CAT'S EYE RADIATION WITH BOUNDARY ELEMENTS: COMPARATIVE STUDY ON TREATMENT OF IRREGULAR FREQUENCIES

May 5, 2005 8:51 WSPC/130-JCA 00256

Journal of Computational Acoustics, Vol. 13, No. 1 (2005) 21–45c© IMACS

CAT’S EYE RADIATION WITH BOUNDARY ELEMENTS:COMPARATIVE STUDY ON TREATMENT

OF IRREGULAR FREQUENCIES

STEFFEN MARBURG

Institut fur Festkorpermechanik, Technische Universitat01062 Dresden, Germany

SIA AMINI

School of Computing, Science and Engineering, University of SalfordGreater Manchester, M5 4WT, UK

Received 8 July 2004Revised 6 September 2004

This paper reviews a number of techniques developed to overcome the well-known nonuniquenessproblem in boundary integral formulations of acoustic radiation. Although the problem has receivedmuch attention, comparative studies are hardly known in this field. Furthermore, the problem hasoften been studied using an unsuitable example, namely a simple radiating sphere. In this case, oftenthe addition of one collocation point in the centre of the sphere suffices to remove the nonuniquenessproblem for a large range of wavenumbers. In contrast to the radiating sphere, the radiating cat’seye structure is considered in this paper. Solution of the discretized ordinary Kirchhoff–Helmholtzintegral equation, also known as the Surface Helmholtz Equation, reveals a large number of so-calledirregular frequencies, i.e. frequencies where the BEM fails. The paper compares the performance ofdifferent methods in alleviating this failure. The CHIEF method and its variation due to Rosen et al.are found to encounter difficulties at high frequencies. A much better performance is obtained bycombining the Kirchhoff–Helmholtz integral equation with its normal derivative. In particular themethod of Burton and Miller and a modification of it which avoids evaluating the hypersingularoperator at nonsmooth points are tested. Both methods seem to provide reliable solutions. Themodified method encounters minor failures in the frequency response function at a geometric sin-gularity, although performing surprisingly well in many cases. More tests need to be carried out toassess fully the effectiveness of this method which allows easy use of continuous quadratic elements.However, it is the Burton and Miller formulation which appears to be the most reliable for acousticradiation analysis. The use of CHIEF and its variations cannot be recommended.

Keywords: Boundary element methods; external acoustics; irregular frequencies; CHIEF; Rosen’smethod; Burton and Miller method; discontinuous boundary elements; numerical integrationschemes.

1. Introduction

Boundary element methods have been used for acoustic problems for over three decades.Although a number of interesting text books have been published since 1980 [Refs. 6, 8, 11,20, 42, 44], there are still crucial issues that are insufficiently clarified in publications so far.

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22 S. Marburg & S. Amini

This includes issues such as efficient numerical integrations and their required accuracy andchoice of methods to avoid irregular frequencies at high wavenumbers. This latter problemis also known as the nonuniqueness problem for direct formulations or the nonexistenceproblem, in the indirect formulations of the acoustic radiation and scattering.

Recently, the author (SM) published two papers24,25 on discretization requirements forthe boundary element collocation method, in particular a comparison between continuousand discontinuous elements with respect to accuracy and efficiency. One of the key conclu-sions of these papers is the unsuitability of common continuous linear elements. In general,discontinuous elements performed better than continuous elements. This required optimallocation of nodes on the element.

In this paper, we will illustrate the problem of irregular frequencies that occurs withconventional boundary element methods. The investigation is essentially based on evaluationof a cat’s eye structure which has been investigated earlier as an scatterer cf.23,26,34,35 Itis the radiation problem which is considered in this paper. We study the reliability of anumber of methods attempting to overcome the problem at irregular frequencies. In Sec. 2we introduce briefly the boundary integral equations for the acoustic radiation problem,highlighting the nonuniqueness issue and outlining some of the remedies. In Sec. 3 we lookbriefly at the example of a pulsating sphere to provide the motivation for considering amore realistic example. This is done in Sec. 4 where the acoustic radiation from a cat’s eyestructure is considered. Here we discuss more fully the effectiveness of various methods foralleviating the irregular frequency problem. A brief conclusion is provided in Sec. 5.

2. Description of Analysis

2.1. Boundary element technique

Let Γ denote the surface of our radiating structure Ω+ and Ω− its exterior and interiordomains respectively. We assume harmonic time dependence of the sound pressure p(x, t)in the form

p(x, t) = p(x)e−iωt

. (1)

It is therefore the Helmholtz equation with Neumann boundary condition and an appropri-ate radiation condition which we need to solve for p(x). That is

(∇2 + k2)p(x) = 0, x ∈ Ω+, (2)

together with,

∂p(x)∂n(x)

= f(x), x ∈ Γ; and limr→∞ r

(∂p

∂r− ikp

)= 0, (3)

where k = ω/c is the wavenumber, the ratio of circular frequency over speed of sound,n(x) represents the outward normal at the surface point x and, in the radiation conditionx = (r, θ, φ) with r = |x|.

