4-2.\

Numerical solution and a posteriori error estimation of exterioracoustics problems by a boundary element method at highwave numbers

P. Geng, J. T. Oden, and L. DemkowiczTexas Institute for Computational and Applied Mathematics. The University of Texas at Austin. Austin.Texas 787/2

(Received 10 August ]995: accepted for publication 27 February 1996)

This paper is concerned with the application of boundary element methods to the exterior acousticalproblem at high wave numbers. The major issues here are to establish a strong theoreticalfoundation for the application of the Burton-Miller integral equation and develop a practical wayfor its numerical implementation. Unlike many conventional approaches. the problem in this studyis formulated by the Galerkin method. Through an analysis on its ellipticity. the Burton-Millerequation is proven to be well-posed in the H,n-Sobolev space and its approximation can attain aquasioptimal convergence rate. The Galerkin method avoids sensitive properties of hypersingu]arintegration. simplifies the numerical implementation. and improves the quality of the numericalsolutions. especially at high wave numbers. An L2-norm residual error estimation technique is alsoimplemented for an adaptive scheme for these problems. The numerical implementation iscompleted on parallel distributed-memory machines as well as conventional sequential machines.The validation of the method at high wave numbers is done through tests on a series of numericalexamples. © /996 Acoustical Society of America.

PACS numbers: 43.40.Rj [CBB]

INTRODUCTION

Much of structural acoustics is concerned with the clas-sical problem of interaction of the motion of a solid structurewith that of a fluid into which the structure is immersed.Small perturbations in the velocity or pressure fields of thefluid create waves that impinge upon the structure and scatterinto the fluid-structure domain while small motions of thestructure may radiate energy into the surrounding fluid me-dia. A major issue involved in the study is the propagation ofwaves in the fluid media. In classical models. the fluid istreated as a quiescent. inviscid medium capable of transmit-ting acoustical wave in a uniform pressure field that is char-acterized by the Helmholtz equation defined on an un-bounded exterior domain.

A classical technique for handling the exterior problemcharacterizing the fluid is to convert the exterior differentialHelmholtz equation into an integral equation. Then a dis-cretization of the integral equation by the boundary elementmethod (BEM) allows one to coniine the numerical approxi-mations to the fluid-structure boundary-the so-called wetsurface. Consequently. the dimensionality of the problems isreduced by one and the exterior boundary issues are avoided.

Unfortunately. the integral form of the Helmholtz equa-tion is not perfectly equivalent to the original differentialequation on the exterior domain and. thus at certain frequen-cies. the so-called forbidden frequencies. the solution of theintegral equation is not unique. Several boundary integralformulations have been proposed to overcome this nonu-niqueness problem: see Refs. 1-6, and the CHIEF (com-bined Helmholtz integral equation formulation) method. pro-posed by Schenck.4 is a widely used method in engineeringcomputations. In the CHIEF method. the system of linear

equations is combined with a few additional equations gen-erated from the Helmholtz integral equation with the sourcepoints in the interior of the closed boundary. The method canbe used to analyze a class of acoustical scattering problems.but because the stability of the method depends strongly onthe number and locations of the source points chosen in thecomputation, the method may not be reliable and efficient athigh wave numbers.

A method which seems to be valid for all cases wasproposed by Burton and Miller in 1971.1 However. directimplementation of the Burton-Miller integral formulationusing the conventional collocation scheme leads to the prob-lem of so-called hypersingular integrals and the method itselfis very complicated and often yields very poor results.

Unlike the conventional approach. the numerical imple-mentation in this study is based on a Galerkin scheme. TheGa]erkin method for the boundary integrals ha~ been treatedextensively by Wendland and Hsiao.1.ll They proved that alarge class of integral equations are strongly elliptic type forwhich the Galerkin method shows an optimal convergencerate. In this paper. we give a summary on their work anddemonstrate that the Galerkin weak form can avoid the sen-sitive properties of hypersingular integrals involved in theoriginal formulation and the numerical solution can attain anoptimal convergence rate.

h is also worth mentioning that the boundary elementprocedures described have been implemented in a speciallip-hierarchical data structure proposed in Refs. 9 and ]0.This data structure allows the mesh size and order of poly-nomial interpolation to be varied over the mesh during com-putation. In addition, an a posteriori error estimate originallyproposed in Ref. 9 is further studied here. The error estima-tion is based on the L2 norm of the residual which is simple

335 J. Acoust. Soc. Am. 100 (1). July 1996 0001-4966/96/100(1 )/335/11/$6.00 © 1996 Acoustical Society of America 335

We refer here to (4) as the normal Helmholtz integral equa-tion and (5) as the hypersillgular illfegral equation.

In 1970. Burton and Millerl proved that neither (4) nor(5) but their complex linear combination is equivalent to theOIiginal differential equation (I) accompanied by the Som-merfeld radiation condition (3).

Using (2), the Burton-Miller fonnulation can be written

to calculate and yields good results in experiments. All ofthese features provide a good basis for the development ofadaptive methods.

