Implementation and computational aspects of a 3D elastic wave modelling by PUFEM

I m pl e m e n t a ti o n a n d

c o m p u t a ti o n al a s p e c t s of a 3 D

e l a s ti c w a v e m o d e lli n g b y

P U F E M

M a h m o o d , M S, L a g h r o u c h e , O, T r e v ely a n , J a n d Ka ci mi, AEl

h t t p :// dx. d oi.o r g / 1 0 . 1 0 1 6 /j. a p m . 2 0 1 7 . 0 5 . 0 1 3

T i t l e

I m pl e m e n t a ti o n a n d c o m p u t a ti o n al a s p e c t s of a 3 D e l a s ti c

w a v e m o d elli n g b y P U F E M

A u t h o r s

M a h m o o d , M S, L a g h r o u c h e , O, T r e v ely a n , J a n d Ka ci mi,

AEl

Typ e

Ar ticl e

U RL

T hi s v e r si o n is a v ail a bl e a t :

h t t p :// u sir. s alfo r d . a c . u k /i d/ e p ri n t/ 4 2 3 5 2 /

P u b l i s h e d D a t e

2 0 1 7

U S IR is a d i gi t al c oll e c ti o n of t h e r e s e a r c h o u t p u t of t h e U n iv e r si ty of S alfo r d .

W h e r e c o p y ri g h t p e r m i t s , f ull t e x t m a t e r i al h el d i n t h e r e p o si t o r y is m a d e

f r e ely a v ail a bl e o nli n e a n d c a n b e r e a d , d o w nl o a d e d a n d c o pi e d fo r n o

n-c o m m e r n-ci al p r iv a t e s t u d y o r r e s e a r n-c h p u r p o s e s . Pl e a s e n-c h e n-c k t h e m a n u s n-c ri p t

fo r a n y f u r t h e r c o p y ri g h t r e s t r i c ti o n s .

ContentslistsavailableatScienceDirect

Applied

Mathematical

Modelling

journalhomepage:www.elsevier.com/locate/apm

Implementation

and

computational

aspects

of

a

3D

elastic

wave

modelling

by

PUFEM

M.S.

Mahmood

c,a

_,

_O.

_Laghrouche

a,∗

_,

_J.

_Trevelyan

b

_,

_A.

_El

_Kacimi

a

a_{InstituteforInfrastructureandEnvironment,Heriot-WattUniversity,EdinburghEH144AS,UK} b_{SchoolofEngineeringandComputingSciences,DurhamUniversity,DurhamDH13LE,UK} c_{SchoolofComputing,ScienceandEngineering,SalfordUniversity,ManchesterM54WT,UK}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received30August2016 Revised7February2017 Accepted7May2017 Availableonline12May2017

Keywords:

PUFEM Planewavebasis Elasticwaves Pressurewaves Shearwaves 3Dscattering

a

b

s

t

r

a

c

t

Thispaperpresentsanenrichedfiniteelementmodel forthreedimensionalelasticwave problems, inthe frequency domain,capable ofcontainingmanywavelengths pernodal spacing.Thisis achievedbyapplyingtheplane wavebasisdecompositiontothe three-dimensional (3D)elasticwaveequationand expressingthedisplacementfieldas asum ofbothpressure(P)andshear(S)planewaves.Theimplementationofthismodelin3D presentsanumberofissuesincomparisontoits2Dcounterpart,especiallyregardinghow S-wavesareusedinthebasisateachnodeandhowtochoosethebalancebetweenPand S-wavesintheapproximationspace.Variousproposedtechniquesthatcouldbeusedfor theselectionofwavedirectionsin3Darealsosummarisedandused.Thedeveloped ele-mentsallowustorelaxthetraditionalrequirementwhichconsiststoconsidermanynodal pointsperwavelength,usedwithloworderpolynomialbasedfiniteelements,and there-foresolveelasticwaveproblemswithoutrefiningthemeshofthecomputationaldomain ateachfrequency.Theeffectivenessoftheproposedtechniqueisdeterminedby compar-ingsolutionsforselectedproblemswithavailableanalyticalmodelsortohighresolution numericalresultsusingconventionalfiniteelements,byconsideringtheeffectofthemesh sizeandthenumberofenriching3Dplanewaves.Bothbalancedandunbalancedchoices ofplanewavedirectionsinspaceonstructuredmeshgridsareinvestigatedforassessing theaccuracyandconditioningofthis3DPUFEMmodelforelasticwaves.

© 2017 Published by Elsevier Inc.

1. Introduction

Growingresearchactivitieshavebeentakingplaceinvariouswavenumericalmodellingfieldssuchasacoustics,surface waterwaves,radarwaves,seismology,geophysicsandbiomedicalultrasound.Therelatedproblemsinthefrequencydomain aremodelledusingmainlytheHelmholtzequation [1–3],Maxwellequation [4]andNavierequation [5–8]dependingonthe wavepropagationmediumandthetypeofapplication.

Anumberofdifferentnumericalmethodshavebeenusedtosolvesuch problemsandthemostcommonlyusedisthe finiteelement method(FEM)dueto itsflexibility inhandlingcomplexgeometriesanditsabilityto modeldifferenttypes ofmedia. However, theuse oflow orderpolynomial basedfiniteelementsforsolving wave problemsrequires finemesh

∗ _{Correspondingauthor.Tel:+44(0)1314513100.}

E-mail addresses: [email protected] (M.S. Mahmood), [email protected](O. Laghrouche),[email protected] (J. Trevelyan),

[email protected](A.ElKacimi).

http://dx.doi.org/10.1016/j.apm.2017.05.013

gridswithmanynodalpointsperwavelength.Atmedium andhighfrequencies, thisleadstolarge memoryrequirements andhighCPUtime.Moreover,becauseoftheso-calledpollutionerror,evenfinermeshgridswouldberequiredtomaintain anacceptableaccuracy.

Thelasttwodecadessaw thedevelopmentofnumericaltechniquesaimingatmodellingwaveproblemswithimproved accuracyandreducedcomputationalcost,incomparisontotheconventionalFEM.Thesetechniquesinvolvethe incorpora-tionofa prioriknowledge ofthewavefield inthenumericalmodel.Indeed,thewave field isexpandedintosetsof ana-lyticalfunctions,forexample,intheformofplanewavespropagatingindifferentdirections.ForthecaseoftheHelmholtz equation, various methodshave beensuccessful inefficiently solving wave problems withcoarse mesh grids inboth 2D and3D withreducedcomputationaleffortandbetteraccuracy, incomparisontostandardpolynomial basedFEM.Among thesemethodsare the least-squares method [9],thepartition of unityfinite elementmethod (PUFEM) [10–16],the ultra weak variationalformulation (UWVF) [17,18],Plane wave discontinuousGalerkin (PWDG) [19], the generalizedfinite ele-mentmethod [20]andthediscontinuousenrichmentmethod(DEM) [21].Inthecaseofthe boundaryelementmethod,a partitionofunityboundaryelementmethod(PUBEM)hasbeendevelopedfortwoandthreedimensionalHelmholtz prob-lems [15,22]andanisogeometricwave-enrichedmethodwithin thecontextofboundaryelements(XIBEM)waspresented fortwoandthreedimensionalHelmholtzproblems [23,24].Othersolutionswerealsoconsideredsuchastheoscillatedfinite elementpolynomials [25]andinamorerecentworkthestablediscontinuousGalerkinmethod[26].Thevariationaltheory ofcomplexrays(VTCR) [27]andthephasereductionfiniteelementmethod(PR-FEM) [28]areotherdedicatedstrategiesfor solvingshortwaveproblemswithcoarsemeshgrids.