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Cat’s Eye Radiation with Boundary Elements 23

The analysis is essentially based on the Kirchhoff–Helmholtz integral solution for thetime harmonic sound pressure p, given by

c(x)p(x) −∫

Γ

∂G(x, y)∂n(y)

p(y) dΓ(y) +∫

Γiωρ0 G(x, y) vf (y) dΓ(y) = 0. (4)

The fluid particle velocity vf is related to the sound pressure by

vf (x) =1

iωρ0

∂p(x)∂n(x)

. (5)

In Eq. (5), i is the imaginary unit (i2 = −1) and ρ0 is the average density of the fluid.The coefficient c(x) in Eq. (4) is related to the so-called solid angle subtended and dependson the position of the point x. For an internal point, x ∈ Ω−, this coefficient becomes 0;whereas c(x) = 1 if x ∈ Ω+. When x is on the boundary Γ the value of c is found to bebetween 0 and 1. If x is located on a smooth part of the boundary, then c(x) = 0.5.

In the case x ∈ Γ, the relationship (4) is known also as the surface Helmholtz equation.Let us write this in a simplified operator notation as follows:

−c(x)p(x) + (Mkp) (x) = iωρ0 (Lkvf ) (x), x ∈ Γ, (6)

with obvious definitions for the single and double layer Helmholtz potential operators Lk

and Mk respectively. It is this boundary integral equation which is generally solved to findthe missing boundary data p(x). Knowledge of both the boundary data p and vf , so-calledthe Cauchy data, enables us to evaluate the sound pressure at any field point x ∈ Ω+ usingEq. (4) with c(x) = 1. That is:

p(x) = (Mkp) (x) − iωρ0 (Lkvf ) (x), x ∈ Ω+. (7)

By formally differentiating Eq. (6) in the direction of the outward normal n(x), we obtainanother relationship between the Dirichlet and Neumann boundary conditions, namely

(Nkp) (x) = iωρ0

(12

+ MTk

)vf (x), x ∈ Γ, (8)

the so-called differentiated surface Helmholtz equation. In (8) we are assuming that x ison a smooth part of the surface, i.e. avoiding edges and corners. Mathematically, this isnecessary since on edges and corners the normal derivative n(x) is not uniquely defined.

For scattering problems, where vf is the unknown boundary data, some authors advocatethe use of Eq. (8) in place of (6), as we now have to solve a second kind equation. Equation (8)is needed later to introduce the Burton and Miller and related formulations.

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The function G(x, y) represents a free-space Green’s function or the fundamental solu-tion of the Helmholtz equation and is given by

G(x, y) =ei k r

4πrwith r = |x − y| , (9)

for three-dimensional problems. It is perhaps worth mentioning that the operator Nk is theso-called hypersingular operator and is defined as

(Nkp) (x) = =∫

Γ

∂2G(x, y)∂n(x)∂n(y)

p(y) dΓ(y), (10)

where the integral sign =∫

indicates that the integral is to be interpreted in the sense of itsHadamard finite part.

Discretization of Eq. (6), that is, Eq. (4) for x ∈ Γ, by collocation method based onpiecewise polynomial approximation of the boundary data p and vf yields a linear systemof equations that can be written in the form

Hp = Gvf . (11)

G and H represent the global system matrices resulting from integration over the surfaceelements and p and vf are vectors that contain the nodal values of the sound pressure andthe particle velocity. We assume that values of the particle velocity are prescribed. Solutionof the system of Eq. (11), generally obtained by an iterative scheme of conjugate gradienttype, gives the discrete values of the sound pressure on the surface as

p = H−1Gvf . (12)

If n is the number of collocation points xj on the surface, our n unknowns will bepi ≈ p(xj). Matrix H is square and formed as

H =

−c1 + h11 h12 · · · h1n

h21 −c2 + h22 · · · h2n...

......

hn1 hn2 · · · −cn + hnn

. (13)

Entries hjk involve boundary integrations of the double layer potential whereas cj = c(xj).

2.2. The problem of irregular frequencies

2.2.1. General remarks

The exterior Helmholtz problem with appropriate boundary and radiation conditions hasa unique solution for all frequencies. However, it is well-known that the direct boundaryintegral equation (6), will fail to have a unique solution for a countable set IΓ of values of k;see for example Refs. 21, 27, 43, 46. This problem of irregular frequencies can be explainedin various equivalent ways. For example, it can be seen as the consequence of using a free-space Green’s function, which by nature, takes no account of the boundary Γ. A number of

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methods and formulations have been proposed over the last 3–4 decades for overcoming thisproblem. An excellent survey on these methods is presented by Rego Silva.30 More recently,Ochmann28 and Visser41 have discussed various aspects of different solution methods. Inthis paper we discuss two categories of these remedies in detail and briefly outline someothers.