The paper is organized as follows. We first describe themathematical model for the problem of interest and give ananalysis on its mathematical properties together with an aposteriori error estimation scheme: next we discuss basicprinciples followed in the problem discretization: and. fi-nally. we present the results of numerical experiments.

I. PROBLEM STATEMENT

I ap(x) a r [ a<t)(kr)2' an(x) - an(x) Jrc p(y) an(y)

apinc(x)

an(x) .

ap(y) <l>(kr)]dS(y)an(y)

(5)

and

exp(ikr)<l>(kr)= A in three dimensions,

7Tr

i<l>(kr)= 4' Hb(kr) in two dimensions

(6)=f(x).

f(x)=pinc(x)+ r g(y)<l>(kr)ds(y) + a ~( ) [pine(x)Jr dn x

+ frg(Y)<l>(kr)dS(Y)] - ig(x).

Equation (6) defines the mathematical model used in thisinvestigation to solve the problem of the acoustic wave scat-tering on the exterior domain De.

as

or

where a is a complex constant whose imaginary part cannotbe zero and

1 r [a<l>(kr) ]2" [I +aG(x)]p(x)- Jr an(y) G(y)<l>(kr) p(y)ds(y)

a r [a<l>(kr) )- a an(x) lr an(y) G(y)<l>(kr) p(y)ds(y)

II. THEORETICAL CONSIDERATIONS

In this section. we discuss several theoretical issues in-volved in the Burton-Miller integral formulation. The dis-cussion involves certain basic concepts in functional analysisand integral equations, and we here refer to Refs. II and 12for further references. The major point in this section is thatthe Burton-Miller formulation (6) is a strongly elliptic equa-tion. and the corresponding numerical solution by a suitablefamily of Galerkin approximations is stable and convergent.Most importantly, the proof of convergence and stabilityonly requires that the boundary be of Lipschitz-type. i.e ..there is, in this context, full theoretical justification of usingthe boundary element method to solve problems with nons-mooth boundaries.

For convenience. we rewrite (6) in an operator form:Find p E Hln(r) such that

t< 1+ aG)lp - M /cP+ LkGp- a(NkP- M[Gp) =f (7)

(I)

(3)

(2)

ap(y) <1>(kr)]dS(y)=pine(X).an(~ ~)

apl =Gp+g.(In r

[apS(x) ]

Jim p -a-- ikpS(x) =0,p .... oc P

and the boundary conditions

12" p(x)- fr[p(y) a<l>(kr)

where k = wlc is the acoustic wave number with c and w, thespeed and angular frequency of sound, p=lxl represents thedistance between origin and the position x, n denotes theoutward normal unit vector on the surface r, pS = p - pinc isthe scattered wave pressure. and pine is the pressure of inci-dent wave. The boundary condition (3) is the Sommerfeldradiation condition and in (2). G and g are two given func-tions on the boundary r.

Equations (J) and (3) can be converted into an integralequation on the boundary r. i.e ..

In (4). x and yare two points on the boundary. r=ly-xl. andn(·) denotes the outward normal unit vector at the corre-sponding position. Here, <l> represents the fundamental solu-tion of the Helmholtz equation and is of the form

The problem under consideration is the exterior acousti-cal scattering problem. To be precise, we wish to find thepressure distribution p = p(x) in a domain De exterior to aclosed bounded surface r. The scalar field p satisfies theexterior Helmholtz equation

where Hb is the Hankel function of the first kind of zeroorder and i= Fl. Applying the directional derivativea/an(x) to both sides of (4) leads to another form of theHelmholtz integral equation:

Ap=f,

where

A = t< 1+ aG)I-Mk+LkG-a(Nk-M[G).

(8)

336 J. Acoust. Soc. Am.. Vol. 100. No.1. July 1996 Gang at al.: BEM at high wave numbers 336

(17)

(16)

(I5)

lIu-uhllv~c inf Ilu-uhllv.vh EVil

Find u E V such that

a(u.u)=l(u) 'rIu E V.

where y= 1/2 on the smooth boundary and takes on differentvalues at edges or corner nodes. In the conventional colloca-tion method. an extra consideration must be taken to solveproblems with nonsmooth boundaries (for example, see Ref.14): and special treatment is also needed to evaluate normalderivative a/all at edges or corner nodes. However, math-ematically. the measure of all edges and corner nodes is zero.which implies that

f [ a<l>(kr) ap(y) ] .yp(x)- r p(y) an(y) all(Y) <I>(kr) dS(y)=plDC(x).

where h is the mesh size and m is the order of polynomialson each element. Thus (17) in fact indicates that the error ofnumerical solutions will also converge with a rate ofO(hm+In), and this property is usually referred to as theoptimal convergence rate.