SomeofthesetechniqueshavebeenextendedtodealwithHelmholtzproblemsinnon-homogeneousmedia [1–3]and flowacoustics [29].Theywerealsoextendedtodealwithtwo-dimensionalelasticwaveproblems,forexample,PUBEM [30], UWVF [31],DEM [32]andPUFEM [5–7].Otherapplicationsincludefluid-solidinteraction [33],evanescentwaveproblemsat solid-fluidinterface [34],solid-solidinterface [35],acousticfieldsincavitieswithabsorbingmaterials [36]andsimulationof Biotswavesinporoelasticmaterials [37].Thesetechniqueshaverecentlybeenreviewedin [38]fortheHelmholtzequation andfurtherreferencesareindicated.

Inthecurrentwork,thePUFEMconceptisextendedtothree-dimensionaltime-harmonicelasticwaveproblems.Tothe authors’knowledge, only the works [39,40]have dealt withthree-dimensional plane wave enriched elementsfor elastic waveproblemswithinDEMandUWVFframeworks,respectively.InbothDEM [39]andUWVF [40]theapproximatingplane wavesare definedattheelementlevelandhenceinter-elementcontinuitymustbeenforcedintheformulation.However, inthepresentedPUFEMelasticmodel,theplanewavesdefinedatthenodesoftheelementsaremultipliedbytheLagrange polynomialshapefunctionsandhencecontinuityisautomaticallyensured.

Inthismodel,thecomponentsofthedisplacementfieldare firstexpressedassumsofbothpressure(P)andshear(S) wavecontributions,like inthetwo-dimensionalcase [5].The maindifferencewiththethree-dimensionalcaseconsistsin thedirectionofS-waveinduceddisplacement,whichisorthogonaltothedirectionofpropagation.Whileintwodimensions theorthogonaldirectioniswelldefinedviaasinglevector,inthree-dimensionstheorthogonaldirectioniscontainedwithin theorthogonalplanetothepropagationdirectionandthereforetwovectorsarerequiredforitsresolution.Furthermore,in twodimensions, thedirectionsofthe approximatingplanewavesare easily definedwhetherina uniformornonuniform distribution.This is not the casein three dimensionswhere the distributionof plane waves in space, either uniformor nonuniform,isnotstraightforward.

Indeed,intheimplementationofthethree-dimensionalversionofPUFEMtheuniformdistributionofdirectionsinspace isachallengingissue,unlikeinthetwo-dimensionalcase.Infact,theuniformdistributionofasetofpointsonaspherehas attractedtheattentionofresearchersfromvariousfieldsincludingmeteorologyandquantumtheory.Inthispaper,different approachesforuniformdistributionofdirectionsinspacearesummarised andthensome ofthemare usedtoobtain sets of linearlyindependent directions. Moreover, to keep the conditioning within acceptable limits, unbalanced numbers of directionsforthepressureandshearwavesareused.

Theoutlineofthepaperisasfollows.In Section2,themodelproblemandtheusednotationsarepresented. Section3is dedicated to thevariational formulation and Section 4presents the PUFEM approximated solution. In Section 5, various approaches to uniformly distribute sets of plane waves in space are presented. The PUFEM discrete model is given in

Sections6and 7 presentsavalidation ofthe modelagainst problemswithanalytical solutionssuch asprogressiveplane wavesinanelasticinfinitemediumandelasticwavescatteringbyasphericalrigidbody.Forthelatterproblem,a compari-sonbetweenPUFEMandthecommonlyusedloworderFEMisalsoattempted.Finallysomeconcludingremarksaredrawn in Section6.

2. Problem statement

Let

be the spatialdomain inR3 _{occupiedby an elastic medium. Letusdenote by (e}

1, e 2, e 3) thecartesian vector

system,and x =x1e 1+x2e 2+x3e 3 a genericpoint inR3.Thepropagation ofelastic wavesina3D homogeneousisotropic

mediumisgovernedbythewaveequation [41]

ρ∂

tt U−

∇

·

σ

(

U

)

=

ρ

F, (1)

classicalHooke’slaw

σ

(

U

)

=

λ

∇

·UI+

μ

∇

U+

(

∇

U

)

, (2)

where I isthe identitymatrixinR3×3 _{(3× 3matrix),}

_λ

_and

_μ

_{aretheLamé parametersoftheelastic material,assumed}

constant,and

∇

U =

(

∇

U1,

∇

U2,

∇

U3

)

.Wewilldenotehereby‘’thetransposeofagivenvectorortensor.Thedotproduct

‘·’of

∇

withatensorfield A =

(

A 1,A 2,A 3

)

inR3×3 isdefinedby

∇

·A=

_∇

_·A 1

∇

·A2

∇

·A3

. (3)

However,foravectorfieldinR3_{,‘·’istheusualdotproduct.}

Thecaseofharmonicmotionhaspracticalinterest.Supposethat F =exp

(

i

ω

t

)

f ,where

ω

isthecircularfrequencyand i=√−1istheimaginaryunitnumber.If U =exp

(

i

ω

t

)

u isasolutionof (1)thenthestrongformoftheprobleminterms of u canberewrittenasfollows

−

ρω

2_u−

_∇

_·

_σ

₍

_u

₎

₌

_ρ

_{f, in}

_{, (4)}

withthefollowingboundarycondition

σ

(

u

)

n=i

((

λ

+2

μ

)

k_P

(

u·n

)

n+

μ

k_S

(

u·t

)

t

)

+g, on

, (5)

where n and t denote,respectively,theoutwardunitnormalandtangentvectorstotheboundary

(

₌

∂

)and g isthe sourceterm. Detailsonhowthisboundaryconditionisdeduced aregiveninreference [31].Notethat (5)isan absorbing boundaryconditionofthefirstorderforelasticwaveequationswhen g =0on

.Theaimofthisworkconsiststovalidate thepresentednumericalmodel andthereforethe sourceterm g isintroducedanalytically inexpression (5)toenablethe solutionoftheproblem.