The problem of irregular frequencies in (6) arises because, for k ∈ IΓ, both the oper-ators (−c + Mk) and Lk will be singular. For values of k close to the countable set IΓ

the operators and therefore the matrix approximations to them will be ill-conditioned.Furthermore, it can be shown that the number of elements in IΓ which are less than avalue K are O(K3). Therefore, the density of these irregular wavenumbers or frequenciesincreases as k increases. For example for a unit sphere IΓ = k|jn(k) = 0, n = 0, 1, 2, . . . =. . . , 20.12, 20.18, 20.20, 20.37, 20.54, 20.98, . . ., where jn is the spherical Bessel function oforder n; see Ref. 2. In general, the set IΓ corresponds to the eigenvalues of the interiorDirichlet problem. Similarly, if we choose to use the boundary integral equation (8), therewill also be a countable set, JΓ, of irregular frequencies, but these now correspond to theeigenvalues of the interior Neumann problem. Practical aspects of solution behavior closeto irregular frequencies were described and discussed by Juhl.17 In the following three sub-sections we briefly discuss some of the methods for alleviating the problem with irregularfrequencies.

2.2.2. CHIEF33 and its variant by Rosen et al.31

First we examine the Combined Helmholtz Integral Equation Formulations (CHIEF) origi-nally proposed by Schenck.33 The basic idea is to add to (11), i.e. the n point discretizationof (4), an m point discretization of the interior Kirchhoff–Helmholtz relation (4), that is withx ∈ Ω− and c(x) = 0. We need to choose m collocation points located in the enclosed cavity.These points are usually referred to as CHIEF points. The system matrix H correspondingto the CHIEF method takes the form

H =

−c1 + h11 h12 · · · h1n

h21 −c2 + h22 · · · h2n...

......

hn1 hn2 · · · −cn + hnn

hn+11 hn+12 · · · hn+1 n...

......

hn+m 1 hn+m 2 · · · hn+m n

. (14)

The rectangular system matrix reflects the fact that we now have an over-determined linearsystem of algebraic equations for the n-dimensional vector of unknowns, p. This system issolved in a least squares sense, where the unknown solution is formally given by

p =(HHH

)−1HHGvf , (15)

with superscript H denoting Hermitian, i.e. transposed conjugate complex matrix.

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There are a large number of publications on the use of this popular method; see forexample Wu and Seybert46 and references therein. A number of variations to the CHIEFmethod has been proposed in the literature. Here we will describe and test the variationdue to Rosen et al.31 Rosen et al. proposed to solve a square system of linear equations,having the same right hand side as CHIEF but with the new matrix H below:

H =

−c1 + h11 h12 · · · h1n h1 n+1 h1 n+2 · · · h1 n+m

h21 −c2 + h22 · · · h2n h2 n+1 h2 n+2 · · · h2 n+m...

......

......

hn1 hn2 · · · −cn + hnn hn n+1 hn n+2 · · · hn n+m

hn+11 hn+12 · · · hn+1 n 1 0 · · · 0hn+21 hn+22 · · · hn+2 n 0 1 · · · 0

......

......

......

hn+m 1 hn+m 2 · · · hn+m n 0 0 · · · 1

. (16)

The new linear system is now solved for the (n + m)-vector (p,0). In Ref. 31, the entries ofthe upper right block matrix are chosen as

hlk = h∗kl with l = 1, . . . , n and k = n + 1, . . . , n + m. (17)

Hence, the upper right block matrix is the complex conjugate of the lower left block.We will refer to this formulation as the conjugated variant of the Rosen et al. method. Wealso test a modified version of this where

hlk = hkl with l = 1, . . . , n and k = n + 1, . . . , n + m. (18)

Thus, the upper right block matrix is just the transpose of the lower left block. This willbe referred to as the unconjugated variant of the Rosen et al. method.

2.2.3. Burton and Miller’s method9 and our modification

The second category of methods for overcoming the irregular frequencies consist of takinga linear combination of the surface Helmholtz equation (6) and its differentiated form (8)to give (

−12

+ Mk + iηNk

)p(x) = iωρ0

(Lk + iη

(12

+ MTk

))vf (x), x ∈ Γ. (19)

The coupling parameter η ∈ R. A similar approach was originally proposed by Panich29

and by Brakhage and Werner7 in the context of indirect boundary integral equations forthe Helmholtz equation; see also Kussmaul.22 The direct formulation (19) which we use inthis paper is due to Burton and Miller.9

In a paper by Meyer et al.,27 based on computational experiments, the authors recom-mended the choice of the coupling parameter as η = 1/k. It was shown later, using rigorousmathematical analysis, that this choice of the coupling parameter almost minimizes the

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condition number of the operators on the left and the right hand sides of (19), when Γ is asphere; cf. Ref. 2 and references therein.

The hypersingular operator Nk is a pseudo-differential operator of order +1; mean-ing that it behaves essentially as a first order differentiation operator. Because of this, inEq. (19), the unknown function p(x) requires higher continuity than that for Eq. (6). Morespecifically, the operator Nk requires C1 continuity of the function at collocation points.As discussed already,24,25 it is quite common and indeed advantageous to use discontinuousboundary elements.

