux

)x

) appear in the elasticwave equation. Here a(x) is a known function that describes a materialproperty such as the shear modulus. These terms could be approximated byapplying D twice. However, this approach leads to di�culties with odd-evenmodes, meaning that the null space of D(aDu)

j

contains highly oscillatorygrid functions. Furthermore, because of the boundary modification, thetruncation error of D is not smooth near the boundary, leading to additionalloss of accuracy during the second application of D.

In [11], we constructed a di↵erence operatorG(a)u approximating (aux

)x

,which does not have problems with odd-even modes. This operator satisfiesthe SBP identity

(v,G(a)u)h1 = �(Dv, aDu)

h1+(v, P (a)u)hr1� v1a1S

b

u1+ v

n

a

n

S

b

u

n

. (41)

Here, Sb is a di↵erence operator approximating the first derivative on theboundary, to full order of accuracy (p). The operator is of the form S

b

u1 =(1/h)

Pm�1k=0 s

k

u

k

, where sk

are constant stencil coe�cients. Note that Sb

u1

makes use of the ghost point value u0. The operator S

b

u

n

is similar, anduses the ghost point value u

n+1. The positive semi-definite operator P (a)is small and non-zero for odd-even modes. Note that the scalar product(v, u)

hr1 is weighted di↵erently than (u, v)h1, see [11] for details.

The operator G(a) derived in [11] is fourth order accurate in the interiorand second order near the boundary. It is designed to be compatible withthe operator D in (41), which is the standard SBP 4th/2nd order accurateapproximation of the first derivative. Because the elastic wave equationis solved in second order formulation, two orders of accuracy are gainedin the solution, which becomes fourth order accurate in maximum norm.Extensions to even higher order is possible, but not pursued here.

14

We remark that there is a related SBP method for discretizing (aux

)x

,which does not use ghost points and instead enforces the boundary condi-tions by a penalty techniques, see e.g. [14, 15, 16].

The spatial operator of the elastic wave equation (38) consists of termssuch as @

i

(N ij

@

j

u). Here N

ij is a 3⇥3 matrix with elements n

ij

pq

. Wheni = j, the term is approximated by G(N ii)u, defined as

G(N ii)u :=

0

BB@

(G(N ii)u)1

(G(N ii)u)2

(G(N ii)u)3

1

CCA , (G(N ii)u)p

=3X

q=1

G(nii

pq

)u(q), (42)

for p = 1, 2, 3, where G(nii

pq

)u(q) is the scalar di↵erence operator describedabove.

For vector valued grid functions, we approximate @j

u using the operatorD

j

u, which is defined component-wise. This operator is used to approximate@

i

(N ij

@

j

u) when i 6= j.The vector version of the SBP identities (39) and (41) are

(u, Dv)h1 = �(Du,v)

h1 � u

T

1 v1 + u

T

n

v

n

, (43)

(v, G(N)u)h1 = �(Dv, NDu)

h1 + (v, P (N)u)hr1

� v

T

1 N1Sb

u1 + v

T

n

N

n

S

b

u

n

, (44)

as can be seen by component-wise application of the corresponding scalaridentities.

4.2. Spatial discretization

The spatial operator (38) is discretized as

L

h

u

i,j,k

=1

J

i,j,k

⇥G1(N

11)ui,j,k

+G2(N22)u

i,j,k

+G3(N33)u

i,j,k

+D1(N12D2u

i,j,k

) +D1(N13D3u

i,j,k

)

+D2(N21D1u

i,j,k

) +D2(N23D3u

i,j,k

)

+D3(N31D1u

i,j,k

) +D3(N32D2u

i,j,k

)⇤, (45)

for i = 1, . . . , n1, j = 1, . . . , n2, and k = 1, . . . , n3. Here, Dm

is the standardSBP finite di↵erence operator acting in one of the curvilinear coordinatedirections m = 1, 2, 3. Similarly, G

m

(N) denotes the second derivative op-erator (42) acting in direction m. The discrete scalar product is defined

15

by

(u,v)h

= h1h2h3

n1X

i=1

n2X

j=1

n3X

k=1

!

(1)i

!

(2)j

!

(3)k

J

i,j,k

u

T

i,j,k

v

i,j,k

.

The matricesN ij are defined by (27). If analytical expressions for the deriva-tives of the coordinate mapping are available, they simply need to be evalu-ated at the grid points. However, if they are not available, they can insteadbe approximated by su�ciently accurate numerical di↵erentiation,

@x

(q)

@r

(p)

�����i,j,k

⇡ D

p

x

(q)i,j,k

, p = 1, 2, 3, q = 1, 2, 3. (46)

In this case, the covariant base vectors follow from (22), with @x

(q)/@r

(p)

replaced by D

p

x

(q). The metric coe�cients ⇠pq

and the discretized Jacobian,J

i,j,k

are given by formula (23).

Remark 1. The di↵erence operators can be evaluated undivided, i.e., the

grid sizes, h1, h2, and h3 can be set to one when evaluating Du and G(N)u,

if also the metric di↵erence approximations D

p

x

(q)i,j,k

are treated in the same

way. This is because the grid sizes (h

i

) cancel in the expression for L

h

u.

Remark 2. The metric identities (26) are in general not satisfied exactly

when the metric derivatives are approximated by the di↵erence formula (46).When first order hyperbolic systems are discretized on curvilinear grids, not

satisfying the corresponding metric identities implies that the numerical ap-

proximation of derivatives of constant functions can be non-zero. This, in

turn, can lead to spurious numerical e↵ects when trying to preserve a con-

stant state over long times, unless special free stream preserving metric dis-

cretizations are used [24, 25, 26]. Because we here solve a hyperbolic system

in second order formulation, L

h

u is identically zero for constant u, also when

the di↵erence formula (46) is used to approximate the metric derivatives. In

practical calculations we have always found that these simpler di↵erence for-

mulas are adequate.

The discrete analog of (15) is

(v,Lh

u)h

= �S

h

(v,u) +B

h

(v,u),

where Sh

is symmetric and positive semi-definite, and B

h

contains boundaryterms. To verify this equality we may proceed in the following way. We firstmultiply (45) by !

(1)!

(2)!

(3)Jv

T from the left, sum over all grid points,

16

and finally apply the one-dimensional identities (43) and (44) along eachcurvilinear coordinate direction. This results in

(v,Lh

u)h

= �(D1v, N11D1u)

h

� (D1v, N12D2u)

h

� (D1v, N13D3u)

h

� (D2v, N21D1u)

h

� (D2v, N22D2u)

h

� (D2v, N23D3u)

h

� (D3v, N31D3u)

h

� (D3v, N32D2u)

h

� (D3v, N33D3u)

h

� (v, P1(N11)u)

hr

� (v, P2(N22)u)

hr

� (v, P3(N33)u)

hr

+B

h

(v,u). (47)

All terms on the right hand side, except the last one, define S

h

(v,u). Theboundary term is given by

B

h

(v,u) = h2h3

n2X

j=1

n3X

k=1

!

(2)j

!