For problems with nonsmooth boundaries. Eq. (4)should be written as

where u is the exact solution and uh is the correspondingnumerical solution in Vh

, and c is a positive constant whichis independent of h. uh. and 14. In (17). infvhevhllu - uhllv isthe interpolation error (the error of the best solution whichwe possibly obtain in Vh) in the norm of V. For V=H1n(f).

inf Ilu-uiJllv=O(hlll+ln).

vh E vh

In Ref. 8, Wendland gives the following theorem for theGalerkin approximation of a strongly elliptic equation

Theorem 3.1: For a uniquely solvable strongly ellipticequation, the numerical solution of the Galerkin approxima-tion is stable for almost all h>O. i.e .. there exists a positiveho such that the approximate equation (16) is uniquely solv-able for O<h<ho. and furthermore, there holds

where V is a Hilbert space (V=Hln(f) in this case), a(u.u)= (Au. u) is the bilinear or sesquilinear form correspondingto the left-hand side of the original equation. l( u) = (f. u) isthe linear form corresponding to the right-hand side of theoriginal equation. and ( ... ) denotes L2 inner product.

For the approximation of (15). let {Vh} be a sequence of

approximating finite dimensional subspaces of V (h typicallyrepresenting the element mesh size parameter). In this article.h -+0 means the mesh size approaches zero or the order ofinterpolation polynomials on each element approaches 00.

We assume that the family Vh of approximating spacesis constructed so that as h -+0. Vh approaches to the realspace V and a better numerical solution is expected. For agiven subspace VhC V. the Galerkin approximation of (15)can be written as

find uh E Vh such that

(9)

(14)

(13)

(12)

(I 1)

(10)

and C is a compact operator on H1n(f). Here, r is requiredto be at least Lipschitz (r E CY and y>O).

In (l3).llull In represents the norm of u in H1n(f) and isdefined by

Ilul17n=J lu(xW ds(x)r

J I lu(x)-u(y)12+ 1 Im+1n ds(y)ds(x).r r x- Y

where D is positive definite. i.e .. there exists a {3>O such that

Ir u Du ds~{3llullin' 'rIu E H1n(f)

the adjoint of the double-layer potential

MIu(x)= an~x) fr <I>(kr)u(y)ds(y),

and the hypersingular operator

a f a<l>(kr)Nku(x)= an(x) r ~u'..\ u(y)ds(y).

Using the methods established in Ref. 13, we can readilyprove that Eg. (8) defines a strongly elliptic equation on theHilbert space HI/2(f). i.e., we can express A as the sum

A=D+C.

where m represents the dimensionality of the surface r(m =] on one-dimensional curves and m =2 on two-dimensional surfaces). The space H1n(f) is called a frac-tional Sobolev space, and. mathematically. it is defined asthe completion of C"'(f) in the norm of (14).

The strong ellipticity of (8) implies that the Fredholmalternative is applicable and the existence of solution followsfrom uniqueness. The uniqueness of the solution of (6) forthe Neumann boundary condition was proven by Burton andMiller in Ref. I and the proof for the Dirichlet boundarycondition is similar. However. for the general boundary con-dition given in (2). the original Helmholtz equation (I). aswell as the boundary integral equation, may not be uniquelysolvable at certain wave numbers or frequencies. which arethe result of certain resonant frequencies introduced throughthe boundary condition (2). Thus except for the resonant fre-quencies. the boundary integral equation (6) is uniquely solv-able on the boundary r.

In the Galerkin method. we convert the original equation(6) into an equivalent variational formulation, i.e .. we con-sider the following problem:

the double-layer potential

r a<l>(kr)Mku(x)= Jr ~--{..\ u(y)ds(y).

I is the identity operator, and Lk' M k' Mf, and Nk are linearoperators defined in Ref. 13 as follows:the single-layer potential

Lku(x)= fr <I>(kr)u(y)ds(y).

337 J. Acoust. Soc. Am., Vol. 100. No.1. July 1996 Gang et al.: BEM at high wave numbers 337

III. THE VARIATIONAL FORMULATION ANDIMPLEMENTATION

Ir yp(x)q(x)ds(x)= fr ~ p(x)q(x)ds(x).

In this section. we present a variational fonnulation of(6). A special problem arising in the implementation of theBunon-MiJler formulation is that the integral

(22)

f a2<1>(kr)l(y) an(x) an(y) ds(y)

= Jr{[n(y) xV yp(y)] ·[n(x) x V x4>(kr)]

+k2n(x)· n(y)<I>(kr)p(y) }ds(y).

Instead of one singular term, the integrals on the right-handside of (20) and (2 I) contain the difference of two singularterms, and as r=ly-xl-+o. the difference is a finite boundedfunction. and the integrals becomes computable. However,such approaches require the subtraction of two nearly equallarge numbers numerically, which will inevitably cause largenumerical error in final results. The approaches based on thefinite part integral method cannot, in general produce satis-factory results. specially, at high wave numbers.

In Ref. I, Burton and Miller also presented the formula-tion

where

1 1'I'(r) = -2 log - in two dimensions.

1T r

]'I'(r) = -4 in three dimensions

1Tr

and

(19)

(I8)a J a<l>(kr)

an(x) r an(y) p(y)ds(y)

in (6) cannot be evaluated easily in a meaningful way. Sym-bolically. it is customary to write (18) in the form

( a24>(kr)J r an(x) an(y) p(y)ds(y).

even on a nonsmooth boundary, q representing a test func-tion. in the Galerkin method; thus we can let y= 1/2 in allcases. Furthermore, in the Galerldn method. we only com-pute the function value and its normal derivative at integra-tion points which are always located inside each element,and, thus the evaluation of the function value and its normalderivative at edges or corner nodes is avoided automatically.There is thus no need for any special numerical treatment forproblems with non smooth boundaries if the Galerkin methodis used.

and

a2<1>(kr) ( I )an(x) an(y) = O? in three dimensions.

