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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2002; 53:455–472 (DOI: 10.1002/nme.292) Three-dimensional time-harmonic Green’s functions for a triclinic full-space using a symbolic computation system Marijan Dravinski ; and Yuqing Niu Department of Aerospace and Mechanical Engineering; University of Southern California; Los Angeles; CA 90089-1453; U.S.A. SUMMARY Time-harmonic Green’s functions for a triclinic anisotropic full-space are evaluated through the use of symbolic computation system. This procedure allows evaluation of the Green’s functions for the most general anisotropic materials. The proposed computational algorithms are programmed in a MATLAB environment by incorporating symbolic calculations performed using Maple Computer Algebra System. Extensive testing of the numerical results has been performed for both displacement and stress leds. The tests demonstrate the accuracy of the proposed algorithm in evaluating the Green’s functions. Copyright ? 2001 John Wiley & Sons, Ltd. KEY WORDS: harmonic Green’s functions; anisotropic material 1. INTRODUCTION Scattering of elastic waves by subsurface inclusions has been subject of many studies in non-destructive evaluation of materials and modelling of ground motion amplication atop a sediment-lled valleys due to earthquakes [1; 2]. These studies mainly deal with two and three- dimensional models involving isotropic materials. However, modelling the scattering of waves in anisotropic media leads to an additional diculty associated with accurate and ecient evaluation of the corresponding Green’s functions. The standard Fourier transform approach requires integration over an innite four-dimensional space which is dicult to accomplish numerically [3; 4]. Consequently, these studies have been conned mainly to two-dimensional problems [5; 6] and lower level of anisotropy [7]. Recently, several researchers approached the solution of the problem of the Green’s func- tions in an anisotropic medium by using the Radon transform (e.g. References [8; 4]). The Radon transform has been studied extensively in computational tomography [9]. The key fea- ture of this transform is that it reduces a three-dimensional wave equation to a one-dimensional Correspondence to: Marijan Dravinski, Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089 1453, U.S.A. E-mail: [email protected] Received 31 August 2000 Copyright ? 2001 John Wiley & Sons, Ltd. Revised 13 February 2001

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 53:455–472 (DOI: 10.1002/nme.292)

Three-dimensional time-harmonic Green’s functions fora triclinic full-space using a symbolic computation system

Marijan Dravinski∗;† and Yuqing Niu

Department of Aerospace and Mechanical Engineering; University of Southern California; Los Angeles;CA 90089-1453; U.S.A.

SUMMARY

Time-harmonic Green’s functions for a triclinic anisotropic full-space are evaluated through the use ofsymbolic computation system. This procedure allows evaluation of the Green’s functions for the mostgeneral anisotropic materials. The proposed computational algorithms are programmed in a MATLABenvironment by incorporating symbolic calculations performed using Maple Computer Algebra System.Extensive testing of the numerical results has been performed for both displacement and stress =leds.The tests demonstrate the accuracy of the proposed algorithm in evaluating the Green’s functions.Copyright ? 2001 John Wiley & Sons, Ltd.

KEY WORDS: harmonic Green’s functions; anisotropic material

1. INTRODUCTION

Scattering of elastic waves by subsurface inclusions has been subject of many studies innon-destructive evaluation of materials and modelling of ground motion ampli=cation atop asediment-=lled valleys due to earthquakes [1; 2]. These studies mainly deal with two and three-dimensional models involving isotropic materials. However, modelling the scattering of wavesin anisotropic media leads to an additional diDculty associated with accurate and eDcientevaluation of the corresponding Green’s functions. The standard Fourier transform approachrequires integration over an in=nite four-dimensional space which is diDcult to accomplishnumerically [3; 4]. Consequently, these studies have been con=ned mainly to two-dimensionalproblems [5; 6] and lower level of anisotropy [7].

Recently, several researchers approached the solution of the problem of the Green’s func-tions in an anisotropic medium by using the Radon transform (e.g. References [8; 4]). TheRadon transform has been studied extensively in computational tomography [9]. The key fea-ture of this transform is that it reduces a three-dimensional wave equation to a one-dimensional

∗Correspondence to: Marijan Dravinski, Department of Aerospace and Mechanical Engineering, University ofSouthern California, Los Angeles, CA 90089 1453, U.S.A.