Itiswellknownfromtheelasticwavetheory [41]thatadisplacementfieldsolutionof (4)isasuperpositionoftwotypes ofpurelyoscillatorywaves:P-wave(primary,orcompressional)andS-wave(secondary,orshear).Thiscanbedemonstrated bywriting

∇

·

σ

(

u

)

=

(

λ

+2

μ

)

∇

(

∇

·u

)

−

μ

∇

×

∇

×u, (6)

ThePandSwavestravelwithspeeds

cP =

λ

+2

μ

ρ

and cS =

μ

ρ

. (7)

Thecorrespondingcompressionalwavenumberk_P andshearwavenumberk_S aregivenby

kP =_c

ω

P and kS =

ω

cS . (8)

Theselasttwo physicalquantitiesareusefulforthenumericalmodellingapproachthatwewilldiscussandinvestigatein thispaper.

3. Variational formulation

SincePUFEMisafiniteelementtypemethod,wewillneedtoestablishavariationalformulationforthetime-harmonic problem (4)and (5).WefirstintroducethefollowingSobolevspaces

V=

{

v∈H1

₍₎

3_;v=0on

}

_{. (9)}

Unless otherwise specified, standard notations of Sobolev spaces are adopted. Then the weak variational formulation of PUFEMcanbedescribedas:Find u ∈Vsuchthat

a

(

u,v¯

)

=b

(

v¯

)

,

∀

v¯∈V, (10)

wherev ¯isthecomplexconjugateofatestfunction v in V and a, b arebilinearandlinearformsdefinedrespectivelyas:

a

(

u,v¯

)

=−

ω

2

_ρ

u·v¯d

+

σ

(

u

)

:

∇

v¯d

−

σ

(

u

)

n·v¯d

,

b

(

v¯

)

=

ρ

f·v¯d

. (11)

Theproductoftwotensorfields A =

(

A 1,A 2,A 3

)

and B =

(

B 1,B 2,B 3

)

inR3×3isgivenby

Thecurloperator

∇

×isdefinedforavectorfield v =

(

v

1,

v

2,

v

3

)

by

∇

×v=

_∂

2

v

3−

∂

3

v

2

∂

3

v

1−

∂

1

v

3

∂

1

v

2−

∂

2

v

1

. (13)

Underthepreviousnotationsanddefinitions,thetensor

σ

canbewrittenasfollows

σ

(

u

)

=

λ

∇

·uI+

μ

[

∇

×u]×+2

μ

∇

u, (14)

wherethematrix[

∇

×u ]× isdefinedby

[

∇

×u]×=

₀

_∂

1u2−

∂

2u1

∂

1u3−

∂

3u1

∂

2u1−

∂

1u2 0

∂

2u3−

∂

3u2

∂

3u1−

∂

1u3

∂

3u2−

∂

2u3 0

, (15)

with u ₌

(

u1,u2,u3

)

.Letusnoticethattheproduct

σ

(

u

)

:

∇

v ¯ in (11)canbe written,using (14),undera forminvolving

theoperators

∇

,

∇

·and

∇

×:

σ

(

u

)

:

∇

v¯=

λ

(

∇

·u

)(

∇

·v¯

)

+2

μ

∇

u:

∇

v¯−

μ

(

∇

×u

)

·

(

∇

×v¯

)

. (16)

Thenthankstotheboundarycondition (5),werewrite a and b ofexpression (11)inthefollowingformulations:

a

(

u,¯v

)

=−

ω

2

_ρ

u·v¯d

+2

μ

∇

u:

∇

v¯d

−

μ

(

∇

×u

)

·

(

∇

×v¯

)

d

−

λ

(

∇

·u

)(

∇

·v¯

)

d

−i

((

λ

+2

μ

)

kP

(

u·n

)(

v¯·n

)

+

μ

kS

(

u·t

)(

v¯·t

))

d

,

b

(

v¯

)

=

ρ

f·v¯d

+

g·v¯d

. (17)

From the mathematical point of view, according to the boundary condition (5) on

, the problem in (10) and (17) is well-posed and an existence and uniqueness resultcan be established by virtue of Fredholm’s alternative theorem and continuation arguments [42,43], under weak assumptions, i.e., let us consider that

is a bounded Lipshitz domain,

g ∈H−12

()

×H−12

()

,andthat f ∈ V ∗,where V ∗denotesthetopologicaldualspaceof V .

4. PUFEM approximated solution

We now need to approximate the problem stated in expression (10) by the PUFEM, based on plane wave basis and polynomialshapefunctions.Forthispurpose,letusconsiderafiniteelementmeshcontainingnnodes,denotedz, z=1,n. Wedenoteby{Nz }thepartitionofunityby polynomialfiniteelementshapefunctions, andrespectivelybym_P andm_S the numbersofapproximatingPandSplanewaves.Followingthetwo-dimensionalPUFEMapproximation [5],thedisplacement field u isapproximatedbysuperimposingpressureandsheardisplacementsasfollows

uh =uP h +uS h . (18)

However,inthreedimensionstheshearwaveinduceddisplacementisresolvedintotwocomponentsasfollows

uS _h =u_h S,1+uS_h ,2. (19)

Therefore,using the plane wave enrichment approach,the approximated displacement field u _h is then expressed in the followingform

u_h = z=1,n

l=1,m P

Nz AP z,l exp

(

ikP x·dl P

)

dl P+

z=1,n

l=1,m S

Nz ASz,,1l exp

(

ikS x·dl S

)

dlS,,1⊥

+

z=1,n

l=1,m S

Nz ASz,,2l exp

(

ikS x·dl S

)

dlS,,2⊥. (20)

TheorthogonalunitS-vector d _S ,⊥,totheunit P-vector d ,isresolvedintotwoorthogonalvectorsdenotedby d lS,,1⊥ and d lS,,2⊥ andaredefinedby

dlS,,1⊥=d×P and d l,2

S,⊥=d

l,1

S,⊥×d×P , (21)

suchthat

dl _S

22=

d

l,1

S,⊥

22+

d

l,2

S,⊥

22, (22)

andthen

dl,1

S,⊥=

d×_P

dl S

2

and dl,2

S,⊥=

dlS,,1⊥×d×P

dl

S

2

Table1

Examplesofsetsofdirections(points)relativelyuniform(equallyspaced)onunitspherefordifferent ap-proaches.

Method Numberofdirections

Regularicosahedron[47] 12 42 162 642 Kuriharadistribution[44] 6 18 38 66 102 Cubeboundarymeshing[15] 8 26 58 98 154

CorrectedBeverlydistribution[48] 6 12 22 34 46 58 84 106 128 156

with

·

2 denoting the L2-norm. Note that, in3D, there is an infinite numberof vectorsthat are orthogonal to a given

vector.Asaresult,findingtheperpendicularvector d ×_P to d l

P canbechosenindifferentways,oneofthembeing

d×_P =

_{0 −1 1}

1 0 −1

−1 1 0

dl _P , (24)

where d l _P =

(

sin

θ

_l cos

φ

_l ,sin

θ

_l sin

φ

_l ,cos

θ

_l

)

,with

θ

_l and

φ

_l beingthetwoanglesofthesphericalcoordinatesystemwith0

≤

θ

l ≤

π

and0≤

φ

l ≤2

π

.Inthecaseof d lP withthesamecomponentssuchas d lP =

(

a,a,a

)

thentheperpendicularvector

isgivenby d ×_P =

(

−a/2,−a/2,a

)

.