The integral kernels a2<1>(kr)/an(x) an(y) are said to be hy-persingular. and the integral in (I9) does not exist in thenormal sense.

The integral (18) can be evaluated in the sense of thefinite part integral; and the following two formulations aregiven in Refs. 15, 16, respectively,

a J a<l>(kr)an(x) r '_1 .. \ p(y)ds(y)

([ a24>(kr) a2'fr(r)]= Jr an(x) an(y) p(y)- an(x) an(y) p(x) ds(y)

(20)

in two dimensions

We can easily verify that as r-+O

a2(ll(kr)

an(x) an(y)

and

a f a<l>(kr)an(x) r an(y) p(y)ds(y)

i a2<1>(kr)= '_1 •.\ ,_1 ..\ [p(y)-p(x)]ds(y)r .

- p(x) Ir n(y)· n(x)(ik)2<1>(kr)ds(y), (21)

The transformation in (22) reduces the order of the singular-ity by one (to I/r in two dimensions and l/r2 in three dimen-sions), and the integral is computable in the sense of Cauchyprincipal value integrals. A numerical implementation basedon (22) can be found in Ref. 17. However, the Cauchy prin-cipal value integral is still inconvenient in numerical imple-mentations and may also yield excessive error at high wavenumbers.

The method used in the present investigation beginswith (22). Using the Galerkin method, we multiply bothsides of (22) by a test function q (the conjugate of q) andintegrate again over r. This yields

( ( a24>(kr)J r q{x) J l(y) an(x) an(y) ds(y)ds(x)

= f q{X)f {[n(y) xVyp(y)], [n(x)xV ,,4>(kr)]r r

+k2n(x). n(y)<I>(kr)p(y)}ds(y)ds(x). (23)

It is easy to verify (see Refs. 18 and 19 for details) that forpiecewise smooth test functions q and trial functions p,

Jr if{x) fr[ n(y) x V yp(y)], [n(x) x V x4>(kr) ]ds(y)ds(x)

= - fr fr 4>(kr)[n(y) xVyp(y)]

. [n(x) X V x'll.x)]ds(y)ds(x).

Then, (23) can be written as

338 J. Acoust. Soc. Am., Vol. 100, No.1, July 1996 Geng et al.: BEM at high wave numbers 338

leads to a system of linear equations for pi (i = 1.. .. ,N). Le.,N

L a/ipi=bt• 1= 1, ... ,N (28)i=1

(32)IimIIC(J-rIh)11 =0,11-0

where

where 11·11 represents the norm on the space$(L2(f),L 2(f). •

Lemma 5.2: Assume (30) is uniquely so]vable: then forthe compact operator C and the L2.projection rIh' the in-verse operators (1+CrIh) -I exist and are uniformlybounded for all sufficiently small h. Le., there exist positivenumbers ho and M such that, for all h~ho.

11(J+CrIlI)-III~M. (33)

•Lemmas 5.1 and 5.2 can further lead to the following

result:Lemma 5.3.' Suppose that (30) is uniquely solvable, then

there holds

We have the following two classical results for a com-pact operator C and the L2.projection IIh (see, for example,Refs. II and 20):

Lemma 5.1: Suppose that operator C: L2(f)-+L2(f) iscompact. Then there holds

where I is an identity operator. f= 2pinc is the right-handside function. and

r [ a<l>(kr) ap(y) ]Cp= -2 Jr p(y) an(y) an(y) <I>(kr) ds(y).

On a smooth boundary r. C is a compact operator on L 2(f),and (30) becomes a Fredholm equation of the second kind.

Let VhCL 2(f) be the space on which the problem issolved, ph be the corresponding approximate solution, and ehbe the error,

eh=p_ph. (31)

Let rIh: L2(f)-t Vh be the orthogonal projection onto thesubspace Vir, Le., for any pEL 2(f),

(rIhP,Uh)=(P,Uh) 'VUhE Vh.

(26)

(27)N

q(x)= L qi7Ji(X).i=1

r r a2<1>(kr)Jr q{ x) J l(y) an(X)an(y) ds(y)ds(x)

= IrIr <I>(kr){-[n(y) xVyp(y)]-[n(x) xVxlJ"lx)]

+en(x)· n(y)q{x)p(y)}ds(y)ds(x). (24)

Multiplying both sides of (6) by q, integrating over rand replacing the corresponding terms by (24), we obtain thefollowing variational formulation of (6):

Find p E Hln(f) such that

~ Ir[ 1+ aG(x)]p(x)q{x)ds(x)

r r [a<l>(kr) ]- Jr Jr an(x) G(y)<I>(kr) p(y)q{x)ds(y)ds(x)

- a Ir Ir {- <I>(kr)[n(y) x V yp(y)]- [n(x) xV xlJ"lx)]

[ a<l>(kr)]}+ k2n(x)· n(y)<I>(kr)- an(x) G(y) p(y)q{x)

X ds(y)ds(x) = I/(X)q{X)dS(X), (25)

for any q EH In(f).Equation (25) contains only weakly singular and regular

integrals which can be treated easily in numerical implemen-tations. Approximation of p and q in (25) by the linear com-binations of the same basis functions:

N

p(x)= L pi7Ji(X),i=1

IV. A POSTERIORI ERROR ESTIMATION

where A={adNXN is a nonsymmetric, complex and densematrix and b={bt}NXI are complex vectors. Solving (28) or(29) gives the approximate pressure solution vector p on theboundary r.

