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1 Science and Technology Explicit Evaluation of Hypersingular Boundary Integral Equation for 3-D Helmholtz Equation Discretized with Constant Triangular Element Toshiro MATSUMOTO, Changjun ZHENG,, Shinya HARADA and Toru TAKAHASHI Department of Mechanical Science and Engineering, Nagoya University Vol.4, No.3, 1 Furo-cho, Chikusa-ku, Nagoya , Japan t.matsumoto@nuem.nagoya-u.ac.jp On leave from Department of Modern Mechanics, University of of China Hefei, China Abstract It is well known that the solution of an exterior acoustic problem governed by the Helmholtz equation is violated at the eigenfrequencies of the associated interior problem when the boundary element method (BEM based on the conventional boundary integral equation (CBIE is applied without any special treatment to solve it. To tackle this problem, the Burton-Miller formulation using a linear combination of the CBIE and its normal derivative (NDBIE emerges as an effective and efficient formula which is proved to yield a unique solution for all frequencies if the imaginary part of the coupling constant of the two equations is nonzero. The most difficult part in implementing the Burton-Miller formulation is that the NDBIE is a hypersingular type, and it is often regularized by using the fundamental solution of the Laplace s equation. But various regularization procedures in the literature give rise to integrals which are still difficult and/or extremely time consuming to evaluate in general. However, when constant triangular elements are used to discretize the boundary, all the strongly-singular and hypersingular integrals can be evaluated in finite-part sense explicitly without any difficulty, and the numerical computation becomes more efficient than any other singularity-subtraction technique. Therefore, in this paper, these singular integrals are evaluated rigorously for triangular constant element as finite parts of the divergent integrals by canceling out the divergent terms which appears in the limiting process explicitly. The correctness of the formulation is also demonstrated through some numerical test examples. Key words : Boundary Element Method, Helmholtz Equation, Acoustic Problem, Fictitious Eigenfrequency, Burton-Miller Method, Constant Element, Hypersingular Integral, Finite Part 1. Introduction Received 14 May, 1 (No. 1-3 [DOI: 1.199/jcst.4.194] Copyright c 1 by JSME The boundary element method (BEM has advantages compared with the finite element method (FEM due to the accuracy on the boundary, the reduction of the dimension, and the incomparable superiority for solving infinite or semi-infinite problems. Hence, it has been widely used in solving radiation and scattering problems which are usually concerned with the infinite or semi-infinite problem domains. However, there is such a drawback that using only the conventional boundary integral equation (CBIE fails to yield unique solutions for exterior acoustic problems governed by the Helmholtz equation at the eigenfrequencies of the associated interior problems (1. These eigenfrequencies are called fictitious eigenfrequencies because they do not have any physical significance, and just arise from the breakdown of the boundary integral representation for the exterior problems. In order to tackle this defect, 194

2 Vol.4, No.3, 1 several methods have been proposed in the last several decades. The Combined Helmholtz Integral Equation Formulation (CHIEF ( can successfully conquer the difficulty by adding some additional constraints in the interior domain, which leads to an over-determined system of equations. This method is very simple to implement but the main difficulty is that there is no method so far to determine the suitable number and positions of the interior points analytically. A more sound and effective alternative to circumvent the fictitious eigenfrequency problem is the Burton-Miller formulation (1 which is a linear combination of the CBIE and the normal derivative boundary integral equation (NDBIE. It has been proved that this composite BIE formulation yields unique solutions for all