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HAL Id: hal-01218784

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Boundary Element Method for the Solution of the

Helmholtz Equation

Hélène Barucq, Abderrahmane Bendali, M Fares, Vanessa Mattesi, Sébastien

Tordeux

To cite this version:

Hélène Barucq, Abderrahmane Bendali, M Fares, Vanessa Mattesi, Sébastien Tordeux. A Symmetric

Trefftz-DG Formulation based on a Local Boundary Element Method for the Solution of the Helmholtz

Equation. [Research Report] RR-8800, INRIA Bordeaux. 2015, pp.31. �hal-01218784�

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RESEARCH

REPORT

N° 8800

October 2015

A Symmetric Trefftz-DG

Formulation based on a

Local Boundary Element

Method for the Solution

of the Helmholtz

Equation

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(4)

RESEARCH CENTRE

BORDEAUX – SUD-OUEST

Solution of the Helmholtz Equation

H. Baru q

∗†

,A. Bendali

‡§

, M. Fares

, V. Mattesi

∗‡†

, S. Tordeux

∗†

Proje t-TeamMagique-

3

D

Resear hReport n°8800O tober201531pages

Abstra t: A symmetri TretzDis ontinuous Galerkin formulation,for solvingthe Helmholtz equation with pie ewise onstant oe ients, is built by integration by parts and addition of onsistentterms. The onstru tionofthe orrespondinglo alsolutionstotheHelmholtzequation isbasedon aboundaryelementmethod. Thenumeri alexperiments,whi hare presented, show anex ellentstability relativelyto thepenaltyparameters,and moreimportantlyanoutstanding abilityofthemethodtoredu etheinstabilitiesknownasthepollutionee t intheliteratureon numeri alsimulationsoflong-rangewavepropagation.

Key-words: Helmholtz equation, pollution ee t, dispersion, Tretz method, Dis ontinuous Galerkin method, integral equations, ultra-weak variational formulation, Diri hlet-to-Neumann operator,BoundaryElementMethod.

EPCMagique-3D,Pau(Fran e)

UniversityofPau(Fran e)

CERFACS,Toulouse(Fran e)

§

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méthode d'éléments de frontière, pour la résolution de l'équation d'Helmholtz

Résumé: UneformulationsymétriquedetypeTretzGalerkinedis ontinue,pourlarésolution numériquedel'équation deHelmholtzà oe ients onstantsparmor eaux,est onstruitepar intégrationsparpartiesetajoutsderelationsvériéespar onsistan e. La onstru tiondes solu-tionslo ales orrespondantesdel'équation deHelmholtzest baséesurlaméthodedeséléments defrontière. Lesexpérien esnumériques, présentéesdans e rapport, montrent une ex ellente stabilité relativement aux paramètresde pénalisation, et surtout une remarquable apa ité de laméthode à réduire les instabilitésnumériques, appelées aussi pollution numérique dans la littératuresurlessimulationsnumériquesdepropagationd'ondessurdelongues distan es. Mots- lés : Équationde Helmholtz, pollutionnumérique, dispersion, méthode detype T re-tz,méthodeGalerkineDis ontinue,équationsintégrales,formulationvariationnelleultrafaible, opérateurdeDiri hlet-to-Neumann,méthodeélémentsdefrontière.

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Contents

1 Introdu tion 5

2 The symmetri Tretz DG method 6

2.1 TheHelmholtzboundary-valueproblem . . . 7

2.2 Thevariationalformulation . . . 8

2.2.1 Theinteriormesh . . . 8

2.2.2 Interiorandboundaryfa es . . . 9

2.2.3 Tra esandGreenformula . . . 9

2.2.4 Generalvariationalformulationofthesymmetri Tretz-DGmethod . . . 10

2.3 ComparisonwithpreviousTretz-DGformulations . . . 12

2.3.1 ComparisonwithInteriorPenaltyDGMethods . . . 12

2.3.2 ComparisonwithDGmethodsbasedonnumeri aluxes . . . 13

2.3.3 Theupwindings heme. . . 13

3 The BEMsymmetri Tretz DGmethod 14 3.1 Theboundaryintegralequationwithin ea helementoftheinteriormesh. . . 14

3.2 TheBEMsymmetri TretzDGmethod. . . 15

3.2.1 Thelo alboundaryelementmethod . . . 15

3.2.2 Approximationofthedualvariables . . . 15

3.2.3 TheBEM-STDGmethod . . . 17

3.2.4 Theassemblypro ess . . . 17

4 Validationof the numeri almethod 18 4.1 Theboundary-valueproblem . . . 18

4.2 ApproximationoftheDtNoperatoronrenedmeshes . . . 20

4.3 ValidationoftheBEM-STDGmethod . . . 21

4.3.1 Adu tproblemofsmallsize . . . 21

4.3.2 Approximationofanevanes entmode . . . 22

4.4 Long-rangepropagation . . . 23

4.4.1 Lowestpolynomialdegree . . . 24

4.4.2 Higherpolynomialdegrees. . . 24

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1 Introdu tion

Usual nite element methods, when used forsolvingthe Helmholtzequation overseveral hun-dredsofwavelengths,arefa edwith thedrawba kgenerally alled pollutionee t. Roughly speaking,itis ne essaryto augmentthedensityofnodesto maintainagivenlevelofa ura y, when in reasingthesizeof the omputationaldomain. Thisin turnrapidly ex eedsthe apa -itiesin storageand omputingevenin theframeworkofmassivelyparallel omputerplatforms ( f., forexample,[29,15,33℄andthereferen estherein).

Several approa heshavebeenproposed to urethis aw. Atrst, forsu h kindsof numer-i alsolutions, it be ame well-established that Dis ontinuous Galerkin (DG)methods are more e ientthan standardFiniteElement Methods (FEM), also alled ContinuousGalerkin (CG) methodsin this ontext. Thise ien yseemstobedueinparttothelessstronginter-element ontinuity hara terizingthesemethods ( f., forexample, [1,2℄). Indeedthis was onrmed in [32℄ where itis shownthat it ispossibleto keepthee ien y of the DGmethods by allowing dis ontinuities onlyat the interiorof theelementsin termsof bubblefun tions with penalized jumps.

Anotheradvantageoftheabovekindofmethodsliesinthepossibilitytouseshapefun tions, moreadaptedto theapproximationofthesolutiontotheinteriorPartialDierentialEquations (PDE) of the problem, but, ontrary to polynomials, with poor properties for enfor ing the usual inter-element ontinuity onditionsoftheFEM.Inthis respe t, Tretzmethods, that is, methodsforwhi hthelo alshapefun tions arewavefun tions,i.e.,solutionsto theHelmholtz equation( f.,forexample,[22,38℄andthereferen estherein),wereintensivelyusedtoalleviate theaforementioned pollutionee t. The ombinationofaTretzandaDGmethodtherefore resultedonnumerousapproa hesforsolvingwaveequationproblems alledTretzDGmethod (TGD) (see,forexample,[20,24, 23,22℄andthereferen estherein).

A tually, Tretz methods without strong inter-element ontinuitywere used for sometime inthe ontextoftheso alledUltraWeakVariationalFormulation(UWVF)devisedbyDesprés [13, 10℄. It was dis overedlater that this formulation an be re ast in the ontext of aTDG method[16,7,20℄atleastforthetwolatterreferen eswhenusingexpli itlo alsolutionstothe Helmholtzequations.

Some riti ismshavebeenhoweveraddressedtotheDGmethods. Theymainly on ernthe in reaseof the oupled degreesof freedomand asuboptimal onvergen e oftheir approximate uxes. Hybridized versions of the DG (HDG) methods were proposed in response to these hallenges [12℄. Howeverat the authors knowledge, HDG methods havenot beenused yet in the framework of aTretz method but only with usual lo al polynomial approximations [18℄, ex ept in a re ent paper [36℄, where these methods were ombined in an elaborate way with geometri al opti s at the element level to e ientlysolvethe Helmholtz equation in the high frequen yregime. Sin ethelo alshapefun tionsareonlyasymptoti solutionstotheHelmholtz equationthen,su hakindofmethod an be alledquasi-TretzHDG.