†E-mail: [email protected]

Received 31 August 2000Copyright ? 2001 John Wiley & Sons, Ltd. Revised 13 February 2001

456 M. DRAVINSKI AND Y. NIU

diJerential equation. Furthermore, the inverse Radon transform can be expressed in terms ofintegrals over the surface of a unit sphere which are easy to evaluate numerically [9]. Con-sequently, the use of the Radon transform provides an eDcient way to evaluate the Green’sfunctions for anisotropic media. Recently, Dravinski and Zheng [10] utilized this approachto develop an eDcient algorithm for evaluation of the Green’s functions for an orthotropicfull-space. This paper considers the Green’s functions for the most general anisotropic ma-terial, i.e. for a triclinic material. For such material the standard numerical approach usedfor more symmetric anisotropic materials does not seem practical [10]. The matrices used todescribe triclinic materials involve 21 material constants and three components of the prop-agation vector leading to very large elements of the resulting matrices which is diDcult tohandle on a digital computer. However, the use of symbolic computations eJectively remediesthis problem thus leading to eDcient algorithm for evaluation of the Green’s functions evenfor the most general anisotropic materials.

It should be pointed out that in the integral equation formulation for scattering of elasticwaves in anisotropic media it is suDcient to evaluate the corresponding full-space Green’sfunctions [11]. Consequently, the Green’s functions considered in this paper can be of usefor a large class of problems of interest in nondestructive testing and strong ground motionseismology.

1.1. Statement of problem

The equations of motion for the Green’s functions are given by [8]

{Mip + �!2�ip}gpk(x; !)=− �ik�(x); i; p; k=1; 2; 3; x∈Rn (1)

where unless stated diJerently, summation over repeated indices is understood. Underlinedindices indicate that the summation convention is being suppressed. Furthermore,

Mip = cijpq@@xj

@@xq

; i; j; p; q=1; 2; 3 (2)

and cijpq are components of the stiJness tensor. For simplicity, the factor e−i!t has beenomitted. Throughout the paper the n-dimensional real and complex vector spaces are denotedby Rn and Cn, respectively.

The stress–strain relations for a homogeneous linearly elastic anisotropic material are givenby the generalized Hooke’s law

�ij = cijpqepq (3)

where epq are the strain components de=ned by

epq = 12(up;q + uq;p) (4)

up;q =@up@xq

(5)

with up denoting the displacement components. Due to symmetry conditions

cijpq = cjipq = cijqp = cpqij (6)

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THREE-DIMENSIONAL TIME-HARMONIC GREEN’S FUNCTIONS 457

there are only 21 independent material constants. Since the strain energy density function

W = 12cijpqeijepq (7)

should be a positive quantity, cijpq must be positive de=nite, i.e.

cijpqeijepq¿0 (8)

for any non-zero real symmetric tensor eij .For time-harmonic displacement Green’s functions gpk , corresponding stress =eld is given

by Equation (3), i.e.

hijk(x; !)= cijpqgpk; q(x; !) (9)

2. ANALYTICAL SOLUTIONS

By using three-dimensional Radon transform, de=ned by the following pair of equations [9]:

�{f(x)}= f̂(s; n)=∫f(x)�(s− n · x) dx (10)

f(x) =− 18�2

∫|n|=1

[@2f̂(s; n)@s2

]s=n·x

dS(n) (11)

Wang and Achenbach [8] showed that the displacement Green’s functions can be written inthe following form:

gpk(x; !)= gSpk(x) + gRpk(x; !) (12)

where the superscripts S and R denote the singular (static) and regular (dynamic) parts,respectively. The regular and singular parts are given by [8]

gRpk(x; !) =i

8�2

∫|n|=1

|n·x|¿0

3∑m=1

kmEpmEkm�c2m

eikm|n·x| dS(n) (13)

dS(n) ∈ DR = {06b61; 06�62�}

gSpk(x) =1

8�2r

∫|d|=1

M−1pk (d) dQ(d) (14)

dQ(d) ∈ DS = {06’62�}

where Epm are the components of the eigenvector of (n)