frequencies if the imaginary part of the coupling constant of the two equations is nonzero. The advantage of this method is that there is no need for making difficult decisions of interior points. The main difficulty is the evaluation of the hypersingular integrals involving a double normal derivative of the fundamental solution. Although various singularity subtraction techniques have been proposed to evaluate the hypersingular integrals to regularize them, most of them are still cumbersome and require extremely complicated numerical procedure in general (3 (1. One of the authors has also been treating this problem and has presented a series of methods (11 (14 in which the fundamental solution of Laplace s equation is subtracted from the fundamental solution of the Helmholtz equation and added back again. Also, the Taylor series expansion of the boundary density function such as sound pressure has been used to regularize the resulting singular integrals originated from the added-back term related to the fundamental solution of Laplace s equation. The methods presented have been applied to the boundary element analyses of three-dimensional acoustic fields discretized using quadratic boundary elements, (11, (1, (14 and have been extended to design sensitivity analyses of three-dimensional acoustic problems. (13 Although a continuity of the gradients of the sound pressure is required as the regularized boundary integral formulation for thin-plate bending problem, the approach based on the regularization scheme is useful because the resulting integral expressions can be evaluated using the standard numerical integration schemes such as the Gaussian quadrature formula (15 and can be applied easily to higher-order curved elements. Direct evaluations of such strongly-singular and hypersingular integrals based on higerorder interpolation functions, in the sense of finite-part, have also been presented (16 but the method based on regularization is more easier to implement in the conventional boundary element method. However, discretization using triangular constant elements are now prevailing in large-scale models after the recent development of the fast-multipole boundary element method (FMBEM. (17 (19 Since the collocation point lies at the center of geometry of the triangle, the collocation point always lies on a smooth part of the boundary when the boundary is discretized with the triangular constant elements. Shen and Liu (19 are still using the singularity-subtraction technique based on the regularization even for FMBEM based on the constant triangular element discretization. Therefore, multipole expansion and other translation formulas for the fundamental solution of Laplace s equation are still needed to be implemented as well as those of the Helmholtz equation. For the case of the triangular constant elements, all the strongly-singular and hypersingular integrals related to the Helmholtz equation may be evaluated analytically in the sense of finite-part, but they have not been shown yet. Therefore, in this paper, we evaluate all the strongly-singular and hypersingular integrals, arising in the Burton-Miller formulation for the three-dimensional Helmholtz equation, directly in the sense of finite-part, by using the approach used to derive the conventional boundary integral equations. ( We consider the integral representations required in the Burton-Miller formulation and assume the collocation point lies at the center of geometry of a triangular constant element. Then, we attach an infinitesimal hemisphere circumventing the collocation point and evaluate analytically all the boundary integrals for the triangular element and the hemisphere under some continuity condition of the boundary density function. 195