Insteadof DGmethods, someauthors preferto use aLagrangemultiplier oraleast-square te hniquetoenfor ethe ontinuity onditions( f. [3,17,43℄). Thisisnottheapproa hretained in thispaper.

Ontheotherhand,itisgenerallyadmittedthatBoundaryIntegralEquations(BIEs)leadto less pollutionee ts thanFEMsevenifattheauthorsknowledgenoformalstudy onrming su hapropertyseemsto havebeenalreadyprovided. Su hagood behaviorisprobablydueto thefa tthatBIEs anbeseenasparti ularTretzmethodswhensu haninterpretationistaken to the extreme. It is hen etempting to use the free spa e Green kernel in an approximation pro edure for the interior Helmholtz equation to redu e the pollution ee ts. This way to pro eed has been already onsidered in [8℄. However it seems hard to extend it to problems

(9)

involvingvarying oe ientsor realisti geometries andboundary onditions. Theaim of this study is pre isely to mix two approa hes: DG methods and BIEs, to devise a TDG method whi h ane ientlyhandleparti ularHelmholtzequationswithvarying oe ients. Spe i ally, either the oe ientsare pie ewise onstantorthey anbe approximatedin this mannerona su ientlyrenedde omposition ofthe omputationaldomain, alledinteriormesh in therest ofthispaper.

The method anbe viewed globallyas aDG method at thelevelof the interior mesh and asaBIElo allyattheelementlevel. A tually,BIEsareusedonlyto omputethe Diri hlet-to-Neumann(DtN)operatorwithinea helementoftheinteriormesh. Asshownbelow,thequality oftheoverall solutionstrongly depends onthea ura y oftheapproximationof thisoperator. Spe i numeri al pro edures have therefore been developed to in rease the a ura y of this approximation. Su hatreatment anberelatedtosimilarte hniquesdevelopedin [26,14℄.

The method proposed in this study ownsother additionalinterestingproperties. As a DG method,itisformulatedasasymmetri DGmethod, thatis,asasymmetri variational formu-lation ofthe orrespondingboundary-valueproblem. Itsderivation followsthe pathdevisedin [4℄ (see also[37, p. 122℄)for designingSymmetri InteriorPenalty(SIP) methods but in a bit dierentway,morestraightforwardinouropinion. Additionally,whenthepenaltyterms enfor -ingthe ontinuityofthenormaltra es(really thedualvariables)aredis arded, thissymmetry hereyields animportantgain. Thestorageof the boundary integraloperators involved in the formulationisthenavoided: the ontributionoftheBIEsthenbeingelement-wiseonly. Itisalso worthnotingthatallthedegreesoffreedomofthedis reteproblemtobesolvedarelo atedon theskeletonofthemesh,thatis,theboundariesoftheelements. Su hafeatureis hara teristi totheredu tionofunknownsyieldedbyHDGmethodsevenifheretherestillremainsunknowns onbothsidesoftheinterfa es. Lastbutnotleastperhapsthemostimportantadvantageofthe proposedapproa hliesonthe hoi eof thelo alshapefun tions whi h a ountforallkindsof waves: evanes ent,propagative,et . Thisisin ontrastwithusualTretzmethodswhi hlo ally useplane, ir ular/spheri alwaves,multipoles, et . ( f., forexample,[3,23, 34, 20,10℄ to ite afew). It should benotedalsothat,evenifthemethod, whi h is onsidered here,isofTretz type,thelo alapproximationsaredonebymeansof aBoundaryElementMethod(BEM)( f., for example, [40, 6℄). As a result, these approximations are ultimately performed in terms of pie ewisepolynomialfun tionsonaBEMmesh. In ontrastthentousualTretzmethods,

h

or

p

renementsareassimpleande ientasinastandardFEM.Thisiswhythismethodis alled theBEMSymmetri TretzDG method inthesu eeding textand more on iselydenoted by BEM-STDG.

The paperis organizedas follows. In Se tion 2, after stating the boundary-valueproblem, werstderivethevariationalformulationofthesymmetri TDGmethod and showhowit an be onne ted to previous DG formulations. Se tion 3 develops the BEM pro edure used to dene the Tretz method. Se tion 4 is devoted to the numeri al validation of the method in twodimensionsandtothe omparisonofitsperforman eswithastandardInteriorPenaltyDG (IPDG) method based onelement-wise polynomial approximations. A nal briefse tion gives some on ludingremarksandindi atesfurther studiesthat anextendthisone.

2 The symmetri Tretz DG method

Afterstatingthewavepropagationproblem,wedes ribethemostgeneralDGformulation on-sideredinthisstudy.

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2.1 The Helmholtz boundary-value problem

The DG variationalformulationsof theHelmholtz equation ( f., for example, [22, 20,24℄) are generallyobtainedbywritingthewaveequationintheformofarst-orderPDEssystem. Most ofthestudiesdedi atedtothesolutionofthisproblembythiskindofte hniques(inadditionto thepreviousreferen es,see, forexample,[10, 7,42, 33℄)dealwiththeHelmholtzequationwith onstant oe ients. Ifa ousti sistakenasthe on reteshapetotheproblembeingdealtwith, thisamountstoassumingthat theequationsgoverningthea ousti u tuationsofpressureand velo ity orrespondtothepropagationofana ousti waveinanidealstagnantanduniformuid ( f.,forexample. [39,Chap. 2℄). Wefollowhereamoregeneralpathand onsiderasin[28℄that thepropagation is relatedto anidealstagnantuidbut notne essarilyuniform. Thea ousti systemforsu h a onguration anbewrittenasfollows( f., forexample,[31, Eqs. (64.5)and (64.3)℄)



1

c

2

̺

t

p + ∇ · v = 0,

̺∂

t

v

+ ∇p = 0,

(1) where

c

and

̺

arethespeedofsoundandthedensitywithinthestagnantuidand

p

and

v

are respe tively thea ousti u tuationsof the pressureand the velo ity. Hereafter data

c

and

̺

are assumed to be pie ewise onstant. Asthis will be learbelow,the handlingof therelated dis ontinuitiesisanimportantpartoftheDGformulation.

To be onsistent with the notation used in previous works [22, 20, 24, 42, 33℄, we denote thephasorsof respe tivelythe pressureand thevelo itybyadierentsymbol:

u

for

p

dened a ordingto thefollowingidentitiesand hara terizations

p(x, t) = ℜ e

−iωt

u(x)



,

v

(x, t) = ℜ e

−iωt

σ

(x)



.

(2) Intheabovedenitions,

ℜz

is thereal partofthe omplexnumber

z

,and

ω > 0

istheangular frequen y. Thesolutionof(1)ishen eredu edto



c

2

̺

u + ∇ · σ = 0,

−iω̺σ + ∇u = 0.

(3)

Wenowassumethattheequationsaresetinaboundedpolygonal/polyhedraldomain

Ω ⊂ R

d

(

d = 2, 3

)anddenoteby

∂Ω

itsboundary. Using thepie ewise onstantwavenumber

κ = ω/c

, and onsidering a non overlapping de omposition

∂Ω

D

,

∂Ω

N

, and

∂Ω

R

of

∂Ω

, we re astthe abovesystemasthe followingHelmholtz equation withvarying oe ientssupplementedwith typi alboundary onditions

·

1

̺

u +

κ

2

̺

u = 0

in

Ω,

u = g

D

on

∂Ω

D

,

1

̺

u · n = g

N

on

∂Ω

N

,

1

̺

u · n − iY u = g

R

on

∂Ω

R

.

(4)

Thethirdboundary onditionisexpressedin termsofafun tion

Y

yieldingthesurfa e ompli-an eof

∂Ω

R

uptoamultipli ative onstant,assumedtobealsopie ewise onstant. Thesour es produ ing thewaveareembodiedin theright-handsides

g

D

,

g

N

and

g

R

. Wehavedenotedby

n

theunitnormalon

∂Ω

dire tedoutwards

(seeFig.1).

Underminimal assumptions onthegeometry of

, on

κ

,

̺

, and

Y

, ontheright-handsides

g

D

,

g

N

and

g

R

,andassumingfurthermoreforexamplethat

ℜY ≥ ν > 0

onapartof

∂Ω

R

with anonvanishinglength/area,itiswell-knownthatproblem(4)admitsoneandonlyonesolution in anadequatefun tionalsetting( f.,forexample,[35,44℄).