In this section. we discuss a method which enables us toevaluate the error of the solution after the problem is solved.For convenienc~ we write the normal He]mholtz integralequation (4) in an operator form

Find pEL 2(f) such that ((J+C)pll,Uh)=(f,Uh) 'VUIIE Vh,

which means that

(34)

(35)

IICehllo= µ(h)llellllo,

where µ(h) is a function of h such that

Jim µ(h)=O,h-O

eh is the error defined in (31). ph is the Galerkin approximatesolution from a finite dimensional space VhCL2(f), and 11·110represents the L 2 norm.

Proof We first define Uh as

Uh=f-Cph. (36)

Let rIh : L2(f)-tL 2(f) be the orthogonal L 2projection ontoVh. The Galerkin approximate solution ph satisfies the equa-tion

(29)

(30)

Ap=b.

(I+C)p=f,

or

339 J. Acoust. Soc. Am., Vol. 100, No.1, July 1996 Geng at al.: BEM at high wave numbers 339

or(43)

Ilhph=fIJ- IlhCph.

and then from (36)

fIhpJa=fIhUh. (37)

Applying fIh to both sides of (31). and substituting it into(37) leads to

fIheJa= fIh(p- ph)=fIh(p- Uh)' (38)

In addition,Ceh= C(p- ph)= f-p- Cph= - p+ f-Cph= -(p- Uh)'

(39)

Therefore,

(/+ CfIh)(p- Uh)= p- Uh+ CIlh(p- Uh)

=-Ceh+CfIh(p-Uh)

= - Ceh+ CIlheh

= - C(/ - fIh)eh. (40)

Replacing p- Uh with -Ceh on the left-hand side of (40)leads to the relation

(/ + CfIh)Ceh= C(I- fIJa)eh

or

where

From Lemma 5.1 and Lemma 5.2.

limIlC(/- Ilh)\I=Oh-O

For the boundary integral equation, IIrhllois readily comput-able after solving for p II ; thus Theorem 5.1 in fact indicatesthat Ihllo can be used to estimate the solution error of thenormal Helmholtz integral equation.

As noted earlier, for the Burton-Miller formulation, itssolution belongs to the space H1n(f) and, according toHsia021 the residual rh must be computed through the hyper-singular integral formulation (5). i.e ..

I apinc(x) ap(x) a f [ acI>(kr)r'=2 -+2-an(x) an(x) an(x) fc p(y) an(y)

ap(y) ]- an(y) cI>(kr) ds(y).

and then

(44)

where Cl and c2 are two positive constants. pJa is the approxi-mate solution of the Burton-Miller formulation, and II·II-Inrepresents the norm in the spaee H-1n(f). the dual ofHln(f). Because the H-tn norm is not easily cornputable inpractice, Ilrhll_In is not readily usable as an error estimator.

A simple approach used in this study is that technically.we can substitute the approximate solution of the Burton-Miller formulation into (41) (the residual equation of thenormal Helmholtz integral equation) to obtain a residual rh•and then the L2 norm of rh• i.e .. IIrhllo, is used as an errorestimation of Ilehllo.Numerical experiments consistently sug-gest that

and IIU+CIlh)-111 is uniformly bounded by (33). Thus wefinally have (35) and complete the proof. •

For any approximate solution ph, we define the corre-sponding residual of the Fredholm equation (30) as

rh=f-ph-Cph (41)

or

rh=eh+Ceh.

Moreover, from (34),

IlrJallo=Ileh+ Cehllo= (l + µ(h»llehllo,

i.e.,

(1-Iµ(h)I)llehllo~lIrhllo~ (I + 1µ(h)I)llehllo.

and, from Lemma 5.1, we have

lim µ(h)=O.h-O

(42)

V. NUMERICAL EXPERIMENTS

The experiments reported here focus on the performanceof the Gaierkin-Burton-Miller method at high wave num-bers and on the effectivity of a posteriori error estimation.The tests are performed on conventional sequential machinesas well as parallel distributed-memory machines. The detailsof the parallel implementation and its performance can befound in Ref. 22. Some numerical results obtained throughparallel computation at very high wave numbers are alsogiven in Ref. 22.