3 Vol.4, No.3, 1 As in the due course of obtaining free-terms in the usual boundary element formulation, ( the Cauchy principal value and the finite-part of the strongly-singular and hypersingular integrals of Burton-Miller s formulation can be obtained rigorously after canceling the divergent terms explicitly and taking the limit of tending the radius of the hemisphere to zero. The obtained boundary integral expression includes neither the fundamental solution of Laplace s equation nor the tangential derivative of the sound pressure, which exist in the formulation based on regularization, and can be easily implemented to FMBEM. Finally, we show some numerical test examples to demonstrate the correctness of the derived boundary integral representation.. Formulations We consider the following Helmholtz equation which is the governing equation of the propagation of time-harmonic acoustic waves in a homogeneous and isotropic medium p(x + k p(x =, (1 where is the Laplace operator, p(x is the sound pressure at point x, and k = π f /C is the wavenumber, f is the frequency, C is the acoustic velocity. The boundary conditions for the Helmholtz equation are written as p(x = p(x on Γ p, ( q(x = p n (x = p,i(x n i (x = i ρω v(x = q(x on Γ q, (3 p(x = z v(x on Γ z, (4 where n(x denotes the outward normal direction at x to the boundary, p,i (x and n i (x, (i = 1,, 3 are the Cartesian components of gradients of p(x and the unit outward normal vector, respectively. Note that Einstein s summation convention is applied for repeated indices all through the paper. Also, i denotes the unit imaginary number, ρ the density of the medium, ω = π f the circular frequency, v(x the particle velocity, z the acoustic impedance. The quantities with overscribed bar are assumed to be given on the boundary. In the case of acoustic radiation, the sound pressure p(x must also satisfy the following radiation condition: lim x ( x + x p(x ikp(x =. (5 The integral representation of the solution to the Helmholtz equation is p(x + q (x,yp(ydγ(y = p (x,yq(ydγ(y, (6 Γ Γ where x is the collocation point, y is the source point, and p (x,y is the fundamental solution, which, for three-dimensional acoustic problems, is given as p (x,y = eikr 4πr, (7 where r = x y, and q (x,y is the normal derivative of p (x,y, i.e., q (x,y = p (x,y n(y = eikr 4πr (1 ikr r n(y, (8 where n(y denotes the unit outward normal vector to the boundary at point y. The directional derivative of the integral representation (6 in a direction n(x defined at x becomes q(x + q (x,yp(ydγ(y = p (x,yq(ydγ(y, (9 Γ where ( = ( / n(x, and Γ p (x,y = eikr 4πr (1 ikr r n(x, (1 196

4 Vol.4, No.3, 1 [ ] q (x,y = eikr (3 3ikr k r r r 4πr 3 n(x n(y + (1 ikrn i(xn i (y. (11 Both Eqs. (6 and (9 are valid for any point x within the domain, and the case in which point x approaches a point on the boundary is considered next. In this study, the boundary is assumed to be discretized into constant triangular elements, and the boundary point to which the internal point x approaches is assumed to be the center of geometry of one of the triangles. Since the fundamental solution and its derivatives have singularities for r = x y =, the problem domain is assumed to be augmented by an infinitesimal hemisphere of radius ɛ centered at x to circumvent the neighborhood of x when evaluating the boundary integrals for the triangular element Δ x, as shown in Fig.1 (a. The boundary integrals of Eqs. (6 and (9 are evaluated for Γ ɛ,, and the remaining part Γ Γ ɛ, separately, where Γ ɛ is the boundary from which the circular area of radius ɛ is excluded and is the boundary of the hemisphere, and then, the radius ɛ is taken to zero to obtain the result expressions. The integrals for Γ Γ ɛ are the summation of the integrals for all the elements except Δ x and they can be evaluated numerically by means of a standard numerical integration method such as the Gaussian quadrature formula. (15 Therefore, our main concern is to evaluate the explicit forms of the integrals for Γ ɛ and and to obtain their limits for ɛ. Fig. 1 (a (b A hemisphere attached to the triangular element Δ x and a spherical coordinate of the hemisphere..1. CBIE for constant triangular element First, we evaluate the related integrals of the Eq.(6. The following integrals are defined for the triangular element Δ x : H 1 = p (x,yq(ydγ(y, (1 H = q (x,yp(ydγ(y, (13 I 1 = p (x,yq(ydγ(y, Γ ɛ (14 I = q (x,yp(ydγ(y. Γ ɛ (15 For the boundary of the hemisphere, we define a polar coordinate as shown in Fig.1 (b, where n(x is the unit outward normal vector to the triangular element at the collocation point x, s(x and τ(x are two unit tangential vectors at x perpendicular to each other and to n(x. Note that s(x, τ(c, and n(x form a system of local orthonomal base vectors at x. 197