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TheHelmholtzequationwithvarying oe ientsinsystem(4)isexa tlythewaveequation onsideredin[28℄. Theboundary onditionhasbeentakenthereinthefollowingform

1

̺

n

u − iηu = Q



1

̺

n

u − iηu



+ g

(5)

with

η = κ/̺

and

Q

and

g

givenhereby

Q = −1,

g = −2iηg

D

,

on

∂Ω

D

,

Q = +1,

g = 2g

N

,

on

∂Ω

N

,

Q = (1 − Y/η) / (1 + Y/η) , g = (1 + Q)g

R

,

on

∂Ω

R

,

(6)

thus expressingthethree boundary onditionsin (4)inasingleone. Thisismorethananother way of writing the boundary onditions. It makes it possible to express the in oming wave

(1/̺)∂

n

u − iηu

intermsofaree tionoftheoutgoingwave

Q (−(1/̺)∂

n

u − iηu)

andasour e term

g

.

Itis worthnotinghoweverthat Helmholtzequationisinvolvedinother kindsofwave prop-agationproblems. An importantexampleoftheseisrelatedto seismi waveswhereattenuation ee tsmustbea ountedforin additionto thepropagationfeatures. TheHelmholtzequation governingthis kindofwavesisinthefollowingform[41℄

∆u +

̺ω

2

E

u = 0

(7)

where

̺

and

ω

arethedensityandtheangularfrequen yand

E

isthe omplexmodulus. Clearly thisequation anbeput intheabovesettingbysubstituting

1/c

2

for

̺/E

and

̺

for1insystem (3). Thisleadsthustoa omplexwavenumber

κ

. Sin eweare interestedmainly inthis paper ona uratelya ountingforlong-rangepropagation,welimitourselvesbelowtoreal oe ients.

2.2 The variational formulation 2.2.1 The interior mesh

Atrst,we onsideranonoverlappingde omposition

T

of

inpolyhedral/polygonalsubdomains ofthe omputationaldomain

, alledtheinteriormeshassaidintheintrodu tion. Considering thattheelements

T ∈ T

areopensetsof

R

d

,wethereforeassumethat

Ω =

[

T ∈T

T ,

T ∩ L = ∅

if

T 6= L.

(12)

T

Figure2: Interiormeshin 2D.

It is worth re alling that this interior mesh an be quite arbitrary. Su h a mesh in the two-dimensional aseisdepi tedinFig.2.

Wealwaysassume thatthe oe ients

̺

and

κ

of theHelmholtzequation arereal positive onstantswithin ea h

T ∈ T

and denotedthereby

̺

T

and

κ

T

respe tively.

2.2.2 Interior and boundary fa es

Wepasstothe denitionof interiorandboundaryedges/fa esonwhi h isbasedthesettingof anyDGmethod. Interioredges/fa es

F

arepartoftheboundary

∂T

of

T ∈ T

sharedbyanother

L ∈ T

. Theyaredenedasfollows

F = ∂T ∩ ∂L

whenthelength/areaof

F

is

> 0.

(8)

Someotherdenitions hara terize

F

byrequiringthatit ontainsatleast

d

points onstitutinga nondegeneratedsimplex(segmentandtriangleinthetwo-andthethree-dimensional ase respe -tively)[4℄. Boundaryedges/fa es

F

aredenedsimilarly byrepla ing

L

with theexteriorof

. Weuseasetnotation

F

I

and

F

torefertothe olle tionsofinteriorandboundaryedges/fa es respe tively. Clearly the interioredges/fa es

F

onstitute a non-overlappingde omposition of thefollowing urve/surfa e

Γ =

[

F ∈F

I

F.

Inthesameway,theboundaryedges/fa esyieldanon-overlappingde ompositionof

∂Ω

∂Ω =

[

F ∈F

F.

InFig.2,

Γ

isdepi tedingreywhile

∂Ω

isinbla k. 2.2.3 Tra es and Greenformula

Assumingthat thesolution

u

toproblem (4) andthe testfun tion

v

arepie ewise smooth,we anusetheusualGreenformulatoget

X

T ∈T

Z

T



1

̺

T

u · ∇v −

κ

2

T

̺

T

uv



dx

=

X

T ∈T

Z

T



1

̺

T

u · ∇v + v∇ ·

1

̺

T

u



dx

=

X

T ∈T

Z

∂T



1

̺

u



|

∂T

· n

T

v

T

ds

(9)

(13)

with

v

T

= v|

∂T

(10)

thetra esbeingtakenfromthevaluesof

v

inside

T

. Ve tor

n

T

istheunitnormalto

∂T

dire ted outwards

T

. A tually below, we antakeadvantageof thefa t that thenormal omponent of

1/̺∇u

is ontinuous a ross

F

to write the sum of the integrals on ea h

∂T

in the following manner

X

T ∈T

Z

∂T



1

̺

u



|

∂T

· n

T

v

T

ds =

X

F ∈F

I

Z

F

1

̺

u · [[v]]ds +

X

F ∈F

Z

F

1

̺

u · nvds

(11)

usingthewidespreadnotation( f.,forexample,[5℄)forthejumpof

v

a ross

F

[[v]] = n

T

v

T

+ n

L

v

L

,

(12)

T

and

L

beingthetwoelementsofthemeshsharingedge/fa e

F

.

ItisapartofthederivationofthevariationalformulationoftheDGmethod toexpressthe ontinuityofthenormal omponentof

1/̺∇u

a rossanyedge/fa e

F

fromthemeanofitstra es onbothsidesof

F

{{

1

̺

∇u}} =

1

2



1

̺

∇u



|

T

+



1

̺

∇u



|

L



(13) thus arrivingto

X

T ∈T

Z

T

1

̺

T

u · ∇v − κ

2

T

uv



dx =

Z

Γ

{{

1

̺

∇u}} · [[v]]ds +

Z

∂Ω

1

̺

u · nvds.

(14)

Theaboveexpressionof

{{1/̺∇u}} · [[v]]

mustbeunderstoodinthemeaningofthenormaltra es sin eonlythesequantities anreallybedenedintheweakformulationofproblem(4)andare involvedin theTDGmethod.

2.2.4 General variational formulation ofthe symmetri Tretz-DGmethod

Inthesameway,assumingnowthattestfun tion

v

isanelement-wisesolutiontotheHelmholtz equation

∆v + κ

2

T

v = 0

in

T

(T ∈ T ) ,

andusing thefa t this on ethat it isthe unknown

u

whi his ontinuousa rosstheinterfa es

F

, we anwrite

X

T ∈T

Z

T

1

̺

T

u · ∇v − κ

2

T

uv



dx =

Z

Γ

u[[

1

̺

∇v]]ds +

Z

∂Ω

u

1

̺

v · n ds

(15)

whereasusual( f., forexample,[5℄)thejump

[[1/̺∇v]]

isdenedby

[[

1

̺

∇v]] =



1

̺

∇v



|

T

· n

T

+



1

̺

∇v



|

L

· n

L

.

(16)

Inthesamewayasabovefor

1/̺∇u

,wesubstitutethemeanvalue

(14)

forthetra eof

u

anduse(14)toobtainthefollowingvariationalequationsetontheedges/fa es oftheinteriormesh

Z

Γ



{{u}}[[

1

̺

∇v]] − {{

1

̺

∇u}} · [[v]]



ds +

Z

∂Ω



u

1

̺

∇v · n −

1

̺

∇u · nv



ds = 0.

(18)

Todesignasymmetri formulation,wepro eedasin[4,15℄(seealso[37,p. 122℄). Wemake useofthefollowinginterioridentities

[[u]] = 0

and

[[

1

̺

∇u]] = 0

(19)

andtheboundary onditionstoaddsome onsistenttermsto Eq. (18)thusarrivingto

Z

Γ



{{u}}[[

̺

1

∇v]] + [[

1

̺

∇u]]{{v}} − {{

1

̺

∇u}} · [[v]] − [[u]]{{

1

̺

∇v}}



ds

Z

∂Ω

D



u

̺

1

∇v · n +

1

̺

∇u · nv



ds

+

Z

∂Ω

N

∪∂Ω

R



u

1

̺

∇v · n +

1

̺

∇u · nv



ds + 2

Z

∂Ω

R

(−iY ) uvds

= −2

Z

∂Ω

D

g

D

1

̺

∇v · nds + 2

Z

∂Ω

N

g

N

vds + 2

Z

∂Ω

R

g

R

vds.

