Perfectly matched layers for flexural waves: An exact analytical model

journalhomepage:www.elsevier.com/locate/ijsolstr

Perfectly

matched

layers

for

flexural

waves:

An

exact

analytical

model

M.

Morvaridi

a

_,

_M.

_Brun

b,∗

a Dipartimento di Ingegneria Civile ed Architettura, Università di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy

b Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Università di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 20 May 2016 Revised 15 October 2016 Available online xxx Keywords: Flexural waves Geometric transformation Perfectly matched layers Cloaking

Harmonic analysis Transient analysis

a

b

s

t

r

a

c

t

InthispaperwepresentananalyticalmodelofPerfectlyMatchedLayersforflexuralwaveswithin elon-gatedbeamstructures.Themodelisbasedontransformationopticstechniquesanditisshowntowork bothintimeharmonicandtransientregimes.Acomparisonbetweenflexuralandlongitudinalwavesis detailedand it isshownthatthe bending problemrequires specialinterfaceconditions.Aconnection withtransformationofeigenfrequenciesandeigenmodesisgivenandtheeffectoftheadditional bound-aryconditionsintroducedattheborderofthePerfectlyMatchedLayerdomainisdiscussedindetailed. SuchamodelisparticularlyusefulforFiniteElementanalysespertainingpropagatingflexuralwavesin infinitedomain.

© 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Inengineeringapplications thenecessityto modelunbounded domains is oftenrequired.This isparticular importantin model-ingofsoil-structureinteraction(LeeandTassoulas,2011;Tassoulas andKausel,1983;Clouteauetal.,2013),fluid-solidinteraction(Hu etal.,2016;Romeroetal.,2015),ground-bornenoiseandvibration emitted by transportation systems (Clouteauet al., 2000; Steele, 2001),geophysics(Gelietal.,1988; Robinovichetal., 2011), non-destructive evaluation methods (Kim and Kim, 2009; Parvanova etal., 2014), fluid-dynamicsandtrafficflow (LoandWang,2005) andgeneralproblemsofwave propagation(electromagnetic, elas-tic, acoustics, seismic). The list includes also hydro- and aerody-namic problems (external flows, duct flows, reacting flows, jets, boundarylayers,free surfaceswithaerospace,marine/naval, auto-motive,meteorological,industrialandenvironmentalapplications), flows in porous media, filtration (with applications to oil recov-ery), magneto-hydrodynamicflows,plasma(e.g.,solarwind)just tonameafew.

In order to keep the computationfeasible there is the neces-sitytotruncate themodels withinafinitecomputationaldomain. This can be done by the boundary integral methods, infinite el-ements, non-reflecting boundary conditions andabsorbing layers. Theboundaryelementmethod(see,forexample,themonographs (Bonnet, 1999; Wrobel, 2002; Aliabadi, 2002; Beer, 2001; Gaul etal., 2003) canbe used directlyforexterior problemsovera

fi-∗ _{Corresponding author. Fax: +390706755418.}

E-mail address: mbrun@unica.it (M. Brun).

nite region. It is an efficient numerical techniqueformulated for bothstatic anddynamicproblems,whichis computationallycost effectiveinview ofthefact that itreduces thedimensionalityof the problem and only the boundary of the domain needs to be discretized.Onthecontrary,itismoredifficulttoimplementwith respecttoFiniteElementandFiniteDifferencealgorithmsandthe couplingwithdifferentnumericalschemes requiresspecial atten-tion.Also,someoftheadvantagesofthemethodarelostand addi-tionaldifficultiesarisefornon-linearproblemsinplasticity(Telles and Brebbia, 1979; Polizzotto, 1988; Maier et al., 1995; Bertoldi et al., 2005) and finite elasticity (Phan-Thien, 1988; Novati and Brebbia,1982;Polizzotto,2000;Brunetal.,2003a,b).

Infiniteelementschemes(BettessandZienkiewicz,1977; Bett-ess,1992)representthedomaininits entiretybyusingelements of infinite extent where the shape functions include outwardly propagatingwave-likefactors.Theformulationmayrelyona trun-catedmultipoleexpansion (Burnett, 1994; Leis, 1986),that incor-poratesfrequencydependentinterpolationfunctionsalongthe ra-dial(outward)direction(Gerdes,2000;Ihlenburg,1998).However, infiniteelementshaveproblemsofaccuracyandunwanted bound-ary reflections in the case of the propagation of guided or bulk waves (Liu andQuek, 2003; Cremers and Fyfe,1995; Astley and Hamilton,2006).Also,aregionmuchlargerthantheregionof in-terestmustbeimplementedinordertoachieveaccuracy.

Absorbing boundary conditions (ABCs) and perfectly matched layers (PMLs) permit outward propagating waves and must sup-press spurious reflections at least to an acceptable level. ABCs werefirstintroducedinLysmerandKuhlemeyer(1969),whereitis shownthat, forsecondorderwave equationsandlinearized

shal-http://dx.doi.org/10.1016/j.ijsolstr.2016.10.024 0020-7683/© 2016 Elsevier Ltd. All rights reserved.

Fig. 1. Transient Finite Element analysis. Transverse displacement at t = 0 . 5 s in a beam subjected to a transient force F ( t ) applied at X = 0 m. F ( t ) is shown in Fig. 8 . Dashed black line shows the displacement in a large domain without PMLs, contin- uous gray line shows the displacement in a short domain with PMLs.

low water equations, exact conditions are expressed in term of integro-differential equations which are then approximated by a hierarchicalsystemofdifferentialequations(Nataf,2013).

Instead ofad-hoc boundaryconditionsPMLis an area border-ing the computational domain where waves are damped so that propagatingwavesbecomeevanescent.Thekeypropertyisalways theabsenceofreflectionattheinterfacebetweenthephysicaland theabsorbingdomains. Theproblemofreflectionattheinterface betweenthe physical domain and the absorbing one wassolved inBerenger(1994)forproblemsgovernedbyHelmholtzequations. ThePMLshavebecomethemostpopularabsorbingconditionsfor finite-differencetime-domainandfiniteelementwavesimulations and many examples demonstrate their superior performance as comparedto ‘sponge-layer’absorbing boundarycondition(Cerjan et al., 1985; Sochacki et al., 1987), paraxial conditions (Higdon, 1991;Quarteronietal.,1998),asymptoticlocalornon-local opera-tors(Givoli,1991;HagstromandHariharan,1998).

PMLs correspond to a coordinate transformationin whichthe coordinatenormaltotheartificialboundaryismappedtocomplex valuesleadingtodecayingamplitudebehavior(ChewandWeedon, 1994).Theycanbealsointerpretedasaviscous anisotropic mate-rial in the boundary region (Rappaport, 1996). It is highly effec-tive in absorbing waves over wide ranges offrequency and inci-denceangles, isnumerically stableandneeds relatively thin lay-ers.PMLwasfirstdevelopedforelectromagneticwaves(Berenger, 1994;ChewandWeedon,1994),andthenextendedtothefieldsof acoustics(QiandGeers,1998),seismology(KomatitschandTromp, 2003; Kristek et al., 2009), dispersive waves (Lancioni, 2012) as well asto elasticwaves (Hastings et al., 1996; Songetal., 2005; Nataf, 2005; Zheng and Huang, 2002). Surprisingly, applications toflexuralwaves,governed byfourth-order differentialequations, arelimitedto a recentresult, whichis focused onthe numerical implementation(Farzanian andArbabi,2014). Thepurposeofthe workistofillthisgap.