5 Vol.4, No.3, 1 Then, we can evaluate the integral H 1 for as H 1 = π π/ e ikɛ π π/ = ɛeikɛ 4π ( ɛe ikɛ + 4π 4πɛ p,i(y n i (y ɛ sin θ dθ dφ ( p,i (y p,i (x n i (ysinθdθdφ π π/ n i (ysinθdθdφ p,i (x. (16 We assume, reasonably, that the sound pressure p(x satisfies the following Hölder C 1,α continuity condition for non-negative constants A and α: p,i (y p,i (x Ar α. (17 Then, we obtain lim H 1 =. (18 ɛ H can be evaluated in the same way. Since r/ n(y = 1 over the surface of the hemisphere, we obtain π π/ ] H = [ eikɛ (1 ikɛp(y ɛ sin θ dθ dφ 4πɛ π π/ = eikɛ (1 ikɛ 4π [ + eikɛ (1 ikɛ ( p(y p(x sin θ dθ dφ π/ ] sin θ dθ p(x 1 p(x, (ɛ. (19 Next, in order to evaluate the boundary integrals over Γ ɛ,wedivideγ ɛ into three parts as shown in Fig.. We find p(y = p(x and q(y = q(x onγ ɛ for the constant elements, thus Fig. Variables used for evaluating integrals for Γ ɛ. ( 3 θ m R(θ e ikr ( 3 θ m R(θ e ikr I 1 = lim r dr dθ q(x = dr dθ q(x ɛ m=1 θ1 m ɛ 4πr m=1 θ1 m 4π 198

6 Vol.4, No.3, 1 = [ 3 θ m m=1 = i k θ m 1 ( 1 1 π i ( ] 1 e ikr(θ dθ q(x 4πk 3 θ m m=1 θ m 1 e ikr(θ dθ q(x. ( Since r,i (y n i (yonγ ɛ, we find r/ n(y =, and we obtain [ ] I = Γɛ eikr 4πr (1 ikr r n(y dγ(y p(x =. (1 Therefore, the CBIE for constant triangular element can be written as 1 p(x + q (x,yp(ydγ(y = p (x,yq(ydγ(y Γ\Δ x Γ\Δ x + i ( θ m e ikr(θ dθ q(x, ( k π m=1 θ1 m where Γ\Δ x denotes the boundary Γ excluding the boundary element Δ x in which the collocation point x is located... NDBIE for constant triangular element As for Eq. (9, which is the directional derivative of the integral representation (6, the following integrals for the triangle element Δ x are to be investigated; J 1 = p (x,yq(ydγ(y, (3 J = q (x,yp(ydγ(y, (4 K 1 = p (x,yq(ydγ(y, (5 Γ ɛ K = q (x,yp(ydγ(y. (6 Γ ɛ In order to evaluate the integrals over the hemisphere, we define the orthonormal bases τ i (x, s i (x and n i (x as shown in Fig.1, and decompose r,i (y and u,i (x in terms of these bases, as follows: r,i (y = n i (y = (sin θ cos φτ i (x + (sin θ sin φs i (x + (cos θn i (x, (7 p,i (x = p τ (xτ i(x + p s (xs i(x + p n (xn i(x. (8 Then, J 1 can be evaluated as follows: J 1 = π π/ e ikɛ π π/ + 4πɛ (1 ikɛ r [ p,i (y p,i (x ] n i (y ɛ sin θ dθ dφ n(x e ikɛ 4πɛ (1 ikɛ r n(x p,i(xn i (y ɛ sin θ dθ dφ. (9 Let us assume that p(y satisfies C 1,α Hölder continuity, an equivalent form of Eq. (17 p(y p(x r i p,i (x < Ar 1+α. (3 Then, the first term of the right-hand side of Eq. (9 vanishes, and we have J 1 = eikɛ (1 ikɛ 4π = eikɛ (1 ikɛ 4π π π/ π π/ p,i (x n i (ysinθcos θ dθ dφ [ (sin θ cos φ p τ (x + (sin θ sin φ p s (x 199

7 Vol.4, No.3, 1 +(cos θ p ] n (x sin θ cos θ dθ dφ ( π π/ = eikɛ (1 ikɛ sin θ cos θ cos φ dθ dφ 4π + eikɛ (1 ikɛ 4π + eikɛ ( π π/ ( π π/ p τ (x p sin θ cos θ sin φ dθ dφ s (x p sin θ cos θ dθ dφ n (x (1 ikɛ 4π = 1 p 6 n (x + O(ɛ. (31 J can also be written in the form as J = q (x,y { p(y p(x r i (yp,i (x } dγ(y ( ( + q (x,ydγ(y p(x + q (x,yrr,i (ydγ(y p,i (x. (3 Since r/ n(x = cos θ, r/ n(y = 1, and r = ɛ over the surface of the hemisphere, we have q (x,y = eikɛ 4πɛ ( + ikɛ + 3 k ɛ cos θ. (33 By assuming Eq. (3 for p(y again, we find that the first term of the right-hand side of Eq. (3 vanishes as ɛ. Then, its second term becomes ( π π/ e ikɛ J 4πɛ ( + ikɛ + 3 k ɛ cos θɛ sin θ dθdφ p(x = eikɛ 4πɛ ( + ikɛ + k ɛ ( π π/ sin θ cos θ dθdφ p(x = 1 ɛ p(x + O(ɛ (34 Note that J is a divergent term proportional to 1/ɛ and its behavior will be considered later with the divergent terms which are generated by evaluating other integrals. For constant elements, we observe p/ τ = p/ s = at any point on the element. Therefore, by using Eqs.(7 and (8, we find r,i (yp,i (x = [ (sin θ cos φτ i (x + (sin θ sin φs i (x + (cos θn i (x ] p n (xn i(x = p (x cos θ (35 n Hence, the third term of J becomes as follows: J 3 π π/ e ikɛ 4πɛ 3 ( + ikɛ + k ɛ ɛ r,i (yp,i (x ɛ sin θ dθ dφ π π/ = eikɛ 4π ( + ikɛ + k ɛ r,i (yp,i (xsinθcos θ dθ dφ = 1 p 3 n (x + O(ɛ. (36 Therefore, we obtain J = 1 ɛ p(x 1 p 3 n (x + O(ɛ. (37 Next, we evaluate K 1 and K by employing the variables defined in Fig.. r/ n(x = onγ ɛ,wehave K 1 = Γ ɛ 1 4πr (1 ikr r q(ydγ(y =. (38 n(x Since