Here we propose a modification of the Burton and Miller method that avoids com-putation of the differentiated surface Helmholtz equation part of the Burton and Millerformulation at points with no C1 continuity. Our method here takes its inspiration fromthe paper by Harris and Amini15 where they take the coupling parameter η as η(x), i.e.dependent on the surface point. In Ref. 15 the authors allow η(x) ≡ 0 for x over a largeproportion of Γ or equivalently for a large number of collocation points. They show that aslong as η is set to zero in a sensible manner, this modification does not affect the stabilityof the Burton and Miller formulation with respect to the irregular frequencies. Here we usethe standard continuous 9 noded quadratic elements on quadrilaterals. Essentially, we setthe coupling parameter η in (19) equal to zero at all collocation points except the ones atthe centre of each element, where we set η = 1/k. This is the same as discretising the sur-face Helmholtz equation or the standard Kirchhoff–Helmholtz equation (6) at all collocationpoints and adding to appropriate rows of this discrete system i/k times the discretization ofthe differentiated equation (8) at the centre of each element, where C1 continuity is assured.Our method is similar, but not the same as the scheme of Francis,14 who cite the paper15

as their influence.

2.2.4. Other methods

There are many other attempts in the literature in eliminating the irregular frequencies. Animportant category is what is generally referred to as modified Green’s function methods.These were first investigated by Ursell40 and Jones,16 where they addressed the problem bymodifying the choice of the free-space Green’s function G. The choice of the function G issomewhat arbitrary, as there are an infinite number of functions satisfying

(∇2 + k2)G(x, y) = −δ(x − y). (20)

Ursell40 suggested the use of a modified Green’s function G1, with

G1 = G + W, (21)

where G is the standard fundamental solution (9) and W (x, y) is an infinite series of spher-ical wave functions. Ursell was then able to show that using this modified Green’s functionthe standard surface Helmholtz equation (6) would not suffer from singularities at anyfrequencies.

In order to reduce the burden of computation of G1 and make this result of some practicaluse, Jones16 suggested replacing W in G1 by WM , where WM is simply the partial sum of

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the first M terms in W . Jones proved that, for any given Γ and a real value K, it is possibleto choose M = M(Γ,K) such that there are no irregular frequencies in the wavenumberrange of 0 < kR < KR. As mentioned earlier, the number of irregular frequencies below agiven value of k grows as O(k3). Unfortunately, it can be shown that even for modest valuesof k, we still require a large number of terms in WM , specially for boundaries which have alarge aspect ratio. These modified Green’s function methods are computationally expensivefor moderate to high frequency radiation and scattering problems. This is the main reasonwhy these elegant methods have not gained favor with practitioners.

Finally, a recent paper by Cremers et al.12 proposes a multi-domain boundary-elementtechnique utilizing infinite boundary elements. The basic intention of the authors was toinvestigate a substructure technique with infinite subdomains separated by infinite bound-ary elements. The method proposed suffers from inefficient integration techniques for theinfinite elements but there is a positive side-effect that when using infinite boundary ele-ments no effect from irregular frequencies is observed. It is likely that future developmentsmay overcome the integration difficulties and render the scheme efficient.

2.2.5. Remarks

In the remaining of this paper we report on the performance of the first two categories ofmethods, namely the two variants of CHIEF31,33 and two variants of the Burton and Millermethod.9,15 We present results for different boundary elements and additionally investigatethe influence of accurate and lumped numerical integration. This might be useful becausethe ill-conditioning of the system will magnify any additional perturbation.

3. The Pulsating Sphere

In a number of papers on boundary element methods for acoustic radiation the pulsatingsphere is used as an example. Most likely, one of the main reasons is that a simple analyticsolution is available for this example. Some authors have remarked that very coarse dis-cretization is sufficient to achieve accurate numerical solutions in the low frequency range,i.e. kR < 10, where R is the radius of sphere; see for example Refs. 10, 13, 30, 36, 45.Discretization of this problem, having a constant solution on the boundary, should give theexact solution, even with one constant element, provided the boundary surface and bound-ary integrals are calculated exactly. Consequently, we should expect an excellent solutionby using constant approximation of the sound pressure and the particle velocity. This, how-ever presupposes high degree of geometry approximation, such as a nine-noded (quadratic)quadrilateral elements, and the accurate computation of the matrix elements.

The pulsating sphere has also been used as an example to show the effect of irregularfrequencies. Because of the simple nature of the problem, many of the higher modes are notexcited. Often, the only irregular frequencies observed in this case correspond to zeros ofj0(kR), that is kR = nπ, with n = 1, 2, 3, . . . . However, we know that the set of irregularfrequencies for a sphere is IΓ = k|jn(kR) = 0, n = 0, 1, 2, . . ., where the density of irreg-ular frequencies increases with increasing wavenumber k. Therefore, the pulsating sphere is

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perhaps a good example for an initial testing of a new code but should not be used to checkconvergence rates or stability with respect to irregular frequencies.

