Scattering of acoustic and elastic waves using hybrid multiple multipole expansions—Finite element technique

Scattering of acoustic and elastic waves using hybrid multiplemultipole expansions—Finite element technique

Matthias G. ImhofEarth Resources Laboratory, Department of Earth, Atmospheric, and Planetary Sciences, MassachusettsInstitute of Technology, 42 Carleton Street, Cambridge, Massachusetts 02142-1324

~Received 24 May 1995; accepted for publication 24 April 1996!

In this paper, two different methods to solve scattering problems in acoustic or elastic media arecoupled to enhance their usefulness. The multiple multipole~MMP! expansions are used to solve forthe scattered fields in unbounded homogeneous regions. The finite element~FE! method is used tocalculate the scattered fields in bounded, heterogeneous regions. Applying the boundary conditions,the different regions and methods are coupled together in the least squares sense as the MMPmethod requires. By construction, the scattered field in the homogeneous regions is decomposedinto P andSmodes for the elastic case. In some examples, the scattered fields are calculated andcompared to the analytical solutions. Additionally, the seismograms are calculated for a fewscattering problems with one or several heterogeneous scatterers of complex geometries. The hybridMMP–FEM technique proves to be a rather general and useful tool to solve complex,two-dimensional scattering problems. ©1996 Acoustical Society of America.

PACS numbers: 43.20.Fn, 43.20.Gp@ANN#

INTRODUCTION

Wave scattering problems have been investigated by dif-ferent techniques. Analytical solutions to the integral equa-tions do not generally exist, except for some very simplegeometries. Analytical mode expansion is limited to geom-etries such as circular cylinders, where the modes decouple.1

Therefore, numerical schemes seem to be the most directprocedure for arbitrary geometries. Numerical boundary in-tegral techniques,2 the T-matrix method,3,4 and MMPexpansions5–7 are examples thereof. Unfortunately, they alldepend on either Green’s functions or other solutions to thewave equation, which tend to be hard or impossible to findfor heterogeneous or anisotropic media. These methods arenormally limited to scattering between homogeneous scatter-ers embedded in a homogeneous background. On the otherhand, these methods do not encounter problems with un-bounded domains. No artificial radiating boundary condi-tions have to be enforced. In fact, the scattered fields can beevaluated anywhere.

In cases where the medium is heterogeneous, finiteelement8–12 ~FE! or finite differences12–14 ~FD! techniquesare routinely used to calculate the scattered wavefields. Incontrast to the boundary methods mentioned above, FE andFD do encounter serious problems with unbounded domains.The domain has to be truncated and radiating boundary con-ditions have to be enforced.15 In addition, they are limited bycomputer memory and runtime considerations. For manyproblems the distances between inhomogeneities, source,and receivers are rather large and thus might result in pro-hibitive computation times and memory requirements.

Many scattering problems exist which fall in betweenthese two classes. These problems involve heterogeneous re-gions which are bounded and embedded in a homogeneousbackground. Applications of this can be found in geophysicalexploration, earthquake engineering, ultrasonic nondestruc-tive testing and medial imaging, and underwater acoustics.

There is an interest in combining methods16–19 for un-bounded, homogeneous domains with methods which canhandle heterogeneous regions of limited extent. In thepresent paper, such a combination is made between MMPexpansions and the FE method~FEM!. As a bonus in theelastic case, the scattered fields in the homogeneous domainare, by construction, decomposed intoP andS waves.

We will apply the hybrid technique to both acoustic andelastic in-plane scattering problems where one or multipleheterogeneities are embedded in a homogeneous medium.Both source and receiver are in the homogeneous region,although all the ideas presented here will also hold if thesource and/or the receiver are located in the heterogeneity.Furthermore, we could use the combined MMP–FEM tech-nique to construct radiating boundary conditions. But in thiswork, we will not investigate this usage of the technique. Wewill not present the case of anti-plane wave motion (SH)because it can easily be derived from the acoustic case.

The present paper is structured as follows: First, we re-view both MMP expansions and the finite element methodfor the acoustic case. Next, we combine the methods for theacoustic case. Then, we extend MMP and FEM to the elasticcase and present the combination. Finally, we discuss a fewdetails of the implementation on a digital computer, presentsolutions to some scattering problems, and compare them toanalytical solutions where available.

I. ACOUSTIC THEORY

We would like to model how an incident wave fieldPinc~x,v! of angular frequencyv scatters from an object. Thesituation is depicted in Fig. 1. The scatterVI is heteroge-neous and embedded in a homogeneous backgroundVO. Forthe sake of simplicity, we will suppress the time factoreivt

in all following expressions. Where necessary, the super-scriptsO, B, and I will respectively denote quantities whichbelong to the homogeneous region on the outside, lie on the

1325 1325J. Acoust. Soc. Am. 100 (3), September 1996 0001-4966/96/100(3)/1325/14/$10.00 © 1996 Acoustical Society of America

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.63.180.147 On: Mon, 24 Nov 2014 02:20:23

boundary between the domains, or are inside the heteroge-neous region. Quantities marked with a tilde are either trans-formed quantities~e.g., LU or QR decomposed! or localquantities for a particular little regionV, e.g., a finite ele-ment. The context normally allows one to infer the correctmeaning.

A. Homogeneous regions: Multiple multipoleexpansions (MMP)

In a homogeneous regionVO, expansions for the scat-tered pressure fields are made with exact solutions to thehomogeneous wave equation

PO~x!5(j51

JO

pjOPj

O~x!, ~1!

where allPjO~x! satisfy the homogeneous Helmholtz equa-

tion

~¹21kO2 !Pj

O~x!50 ~2!

with the wave numberkO5v/aO and the wave velocityaO

in the homogeneous region. The factorspjO are complex val-

ued weighting coefficients for the different expansion func-tionsPj

O. As the name of the method implies, several multi-pole solutions centered at different locations are often usedas expansion functions. These functions have a local behav-ior and thus are able to model wavefields scattered fromcomplex geometries.6 In general, MMP expansions have asmaller number of unknowns than comparable methods.Hence, usingM multipoles with their respective origins lo-cated atxm , we commonly write the scattered wavefields inthe homogeneous region as

PO~x!5 (m51

M

(n52N

N

pmnO einuH unu

~1!~kOux2xmu!. ~3!