InChangetal.(2014)theconstructionofPMLsforelasticwave propagationwaslinkedtoconformalmappingtechniquesadopted forthedesignofinvisibilitycloaks(Miltonetal.,2006;Brunetal., 2009;NorrisandShuvalov,2011),atechniqueimplementedinthe numericalsimulationgiveninBrunetal.(2009).Herewe imple-menta similar approachforthe problemofflexuralwaves (Brun etal.,2014a;Colquittetal.,2014;Jonesetal.,2015).

The model presented isstudied in thetime-harmonic regime. However, the simplecomputation givenin Fig. 1 showsthat the proposed PMLperforms well alsoin the transientregime. Inthe paperwe comparethe caseoflongitudinal andtransverse waves withinan elongatedbeaminorderto stresstheadditionalissues

Fig. 2. Beam structure. The displacement at point X and time t is U = (U , V , W) .

associatedtotheflexuralcase.Also,wegiveparticularimportance tothephysicalinterpretationofthePML.

Thestructureofthepaperisasfollows.InSection2wepresent the transformation optics technique for longitudinal waves in a thinrod,wedetailthetransformedequation,wediscussthe inter-faceboundaryconditionsandpresentacomparisonwiththe ana-lyticalGreen’sfunction ina infiniterod. InSection 3,we present the model for flexural waves in beam structures. We detail the transformed equationandwe discusstheconditionon the trans-formation in order to automatically satisfy interface conditions. We present different numerical examples concerning invariance ofeigenfrequencies,transformationofeigenmodesandwediscuss theeffectofboundaryconditionsanddescribethetransient exam-pleofFig.1.Finalconsiderationsconcludethepaper.

2. Longitudinalwavesinarod

We start presenting the transformation of coordinates tech-niqueforaproblemgovernedbyasecond-orderdifferential equa-tion.We showthat forarodthetransformedequation maintains its form and the interface conditions are automatically satisfied eliminatinganyproblemofreflectionattheinterface.

2.1. Equationofmotion

WeconsiderathinrodhavingYoung’smodulusE,mass-density

ρ

andcross-sectionalareaA.TherodisshowninFig.2.

Thelongitudinal componentUofthe displacementvector U=

(

U,V,W

)

,functionofthepositionXandtimet,satisfiesthe equa-tionofmotion(Graff,1975)

[EAUX

(

X,t

)

]X=

ρ

AUtt

(

X,t

)

, (1)

wheresubscriptsindicatederivative withrespecttothe indicated variable,i.e.UX =

∂

U/

∂

X andUtt=

∂

2U/

∂

t2.TheaxialforceisN=

EAUX.ForahomogeneousrodthelongitudinalstiffnessEAandthe

lineardensity

ρ

Aareconstant.

In the time-harmonic regime the displacement is U

(

X,t

)

= U

(

X

)

e−iωt_,with

_ω

_{theradianfrequency,andthelongitudinal}

com-ponentofthedisplacementsatisfiestheHelmholtzequation

[EAUX

(

X

)

]X+

ρ

A

ω

2U

(

X

)

=0. (2)

2.2. Transformedequation

We introduce a coordinate transformation x=G

(

X

)

, withG(.) injectivefunction, andwe indicatewith g(x) the inversefunction G−1.First-orderderivative inthe originalcoordinateX andinthe transformedonexarerelatedby

d dX = 1 gx d dx. (3)

ImplementingthecoordinatetransformationinEq.(2),we ob-tainthetransformedequationofmotionintroducingatransformed displacement u(x) such that u

(

x

)

=U

(

X

)

. The transformed equa-tionofmotionhastheform



EAux

(

x

)



x+

ρ

A

ω

2.3. Interfaceboundaryconditions

LetusapplythecoordinatetransformationinadomainX>X0,

whereX0 isagivenpoint.Insuchacasetheproblemisgoverned

by the untransformedequation ofmotion (2)for X < X0 andby

thetransformedequationofmotion(4)forX>X0.

Theuntransformedandtransformeddomainshavetosharethe sameinterfacepoint,whichmeansthat thetransformationhasto satisfy the relation X0=g

(

x0

)

=x0. In addition, at the interface

point X0 continuity conditions on the longitudinal displacement

and on the axial force must be satisfied. We note that, if after transformationdisplacementu(x) oraxialforcen(x)changeatthe pointX0,areflectedwavewillbegenerated.Clearly,zeroreflection

isrequiredtohaveperfectmatch.

For therod, inaddition to the imposed equalityu

(

x

)

=U

(

X

)

, wehave n

(

x

)

=EAd dx[u

(

x

)

]= EA gx gx d dX[u

(

x

)

]=EA d dX[U

(

X

)

]=N

(

X

)

. (5)

Therefore, both displacement u and axial force n in the trans-formed point xare equal to displacementU andaxial force N in the originalpoint X.These twoequalities clearlyholdalsoatthe pointX0=x0assuringtheabsenceofreflectionattheinterface.

2.4. TransformationforPerfectlyMatchedLayersinarod

In addition to the perfect match, the transformation must damp the incoming waves; this target can be achieved by ap-plying a complextransformationwithg

(

x

)

=x+ih

(

x

)

,wherethe real function h(x) _{≥ 0} and h

(

x0

)

=0. We note that, employing

such a transformation, the generic wave exp[ikX] is transformed intothewaveexp[k

(

−h

(

x

)

+ix

)

],whichdecaysexponentiallyfast. Such a model, based on coordinate transformation, include pre-viously proposed PMLs obtained introducing artificial dissipation in the form of complex linear stiffnessand density, respectively. In Sacks etal.(1995) complexmaterial parameters were used to build PMLs forelectromagnetic problems,whichare governed by Helmholtzequations,whiletheconformalmappingtechniquewas applied in Chang et al. (2014) in order to define PMLs for the plane elastodynamic problem governed by a system of second-order PDE. Here we consider the general transformation g

(

x

)

= x+i

α

(

x− x0

)

n, where

α

and n are two parameters that can be

variedinordertotunethewavedamping.Forthepurposeof illus-tration,weshowacomparisonbetweenananalyticalsolutionand anumericalimplementationfortheinfinitebodyGreen’sfunction. Fortherodproblemthetime-harmonicGreen’sfunction express-ingthedisplacementinXduetoaunitforceappliedinXc

vibrat-ingharmonicallywithfrequency

ω

isgivenby

Ug

(

X,Xc;

ω

)

=−

1

2ksink

|

X− Xc

|

, (6)

wherek₌

ω



ρ

_/E(see,forexample,Graff (1975)).