(20)

Tostabilizetheformulation,inviewofalreadyknownDGmethods[5,20,15℄,wenallyadd onsistentpenalty termsexpressedbymeansofgivenfun tions

α

,

β

,

γ

and

δ

dened on

Γ

and

∂Ω

. In this way, wearrive to thefollowingmost generalvariationalformulation on whi h are basedtheTDGmethods onsideredinthispaper

a(u, v) = Lv

(21)

where

a

isthefollowingsymmetri bilinearform

a(u, v) =

Z

Γ



{{u}}[[

̺

1

∇v]] + [[

1

̺

∇u]]{{v}} − {{

1

̺

∇u}} · [[v]] − [[u]] · {{

1

̺

∇v}}



ds

+

Z

Γ



α[[u]][[v]] + β∇

[[u]] ⊙ ∇

[[v]] + γ[[

̺

1

∇u]][[

1

̺

∇v]]



ds

Z

∂Ω

D



u

1

̺

∇v · n +

1

̺

∇u · nv



ds

+

Z

∂Ω

N

∪∂Ω

R



u

1

̺

∇v · n +

1

̺

∇u · nv



ds − 2

Z

∂Ω

R

iY uvds

+

Z

∂Ω

D

(αuv + β∇

u∇

v) ds +

Z

∂Ω

N

δ

1

̺

∇u · n

1

̺

∇v · nds

+

Z

∂Ω

R

δ



i

Y

1

̺

∇u · n

1

̺

∇v · n +

1

̺

∇u · n v + u

1

̺

∇v · n − iY uv



ds,

(22)

andwheretheright-handsideisdenedby

Lv =

Z

∂Ω

D



−2g

D

1

̺

∇v · n + αg

D

v + β∇

g

D

v



ds

+

Z



2g

N

v + δg

N

1

∇v · n



ds +

Z

2g

R

v + δg

R



i

1

∇v · n + v



ds.

(23)

(15)

In the above expressions,

u

is the tangential gradient of

u

whereas

[[u]] ⊙ ∇

[[v]]

is denedby

[[u]] ⊙ ∇

[[v]] = ∇

(u

T

− u

L

) · ∇

(v

T

− v

L

)

= ∇

(u

L

− u

T

) · ∇

(v

L

− v

T

)

onanyinterioredge/fa e

F

sharedbyelements

T

and

L

.

2.3 Comparison with previous Tretz-DG formulations

A thorough review of Tretz methods for solving the Helmholtz equation has been re ently performed in [22℄. We limit ourselves here to a omparison with methods of DG type. The following lear denition of su h akind of methods isgiven in this referen e: DG [...℄ [are℄ methodsthat arrive atlo al variationalformulationby applying integration bypartstothe PDE tobeapproximated.

2.3.1 Comparison with InteriorPenalty DG Methods

Interior PenaltyDG (IPDG) methods are mostly introdu ed as aboveby integration byparts at the element level and adding onsistent penalty terms (see for instan e [4, 15, 37℄ and the referen estherein).

A tuallyadaptingtheIPDGintrodu edin[4℄totheHelmholtzequationinvolvedin(4)and onsideringthat

∂Ω

D

= ∂Ω

asin thisreferen e,weobtain

X

T ∈T

Z

T

1

̺

T

u · ∇v − κ

2

T

uv



dx

Z

Γ



{{

1

̺

∇u}} · [[v]] + [[u]]{{

1

̺

∇v}}



ds −

Z

∂Ω

D



u

1

̺

∇v · n +

1

̺

∇u · nv



ds

+

Z

Γ

α[[u]] · [[v]] ds +

Z

∂Ω

D

αuv ds =

Z

∂Ω

D



−g

D

1

̺

∇v · n + αg

D

v



ds.

Usingthefa t that

v

isalsoasolutiontotheHelmholtzequationin

T

andintegratingbyparts on eagain,weget

Z

Γ



u[[

1

̺

∇v]] − {{

1

̺

∇u}} · [[v]] − [[u]]{{

1

̺

∇v}}



ds

+

Z

∂Ω

D

u

̺

1

∇v · nds −

Z

∂Ω

D



u

1

̺

∇v · n +

1

̺

∇u · nv



ds +

Z

∂Ω

D

αuv ds

+

Z

Γ

α[[u]] · [[v]] ds =

Z

∂Ω

D



−g

D

1

̺

∇v · n + αg

D

v



ds.

Usingtheequivalentexpressions

Z

Γ

u[[

1

̺

∇v]]ds =

Z

Γ

{{u}}[[

1

̺

∇v]]ds

and

Z

Γ

[[

1

̺

∇u]]{{v}}ds = 0

andsubstituting

g

D

for

u

intherstintegralon

∂Ω

D

,wedire tlyarrivetoformulation(21)with

β = γ = 0

.

Pro eedinginthesamewayfortheIPDGmethod onsideredin[15℄,wendagainformulation (21)with

Y = −κ

,

g

D

= 0

,

δ = 0

,

∂Ω

N

= ∅

.

Itis learfromtheaboveexamplesthat,uptosome onsistentterms,anyIPDGmethod an beputintheformofvariationalformulation(21)withsuitablevaluesforthepenaltyparameters

(16)

2.3.2 Comparisonwith DG methodsbased on numeri aluxes

Two broad lasses, in whi h an be split the DG methods based on numeri al uxes for the Helmholtz equation, likely rst ome to mind: those whi h are a simple reformulation of the aboveIPDGmethodsandthosewhi h anbelinkedtoanupwindingnumeri als heme. A tually, inthe ontextofthesolutionoftheHelmholtzequation,theupwindingte hniquesareintimately relatedtotheUWVFasthiswasbroughtoutin[16℄. However,intheauthorsopinion,upwinding is stated in the literature in a lear manner only for the Helmholtz equation with onstant oe ients. We found ituseful to re all somefeatures about these te hniquesto more learly set outthedieren ebetweenarealupwinds hemeandasimpleenfor ementofthe ontinuity onditionswhenthePDE oe ientsaredis ontinuous.

Thestartingpointistheuseofeitherofthefollowingte hniquesperformedineveryelement

T

oftheinteriormesh:

ˆ the primal method, as it is alled in [20℄, whi h onsists in integrating by parts the Helmholtzequationwiththeadditionalfeature that

v

isasolutiontothelo alHelmholtz equation,

ˆ the mixed method [24℄, where the integration by parts is arried outon arst-order system,whi hisanequivalentformulationoftheHelmholtzequationwithapairing

(v, τ )

solutionto the omplex onjugatesystem(this analso bedone withoutreferen etothe s alarequation,dire tlyonsystem(3),in[16℄).

Bothofthese approa hesgiverisetothefollowingvariationalequation

Z

∂T

(

σ

b

· n

T

v

T

+ b

un

T

· τ ) ds = 0

(24)

where,withoutfurtherstepsbeingtaken,

σ

b

= σ

T

and

u = u

b

T

(see[42℄ also).

In aseries of papers ( f. [22℄ and the referen es therein), Hiptmair, Moiola, Perugia, and their o-authorsobtainedvariationalformulation(21)withoutthe onsistenttermsaddedtothe aboveIPDG methods toget asymmetri variationalformulation. It is worthmentioning that the variational formulation used in these studies is not symmetri . It an lead however to a symmetri linearsystemiftheinvolvededge/fa eintegralsare al ulatedexa tly.