(3)k

hv

T

i,j,k

eA1,h

er

h

u

i,j,k

ii=n1

i=1

+ h1h3

n1X

i=1

n3X

k=1

!

(1)i

!

(3)k

hv

T

i,j,k

eA2,h

er

h

u

i,j,k

ij=n2

j=1

+ h1h2

n1X

i=1

n2X

j=1

!

(1)i

!

(2)j

hv

T

i,j,k

eA3,h

er

h

u

i,j,k

ik=n3

k=1, (48)

with

eA1,h

er

h

u = N

11S

b

1u+N

12D2u+N

13D3u, (49)

eA2,h

er

h

u = N

21D1u+N

22S

b

2u+N

23D3u, (50)

eA3,h

er

h

u = N

31D1u+N

32D2u+N

33S

b

3u. (51)

Energy conserving boundary conditions, i.e., boundary conditions thatmake B

h

(v,u) = 0, can be imposed either by a homogeneous Dirichletcondition,

v

i,j,k

= 0, (i, j, k) on the boundary,

or by a free-surface condition,

eA

q,h

er

h

u

i,j,k

= 0, (i, j, k) on the boundary.

Here, q = 1, 2, or 3 depending on which side is being considered. Forexample, on the boundary k = 1, we have

N

31i,j,1D1ui,j,1+N

32i,j,1D2ui,j,1+N

33i,j,1S

b

3ui,j,1 = 0, i = 1, . . . , n1, j = 1, . . . , n2.

17

This constitutes a system of three equations for the three unknowns ui,j,0,

s0N33i,j,1ui,j,0 = �N

33i,j,1

m�1X

k=1

s

k

u

i,j,k

�N

31i,j,1D1ui,j,1 �N

32i,j,1D2ui,j,1.

Because N

33i,j,1 is positive definite and s0 6= 0, this system always has a

unique solution. Note that the system only couples the three ghost pointvalues u

i,j,0, for each (i, j). There is no coupling along the boundary.By comparing (49) with (32), we note that the former is an approxima-

tion of the scaled boundary traction, where the scaling factor J |rr

(1)| is the

surface measure. To make this obvious, we can write the first sum of theright hand side of (48) as (omitting the factor h2h3)

n2X

j=1

n3X

k=1

!

(2)j

!

(3)k

J

i,j,k

|rr

(1)|v

T

i,j,k

1

J

i,j,k

|rr

(1)|

eA1,h

er

h

u

i,j,k

�i=n1

i=1

=

n2X

j=1

n3X

k=1

!

(2)j

!

(3)k

hJ

i,j,k

|rr

(1)|v

T

i,j,k

(n · T

h

)i,j,k

ii=n1

i=1, (52)

and similarly for the other two sums. Here, n · T

h

= 1J |rr

(1)|eA1,h

er

h

u is the

discretization of the boundary traction (32).Finally, when energy conserving boundary conditions are imposed, the

semi-discrete energy estimate

1

2

d

dt

((⇢ut

,u

t

)h

+ S

h

(u,u)) = (ut

,F)h

, (53)

follows in the same way as the corresponding estimate for the continuousproblem. This leads to stability if the energy, (⇢u

t

,u

t

)h

+ S

h

(u,u), is pos-itive. Our SBP discretization has the property that S

h

is positive semi-definite with a null space that is a discretized approximation of the nullspace of the continuous operator, i.e., solid body translations and rotations.For example, the odd-even modes, u

j

= (�1)j are not in the null space ofS

h

because of the terms (u, Pj

(N jj)u)hr

, which are positive for such gridfunctions [11]. Also note that solid body translations and rotations are notpossible if u satisfies homogeneous Dirichlet conditions on at least part ofthe boundary. In this case S

h

(u,u) becomes positive definite and the SBPdiscretization is stable.

18

4.3. Time discretization

The equations are advanced in time with an explicit time integrationmethod. As with all explicit time stepping methods, the time step must notexceed the CFL stability limit. With a Newmark time stepping scheme,

u

n+1� 2un + u

n�1

�2t

= L

h

u

n + F

n

, n = 0, 1, . . . , (54)

the expression for the CFL time step limit is

�2t

⇣

4 1. (55)

The spectral radius ⇣ = maxu 6=0

S

h

(u,u)/(u, ⇢u)h

is di�cult to compute fora general heterogeneous material, even when the material is isotropic. Asan approximation we consider the elastic wave equations in a homogeneousmaterial with periodic boundary conditions. A von Neumann analysis of theFourier transformed problem, in the case of second order accuracy, showsthat the spectral radius in an isotropic elastic material is well approximatedby

⇣ ⇡

4

h

2

2µ+ �+ µ+ µ

⇢

=4

h

2

�c

2p

+ 2c2s

�. (56)

Here, cp

and c

s

are the longitudinal and transverse phase velocities, respec-tively.

In a general anisotropic material, the square of the phase velocity, c2, isan eigenvalue of the Christo↵el equation,

c

2r =

1

⇢

3X

j=1

3X

k=1

n

j

n

k

M

jk

r. (57)

In general, the phase velocity depends on the direction of wave propagationn = (n1, n2, n3)T , |n| = 1. In the isotropic case, the three eigenvaluesof the Christo↵el equation are [c2

p

, c

2s

, c

2s

], independently of the direction n.Since h2⇣/4 can be approximated by the sum of the three eigenvalues of theChristo↵el equation in the isotropic case, it is reasonable to assume that thesum of the eigenvalues of (57) would also be a good approximation of h2⇣/4,in the anisotropic case. The sum of the eigenvalues of a matrix equals thetrace of the matrix, i.e., the sum of its diagonal elements. Hence, the sumof the eigenvalues in the direction n = (n1, n2, n3)T is given by

1

⇢

Tr

0

@3X

j=1

3X

k=1

n

j

n

k

M

jk

1

A =1

⇢

3X

j=1

3X

k=1

n

j

n

k

Tr

⇣M

jk

⌘, (58)

19

where Tr(M) denotes the trace of M . The expression (58) is a quadraticform, whose maximum over all directions n equals the maximum eigenvalue,

max

= max, defined by

det(T � I) = 0, T =1

⇢

0

BB@

Tr(M11) Tr(M12) Tr(M13)

Tr(M21) Tr(M22) Tr(M23)

Tr(M31) Tr(M32) Tr(M33)

1

CCA .

Calculating the largest eigenvalue of this symmetric 3⇥3 matrix is inexpen-sive. Furthermore, the calculation only needs to be done once, before thestart of the time stepping, because the material properties do not change intime. We then use

max

as an approximation of h2⇣/4, resulting in the timestep restriction

�2t

h

2

max

C

cfl

. (59)

If the material model has heterogeneous properties, the procedure is re-peated at each grid point of the mesh, and the largest value of

max

is usedin (59).

The Newmark scheme (54) is only second order accurate in time. Thecalculations shown in this paper use a predictor-corrector modification toobtain fourth order accuracy. It turns out that the fourth order scheme hasa somewhat larger stability limit for the time step [11], but the procedure toestimate the largest eigenvalue ⇣ remains the same. Unless otherwise noted,we use C

cfl

= 1.3 in the numerical experiments in this paper.