Noticethat

∇

_×u p _{h =}0and

∇

_·u S_h ,1₌

∇

_·u S_h ,2₌0.Inexpression (20),hdescribesthecomputationalmeshsize.For sim-plicityofthenotations,thedependencyofthenumericalsolution u _h onthenumbersofapproximatingPandSplanewaves,

m_P andm_S ,isnotindicated.

The above approximation (20) can be derived fromthe Helmholtz decomposition theorem, as discussed in [30]. The unknownsarenolongerthenodalvaluesofthedisplacement u _h ,butarenowtheamplitudesA_zP _,l,AS_z,,_l 1andAS_z,_,2_l attachedto anodezandcorrespondingtoPandSplanewavestravellinginthedirections d l _Pand d l _S,respectively.

5. Choice of plane wave directions in space

Intwodimensions,thechoiceoftheplanewavesandtheiruniformspacingorclusteringarounddirectionsofpreference isatrivialproblem.Themovetothethreedimensionalcasepresentshoweversomedifficulties.Infact,inthreedimensions, it isneitherintuitive noran easy tasktodefine m uniformdivisionsofthe4

π

solid angle,mbeinganyintegernumber, a part froma few known solutions. In previous work [15], the authors used a uniformboundary meshing ofa cube. A reasonablywell spaced setofpointsis definedbythe vectorsjoiningthecentre ofthecube toeach node on thecube’s boundary.However,thisapproachisalsolimitedtoafewspecialcasesofmforwhichaboundary-meshedcubeisavailable. Specifically,thealgorithmislimitedtovaluesofm=p3₋

₍

_p−2

₎

3 _{forallp>1,thusgivingm=8,26,56,98,...etc.}

Theuniformdistributionofa setofpointson asphereattractedtheattentionofresearchersfromvariousfields.Early workinmeteorologicalmodelling,forexamplebyKurihara [44],resultedinalgorithmsthatproducedistributionsbasedon the mappingof a regular grid onan equilateraltriangleto each octant ofthe sphere.However, this islimited to values of m₌4p2_{+4p−2,where p+1 isthe number ofgrid pointsfromthe north poleto the equator.The icosahedral grid}

ofWilliamson [45] usestheverticesofthe regularicosahedron withthe possibilityofsubdividingeach triangularfacein a similar manner to the Kurihara grid [46] or in a symmetricalmanner such as performed in reference [47]. All these possibilitiesproduce distributions withlimitations onthe values ofm. Forexample,in reference [47], itis givenasm₌

10p2_{+2 where p=2}q _{withq ≥0 beingthe number ofequal intervalsinto which each side of theoriginal icosahedral}

trianglesisdivided.

With the motivation to define sets of plane waves forming bases forthe DEM, Grosu andHarari [48] corrected the techniqueofBeverly [49]findingequallyspacedpointsonlatitudinallinesofaspherewhicharethemselvesequallyspaced alongthelongitudinallines.Thistechniqueprovidessetsofapproximatelyuniformdistributionsofpointsonaspherewith morepossibilities,incomparisontothepreviouslycitedapproaches(see Table1).

Inthe caseoftheUWVF [40],a librarywasusedinwhich theselection ofm directionsinspace, with4≤ m≤ 130, wasbasedonminimising thedistancesbetweenpointsona surfaceofasphere.The recursivezonalequal-area partition proposedbyLeopardi [50]offersthepossibilitytogenerateanynumbermofpointsonthespherethroughthepartitioning intoregionsofequalareaandsmallequivalentdiameter.Inthisapproach,itisrequiredtofindthecolatitudesofpolarcaps (zonesbetweenadjacentlatitudinallines), thendeterminethe collaranglewhichprovidesthenumberofregions ineach collar.

More recently,Peake etal. [51] presenteda methodof uniformlyspacinga set ofpointson a spherebased on static equilibriumof chargedparticles and hencenamed it theCoulomb force method.This work ismotivated by theneed to use uniformwave directionsin spacewithin PUBEM. It isshownthat the performance ofthe enriched elements, within PUBEM,isnotsensitivetosmallvariationsintheuniformityoftheplanewavedirectiondistribution.Theinterestedreaders inmethodsdealingwithuniformdistributionofpointsonaspherearereferredtothereferencesincludedin [51].

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

Fig.1. Distributionexamplesof58pointsonthesurfaceofasphere:Cube-boundarymeshing(CM)[15](left),CorrectedBeverly(CB)[48](middle)and Leopardimethods[50](right).

consiststouse theCube-boundaryMeshing (CM) which waspreviouslyused inPUFEM for3D Helmholtz problems [15]. Thisallows anumberofdirectionsm=p3₋

₍

_p−2

₎

3 _{forallintegerp>1,whichgivesm=8,26,56,98,...etc.Forevenp}

numbers,twofurtherdirectionsareaddedpointingtothenorthandsouthpolesleadingtom=10,26,58,98,...etc. The secondapproachistheCorrectedBeverly(CB)method [48]whichconsiststofindequallyspacedpointsonlatitudinallines ofa spherewhich themselvesare equally spacedalong the longitudinallines. Theapproach sets thechord betweentwo adjacentpointsonthenthlatitudinallineequaltothechordbetweenthenthand

(

n₊1

)

thlatitudinallines.Thenumber

Nn ofpointsatthenthlatitudinallineisthengivenby

Nn =

π

/sin−1 sin_sin

(

₍

φ

_n0

_φ

/2

)

0

)

, (25)

where

φ

0isaconstantanglebetweentwoconsecutivelatitudinallines.

Fig.1 showsdistribution examples of 58 pointsobtainedby the cube-boundary meshing (left), the corrected Beverly method(middle)andtheLeopardimethod(right).Itisobviousthatthecube-boundarymeshingdistributionprovidespoints relativelymorespacedaroundtheequator incomparisontotheregions aroundthesouthandnorthpoles,whilethetwo othermethods providerelativelyuniformdistributions. Moreover,thecorrected Beverlymethodandthe Leopardimethod leadtoverysimilardistributionsform=6,12,22,34...etc.Ifthesouthcap,orthenorthcap,oftheLeopardimethodis rotatedbyacertainangle,bothmethodsleadtothesamedistribution.