In Fig. 3a we compare the analytical solution for the infinite body Green’s function with a numericalsolution with

α

=1 and n₌1_,implementedinComsolMultiphysics® _{onafinitedomainof}

totallengthof20mandcenteredatX=0.TwoPMLdomainshave been implemented in the boundary regions 8m ≤ |X| ≤ 10m so thatX0=±8m,theradianfrequencyis

ω

=

π

/2.Clamped

bound-ary conditionsare applied atX=±10,they are knownto gener-atelargeramplitudereflectedwaveswithrespecttoothertypeof

anaffinetransformationwith

α

₌1_,n₌1andanon-affine trans-formationwith

α

=3,n=4.Theresults,givenfor

ω

=

π

and re-portedonlyinthe regionX ≥ 0,show againexcellent agreement in the central region and the increased damping for the second choiceofmaterialparameters.

InconclusionofthisSectionwenotethatthetransformationis frequencyindependentand,therefore,thePMLsworkequallywell atdifferent frequencies subjectedto the usual limitations on the meshsizewithrespecttothewavelength.

3. Flexuralwavesinabeam

Inthis Section we apply the transformationcoordinates tech-niqueforflexuralwavesinaslenderbeam,governedbya fourth-orderdifferentialequation.Wegiveaphysicalinterpretationofthe transformedequationsasinBrunetal.(2014a)andColquittetal. (2014). We also show that, under coordinate transformation, the transformed medium possessesthe same eigenfrequenciesas the originalone,apropertythatcanbeusedinordertocheckto cor-rectnessofthe transformationinfinitedomains withevident ad-vantagesontheimplementation.

3.1. Equationofmotion

Weconsidertime-harmonictransversedisplacementsV

(

X,t

)

= V

(

X

)

e−iωt _{inaslenderbeamstructure asin Fig.2.Thebeamhas}

cross-sectionalarea A,second-momentofinertiaJ,Young’s modu-lusEanddensity

ρ

.TheequationofmotionforV(x)is

[EJVXX

(

X

)

]XX−

ρ

A

ω

2V

(

X

)

=−TX

(

X

)

−

ρ

A

ω

2V

(

X

)

=0. (7)

whereT

(

X

)

=MX

(

X

)

=−[EJVXX

(

X

)

]X istheshear forceandM(X)

the bending moment. The transverse displacement component W(X) is governed by an analogousfourth-order differential equa-tion.

3.2.Transformedequation

Again,weintroduce thetransformationx=G

(

X

)

,withinverse transformation X₌g

(

x

)

. Then, the transformed equation has the form

[t

(

x

)

+n

(

x

)

v

x

(

x

)

]x+

ρ

A

(

x

)

ω

2

v

(

x

)

=0, (8)

wherev(x)isthetransformedtransversedisplacement,thatwe as-sume equal to V(X). The transformed shear and axial forces are givenby t

(

x

)

=mx

(

x

)

=−



EJ

(

x

)

v

xx

(

x

)



_x, n

(

x

)

= 3g2xx

(

x

)

− gxxx

(

x

)

gx

(

x

)

g5 x

(

x

)

EJ, (9)

respectively,wherem(x)isthetransformedbendingmoment.The bendingstiffnessandlineardensitytransformasfollows

EJ

(

x

)

=EJ

g3

x

,

ρ

A

(

x

)

=gx

ρ

A. (10)

Wenotethattheequationofmotion(8)representsan inhomo-geneousbeaminpresenceofaxialstress (seeBrunetal.(2014b); Colquittetal.(2014)).

Fig. 3. Time-harmonic infinite body Green’s function in rod structures. Results are given for k = 1 m −1 _{. (a) Comparison between analytical solution (6) for an infinite rod and}

numerical solution implemented in Comsol Multiphysics ®_{for a finite system, with | X | ≤ 10m and radian frequency ω = π/ 2 . The PMLs have been implemented employing}

a transformation with α= 1 and n = 1 . (b) Comparison between numerical solutions with transformation parameters α= 1 , n = 1 and α= 3 , n = 4 . Results are given for ω = πand are shown only in the region 0≤ X ≤ 10m.

3.3.Interfaceconditions

At the interface X0=x0 between untransformed and

trans-formeddomainsthefollowingessentialconditions



V

(

X0

)

=

v

(

x0

)

,

VX

(

X0

)

=

v

x

(

x0

)

, (11)

andnaturalconditions



M

(

X0

)

=m

(

x0

)

,

T

(

X0

)

=t

(

x0

)

+n

(

x0

)

v

x

(

x0

)

, (12)

mustbesatisfied.

Expressingtheinterface conditionsinthetransformeddomain asafunctionoftheoriginalvariableX,oftheoriginaldisplacement V(X)andoftheinversetransformationfunctiong(x),weobtain af-tersimplealgebraicmanipulations

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

v

(x0)=V(X0),

v

x(x0)=gx(x0)VX(X0), m(x0)=−EJ g2 x(x0)VXX(X0)+gxx(x0)VX(X0) g3 x(x0) , r(x0)=t(x0)+n(x0)

v

x(x0)=EJ 3g2 xx(x0)− gxxx(x0)gx(x0) g4 x(x0) V(X0) −EJg3x(x0)VXXX(X0)+3gx(x0)gxx(x0)VXX(X0)+gxxx(x0)VX(X0) g3 x(x0) . (13)

Then,theconstraints

gx

(

x0

)

=1, gxx

(

x0

)

=0, gxxx

(

x0

)

=0, (14)

inEq.(13) assurethat interface conditions (11) and(12) are sat-isfiedindependently on the general choice of the transformation G(X)orofitsinverseg(x).ThethreeconditionsinEq.(14)mustbe complementedbytheadditionalconditionX0=g

(

x0

)

=x0,which

identifies the same interface point between untransformed and transformeddomains.

We notethattransformed bendingstiffnessandlineardensity, defined in Eq. (9), are homogeneous only for affine transforma-tion. However, the only admissible affine transformation for the beam case is the identity in view of the constraints g

(

x0

)

=X0

andgx

(

x0

)

=1, which means that an inhomogeneous material is

neededinthetransformeddomain. 3.4.Eigenfrequencyanlaysis

Here we compare eigenfrequencies andeigenmodes fora ho-mogeneousbeamdefinedinthedomain−L≤ X≤ Landfora sec-ondbeamstructurewhereweapplyatransformationontheright

half of the structure 0 ≤ X ≤ L which transforms into the do-main 0 ≤ x ≤ l, as shownin Fig. 4a. We consider a polynomial transformationg(x) subjectedtotheconstraintsasinEq.(14)and x0=g

(

x0

)

=X0 attheinterface point x0=X0=0.In addition,we

imposeg

(

l

)

₌L_,which definesthelengthofthe transformed do-mainandtheadditionalconditions

gx

(

l

)

=1, gxx

(

l

)

=0, gxxx

(

l

)

=0, (15)

assuring, inview ofrelations (13),thedirect identificationofthe sameboundaryconditionsonrotation,momentandverticalforce attheboundarypointx=l,asdemonstratedintheprevious Sec-tion. The corresponding transformation is the monotonically in-creasingsepticpolynomial

g

(

x

)

=x+35L− l l4 x 4_{− 84}L− l l5 x 5₊₇₀L− l l6 x 6_{− 20}L− l l7 x 7_{. (16)}

Coordinatetransformation (16)doesnot dependon theboundary conditions.Forspecificboundaryconditionsitisnot necessaryto impose all conditions (15); for example, in the case of a simple support, whereV

(

L

)

=0,VXX

(

L

)

=0,VX(L) = 0and VXXX(L) = 0,

onlytheconditionsgx

(

l

)

=1,gxx

(

l

)

=0areneededtoassurethat

thebendingmomentm(l)iszeroinaddition tothedisplacement v(l)(see Eq.(13)). Nevertheless,we haveproposedtransformation (15)which includes all possible boundary conditions.