5. Super-grid boundary conditions

We truncate unbounded or semi-bounded domains by using the super-grid approach [27, 12]. In this technique, damping layers are added outsidethe domain of interest. The idea is to mimic a very large physical domain,where reflections from the boundary would need a very long time to returnto the domain of interest. Similar to our treatment of curvilinear domains,a coordinate mapping is used in the layers. The elastic wave equation is dis-cretized on a regular grid in parameter space, and the mapping correspondsto stretching the grid to cover a very large physical domain. In parameterspace, the mapping acts by gradually slowing down and compressing thewaves as they progress through the layer. A high order artificial dissipa-tion operator is applied to damp out waves that become poorly resolveddue to the coordinate mapping. The simple combination of a real-valued

20

stretching function and artificial dissipation makes super-grid very straight-forward to implement. This should be compared to the complexity of thePML method [17], where additional di↵erential equations must be solved forauxiliary functions in the layers.

A very important property of the super-grid method is that, if the un-derlying scheme is stable on a curvilinear grid, it will also be stable with theartificial dissipation [12]. By using su�ciently smooth stretching functionsand high order artificial damping terms, we demonstrated in [12] that, if thelayers are su�ciently wide, artificial reflections can be made to converge tozero at the same rate as the interior scheme. In that paper we consideredthe isotropic elastic wave equation in heterogeneous materials. Here, wegeneralize the approach to the anisotropic case, where it is known that thePML technique can lead to stability problems [18].

For simplicity, we describe the super-grid technique for a Cauchy (whole-space) problem (�1 < x

(k)< 1), with super-grid layers on all sides of the

computational domain. This approach is straightforward to generalize tomore general configurations by omitting the layers on some sides of thedomain. The stretching functions are one-dimensional, i.e., x(k) = x

(k)(r(k))for k = 1, 2, 3. Only the diagonal terms of the metric tensor ⇠

ij

are non-zeroand the curvilinear transformation is of the form

@

k

= �

(k)(r(k))@k

, �

(k) =1

@x

(k)/@r

(k), k = 1, 2, 3, ⇠

ij

=

(�

(i), i = j,

0, i 6= j.

(60)The Jacobian of the transformation satisfies J�1 = �

(1)�

(2)�

(3).An artificial dissipation term of order 2p is added in the super-grid layers.

On the semi-discrete level, the elastic wave equation with super-grid layersbecomes

⇢

d

2u

dt

2= L

h

u+ F� "(�1)pQ2p

✓du

dt

◆, " = �2ph

2p�1 cmax

C

cfl

. (61)

Here, �2p is a constant that depends on the order of the dissipation and C

cfl

is the CFL number that determines the time step for the fully discretizedwave equation. For the isotropic elastic wave equation, we use c

max

=qc

2p

+ 2c2s

, where c

p

and c

s

are the compressional and shear wave speeds,

respectively. Section 4.3 discusses estimation of the wave speeds for the fully

21

anisotropic equations. The dissipation term in (61) is of the form

Q2pv =

0

BB@

P3k=1 �

(k)Q

(k)2p (�

(k)⇢)v(1)

P3k=1 �

(k)Q

(k)2p (�

(k)⇢)v(2)

P3k=1 �

(k)Q

(k)2p (�

(k)⇢)v(3)

1

CCA . (62)

Each term in the sums of (62) acts along one of the coordinate directions,and �(k) is a smoothly varying dissipation coe�cient. The damping is scaledby density (⇢) to make it balance the left hand side of (61). This allows �2pto be constant when ⇢ varies in space.

We will use either fourth or sixth order artificial dissipation, correspond-ing to p = 2 or p = 3. When a fourth order (p = 2) artificial dissipation isused, each one-dimensional term is discretized according to

Q

(k)4 (�(k)⇢)v := D

(k)+ D

(k)�

⇣�

(k)⇢D

(k)+ D

(k)� v

⌘. (63)

The sixth order (p = 3) artificial dissipation is discretized according to

Q

(k)6 (�(k)⇢)v := D

(k)+ D

(k)� D

(k)+

⇣�

(k)⇢D

(k)� D

(k)+ D

(k)� v

⌘, (64)

where cell-centered averages are used for the coe�cients, e.g., �j+1/2 = (�

j

+

�

j+1)/2. Here D

(k)± denote the standard first order forward and backward

divided di↵erence operators in direction k.When (61) is discretized in time by the explicit predictor-corrector method,

we want the stability restriction on the time step to be determined by the bythe largest wave speed in the interior of the domain, and not by the amountof dissipation in the super-grid layers. For C

cfl

= 1.3, this turns out to betrue for �4 0.02 and �6 0.005, respectively. If the damping coe�cientexceeds these values, C

cfl

must be reduced to make the time-stepping stable,see [12].

In [12] we used the energy method to prove stability of the discretizedelastic wave equation with super-grid stretching and artificial dissipation.This can be done without the SBP boundary modifications at the dampinglayer boundaries. Instead, a su�cient number of ghost points are introducedsuch that the centered finite di↵erence operators can be applied up to theouter boundary of the damping layer. Homogeneous Dirichlet conditionsare imposed at all ghost points. This procedure leads to a SBP-like stabilityestimate, see [12] for details.

In practice, the solution at the ghost points are set equal to zero beforeeach time step. The material properties for the ghost points is usually

22

extrapolated from the interior. Alternatively, they can be defined directlyfrom the material model if it is defined on a su�ciently large domain.

The stretching function �(r) and the damping functions �(r) are con-structed from an auxiliary function (⇠) (the blue curve in Figure 2), whichsmoothly transitions from one to zero and then back to one,

(⇠) =

8>>>>>><

>>>>>>:

1, ⇠ 0,

P (1� ⇠/`), 0 < ⇠ < `,

0, ` ⇠ 1� `,

P ((⇠ � 1)/`+ 1), 1� ` < ⇠ < 1,

1, ⇠ � 1.

(65)

Here we use the polynomial function P (⌘) = ⌘

6(462 � 1980⌘ + 3465⌘2 �

3080⌘3 + 1386⌘4 � 252⌘5), which satisfies P (0) = 0, P (1) = 1, and makes (⇠) five times continuously di↵erentiable. The one-dimensional stretchingand damping functions are defined by

�(r) = (1� (1� "

L

) (r)) , �(r) = (r)

�(r), 0 < "

L

⌧ 1. (66)

This means that �(r) = 1, and �(r) = 0 for ` r 1 � `. Note that"

L

is given a small positive value to prevent the coordinate stretching frombecoming singular. It is not related to the damping coe�cient " in (61).Throughout the numerical experiments in this paper, we use "

L

= 10�4.Examples of the functions and � are plotted in Figure 2.

5.1. Three-dimensional considerations

Along the sides of a three-dimensional domain, where only one super-grid damping layer is active, we use a one-dimensional damping function, asdescribed above. For example,

�

(1)(r(1), r(2), r(3)) = �(r(1)), 0 r

(1) `, ` (r(2), r(3)) 1� `.