Mip(n)Epm= !mEim; m=1; 2; 3 (15)

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

458 M. DRAVINSKI AND Y. NIU

Mip(n) denotes the ChristoJel matrix [12]

Mip(n)= cijpqnjnq (16)

and underlined indices indicate that the summation convention is being suppressed. cm and kmare the phase velocities and wave numbers de=ned by

cm =

√!m�

(17)

km =!cm

For a triclinic material the elements of the ChristoJel matrix are given explicitly by [12]

M11(n) =C11n21 + C66n22 + C55n23 + 2C16n1n2 + 2C15n1n3 + 2C56n2n3

M12(n) =C16n21 + C26n22 + C45n23

+ (C12 + C66)n1n2 + (C14 + C56)n1n3 + (C46 + C25)n2n3

M13(n) =C15n21 + C46n22 + C35n23

+ (C14 + C56)n1n2 + (C13 + C55)n1n3 + (C36 + C45)n2n3

M22(n) =C66n21 + C22n22 + C44n23 + 2C26n1n2 + 2C46n1n3 + 2C24n2n3

M23(n) =C56n21 + C24n22 + C34n23 + (C25 + C46)n1n2

+ (C45 + C36)n1n3 + (C23 + C44)n2n3

M33(n) =C55n21 + C44n22 + C33n23 + 2C45n1n2 + 2C35n1n3 + 2C34n2n3

where the conversion convention from the fourth-order stiJness tensor, cijpq, to the second-order tensor, Cij , is assumed to be: 11→ 1; 22→ 2; 33→ 3; 23→ 4; 13→ 5, and 12→ 6.

Throughout the paper, the unit vector n=(n1; n2; n3) denotes the direction of wavepropagation. Unit vectors n and d are related through (Figure 1)

n=√

1− b2d+ be (18)

e=x|x| (19)

with

d=(1− e23)−1=2(e2 cos’+ e1e3 sin’;−e1 cos’+ e2e3 sin’;−(1− e23) sin’) (20)

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

THREE-DIMENSIONAL TIME-HARMONIC GREEN’S FUNCTIONS 459

Figure 1. Domain of integration DR = {06’; 06b61} for the regular parts of theGreen’s functions and domain of integration DS = {06’62�} for the singular parts.

d; e; and n are unit vectors, with e · d=0 [11].

The integration for the regular part, gRpk ; takes place over a half of a unit sphere, |n|=1;with domain which can be expressed in terms of variables b and ’ as (Figure 1)

DR : {06b61; 06’62�} (21)

For the singular part, gSpk ; the integration takes place over a unit circle, |d|=1; (Figure 1)so that the corresponding domain of integration is de=ned as

DS : {06’62�} (22)

The stress Green’s functions also can be separated into a singular and regular parts [8]

hijk(x; !)= hSijk(x) + hRijk(x; !) (23)

where the regular part is given by [8]

hRijk(x; !) = −cijpq4�2

∫|n|=1n·x¿0

3∑m=1

k2mnqEpmEkm2�c2m

eikm|n·x| dS(n) (24)

dS(n) ∈ DR = {06b61; 06�62�}

It is not diDcult to show that the singular part of the stress =eld can be expressed as

hSijk(x) =cijpq8�2

∫|n|=1

M−1pk (n)nq�′(n·x) dS(n) (25)

=− cijpq8�2r2

∫ 2�

0Gbpqk(’) d’ (26)

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

460 M. DRAVINSKI AND Y. NIU

where

Gbpqk =

{@@bGpqk(

√1− b2d+ be)

}b=0

(27)

Gpqk(n) = nqM−1pk (n) (28)

The form of the singular stress =led given by Equation (26) is found to be more convenientfor numerical evaluation that the one given in Reference [8].

The key feature of the Green’s functions obtained through the Radon transform is that theintegration required to calculate them takes place over =nite domains: a unit sphere or a unitcircle. The domain of integration for the regular parts is over a unit semi-sphere while forthe singular parts is over a unit circle.

This concludes the summary of the analytical results. Integration algorithms for evaluationof the Green’s functions are considered next.