Wealsonotethat,apartfromsatisfactionofconditionsat inter-faceandboundarypoints,thereiscompletefreedominthechoice of thetransformation g(x) within a properlydefined set of func-tions.

Restricting theattentionto a simplysupported beam, the ho-mogeneousproblemgovernedbytheEq.(7)hassolution

V

(

X

)

=A1eiβX+A2e−βX+A3e−iβX+A4eβX, (17)

where

β

4₌

₍

_ρ

_A

₎

_/

₍

_EJ

₎

_ω

2_{; such a solution, supplemented by}

the boundary conditions V

(

−L

)

=V

(

L

)

=0, VXX

(

−L

)

=VXX

(

L

)

=

0, gives the well known result that eigenfrequencies are

ω

=

(

p

π

)

2_/

₍

_4L2

₎



₍

_EJ

₎

_/

₍

ρ

_A

₎

_{(ppositive integernumber) and the}

cor-respondingeigenmodesareV

(

X

)

=sin[p

π

(

X+L

)

/

(

2L

)

].

Fortheprobleminwhichtheregion0_{≤ X≤ L}hasbeen trans-formed intothe region0 ≤ x ≤ lby meanof thetransformation (16),thesolution(17)isstillvalidwithin thedomain−L≤ X≤ 0, while in the domain 0 ≤ x ≤ l the problemis governed by the transformedequationofmotion,giveninEq.(8).Insuchadomain theinhomogeneousbendingstiffnessandlineardensityare

EJ₌

EJ 1+140L−l l4 x3− 420 L−l l5 x4+420 L−l l6 x5− 140 L−l l7 x6



3,

ρ

A=



1+140L− l l4 x 3₋₄₂₀L− l l5 x 4₊₄₂₀L− l l6 x 5₋₁₄₀L− l l7 x 6

ρ

A, (18) respectively.

Fig. 4. Eigenmodes of homogeneous beam of length 2 L and inhomogeneous beam of length L + l. (a) Simply supported beams: the inhomogeneous beam has been obtained from transformation (16) , the half-length L = 1 m is transformed into the length l = 0 . 2 m. (b) First eigenmodes for the two structures at ω = π2 / _{(4 L} 2₎_{(EJ) / (ρA) . (c)}

Second eigenmodes at ω = 4 π2 / _{(4 L} 2₎_{(EJ) / (ρA) . (d) Third eigenmodes at ω = 9 π}2 / _{(4 L} 2₎_{(EJ) / (ρA) .}

Fig. 5. Eigenmodes of homogeneous beam of length 2 L and inhomogeneous beam of length L + l, L = 1 m and l = 0 . 2 m. (a) Clamped-free boundary conditions: the first eigenmode is shown, the eigenfrequency is ω = (0 . 597 π)2 / _{(4 L} 2₎_{(EJ) / (}ρ_{A) . (b) Clamped-clamped boundary conditions: the first eigenmode is shown, the eigenfrequency}

is ω = (1 . 505 π)2 / _{(4 L} 2₎_{(EJ) / (ρA) .}

The equation of motion for the transformed domain has the generalsolution

v

(

x

)

=B1eiβg(x)+B2e−βg(x)+B3e−iβg(x)+B4eβg(x), (19)

The system of two equations of motion is supplemented by the four boundary conditions V

(

−L

)

=

v

(

l

)

=0, VXX

(

−L

)

= v xx

(

l

)

=0

and by the four interface conditions givenin Eqs. (11) and(12), whereX0=x0=0.

Eigenfrequenciesandeigenmodesareobtainedfromthe eigen-values and eigenvectors of the system of equations Ma=0

given by the 8 boundary and interface conditions, where a= [A1A2A3A4B1B2B3B4]T is the vector of the unknown

ampli-tudes andthematrixM collectsthecoefficientsofthe equations. Then,thecondition

detM=256

(

EJ

)

2

β

10sin

(

2

β

L

)

sinh

(

2

β

L

)

=0, (20)

givesexactly thesameeigenfrequenciesofthe homogeneous sys-tem, namely

ω

=

(

p

π

)

2_/

₍

_4L2

₎



₍

_EJ

₎

_/

₍

_ρ

_A

₎

_{(p positive integer) and}

the trivial one

ω

=0. The first 3 eigenmodes for the homoge-neous and not homogeneous systems are given in Fig. 4b–d. In particular, we note that the solution in the transformed domain is v

(

x

)

=V[g

(

x

)

]=V

(

X

)

.

In Fig.5we report thefirst eigenmodefor thetwo structures for different boundary conditions and the same transformation given inEq. (16): clamped-free inpart (a) andclamped-clamped inpart(b).Again, theeigenfrequencies forthehomogeneous and inhomogeneoussystemscoincideandtheeigenmodesinthe trans-formeddomainaresuchthat v

(

x

)

=V

(

X

)

.

We note that, while the coincidence of eigenfrequencies can beexpected, itdependsontheboundaryconditionswhich,inthe caseofflexuralwaves,arenotpreservedbyageneral transforma-tion, asshownpreviously.We alsostress that, tothe bestofour

knowledge,suchacomparisonhasneverbeenconsideredbeforeto checktheconnectionbetweenthesolutionsbeforeandafter trans-formation.

3.5.PerfectlyMatchedLayersinabeam

In order to define the Perfectly Matched Layer we consider a complex transformation such that g

(

x

)

=gR

₍

_x

₎

_+igI

₍

_x

₎

_{, where}

gR_{(x) and g}I_{(x) stand for the real and imaginary parts. The}

con-straintsofEq.(14)attheinterfacepointX0=x0plusthecondition

g

(

x0

)

=x0implythat

⎧

⎪

⎪

⎨

⎪

⎪

⎩

gR

₍

_x 0

)

=x0, gI

(

x0

)

=0, gR x

(

x0

)

=1, gIx

(

x0

)

=0, gR xx

(

x0

)

=gIxx

(

x0

)

=0, gR xxx

(

x0

)

=gIxxx

(

x0

)

=0. (21)

Therefore,ifweconsiderapolynomialtransformation,therealand imaginarypartsgR_(x)andgI_{(x),respectively,mustbeatleast}

poly-nomials of degree 4 in

(

x_{− x}0

)

and, for the imaginary part, the

lowestnon-zerotermhasatleastpower4. 3.6.Additionalboundarycondition

IntheimplementationofthePerfectlyMatchedLayerwithina FiniteElementcode theinfinite domainis substitutedby a finite domain,whichintroducesanadditionalboundaryconditionatthe boundaryx=x1ofthePerfectlyMatchedLayerdomain.Ingeneral,

thisboundaryconditionperturbstheinfinitedomainsolution. By looking at the solution (19) of the transformed problem, wenote thatthepropagatingsolutionB1eiβg(x)+B2e−βg(x)is

B3e−iβg(x)+B4eβg(x) isgeneratedatthefictitiousboundaryx=x1.