The functionsH unu(1) are the Hankel functions of the first kind

and ordern radiating outward. Each summation over theindexn builds up one multipole. Since the Hankel functionshave a singularity at their origin, the centers of expansions

xm may not be located in the homogeneous regionVO. Foreach multipole, all orders between2N<n<1N are nor-mally used as basis functions.

Additional expansion functions such as plane waves,beams, or other special modes can be included. Although wewill not make use of these additional expansion functions inthe present work, we prefer the notation~1! to ~3! for itsgenerality and simplicity. Equations for the weighting coef-ficients pj

O are obtained by enforcing boundary conditionsfor the pressure and the normal displacement on discretematching pointsmi on the boundaries between domains. Thenormal direction pointing outward frommi is denoted byni .Then, the boundary conditions between two domainsVO andVX are

(j51

JO

pjOPj

O~mi !1Pinc~mi !5PX~mi !, ~4!

21

kO2 lO ni•(

j51

JO

pjO“Pj

O~mi !121

kO2 lO ni–“P

inc~mi !

521

kX2lX ni–“P

X~mi !, ~5!

where it is assumed that the only source is the incident fieldPinc propagating in the homogeneous domainVO. The pa-rameterl5a2r is the Lame´ constant for a region with den-sity r and velocitya. Hence, we can construct a linear equa-tion system

S POUOD •pO5S PX2Pinc

UX2UincD , ~6!

where the submatricesPi jO and Ui j

O contain PjO and

~21/kO2 lO!ni•¹Pj

O evaluated at the matching pointsmi . Ingeneral, expansions of the form~1! are not orthogonal. Thus,more matching points than expansion functions have to beused and the resulting overdetermined linear system~6! hasto be solved in the least square sense minimizing the overallerror in the boundary conditions. Thus, by construction, thematricesPO andUO are rectangular and dense.

B. Heterogeneous regions: Finite elements (FE)

Neglecting source terms, waves propagating in an het-erogeneous regionVI are governed by the general Helmholtzequation

“•$r21“P%1r21k2P50, ~7!

where r5r~x! and k5k~x! respectively denote spatiallyvarying density and wave number.

To solve this equation, we partition the heterogeneousdomainVI into small and nonoverlapping elementsV8–10

depicted in Fig. 1. Choosing quadrangular elements, we in-terpolate the pressure inside the elementV by bilinear shapefunctions9 Nj ~x! which vanish by definition outside their el-ement V. Thus, the pressure inside the elementV is ex-pressed by

P~x!5(j51

4

p j N j~x!. ~8!

FIG. 1. The generic scattering problem to be solved by the hybrid MMP-FEM technique. A heterogeneous scattererVI is embedded in a homoge-neous backgroundVO. In the acoustic case, the incident field isPinc and thescattered field isPO. In the elastic case, the incident field iswinc and thescattered fieldwO. The triangles symbolize expansion centers for the MMP.

1326 1326J. Acoust. Soc. Am., Vol. 100, No. 3, September 1996 Matthias G. Imhof: Scattering of acoustic and elastic waves


The complex valued weighting coefficientsp j are the pres-sure values at the element’s cornersxj . They are also knownas node points and thus the coefficientsp j are called nodevariables.

Consequently, the continuous pressure in the Helmholtzequation~7! is replaced by the interpolation~8!. ApplyingGalerkin’s method, we multiply the resulting expression bythe test functionNi~x! and integrate over the elementV. Weobtain

(j51

4 H E EV

r21“Ni–“Nj dA2E E

Vr21k2Ni Nj dAJ p j

2EG

r21Ni

]P

]ndl50, ~9!

where the divergence theorem has been used to transform thefirst integral. All integrals are over the surfaceV or aroundthe boundaryG of the elementV, respectively. Evaluating~9! for all four Ni~x! will yield a set of four equations for thefour unknown node variablesp j .

These three integrals define the local stiffness matrixS,the local mass matrixM and the local force vectorf:

Si j5E EV

r21“Ni•“Nj dA, ~10!

M i j5E EV

r21k2Ni Nj dA, ~11!

f i5EG

r21Ni

]P

]ndl. ~12!

For interior elements, the boundary integral~12! is not zero,but its contributions will exactly cancel with like terms com-ing from neighboring elements. One only need recall that thetermr21(]P/]n) is proportional to the normal displacement.But, both the normal displacement and the pressure are con-tinuous across boundaries. Therefore, the line integral has tobe taken into account only on the domain boundary since it isnot cancelled by another term there.

If the elementV is adjacent to a void domain, theboundary integral~12! will vanish, sinceNi50. If the ele-ment is adjacent to a rigid domain, the integral~12! alsovanishes because]P/]n50. In all other cases, the boundaryintegral~12! has to be included. Assuming that]P/]n can beapproximated by a function similar toNi along the boundary,we replace~12! by

f i5(j51

4

F i j

]P~xj !

]nwhere F i j5E

Gr21Ni Nj dl. ~13!

If the densityr and the wave numberk are treated as con-stants within each element,S, M , and F can be evaluatedexactly. Once the contributions of the various elements aredetermined, the global system of equations is formed bymapping the local node numbers onto the global node num-bers, giving rise to the global pressure vectorp, and combin-ing all of the subsystemsS, M , and f into their global coun-terpartsS, M , and f.9 Both matricesS andM are sparse,

banded and symmetric. Each row of the global matrix systemcan then be reduced to

(j51

J H ~Si j2Mi j !pj2Fi j

]P~xj !

]n J 50, ~14!

whereJ is the total number of node variables or in the sim-pler matrix form

K–p2f50. ~15!