3. ALGORITHMS FOR EVALUATION OF THE GREEN’S FUNCTIONS

Throughout the paper, one- and two-dimensional Gauss–Legendre composite quadratures arebeing used. The are de=ned by [13]

∫ b

af(x) dx ≈Qx ≡ (b− a)wTf

wT ≡ [w1; :::; wNx ]; fT ≡ [f(x1); : : : ; f(xNx)] (29)

∫ b

adx

∫ d

cf(x; y) dy ≈Qxy ≡ (b− a)(d− c)wTF\

F≡ [f(xi; yj)]; 16i6Nx; 16j6Ny (30)

\T = [,1; :::; ,Ny ]

where the superscript T denotes the transpose and

Nx = nxmx

Ny = nymy (31)

Here, nx and mx denotes the number of panels and number of points in each panel, respectively,in the x-direction with corresponding values ny and my in the y-direction. It should be pointedout that the quadrature rules given by Equations (29) and (30) are written in such a way thatthe integrand function is computed only once at each integration node [13].

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

THREE-DIMENSIONAL TIME-HARMONIC GREEN’S FUNCTIONS 461

3.1. Displacement Green’s functions

3.1.1. Regular part. As it can be seen from Equation (13) evaluation of the regular part, gRpk ;requires solving an eigenvalue problem for matrix (n) and a double integration over a unithalf-sphere, DR. The corresponding algorithm is then given by

function [gRpk]=GR(x;C; �; !;mb; nb; m’; n’)

e=x=|x|% Gauss–Legendre weights and abscissas[w’i ; ’i]=GaussWeights&Abscissas(0; 2�;m’; n’); 16i6m’n’[wbj ; bj]=GaussWeights&Abscissas(0; 1; mb; nb); 16j6mbnb% Forming the integrands for a 2D quadraturefor i=1 : m’n’

d=(e2 cos’i + e1e3 sin’i;−e1 cos’i + e2e3 sin’i;−(1− e23) sin’i)=√

1− e23for j=1 : mbnb

n= d√

1− b2j + bje

Mkp =∑3

s; q=1 ckspqnsnq; k=1 : 3;p= k : 3

% Eigenvalues and eigenvectors of M(n)[E;D]= eig((n))

cm=√D(m;m)=�; km=!=cm; m=1 : 3

FRpk(j; i)=

∑3m=1

kmEpmEkmc2m

e√−1·km(|x|bj); p=1 : 3; k=p : 3

endend% Double integration over domain DR

gRpk =√−14��!2 (w

b)TFRpkw

’; p=1 : 3; k=p : 3

3.1.2. Singular part. The singular part of the displacement Green’s functions, gSpk ;(see Equation (14)) requires evaluation of an inverse of M(n) and a 1D-integration over a unitcircle, DS. The singular part turns out to be a real number. In this paper the symbolic calcu-lations are denoted by sym{}: So the algorithm for evaluation of gSpk is given by

function [gSpk]=GS(x;C; �; m’; n’)

% Gauss–Legendre weights and abscissas[w’i ; ’i]=GaussWeights&Abscissas(0; 2�;m’; n’); 16i6m’n’% Symbolic calculationssym{e}=x=|x|

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

462 M. DRAVINSKI AND Y. NIU

sym{Mkp}=∑3

s; q=1 ckspqnsnq; k=1 : 3;p= k : 3

sym{d}=(e2 cos’+ e1e3 sin’;−e1 cos’+ e2e3 sin’;−(1− e23) sin’)=√

1− e23sym{n}= d√1− b2 + besym{M−1(b; ’)}=M−1(n)% Forming the integrands for a 1D quadraturefor i=1 : m’n’

[M0]= [M−1(b=0; ’i)]fSpk(i)=M0

pk ; p=1 : 3; k=p : 3

end% 1D Quadrature

gSpk =1

4��r(w’)TfSpk ; p=1 : 3; k=p : 3

The symbolic calculations can be accomplished in several ways. First approach uses asymbolic computation system for both symbolic and numerical computations. This method isconvenient when there is no need to utilize the programs previously developed using standardprogramming languages (e.g. free-=eld calculations for scattering problems). Second approachrequires a standard programming language with a symbolic kernel (e.g. MATLAB with aMaple symbolic system).Third approach involves performing the symbolic calculations sepa-rately using a symbolic computational system and integrating these results into the standardprogramming language. The last approach has been used in the present study.