This is confirmed by our first example. It considers the pulsating sphere. The entiresphere is discretized into 24 elements, i.e. three per octant. These quadrilateral elementsapproximate the geometry by using quadratic polynomials. We use the data R = 1 m,ρ = 1.3 kg/m3, and c = 340 m/s, for sphere radius, fluid density, and speed of sound,respectively. We assume unit particle velocity (1 m/s) independent of the location on thesurface and frequency. We consider the frequency range up to 2000 Hz. This is equivalent tothe relative wavenumber kR ≈ 11.75π. Since the integrand is a highly oscillatory function forhigh frequencies, we need a high order of integration. Here, all integrals were evaluated usingGauss–Legendre quadrature rule with 30 integration points per direction, i.e. 900 points perelement. The analytic expression for the sound pressure magnitude at the surface can bewritten as

p(R) = ρcvfk R√

1 + k2 R2(22)

where vf stands for the uniform particle velocity at the surface of the sphere.In Fig. 1, the numerical solutions are compared for different boundary element methods.

The surface value of the sound pressure (left subfigures) is an average over all the nodes.However, as expected, we observed very little difference in the nodal values. The relativeerror of the solution is based on this average. The solution in the top subfigure was calculatedusing the Kirchhoff–Helmholtz integral equation or the surface Helmholtz equation (6).The two graphs in the middle subfigures correspond to the Combined Helmholtz IntegralEquation Formulation (CHIEF)33,46 where one and 43 CHIEF points were used. The lowestsubfigures present the results for the uniquely solvable method of Burton and Miller.9,27,30

In the upper subfigures of Fig. 1 we can clearly identify irregular frequencies since polesare observed for these wavenumbers. The addition of only one CHIEF point in the centreof the sphere is sufficient to get rid of all spurious modes. The results show also a smalldecrease in the error as the wavenumber is increased. Similar behavior can be seen using43 CHIEF points that were evenly distributed inside the radiator. The solution appears tohave smaller errors than in the case with only one CHIEF point. However, a very sharp errorpeak is observed for the frequency of 1626 Hz (kR = 9.56π) and only for that. Our numericalresult gives a very poor result for this single frequency. When looking at the solution bythe Burton and Miller formulation we note an error of one order of magnitude higher thanthat for the CHIEF solution. As mentioned before, for the pulsating sphere, there is nointerpolation error, even with our piecewise constant approximation. Therefore, any errorin the solution (except near the irregular frequencies for the CHIEF and ordinary BEM) isdue to numerical integration and surface approximation. We note that the Burton and Millersolution is continuous over the frequency range, showing its validity for all wavenumbers.

In Fig. 2, similar results are shown for continuous quadratic elements. The results forthe surface Helmholtz formulation, i.e. the ordinary BEM with no treatment at irregularfrequencies, are shown in the upper subfigures, whilst the results for the CHIEF method,

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Fig. 1. Pulsating sphere, 24 constant elements, surface solution (left) and its relative error (right), comparisonof solutions by ordinary BEM, by CHIEF and by method of Burton and Miller (normalized wavenumberkR = 10π equivalent to frequency f = 1700 Hz).

using 43 internal points, are plotted in the lower subfigures. Essentially, we observe someadditional effects. The error in the ordinary BEM case is less smooth between the irregularfrequencies, than the piecewise constant example. Similarly, the errors in the CHIEF methodis larger than that for the constant case.

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Fig. 2. Pulsating sphere, 24 quadratic elements, surface solution (left) and its relative error (right), com-parison of solutions by ordinary BEM and by CHIEF (normalized wavenumber kR = 10π equivalent tofrequency f = 1700 Hz).

As mentioned before, to investigate the convergence of a code and effectiveness of theunderlying method for overcoming the irregular frequencies, it will be better to use a spher-ical scatterer than a pulsating sphere. It has the same benefit that an analytic solution isavailable, but because the solution is now oscillatory, one can test the effect of changingthe order of interpolation too. Furthermore, we are likely to observe many more irregularfrequencies for this example as more modes are excited. Examples of rigid scattering fromspheres were given in Refs. 14, 18, 39.

4. The Cat’s Eye

4.1. Problem and model description

After having seen that the sphere might not be a representative example, the idea of inves-tigation of a cat’s eye structure is essentially based on two considerations:

• As will be shown later, this radiator allows construction of a smooth solution that willmake it easy to identify solution failures caused by the ill-conditioning of the integraloperator, i.e. the irregular frequencies.

• The cat’s eye structure is a more complicated shaped than a sphere. Hence, we expectmore irregular frequencies.

The cat’s eye has been analyzed in a number of papers in recent years, cf.23,26,34,35 Asshown in Fig. 3, it is a sphere with the positive octant cut out. Herein, we investigate

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Centre point Backside point

Fig. 3. Boundary element model (1920 elements) of cat’s eye structure, identification of observed points.

the radiation problem with a vibrating surface where it coincides with the spherical one.The plain surfaces of the missing octant remain calm. In Fig. 3, the vibrating and thenonvibrating domains are distinguished by their brightness. The idea is that the soundpressure at points in the centre of the backside should behave asymptotically (as frequencyincreases) like the one at the surface of the sphere. This implies that the noise transferfunction, i.e. sound pressure divided by surface velocity in terms of frequency, returns avery smooth function similar to the solution of the radiating sphere where it is easy toidentify frequencies where the solution fails.