2.3.3 The upwinding s heme

Itisalsoshownintheabovepapers(seealso[7℄)that,fortheHelmholtzequationwith onstant oe ients,theUWVF anbere astintheframeworkoftheaboveTDGmethodforparti ular values of

α

,

β

,

γ

and

δ

. Formulation (21) an hen e be viewed as a symmetri variational extensionoftheUWVF methodifthespe i propertiesoftheUWVF,relatedtothefa tthat it an be posed in terms of a perturbation of the identity by a norm dimunishing operator, are dis arded [10℄. However, onemust be aware that then this formulation an no longer be onsideredasanupwindings heme. Inthesameway,theextensiongivenin [28℄for boundary-valueproblem(4)forpie ewise onstant oe ients, anstillbeunderstoodasaUWVFor anbe re astasTretzDGmethodbutnotexa tlyasanupwindings heme. A tually,thisextension an beinterpretedasa enteredmethodfordesigningalo alhomogeneouspropagationenvironment rst and using a upwinding s heme then. A similar handling of dis ontinuous oe ients is standardin the numeri alsolution oftime domain hyperboli systems. A ni epresentationof thiste hniqueisgivenin [21℄. Indeed,itisshownin [9℄that themedium, in whi hthewaveis propagating, an be set arbitrarily before performing theupwinding s heme while keepingthe

(17)

generalproperties of the UWVF. A lear onne tion, at least for an homogeneousmedium of propagation,betweentheUWVFandanupwindings hemebasedonawaytoexpressmat hing onditions(19) equivalentlyasabalan esheetofthein omingandoutgoingwaves rossingan edge/fa e,isgivenin[16℄.

3 The BEM symmetri Tretz DG method

Werstuseaboundaryintegralequationapproa htoexpressthedualvariables

p

T

=



1

̺

T

∇u



|

T

· n

T

and

q

T

=



1

̺

T

∇v



|

T

· n

T

(25)

fromthetra es

u

T

and

v

T

of

u

and

v

on

∂T

respe tively. TheBEM-STDGmethod anthenbe fullyderivedfromaboundaryelementapproximationof

u

T

,

v

T

,

p

T

,and

q

T

for

T ∈ T

.

3.1 Theboundaryintegralequationwithinea helementoftheinterior mesh

Forthemoment,weassumethattheinteriorDiri hletproblemiswell-posedwithinany

T ∈ T

. A geometri al riterionensuringthispropertyisgivenbelow. Asaresult,thesingle-layerboundary integraloperatordened forsu ientlysmooth

p

T

by

V

T

p

T

(x) =

Z

∂T

G

T

(x, y)p

T

(y)ds

y

(x ∈ ∂T )

(26)

is invertible. From the well-known integral representations of the solutions to the Helmholtz equationwith onstant oe ients,itthenresultsthattheabovetra es

u

T

and

p

T

= 1/̺

T

∇u·n

T

arelinkedasfollows

V

T

p

T

=

1

̺

T

1

2

− N

T



u

T

(27)

where

N

T

isthedouble-layerboundaryintegraloperator

N

T

u

T

(x) = −

Z

∂T

n

T

(y)

G

T

(x, y)u

T

(y)ds

y

(x ∈ ∂T ) .

(28)

Thekernel

G

T

(x, y)

involvedintheaboveformulasisthat orrespondingtotheoutgoingsolutions totheHelmholtzequationwithwavenumber

κ

T

. Forallthesepropertiesrelatedto thesolution ofHelmholtzequation byboundaryintegralequations,wereferforinstan e to[35,25,6℄.

Wenowturnourattentiontotheabovementionedgeometri riterion. Itisstatedasfollows. Geometri riterion. Assumethat thereexistsaunitve tor

υ

su hthat

sup (x − y) · υ ≤ λ

T

/2

for all

x

and

y

in

T,

(29)

where

λ

T

= 2π/κ

T

is the wavelength within

T

. Then, the boundary-value problem for the Helmholtzequation withDiri hletboundary onditionandwavenumber

κ

T

iswellposedin

T

.

Set

ℓ = sup (x − y) · υ

. With no loss of generality, we an assume that

T ⊂ ]0, ℓ[ ×

Q

i=2,...,d

]0, ℓ

i

[

. From the

min max

prin iple, it an be argued that the rst eigenvalue

χ

2

of theLapla eoperatorwithaDiri hletboundary onditionsatises

χ

2

≥ π

2

/ ℓ

2

+ ℓ

2

2

+ · · · + ℓ

2

d



, thus establishingthe riterionsin e

ℓ ≤ λ

T

/2

,andtherefore

κ

T

≤ π/ℓ < χ

.

(18)

3.2 The BEM symmetri Tretz DG method 3.2.1 The lo alboundaryelementmethod

A tually,onlytheinterfa es

F ∈ F

I

sharedbytwoelementsoftheinteriormeshorthose

F ∈ F

limiting the exteriorof the omputational domain, in other words the skeleton of theinterior mesh, havetobemeshed. Thisis perhapsarstimportantfeatureofthemethod: itisaTDG method but also turns out to be aBEM at the element level. It is thereforepossible to arry outarenementoftheskeletonmesh, thatis,themesha ountingforthea ura yofthelo al approximatingfun tions,withoutanymodi ationoftheinteriormesh.

A tually, itis possibleto usea BEM with nomat hing onditionand thus to benet from theadvantageofmeshingthevariousfa es

F

ea hindependentlyoftheother. However,wehave observed from several numeri al experimentsthat ahigher a ura y is rea hed for ontinuous approximationsof

u

T

and

v

T

respe tively,of oursewithnointer-element ontinuity ondition. Thisisnotatallrestri tiveinthetwo-dimensional asebutmakesitne essarytomeshea hfa e

F

a ordingto theusualmat hing onditionsofa ontinuousFEMwithin theboundary

∂T

of

T

in threedimensions( f.,forexample,[11,30℄). Theresultingmeshis alledtheskeletonmesh in thesu eedingtext. Anyfun tion

u

T

or

v

T

issoughtasapolynomialfun tion ofdegree

m

, that is,in

P

m

,within ea helementof theskeletonmesh, ontinuouson

∂T

but withnofurther ontinuity onditionas said above. A learideaon the ontinuity onditionsthat are imposed onthe onsideredelement-wiseBEMis giveninFig.3. For larity,theboundarynodesonthe various fa es are represented inside the elements of the interiormesh. A same marker forthe nodesisusedtoindi ate the ontinuity onditionsimposedontheboundarytra esoftheshape fun tions.

Figure3: Skeletonmeshandnodesusedin the2D ase.

3.2.2 Approximation ofthe dual variables

The involvementof the BEM at the level of the TDG method is ompletely embodied in the approximationoftheDtNoperatorexpressingthedualvariable

p

T

intermsof

u

T

bysolvingEq. (27). Thea ura yofthisapproximationis ru ialfortheredu tionofthepollutionee t. To enhan ethesharpnessofthispro edure,wehaveadoptedthefollowingstrategy:

ˆ

u

T

isapproximatedontheskeletonmeshand

p

T

onarenedmeshobtainedbysubdividing ea hoftheelementsoftheformer;

ˆ ontraryto

u

T

,

p

T

is ontinuous within ea h edge/fa e only, but notat the jun tions of theedges/fa es;

Letus denote by

[u

T

]

,

h

p

#

T

i

,

[v

T

]

,

h

q

#

T

i

the olumn-wise ve tor whose omponents arethe nodalvaluesof

u

T

,

p

T

,

v

T

,

q

T

. Wedenotealsoby

h

u

#

T

i

and

h

(19)

onthe augmentedset of nodes obtainedeither byinterpolating

u

T

and

v

T

respe tively onthe renedmeshorbydoublingnodeswhere

p

T

or

q

T

arenot ontinuoussothat

h

u

#

T

i

,

h

v

T

#

i

,

h

p

#

T

i

, and

h

q

#

T

i

are allofthesamelengthandhave omponentsallreferringtothe samenodes. The

omponentsof

h

u

#

T

i

areexpressedintermsofthoseof

[u

T

]

bymeansofanexpli itmatrix

[P

T

]

h

u

#

T

i

= [P

T

] [u

T

] .

(30)

Letusthendenethematri es

h

M

T

#

i

,

h

V

T

#

i

,and

h

N

T

#

i

throughthefollowingidenti ations

h

q

#

T

i

h

M

T

#

i h

p

#

T

i

=

Z

∂T

p

T

q

T

ds,

h

q

#

T

i

h

V

T

#

i h

p

#

T

i

=

Z

∂T

(V

T

p

T

) q

T

ds,

h

q

#

T

i

h

N

T

#

i h

p

#

T

i

=

Z

∂T

(N

T

p

T

) q

T

ds.