8 Vol.4, No.3, 1 Also, we observe n(x = n(y and r/ n(y = onγ ɛ and we have q (x,y = eikr (1 ikr. (39 4πr3 Thus, we obtain ( K = q (x,ydγ(y p(x = = = Γ ɛ 3 θ m R(θ m=1 θ1 m 3 θ m m=1 θ m 1 1 ɛ + ik 3 e ikr (1 ikr rdrdθ ɛ 4πr3 p(x ( e ikɛ 4πɛ eikr(θ dθ p(x 4πR(θ θ m e ikr(θ 4πR(θ dθ p(x + O(ɛ. (4 m=1 θ m 1 All the divergent terms are cancelled out in the summation of J 1, J, K and K, and after taking the limit ɛ, the resulting form of NDBIE for constant triangular element discretization can be written using only the integrals which can be evaluated in the standard sense, as follows: 1 q(x + q (x,yp(ydγ(y = p (x,yq(ydγ(y Γ\Δ x Γ\Δ x ( ik 3 θ m e ikr(θ m=1 θ1 m 4πR(θ dθ p(x. (41 Finally, the Burton-Miller formulation for constant triangular element can be written as 1 p(x + q (x,yp(ydγ(y + α q (x,yp(ydγ(y Γ\Δ x Γ\Δ x = α Γ\Δ q(x + p (x,yq(ydγ(y + α p (x,yq(ydγ(y x Γ\Δ x + i θ m e ikr(θ dθ q(x α ik k π m=1 θ1 m 3 θ m e ikr(θ m=1 θ1 m 4πR(θ dθ p(x, (4 where α is a coupling constant that can be chosen as i/k. (1 3. Numerical examples Two numerical examples are presented here. The purpose of the first example is to verify the formulations obtained in the previous section. The second example verifies the formulations and also shows the validity of the Burton-Miller formulation to conquer the fictitious eigenfrequency problem of exterior acoustic problems Interior sound pressure of a cubic box A field of a cubic box with an edge length of L = 1 [m], as depicted in Fig.3, is chosen as an interior example to verify the present formulations. The acoustic medium in the box is assumed to be air, with a mass density of ρ = 1.[kg/m 3 ] and the velocity of sound wave propagation c = 34 [m/s]. The sound pressures on the left and right side surfaces are given as Pa and 1 Pa, respectively. The other four surfaces are assumed to be rigid walls, that is, the particle velocities are given as zero on these surfaces. Under these boundary conditions, the sound pressure in the box varies only in y-direction and is given by 1 sin(ky p(y =. (43 sin(kl The whole surface of the cubic-box is discretized into 384 constant triangular elements as shown in Fig.4. Sound pressure was calculated at nine interior points for y=.1,.,.3, 1

9 Vol.4, No.3, 1 Fig. 3 Cubic-box field model.,.9 [m] with x =.5 [m] and z =.5 [m]. The obtained results are compared with the analytical solution given by Eq.(43 in Fig.5. The numerical solutions based on the present formulation show good agreements with the analytical solution, and both Eqs. ( and (41 are turned out to be valid for acoustic problems for interior fields. Fig. 4 Boundary mesh of the cubic-box field model. 3.. Example for an exterior problem of a pulsating sphere An exterior field with a pulsating sphere is chosen as the next example to verify the present formulations. The radius of the sphere is assumed to be a = 1 [m], and the velocity boundary conditions are specified so that the sphere surface vibrates with a uniform radial velocity v = 1.[m/s]. The same properties as in the cubic box model are used for the acoustic medium. The analytical sound pressure is given by p(r = iωρva (1 + ikar e ik(r a, (44 where r is a radial distance from the center of the sphere to the field point of interest. In the numerical analysis, the surface of the sphere is discretized into constant triangular elements. We shown in Fig.6 the sphere model whose surface is discretized into 49 elements. Sound pressure is calculated for some range of frequency at an exterior field point with r = [m]. In Fig.7, the results for the sound pressure level (SPL obtained by the conventional boundary integral equation (CBIE, the normal derivative boundary integral equation (NDBIE, and the combined boundary integral equations based on Burton-Miller s method (Burton-Miller are compared each other with the analytical solution..