We assume material data of air as density ρ = 1.3 kg/m3 and speed of sound c =340 m/s. The spherical radius is taken to be R = 1.0 m and the particle velocity for thevibrating surface as v = 1 m/s. We use different types of boundary elements. They consistof discontinuous constant elements (P0), discontinuous linear ones with collocation pointsat zeros of Legendre polynomials (P1L), continuous linear P1c, and continuous quadraticelements P2c. These elements have been discussed in two recent papers.24,25 For the P1L

and P2c interpolations we use 1920 elements and for P0 and P1c we use 7680 elements. Weconsider a frequency range of up to 1700 Hz which corresponds to a normalized wavenumberof kR = 10π. A frequency step size of 0.5 Hz is applied.

In case of CHIEF and Rosen et al. we used 240 CHIEF points. Considering that we usearound 7680 collocation points on the surface nodes, the number of CHIEF points used arerelatively small.

The arising linear system of equations is solved by using the Generalized MinimumResidual algorithm, cf. Saad.32 Iterative techniques have been investigated in quite a numberof papers by Amini and co–workers1,3−5 who investigated applicability of different iterativemethods for exterior acoustic problems, in particular for the Burton and Miller formulation.These iterative techniques have become a reliable tool for the efficient solution of large scaleacoustic problems as shown in recent papers.26,35 For the cat’s eye investigation, we usean unpreconditioned GMRes. In case of the least squares solution of the over-determinedsystem arising in conventional CHIEF, the GMRes is applied to the normal equations. We

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Table 1. Cat’s eye: Data of the boundary element models.

Subdivision Type of Number of Number Number of elements perper line elements elements of nodes wavelength for (kR = 10π)

16 P2c 1920 7682 1.7316 P1L 1920 7680 1.73

32 P1c 7680 7682 3.4632 P0 7680 7680 3.46

Table 2. Cat’s eye: Coordinates of the four pointsunder consideration.

Name x y z

Centre point 0 0 0

Backside point −1/√

3 −1/√

3 −1/√

3

Point at 2R −2/√

3 −2/√

3 −2/√

3

Point at 10R −10/√

3 −10/√

3 −10/√

3

set a stopping criterion on the residual of 10−8. This is two to three orders of magnitudesmaller than the level of discretization error. Thus, the additional error introduced by theiterative solver is negligible.

Although the solution is available at all surface nodes, we limit our interest to the soundpressure at four discrete points. These points are identified in Table 2 and in Fig. 3. Thesound pressure in terms of frequency will be called a noise transfer function.

4.2. Ordinary Kirchhoff–Helmholtz integral equation

We start our investigation by testing the performance of the ordinary Kirchhoff–Helmholtzintegral equation, i.e. the surface Helmholtz equation (6).

Figure 4 contains noise transfer functions for the backside point and the four differentelement types. Basically, we expect a smooth solution. However, it is often interrupted.The solution fails at irregular frequencies. This phenomenon starts at about 200 Hz. Hardlyany smooth regions are identified above 600 Hz. Consequently, it appears impossible toactually distinguish between regular and irregular frequencies. We note that even at mostirregularities, the solution seems to remain in the vicinity of the correct solution. This isbecause our linear systems often do not become singular at the frequencies considered, butsimply ill-conditioned, where errors are merely magnified. Nevertheless, the performance ofthe method is clearly not acceptable.

A comparison between solutions for different element types indicates more moderatefailures in case of constant and continuous linear elements, especially in the higher frequencyrange. Discontinuous linear and continuous quadratic elements seem to produce smoothersolution in the lower frequency range.

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Fig. 4. Radiating cat’s eye: Sound pressure at backside point evaluated by ordinary BEM using differentelement types.

Fig. 5. Radiating cat’s eye: Sound pressure at different points and number of iterations (= matrix vectorproduct evaluations) for ordinary BEM using constant elements.

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Figure 5 contains the noise transfer functions for constant elements at the other threepoints, namely, Centre, 2R and 10R. Clearly, the solutions at field points above the back-side point (upper subfigures) fails analogously to the backside point, cf. Fig. 4. However, thecentre point solution appears much worse. This is likely to be caused by the geometric singu-larity and certainly less acceptable than the moderately failing solutions at the other points.

It is remarkable that the GMRes solution requires less than 250 matrix vector productevaluations even for higher frequencies and even in the vicinity of irregular frequencies,cf. lower right subfigure of Fig. 5.

4.3. CHIEF33

Noise transfer functions for the conventional CHIEF, i.e. solution in a least squares sense,are plotted in Fig. 6. Solution is chosen for two points, namely at backside point and at

Fig. 6. Radiating cat’s eye: Sound pressure at two points evaluated by CHIEF using different element types.