6. PUFEM discretization

Inthiswork,thedisplacementfieldisapproximatedviapropagatingplanewavesinspaceinwhichbothbalanced(m_{P =} m_S )andunbalanced(m_P =m_S )choicesofplanewavesareconsideredinexpression (20)oftheapproximateddisplacement field.Letusset

p_l :=exp

(

ikPx·dl _P

)

dPl s1l :=exp

(

ikSx·dl _S

)

dl_S,_,1_⊥, s2_l :=exp

(

ikSx·dl _S

)

dl_S,_,2_⊥. (26)

Afiniteelementapproximationofthevariationalproblem (17)isderivedbyusingN_z p ¯_{l ,}N_z s ¯1

l andNz s ¯2l : −

ω

2

_ρ

uh ·

(

Nz pl

)

d

+

λ

(

∇

·uh

)(

∇

·

(

Nz pl

))

d

+2

μ

∇

uh :

∇

(

Nz p¯l

)

d

−

μ

(

∇

×uh

)

·

(

∇

×

(

Nz pl

))

d

−i

r

((

λ

+2

μ

)

kP

(

uh ·n

)(

Nz pl ·n

)

+

μ

kS

(

uh ·t

)(

Nz pl ·t

))

d

=

ρ

f·

(

Nz pl

)

d

+

g·

(

Nz pl

)

d

, l=1,mP , (27)

−

ω

2

_ρ

uh ·

(

Nz s

1

l

)

d

+

λ

(

∇

·uh

)(

∇

·

(

Nz s

1

l

))

d

+2

μ

∇

uh :

∇

(

Nz s¯

1

l

)

d

−

μ

(

∇

×uh

)

·

(

∇

×

(

Nz s

1

l

))

d

−i

r

((

λ

+2

μ

)

kP

(

uh ·n

)(

Nz s1l ·n

)

+

μ

kS

(

uh ·t

)(

Nz s1l ·t

))

d

=

ρ

f·

(

Nz s

1

l

)

d

+

g·

(

Nz s

1

−

ω

2

_ρ

uh ·

(

Nz s

2

l

)

d

+

λ

(

∇

·uh

)(

∇

·

(

Nz s

2

l

))

d

+2

μ

∇

uh :

∇

(

Nz s¯

2

l

)

d

−

μ

(

∇

×uh

)

·

(

∇

×

(

Nz s

2

l

))

d

−i

r

((

λ

+2

μ

)

k_P

(

u_h ·n

)(

Nz s2l ·n

)

+

μ

kS

(

uh ·t

)(

Nz s2l ·t

))

d

=

ρ

f·

(

Nz s

2

l

)

d

+

g·

(

Nz s

2

l

)

d

, l=1,mS . (29)

The mainfeatureofthe discretescheme (27),(28)and (29),thanksto thestress tensorform (14)andthenumerical ap-proximationofthedisplacementfield (20),isthattheresultingmatricescanbeeasilycomputedasinthecaseofthescalar Helmholtzequation [12],intermsoftheoscillatoryshapefunctions

NP _z,l :=exp

(

ikP x·dl P

)

Nz and NS z,l :=exp

(

ikS x·dl S

)

Nz . (30)

Indeed,asNz p _l =N_zP _,l d l _P ,Nz s 1

l =NzS ,ld l,1

S,⊥andNz s 2l =NzS ,ld l,2

S,⊥,themassmatrixcoefficientsaresuchthat

(

Nz pl

)

·

(

Nz pl

)

d

=d l P ·dl P

N P z ,l N¯P z,l d

,

(

Nz s

1

l

)

·

(

Nz pl

)

d

=dl S,,1⊥·dl P

N S z ,l N¯zP ,l d

,

(

Nz s

2

l

)

·

(

Nz pl

)

d

=dl S,,2⊥·dl P

N S z ,l N¯zP ,l d

,

(

Nz s

1

l

)

·

(

Nz s1l

)

d

=dSl ,,1⊥·dlS,,1⊥

N S z ,l N¯zS ,l d

,

(

Nz s

2

l

)

·

(

Nz s2l

)

d

=dSl ,,2⊥·dlS,,2⊥

N S z ,l N¯zS ,l d

,

(

Nz s

1

l

)

·

(

Nz s2l

)

d

=dSl ,,1⊥·dlS,,2⊥

N S z ,l N¯zS ,l d

,

(

Nz s

2

l

)

·

(

Nz s1l

)

d

=dSl ,,2⊥·dlS,,1⊥

N S

z ,l N¯zS,l d

. (31)

Theothermatricesinvolvedin (27),(28)and (29)canbewritten,inasimilarwayasforthemassmatrix,intermsofthe productofthenewoscillatoryshapefunctions.Thenumericalevaluationoftheaboveintegralsisperformedwithhigh-order Gauss–Legendrescheme.

Anodalpointz,withz₌1_,n_,hasm_{P +}2m_S degreesoffreedom.Sothetotal numberofdegreesoffreedomisn

(

m_{P +}

2m_S

)

, andtheunknown amplitudesAP _z,l, AS_z,_,1_l andA_zS,_,2_l forz=1,n canbe storedinn blocksofvectorsin Rm P+2m S ofthe form

(

A P z,A S z 1,A z S 2

)

.Theglobalunknownamplitudevectorcanbestoredasfollows

A=

(

AP ₁,AS 1

1

,AS 2

1

,...,AP _n ,AS 1

n ,AS n 2

)

, (32)

andtheglobalmatrixresultingfrom (27),(28)and (29)hasthefollowingform

W=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

WP_1,,P₁ WP,S 1

1,1 W

P,S 2

1,1 . . . W

P,P

1,n W1P,,S n 1 WP1,,S n 2

WS 1,P

1,1 W

S 1,S 1

1,1 W

S 1,S 2

1,1 . . . W

S 1,P

1,n W1S 1,,n S 1 WS 11,n ,S 2

WS 2,P

1,1 W

S 2,S 1

1,1 W

S 2,S 2

1,1 . . . W

S 2,P

1,n W1S 2,,n S 1 WS 12,n ,S 2

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

WP,P n,1 W

P,S 1

n,1 W

P,S 2

n,1 . . . W

P,P

n,n WnP,,S n 1 WPn,,S n 2

WS 1,P

n,1 W

S 1,S 1

n,1 W

S 1,S 2

n,1 . . . W

S 1,P

n,n WnS 1,,n S 1 WS n1,n ,S 2

WS 2,P

n,1 WS n2,1,S 1 WnS 2,,1S 2 . . . WnS 2,n ,P WnS 2,,n S 1 WS n2,n ,S 2

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

, (33)

wheretheblocks W Pz,,z P , W S z1,z ,S 1, W S z,2z ,S 2 aresquarematricesinRm P×m P,Rm S×m S _andRm S×m S,_{respectively,withz},_z=₁,_n.The

blocks W P,S1

z,z , W Pz,,z S2, W Sz,1z ,Pand W Sz2,z ,Pareeventually,ifmP =mS ,rectangularmatricesinRm P×m S _orRm S×m P_{.Last,theblocks} W S 1,S 2

0 1

2 3

4

0 1 2 3 4 0 1 2 3 4

0 1

2 3

4

0 1 2 3 4 0 1 2 3 4

0 1

2 3

4

0 1 2 3 4 0 1 2 3 4

Fig.2. Consideredmeshgrids:(left)h1,(middle)h1 ₂ and(right)h1 ₄ .

Table2

Valuesofh/λS for variousmeshsizes andcircularfrequencies.