10 Vol.4, No.3, 1 Fig. 5 Results for the sound pressure at the center of the cubic-box. Fig. 6 Pulsating sphere model the surface of the sphere is discretized into 49 triangular constant elements. Fig. 7 Results for the sound pressure level at an exterior point for r = of the pulsating sphere model discretized with 49 elements. 3

11 Vol.4, No.3, 1 The corresponding internal problem giving the fictitious eigenfrequencies is that with a Dirichlet boundary condition. The analytical solution to this problem becomes p a j (π fr/c/ j (π fa/c, where j denotes the spherical Bessel function of order zero, and p a the sound pressure at r = a. The fictitious eigenfrequencies f [Hz] result in the roots of sin(π f /c =, i.e., f = 17, 34, 51, 68, 85,. Also, for NDBIE, the analytical solution of the corresponding internal problem becomes v a j 1 (π fr/c/ j 1 (π fa/c, where j 1 denotes the spherical Bessel function of order one, and v a the particle velocity at r = a. Hence, for the present example, the fictitious eigenfrequencies can be obtained, finally, as the roots of the following equation: π f c ( ( π f π f cos sin =, (45 c c and they are calculated as f = 43.15, 418.3, 59.5, , ,. From Fig. 7, it can be seen that the sole use of either CBIE or NDBIE solutions follow the analytical solution closely except in the vicinity of the fictitious eigenfrequencies. But the Burton-Miller formulation solution agree well with the analytical solution over the entire frequency range. In the frequency range higher than 8 Hz, we observe inaccurate results for the frequencies other than fictitious ones. Those are possibly be fictitious eigenfrequencies caused by modeling the sphere with a polygon consisting of triangular elements. We show in Fig.8 the results for SPL at the same exterior point, but obtained with 44 triangular elements. We find that in the results obtained with the finer mesh the errors observed in Fig.7 in the higher frequency range have actually disappeared. Fig. 8 Results for the sound pressure level at an exterior point for r = of the pulsating sphere model discretized with 44 elements. 4. Conclusions The strongly singular integrals and hypersingular integrals of the boundary integral equations used in Bourton-Miller s approach for three-dimansional steady-state acoustic problems have been evaluated explicitly, in the sense of finite part, for the triangular constant element discretization by using the approach used to derive the conventional boundary integral equations. The resulting integrals for the element on which the collocation point of the fundamental solution lies consist only of the integrals for regular functions of angular variables and can be evaluated numerically by means of the standard Gaussian quadrature formula. 4

12 Vol.4, No.3, 1 Some numerical results obtained using the present formulation were shown and the validity of the derived expressions were demonstrated. The present formulation is not based on the regularization of the singular integrals using the fundamental solution of Laplace s equation. Therefore, the present integral representation is more tractable in applying to the fast-multipole boundary element method. References ( 1 Burton, A.J. and Miller, G.F., The Application of Integral Equation Methods to the Numerical Solution of Some Exterior Boundary-Value Problems, Proceedings of the Royal Society A, Vol.33 (1971, pp ( Schenck, H.A., Improved Integral Formulation for Acoustic Radiation Problems, Journal of the Acoustical Society of America, Vol.44, No.1 (1968, pp ( 3 Chien, C.C., Rajiyah, H. and Atluri, S.N., An Effective Method for Solving the Hypersingular Integral Equations in 