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Fig. 7. Radiating cat’s eye: Sound pressure at two points evaluated by CHIEF using constant elements.

10R and for three types of polynomials. As expected, the solution appears much smootherwith the CHIEF method. However, it still contains numerous discontinuities. We recall thatwe are using 240 CHIEF points here. Theoretically, only one CHIEF point (at the correctposition) is sufficient. In practice many more are required to obtain a reliable solution.

We can state that the solution is smooth up to a frequency of about 600 Hz. Furthermore,as one might expect the solution at the field point (right subfigures) is somewhat smootherthan the surface solution. Use of quadratic element seems to magnify the effect of failuresat certain frequencies. This is to be expected. The reason is that the more accurate ourapproximation is the more closely it behaves like the continuous boundary integral operator.That is, the severity of failure at the irregular frequencies increases but the width aroundthese frequencies, where the failure occurs, narrows.

These observations are confirmed by the noise transfer functions for the other two points,cf. Fig. 7. Again, the effect of irregular frequencies is magnified for the geometric singularityin the centre point.

4.4. CHIEF variant of Rosen et al.31

Two variants of the Rosen et al.31 modification of CHIEF were tested here; namely, the con-jugated and the unconjugated versions. The resulting noise transfer functions for the originalconjugated version are shown in Fig. 8. Although smooth up to about 500 Hz, the solutionin the upper frequency range appears hardly better than those without any CHIEF points,cf. Figs. 4 and 5. In contrast to the conjugated version, the unconjugated version returnsmuch smoother curves, cf. Fig. 9. Nevertheless, their smoothness and reliability is clearlyworse than that for the conventional CHIEF, in particular at the geometric singularity atthe centre point. The gain in efficiency obtained by solving a square system of equationsinstead of a least square problem should not be at the expense of reliability and accuracy.

4.5. Method of Burton and Miller9

Situation is completely different when using the alternative approach originally proposedby Burton and Miller 9. Although Eq. (19) is computationally more challenging it yields

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Fig. 8. Radiating cat’s eye: Sound pressure at different locations evaluated by Rosens method (conjugated)using constant elements.

Fig. 9. Radiating cat’s eye: Sound pressure at different locations evaluated by Rosens method (unconjugated)using constant elements.

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a smooth solution, as can be seen in Fig. 10. Noise transfer functions are plotted for bothtypes of discontinuous elements. Both solutions coincide well except for the centre point.Deviations are likely to have been caused by different quality of approximation close to thegeometric singularity which is beyond the scope of this paper.

4.6. Modified Burton and Miller

It was pointed out earlier that by setting the coupling parameter to zero at all but thecentre nodes where the C1 continuity is assured we were able to apply the Burton andMiller method to continuous quadratic elements. Consequently, the hypersingular formu-lation appears for approximately one quarter of all collocation points. This simple andefficient modification of the original method produces a surprisingly smooth solution for thebackside point and for the field points, cf. Fig. 11. Some traces of discontinuities appear forthe centre point. Compared to the variants of CHIEF that have been discussed above, theyare much less pronounced even at this geometric singularity.

4.7. The effect of numerical integration errors

Having studied the performance of different boundary integral formulations for the soundradiation from a cat’s eye structure, we now turn our attention to investigating the effectof numerical integration errors on the appearance of irregular frequencies. Here we employan adaptive integration scheme similar to the one suggested by Telles.37 Singular integralsand quasi-singular or nearly-singular integrals, i.e. those where the collocation point isclose to the integration element, are evaluated in a more sophisticated way than integralsover elements in the far field of a collocation point. This self-adaptive technique uses thepolynomial transformation which was proposed by Telles.37,38 Polynomial transformation isnot required for elements where a ratio between distance from collocation point and elementsize becomes greater than a certain value given in Ref. 37. At a distance of about three timesthat value, Gauss–Legendre quadrature with a small number of nodes is sufficient. Here wewish to investigate the effect of numerical quadrature errors in the far-field integrals on theglobal behavior of our methods.

A number of researchers, e.g. Juhl,19 suggest the use of a simple integration strategy.Their investigations showed that for constant elements convergence in terms of elementsize is indeed observed using a one-point integration scheme on nonsingular elements. Theirscheme is even less accurate than that referred to as the one-point scheme in this paper.Additionally, we also consider in this section a two-point rule, i.e. far-field integration carriedout using a two by two integration scheme, whereas all preceding examples in this paperutilized a three-point rule, i.e. a three by three integration scheme for elements in far field.We limit our investigation to constant elements. Note that the geometry approximationis carried out in a superparametric way, i.e. approximation of geometry uses quadraticelements.

The study is limited to the conventional CHIEF in combination with constant elements.Result are similar for other variants of CHIEF and for the ordinary Kirchhoff–Helmholtz

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Fig. 10. Radiating cat’s eye: Sound pressure at different locations evaluated by method of Burton and Millerusing different discontinuous elements.