If the one-dimensional damping function is used where several super-gridlayers meet (at edges or corners of the computational domain), it is necessaryto reduce the damping coe�cient (�2p) to avoid making the explicit time-stepping scheme unstable, see [12]. However, this reduces the strength ofthe damping where only one super-grid layer is active, which leads to largerartificial reflections.

23

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

r

Figure 2: The stretching function �(r) (red) and the auxiliary function (r) (blue), whichcontrols the strength of the damping. In this case, the width of each super-grid layer is` = 0.2.

A simple remedy is provided by introducing the linear taper function ⌧ ,

⌧(r) =

8><

>:

↵, r < 0,

↵+ (1� ↵)r/`, 0 r `,

1, r > `.

For example, along the edge 0 r

(1) `, 0 r

(2) `, we define the

two-dimensional damping functions(�

(1)(r(1), r(2), r(3)) = �(r(1))⌧(r(2)),

�

(2)(r(1), r(2), r(3)) = ⌧(r(1))�(r(2)),` r

(3) 1� `,

where �(r) is the one-dimensional damping function (66). Using this con-struction, the strength of the damping is determined by

I2(r(1)

, r

(2)) := (�(1) + �

(2))� = ⌧(r(2)) (r(1)) + ⌧(r(1)) (r(2)),

24

where (r) is the auxiliary function (65). When ↵ = 1/3, this constructionsatisfies max I2 = 1. Away from the edge, the strength of the damping is thesame as in the one-dimensional case because (r) = 0 and ⌧(r) = 1 for r � `.Therefore, I2(r(1), r(2)) = (r(2)) for r

(1)� ` and I(r(1), r(2)) = (r(1)) for

r

(2)� `. At the edge, ⌧(0) = 1/3 and (0) = 1, giving I2(0, 0) = 2/3.

The function I2(r(1), r(2)) has a local maxima along the diagonal r

(1) =r

(2)⇡ 0.31`, where I2 ⇡ 0.983. The tapering approach is straightforward to

generalize to the other edges of the computational domain.Near a corner where three super-grid layers meet, the strength of the

damping equals I3 := (�(1) + �

(2) + �

(3))�. For r

(k) `, we generalize the

tapering approach by defining8>><

>>:

�

(1)(r(1), r(2), r(3)) = �(r(1))⌧(r(2))⌧(r(3)),

�

(2)(r(1), r(2), r(3)) = ⌧(r(1))�(r(2))⌧(r(3)),

�

(3)(r(1), r(2), r(3)) = ⌧(r(1))⌧(r(2))�(r(3)).

This construction also satisfies max I3 = 1. The strength of the dampinghas a local maxima along the space-diagonal r(1) = r

(2) = r

(3)⇡ 0.37` where

I3 ⇡ 0.823. Also note that the two-dimensional strength is recovered alongedges of the three-dimensional domain (where two super-grid layers meet),e.g. I3(r(1), r(2), r(3)) = I2(r(1), r(2)) for r(3) � `.

The tapering approach is of significant practical importance in three-dimensional calculations, where up to three super-grid layers can meet atcorners. This is because the tapering keeps the maximum strength of thesuper-grid damping approximately equal along sides, edges, and corners ofthe computational domain. With the tapering approach, if the dampingcoe�cient �2p makes the time stepping stable with super-grid damping inonly one direction, the same value will also work when three super-grid layersmeet at a corner. Without the tapering approach, the time stepping wouldbecome unstable unless the damping coe�cient is reduced to approximately�2p/3, leading to less e�cient damping. To compensate, the layers wouldneed to be much thicker. Since the super-grid layers are added outside thedomain of interest, this would significantly increase the total number of gridpoints in a three-dimensional case, thus making the calculation much moreexpensive.

6. Numerical experiments

All simulations reported here were performed with the open source codeSW4, version 1.1 [28]. While most of the numerical experiments below

25

were made with constant materials, our theory and numerical method isapplicable to any heterogeneous anisotropic material. The implementationin SW4 allows the anisotropic material properties to be di↵erent at eachgrid point. As a special case of a heterogeneous material, SW4 can handlea material model with piecewise constant properties, see Section 6.4.

We consider wave propagation in Indium Arsenide (InAs), which is acrystal with cubic symmetry. When the coordinate axes are oriented alongthe bonds of the cubic crystal, its density and sti↵ness matrix are given by(see [29])

⇢ = 5.67·103, C =

0

BBBBBBBBBB@

83.29 45.26 45.26 0 0 0

45.26 83.29 45.26 0 0 0

45.26 45.26 83.29 0 0 0

0 0 0 39.59 0 0

0 0 0 0 39.59 0

0 0 0 0 0 39.59

1

CCCCCCCCCCA

·109, (67)

using SI-units.We start by recalling some fundamental aspects of anisotropic wave prop-

agation, see e.g. [20] for details. The properties of wave propagation in a ho-mogeneous anisotropic material are often quantified by its phase and groupvelocities, as well as its slowness surfaces. A plane wave propagating in unitdirection n = (n1, n2, n3)T can be described by

u(x, t) = re

i(k·x�!t) = re

i⇠(n·x�ct), k = ⇠n, ⇠ = |k|, s =

k

!

.

Here, r is the polarization vector, k = (k1, k2, k3)T is the wave vector, ! theangular frequency, ⇠ the (angular) wave number, and s is the slowness vector.Assuming homogeneous material properties, inserting the plane wave ansatzinto the elastic wave equation (1) with F = 0 and using (5), leads to thedispersion relation (57), i.e., the Christo↵el equation. The phase velocity,

c =!

⇠

=1

|s|

,

equals the (positive) square root of an eigenvalue c

2 of the Christo↵el equa-tion. Note that the matrix in (57) is symmetric and positive definite, so c

2 isreal and positive. The slowness surface corresponding to phase velocity c isdefined by s = n/c. In spherical coordinates, (r,�, ), we have n = n(�, )

26

−5 0 5

x 10−4

−5

0

5x 10

−4

θ

L

S1

S2

Figure 3: Slowness curves for InAs for wave propagation in the (x(1), x

(2)) plane. Unitsare in [s/m].

and the slowness surface is given by r(�, ) = 1/c(�, ). The group veloc-ity, in which direction energy propagates, is a vector with three componentsdefined as

v

g

= (@!/@k1, @!/@k2, @!/@k3)T

.

One can show that the group velocity vector is orthogonal to its correspond-ing slowness surface [18].

In general the Christo↵el equation has three eigenvalues, correspondingto three slowness surfaces. A slowness curve is a cross-section of a slownesssurface. In Figure 3 we show the slowness curves for InAs in the (x(1), x(2))plane. The curves are plotted in polar coordinates (r, ✓). For each an-gle ✓, we solve the Christo↵el equation in the direction (n(1)

, n

(2), n

(3)) =(cos ✓, sin ✓, 0), resulting in three eigenvalues c

2k

(✓), k = 1, 2, 3. The corre-sponding radii in polar coordinates are r

k

(✓) = 1/ck

(✓).The innermost curve (black color, labeled “L”) corresponds to the quasi-

longitudinal wave. It has the largest phase velocity, which only varies mildlywith the direction of wave propagation. The second fastest wave is shown inblue color and labeled “S1”. This is a quasi-transverse wave with the samephase velocity in all directions, corresponding to a circular slowness curve.The other quasi-transverse wave (red color, labeled “S2”) has the slowestphase velocity, with the minima c

min

= 1.831 · 103 at ✓ = ⇡/4, 3⇡/4, 5⇡/4,and 7⇡/4.