The matrixK is defined asS2M . The vectorp contains allthe unknown, global nodal valuespj .

C. Coupling the regions

By reordering equations and unknowns, the vectorp ob-tained from the finite elements containing the node variablescan be split into two subvectorspI andpB. The node vari-ables from inside the domainVI are collected in the vectorpI . The subvectorpB accommodates the node variableswhose node pointsxj lie on the boundary]VB. Since theboundary]VB belongs to both domains, the scattered wave-field PO expanded into~1!, Pinc, and PB have to matchacross the boundary. Hence, we replace the node variablespjB by

pjB5 (

k51

JO

pkOPk~xj !1Pinc~xj !. ~16!

Furthermore, we can also find]P~xj !/]n by evaluating

]P~xj !

]n5 (

k51

JO

pkOnj•“Pk~xj !1nj•“P

inc~xj !. ~17!

Combining~14!, ~16!, and~17! yields the hybrid matrix sys-tem

(j51

JI

$Ki j pj%1 (j5JI11

J H Ki j (k51

JO

pkOPk~xj !2Fi j (

k51

JO

pkOnj

•“Pk~xj !J 5 (j5JI11

J

$Fi j nj•“Pinc~xj !2Ki j P

inc~xj !%,

~18!

where the node pointsxj lie in the interior of the heteroge-neous FE domainVI for 1< j<JI . The node pointsxj arelocated on the boundary]VB for JI11< j<J. As before,Jis the total number of node points. Finally,JO is the totalnumber of functions used for the MMP expansion of theoutside field. The complete hybrid system can be written in amore compact form as

S AII AIO

AOI AOOD •S pIpOD 5S fIfOD , ~19!

where the first term of~18! maps intoAII or AOI dependingon whether the node point of the corresponding test functionNi lies inside the heterogeneity~AII ! or on the boundary~AOI!. Similarly, the second term and the right hand sidemap intoAIO or AOO and fI or fO, respectively. It is impor-tant to distinguish between the submatricesAII ,...,AOO whichhave different physical interpretations, mathematical forms



and numerical properties. The submatrixAII is a sparse, di-agonally dominant and symmetric matrix which stems fromthe heterogeneous region where wavefields are modeled byFE. Thus, the wavefields inside only approximate the waveequation~7!. The residuals between the true wavefield andthe approximate solution are going to be spread over all nodevariablespI .

Contrarily, the submatrixAOO is dense and rectangular.Each column stems from a MMP expansion function whileeach row originates from the boundary conditions to be sat-isfied at a particular node point on the boundary. Since theMMP expansions are non orthogonal, the coefficientspO

have to be solved for in the least squares sense.The submatricesAIO andAOI are sparse and rectangu-

lar. These submatrices actually couple the wavefield~FE!inside the heterogeneity to the wavefield~MMP! in the ho-mogeneous background. Altogether, we end up with the rect-angular system~19! where the upper half has completelydifferent properties than the lower half. Therefore, we solvethe system~19! in two steps: first, the interior node variablespI are successively Gaussian eliminated20 reducing the sub-matrix AII to the triangular submatrixAII . This replaces thecoupling matrixAOI by a null matrix0. BecauseAII is di-agonally dominant, the Gaussian elimination can be per-formed without additional pivoting. The elimination of theinterior node variablespI respectively transformsAIO andAOO into AIO andAOO. We obtain a new, transformed linearsystem:

S AII AIO

0 AOOD •S pIpOD 5S fIfOD . ~20!

Actually, this step corresponds to a partialLU decomposi-tion or similarly, to a static condensation9 where the nodevariablespj for jP$1,JI% in the heterogeneous domain aresuccessively eliminated. The complete heterogeneous do-main collapses into one single ‘‘super-element’’ coupling thenodal points on the boundary. This ‘‘super-element’’ directlycouples the incident field to the MMP expansion.

Second, the remaining system,AOO–pO5fO, is solved in

the least squares sense usingQR decomposition.21 If desired,the values of the node variablespI can later be found byback-substitution:

AII–pI5 fI2AIO

–pO. ~21!

In practice, the system~19! is solved by a combined, row-orientedLU–QR algorithm. The scheme is similar to tradi-tional Givens updating.20 But for each new equation added toupdate the system, the firstJI Givens rotations are replacedby Gaussian eliminations instead. As a result, the firstJI

equations are onlyLU composed and used to build the tri-angular submatrixAII and the transformedAIO. For the re-maining equations, the firstJI unknowns can be eliminatedusing AII AIO. The rest are used for Givens updating whichbuilds AOO directly.

II. ELASTIC THEORY

A. Homogeneous regions: Multiple multipoleexpansions

In a homogeneous regionVO, expansions for the scat-tered displacement fieldswO~x!5„u~x!,v~x!… are made withexact solutions to the homogeneous wave equation:7

wO~x!5(j51

JO

$f jwjF~x!1c jwj

C~x!%, ~22!

where

wjF~x!5“F j~x!, ~23a!

wjC~x!5“3@ yC j~x!#. ~23b!

Each expansion functionFj or Cj satisfies a Helmholtzequation:

~¹21kO2 !F j50, ~24a!

~¹21 l O2 !C j50. ~24b!

The constantskO and lO are respectively the wave numbersof the P and theS wave in the homogeneous region. Thefactorsfj andcj are complex valued weighting coefficientsfor the different expansion functionsFj andCj . Similar tothe acoustic MMP expansions~3!, the displacement fieldwO~x! is expanded into multiple multipoles7

wO~x!5 (m51

M

(n52N

N

$fmn“Fmn~x!

1cmn“3@ yCmn~x!#% ~25!

5 (m51

M

(n52N

N

$fmn“einuH unu

~1!~kOux2xmu!