3.2. Stress Green’s functions

3.2.1. Regular part. Based on Equation (24), evaluation of the regular part of the stressGreen’s function, hRijk ; requires solving an eigenvalue problem for matrix M(n) and doubleintegration over a unit semi-sphere, DR. The corresponding algorithm is given by

function [hRijk]=HR(x;C; �; !;mb; nb; m’; n’)

e=x=|x|% Gauss–Legendre weights and abscissas[w’i ; ’i]=GaussWeights&Abscissas(0; 2�;m’; n’); 16i6m’n’[wbj ; bj]=GaussWeights&Abscissas(0; 1; mb; nb); 16j6mbnb% Forming the integrands for a 2D quadraturefor i=1 : m’n’

d=(e2 cos’i + e1e3 sin’i;−e1 cos’i + e2e3 sin’i;−(1− e23) sin’i)=√

1− e23for j=1 : mbnb

n= d√

1− b2j + bje

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THREE-DIMENSIONAL TIME-HARMONIC GREEN’S FUNCTIONS 463

Mkp =∑3

s; q=1 ckspqnsnq; k=1 : 3;p= k : 3

% Eigenvalues and eigenvectors of M(n)[E;D]= eig((n))

cm=√D(m;m)=�; km=!=cm; m=1 : 3

Fpqk(j; i)=∑3

m=1 k4mnqEpmEkme

√−1·km(|x|bj); p; q; k=1 : 3end

end% Double integration over domain DR

FRpqk =(wb)TFpqkw’; p; q; k=1 : 3

hRijk =− 14��!2

∑3p; q=1 cijpqF

Rpqk ; i; k=1 : 3; j= i : 3

3.2.2. Singular part. Based on Equation (26) the algorithm for evaluation of the singularpart of the stress Green’s functions, hSijk ; requires evaluation of Gb

pqk(’) (Equation (27)) andsingle integration over a unit circle DS: Corresponding algorithm is given as follows:

function [hSijk]=HS(x;C; �; m’; n’)

% Gauss–Legendre weights and abscissas[w’i ; ’i]=GaussWeights&Abscissas(0; 2�;m’; n’); 16i6m’n’% Symbolic calculationssym{e}=x=|x|sym{Mkp}=

∑3s; q=1 ckspqnsnq; k=1 : 3;p= k : 3

sym{d}=(e2 cos’+ e1e3 sin’;−e1 cos’+ e2e3 sin’;−(1− e23) sin’)=√

1− e23sym{n}=d√1− b2 + besym{M−1(b; ’)}=M−1(n)sym{Gpqk(b; ’)}= nqM−1

pk (b; ’)

sym{Gbpqk(b; ’)}=

@@bGpqk(b; ’)

sym{G0pqk(’)}=Gb

pqk(b=0; ’)

% Forming the integrands for a 1D quadraturefor i=1 : m’n’fSpqk(i)=G

0pqk(’i); p; q; k=1 : 3

end% 1D QuadratureIpqk =(w’)TfSpqk ; p; q; k=1 : 3

hSijk =− 14�r2

∑3p; q=1 cijpqIpqk ; i; k=1 : 3; j= i : 3

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

464 M. DRAVINSKI AND Y. NIU

This completes the algorithms for evaluation of both stress and displacement time-harmonicGreen’s functions for a triclinic full-space. Numerical results are considered next.

4. NUMERICAL RESULTS

Three tests are performed in order to assess the accuracy of the proposed algorithm.• Test 1 involves calculating isotropic Green’s functions for which closed form analytical

results are available.• Test 2 compares the Green’s functions for transversely isotropic material in the x1–x2 plane

with those of Dravinski and Zheng [10] who evaluated transversely isotropic Green’s func-tions based on the analytical results of Wang and Achenbach [8].