Therefore,apossibleapproachinordertoeliminatethe perturba-tionintroducedbytheadditionalboundaryconditions,istodefine ad-hocboundaryconditionsatx=x1thatwouldeliminatethe

re-flectedsolution,namelyboundaryconditionsleadingtoB3=B4=

0.Thefieldsatx=x1canbewritteninthepartitionedform



_A 11 A21 A12 A22





_b 1 b2



=



c_c1 2



(22) where b1=



B1 B2

, b2=



B3 B4

, c1=



v

(

x1

)

v

x

(

x1

)

, c2=



m

(

x1

)

r

(

x1

)

, (23) and A11=



eiβg(x1) _e−βg(x1) i

β

gx

(

x1

)

eiβg(x1) −

β

gx

(

x1

)

e−βg(x1)



, A12=



e−iβg(x1) _eβg(x1) −i

β

gx

(

x1

)

e−iβg(x1)

β

gx

(

x1

)

eβg(x1)



, A21=EJ

β

⎡

⎢

⎢

⎣

eiβg(x1 ) _[

β

_g2 x(x1)− igxx(x1)] g3 x(x1) − e−βg(x1 ) _[

β

_g2 x(x1)− gxx(x1)] g3 x(x1) ieiβg(x1 )

η

1(x) g4 x(x1) e−βg(x1 )

η

2(x) g4 x(x1)

⎤

⎥

⎥

⎦

, A22=EJ

β

⎡

⎢

⎢

⎣

e−iβg(x1 ) _[

β

_g2 x(x1)+igxx(x1)] g3 x(x1) − e βg(x1 ) _[

β

_g2 x(x1)+gxx(x1)] g3 x(x1) −ie−iβg(x1 )

η

1(x) g4 x(x1) − e βg(x1 )

η

2(x) g4 x(x1)

⎤

⎥

⎥

⎦

, (24) with

η

1(x)=

β

2g4x(x1)+6gxx(x1)2− 2gx(x1)gxxx(x1),

η

2(x)=

β

2g4x(x1)− 6gxx(x1)2+2gx(x1)gxxx(x1). (25) If we substitutethe solution b1=A−111[c1− A12b2]of the firstpair of equationsin(23),intothesecondpairofequations,weobtain

(A22− A21A−111A12)b2=c2− A21A−111c1. (26) Thesolutionofthesystemoftwoequations(26)iszero,i.e.B3=B4= 0,if

c2=A21A−111c1, (27)

providedthat

det[A22− A21A−111A12]= 0and det[A11]=0. (28) Thetwoconditionsin(27)expressthenaturalboundary condi-tionsm(x1)andr(x1) asafunctionoftheessentialboundary

con-ditionsv(x1)andvx(x1).Theexplicitexpressionsare

m

(

x1

)

=EJ−i

β

2_g3 x

(

x1

)

v

(

x1

)

+[−gxx

(

x1

)

+

(

1− i

)

β

g2x

(

x1

)

]

v

x

(

x1

)

g4 x

(

x1

)

, r

(

x1

)

=EJ

(

1+i

)

β

3_g5 x

(

x1

)

v

(

x1

)

+[

η

1

(

x1

)

+

(

i− 1

)

β

2g4x

(

x1

)

]

v

x

(

x1

)

g5 x

(

x1

)

, (29)

where

η

1isgiveninEq.(25).Wenotethatthetwodeterminantsin

Eq.(28)can be always set different from zero for every

β

by modu-latingthequantityg(x1).

The Perfectly MatchedLayer andthe optimalboundary condi-tions giveninEq.(29) havebeen implementedin theFinite Ele-mentcodeComsolMultiphysics®_{.Inparticular,weconsiderthe}

in-finitebodytime-harmonicGreen’sfunction,whichhasthe analyt-icalexpression Vg

(

X,Xc;

ω

)

= 1 4EJ

β

3[e−β| X−Xc|+_sin

(

β|

_X− X c

|

)

], (30)

as in Brun et al. (2012). The analytical expression is compared with numerical simulations. We considered the following struc-turalparameters:EJ=1MPa,

ρ

A=1kg/m,XC=0m,X0=x0=8m

andx1=10m.Theimplementedinversetransformationis

g

(

x

)

=x∓35

(

x1− 2x0

)

(

x1− x0

)

4

(

x∓ x0

)

4+ 84

(

x1− 2x0

)

(

x1− x0

)

5

(

x∓ x0

)

5 ∓ 70

(

x1− 2x0

)

(

x1− x0

)

6

(

x∓ x0

)

6+ 20

(

x1− 2x0

)

(

x1− x0

)

7

(

x∓ x0

)

7 +i

(

x∓ x0

)

α, (31)

where_{∓ stands}forthePMLdomains atx_∈( _±x0,±x1) and

α

=

5.Transformation(31)hasbeenobtainedapplyingconditions(21), wherex0 standsfor±x0andconditions

gR

₍

_±x

1

)

=±2x0, gRx

(

±x1

)

=1, gRxx

(

±x1

)

=0, gRxxx

(

±x1

)

=0 (32)

ontherealpartofthetransformation.Additionalconditionsonthe imaginarypartofthetransformation gI_atx=±x

1 havenotbeen

appliedsincetheyleadtolargeramplitudereflectedfields. ThedeformedshapesaregiveningraylinesinFig.6afor differ-entdiscretizations,whilethedashed blacklineindicatesthe ana-lyticalsolutionasinEq.(30).Thecomparativeanalysisshowsthat theresultsconvergetowards theanalyticalsolutioninthecentral regionincreasingthenumberofelements.InthePMLregionsitis evidentthedampingofthewave.

Wedefinethequalityfactor,themeasure

Q=  ₊X0 −X0



V

(

X

)

− Vg

(

X,0;

ω

)

Vg

(

0,0;

ω

)

2 dX, (33)

where V(X) is the solution in the untransformed domain X∈

(

−X0,X0

)

.QisaquantitativedescriptionofthequalityofthePML,

whichtends to0forperfectPMLs,indicating theabsenceof per-turbationwithin thecentral domain X∈

(

−X0,X0

)

.In Fig. 6bthe

qualityfactorQisshownasafunctionofthesizesoftheelements indoublelogarithmicscale.Forsimplicity,ineachcomputationwe considered elementsof thesame size s. The resultsshow an ex-cellent convergence of the numerical results toward the analyti-calsolution.Thelinearregression,indicatedwithadashed linein Fig.6b,indicatesthatthequalityfactor_Qgoestozeroas6.78s8.14_.

3.7. PerfectlyMatchedLayerswithstandardboundaryconditions In Section 3.6 we detailed how to implement perfect PMLs proposing an optimal solution for the additional boundary con-ditionsintroduced inthefinitedomain implementednumerically. Suchamodelgivesexcellentresults,buthastwolimitations:first, itisdifficultto implementin astandardFinite Elementcodeand second,boundaryconditionsarefrequencydependent.Specifically, relations (29) between different boundary conditions depend on the parameter

β

∝√

ω

andthe frequency dependencelimits the applicabilityoftheproposedmodeltotransientproblems.