1cmn“3@ yeinuHunu~1!~ l Oux2xmu!#%, ~26!

wherexm is the location or expansion center of themth mul-tipole. Although we will not use other expansion functionsthan applied in~25! and~26!, we prefer the notation~22! forits simplicity and generality. To go from~25! to ~22!, thedouble indexmn is simply replaced by the single indexj andvice versa. Thus, we will use~22! but mean~25!. Equationsfor the weighting coefficientsfj andcj are obtained by en-forcing boundary conditions along discrete matching pointsmi on the boundaries between domains. The boundary con-ditions between two domainsVO andVX are the continuityof displacement and stresses in normal and tangential direc-tions ni and t i :

(j51

JO

ni•$f jwjF~mi !1c jwj

C~mi !%1ni•winc~mi !

5ni•wX~mi !, ~27!

(j51

JO

t i•$f jwjF~mi !1c jwj

C~mi !%1 t i•winc~mi !

5 t i•wX~mi !, ~28!



(j51

JO

ni•$f jsjF~mi !1c jsj

C~mi !%•ni1ni•sinc~mi !•ni

5ni•sX~mi !•ni , ~29!

(j51

JO

t i•$f jsjF~mi !1c jsj

C~mi !%•ni1 t i•sinc~mi !•ni

5 t i•sX~mi !•ni . ~30!

The quantitiessjF~mi!, sj

C~mi!, sinc~mi!, andsX~mi! denotethe stress tensors evaluated atmi due to the respective dis-placementswj

F~mi!, wjC~mi!, wj

inc(mi), andwjX~mi!. Accord-

ingly, we build a linear equation system for the unknowncoefficientsfj andcj ;

S UF UC

VF VC

SnF Sn

C

StF St

C

D •S fcD5S UX2Uinc

VX2V inc

SnX2Sn

inc

StX2St

incD , ~31!

where the submatrices contain Eqs.~27!–~30! evaluated atall matching pointsmi . In general, expansions of the form~22! are not orthogonal. Thus, more matching points thanexpansion functions are used and the resulting overdeter-mined linear system~31! is solved in the least-square senseminimizing the overall error in the boundary conditions.

B. Heterogeneous regions: Finite elements

In a Cartesian coordinate frame (x,z) with unit vectorsxand z, the coupled equations of motion for the displacementw~x!5„u~x!,v~x!… are

v2ru~x!1$c11u~x! ,x1c12v~x! ,z% ,x

1c33$u~x! ,z1v~x! ,x% ,z50, ~32a!

v2rv~x!1c33$u~x! ,z1v~x! ,x% ,x

1$c12u~x! ,x1c22v~x! ,z% ,z50; ~32b!

where we neglected eventual source terms. The densityr andthe elastic constantsc115c225l12m, c125l and c335mare all spatially varying. The parametersl and m are thespatially varying Lame´ constants. The subscripts,x and ,zdenote partial derivatives with respect tox or z. Similar tothe acoustic case~8!, the heterogeneous region is partitionedinto small elementsV. The components of the displacementinside the elementsV are approximated by bilinear interpo-lation functionsNj ~x!:8,9,11

u~x!5(j51

4

u j N j~x!, ~33a!

v~x!5(j51

4

v j N j~x!. ~33b!

The complex valued weighting coefficientsu j andv j are thecomponents of the displacements at the element’s cornersxj .

In the equations of motion~32!, we replace the continu-ous displacementsu andv by their interpolations~33!. Ap-

plying Galerkin’s method for each elementV, we multiply

the resulting expression by the test functionNi~x! and inte-grate over the elementV. Hence, we obtain

(j51

4 H E EV

~c11Ni ,xNj ,x1c33Ni ,zNj ,z!dA

2E EV

rv2Ni NjdAJ u j1(j51

4 H E EV

~c12Ni ,xNj ,z

1c33Ni ,zNj ,x!dAJ v j2EGNi x•s•n dl50, ~34a!

(j51

4 H E EV

~c12Ni ,zNj ,x1c33Ni ,xNj ,z!dAJ u j1(

j51

4 H E EV

~c33Ni ,xNj ,x1c22Ni ,zNj ,z!dA

2E EV

rv2Ni Nj dAJ v j2EGNi z•s•n dl50, ~34b!

where we used the divergence theorem to transform some ofthe volume integrals into line integrals. The quantitys5s~x! denotes the stress tensor along the boundary. Evalu-ating ~34! for all Ni~x! yields a set of linear equations for theunknown node variables,u j and v j . Again, Eqs.~34! definethe stiffness matricesSrs, mass matricesM rs and the forcevectors f r where the superscriptrP$1,2% denotes whetherquantity is obtained from Eq.~34a! or ~34b!. The superscriptsP$1,2% indicates whether the matrix governs the respectivecoefficientsu j or v j :

M i j115M i j

225E EV

rv2Ni Nj dA, ~35!

M i j125M i j

2150, ~36!

Si j115E E

V~c11Ni ,xNj ,x1c33Ni ,zNj ,z!dA, ~37!

Si j225E E

V~c33Ni ,xNj ,x1c22Ni ,zNj ,z!dA, ~38!

Si j125Sj i

215E EV

~c12Ni ,xNj ,z1c33Ni ,zNj ,x!dA, ~39!

f i15E

GNi x•s•n dl, ~40!

f i25E

GNi z•s•n dl. ~41!

If the densityr and the elastic constantsc11, c22, c12, and

c33 are treated as constants within each elementV, all inte-grals~35!–~39! can be evaluated exactly. For elements whichare in the interior, the boundary integrals~40! and ~41! arenot zero, but their contributions will exactly cancel with liketerms coming from neighboring elements because displace-ments and stresses are continuous across elements. There-fore, the line integrals have to be taken into account only on



the domain boundary. Assuming thatx•s•n and z•s•n canbe approximated by functions similar toNi along the bound-ary, we replace~40!, ~41! by

f i15(

j51

4

F i j x•s~xj !•nj , ~42!

f i25(

j51

4

F i j z•s~xj !•nj ; ~43!

where

F i j5EGNi Nj dl, ~44!

which can also be evaluated analytically. Finally, mappingthe local node numbers into global node numbers yields theglobal matricesSrs, M rs, fr and the global nodal vectorsuandv. Writing K rs5Srs2M rs, the global matrix system re-duces to the simpler system

K11•u1K12

•v2f150, ~45a!