27

6.1. The whole-space problem with super-grid layers

We shall solve the anisotropic elastic wave equation numerically and westart by studying a Cauchy (whole-space) problem, truncated by super-gridlayers on all sides of the domain. We take the domain of interest to be(x(1), x(2), x(3)) 2 [1.6 · 103, 10.4 · 103]3. In the unit cube of parameter space,the super-grid layers have thickness ` = 1.6/12 ⇡ 0.133. In the figures below,cross-sections of the solution are plotted as function of scaled parametercoordinates, to equal (x(1), x(2), x(3)) within the domain of interest. In thisscaled parameter space, the super-grid layers have thickness 1.6 · 103.

The solution is driven by an isotropic point moment tensor source,

f(x, t) = g(t)M0

0

BB@

1 0 0

0 1 0

0 0 1

1

CCAr�(x� x

s

), M0 = 1017, (68)

located at x

s

= (6, 6, 6) · 103. Here, r� is the gradient of the Diracdistribution. The source time function is the Gaussian,

g(t) =1

p

2⇡�e

(t�t0)2/2�2, � =

1

16, t0 = 0.375. (69)

We estimate the dominant frequency in the Gaussian by f0 = 1/(2⇡�) ⇡

2.55 and the highest significant frequency by f

max

⇡ 2.5f0 ⇡ 6.37. The pointmoment tensor source term is discretized in space by using the techniquedescribed in [30].

In Figure 4 we show the magnitude of the displacement in the x

(3) =6 · 103 plane at time t = 1.5. The outermost wave front corresponds tothe quasi-compressional wave. Corresponding to the shape of its slownesscurve (labeled “L” in Figure 3), it propagates slightly faster along the di-agonal than along the coordinate axes. The waves closer to the center ofthe figure are of quasi-shear type, which are generated by the moment ten-sor source (68), even though it is isotropic. The complex wave fronts are aresult of the directional variation in phase velocity. Because the motion isgenerated by a point source, all wave fronts are initially circular, but thequasi-shear waves quickly develop a more complicated structure. In partic-ular, note the swallow tail-shapes of the the slowest quasi-shear wave. Theyare due to the inflection points in the “S2” slowness curve in Figure 3.

The anisotropic properties of InAs make it challenging to truncate thecomputational domain in a stable and accurate way. Recall that the groupvelocity vector is orthogonal to the slowness curve. The slowness curve of

28

0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

Figure 4: Magnitude of the displacement at time t = 1.5 in the plane x

(3) = 6000. Thecontour levels are spaced between 0.0375 (dark blue) and 1.2 (red) with step size 0.0375.

the quasi-shear wave “S2” has several segments where one component of theslowness and group velocity vectors have opposite signs. According to thetheory by Becache et al. [18], such materials violate the geometric stabilitycondition [19] for a perfectly matched layer (PML), making the numericalapproximation unstable.

We have theoretically shown that our discretization is stable on a curvi-linear grid, and that the artificial dissipation makes the discrete energy decayin time (see [12] for details). This implies that there are no exponentiallygrowing solutions of the anisotropic elastic wave equation with super-gridlayers, implying that the approximation is stable. For isotropic elastic ma-terials we have numerical evidence that the super-grid method can be madeas accurate as the interior scheme [12]. We proceed by numerically investi-gating whether these properties generalize to the anisotropic case.

Because it is impractical to store the numerical solution at all points inspace and time, we limit our investigation to study the convergence of thetime-dependent solution at fixed locations in the outer parts of the domainof interest. For each grid size, we record the solution (as function of time)at nine locations, on a uniform 3⇥ 3 grid,

x

(1)r

= 2 · 103, x

(2)r

= (2.0, 3.6, 5.2) · 103, x

(3)r

= (2.0, 3.6, 5.2) · 103. (70)

As an example, Figure 5 shows the Cartesian components of the solution fort 2 [0, 6], at the location x

r

= (2.0, 3.6, 5.2)T · 103. The di↵erence betweensolutions computed with grid sizes h = 20 and h = 10 is shown on the rightside of the same figure. Note that the di↵erence is significantly smaller than

29

0 1 2 3 4 5 6−0.2

0

0.2U

x

0 1 2 3 4 5 6−0.1

0

0.1

Uy

0 1 2 3 4 5 6

−0.05

0

0.05

Time

Uz

0 1 2 3 4 5 6−2

0

2x 10

−4

Ux−

diff

0 1 2 3 4 5 6−2

0

2x 10

−4

Uy−

diff

0 1 2 3 4 5 6−2

0

2x 10

−4

Uz−

diff

Time

Figure 5: Solution of the Cauchy (whole-space) problem at (x(1), x

(2), x

(3)) = (2.0, 3.6, 5.2)·103 as function of time, computed with grid size h = 20 (left). Di↵erence between thenumerical solutions computed with grid size h = 20 and h = 10 (right).

the solution itself, indicating that it is well-resolved on the grid.We assume that the numerical solution, u

h

, is a p

th order accurate ap-proximation of the solution of the continuous problem, u, and that therelation

u

h

⇡ u+ h

p

r, (71)

holds, where r is a function that can be bounded independently of the gridsize, h. It follows from (71) that u2h ⇡ u + 2phpr and u4h ⇡ u + 4phpr.Therefore,

⇥ :=ku4h � u

h

k

t

ku2h � u

h

k

t

⇡

4p � 1

2p � 1= 2p + 1,

and we can estimate the convergence rate by p ⇡ log2(⇥ � 1). Here, kfkt

denotes the discrete L2-norm of f(t). We remark that the expansion (71)is only valid when the numerical solution is resolved on the computationalgrid. For wave propagation problems, the resolution requirements for afinite di↵erence discretization can be quantified in terms of the number ofgrid points per shortest wave length, P = L

min

/h, see [31]. Based on thelargest significant frequency of the Gaussian (f

max

⇡ 6.37), and the slowestshear velocity (c

min

= 1831), we estimate the shortest shear wave lengthto be L

min

= c

min

/f

max

⇡ 1831/6.37 ⇡ 287.6. For a fourth order accuratedi↵erence scheme, adequate resolution can be expected if P � 6, see [31] fordetails. In the numerical experiments below, we use the grid sizes h = 40, 20,and 10, corresponding to P = 7.19, 14.38, and 28.76 grid points per shortestsignificant wave length. The grid sizes correspond to 301, 601, and 1201

30

x

(2)r

x

(3)r

ku4h � u

h

k

t

ku2h � u

h

k

t

ratio (⇥) rate (p)