Fig. 11. Radiating cat’s eye: Sound pressure at different locations evaluated by modified Burton and Millermethod using continuous quadratic elements.

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integral equation. Figure 12 shows substantial differences between results using a one-pointintegration rule compared to the other schemes. While a two by two scheme will be sufficientover the entire frequency range, a one-point rule is not, even if the self-adaptive scheme isused. The one-point rule magnifies the effect of irregular frequencies and, hence, failuresappear more pronounced.

This effect becomes stronger if higher order elements are used. Two reasons are identi-fied for this. Firstly, higher order elements use higher order polynomials for interpolationrequiring higher order integration schemes. Secondly, the user aims to use a larger element,resulting in a more oscillating behavior of the kernel, which is another reason why higherorder integration rules should be used. Therefore, all examples except those mentionedabove utilized the three by three integration scheme for the far field integrals.

Fig. 12. Radiating cat’s eye: Sound pressure at two points evaluated by CHIEF using constant elements andthree different numerical integration schemes.

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4.8. Iterative solution of linear system of equations

It is well-known that at or near irregular frequencies the system matrix H will become ill-conditioned. We solve the linear systems of equations by the GMRes algorithm. As discussed,the lower right subfigure of Fig. 5 contains the number of matrix-vector products requiredfor solution in the case of the ordinary Kirchhoff–Helmholtz integral equation. Thoughmoderate in general, this number varies with frequency. Furthermore, its average increaseswith frequency. We can state that in general large number of iterations correspond tosolution failures, i.e. irregular frequencies. However, solution failures do not always resultin a large number of iterations. Hence, this cannot be used as an indicator for irregularfrequencies.

Fig. 13. Radiating cat’s eye: Number of matrix vector product evaluations required for GMRes solution andresidual of 10−8.

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Figure 13 presents the number of matrix-vector products required for the methods dis-cussed in previous subsections. Left subfigures presents the data for the CHIEF methodwith constant and also with continuous quadratic elements as well as the unconjugatedmethod of Rosen et al. for constant elements. The right subfigures present these countsfor the method of Burton and Miller and our modification of Burton and Miller for thecontinuous quadratic elements.

It can be seen that compared to the method of Rosen et al., the CHIEF is computation-ally more expensive. Quadratic elements require more iterations than constant elements.Method of Rosen et al. is cheaper because it requires only one matrix vector product periteration whereas CHIEF needs two. The condition number of the CHIEF system matrix issquared when normal equations are solved.

The Burton and Miller method is computationally the most expensive among the fourdiscussed here. Again, we observe that higher order elements, for example, the linear ele-ments as opposed to the constant elements, require some more iterations. However, the mod-ified Burton and Miller formulation performs much more efficiently than the original one.

5. Conclusion

This comparative study has been motivated by the observation that radiation from a pul-sating sphere can be accurately approximated by very few constant elements. The effectof irregular frequencies in this case can be removed by using either one CHIEF point orthe method of Burton and Miller. Irregular frequencies which are excited are equidistantlydistributed at normalized wave numbers kR = nπ.

A more complicated shape such as the cat’s eye shows many more irregular frequencies.In general a reasonable solution cannot be obtained without careful treatment of that phe-nomenon. The most commonly used method for removing the effects of irregular frequenciesis the CHIEF method where additional internal collocation points are used to create an over-determined system of equations. In our example, with a smooth noise transfer function, evenwith 240 CHIEF points, still numerous discontinuities remain. The method of Rosen et al.gives rise to a square system matrix. However, the solution is even less reliable than theconventional CHIEF. Smooth noise transfer functions are obtained when using the methodof Burton and Miller. Our modified Burton and Miller formulation for application with con-tinuous quadratic elements also provides us with a smooth solution. Some discontinuitieshave been discovered for the noise transfer function at the geometric singularity of the cen-tre point. Iterative solution using GMRes performs more efficiently for the modified Burtonand Miller method compared to the original one. A case study on numerical integrationshowed that the one point integration scheme even just for far field integrals is insufficientwith regard to irregular frequencies. Constant elements perform well when using a two bytwo integration scheme whereas higher order elements require appropriately higher orderintegration schemes.

More generally, we can conclude that the ordinary Kirchhoff–Helmholtz integral equationshould not be solved for radiation or scattering problems without any particular treatment of

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irregular frequencies. The CHIEF method performs well at low frequencies, but in general anaccurate solution can never be guaranteed. Method of Rosen et al. is not recommended sinceit performs even less reliably than the conventional CHIEF. Hence, the authors recommendthe hypersingular formulation of Burton and Miller. Results of the modified Burton andMiller method are very encouraging. We intend to study this method in more detail andhope to report on our findings in the near future.

Acknowledgment

The authors wish to acknowledge that the computation was run on the SGI Origin 3800 atthe Zentrum fur Hochleistungsrechnen of the Technische Universitat Dresden.

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CAT'S EYE RADIATION WITH BOUNDARY ELEMENTS: COMPARATIVE STUDY ON TREATMENT OF IRREGULAR FREQUENCIES

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