• Test 3 compares triclinic Green’s functions with the ones obtained through appropriaterotation of the transversely isotropic ones. Namely, relative to a Cartesian co-ordinate system{xi}; i=1 : 3 the components of the stiJness tensor for a transversely isotropic material inthe x1–x2 plane are de=ned by

C=

C11 C12 C13 0 0 0

· C22 C13 0 0 0

· · C33 0 0 0

· · · C44 0 0

· · · · C44 0

· · · · · C11 − C12

2

(32)

where the dots denote the symmetric elements of the upper triangular part of C.Suppose the co-ordinate system {xi} is rotated into the system {x′i} according to

x′i = ‘(1)ij xj; i; j=1 : 3 (33)

where ‘(1)ij denotes the rotation about the x2-axis for a positive angle 2. Then with respectto the new co-ordinate system {x′i} the material is now monoclinic with the x′1–x

′3 plane of

symmetry, and the elastic tensor is de=ned by

c′ijmn= ‘(1)ip ‘(1)jq ‘

(1)mr ‘

(1)ns cpqrs (34)

An additional rotation for an angle 2 about the x′3-axis

x′′i = ‘(2)ij x′j (35)

produces a triclinic material with the elasticity tensor de=ned by

c′′ijmn= ‘(2)ip ‘(2)jq ‘

(2)mr ‘

(2)ns c

′pqrs (36)

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

THREE-DIMENSIONAL TIME-HARMONIC GREEN’S FUNCTIONS 465

or

C′′ =

C ′′11 C ′

12 C ′′13 C ′′

14 C ′′15 C ′′

16

· C ′′22 C ′′

23 C ′′24 C ′′

25 C ′′26

· · C ′′33 C ′′

34 C ′′35 C ′′

36

· · · C ′′44 C ′′

45 C ′′46

· · · · C ′′55 C ′′

56

· · · · · C ′′66

(37)

Therefore, the displacement Green’s functions for a triclinic material, g′′ik ; can be calculatedfrom the Green’s functions for a transversely isotropic material, gpk ; according to

g′′ij = ‘(2)im ‘

(2)jn ‘

(1)mp ‘

(1)nq gpq; i; j; m; n; p; q=1 : 3 (38)

Consequently, the Test 3 involves the following steps:

• Evaluation of the triclinic Green’s functions using transversely isotropic Green’s functionsas speci=ed by Equation (38),

• calculation of the triclinic Green’s functions directly by the algorithms proposed earlierusing the elastic tensor de=ned by Equation (36) and

• comparison of the results from the last two steps.

Unless stated diJerently, throughout the testing the following parameters are assumed:

2=10◦

T=60◦; U=45◦; 16r=a615

x= r(sin T cosU; sin T sin U; cosT) (39)

!=1 (1=sec);�=1

mb =m’= nb= n’=10

where a=1 is a length parameter.The material constants of a transversely isotropic material in the x1–x2 plane are chosen

to be

C=

1:97 0:97 0:9178 0 0 0

· 1:97 0:9178 0 0 0

· · 22:73 0 0 0

· · · 1 0 0

· · · · 1 0

· · · · · 0:5

(40)

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

466 M. DRAVINSKI AND Y. NIU

Then according to Equation (36) the stiJness tensor C′′ for a triclinic material (for 2=10◦)is given by

2:0416 0:9689 1:4529 0:0148 −0:2525 −0:0077

· 1:9717 0:9360 0:0287 +0:0059 −0:0050

· · 21:5523 0:5714 −3:2408 −0:0941

· · · 1:0020 −0:0969 −0:0871

· · · · +1:5346 +0:0312

· · · · · +0:5156

(41)

For estimating the Green’s functions accuracy it is convenient to de=ne the following errorcriteria:

err(G∗;G) =‖G∗ −G‖2

‖G∗‖2 (42)

G;G∗ ∈ Cn×n

where G∗ denotes the Green’s function matrix based on analytic results (Tests 1 and 2) ortheir rotations (Test 3) while G represents numerical Green’s functions obtained directly bythe algorithms proposed in this paper. Cn×n denotes the n-dimensional complex vector spaceand ‖ ‖2 represents the 2-norm of a matrix.