A. Applicability of the method at high wave numbers

In this experiment, the performance of the boundary el-ement method for solving problems at high wave numbers isexplored. Two indices involved are the relative L2 errorIlehlloand the relative L2 residualllrhilo which are computedthrough the formulae

Therefore. we have the following theorem on the behavior ofIlrhllo as h .......O:

Theorem 5.1: Suppose that the Fredholm equation (30)is uniquely solvable. Then there holds

340 J. Acoust. Soc. Am., Vol. 100, No.1. July 1996

(45)

and

Gang et aJ.: BEM at high wave numbers 340

1000

100

f. f.

g g.. ..~ !;; !i! !" ".. ..i i 10

§i ;N N-' -' I ~ ..---" ~~<; e~ .... ."' >1i i!~ 1 N

-'

l/. .10 IS 20 2S ~ 3S ~ 45 50 10 15 20 2S ~ as ~ 4S 50

ka lea

HG. 1. Relationship between the errors and wave number for 64 quadraticelements (solutions of nonnal Helmholtz integral equation with the bound-ary condition of dpldn=p). Clearly. the solutions are unstable at or nearforbidden frequencies. - L2 norm of estimated relative error. _m L2 nonnof relative error.

HG. 2. Relationship between the errors and wave number for 64 quadraticelements (solutions of Burton-Miller fonnulation with the boundary con-dition of dpldn '" pl. The solutions are stable at all frequencies. - L 2-nonnof estimated relative error, ---- L2 nann of relative error.

(48)

(49)

(50)

Ilehllv=IIPe-phllv~c inf IIPe-vhllv,uh e vh

inf IIPe - vhllvuh e yh

inf IIPe - uhIIL2(f)~max{p~m+ l)(x)}hm+ 1,

uheVh xer

where 1I·llv is the norm corresponding to the space V wherethe problem is defined, and

tory results. For example, as shown in Fig. 2. the estimatederror is almost equal to true error when ka <30 and throughthe range of ka =30 to 50, the maximum difference betweentrue error and estimated error is less than 20 percent. Resultsfor the case of E=O (Le., dpldn =0. rigid scattering) aresimilar and are, therefore, not presented.

Example 2: The relationship between kh and the relativeL2 error.

Theorem 3.1 indicates that the approximate solution phof the integra) equations converges at the rate of

where p~m+ t) represents m + I derivative of the exact solu-tion P e' and m is the order of clements. For the harmonicvibration problems, we can assume that the exact solution Peis oscillating in the sense that

p~m+l)(J(km+l.

is the interpolation error-the error of the best possible so-lution in the space of Vh; and. if the exact solution p e issmooth enough, according to the Aubin-Nitsche lemma,24we can obtain also the L2 estimate

Because

Then, if e is the constant in (48). the solution error willdepend only on kh, i.e.,

(46)

(47)

11':112= f rlrll12ds(f)o r I '7 I I'n,,'

where p e and pll represent the exact and numerical solutions.respectively. and ,11 is the residual computed by (4 I),

The acoustic scattering problems of a plane harmonicwave impinging on a cylinder (in two dimensions) or asphere (in three dimensions) with the boundary condition

ap-=€p on ran

are used here as test problems. The analytic solutions exist(see Ref. 23 for details) in both two- and three-dimensionalcases, and we are able to compare them with numerical re-sult ...

Example 1: Comparison of the Burton-MilJer integralequation and the normal Helmholtz integra) equation.

The two-dimensional acoustics scattering test problemwith boundary condition given in (47) is solved by using twodifferent formulations: the Burton-Miller formulation andthe normal Helmholtz integral equations, through a range ofwave numbers with the boundary condition (47) with €= Iand O.

Figures I and 2 show results obtained by using the nor-mal Helmholtz integral equation and the Burton-Miller for-mulation for a uniform mesh of 64 quadratic elements with€= I. In both cases, the problem is solved for a range ofnondimensional wave numbers ka from 10 to 50 with tlka=0.1 where a is the radius of the cylinder.

In the two figures, the error increases as the wave num-ber increases. Figure I c1ear]y indicates th'at the normalHelmholtz integral equation is unable to provide stable solu-tions at high wave numbers while the solutions (Fig. 2) ofthe Burton-Miller formulation are stable throughout the en-tire range of wave numbers tested. Figures I and 2 also in-dicate that the L2 residual error estimates produce satisfac-

341 J. Acoust. Soc. Am.• Vol. 100, No.1. July 1996 Geng et al.: BEM at high wave numbers 341

10

- ""=3.5

kh03

.........._._ .-. _ _..-. _.5

- -o

o 50 100 150 200 250 300 ~ .j(]() 450 500kI

FIG. 3. The relationship between kh and L 2-relative error for the solution ofacoustic wave scattering on a rigid cylinder. The uniform mesh of quadraticelements are used and the mesh size h = 2 Tralno. of elements.

21TAkh= b fl' I .num er 0 e ements per untt eng'

where A is the wavelength. kh in fact is a measure of thenumber of elements per wavelength. and (50) in fact states awell-known rule of thumb in engineering computation: thesolution error depends on the number of elements per wavelength.

However, Theorem 3.1 states only that c in (48) is in-dependent of p •. ph. and h; it may still depend on k. If cincreases as the wave number k increases, the rule of thumbcannot be applied and this will deteriorate the efficiency ofthe method in the range of high wave numbers. We test thisdependence experimentally.

