Combined Integral Equation Formulation and Null-field Method for the

Combined integral equation formulation and null-field method for the exterior acoustic problem B. Stupfel,A. Lavie,and J. N. Decarpigny Laboratoire d'Acoustique, L. ,4.253au CNRS,InstitutSupbrieur d'Electronique duNord, 41, Boulevard Vauban, 59046Lille Cedex, France

(Received30August1987;accepted forpublication 8 November1987) The numericalimplementation of a combinedintegralequationandnull-fieldmethodusedto solvetheexteriorNeumannproblem[D. S.Jones, Q. J. Mech.Appl.Math.XX¾II, 129 (1974)] ispresented here.The exteriorHelmholtzintegralequationissolvedon theradiating or scattering surface,and theirregularfrequencies areeliminatedup to a givenirregular frequency f•t throughthe useof M additionalnull-fieldequations. An impedance matrix, definedon the objectsurface,is thenobtainedthat canbe usedasan exactradiationcondition in a finite-element code.The programandthenumericalexamples presented hereare

specialized to axisymmetrical problems. A purelynull-field methodisimplemented andsimple rulesaredefinedthatdisplayitsfailurewhenappliedto highaspect-ratio objects and (or) in thehigh-frequency range.Similar,butlessrestricting, rulesareusedto specify thenumerical limitationsof Jones'technique. Besides, a fewtheoretical considerations clarifytheroleplayed bytheadditional null-field equations in theelimination of theirregularfrequencies andhelpin performing accurate high-frequency computations for surfaces suchasthecircularconeand the finite circular cylinder. PACS numbers: 43.20. Rz, 43.20.Tb

INTRODUCTION The numerical

determination

of the farfield and/or

nearfieldpressurein the fluid surroundingan activeor passivetransduceris oneof the majorproblemsin acoustics. It can be solvedentirely through the finite-elementmethod (see,e.g.,Refs.1-4). However,if thismethodisparticularly well suitedto model the transducer,the structureof which is

oftencomplexandcomposite, thesolutionof thewholecoupledfluid-structure problem,whentheoveralldimensions of the transducerare largecomparedwith the wavelengthof the radiated or scatteredwaves,involveslinear equations systems of largesize,the solutionof whichnecessitates the useof powerfulcomputers.The sizeof thesesystemscanbe considerablyreducedif the radiationconditionis exactly accountedfor on the surfaceof the devicethrough,e.g., an integralequation.Thisimpliesthecouplingbetweenthefinite-elementmethodandan integralequationformulation. Here, we describethe developmentof a computerprogram,solvingthethree-dimensional Helmholtzequationassociatedwith a prescribedNeumannboundaryconditionon the radiatingor scatteringsurface& througha combined integralformulationand null-fieldmethod.This program and, consequently,the numericalresultspresentedhere, are presentlylimitedto thesolutionof axiallysymmetricalproblems.One of its importantfeaturesis that S, as well as the known and unknownfunctionson S (namely the normal component of the surfacevelocityandthe pressure),areapproximatedby second-order polynomials, identicalto those used in the finite-element code ATILA.4-6

The solutionof the 3D Helmholtz equation,

for the pressurep(r)in the infinitedomainI/e exteriorto S with prescribedboundaryconditionson S, is a well-known problemin acoustics andcanbe performedmainlythrough the useof threemethods(not mentioningthe finite-element method): (i) seriesexpansionofp (r) in functionssolution of (1) and satisfyingthe radiationcondition,(ii) null-field (or extendedboundarycondition)method,leadingto the Tmatrix methodfor a scatteringproblem,and (iii) integral equation formulation. Thefirstmethodisstandard whenSis a coordinatesurfaceof oneof the elevencoordinatesystems

in which( 1) isseparable7; if not,it isstilltheoretically justified,8independently ofthevalidityoftheRayleigh hypothesis,thoughit hasbeenveryseldomly used. øThenull-field approach•ø allowsthedetermination ofthesurface pressure (Neumann conditions) or of the surface normal velocity

(Dirichlet conditions).It is widelyusedin acoustics in the T-matrixformulationto computethe directivitypatternsin scatteringproblems(see,e.g.,Refs. 10-14); however,in its usualform, i.e., whensphericalwavefunctions [solutionsof (1) in the sphericalcoordinatesystems]are used,it is known to be moderatelyaccuratefor high aspect-ratiosur-

faces. •4-•6Theintegralequation formulation offerstwopossibilities,dependingontheparticularintegralrepresentation chosenforp (r) (see,e.g.,Reft 17): Eitherthecorresponding integralequationis uniquelysolvableon & or it possesses indeterminatesolutionsfor discretefrequencies,sometimes calledirregularfrequencies. We donot considerintegralrepresentations leadingto anintegralequationthatmayhaveno solutionfor thesefrequencies, suchasthesimplelayerrepresentation.For a Neumannboundaryproblem,the solution

ofanintegralequation ofthefirsttype18-2• isnotstraightforwardsinceit involves(i) highlysingularkernelsand (ii) the