K21•u1K22

•v2f250. ~45b!

C. Coupling the regions

After reordering the equations and unknowns, the solu-tion vectorsu andv obtained from the finite elements can besplit into two subvectors,uI ,uB andvI ,vB. The node variablesui and v i from inside the domainVI are collected in therespective subvectorsuI and vI . The subvectorsuB and uB

accommodate the node variables whose node pointsxj lie onthe boundary]VB. Since the boundary]VB belongs to bothdomains, the wavefieldswO expanded into~22!, the sourcefield winc, andwB have to match along the boundary. There-fore, we replace the node variablesuj

B andv jB by

ujB5 (

k51

JO

$fkx•wkF~xj !1ckx•wk

C~xj !%1 x•winc~xj !,

~46a!

v jB5 (

k51

JO

$fkz•wkF~xj !1ckz•wk

C~xj !%1 z•winc~xj !,

~46b!

where we use the unit vectorsx andz to extract the displace-ment componentsu~x! andv~x! in the x, respectively thezdirection. Also, we find the stress tensors~xj ! along theboundary by evaluating

s~xj !5 (k51

JO

$fkskF~xj !1cksk

C~xj !%1sinc~xj !. ~47!

Combining ~45!, ~46!, and ~47! yields the coupled MMP–FEM system. For the sake of clarity, we will expand it ex-plicitly:

(j51

JI

$Ki j11uj%1(

j51

JI

$Ki j12v j%1 (

k51

JO

(j5JI11

J

$Ki j11x•wk

F~xj !

1Ki j12z•wk

F~xj !2Fi j x•skF~xj !•nj%fk

1 (k51

JO

(j5JI11

J

$Ki j11x•wk

C~xj !1Ki j12z•wk

C~xj !

2Fi j x•skC~xj !•nj%ck

5 (j5JI11

J

$Fi j x•sinc~xj !•nj2Ki j11x•winc~xj !

2Ki j12z•winc~xj !%, ~48a!

(j51

JI

$Ki j21uj%1(

j51

JI

$Ki j22v j%1 (

k51

JO

(j5JI11

J

$Ki j21x•wk

F~xj !

1Ki j22z•wk

F~xj !2Fi j z•skF~xj !•nj%fk

1 (k51

JO

(j5JI11

J

$Ki j21x•wk

C~xj !1Ki j22z•wk

C~xj !

2Fi j z•skC~xj !•nj%ck

5 (j5JI11

J

$Fi j z•sinc~xj !•nj2Ki j21x•winc~xj !

2Ki j22z•winc~xj !%. ~48b!

Again, 1< j<JI means that the node pointxj is in the inte-rior VI of the FE domain. Contrarily, forJI11< j<J thenode pointsxj are located on the boundary]VB. As before,Jis the total number of node points. Finally,JO is the totalnumber of functions used for the MMP expansion of theoutside field. The resulting hybrid system of linear equationscan be written in a more compact form as

S K11,I K12,I F1,I C1,I

K21,I K22,I F2,I C2,I

K11,O K12,O F1,O C1,O

K21,O K22,O F2,O C2,O

D •S uIvIfcD 5S f1,I

f2,I

f1,O

f2,OD . ~49!

Equation~48a! maps into the first or third row of~49! de-pending on whether the node point (x) i of the correspondingtest functionNi lies inside the heterogeneity or on the bound-ary. Equation~48b! maps exactly the same way into the sec-ond or the fourth row of~49! depending on the location ofthe corresponding node point. Furthermore, the first term inboth ~48a! and~48b! maps into the first column of~49!. Thesecond terms map into the second column and so on for theother terms and the right-hand side of~48!.

The hybrid system~49! is very similar to the acousticsystem~19!. As in the acoustic case, the submatricesK rs,I ,wherer ,sP$1,2%, result from the interior problem. They aresparse and square. All other submatrices are rectangular.Moreover,Fr ,O andCr ,O are dense. By construction,K11,I

andK22,I are diagonally dominant. Therefore, we can elimi-



nate the node variablesuI and vI by LU decomposition orGaussian elimination without additional pivoting. We obtaina new linear system

S K11,I K12,I F1,I C1,I

0 K22,I F2,I C2,I

0 0 F1,O C1,O

0 0 F2,O C2,O

D •S uIvIfcD 5S f1,I

f2,I

f1,O

f2,OD , ~50!

whereK11,I and K22,I are now triangular matrices. The sub-matricesK rs,O where r ,sP$1,2% vanish because we elimi-nated the node variablesuI andvI from the lower half of thesystem. In fact, this step corresponds to static condensationof the inner node variablesuI andvI . The whole FE domainis collapsed into one ‘‘super-element’’ coupling the nodevariables on the boundary. In fact, the ‘‘super-element’’couples the incident field directly to the MMP expansion.The lower half~51! of the transformed system~50! is nowsolved in the least squares sense by QR decomposition be-causeFr ,O andCr ,O are still rectangular matrices:

S F1,O C1,O

F2,O C2,OD •S fcD 5S f1,Of2,OD . ~51!

If desired, the node variables in the heterogeneous, interiorFE regionVI are easily recovered by back-substitution intothe upper half of~50! which is already in triangular form:

S K11,I K12,I

0 K22,I D •S uIvI D 5S f1,I2F1,I•f2C1,I

•c

f2,I2F2,I•f2C2,I

•c D . ~52!