4.1. Test 1

When the material is assumed isotropic the proposed algorithms should produce the resultsknown in the literature (e.g. Reference [14]). For the parameters de=ned by Equation (39)this test produced the following error estimates for the displacement and the stress =elds

err(g∗; g) = 5:22× 10−16

err([h∗ij1]; [hij1]) = 1:09× 10−15

err([h∗ij1]; [hij2]) = 1:16× 10−15

err([h∗ij1]; [hij3]) = 1:34× 10−15

(43)

The Test 1 error estimates demonstrate very good accuracy of the proposed algorithm inevaluating both displacement and stress =elds.

4.2. Test 2

This test compares transversely isotropic Green’s functions (Equation (40)) obtained in thisstudy with those based on the analytical results of Wang and Achenbach [8] as calculated byDravinski and Zheng [10]. However, for the sake of consistency the Newton–Cotes quadratures

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

THREE-DIMENSIONAL TIME-HARMONIC GREEN’S FUNCTIONS 467

Table I. Test 2. Comparison of the displacement Green’s functions for atransversely isotropic material.

g11 g22 g33

100gij (3.305547, 7.758555i) (3.305547, 7.758555i) (1.248244, 1.540302i)100g∗ij (3.305547, 7.758555i) (3.305547, 7.758555i) (1.248244, 1.540302i)

g23 g13 g12

100gij (0.188524, 0.019895i) (0.188524, 0.019895i) (2.975494, 0.806068i)100g∗ij (0.188524, 0.019895i) (0.188524, 0.019895i) (2.975494, 0.806068i)

Table II. Test 2. Comparison of the stress Green’s functions for atransversely isotropic material.

h111 h221 h331

hij1 (−0.064110, −0.020266) (−0.032648, 0.007700) (0.018410, 0.004728)h∗ij1 (−0.064110, −0.020266) (−0.032648, 0.007700) (0.018412, 0.004728)

h231 h131 h121

hij1 (−0.038056, −0.000851) (−0.052650, −0.014208) (−0.065730, −0.015422)h∗ij1 (−0.038056, −0.000851) (−0.052650, −0.014208) (−0.065730, −0.015422)

Table III. Test 2. Comparison of the stress Green’s functions for atransversely isotropic material.

h112 h222 h332

hij2 (−0.032648, 0.007700i) (−0.064110, −0.020266i) (0.018410, 0.004728i)h∗ij2 (−0.032648, 0.007700i) (−0.064110, −0.020266i) (0.018410, 0.004728i)

h232 h132 h122

hij2 (−0.052650, −0.014208i) (−0.038056, −0.000851i) (−0.065730, −0.015422i)h∗ij2 (−0.052650, −0.014208i) (−0.038056, −0.000851i) (−0.065730, −0.015422i)

used by Dravinski and Zheng [10] have been replaced by the corresponding Gauss–Legendreformulas.

For r=1 (see Equation (39)) the results of the calculations are given by Tables I–IV.Based on these results the errors for Test 2 are found to be

err(g∗; g) = 2:92× 10−16

err([h∗ij1]; [hij1]) = 1:97× 10−16

err([h∗ij2]; [hij2]) = 1:56× 10−16

err([h∗ij3]; [hij3]) = 1:72× 10−15

(44)

which illustrates very good accuracy of the proposed algorithm.

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468 M. DRAVINSKI AND Y. NIU

Table IV. Test 2. Comparison of the stress Green’s functions for atransversely isotropic material.

h113 h223 h333

hij3 (−0.001776, 0.000757i) (−0.001776, 0.000757i) (−0.016134, −0.003190i)h∗ij3 (−0.001776, 0.000757i) (−0.001776, 0.000757i) (−0.016134, −0.003190i)

h233 h133 h123

hij3 (−0.018885, −0.003021i) (−0.018885, −0.003021i) (−0.003471, −0.000016i)h∗ij3 (−0.018885, −0.003021i) (−0.018885, −0.003021i) (−0.003471, −0.000016i)

Figure 2. Test 2: Comparison of the transversely isotropic displacement Green’s function G11 calculatedby the proposed method (solid and dot-lines) with the results of Dravinski and Zheng [10] based onanalytical solution of Wang and Achenbach [8] (open circles and stars). Position parameter: 16r615

(Equation (39)), a=1 is a length parameter.

It should be noted that this test has been performed for a wide range of observation points(16r615). Typical results for one displacement and one stress Green’s functions are depictedby Figures 2 and 3.