3-D acoustics, Journal of the Acoustical Society of America, Vol.88, No. (199, pp ( 4 Krishnasamy, G., Schmerr, L.W., Rudolphi, T.J. and Rizzo, F.J., Hypersingular Boundary Integral Equations: Some Applications in Acoustic and Elastic Wave Scattering, Transaction of the ASME, Journal of Applied Mechanics, Vol.57 (199, pp ( 5 Liu, Y.J. and Rizzo, F.J., A Weakly Singular Form of the Hypersingular Boundary Integral Equation Applied to 3-D Acoustic Wave Problems, Computer Methods in Applied Mechanics and Engineering, Vol.96 (199, pp ( 6 Tanaka, M., Sladek, V. and Sladek, J., Regularization Techniques Applied to Boundary Element Methods, Applied Mechanics Reviews, Vol.47, No.1 (1994, pp ( 7 Hwang, W.S., Hypersingular Boundary Integral Equations for Exterior Acoustic Problems, Journal of the Acoustical Society of America, Vol.11, No.6 (1997, pp ( 8 Matsumoto, T. and Tanaka, M., Regularized Boundary Integral Formulation for Thin Elastic Plate Bending Analysis, Singular Integrals in Boundary Element Methods, Sladek V. and Sladek J. (eds, Chap.8, Computational Mechanics Publications, Boston, (1998, pp ( 9 Yang, S.A., An Integral Equation Approach to Three-Dimensional Acoustic Radiation and Scattering Problems, Journal of the Acoustical Society of America, Vol.116, No.3 (4, pp (1 Li, S., Huang, Q., An Improved Form of the Hypersingular Boundary Integral Equation for Exterior Acoustic Problems, Engineering Analysis with Boundary Elements, Vol.34 (1, pp (11 Tanaka, M., Matsumoto, T. and Arai, Y., A Boundary Element Analysis for Avoiding the Fictitious Eigenfrequency Problem in Acoustic Field (nd Report, Revised Version, Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.7, No.719 (6, pp (1 Arai, Y., Tanaka, M. and Matsumoto, T., A New Boundary Element Analysis of 3-D Acoustic Fields Avoiding The Fictitious Eigenfrequency Problem, Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.73, No.79 (7, pp (13 Arai, Y., Tanaka, M. and Matsumoto, T., A New Design Sensitivity Analysis of Acoustic Problems Based on BEM Avoiding Fictitious Eigenfrequency Issue, Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.73, No.79 (7, pp (14 Tanaka, M., Arai, Y. and Matsumoto, M., New Boundary Element Analysis of Acoustic Problems with The Fictitious Eigenvalue Issue, Boundary Elements and Other Mesh Reduction Methods XXIX, Brebbia, C.A., Poljak, D. and Popov, V. (eds, WIT Press, Southampton/UK, (7, pp

13 Vol.4, No.3, 1 (15 Stroud, A. H. and Secrest, D., Gaussian Quadrature Formulas, Englewood Cliffs, NJ: Prentice-Hall, (1966. (16 Guiggiani, M., Formulation and Numerical Treatment of Boundary Integral Equations with Hypersingular Kernels, Singular Integrals in Boundary Element Methods, Sladek V. and Sladek J. (eds, Chap.3, Computational Mechanics Publications, Southampton, UK, (1998, pp (17 Greengard, L., Huang, J.F., Rokhlin, V. and Wandzura, S., Accelerating Fast Multipole Methods for the Helmholtz Equation at Low Frequencies, IEEE Computational Science & Engineering Archive, Vol.5, No.3 (1998, pp (18 Amaya, K. and Aoki, S., Effective Boundary Element Methods in Corrosion Analysis, Engineering Analysis with Boundary Elements, Vol.7, No.5 (3, pp (19 Shen, L., Liu, Y.J., An Adaptive Fast Multipole Boundary Element Method for Three-Dimensional Acoustic Wave Problems Based on the Burton-Miller Formulation, Computational Mechanics, Vol.4 (7, pp ( Brebbia, C.A. and Dominguez, J., Boundary Elements An Introductory Course, McGraw-Hill, New York, (1989. (1 Kress, R., Minimizing the Condition Number of Bounday Integral Operators in Acoustic and Electromagnetic Scattering, Quarterly Journal of Mechanics and Applied Mathematics, Vol.38 (1985, pp