ω 1 10 15 20

h1 0.32 3.18 4.77 6.37 h1 ₂ 0.16 1.59 2.39 3.18

h1

4 0.08 0.79 1.19 1.59

Galerkinweighting is used inthis work andhence theglobal matrixof the resultingsystem issymmetric andblock banded. It is stored using a steering vector to locate the elements andthe solution of the final system is obtained via a directsolver [52] based on LDLT decomposition where LT is the transposeofthe lower triangular matrixL andD is a diagonalmatrix.

7. Numerical analysis

Inthis section,the developed numericalmodel forselected elasticwave problems is investigatedwithrespect to the computationalmeshsizehandtheplanewaveenrichment,intermsofthenumbersm_P andm_S oftheapproximatingPand Splanewaves.Intheconvergenceanalysisconsideredbelow,specialcareistakensothattheincidentwavedirection d inc _P

isnottooclosetotheplanewavedirectionsusedintheapproximatingmodel.Thecomputationaldomainisdefinedby

=

{

x=

(

x1,x2,x3

)

;1≤x1,x2,x3≤3

}

. (34)

Allparameters a,

λ

,

μ

and

ρ

aretakenequalto1withtheirrespectivecorrespondingunits.Thefiniteelementsusedhere are 8-node hexahedral elements, for which thegeometry is interpolatedvia linearLagrange polynomials. Three uniform mesh grids are considered. They are denoted by h1, h1

2

and h1 4,

see Fig. 2. The coarser mesh h1 has 8 elements. It is

hierarchicallyrefinedto obtainh1

2 andh14 mesh gridswith64 and512elements, respectively.The EuclideannorminR

3

oftherealpartofthecomputeddisplacementfield u _h isdenotedby|Re(u _h )|andtheaccuracyofthenumericalsolutionis measuredbythefollowingL2 error

ε

2=

||

u_h −u

||

_L 2()

||

u

||

L 2()

×100% (35)

where u representstheanalyticalsolutionoftheconsideredproblem.

Toquantifythenumberofdegreesoffreedom(DOF) perPandS wavelengths,respectively,the parameters

τ

_P and

τ

_S areintroducedinthefollowingforms [5]

τ

P = _k

π

P

(

nmP

)

1/3 _and

_τ

S =_k

π

S

(

n mS

)

1/3_{, (36)}

wheren,asmentionedbefore,isthenumberofnodalpointsinthemeshgrid.

Letusdenote by

α

the average rateofconvergence withrespect to hsuch that the L2 error

ε

2=O

(

hα

)

,where

α

is

tobeevaluatednumericallyvia

α

=

(

log

ε

₂(1/4)−log

ε

₂(1/2)

)

/

(

logh1/4−logh1/2

)

,whenconsideringmeshgridsh1/4andh1/2,

forexample. Table2showsthenumberofshearwavelengths

λ

_S perelementsizehoftheconsideredmeshgrids(Fig.2), forcircularfrequencies

ω

=1,10,15and20.Thecomparisonofthemeshsizewiththeshearwavelengthrevealsthatfor lowcircularfrequency

ω

,eachfiniteelement containsafractionof

λ

_S whereas athigher

ω

,each finiteelementbecomes multi-wavelengthsized.Forexample,forthemeshgridh1

2,

Table3

Progressiveplanewaveproblem:L2errorin%forω=10.

(mp,ms) (10,10) (26,26) (58,58) (98,98)

h1 83.6 14.4 0.964 0.0377 h1

2 3.93 0.50 0.0032 0.0019

h1

4 0.237 0.0123 0.00002 0.00002

α 4.2 5.1 7.8 5.4

Table4

Progressiveplanewaveproblem:L2errorin%forω=15.

(mp,ms) (10,10) (26,26) (58,58) (98,98)

h1 109.4 105.5 2.31 0.213 h1

2 44.4 1.13 0.097 0.019

h1

4 1.59 0.205 0.00066 0.00017

α 3.1 4.5 5.9 5.1

Table5

Progressiveplanewaveproblem:L2errorin%forω=20.

(mp,ms) (10,10) (26,26) (58,58) (98,98)

h1 113.6 103.4 76.3 40.2 h1

2 102.5 26.5 0.973 0.310

h1

4 3.91 0.711 0.0076 0.00089

α 2.4 3.6 6.7 7.7

theempiricalexpressiongivingn_{gauss =}10_×

h_/

λ

_S

₊2_,toensureenoughintegrationpointsareused.Itispossibleto de-velopsemi-analyticalintegrationschemessuchasdone in [6,53]toavoidtheburdenofthehighorderscheme,butthisis nottheobjectiveofthecurrentwork.AllcomputationsarecarriedoutinFortranwithdouble-precisioncomplexnumbers onWorkstationIntelCorei73.5GHzwith32GBRAM.

7.1. Progressiveplanewavetestproblem

Letusfirstconsiderasimplemathematicalmodelconsistingofaprogressiveplanewaveinathreedimensionalinfinite elasticmedium.Thisproblemhasananalyticalsolutionandtheexpressionofthedisplacementfieldisgivenby

u=exp

(

ikPx·dinc P

)

dinc P +exp

(

ikS x·dS

)

d1S,⊥+exp

(

ikS x·dS

)

d2S,⊥, (37) where d inc _P ischosensuchthat

(

dinc

)

=

(

sin

β

1cos

β

2,sin

β

1sin

β

2cos

β

1

)

,

β

1=33.33o ,

β

2=333.33o , (38)

andthetwocomponents d 1

S,⊥ and d 2S,⊥ofvector d S orthogonaltothedirection d inc P aregivenby

d1

S,⊥=

(

dinc P

)

× and d2S,⊥=d1S,⊥×dinc P , (39)

where

(

d inc _P

)

×isdefinedinexpression (24).

Thesourceterm g ofexpression (5)iscalculatedfromthetraction

σ

(u )n on

,evaluatedwiththeanalyticalsolution u

giveninexpression (37).Notethatthisisonlypossiblewhenexactboundaryconditionsareprescribedon

.

Tables 3–5 show theL2 errorin % aswell asthe average rateof convergence

α

for the threeconsidered mesh grids

h1,h1

2 andh14 whenthenumbers (mP ,mS )oftheapproximating planewavesare increasedforthe valuesofthecircular

frequency

ω

=10,15and30,respectively.Notethatnoneofthesetsofapproximatingplanewavescontainstheconsidered directionoftheimposedprogressiveplanewaveofexpression (38).