Equation(27)thenyieldsthatnodalvalues

h

p

#

T

i

and

h

q

T

#

i

areexpressedatthelevelofinterior element

T

by

h

p

#

T

i

=

h

D

T

#

i h

u

#

T

i

,

h

q

T

#

i

=

h

D

#

T

i h

v

T

#

i

,

h

D

T

#

i

=

1

̺

T

h

V

T

#

i

1



1

2

h

M

T

#

i

h

N

T

#

i

.

(31)

It is at this levelthat the well-posedness ofthe interiorDiri hlet problem forthe lapla ian entersintothepi ture. Itensurestheinvertibilityofmatrix

h

V

T

#

i

. Using(30),wethus gettheapproximationoftheDtNoperator

h

p

#

T

i

=

h

D

T

#

i

[P

T

] [u

T

] .

(32)

Atthisstage,itisimportantfor laritytore allthat themethodinvolvesthreemeshes: ˆ theinteriormesh

T

usedforsettingtheBEM-STDGmethod;ea h

T ∈ T

mustsatisfythe

abovegeometri riterionyieldingthat thelo alDiri hletproblem iswell-posed;

ˆ theskeletonmeshusedbythelo alBEMtosetupthelo alapproximatingfun tionswhi h aresolutionswithin ea h

T ∈ T

of theHelmholtzequation(lo alwavefun tions);

ˆ the rened mesh within the boundary

∂T

of ea h

T ∈ T

allowing for an a urate ap-proximation of the DtN operator; this mesh is spe ied througha positiveinteger

N

add yielding the way in whi h ea h elementof the skeletonmesh is subdivided; forinstan e, forthenumeri alexperimentsintwodimensionsperformedbelow,

N

add

isthenumberof segmentsinwhi hea hsegmentoftheskeletonmesh issubdivided.

A s hemati view of these three meshesis displayedin Fig. 4. Note that the global nodal values orrespondto the nodesof theskeletonmesh (verti esof theskeletonmesh when using aBEM with lo al shape fun tions that are polynomialsof degree

m = 1

) and that the nodes relatedtotherenedmeshareonlyusedinelement-wise omputations.

(20)

Figure 4: S hemati view of the three kinds of meshes, whi h are used in the BEM-STDG method. The 4polygonals onstitute the interior mesh. The verti es of the skeleton and the renedmeshesaremarkedbylargedotsandsmall ir lesrespe tively. Therenementparameter

N

add

istakenequalto3here.

3.2.3 The BEM-STDGmethod

Colle ting theve tors

[u

T

]

and

[v

T

]

for

T ∈ T

in olumn-wise ve tors

[u]

and

[v]

respe tively, and expressing

h

p

#

T

i

and

h

q

#

T

i

from (32), we form by means of an assembly pro ess, detailed below,thesquarematrix

[A]

and olum-wiseve tor

[b]

throughthefollowingidenti ations

[v]

[A] [u] = a(u, v),

[v]

[b] = Lv.

Wearehen eledto solvethesymmetri linearsystem

[A] [u] = [b] .

Clearly,

[A]

is also a sparse matrixin themeaning that any twodegreesof freedom whi h belongtotwointeriorelementsnotsharinga ommonfa e arenot onne ted.

3.2.4 The assemblypro ess

It is helpful in theassembly pro ess to expressthe abovebilinearand linearforms in termsof lo alforms relatedto ea helement

T

ofthemesh

T

a(u, v) =

X

T ∈T

X

F ⊂∂T

a

F,T

(u, v),

Lv =

X

T ∈T

X

F ⊂∂T

L

F,T

v.

(33)

However,someadditionalnotationandobservationsarerequiredbeforetheexpli itexpressions oftheselo al forms anbeobtained.

When

F

isaninterioredge/fa esharedby

T

and

L

, deningsimilarly asin Eq. (25)by

p

L

and

q

L

thedualvariablesrelatedto

L

,theintegralson

F

involvedin

a(u, v)

anbewritten ina simplerform

Z

F

({{u}}[[a∇v]] + [[a∇u]]{{v}} − {{a∇u}} · [[v]] − [[u]] · {{a∇v}}) ds

=

Z

F

(u

T

q

L

+ u

L

q

T

+ p

T

v

L

+ p

L

v

T

) ds,

(21)

Z

F

(α[[u]][[v]] + β∇

[[u]] · ∇

[[v]]) ds =

Z

F

α (u

T

− u

L

) v

T

+ β∇

(u

T

− u

L

) · ∇

v

T

ds+

Z

F

α (u

L

− u

T

) v

L

ds + β∇

(u

L

− u

T

) · ∇

v

L

ds,

(35)

Z

F

γ[[a∇u]][[a∇v]]ds =

Z

F

γ (p

T

+ p

L

) q

T

ds +

Z

K

γ (p

L

+ p

T

) q

L

ds.

(36)

In this way, generi allydenoting by

L

the elementsharing fa e

F

with urrent element

T

when

F ∈ F

I

,the ontribution

a

F,T

(u, v)

totheglobalbilinearform

a(u, v)

reads

a

F,T

(u, v) =

Z

F

(p

T

v

L

+ u

L

q

T

) ds

+

Z

F

(αu

T

(v

T

− v

L

) + β∇

u

T

· ∇

(v

T

− v

L

) + γ (p

T

+ p

L

) q

T

) ds.

(37)

Theexpressions of

a

F,T

(u, v)

and

L

F,T

v

for

F ∈ F

are obtainedin astraightforwardwayby usingtheappropriateintegrala ordingtotheinvolvedpartof

∂Ω

andsubstituting

p

T

and

q

T

forrespe tively



1

̺

T

∇u



|

T

· n

T

and



1

̺

T

∇v



|

T

· n

T

.

Remark It is very important to note that if

γ = 0

, that is, when the variationalformulation involvesnopenalty onthe mat hing of thedual variables, only

p

T

and

q

T

areinvolved in the expressions of the lo al forms but neither those

p

L

nor

q

L

related to an adja ent element

L

. Boundaryelementmatrix

h

D

T

#

i

anthereforebe omputedonlyat thelevelof theassemblyof element

T

andhasnottobestored.

4 Validation of the numeri al method

We begin with the statement of a problem, whi h involves long-range wave propagation in a typi al way. This problem will provide us with a good guideline for measuring the level of pollution ee t o uringin anynumeri al solutionof theproblem. We will hen ebeable to ompare the performan es of the BEM-STDG method with the usual polynomial IPDG one. Prior to that, we rst give some numeri al results onrming the importan e of an a urate approximationoftheDtN operator,just aswaspreviouslymentionned.

4.1 The boundary-value problem

We onsiderthefollowingexampleinspiredfromthewavepropagationinadu twithrigidwalls aspresentedin [27℄

∆u + κ

2

n

2

u = 0

in

Ω,

u(0, y) = 1, ∂

x

u(2L, y) − iκu(2L, y) = 0, 0 < y < H,

y

u(x, 0) = ∂

y

u(x, H) = 0, 0 < x < 2L,

(38)

setin

(22)

(seeFig.5)where

κ

is onstantand

n

isthepie ewise onstantfun tion givenby

n =



1

for

|x − L| > D,

n

0

for

|x − L| < D.

(40)

Comparativelywith the problem onsidered in [27℄, we added a Diri hlet boundary ondition on theinlet boundary. Inthis way, wedealwith the threekinds ofboundary onditions sin e weadditionallyhaveNeumannandFourier-Robinboundary onditionsonrespe tivelytherigid wallsandtheoutletboundary. Moreoverhere,itispossibleto onsideranonhomogeneousdu t by hoosing

n

0

onstantbut

6= 1

.

Contrasted

Layer

Inlet b

oundary

Outlet b

oundary

Lower rigid wall

Upper rigid wall

Figure5: Geometryoftheinhomogeneousdu twithrigidwalls.

Indeed, thesolution to this problem is independent of

y

and anbe expressed in termsof fourparameters:

R

,

T

,

R

D

,and

T

D

asfollows

u(x, y) =

T

D

e

iκn(L−D)x

+ R

D

e

−iκn(L−D)x

,

for

|x − L| < D,

(1 − R) e

iκx

+ Re

−iκx

,

for

x < L − D,

T e

iκx

,

for

x > L + D.