Here, we propose a simplersolution with frequency indepen-dent boundary conditions. In particular, we implement classical boundaryconditionsatx=±x1,namelyclamped,free andsimply

supported.

When these classical boundary conditions are implemented in Comsol Multiphysics® _{theobtained displacement fields,not}

re-ported hereforbrevity, show againan excellent agreement with theanalyticalresultsinthecentralregion.InFig.7wereportthe qualityfactor Qasa functionofthenormalizedfrequencym0

β

=

(

x1− x0

)

β

for

α

=4inEq.(31).ThequalityfactorQhasbeen

com-putedfromtheanalyticalsolutionsfortheinfinitemediumandthe finitemediumwithPMLsinordertochecktheeffectofthe bound-aryconditionsindependentlyontheinfluenceofthediscretization.

Fig. 6. Fig (a) Time-harmonic Green’s function. The analytical displacement of Eq. (30) is given in black dashed line. The numerical results are given in continuous gray lines. Different curves correspond to different size s of the elements given in meter; the elements have constant length within the domain (−x 1 , x 1) = (−10 m , 10 m ) (b) Quality

factor Q as a function of the size s of the element. Results are given in logarithmic scale.

Free Clamped

Simply Supported

Fig. 7. Quality factor Q as a function of the normalized frequency m 0β. Results are

given for the same mechanical parameters of Fig. 6 . Continuous black line cor- responds to simply supported boundary conditions, continuous gray line to free boundaries and dashed black line to clamped boundaries.

Theconvergenceincreaseswithfrequencyandthethreeboundary conditionsgiveequivalentresultswithapreferenceonthesimply supported caseatthelowestfrequencies.Increasingtheexponent

α

intheimaginarypartofg(x)inEq.(31)givesequivalentresults withthe differencethat the numberofasymptotes inthequality factorQcurvesasafunctionof

β

increaseswith

α

.

3.8. Dimensionofthelayer

In order to estimate the error introduced by the layer of dimension m0=

|

x1− x0

|

, we consider an incident plane wave

wI

(

X

)

=eiβX impinging the interface between the homogeneous

domain andthe PML at X0=x0=0 and generatingthe reflected

wave wR

(

X

)

=R1e−iβX+R2eβX andthetransmittedwave wT

(

x

)

=

T1eiβg(x)+T2e−βg(x)+T3e−iβg(x)+T4eβg(x),where the six constants

R1,R2,T1,T2,T3,T4 canbeeasily foundby imposingthefour

in-terface conditions at x₌x0=0 and two boundary conditions at

x=x1=x0+m0. The solution, for different boundary conditions

has the form T1=1, T2=0, indicating the perfect match atthe

interface,andR1=T3,R2=T4,showingthatthereflectedwaveis

generated by theboundary conditionsatx=x1. Inparticular, for

perfect boundary conditionsas inEq. (29)there is no reflection, i.e.R1=T3=R2=T4=0and,inprinciple,theonlyboundary

con-ditions (29)are sufficientto avoidreflectionwithoutthe needto introduce aPML.Forclamped,simply supported andfree

bound-aryconditionsthereflectedamplitudesare

|

R1

|

= f1

(

m0,

β

)

e−2m α 0β,

|

R 2

|

=f2

(

m0,

β

)

e−m α 0β, (34)

wheref1(m0,

β

)andf2(m0,

β

) areO

(

1

)

inm0 and

β

and

α

isthe

exponentoftheimaginarypartofthetransformation(31).Forall boundaryconditions,includingtheperfectones, thedisplacement amplitudedecaysexponentiallyase−mα0,while(Nataf,2013) indi-catesthat forproblems governed by Helmhotz equations the re-flection coefficients decayexponentially ase−2m_0.Results

consis-tentwithNataf (2013) areobtainedifnullconditionsare applied at_±x1 togIandtoitsfirstthreederivatives.

3.9.Transientloadresults

ThePMLswithsimplysupportedboundaryconditionshasbeen tested for a transient load as given in Fig. 1. For the transient regimethefollowingequationsofmotionhavebeensolved numer-ically:

[EJVXX

(

X,t

)

]XX+

ρ

AVtt

(

X,t

)

=0 (35)

intheuntransformeddomainD1and



EJ

(

x

)

v

xx

(

x,t

)



x− n

(

x

)

v

x

(

x,t

)



x+

ρ

A

(

x

)

v

tt

(

x,t

)

=0, (36)

inthetransformedoneD2,whereEJ

(

x

)

and

ρ

A

(

x

)

aregiveninEq.

(10)andn(x) is giveninEq.(9)andthey are thesame asinthe time-harmonicregime.Zeroinitialboundaryconditionshavebeen applied,namely

V

(

X,0

)

=

v

(

x,0

)

=0,Vt

(

X,0

)

=

v

t

(

x,0

)

=0, X∈D1,x∈D2. (37)

Thetimevariationofthepointload

F

(

t

)

= 6  i=1



(

−1

)

i+1_10ie1000(t−0.08i)2



(38)

is shown in Fig. 8 and it has been applied at X=0. Two ge-ometries have been implemented in Finite Elements: a larger onewithhomogeneous propertiesEJ₌1Pa,

ρ

A₌1kg/m andX_∈ [−20m,20m] and a shorter one with the same homogeneous properties in X∈[−4m,4m] and PMLs in |X| ∈ [4m, 7m]. The transformationisgiveninEq.(31)with

α

₌5anditisunchanged withrespecttothetime-harmonicregimesince itinvolvesonlya spatialtransformation.

The two initial boundary value problemshave been solved in Comsolusingabackwarddifferentiationformula;atotalperiodof 5shasbeenanalyzedandstandardconvergenceanalysishasbeen consideredonthetimestepsandelementsize;theinitialstephas been set to 10−4s and elements of uniform size s=5 cm have beenimplemented.

Fig. 8. Time distribution of the point load applied at X = 0 in the transient analysis.

ThetransversedisplacementisgiveninFig.1att=0.5s,when thepropagatingwave hasreachedthefictitiousboundaryatx1=

±7m but not the boundaries X=±20m for the larger domain. In the Figure only the region X∈[−10m,10m] is shown for vi-sualization purposes. The comparison between the two numeri-cal solutions evidences an excellent agreement in the central re-gion where PMLs are not present. Such an example reveals the competitive behavior of the proposed techniquein the transient regime;thisisexpectedsinceonlyaspatialtransformationg(x)is implemented.Nevertheless,acompleteanalysisofthetransient re-sponserequiresadifferenttypeofstudywhichisleftforafuture work.

4. Conclusions

We haveproposed ananalyticalPMLmodelforflexuralwaves. Theexcellent agreementwithanalytical Green’s functionfor infi-nitedomainisdetailed,theerrorinthecaseofnon-perfect addi-tionalboundaryconditions isestimatedandthe influenceof dis-cretizationisalsogiven.