Fromthenumericalresultsitisobviousthath-refinementimprovestheaccuracyoftheschemeforallcasesof frequen-ciesandplane waveenrichments. Increasingthenumbers ofapproximatingplanewaves,forafixed h,doesalsoimprove theaccuracyoftheplanewaveenrichmentmodel.Infact,thisisexpectedasboththemeshrefinementandtheincreasein thenumberofapproximatingplanewavesleadtohighernumbers

τ

_P and

τ

_S ofDOFperPandSwavelengths,respectively. Nevertheless,tworemarksare worthmentioning.Thefirst oneconcernsthecasesofcoarsemeshgrids,incomparisonto the wavelength,withlownumbers ofapproximating plane waves.Insuch cases,theL2 errorishighandthereason lies

in thefact that the discretizationlevel represented bythe numbers ofDOF per P- andS-wavelength,

τ

_P and

τ

_S , isvery low.Forexample,forthemeshgridh1

2 at

ω

=15,theL2 errorisvery high,

ε

2=44.4%,for

(

mP ,mS

)

=

(

10,10

)

forwhich

(

τ

P ,

τ

S

)

=

(

3.1,1.8

)

.Forahigherenrichment,

(

mP ,mS

)

=

(

26,26

)

theL2 errorisa lotlowerincomparisontotheprevious

to

(

τ

_{P ,}

τ

_S

)

₌

(

5_.6_,3_.2

)

_,for

(

m_{P ,}m_S

)

₌

(

58_,58

)

_,theL2errorisevenlower,

ε

2=0.097%,andthenfor

(

mP ,mS

)

=

(

98,98

)

the L2 errorreducesto

ε

2=0.019%.

Thesecond remarkisrelatedtotheaveragerateofconvergence

α

.Fromthe three Tables2–5,forfixed numbers(m_P , m_S )ofapproximatingplanewaves,theresultsshowanexponentialdecreaseoftheL2errorwithrespecttoh-refinement.In

general,theaveragerate

α

increaseswiththeincreaseofthenumbersofapproximatingplanewaves,suchasshownforthe caseof

ω

=20.However,continuingtoincrease thenumberofapproximatingplane waves,fora fixedcircularfrequency, doesnotnecessarilyleadtoanincreaseintherateofconvergence.Thisisthecasefor(mP,mS)=(98,98),at

ω

=10and15

wheretheratehasdecreasedfrom7.8to5.4andfrom5.9to5.1,respectively.Thisismostlikelyduetotheillconditioning issue,whichisaninherentfeatureoftheplanewavebasisfiniteelementapproaches.Thisaspectisinvestigatedinthenext testexampledealingwithelasticwavescatteringbyarigidsphere.

7.2.Wavescatteringtestproblem

Theproblemofharmonicwavesscatteredbya sphericalrigidbody containedinathreedimensionalhomogenousand isotropicinfinite elasticspace isconsidered. An upward P-wavewith amplitude

0,for whichthe thepotential isgiven

by

=

0exp

(

−ikP x3

)

exp

(

i

ω

t

)

, (40)

where

0= _k i_p,impingesonasphericalbodyofradiusa.Theincidentdisplacementfieldisobtainedfrom

uin₌

_∇

_{. (41)}

Thetotaldisplacementfield u isasuperpositionoftheincident u in andscattered u sc fields, u ₌u in ₊u sc .Infact,aP-wave impingingonarigidbodywouldleadtobothscatteredPandSwaves,inaninfiniteelasticmedium.Thetotaldisplacement canbewrittenintermsofitscomponents

u=ur er +uθeθ+uφeφ, (42)

withrespecttothesphericalunitvectors e r , e _θ and e _φ with

ur =1_r

ν=0,∞

0

ε

₁p(0)

(

ν

,r

)

+Aν

ε

₁p(1)

(

ν

,r

)

+Bν

ε

₁s (1)

(

ν

,r

)

P_ν

(

cos

θ

)

, (43)

u_θ=1

r

ν=0,∞

0

ε

₂p(0)

(

ν

,r

)

+Aν

ε

₂p(1)

(

ν

,r

)

+Bν

ε

₂s (1)

(

ν

,r

)

dP_ν

(

cos

θ

)

d

θ

, (44)

u_φ=0, (45)

whereP_ν areLegendrepolynomialsfor

ν

=0,∞.TheconstantsA_ν andB_ν arechosensuchthat u

|

r=a =0representingthe totalreflectionconditionontheboundary

ofthesphericalrigidbody.Thisisobtainedvia

0

ε

₁p(0)

(

ν

,a

)

+Aν

ε

₁p(1)

(

ν

,a

)

+Bν

ε

₁s (1)

(

ν

,a

)

=0,

0

ε

2p(0)

(

ν

,a

)

+Aν

ε

p(1)

2

(

ν

,a

)

+Bν

ε

s ( 1)

2

(

ν

,a

)

=0, (46)

forall

ν

inN.The associatedstress field

σ

(u )is thenobtainedfromthedisplacement field (42)intermsofits spherical components

σ

rr = 2_r

μ

₂

ν=0,∞

0

ε

₃p(0)

(

ν

,r

)

+Aν

ε

₃p(1)

(

ν

,r

)

+Bν

ε

₃s (1)

(

ν

,r

)

P_ν

(

cos

θ

)

, (47)

σ

rθ = 2

μ

r2

ν=0,∞

0

ε

4p(0)

(

ν

,r

)

+Aν

ε

p(1)

4

(

ν

,r

)

+Bν

ε

s (1)

4

(

ν

,r

)

dP_ν

(

cos

θ

)

d

θ

, (48)

σ

θθ = _r

λ

2

ν=0,∞

0

ε

5p(0)

(

ν

,r

)

+Aν

ε

p(1)

51

(

ν

,r

)

P_ν

(

cos

θ

)

+2

μ

r2

ν=0,∞

0

ε

₆p(0)

(

ν

,r

)

+Aν

ε

₆p(1)

(

ν

,r

)

+Bν

ε

₆s (1)

(

ν

,r

)

P_ν

(

cos

θ

)

+2

μ

r2

ν=0,∞

0

ε

7p(0)

(

ν

,r

)

+Aν

ε

p(1)

7

(

ν

,r

)

+Bν

ε

s (1)

7

(

ν

,r

)

dP_ν

(

cos

θ

)

and

σ

φφ = _r

λ

2

ν=0,∞

0

ε

5p(0)

(

ν

,r

)

+Aν

ε

p(1)

5

(

ν

,r

)

P_ν

(

cos

θ

)

+2

μ

r2

ν=0,∞

0

ε

₈p(0)

(

ν

,r

)

+Aν

ε

₈p(1)

(

ν

,r

)

+Bν

ε

₈s (1)

(

ν

,r

)

P_ν

(

cos

θ

)

+2

μ

r2

ν=0,∞

0

ε

9p(0)

(

ν

,r

)

+Aν

ε

p(1)

9

(

ν

,r

)

+Bν

ε

s (1)

9

(

ν

,r

)

dP_ν

(

cos

θ

)

d

θ

. (50)

The symbols

ε

_ip(0),

ε

p(1) i and

ε

s (

1)

i ,fori=1,9, arecoefficientsobtainedformthe variousderivatives.The radialspherical

functionsaredefinedas,see [54]:

ε

p(i )

1

(

ν

,r

)

=A

{

n

ξ

p(i )

ν

(

kP r

)

−kP r

ξ

νp(i )

(

kP r

)

}

,

ε

p(i )

2

(

ν

,r

)

=A

{

ξ

p(i ) ν

(

kP r

)

}

,

ε

p(i )

3

(

ν

,r

)

=A

n2_−n−1

2

(

kS r

)