2.0 · 103 2.0 · 103 4.129 · 10�4 9.244 · 10�6 44.673 5.449

2.0 · 103 3.6 · 103 6.847 · 10�4 3.585 · 10�5 19.099 4.178

2.0 · 103 5.2 · 103 5.368 · 10�4 6.639 · 10�6 80.853 6.319

3.6 · 103 2.0 · 103 6.847 · 10�4 3.585 · 10�5 19.099 4.178

3.6 · 103 3.6 · 103 1.020 · 10�3 4.768 · 10�5 21.387 4.349

3.6 · 103 5.2 · 103 9.681 · 10�4 5.021 · 10�5 19.278 4.192

5.2 · 103 2.0 · 103 5.368 · 10�4 6.639 · 10�6 80.853 6.319

5.2 · 103 3.6 · 103 9.681 · 10�4 5.021 · 10�5 19.278 4.192

5.2 · 103 5.2 · 103 2.202 · 10�3 1.327 · 10�4 16.594 3.963

Table 1: Grid refinement study for the Cauchy (whole-space) problem. All stations are

located on the plane x

(1)r = 2 · 103 and the numerical solutions are computed for 0 t 6.

grid points in each spatial direction. In total, the computational grids haveapproximately 2.7e6, 2.2e8, and 1.7e9 grid points. To integrate the systemin a stable manner on these grids, the explicit time stepping algorithm needsapproximately 103, 206, and 413 time steps per unit time, respectively.

In Table 1 we report the L2 norm of the di↵erences between the numericalsolutions at the nine locations. The observed convergence rates indicate thatthe numerical solution is fourth order accurate, or better. While it wouldhave been desirable to further refine the mesh to grid size h = 5, sucha calculation would have resulted in a computational grid with almost 14billion grid points and 826 time steps per unit time. Unfortunately, we didnot have access to su�ciently large computational resources to perform sucha calculation.

For the isotropic elastic wave equation, the numerical experiments in [12]indicate that, for long times, a sixth order artificial dissipation gives smallererrors than a fourth order dissipation. In [12] we solved Lamb’s problem inan isotropic half-space, which has an analytical solution, allowing the errorin the numerical solution to be evaluated explicitly. Unfortunately, theanisotropic elastic wave equation is very di�cult to solve analytically, and itis necessary to use a di↵erent approach to estimate the long time accuracy ofthe numerical solution. Here we exploit the absence of evanescent modes inthe solution of a Cauchy (whole-space) problem with homogeneous material

31

properties. This means that the analytical solution should be identicallyzero after all waves have propagated past a fixed location in space. Afterthat time, the numerical solution therefore equals the error. We proceed bystudying the norm of the displacement,

|u|(xr

, t) =

vuut3X

k=1

(u(k))2(xr

, t), (72)

at the nine locations (70) for t � t1.We are interested in the accuracy for longer times, and extend the above

simulations to run for 0 t 12. Of the locations evaluated in Table 1, xr

=(2.0, 2.0, 2.0)·103 is the furthest from the source, at a distance d ⇡ 6.928·103.The slowest phase velocity is c

min

= 1.831 · 103, from which we estimate thepropagation time from the source to x

r

to be t

p

6.928/1.831 ⇡ 3.784.The Gaussian source time function (69) satisfies g(t) 10�7 for t � 2t0 =0.75. By combining these estimates, we conclude that the analytical solutionshould reach round-o↵ levels after t � t1 ⇡ 3.784+0.75 = 4.534. (Note thatthe Gaussian time function decays exponentially fast for large times, butis never identically zero.) To test this estimate, we plot the norm of thedisplacement at x

r

= (2.0, 2.0, 2.0) · 103 in Figure 6. On the finest grid, thesolution with sixth order artificial dissipation decays from |u|(x

r

, t) ⇡ 10�2

for t ⇡ 3.9 to |u|(xr

, t) ⇡ 10�9 for t ⇡ 4.1. This rapid decay indicatesthat the analytical solution can be taken to be zero for t � t2, where t2 ⇡

4.1. The fact that t1 > t2 indicates that we underestimated the value ofthe slowest phase velocity in the direction between the source and x

r

. InFigure 6, we also compare the di↵erence between fourth and sixth orderartificial dissipation. On the finest grid, it is obvious that the sixth orderdissipation gives a more accurate numerical solution. However, the fourthorder dissipation gives comparable, or slightly better, accuracy on the twocoarser grids. The numerical solutions at the remaining locations (givenby (70)) show the same qualitative behavior.

Based on these limited numerical experiments, we surmise that the ben-efits of using a sixth order dissipation are very limited. Furthermore, thecode for the sixth order dissipation is slightly slower than its fourth ordercounterpart, because it requires a wider computational stencil and moredata to be communicated after each time step. Hence, the fourth order ar-tificial dissipation appears to be preferable for most practical simulations,which seldom resolve the numerical solution by more than 10 grid points pershortest wave length.

32

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

2 4 6 8 10 12

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

2 4 6 8 10 12

Figure 6: Norm of the numerical solution at xr = (2.0, 2.0, 2.0) · 103 as function of time,for grid sizes h = 40 (red), h = 20 (blue), and h = 10 (black). The artificial dissipation isof order four and six in the left and right sub-figures, respectively.

6.2. Absorption properties of the super-grid layers

To simplify the setup of practical wave propagation simulations we wantto develop a guideline for how to choose the parameters in the super-grid lay-ers. Due to the di�culties in deriving analytical solutions of the anisotropicwave equation, we continue studying the Cauchy (whole-space) problem ina homogeneous material, where the solution is driven by a point moment-tensor source of the form (68)-(69). Because there are no evanescent modesin the solution, the exact solution is identically zero after all waves havepassed the recording stations. By locating the stations at the same dis-tances relative to the source as in section 6.1, our previous estimate andnumerical experiment show that for t � 4.5 the analytical solution is zeroat all stations. Then the numerical solution equals the reflection error.

We use the same general setup as in the previous section, and vary thewidth of the super-grid layers, while keeping the grid size at h = 40. Wequantify the reflection errors by evaluating how large the norm of the numer-ical solution is in the time interval 4.5 t 8.0, relative to its maximumvalue for 0 t 8.0. The idea is that the reflections can be ignored ifthey are smaller than a fraction of the solution itself. Here the norm of thesolution is computed according to (72), and the relative reflection error ateach station is calculated according to

e(xr

) =max4.5t8 |u|(xr

, t)

max0t8 |u|(xr

, t). (73)

In Table 2 we report the largest value of e(xr

) from the nine stations. The

33

N

sg

�4 C

cfl

maxr

e(xr

)

20 2e-2 1.3 6.545e-2

30 2e-2 1.3 2.314e-2

40 2e-2 1.3 1.117e-2

50 2e-2 1.3 4.367e-3

N

sg

�4 C

cfl

maxr

e(xr

)

20 4e-2 0.65 1.929e-2

30 4e-2 0.65 3.227e-3

40 4e-2 0.65 1.697e-3

50 4e-2 0.65 4.631e-4

Table 2: Reflection errors from the super-grid layers for a Cauchy (whole-space) problemwith the damping coe�cient profile (66). The largest relative reflection error, definedby (73), of the nine stations is reported here. The width of the super-grid layers are hNsg,where h = 40.

results indicate that the reflection errors decrease substantially by makingthe layers wider, and by doubling the damping coe�cient �4. However, toavoid instabilities in the explicit time-stepping scheme, the time step mustbe reduced when �4 exceeds ⇡ 0.02. Because of how the damping term isscaled with the grid size, the time step must be approximately inverselyproportional to �4 for �4 � 0.02, see [12] for details. Hence C

cfl

is reducedby a factor of two for �4 = 0.04.