III. IMPLEMENTATION

Because the technique is a mixture of MMP expansionsand the FE method, the finite element method is merged intothe prior MMP codes.6,7 Thus, the method is implemented ona nCUBE2 parallel computer using the programming lan-guage C11. The object oriented design has the advantagethat the coupling as described in~18! and ~48! is basicallyhidden in objects for node variables, finite elements, and theexpansion functions for the exterior. First, the objects for thefinite elements calculate the localM , S, and F matrices.Then, the resulting coefficients are mapped into the globalequation system. Objects for internal node variables simplymap the coefficientsK i j

(rs) for thepjI into the global system of

equations. In contrast, objects for node variables on theboundary automatically evaluate the MMP expansion at thenode point as described in Eqs.~16!, ~17! or ~46!, ~47!,weight the expansion with the appropriateK i j

(rs) or F i j(rs) co-

efficient and map the resulting coefficients forpjO, or fj and

cj into the global system.To reduce numerical noise, the materials are made

slightly lossy by adding a small imaginary componentvI tothe angular frequency. If seismograms are calculated by Fou-rier synthesis, the true amplitude is later recovered by a mul-tiplication with ev I t.

IV. NUMERICAL RESULTS: ACOUSTICS

As a first test, we simply embed a homogeneous regionin a homogeneous full space and illuminate it by an incidentplane wave. The wavefield in the embedded region is mod-eled by FE. The scattered wavefields in the full space areexpanded into a MMP series. The material parameters inboth regions are the same. Namely, the velocitya is 2000m/s and the densityr is 2000 kg/m3. Hence, all coefficientsof the MMP expansion should be zero, while the FE solutionshould simply interpolate the incoming field. Clearly, due tothe discretization of the field in the interior, the solution inthe interior will deviate from the incident field and thus, anadditional scattered field will be induced. The strength of thisinduced field is both a function of the number of elementsper wavelength~EPW! and the angle of incidence of thesource field with respect to the FE region. The embeddedregion consists of 18318 elements, each 434 m in size. TheMMP expansion used is~3! with M54 and N54. Alto-gether, 36 expansion functions are used. Figure 2 shows theexact position of node points and expansion centers. The FEregion is illuminated by incident plane waves with four fre-quencies corresponding to 100, 25, 10, and 5 EPW. Theresulting relative errorsuP2Pincu/uPincu& along the boundaryof the inclusion are shown in Fig. 3 as a function of angle ofincidence and EPW. Clearly, the errors increase with de-creasing EPW until the spurious fields are of similar order ofmagnitude as the source field. Also visible is the apparentanisotropy due to the finite elements. Compared to normalincident, the realive error for plane waves incident in thediagonal direction is slightly reduced. For 10 EPW, we ob-tain a maximal relative error of about 5%.

To test the accuracy of the MMP–FE technique, wecompare the scattering from an acoustic cylinder with thewell-known analytical series solution.1 The velocity insidethe cylinder is 3000 m/s; the velocity outside the cylinder is

FIG. 2. An embedded homogeneous region. Each cross represents a nodepoint xi and each star a MMP expansion centerxm . From each expansioncenter, we set up an expansion(n524

4 pmnO einuH unu

(1)~kOux2xmu!.



2000 m/s. In both regions, the density is kept constant at2000 kg/m3. The radiusa of the cylinder is 44 m. To sim-plify the generation of the FE mesh, a square region largerthan the actual cylinder is discretized by 24 elements in ei-ther direction. Due to the symmetry of the problem, only onemultipole ~3! is used, whereM51 andN520. It is located atthe origin. The geometry is shown in Fig. 4. The size of theelements is 4 m. The experiment is performed for two dif-

ferent incident wavelengths. First, the wavelength is 100 mand the correspondingka52.5. Second, the wavelength is 25m andka510. Magnitude and phase are presented in Figs. 5and 6. For the longer wavelengthka52.5, the analytical andthe hybrid solution agree very well. For the shorter wave-length whereka510, the deviations of the FE–MMP solu-tion from the analytical one are due to the size of the finiteelements. Reducing the element size would reduce the devia-tions. Furthermore, the largest deviations correlate with thesmallest magnitudes as can be observed in both Figs. 5 and6. This is an effect of the least-squares solving procedure.

FIG. 3. Acoustics: Relative boundary error^uP2Pincu/Pincu& as a function ofcontinuous angle of incidence of a planar sourcefield with respect to thefinite elements for different numbers of elements per wavelength~EPW!. 10EPW yield an error of about 5%. The grid anisotropy is clearly visible forsmall wavelengths or correspondingly small EPWs.

FIG. 4. A cylindrical scatterer is illuminated by a plane wave. Outside, thevelocity is 2000 m/s; inside, the velocity is 3000 m/s. The grid represents thefinite elements used. The grid spacing is 4 m and the radius of the cylinderis 44 m. The triangle denotes the expansion center for the multipole.

FIG. 5. Acoustics: A cylindrical scatterer is illuminated by a plane wave propagating in the positive, horizontal direction. Shown is the magnitudeuPu forka52.5 andka510 as a function of angle wherea is the radius of the cylinder andk the wave number.



The solver uniformly minimizes the misfit at each boundarypoint. Thus, if the average misfit ise, any true field valuesmaller thane is lost in the misfit. If better~relative! accu-racy is desired, the solution should be calculated again withthe equations scaled by the reciprocal of the previously ob-tained field. Basically, each row ofAOI andAOO should bescaled by 1/pi . Further details on scaling can be found in aprior paper.6

Last, we calculate the seismogram for a complex geom-etry depicted in Fig. 7. The scatterers are roughly 180 m longand 35 m thick. The velocity and density in the backgroundare 2000 m/s and 2000 kg/m3, respectively. The velocity andthe density in the two scatterers are 3000 m/s and 2000kg/m3, respectively. Each finite element is 333 m in size.For each scatterer, five centers of expansion are used. Ateach centerxm , an expansion of the form

(n528

8

pmnO einuH unu

~1!~kOux2xmu!

is set up. The incident fieldPinc is an explosive line sourcemodulated by a Ricker pulse22 of 50-Hz center frequency.Altogether, 64 receivers will measure the pressure of thescattered field. The resulting seismogram is shown in Fig. 8.