(41)

Parameters

R

,

T

,and

R

D

anbeexpressedin termsof

T

D

through



e

−iκn

0

D

R

D

=

n

n

0

0

+1

1

T

D

e

iκn

0

D

, e

iκL

T =

2n

0

n

0

+1

e

iκ(n

0

−1)D

T

D

,

e

−iκL

R = −

n

0

−1

2

e

−iκ(n

0

+1)D

1 − e

4iκn

0

D



T

D

,

(42)

whi hitselfisgivenby

T

D

=

2e

iκn

0

D

(n

0

+ 1) e

−iκ(L−D)



1 − e

4iκn

0

D (n

0

1)

2

(n

0

+1)

2



− (n

0

− 1) e

iκ(L−D)

(1 − e

4iκn

0

D

)

.

(43)

TotesttherobustenessoftheBEM-STDGmethodrelativelyto long-rangepropagation,we mainlylimitourselvestothesimpler asewhere

n

0

= 1

. Then,only

T

and

R

remainmeaningful andhavethefollowingvalues

T = 1, R = 0.

(44)

Thestru turedinteriormesh, whi his usedfor thesetests, isdepi tedin Fig.6. Thismesh is hara terized by twopositiveintegers

N = 2L

and

M = H

. Inallthese tests,

κ

is takenequal to

π

, so that the unit length is a half-wavelength. This automati ally ensures that the lo al Diri hletproblemfortheLapla eequation iswell-posedinea helementoftheinteriormesh.

(23)

Figure6: Stru turedinteriormeshusedformostofthenumeri alexperiments

ˆ Maximumglobalerror

Err

= 100

max

(x

m

,y

m

)

|u (x

m

, y

m

) − u

m

|

max |u (x, y)|

(45)

where

u

m

isthenodalvalueatnode

(x

m

, y

m

)

ofthesolutiondeliveredbytheBEM-STDG method;

ˆ Erroronthetransmittedwave

Err

T

= 100 |T − T

omp

|

(46) where

T

is the oe ient, given above, hara terizing the solution for

x > L − D

, and

T

omp

isitsapproximate valueobtainedfrom thenumeri alsimulation; ˆ Errorontheree tedwave

Err

R

= 100 |R − R

omp

|

(47) obtainedsimilarlyto

Err

T .

4.2 Approximation of the DtN operator on rened meshes

TheplotsinFig.7depi tthemaximumerrorin%foradu thavingalengthof500wavelengths versusthenumber

N

add

ofsegmentsin whi h issubdividedea hsegmentoftheskeletonmesh. In allthe su eeding text, we hara terize ea h skeleton mesh by the number of nodes per wavelengthinsteadofthemeshsize

h

oftheskeletonmesh. Thereasonsbehindthe hoi eofthis parameterwill be detailed below. For instan e, for the BEM, used in this experiment, whose shapefun tionsare polynomialsofdegree

4

,24nodesperwavelength orrespondto ameshsize

h = 1/3

, that is, 3segments per half-wavelength, and 16nodes perwavelength with

h = 1/2

, thatis,2segmentsperhalf-wavelength.

Parameters

α = β = 1.0 10

2

,

γ = 0

, and

δ = 0

have been spe ied empiri ally. A tually, the method has alowsensitivity relatively to these parametersas soonas

α

and

β

are taken su ientlylarge,greaterthan

1.0 10

2

andlessthan

1.0 10

7

,and

γ

issu ientlysmall,sethere atzero. Itisworthre allingthat this hoi efor

γ

hasastrongimpa tontheassemblypro ess. The plots in Fig. 7 learly demonstratethat a better approximation of the DtN operator greatlyredu es thepollutionee t. Below

N

add

= 3

,there hasbeen absolutelynoadvantage touse24insteadof16nodesperwavelength.

(24)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

10

−3

10

−2

10

−1

10

0

Parameter N

add

for the refinement of the skeleton mesh

Maximum error (%)

BEM−STDG

16 nodes per wavelength

Dimensions of the duct problem

Width: 1 wavelength

Length: 500 wavelengths

Polynomial degree of the local BEM: m = 4

BEM−STDG

24 nodes per wavelength

Figure7: Maximumerrorin %versus

N

add

4.3 Validation of the BEM-STDG method

WerstvalidatetheBEM-STDGmethod ontwoproblemsofsmallsize. Therstone on erns the du t problem onsidered above and the se ond one is related to the approximation of an evanes entwave.

4.3.1 Adu t problem of smallsize

We onsidertheabovedu tproblemforthefollowingdata: ˆ

κ = π,

ˆ lengthofthedu t:

2L = 10

half-wavelengths,widthofthedu t:

H = 2

half-wavelengths, ˆ thi kness of the ontrasted layer: 4 half-wavelengths (

D = 2

) and its refra tive index

relativelytotherestofthedu t:

n

0

= 2

.

Theinteriormeshofthedu tisdepi tedinFig.8. Thetwoverti alstraightlinesdenethe boundaryof the ontrastedlayer.

(25)

0

1

2

3

4

5

6

7

8

9

10

−1.5

−1

−0.5

0

0.5

1

1.5

Abscissa along the lower rigid wall of the duct in half−wavelengths

Real parts of the exact and the BEM−STDG solutions

Exact

BEM−STDG

Figure9: Real partsof theexa tand theBEM-STDGsolutionsfor the onsidered exampleof thedu tproblem.

TheparametersusedfortheBEM-STDGmethodarethefollowing: ˆ Meshsizeoftheinteriormesh outsidethe ontrastedlayer:

h

max

= 1

, ˆ Meshsizeoftheinteriormesh insidethe ontrastedlayer:

h

layer

= 0.5,

ˆ Numberofsegmentsperedgeoftheinteriormeshto gettheskeletonmesh: 16, ˆ NumberofaddedsegmentsfortheapproximationoftheDtNoperator:

N

add

= 4

, ˆ PolynomialdegreeusedintheBEM:

m = 1.

Theplotsin Fig.9depi tthereal partsof theexa tand omputed solutionson thenodes lo atedonthelowerrigidwall

{y = 0}

ofthedu t. Thetwo urves annotbedistinguished.

Thefollowingerrors,whi harealllessthan1%,validatetheBEM-STDGmethod: ˆ Maximumerror:

Err

= 0.4 %

;

ˆ Transmittedwave:

Err

T

= 0.06 %

; ˆ Ree tedwave:

Err

R

= 0.3 %.

4.3.2 Approximation of anevanes entmode

Now,wetesttheabilityoftheBEM-STDGmethodto orre tlyapproximateevanes entwaves. For this ase too, we adapt the onditions leading to an evanes ent mode in [27℄. We thus onsider the same du t geometry than for the previous example with the samewavenumber

κ = π

but wenowassumethatthedu t ishomogeneous,thatis,

n

0

= 1

,andtake

u(0, y) = cos(2πy), 0 < y < 2,

(48)

forthedatainvolvedin theDiri hletboundary onditionontheinletboundary. Toensurethat theexa tsolutionisthese ondevanes entmode

(26)

itisenoughtotakethefollowingtransparentboundary onditionontheoutletboundary



x

u +

3πu



(2L, y) = 0, 0 < y < 2.

(50)

Weusedainteriormeshwith

h

max

= 0.5

and,asin theaboveexample,wetook16segments peredge forthe skeleton mesh,

N

add

= 4

forthe renementof theskeletonmesh for thelo al omputationoftheDtN operator. Onlythemaximumerrorremainsmeaningful

Err

= 0.4 %

(51)

and is similar to the ase of propagativemode. The plot depi ted in Fig. 10 shows that the exponentialde ayofthemodeiswellreprodu edbythesolutionobtainedfromtheBEM-STDG numeri als heme.

0

1

2

3

4

5

6

7

8

9

10

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Abscissa along the rigid wall of the duct

in half−wavelengths

The Exact and the BEM−STDG solutions

Exact

BEM−STDG

Figure10: Exa tand omputedevanes entmodealongthelowerrigidwallofthedu t.