4.3. Test 3

This test compares the triclinic Green’s functions obtained through rotation of transverselyisotropic Green’s functions according to Equation (38) with those obtained directly by theproposed algorithms. For the displacements, the results of this test are depicted by Table V.

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 53:455–472

THREE-DIMENSIONAL TIME-HARMONIC GREEN’S FUNCTIONS 469

Figure 3. Test 2: Comparison of the transversely isotropic stress Green’s function H111 calculated bythe proposed method (solid and dot-lines) with the results of Dravinski and Zheng [10] based onanalytical solution of Wang and Achenbach [8] (open circles and stars). Position parameter: 16r615

(Equation (39)), a=1 is a length parameter.

Table V. Test 3 for the displacement Green’s functions and a triclinic material.

g11 g22 g33

100gij (4.137868, 7.840427i) (2.310710, 7.482374i) (1.374758, 1.734610i)100g∗ij (4.173868, 7.840427i) (2.310710, 7.482374i) (1.374758, 1.734710i)

g23 g13 g12

100gij (0.599823, −0.030760i) (0.642899, 1.093348i) (2.744445, 0.775931i)100g∗ij (0.599823, −0.030760i) (0.642899, 1.093348i) (2.744445, 0.775931i)

For r=1 the Test 3 error estimates are found to be

err(g∗; g) = 3:88× 10−10

err([h∗ij1]; [hij1]) = 9:41× 10−8

err([h∗ij2]; [hij2]) = 3:46× 10−7

err([h∗ij3]; [hij3]) = 3:18× 10−6

(45)

Typical Test 3 results for a range of observation stations are shown by Figures 4 and 5. It isapparent from these =gures that the proposed algorithm accurately evaluates triclinic Green’sfunctions. Similar Test 3 results have been obtained for a range of triclinic materials (i.e. fordiJerent value of rotation angle 2; see Equation (39)).

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470 M. DRAVINSKI AND Y. NIU

Figure 4. Test 3: Comparison of triclinic displacement Green’s function G11 calculatedby the proposed method (solid and dot-lines) with the results obtained by double

rotation of triclinic Green’s functions (open circles and stars).

Figure 5. Test 3: Comparison of triclinic stress Green’s function H111 calculated bythe proposed method (solid and dot-lines) with the results obtained by double rotation

of triclinic Green’s functions (open circles and stars).

Therefore, Tests 1–3 clearly demonstrate the accuracy of the proposed algorithm for eval-uation of the Green’s functions for most general anisotropic media.

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THREE-DIMENSIONAL TIME-HARMONIC GREEN’S FUNCTIONS 471

5. SUMMARY AND CONCLUSIONS

Numerical evaluation of the three-dimensional time-harmonic Green’s functions for a triclinicfull-space is considered for both displacement and stress =elds. Wave propagation in suchmaterial is described in terms of 21 material constants and three components of the propagationvector. This results in very large elements of the ChristoJel matrix needed to model themotion. Using symbolic computations an algorithm for numerical evaluation of the Green’sfunctions has been developed. The algorithm utilizes the fact that through application of theRadon transform the analytical forms for the Green’s functions can be expressed in terms of=nite integrals. The regular and singular parts of the Green’s functions involve integrationover a unit sphere and a unit circle, respectively. These integrals are evaluated using one- andtwo-dimensional Gauss–Legendre quadratures. In addition, the algorithm requires solution ofan eigenvalue problem for the ChristoJel matrix.

Three tests were performed to validate the accuracy of the proposed algorithm. First testveri=es the results for isotropic medium. Second test compares the Green’s functions fortransversely isotropic media obtained in this study with the corresponding analytical results[8; 10]. Third test compares triclinic results obtained through double rotation of the trans-versely isotropic Green’s functions with those calculated directly by the proposed algorithms.Extensive testing of the numerical results con=rmed the accuracy of proposed algorithm.Consequently, this method should facilitate studies on scattering of elastic waves in a generalthree-dimensional anisotropic media of interest in nondestructive evaluation and strong groundmotion seismology.

ACKNOWLEDGEMENTS

The authors would like to thank an anonymous reviewer for constructive comments about thepaper.

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