To demonstrate the effect of heterogeneous scatterers,we replace the constant velocities in the scatterers by randomvelocities described by their mean velocity~3000 m/s!, thestandard deviation~500 m/s! and the spatial autocorrelationR(Dx ,Dz).

23 For simplicity, we choose a Gaussian correla-tion function depending on two perpendicular correlationlengthsa1, a2 and the anglez betweena1 and the horizontaldirection x:

FIG. 6. Acoustics: A cylindrical scatterer is illuminated by a plane wave propagating in the positive, horizontal direction. Shown is the phase arg(P) forka52.5 andka510 as a function of angle wherea is the radius of the cylinder andk the wave number.

FIG. 7. Acoustics: Generic scatterers used to calculate seismograms. Two homogeneous scatterers are embedded in a homogeneous background. The velocityand the density in the background are 2000 m/s and 2000 kg/m3. The velocity and the density in the two scatterers are 3000 m/s and 2000 kg/m3. The trianglesshow the location of the centers for the MMP expansion.



R~Dx,Dz!5expS 2H Dx cosz1Dz sin z

a1J 2

2H Dz cosz2Dx sin z

a2J 2D . ~53!

For the present example, we choosea1518 m,a256 m, andz5210°. The density as well as the geometry and locationsof the scatterers, source, and receivers are the same as in theprior experiment. The resulting seismogram is shown in Fig.9. The difference between the two models and the resultingseismograms~Figs. 8 and 9! is prominent. The comparisonclearly shows the need for numerical modelling techniqueswhich can incorporate small scale variations into the models.

V. NUMERICAL RESULTS: ELASTICS

As in the acoustic case, the first test is to embed a ho-mogeneous region in a homogeneous full space and illumi-nate it with an incident plane wave. The wavefields in theembedded region are modeled by FE, while the scattered

wavefields in the full space are expanded into a MMP series.Because the material parameters in both regions are thesame, all coefficients of the MMP expansion should be zeroand the FE solution should perfectly interpolate the incomingfield. TheP-wave velocity is 2000 m/s, theS-wave velocity1300 m/s and the density 2000 kg/m3. Clearly, due to thediscretization of the fields in the interior, the solution willdeviate from the incident field and thus, additional scatteredfields will be induced. The strength of these induced fields isboth a function of the number of elements per wavelengthand the angle of incidence of the source field. The embeddedregion consists of 18318 square elements, each 434 m insize. The MMP expansion is the same as~26! with M54 andN54. Altogether, 2336 expansion functions are used. Fig-ure 2 shows the exact position of node points and expansioncenters. As source fields, we use plane waves of purelyP orS polarization. Both experiments are performed with at fourdifferent frequencies corresponding to 100, 25, 10, and 5elements perP wavelength~EPW!. Figure 10 shows the re-sulting relative boundary errorsuw2wincu/uwincu& along theinclusion as a function of the angle of incidence. As ex-pected, the errors increase with increasing frequency or ac-cordingly with decreasing EPW. Not surprisingly, the errorsfor incidentSwaves are in general larger than for incidentPwaves of the same frequency. This observation is readilyexplained by the shorter wavelengths of theS phases. As inthe acoustic case, less spurious fields are induced for incidentP or S waves propagating in diagonal direction. For 25EPW, we obtain a maximal relative error of about 5% fornormal incidence.

FIG. 8. Acoustics: The resulting seismogram for the complex geometrydepicted in Fig. 7. The velocities in the scatterers are constant.

FIG. 9. Acoustics: The resulting seismogram for the complex geometrydepicted in Fig. 7 where the velocities in the scatterers are replaced byrandom fields with the same mean velocity~3000 m/s!, a standard deviationof 500 m/s and a Gaussian spatial autocorrelation with correlation lengths ofa1518 m,a256 m and a tilt of210°.

FIG. 10. Elastics: Relative boundary error^uw2wincu/uwincu& as a function ofthe continuous angle of incidence of the source field with respect to thefinite elements for a series of number of elements perP wavelength~EPW!.On the left-hand-side, the incident field is a planarS wave. On the right-hand-side, the incident field is a planarP wave. An incidentSwave is moreaffected by the EPW because its wavelength is only about half as long as thecorrespondingP wave’s. 25 EPW yields a maximal relative error of 5% forthe incidentS-wave.



To test the accuracy of the MMP–FE technique in theelastic case, we compare the scattering from a cylinder withthe analytical series solution.1 Inside the cylinder, theP ve-locity is 3000 m/s, theS velocity is 1700 m/s and Poisson’sratio 0.26. Outside the cylinder, theP velocity is 2000 m/s,theS velocity is 1300 m/s and Poisson’s ratio 0.13. In bothregions, the density is 2000 kg/m3. The radiusa of the cyl-inder is 12 m. To simplify the generation of the FE mesh, asquare region larger than the actual cylinder is discretized by24 elements in either direction. Due to the symmetry of theproblem, we use only one multipole~26! located at the ori-gin, whereM51 andN520. The geometry is similar to the

one shown in Fig. 4. The size of the elements is 1 m. Thewavelength of the incidentP wave is 50 m~ka51.5! and thewavelength of the incidentS wave is 32 m~la52.3!. Themagnitude and phase of theu and thev components areshown in Figs. 11 and 12. For all incident phases, the matchbetween the analytical solution and the results obtained fromthe MMP–FE method is excellent.

Last, we calculate the seismogram for a complex geom-etry depicted in Fig. 13. A scatterer is illuminated by a linesource. The scatterer is roughly 180 m long and 35 m thick.TheP andS velocities and density in the scatterer are 3000m/s, 1730 m/s, and 2000 kg/m3, respectively. Poisson’s ratio

FIG. 11. Elastics: A cylindrical scatterer is illuminated by a plane wave propagating in the positive, horizontal direction. BothP andS modes are used asincident fields. Shown are the magnitudesuuu anduvu for ka51.5 andla52.3 as a function of angle wherea is the radius of the cylinder,k theP wave numberand l theS wave number.