4.4 Long-range propagation

Now, we ome to the main motivation for onsidering this BEM-STDGmethod: its ability to redu e the pollutionee t and hen eto perform orre tnumeri al simulationsof long-range propagation. Towardthisend,we onsiderthe aseoftheabovehomogeneousdu ttogetherwith the stru tured mesh giventhere. We omparethe maximumglobal errorsin % dened earlier versusthelengthofthedu tfortheBEM-STDGmethod withamore onventionalpolynomial IPDGmethod( f.,forexample,[15℄).

It wasnoteasy to nd a ommonbasis for omparing thetwomethods sin e thea ura y of theoverallsolutionof theBEM-STDG methodis mainly basedontwomeshes: theinterior andtheskeletonones,andthepolynomialIPDGmethod usesausualstru tured niteelement mesh in triangles only. Anyway, the following ba kground seems to be a good basis for this omparison:

ˆ usepolynomiallo alapproximationsofthesamedegreeforboththeBEM-STDGandthe polynomialIPDGmethod;

(27)

ˆ assumethatthedegreesoffreedomoftheIPDGmethodarethenodesofthe orresponding Lagrangeniteelementmethod;then hara terizeea hofthesetwomethodsbythedensity ofnodesalongea hedge(numberofnodesperwavelength). Forinstan e,forapolynomial IPDG method onstru ted ona stru tured mesh in isos elesre tangular triangles whose lengthofaright-anglesideis

1/N

h

,andforaskeletonmesh builtonthestru turedmesh giveninFig.6with

N

h

segmentsalongea hedge,thedensity, hara terizingboththetwo methodsforpolynomialshapefun tions ofdegree

m,

willbe

2mN

h

.

Thiserror,as afun tion ofthelengthofthedu t,generallytswellwithastraightline, at leastforlargeenoughlengths. TheLeastSquareGrowRate(LSGR)istheslopeofthisstraight line,whi hisobtainedbytheleastsquaremethod. Itisusedasanindi atorfortheimpa tofthe pollutionee t. Below,wesu essively omparethetwomethodsfromlowdegreepolynomial approximations orrespondingto

m = 1

tohighdegreeones orrespondingto

m = 4

forvarious densitiesofnodes perwavelengthandfordu tswithlengthupto500wavelengths.

4.4.1 Lowest polynomialdegree

Forthe lowestpolynomialdegree

m = 1

, theBEM-STDG method widelyout lasses the usual polynomial IPDG method. The error of the latter even with a double density of nodes per wavelength is 10 times higher. To be able to plot the error urves orresponding to the two methods in Fig.11, wehavehad to usetwoaxesat twodierents ales. Clearly, asindi ated bythereportedLSGR, theimprovementgainedbytheBEM-STDGmethodismainly duetoa mu h betterredu tionof thepollutionee t.

0

50

100

150

200

250

300

350

400

450

500

0

20

40

60

80

Maximum Error (%) −− Dashed Line

0

50

100

150

200

250

300

350

400

450

500

0

2

4

6

8

Maximum Error (%) −− Solid Line

Length of the duct in wavelengths

BEM−STDG

Dens. 32 nodes /

λ

LSGR: 0.01

Polynomial IPDG

Dens. 32 nodes /

λ

LSGR: 0.9

Polynomial IPDG

Dens. 64 nodes /

λ

LSGR: 0.2

Figure11: Maximumerrorin%forpolynomialapproximationsofdegree

m = 1

. Theleft

y

-axis orresponds to the error urvesof theIPDG method and the right

y

-axisto the BEM-STDG method.

4.4.2 Higher polynomial degrees

For polynomial degrees from

m = 2

up to

m = 4

, we have done three ben hmark tests: the nearestdensitiesto respe tivelyone,oneandhalf,and twotimestherule oftumb of12 nodes perwavelength.

(28)

Polynomialdegree Density(nodes/

λ

) Method Error LSGR

m = 2

12 IPDG 72%

4.1 10

1

BEM-STDG 22%

4.3 10

2

16 IPDG 67%

1.3 10

−1

BEM-STDG 5.6 %

1.1 10

2

24 IPDG 13%

2.7 10

2

BEM-STDG 0.8 %

1.5 10

−3

m = 3

12 IPDG 19%

3.7 10

−2

BEM-STDG 1.6 %

3.0 10

−3

18 IPDG 1.7 %

3.5 10

3

BEM-STDG 0.1 %

1.0 10

4

24 IPDG 0.3 %

6.2 10

−4

BEM-STDG 0.02%

−2.6 10

10

m = 4

8 IPDG 1.8 %

3.9 10

3

BEM-STDG 10.4%

2.0 10

2

16 IPDG 0.17%

3.0 10

−4

BEM-STDG 0.02%

4.3 10

6

24 IPDG 0.007%

1.3 10

5

BEM-STDG 0.003%

3.0 10

−12

Table1: Maximumerrorin%foradu tof500wavelengthsandLeastSquareGrowRateofthe errorasafun tionofthelengthof thedu t.

Theresultsarereportedin Tab.1andthe mostfeaturingof theseare depi tedin Fig.12, Fig. 13, Fig. 14, and Fig.15. The negative LSGR for

m = 3

and a density of

24

nodes per wavelengthis ertainlydueto roundingerrors(seealsoFig.13below).

Alltheseben hmarktests,ex epttheone orrespondingtoapolynomialdegree

m = 4

anda densityof8nodesperwavelengthdepi tedin Fig.14, onrmthattheBEM-STDGmethodis abletoredu ethepollutionee tmu hmoree ientlythantheusualpolynomialIPDGmethod. The asewheretheBEM-STDGmethodsu eededlesswellthanthepolynomialIPDGmethod is that where thedensity wasonly of8 nodesperwavelength, hen ebeing lessthan theusual ruleofthumbof12nodesperwavelength. ThissuggeststhattheBEM-STDGmethodrequires aminimaldensityofnodesto bee ient.

It must alsobe noti ed that the BEM-STDGmethod su eeded to pra ti ally ruboutthe pollutionee t upto500wavelengths forpolynomialapproximations

m = 3

and

m = 4

with 24nodesperwavelength(seeFig.13andFig.15), ontrarytotheIPDGmethodforwhi hthis error ontinuestofeatureevenatalowlevelinsome ases.

(29)

0

50

100

150

200

250

300

350

400

450

500

0

10

20

30

40

50

60

70

80

Length of the duct in wavelengths

Maximum Error (%)

BEM−STDG

LSGR: 4.4e−2

Poly. IPDG

LSGR: 4.1e−1

BEM−STDG

Least Square

Poly. IPDG

Least Square

Figure12: Maximumerrorin % forpolynomialapproximationsofdegree

m = 2

andadensity of12nodesperwavelength.

0

50

100

150

200

250

300

350

400

450

500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Length of the duct in wavelengths

Maximum Error (%)

BEM−STDG

LSGR: −2.6e−10

Poly. IPDG

LSGR: 6.3e−4

BEM−STDG

Least Square

Poly. IPDG

Least Square

Figure13: Maximumerrorin % forpolynomialapproximationsofdegree

m = 3

andadensity of24nodesperwavelength.

(30)

0

50

100

150

200

250

300

350

400

450

500

0

2

4

6

8

10

12

Length of the duct in wavelengths

Maximum Error (%)

BEM−STDG

LSGR: 2.1e−2

Poly. IPDG

LSGR: 3.9e−3

BEM−STDG

Least Square

Poly. IPDG

Least Square

Figure 14: Maximumerrorin% forpolynomialapproximationsofdegree

m = 4

and adensity of8nodesperwavelength.

0

50

100

150

200

250

300

350

400

450

500

1

2

3

4

5

6

7

8

x 10

−3

Length of the duct in wavelengths

Maximum Error (%)

BEM−STDG

LSGR: 3.0e−12

Poly. IPDG

LSGR: 1.4e−5

BEM−STDG

Least Square

Poly. IPDG

Least Square

Figure 15: Maximumerrorin% forpolynomialapproximationsofdegree

m = 4

and adensity of24nodesperwavelength.

Figure

Figure 1: Shemati view of the omputational domain.
Figure 2: Interior mesh in 2D.
Figure 3: Skeleton mesh and nodes used in the 2D ase.
Figure 4: Shemati view of the three kinds of meshes, whih are used in the BEM-STDG
+7

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