We have givenparticularimportance to the physical interpre-tation of the transformed equations in order to show that the methodis simple andcan be implemented in standard finite el-ementpackages;theeigenfrequencyanalysismayalsobeusedas asimplecheckofthecorrectnessoftheimplementation.

The PMLs forflexural waves can be particularly useful inthe analysis of elongated structures like bridges and pipelines and comparisons with analytical results for infinitely long structures (Cartaetal.,2014;CartaandBrun,2015).

References

Aliabadi, M. , 2002. The Boundary Element Method, Applications in Solids and Struc- tures, 2. John Wiley Andamp, Sons, Ltd, West Sussex, England .

Astley, R., Hamilton, J., 2006. The stability of infinite elements schemes for transient wave problems. Comput. Methods Appl. Mech. Eng. 195 (29–32), 3553–3571. doi: 10.1016/j.cma.2005.01.026 .

Beer, G. , 2001. Programming the Boundary Element Method: An Introduction for Engineers, 1. Wiley, Chichester, England .

Berenger, J., 1994. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (2), 185–200. doi: 10.1006/jcph.1994.1159 . Bertoldi, K., Brun, M., Bigoni, D., 2005. A new boundary element technique without

domain integrals for elastoplastic solids. Int. J. Numer. Methods Eng. 64, 877– 906. doi: 10.1002/nme.1385 .

Bettess, P. , 1992. Infinite Elements. Penshaw Press, U.K .

Bettess, P., Zienkiewicz, O., 1977. Diffraction and refraction of surface waves us- ing finite and infinite elements. Int. J. Numer. Methods Eng. 11 (8), 1271–1290. doi: 10.1002/nme.1620110808 .

Bonnet, M. , 1999. Boundary Integral Equation Methods for Solids and Fluids. Wiley, Chichester .

Brun, M., Bigoni, D., Capuani, D., 2003. A boundary element technique for incremen- tal, non-linear elasticity. part ii: bifurcation and shear bands. Comput. Methods Appl. Mech. Eng. 192, 2481–2499. doi: 10.1016/S0 045-7825(03)0 0272-X . Brun, M., Capuani, D., Bigoni, D., 2003. A boundary element technique for incremen-

tal, non-linear elasticity. part i: formulation. Comput. Methods Appl. Mech. Eng. 192, 2461–2479. doi: 10.1016/S0 045-7825(03)0 0268-8 .

Brun, M., Colquitt, D., Jones, I., Movchan, A., Movchan, N., 2014. Transformation cloaking and radial approximations for flexural waves in elastic plates. New J. Phys. 16, 093020. doi: 10.1088/1367-2630/16/9/093020 .

Brun, M., Colquitt, D., Jones, I., Movchan, A., Movchan, N., 2014. Transformation cloaking and radial approximations for flexural waves in elastic plates. New J. Phys 16. doi: 10.1088/1367-2630/16/9/093020 .

Brun, M., Giaccu, G., Movchan, A.B., Movchan, N., 2012. Asymptotics of eigenfrequen- cies in the dynamic response of elongated multi-structures. Proc. R. Soc. Lond. A 468 (2138), 378–394. doi: 10.1098/rspa.2011.0415 .

Brun, M., Guenneau, S., Movchan, A., 2009. Achieving control of in-plane elastic waves. Appl. Phys. Lett. 94, 061903. doi: 10.1063/1.306 84 91 .

Burnett, D., 1994. A three-dimensional acoustic infinite element based on a gener- alized multipole expansion. J. Acoust. Soc. Am. 96 (5), 2798–2816. doi: 10.1121/ 1.411286 .

Carta, G., Brun, M., 2015. Bloch-floquet waves in flexural systems with continuous and discrete elements. Mech. Mater. 87, 11–26. doi: 10.1016/j.mechmat.2015.03. 004 .

Carta, G., Brun, M., Movchan, A., 2014. Dynamic response and localization in strongly damaged waveguides. Proc. R. Soc. A. Math. Phys. Eng. Sci. 470 (20140136). doi: 10.1098/rspa.2014.0136 .

Cerjan, C., Kosloff, D., Kosloff, R., Reshef, M., 1985. Absorbing boundary conditions for the numerical simulation of waves. Geophysics 50, 705–708. doi: 10.1190/1. 1441945 .

Chang, Z., Guo, D., Feng, X., Hu, G., 2014. A facile method to realize perfectly matched layers for elastic waves. Wave Motion 51 (7), 1170–1178. doi: 10.1016/j. wavemoti.2014.07.003 .

Chew, W., Weedon, W., 1994. A 3d perfectly matched medium modified maxwell’s equations with stretched coordinates. Microwave Optical Tech. Lett. 7 (13), 599– 604. doi: 10.10 02/mop.4650 071304 .

Clouteau, D., Cottereau, R., Lombaert, G., 20 0 0. A review on the modelling of wheel/rail noise generation. J. Sound Vib. 231 (3), 519–536. doi: 10.1006/jsvi. 1999.2542 .

Clouteau, D., Cottereau, R., Lombaert, G., 2013. Dynamics of structures coupled with elastic media-a review of numerical models and methods. J. Sound Vib. 332 (10), 2415–2436. doi: 10.1016/j.jsv.2012.10.011 .

Colquitt, D., Brun, M., Gei, M., Movchan, A., Movchan, N., Jones, I., 2014. Transfor- mation elastodynamics and cloaking for flexural waves. J. Mech. Phys. Solids 72, 131–143. doi: 10.1016/j.jmps.2014.07.014 .

Polizzotto, C., 20 0 0. A symmetric galerkin boundary/domain element method for finite elastic deformations. Comput. Methods Appl. Mech. Eng. 189 (2), 481–514. doi: 10.1016/S0 045-7825(99)0 0303-5 .

Farzanian, M. , Arbabi, F. , 2014. A pml solution for vibration of infinite beams on elastic supports under seismic loads. J. Seismol. Earthquake Eng. 16 (1), 1–16 . Gaul, L. , Kögl, M. , M.Wagner , 2003. Boundary Element Methods for Engineers and

Scientists: An Introductory Course with Advanced Topics, 1 Springer, Verlag Berlin Heidelberg .

Geli, L. , Bard, P. , Jullien, B. , 1988. The effect of topography on earthquake ground motion: a review and new results. Bull. Seismol. Soc. Am. 78 (1), 42–63 . Gerdes, K., 20 0 0. A review of infinite element methods for exterior helmholtz prob-

lems. J. Comput. Acoust. 8 (1), 43–62. doi: 10.1016/S0218-396X(0 0)0 0 0 04-2 . Givoli, D., 1991. Non-reflecting boundary conditions: review article. Geophysics 94

(1), 1–29. doi: 10.1016/0021-9991(91)90135-8 .

Hagstrom, T. , Hariharan, S. , 1998. A formulation of asymptotic and exact boundary conditions using local operators. Appl. Numer. Math. 27, 403–416 .

Hastings, F., Schneider, J., Broschat, S., 1996. Application of the perfectly matched layer (pml) absorbing boundary conditions to elastic wave propagation. J. Acoust. Soc. Am. 100 (5), 3061–3069. doi: 10.1121/1.417118 .