The above experiment indicates that artificial reflections can be signifi-cantly reduced by increasing the amount of damping in the layer. To makethe time step independent of the damping coe�cient, we could in principleuse a semi-implicit technique, where only the dissipative terms in the super-grid layers are treated implicitly. However, in terms of computational cost,such an algorithm would be significantly more expensive than the explicitapproach, because a linear system of equations must be solved in each timestep. Furthermore, a semi-implicit approach would be more di�cult to loadbalance for parallel computations, as the super-grid layers are only activenear the boundary of the domain. As a more straightforward alternative,we attempt to increase the amount of damping in the middle of each layerby modifying the profile of the damping coe�cient �(r). For this purpose,we introduce a layer transition width 0 < w

tr

1, and modify (66) to be

�(r) = 2(r)

�(r), (74)

34

N

sg

�4 C

cfl

maxr

e(xr

)

20 2e-2 1.3 4.762e-2

30 2e-2 1.3 1.539e-2

40 2e-2 1.3 6.143e-3

50 2e-2 1.3 2.235e-3

N

sg

�4 C

cfl

maxr

e(xr

)

20 4e-2 0.65 1.909e-2

30 4e-2 0.65 3.115e-3

40 4e-2 0.65 1.662e-3

50 4e-2 0.65 4.621e-4

Table 3: Reflection properties of the super-grid layers for a Cauchy (whole-space) problemwith the modified damping coe�cient profile (74) with wtr = 0.5. The largest relativereflection error, defined by (73), of the nine stations is reported here. The width of thesuper-grid layers are hNsg, where h = 40.

where

2(⇠) =

8>>>>>><

>>>>>>:

1, ⇠ (1� w

tr

)`,

P ((`� ⇠)/wtr

`), (1� w

tr

)` < ⇠ < `,

0, ` ⇠ 1� `,

P ((⇠ � (1� `))/wtr

`), 1� ` < ⇠ < 1� (1� w

tr

)`,

1, ⇠ � 1� (1� w

tr

)`.

Note that the stretching function, �(r), is kept the same as before. Aftersome numerical experimentation, we found that the time stepping algorithmremains stable for �4 = 0.02 and C

cfl

= 1.3, if the transition width satisfies0.4 w

tr

1. In a second set of numerical experiments, we set w

tr

= 0.5and repeat the above calculations, see Table 3. We conclude that for �4 =0.02, the reflection errors are reduced by a factor of almost two comparedto Table 2. For �4 = 0.04, the improvements are much smaller, indicatingthat the remaining reflection errors are due to other causes.

To summarize the results of our experiments, we conclude that the re-flection errors from the super-grid layers always decrease when the numberof grid points in the layer is increased. The reflection error also decreaseswhen the damping coe�cient is increased from �4 = 0.02 to �4 = 0.04, butat a high computational cost because the time step must be reduced bya factor of two. A more economical alternative is to keep �4 = 0.02 andmodify the damping coe�cient profile �(r) by setting the transition widthto w

tr

= 0.5. This results in about 1 % reflection error with approximately35 grid points in the layer, and less than 0.25 % error when 50 grid pointsare used in the layer.

35

6.3. A half-space problem

Next, we study the half-space problem subject to a free surface boundarycondition along x

(3) = 0 and take the domain of interest to be

1.6 · 103 (x(1), x(2)) 10.4 · 103, 0 x

(3) 4.4 · 103.

In this case, we locate the source term (68) at xs

= (6, 6, 1) ·103. We choosethe thickness of the super-grid layers to be ` = 1.6/12 in the r

(1)- and r

(2)-directions of parameter space. In the r

(3)-direction, we only add a super-grid layer near the r

(3) = 1 boundary, of thickness ` = 1.6/6. As before,cross-sections of the solution are plotted as function of scaled parametercoordinates, to equal (x(1), x(2), x(3)) within the domain of interest. In thesescaled parameter coordinates, all super-grid layers have thickness 1.6 · 103.The damping coe�cient profile (66) is used in these experiments.

In Figure 7 we show snapshots of the magnitude of the numerical solutionwith grid size h = 20. The solution is shown along the free surface, x(3) = 0,and in the vertical plane x

(1) = 6 · 103. Due to the free surface boundarycondition along x

(3) = 0, the solution has much more structure compared tothe Cauchy problem, and several sets of quasi-compressional, quasi-shear,and surface waves can be identified in the solution. Note that no reflectedwaves are visible within the domain of interest at time t = 3.5 (Figure 7,bottom row).

To further investigate the accuracy of the numerical solution, we studyit as function of time, for 0 t 6, at nine spatial locations on a uniform3⇥ 3 grid near the boundary of the domain of interest,

x

(1)r

= 2 · 103, x

(2)r

= (2, 4, 6) · 103, x

(3)r

= (0, 1, 2) · 103.

The numerical solution is calculated on three grids with sizes h = 40,20, and 10. As before, we estimate the convergence rate by evaluating theL2-norm of the di↵erences. The results are given in Table 4. The estimatedconvergence rates are close to four at all locations. The largest di↵erencesoccur on the symmetry line, x

(2) = 6 · 103, where the solution has themost structure. Surface waves propagate at a slightly slower phase velocitycompared to shear waves, and therefore have a slightly shorter wave length.Hence, the number of grid points per wave length is somewhat reducedcompared to the Cauchy problem, and the numerical solution on the coarsestgrid might only be marginally resolved.

The free surface boundary condition leads to evanescent modes in thesolution, i.e., at every fixed location x

r

, the solution only decays exponen-tially in time as t ! 1. Hence, there is no time t1 after which the analytical

36

0 5000 100000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 12000

0

2000

4000

6000

0 5000 100000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 12000

0

2000

4000

6000

0 5000 100000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 12000

0

2000

4000

6000

Figure 7: Half-space problem: Magnitude of the displacement at times 1.5, 2.5, and 3.5(top to bottom) along the free surface x

(3) = 0 (left) and the x

(1) = 6 ·103 plane (right). Inthe latter figures, the free surface is located along the top edge. The super-grid layers havethickness 1.6 · 103. The contour levels are the same in all plots and are spaced between0.0375 (dark blue) and 1.5 (red) with step size 0.0375.