Figures 3 and 4 shows the computed relationship be-tween L 2 norm relative error lIehllo and ka for several givenvalues of kh in two and three dimensions, respectively. Theexperiments are performed on the same test problem as in theprevious example with the rigid scattering boundary condi-

10

o8 9 10 11 12 13 14 15 16 17 18 19 20 21

kI

FIG. 4. The L2-relative error corresponding kh = 1.79 for the solution ofacoustic wave scattering on a rigid sphere. The unifonn mesh of quadratictriangular elements are used and the mesh size h =(4 Tra2/00. of elements)1f2.

342 J. Acoust. Soc. Am.. Vol. 100. No.1, July 1996

tion: E=O in (47). and a is the radius of the cylinder in twodimensions or the radius of the sphere in three dimensions.

In two dimensions. Fig. 3 shows that the relative errorlIehllo depends only on kh if kh is not too large (kh~2.5),and the highest wave number solved here is ka =490. Inthree dimensions. we only test the case of kh = 1.79, and theresults indicate that the relative L2 error is less than 5% if wecontrol kh to be less than 1.79. The highest wave numbersolved for the three-dimensional test problem is ka =20.

A uniform mesh of quadratic elements is used for thetwo-dimensional test problem, and a uniform mesh of qua-dratic triangular elements is used for the three-dimensionaltest problem. The mesh size h is computed by

21Tah=

the total number of elements

and

~ 41Ta2

h=

for the two- and three-dimensional test problems, respec-tively.

B. Effectiveness of a posteriori error estimation

A comparison of the L2 norm of the error and the L 2

norm of the residual is studied further in this experiment.The effectiveness of the L2 residual estimate is measured byan effectivity index defined by

ft' •. • d Ilrhllo (51)e lectlvlty 111 ex = lIehllo .

Equation (51) gives a global measure of the effective-ness of the error estimation. However, the efficiency of adap-tive methods is determined by the local quality of the errorestimate: the effectivity of error estimation on each indi-vidual element. The local effectivity of the error estimationis measured by

I al ft' •• • d IIrhll; (52)oc e lectlvlty III ex= PiG'where 11·11; represents the L2 norm on the element i.

Theoretically, we can only prove the global result

. Ilrhllo11m r:wl II = I, (53)h-O e 0

when rh and eh are the residual and error corresponding theapproximate solution of the normal Helmholtz integral equa-tion on a smooth boundary r. However, the major goal hereis to establish an error estimate for the Galerkin-Burton-Miller formulation on a boundary which may possess onlypiecewise smoothness.

Example 3: The global and local effectiveness of theerror estimation.

For k a = 15, the two-dimensional acoustics scatteringtest problem with the boundary condition of E= I in (47) issolved by using the Burton-Miller formulation with differentmesh sizes h and element approximation orders m. The cor-

Geng ef al.: BEM at high wave numbers 342

'"..

y

FIG. 5. The effectivity indices of the error estimation for different mesh sizeh and element approximation order m. The two-dimensional acoustic scat-tering test problem is solved by the Burton-Miller formulation with ka =15and E= 1.

responding effectivity indices are summarized in Fig. 5. Theresults verify (53). which states that as the degrees of free-dom increase (h-+O or m-+oo). the L2 norm of residual ap-proaches the L2 norm of true error.

The experimental results also indicate that the L2-nomlresidual possesses good local quality. The local effectivityindex on each element is very close to the global effectivityindex. As an example, Table I present .. a comparison of thelocal L 2 error and L 2 residual for a mesh of 32 quadraticelements obtained by solving the two-dimensional test prob-lem with Burton-Miller formulation at ka = 15 and ka=5.1356. Because of symmetry, Table I lists data for a halfof elements only. Similar tests are also performed for three-dimensional test problems, and the corresponding results canbe found in Ref. 22.

Example 4: A further survey of the effectivity of error

(54)

(56)

x

m= 1,.... 00,

FIG. 6. The radiation tcst problem.

estimation on a problem with nonsmooth boundaryFor the Neumann boundary condition.

ap aHm(kr)eikO-=an an

(55)

it is easily verified that the pressure solution of the corre-sponding radiation problem

(1 ) (1 T) aH~(kr)eiko2'1+Mk+aNk p= 2'+Lk+aMk

is

Comerpoinl

Here, H m is the Hankel function of the first kind of order m.and M k, N k ' Lk' and M[ are operators defined in (9)-(12).For simplicity, we consider a problem containing only onecomer (see Fig. 6).

A uniform mesh of 32 quadratic elements is used andm= 10 and k=5 are chosen. The comer point in Fig. 6 coin-cides with the node shared by the element 4 and 5. Figure 7shows the comparison between the numerical solution andanalytic solution and the global effectivity index. In thiscase, the global effectivity index is 1.0067, being very close

150 200 2SO

_.~~~~~"'.D~~):5.~

50LDg N IN: dogtoos of freedom>

10

1.8

1.6

I.'

j 1.2

~~W 0.8

0.6

0-.

02

TABLE I. Comparison of local L 2 errors and L 2 residuals for a mesh of 32quadratic elements using the Burton-Miller fonnulation.

Local effectivity index

Elem. no.