Higdon, R., 1991. Absorbing boundary conditions for elastic waves. Geophysics 56 (2), 231–241. doi: 10.1190/1.1443035 .

Hu, G., Rathsfeld, A., Yin, T., 2016. Finite element method to fluid-solid interaction problems with unbounded periodic interfaces. Numer. Methods Partial Differ. Eq. 32 (1), 5–35. doi: 10.1002/num.21980 .

Ihlenburg, F. , 1998. Finite Element Analysis of Acoustic Scattering. Springer, Verlag New York .

Jones, I., Brun, M., Movchan, N., Movchan, A., 2015. Singular perturbations and cloaking illusions for elastic waves in membranes and kirchhoff plates. Int. J. Solids Struct. 69–70, 498–506. doi: 10.1016/j.ijsolstr.2015.05.001 .

Graff, K.F. , 1975. Wave Motion in Elastic Solids. Oxford University Press, Oxford . Kim, J., Kim, H., 2009. Finite element analysis of piezoelectric underwater transduc-

ers for acoustic characteristics. J. Mech. Sci. Technol. 23 (1), 452–460. doi: 10. 10 07/s12206-0 08-1126-x .

Komatitsch, D., Tromp, J., 2003. A perfectly matched layer absorbing boundary con- dition for the second-order seismic wave equation. Geophys. J. Int. 154 (1), 146– 153. doi: 10.1046/j.1365-246X.2003.01950.x .

Kristek, J., Moczo, P., Galis, M., 2009. A brief summary of some pml formulations and discretizations for the velocity-stress equation of seismic motion. Stud. Geo- phys. Geod. 53 (1), 459–474. doi: 10.10 07/s1120 0-0 09-0 034-6 .

Lancioni, G., 2012. Numerical comparison of high-order absorbing boundary condi- tions and perfectly matched layers for a dispersive one-dimensional medium. Comput. Methods Appl. Mech. Eng. 209–212, 74–86. doi: 10.1016/j.cma.2011.10. 015 .

Cremers, L., Fyfe, K., 1995. On the use of variable order infinite wave envelope ele- ments for acoustic radiation and scattering. J. Acoust. Soc. Am. 97 (5–6), 2028– 2040. doi: 10.1121/1.411994 .

Lee, J., Tassoulas, J., 2011. Consistent transmitting boundary with continued-fraction absorbing boundary conditions for analysis of soil- structure interaction in a layered half-space. Comput. Methods Appl. Mech. Eng. 200 (13–16), 1509–1525. doi: 10.1016/j.cma.2011.01.004 .

4684. doi: 10.1016/j.cma.2004.12.011 .

Lysmer, J. , Kuhlemeyer, R. , 1969. Numerical comparison of high-order absorbing boundary conditions and perfectly matched layers for a dispersive one-dimen- sional medium. J. Eng. Mech. Div., Proc. ASCE 95, 859–876 .

Maier, G., Miccoli, S., Perego, U., Novati, G., 1995. Symmetric galerkin boundary el- ement method in plasticity and gradient plasticity. Comput. Mech. 17 (1), 115– 129. doi: 10.10 07/BF0 0356484 .

Milton, G., Briane, M., Willis, J., 2006. On cloaking for elasticity and physical equa- tions with a transformation invariant form. New J. Phys. 8, 248. doi: 10.1088/ 1367-2630/8/10/248 .

Nataf, F., 2005. New constructions of perfectly matched layers for the linearized euler equations. C.R. Math. 340 (10), 775–778. doi: 10.1016/j.crma.2005.04.013 . Nataf, F. , 2013. Absorbing boundary conditions and perfectly matched layers in wave

propagation problems. Radon Series on Computational and Applied Mathemat- ics 11, 219–231 .

Norris, A., Shuvalov, A., 2011. Elastic cloaking theory. Wave Motion 48 (6), 525–538. doi: 10.1016/j.wavemoti.2011.03.002 .

Novati, G., Brebbia, C., 1982. Boundary element formulation for geometri- cally nonlinear elastostatics. Appl. Math. Model 6 (2), 136–138. doi: 10.1016/ 0307- 904X(82)90025- 7 .

Parvanova, S., Dineva, P., Manolis, G., et al, 2014. Dynamic response of a solid with multiple inclusions under anti-plane strain conditions by the bem. Comput. Struct. 139, 65–83. doi: 10.1016/j.compstruc.2014.04.002 .

Phan-Thien, N., 1988. Rubber-like elasticity by boundary element method: finite de- formation of a circular elastic slice. Rheol. Acta 27 (3), 230–240. doi: 10.1007/ BF01329739 .

Polizzotto, C., 1988. An energy approach to the boundary element method. part ii: elastic-plastic solids. Comput. Methods Appl. Mech. Eng. 69 (3), 263–276. doi: 10. 1016/0 045-7825(88)90 043-6 .

974. doi: 10.1109/20.497403 .

Robinovich, D., Givoli, D., Bielak, J., et al, 2011. A finite element scheme with a high order absorbing boundary condition for elastodynamics. Comput. Methods Appl. Mech. Eng. 200 (23–24), 2048–2066. doi: 10.1016/j.cma.2011.03.006 .

Romero, A., Tadeu, A., Galvin, P., Antonio, J., 2015. 2.5D coupled bem-fem used to model fluid and solid scattering wave. Int. J. Numer. Methods Eng. 101 (2), 148– 164. doi: 10.1002/nme.4801 .

Sochacki, J., Kubichek, R., George, J., Fletcher, W., Smithson, S., 1987. Absorbing boundary conditions and surface waves. Geophysics 52, 60–71. doi: 10.1190/1. 1442241 .

Song, R., Ma, J., Wang, K., 2005. The application of the non splitting perfectly matched layer in numerical modeling of wave propagation in poroelastic me- dia. Appl. Geophys. 2 (4), 216–222. doi: 10.10 07/s11770-0 05-0 027-3 .

Steele, C., 2001. A critical review of some traffic noise prediction models. Applied Acoustics 62 (3), 271–287. doi: 10.1016/S0 0 03-682X(0 0)0 0 030-X .

Tassoulas, J., Kausel, E., 1983. Elements for the numerical analysis of wave motion in layered strata. Int. J. Numer. Methods Eng. 19 (7), 1005–1032. doi: 10.1002/ nme.1620190706 .

Telles, J., Brebbia, C., 1979. On the application of the boundary element method to plasticity. Appl. Math Model 3 (6), 466–470. doi: 10.1016/S0307-904X(79) 80030-X .

Wrobel, L. , 2002. The Boundary Element Method, 1. Wiley, Chichester, West Sussex, England .

Zheng, Y. , Huang, X. , 2002. Anisotropic perfectly matched layers for elastic waves in cartesian and curvilinear coordinates. Earth Resour. Lab. Ind. Consortia 1–18 . Sacks, Z.S., Kingsland, D.M., Lee, R., Lee, J-F., 1995. A perfectly matched anisotropic

absorber for use as an absorbing boundary condition. IEEE Trans. Antennas Propagation 43 (12), 1460–1463. doi: 10.1109/8.477075 .