FIG. 12. Elastics: A cylindrical scatterer is illuminated by a plane wave propagating in the positive, horizontal direction. BothP andS modes are used asincident fields. Shown are the phases arg(u) and arg(v) for ka51.5 andla52.3 as a function of angle wherea is the radius of the cylinder,k theP wavenumber andl theS wave number.



is s50.26. In the background, theP andS velocities anddensity in the background are 2000 m/s, 1300 m/s, and 2000kg/m3, respectively. Poisson’s ratio iss50.13. Each finiteelement is 232 m in size. In the scatterer, five expansions ofthe form ~26! with M55 andN57 are used. Two differentincident fields are chosen: a compressional and a rotationalline source. Each source is modulated with a Ricker pulse22

of 50-Hz center frequency. Altogether, 64 receivers measurethe vertical displacement component of the scattered field.The resulting seismograms are shown in Figs. 14–17. WhileFig. 14 shows the scattered field due to the compressionalline source, seismogram in Fig. 15 contains only the modeconvertedS waves. Similarly for the rotational line source.While Fig. 16 shows the complete scattered wavefield, theseismogram in Fig. 17 contains only the mode convertedPwaves. For both source fields, energy is mode converted bythe scatterer.

To show the effect of a heterogeneous scatterer, we re-place the constant velocities in the scatterer by random fieldsdescribed by the mean velocities, the standard deviations andthe spatial auto-correlationsR(Dx ,Dx). We choose a Gauss-ian correlation function~53! with two perpendicular correla-

tion lengthsa1518 m,a256 m, and an anglez5210°. ThemeanP velocity is 3000 m/s, the meanS velocity 1730 m/s,and the standard deviation of theP velocity 500 m/s. Every-where within the scatterer, theS velocity is chosen to pre-serve a Poisson’s ratio of 0.26. The density as well as thegeometry and locations of the scatterer, source, and receiversare the same as in the prior experiment. For both a compres-sional and a rotational source, the respective seismogramsare shown in Figs. 18 and 19. Comparing the seismogramdue to the homogeneous scatterers~Figs. 14 and 16! with theones for the heterogeneous scatterer~Figs. 18 and 19! clearlyshows the importance of allowing small scale variations inthe material properties and thus the need for hybrid tech-niques.

VI. SUMMARY

The MMP code has been successfully coupled with theFE method in both acoustic and elastic media. The couplingof the two methods enhances their usefulness for a range of

FIG. 13. Elastics: Generic scatterer used to calculate elastic seismograms.The scatterer is embedded in a homogeneous background. TheP and Svelocities and density in the background are respectively 2000 m/s, 1300m/s, and 2000 kg/m3. TheP andS velocities and density in the scatterer arerespectively 3000 m/s, 1730 m/s, and 2000 kg/m3. The triangles show thelocation of the centers for the MMP expansion.

FIG. 14. Elastics: The vertical component of the total scattering displace-ment for the complex geometry depicted in Fig. 13. The incident field is acompressional line source.

FIG. 15. Elastics: The vertical displacement of theP→S conversion due tothe complex geometry depicted in Fig. 13. The incident field is a compres-sional line source.

FIG. 16. Elastics: The vertical component of the total scattering displace-ment for the complex geometry depicted in Fig. 13. The incident field is arotational line source.



problems. The FE technique allows the simulation of wavepropagation in heterogeneous materials. The MMP expan-sions allow to calculate propagating waves in homogeneous~unbounded! regions in an efficient manner because theycommonly need fewer unknowns to be evaluated and solvedfor than comparable methods.

Steady-state solutions, as well as seismograms obtainedby Fourier synthesis, were calculated for a range of differentproblems for both acoustic and elastic media. Where avail-able, the solutions obtained by the combined MMP–FEMscheme compared favorably with the analytical solutions.Comparisons between homogeneous and heterogeneous scat-terers present major differences for the resulting wavefieldswhich show the need for hybrid schemes which allow includ-ing variations in the material parameters. For the elastic case,we automatically obtain the scattered wavefields decom-posed intoP andS modes. This feature allows us to study

the conversion of modes fromP→S or vice versa due to theheterogeneous scattering region.

The combined scheme compensates the individualweaknesses of MMP and FEM and takes advantage of boththeir strengths. Thus, the method is well-suited to solve two-dimensional scattering problems for a range of problemswhich neither method could handle alone.

ACKNOWLEDGMENTS

This work was supported by the Air Force Office ofScientific Research under Contract No. F49620-94-1-0282.Also, the author was supported by the nCUBE fellowship.The author thanks Dr. C. Hafner, Dr. L. Bomholt, and Pro-fessor H. Schmidt for their helpful comments and discus-sions.

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FIG. 17. Elastics: The vertical displacement of theS→P conversion due tothe complex geometry depicted in Fig. 13. The incident field is a rotationalline source.

FIG. 18. Elastics: The vertical component of the seismogram for the com-plex geometry depicted in Fig. 13 where the velocities in the scatterer arereplaced by random fields with a meanP velocity of 3000 m/s, a standarddeviation of theP velocity of 500 m/s and a Gaussian spatial autocorrelationwith correlation lengths ofa1518 m, a256 m and a tilt of210°. TheSvelocity is chosen to preserve a Poisson’s ratio of 0.26 everywhere in thescatterer. The incident field is a compressional line source.

FIG. 19. Elastics: The vertical component of the seismogram for the com-plex geometry depicted in Fig. 13 where the velocities in the scatterer arereplaced by random fields with a meanP velocity of 3000 m/s, a standarddeviation of theP velocity of 500 m/s and a Gaussian spatial autocorrelationwith correlation lengths ofa1518 m, a256 m and a tilt of210°. TheSvelocity is chosen to preserve a Poisson’s ratio of 0.26 everywhere in thescatterer. The incident field is a rotational line source.



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Scattering of acoustic and elastic waves using hybrid multiple multipole expansions—Finite element technique