Ap(r) + k:p(r) =0, 927

J. Acoust.Soc. Am. 83 (3), March 1988

(1)

determination of an unknown surface distribution which,

0001-4966/88/030927-15500.80

@ 1988 AcousticalSocietyof America

927

unlikethe pressure,is not necessarily continuouson $. The

numerical implementation ofaninte$ralequation ofthesecond type is easier(see,e.g., Ref. 22). However,sincethe numberof irregularfrequencies increases rapidlywith fre-

quency 23andthenumerical solution oftheintegral equation

Whenf is an irregularfrequency, i.e.,whenk = 2rrf/c is an eigenvalue of the associated interiorDirichlet problem

Ar/(r) +k2r/(r) =0,

reV,.,

r/(r) = 0, reS,

(6)

generatesnecessarilyan indeterminateerror, dependingon

then the indeterminate solutions of (4) are of the form

theprecision ofthecomputer andthenatureofthecomputationalgorithms, 24it isgenerally necessary--for mediumor

[p(r) + u(r)], where u(r) isdefined onSi•y

high frequencyproblems--to eliminatetheseindeterminaticns. For Neumann boundary conditions,this has been

y(r)u(r) =fsu(r')On'g(r,r')dr', reS, (7)

r/(r) by achieved byrequiringp(r)toverifyoneorseveral add(tional andis relatedto the eigenfunction equations ensuringtheunicityof solutions. A goodexample ofthisisgivenbySchenck. 22He chooses thefollowing integralrepresentation forp (r):

•7(r)=fsU(r')On'g(r,r')dr', reV.. (8)

p(r) =•sP(r' )3n' g(r,r' )dr' (2) - fs3n'p(r')g(r,r')dr ', reV,,

Schenckhasshownthat if everysolutionof (4) alsosatisfies (5) for at leastonepointr in V,.,thenall the irregularfrequencies are suppressed, providedthat r is not situatedon a nodal surfaceof the corresponding eigenfunctionr/(r). However,froma formalpointof view,thismethodpresents seriousdrawbackssince(i) the nodal surfacesare not usual-

thetimedependence of thepressure beinge - lot.Here,r, r' are arbitrarypoints,the origin0 of the coordinate system beingchosen anywhere in theinteriorregionV,.,providedit

ly knownapriori,(ii) theirdensityincreases withfrequency, and (iii) no ruleshave beengiven on how many points

doesnot belongto S (seeFig. 1); n' istheunit normalto S at

seemsthat thesepointsaredistributedat randominsideV• sothat it isunlikelytheyall fall onnodalsurfaces. For low or moderatelyhigh frequencies,this techniqueis success-

pointr', pointing towards Vt' g(r,r') = eikIr- r,i/4rr Ir - r'I is the freespaceGreen'sfunction(k = to/c). Here, On'denotesnormaldifferentiation at the pointr' in the direction from Vi towardsS and

On'p(r') = itopo(r'),

(3)

whereo(r') isthenormalcomponent of thevelocityat point r' andp is the fluid densitymodulus.For a radiationproblem,o(r') isiknownonS; for a scatteringproblem,the incident pressuremust be addedto the right-handsideof (2) and, in the caseof a perfectlyrigid object,On'p(r') = 0 is

putinplaceof ( 3),p beingthetotal(incident plusscattered) pressure. If r belongs toS,thenweobtainthecorresponding "exteriorHelmholtzintegralequation":

should bechosen in Vi andwheretheyshouldbelocated. It

ful.22,25. 26

Here, we implement numerically another method-thoughbasedon thesameprinciples--whose authoris D. S.

Jones. 27Jones hasestablished thefollowing result, whichwe shallafterwardscallR 1:Ifp ( r ) simultaneously satisfies (4)

andthesetofM 2null-field equations,

fsP(r)On •",, (r)dr =fs0n p(r)•, (r)dr, 0 N givesno betterresults.The linearsysSimilarly,if we take q•. = •., whichdo constitutea com- tem (17a) is theninvertedusingstandardtechniques. Usedasit stands,the null-fieldmethodgiveserroneous pletesetonS,32thenN is limitedby thegreatest available resultsfor smallfrequencies and/or largeM, owingto the number, since undesirable numericalbehaviorof •,. (r) for the corresponding values ofkr and/orrn:If 10• isthegreatest number x/• (2n+ 1•n+' 1

j.(kr).• 2n+l'•2•,2n+1]' kr•