37

x

(2)r

x

(3)r

ku4h � u

h

k

t

ku2h � u

h

k

t

ratio (⇥) rate (p)

2.0 · 103 0 1.266 · 10�3 6.487 · 10�5 19.519 4.211

2.0 · 103 1.0 · 103 4.774 · 10�4 1.702 · 10�5 28.039 4.757

2.0 · 103 2.0 · 103 8.284 · 10�4 4.277 · 10�5 19.368 4.199

4.0 · 103 0 1.903 · 10�3 1.137 · 10�4 16.739 3.976

4.0 · 103 1.0 · 103 1.258 · 10�3 7.359 · 10�5 17.106 4.009

4.0 · 103 2.0 · 103 1.628 · 10�3 9.953 · 10�5 16.355 3.941

6.0 · 103 0 3.832 · 10�3 2.386 · 10�4 16.056 3.912

6.0 · 103 1.0 · 103 3.167 · 10�3 1.978 · 10�4 16.017 3.908

6.0 · 103 2.0 · 103 2.252 · 10�3 1.461 · 10�4 15.414 3.849

Table 4: Grid refinement study for the half-space problem. All stations are on the planex

(1)r = 2 · 103 and the solutions are computed for 0 t 6.

solution is identically zero at x

r

. Unfortunately, this prevents us from us-ing the same technique as in § 6.1 for quantifying the long time reflectionproperties of the super-grid layers.

6.4. Energy conservation test

To verify the theoretically predicted energy conservation property of ourscheme, we perform a computation without source terms, but with uni-formly distributed random noise as initial data. For this calculation, thecomputational domain is 0 x 30000, 0 y 30000, 0 z 17000.Energy conservation is ensured by enforcing periodic boundary conditionsin the x- and y-directions, a free surface boundary condition along z = 0,and a homogeneous Dirichlet condition along z = 17000. The grid spacing ish = 200, which gives 150⇥ 150⇥ 86 grid points. We use a vertically layeredmaterial model, with InAs in the sub-domain 0 < z < 2000 and quartz in2000 < z < 17000. The density and sti↵ness matrix for InAs are given by

38

-1e-14

-5e-15

0

5e-15

1e-14

0 10 20 30 40 50 60 70 80 90

(E(t)-E(0))/E(0)

Time

Figure 8: The relative change in discrete energy as function of time, with random initialdata and a heterogeneous layered anisotropic material model. Here, time t = 90 corre-sponds to 3,200 time steps.

(67) and the following material properties are used for quartz:

⇢ = 2.62·103, C =

0

BBBBBBBBBB@

86.74 6.99 6.99 �17.91 0 0

6.99 86.74 6.99 17.91 0 0

6.99 6.99 107.2 0 0 0

�17.91 17.91 0 57.94 0 0

0 0 0 0 57.94 �17.91

0 0 0 0 �17.91 39.875

1

CCCCCCCCCCA

·109,

using SI-units.For the semi-discrete approximation, the energy is given by (⇢u

t

, u

t

)h

+S

h

(u, u), see (53). By using the same approach as for the isotropic elasticwave equation, see [11], the expression for the fully discrete energy becomes

E

n+1/2h

=

����p

⇢

u

n+1� u

n

�t

����2

h

+ S

h

(un+1,u

n)��2

t

12

�L

h

u

n+1,L

h

u

n

�h

. (75)

In Figure 8 we plot the relative change in discrete energy, (En+1/2�E

1/2)/E1/2

as function of time, for t 2 [0, 90]. This corresponds to 3200 time steps. Our

39

calculation confirms that the discrete energy remains constant modulus verysmall fluctuations, which are on the order of the round-o↵ level in doubleprecision arithmetic.

7. Conclusions

We have presented a fourth order accurate finite di↵erence discretizationof the elastic wave equation in second order formulation for a general, 21 pa-rameter anisotropic, heterogeneous, material. The discretization is definedon a curvilinear grid, by use of a general coordinate transformation. Theproposed method generalizes our previous finite di↵erence method [11] toanisotropic elastic materials and curvilinear grids. The proposed method isenergy conserving and stable under a CFL time-step constraint, and we havedeveloped a practically useful approach for estimating the size of the largeststable time step. We have also generalized the super-grid technique [12]to anisotropic elastic materials, and demonstrated that it leads to a stablenumerical method with very small artificial reflections.

It would be straightforward to extend the proposed method to higherorders of accuracy. Such an extension relies on compatible, higher orderaccurate, summation by parts operators for approximating both first andsecond derivatives with variable coe�cients. In particular, the di↵erenceapproximations must satisfy (39) and (41), respectively. For first derivatives,it is well know that such operators exist with up to eighth order truncationerror in the interior of the domain, with a reduction to order four on theboundary. For second derivatives with variable coe�cients, we have derivedoperators having truncation errors of order six and eight in the interior, withboundary reduction to order three and four, respectively. Because we solvethe elastic wave equation in second order formulation, the solution is twoorders more accurate than the truncation error near the boundary. Theseoperators could therefore be used to device a sixth order accurate schemefor the anisotropic elastic wave equation.

By generalizing the technique developed in [32], it would be straightfor-ward to extend the proposed method to include visco-elastic attenuation.However, the number of material parameters would increase by 21 for eachvisco-elastic mechanism in the model. For the isotropic visco-elastic model,these parameters are usually determined by matching observed attenuationrates of compressional and shear waves [33]. It is unclear if that approachcould be generalized to estimate all the parameters in an anisotropic visco-elastic model.

40

Acknowledgments

This work performed under the auspices of the U.S. Department of En-ergy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. This is contribution LLNL-JRNL-663238.

Appendix A. The M ijmatrices

The divergence of the stress tensor, Lu, can be expressed in terms ofthe symmetric positive definite 6⇥ 6 sti↵ness matrix C and the di↵erentialoperator G

s

in (3),

Lu = G

T

s

CG

s

u, C =

0

BBBBB@

c11 c12 · · · c16

c12 c22 · · · c26

......

. . ....

c16 c26 · · · c66

1

CCCCCA.

The divergence of the stress tensor can also be written in the form (7),where the terms are given by (8)-(10). By identifying each term in the twoexpressions for Lu, we arrive at

M

11 =

0

BB@

c11 c16 c15

c16 c66 c56

c15 c56 c55

1

CCA , M

12 =

0

BB@

c16 c12 c14

c66 c26 c46

c56 c25 c45

1

CCA ,

M

13 =

0

BB@

c15 c14 c13

c56 c46 c36

c55 c45 c35

1

CCA , M

21 =

0

BB@

c16 c66 c56

c12 c26 c25

c14 c46 c45

1

CCA ,

M

22 =

0

BB@

c66 c26 c46

c26 c22 c24

c46 c24 c44

1

CCA , M

23 =

0

BB@

c56 c46 c36

c25 c24 c23

c45 c44 c34

1

CCA ,

M

31 =

0

BB@

c15 c56 c55

c14 c46 c45

c13 c36 c35

1

CCA , M

32 =

0

BB@

c56 c25 c45

c46 c24 c44

c36 c23 c34

1

CCA ,

41

and

M

33 =

0

BB@

c55 c45 c35

c45 c44 c34

c35 c34 c33

1

CCA .

By inspection, the diagonal blocks M11, M22, and M

33 are symmetric andM

ji = (M ij)T for i 6= j. To show that M

11 is positive definite, we takez = (z1, z2, z3)T and y = (z1, 0, 0, 0, z3, z2)T . Now,

z

T

M

11z = y

T

Cy � y

T

y = z

T

z, > 0,

because C is positive definite. The same technique can be used to show thatM

22 and M

33 are positive definite.

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