I23456789

10It1213141516

ka=5.l356

0.99280.99910.99761.00170.99741.00040.99990.99681.00351.01040.99960.99051.01180.98200.98901.0138

ka =15

1.00480.99781.00010.99591.00090.99700.99901.00140.99460.99591.00040.96990.98181.00670.92131.0016

30

20

·30

~ f! ~ r,t· H H t.f1 H f~ Hd t~ H ~.~qttf, ' .. '~ r ! +

~ ~ ~ f.; :..; : r ! ;~: .: :.l~:r ~::; t~ ~t!; ~t ~.'i V V

Global effectivity index0.9992 0.9963

FIG. 7. The comparison between numerical and analytic solutions for theradiation test problem with nonsmooth boundary. The relative L2 error andthe corresponding estimated error are 2.32% and 2.34%. respectively. andthe effectivity index is 1.0067. ---- analytic solution. 0 numerical solution.

343 J. Acoust. Soc. Am.. Vol. 100. No.1. July 1996 Geng et al.: BEM at high wave numbers 343

TABLE n. The comparison of local L2-norm errors and L2-norm residualsfor the radiation test problem with non smooth boundary.

Element no. Local effectivity index Element no. Local effectivity index

I 0.983 17 1.0132 1.028 18 1.0083 1.039 19 1.0084 0.890 20 1.0095 1.061 2] 1.0126 LOtI 22 1.0067 1.021 23 1.0108 0.979 24 1.0109 1.056 25 1.010

10 1.006 26 1.009II LOn 27 1.00812 1.005 28 1.00713 1.012 29 1.01114 1.007 30 1.01215 1.01 I 3t 1.00616 1.006 32 1.045

to I. Table II lists the local effectivity indices on each ele-ment. Except for element 4. the local effectivity indices hereare all very close to 1. The local effectivity index for element4 is about 0.89 which is a little lower than the average effec-tivity indices for the rest of elements. This suggests that thecorner node may have some local influence on the effective-ness of the error estimation.

VI. SUMMARY AND FUTURE WORK

The major difference between the approach used in thiswork and conventional approaches is that the approximationin the boundary element method is based on a Galerkin ap-proximation of the Burton-Miller hypersingular integral for-mulation of the exterior Helmholtz equation. The advantagesof using Galerkin approximation demonstrated in this studyare listed as follows.

(I) To overcome the problem of non uniqueness at for-bidden frequencies, the formulation proposed by Burton andMiller in Ref. I is used. A special difficulty that arises inapplying their method is the occurrence of hypersingular in-tegrals in the boundary integral equation. However. if theGalerkin method is used to approximate the weak form ofthis equation. the resulting formulation contains only regularand weakly singular integrals. Implementation of the newformulation is straightforward and produces stable solutionseven at very high wave numbers.

(2) For strongly elliptic problems. the Galerkin schemehas a well-established theoretical foundation. We proveherein that the problems considered in this study are ofstrongly elliptic type: it follows that the Galerkin approxima-tions are quasioptimal. i.e. for any approximate solution uh

on a given space Vh• there holds

Ilu-t/'llv"'c inf Ilu-u"llv.v" e vh

(3) Using certain fundamental properties of Galerkinmethods and Fredholm integral equations of the second kind,we prove that the L2 nonn of the residual of the normalHelmholtz integral equation approaches L2 norm of error as-ymptotically. i.e"

344 J. Acousl. Soc. Am .. Vol. 100. No.1. July 1996

11r"llo1~Ilehllo = 1.

Because Ilrhllo is computable after solving the boundary ele-ment problem. it can be used as an a posteriori error estima-tion. Numerical experiments further suggest that the errorestimation Ilrhllocan provide a very accurate estimation forthe global error as well as the local error.

As it is well known. a major problem in the applicationof the boundary element methods is that the system of linearequations generated by this method is full and nonsymmetricand has to be solved by a direct dense linear solver. Conse-quently. the 0(113) cost of dense solvers. in combinationwith the 0(112) memory required to store a matrix of dimen-sion 11 limits the size of problem that can be solved using thismethod. Furthermore, because of the involvement of singularintegrals. the formation of the linear system in the boundaryelement model is often much more expensive than the for-mulation of linear systems when other numerical techniquesare used. These facts have limited the range of sizes of prob-lems to which the boundary element method can be applied.

However, the range for which boundary element meth-ods can be used can be greatly expanded through the utiliza-tion of massively parallel computations. Dense linear solverslend themselves particularly well to paraUel computing,achieving very good efficiency even when very large num-bers of processors are used. The parallel computation canalso significantly reduce the computational time in the for-mation of the linear system.

The parallel implementation. including the formation oflinear system, linear dense solver, and error estimation hasbeen initially done in Ref. 22. Future work will concentrateon improvement in the efficiency of the current parallel al-gorithms and on methods to increase capacity. Advances inthese areas should help to make this particular type ofboundary element method an efficient tool for solving large-scale and high-frequency acoustical problems.

ACKNOWLEDGMENT

The support of the Office of Naval Research of thiswork under Contract NOOOI4-89-J-3109 is gratefully ac-knowledged.

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