2

_ξ

p(i )

ν

(

kP r

)

+2kP r

ξ

_νp₊(i 1)

(

kP r

)

,

ε

p(i )

4

(

ν

,r

)

=A

{

(

n−1

)

2

ξ

p(i )

ν

(

kP r

)

−kP r

ξ

_νp₊(i 1)

(

kP r

)

}

,

ε

p(i )

5

(

ν

,r

)

=A

{

−k 2

P

ξ

νp(i )

(

kP r

)

}

,

ε

p(i )

6

(

ν

,r

)

=A

{

−n 2

_ξ

p(i )

ν

(

kP r

)

−kP r

ξ

_νp₊(i 1)

(

kP r

)

}

,

ε

p(i )

7

(

ν

,r

)

=A

{

−cot

(

θ

)

ξ

p(i ) ν

(

kP r

)

}

,

ε

p(i )

8

(

ν

,r

)

=A

{

n 2

_ξ

p(i )

ν

(

kPr

)

−kPr

ξ

_νp₊(i ₁)

(

kPr

)

}

,

ε

p(i )

9

(

ν

,r

)

=A

{

cot

(

θ

)

ξ

p(i )

ν

(

kP r

)

}

, (51)

wherei₌0ori₌1suchthat

ξ

_νp(0)

(

k_P r

)

₌J_ν

(

k_P r

)

isthesphericalBesselfunctionoffirstkindandordernandP0

n

(

cos

θ

)

is

Legendre’spolynomialofordernanddegreem₌0.A₌

(

₋i

)

ν

(

2n₊1

)

and

ξ

_νp(1)

(

k_P r

)

₌H_ν

(

k_P r

)

_,A₌1

ε

s (1)

1

(

ν

,r

)

=−n

(

n+1

)

Hν

(

kS r

)

,

ε

s (1)

2

(

ν

,r

)

=−n

(

n+1

)

Hν

(

kS r

)

+kS rHν+1

(

kS r

)

,

ε

s (1)

3

(

ν

,r

)

=−n

(

n+1

)

{

(

n−1

)

Hν

(

kS r

)

−kS rHν+1

(

kS r

)

}

,

ε

s (1)

4

(

ν

,r

)

=−

n2_−n−1

2

(

kS r

)

2

_H

ν

(

k_S r

)

−k_P rH_ν+1

(

kS r

)

,

ε

s (1)

5

(

ν

,r

)

=−n

(

n+1

)

Hν

(

kS r

)

,

ε

s (1)

6

(

ν

,r

)

=−n

(

n+1

)

{

(

n+1

)

Hν

(

kS r

)

−kS rHν+1

(

kS r

)

}

,

ε

s(1)

7

(

ν

,r

)

=−cot

(

θ

)

{

(

n+1

)

Hν

(

kS r

)

−kS rHν+1

(

kS r

)

}

,

ε

s (1)

8

(

ν

,r

)

=−n

(

n+1

)

Hν

(

kSr

)

,

ε

s (1)

9

(

ν

,r

)

=cot

(

θ

)

{

(

n+1

)

Hν

(

kS r

)

−kS rHν+1

(

kS r

)

}

. (52)

Thestressfieldisthenusedintheevaluationofthesourceterm g ofexpression (5)tobeusedintheboundarycondition oftheelasticwavescatteringproblem.Theseriesusedinexpressions (43)and (44)aretruncatedwhenasufficientnumber

Nt ofterms isincluded.Thisnumber istakensuch that Nt ≥kSR whereR isthe radiusofthe mostdistant pointof the

considered computational domain [55].Forillustrationpurpose, Fig.3showscontour plotsofthe componentsof|Re(u h)|

in the computational domain forthe case of

ω

=40 when a P-wave travelling upward along the x3-axis impingeson a

sphericalrigidobjectofunitradius.ThecentreoftherigidscatterercoincideswiththecentreoftheCartesiansystemand hencetheconsideredcomputationaldomainisinthevicinityofthescatteringobject.Thescatteredfieldcontainsbothbody waves,PandS,whichinterferearoundthehardscatterer.Thefirstandsecondcomponentsareoppositeeachotherandthe thirdcomponenthashighervaluesduetotheincidentwavepropagatingalongx3-axis.Toobtainthecontourplotsof Fig.3,

thecomputednodalvaluesfromtheproblemsolutionareusedinconjunctionwiththemodelofexpression (20).

Thesamenumericalanalysisisfollowedforthiselasticwavescatteringproblemconsideringthesameparametersalready usedforthecaseoftheprogressiveplanewavetestproblembutwithmoreoptionsregardingtheusedsetsof approximat-ing plane waves.Indeed, inthiscase, plane wave distributions obtainedformthe Cube-boundaryMeshing (CM) andthe Corrected Beverly(CB)methodareused.Thetwo approaches,CMandCB,offermoreflexibilityinthechoiceofuniformly distributedplanewavesinspaceandevenwhentheyallowthesamenumberofplanewavestheCM andCBdistributions aredifferent. Tables6–8summarisetheerroranalysisresultsforthemeshgridsh1,h1

Fig.3. Elasticwavescatteringbyarigidsphere,contourplots(a),(b)and(c)offirst,secondandthirdcomponentsofRe(uh)respectively,ω=40.

approximatingplanewavesareincreased,for

ω

=10,15and20,respectively.Theyalsogivetheaveragerateofconvergence computednumericallyfromtheobtainedL2errors.

From Tables6–8,itcanbeseenthat,ingeneral,theobtainedresultsleadtosimilarconclusionsalreadydrawnfromthe caseofthe progressiveplanewave test problem.Basically,the L2 errorimproves asthe numbers(mP ,mS ) ofthe

approx-imatingPandS plane wavesincrease forall casesofmesh grids andfrequencies presentedhere. However, we noticein certaincasesafterreachingalowL2 errorafurtherincreaseinthenumberofenrichingplane wavesdoesnotnecessarily

improvetheaccuracysuch asreflectedinthecaseofh1 4

and

ω

=10whenincreasing(m_P ,m_S )form(58,58)to(106,106). Inallthesecases,theL2errorremainedunchangedat

ε

2=0.00002%.Forfrequencies

ω

=15and

ω

=20,thisisagainseen

forthemeshgridh1

4 when(mP ,mS )increasedfrom(98,98)to(106,106)andforbothenrichmentstheL2errorstagnated

at

ε

2=0.00004%and0.00006%,respectively.Moreover,inthecaseoffrequency

ω

=20andthemeshgridsh1 andh1 2,for

highnumbers ofenrichingplanewavesthequality oftheresultsdeterioratesinsteadofimproving,oratleaststagnating. Thisisseenwhen(m_P ,m_S )increasedfrom(98,98)to (106,106), forwhichtheL2 errorincreasedfrom4.00% to7.19%in

thecaseofmeshgridh1andincreasedslightlyfrom0.0022%to0.0024%forthemeshgridh1

2.Thisisknowntobecaused

bytheillconditioningaspect,whichisinvestigatedhere.