Abstract INTRODUCTION

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1 Iterative local minimum search for eigenvalue determination of the Helmholtz equation by boundary element formulation N. Kamiya,E. Andoh, K. Nogae Department ofinformatics and Natural Science, School of Informatics and Sciences, Nagoya University, Nagoya , Japan Abstract A scheme for local minimum search for eigenvalue extraction of the scalar-valued Heimholtz equation is developed in this paper. In place of the zero-point search of the determinant of coefficient matrix by small increments of unknown eigenvalue, known conventionally, the zero-point of the first derivative of the determinant is found out ileratively.for this purpose, the coefficient matrix is represented in polynomial form in terms of the eigenvalue with help of the Multiple Reciprocity Boundary Element (MRM-BEM) formulation. The eigenvalue problem is fully formulated in complex-valued variables, different from the earlier MRM for the eigenvalue determination by the present authors.two-dimensional formulation and some numerical examples are shown. INTRODUCTION The boundary element method (DEM) is firmly recognized as a powerful alternate to the finite element method (FEM) for various applications in computational engineering problems. For eigenvalue analysis, however, the conventional scheme based on the boundary element formulation is not thought to be sufficient from computational high efficiency despite of its pronounced advantage in sole boundary discretization. The scheme is based on direct zero-point search of the determinant of coefficient matrix by small increments of unknown eigenvalue, for which repeated integral over boundary elements is inevitable because the eigenvalue is included implicitly inside fundamental solution of the Helmholtz equation [1, 2]. Recently, marked progress has been made by several distinct formulations in the BEM, such as the dual reciprocity method (DRM) [3-6], the multiple reciprocity method (MRM) [7-11] and others [11, 12]. The most formulations employ the framework of the real-valued function but intrinsically the problem belongs to the complex-valued function, which indicates that the former is a part of the latter. The present scheme has close inter-relationship with the earlier MRM formulation devised by the present authors for the eigenvalue extraction of the scalar-valued Helmholtz equation [7-9]. The MRM formulation made it possible to express the coefficient matrix in polynomial in terms of the unknown eigenvalue, for which the above-mentioned integral is left unnecessary for each eigenvalue search. In addition, in this formulation, an analytical expression of the derivative of the determinant can be offered. Considering these affirmative

2 230 Computational Acoustics situation, we here propose a new eigenvalue determination scheme using the zero-point of the first derivative of the determinant by way of the Newton iteration. The determinant of die coefficient matrix defined in complex-valued variables using the Helmholtz fundamental solution becomes so-called Gramian determinant and positive definite, and therefore its zero-points are local minima. Local minima and maxima are distinguishable by the second derivative. Some example computations in the two-dimensional space are shown and discussed. FORMULATION The problem under consideration is for the scalar- valued I lelmholtz differential equation V*u+/^u=0 (inq) (l) in the bounded two-dimensional domain Q condition on the boundary F : with the following homogeneous boundary u = 0 (on TJ, g(5=<7u/dn)=0 (on P,) (2) where k is the wavenumber, unknown for the eigenvalue problem and n denotes die unit outward normal on the boundary. By using the fundamental solution to the two-dimensional Helmholtz operator Eq. (1) comes to the following integral equation: (3) 0 (4) where x and are the field and source points respectively and their distance is r. q* is defined as cy* = 3 u*l d n. c in Eq. (4) is the constant depending on the source point taken, u, q, u* and q* are complex and are divided into respective real (Re) and imaginary (Im) parts. Equation (4) becomes Re(u)Re(cf) - Im(u)Im(q*) rim( u) Re( (f) + Re( u) Im( q*) - c m(u) + J ^^ Re(u*) - Re(q) Im(u*) dt = 0 Equations (5), (6) and the boundary condition (2) constitute the eigenvalue problem; for these formulations the conventional eigenvalue search scheme is known [1, 2J. The most pronounced difficulty is that, since u* and q* appear in the integral sign (approximated by integrals on boundary elements), we must recompute them over boundary elements for each value of k. This process is less computational efficient while we need no domain discretization. In place of the fundamental solution, Eq. (3), we may derive the integral equation using its real part

3 Computational Acoustics 231 as (7) cre(u) + J[Re(u)Re(<f) - Re(u*) Re(q)}/r = 0 r clm(u) + f[lm(u) Re(q*) - Re(u*)Im(g)]dT = 0 r We get tlie following equations from Eqs. (6), (9) and (5), (8) respectively, jt[lm(u)im(g*) - Im(q)Im(u*)]dT -0 Now we notice the equation Substituting Eq. (12) into Eq. (10) and Eq. (7) into Eq. (5), respectively, we obtain J Re(u) /o dt =0 i r <? i cre(u)--f Re(u) ^(Ar)-Re(q) ^(Ar) dt = 0 H- p L ^ J The 0-th order Bessel functions /<, and ^ of die first and second kinds respectively are expanded in series, 2; where Equations (13) and (14) come to, alter substitution of Eqs. (15) into (19),

4 232 Computational Acoustics dt=0 c/t=0 Equation (21) coincides with that obtained by the MRM formulation. Therefore, the MRM formulation for the Helmholtz equation plays a role of a part of the whole formulation [14, 15] Ḃy discretizing Eq. (21) by boundary elements we get the following matrix equation where u^, q^ are the vectors of the real parts of u an q on the boundary nodes and the components of the coefficient matrices H^, G^ are computed on the boundary elements using the integrands indicated in Eq. (21) multiplied by the interpolation function. It is worthwhile mentioning that the last integration does not contain the unknown k. 11^ and G^ are represented as the polynomials in terms of k* Hre = ri re + k. lire + K Hrc +... (23) From Eq. (20), we similarly obtain Ilinillrf- = (jfinicfiv where (25) Equations (22) and (24) are for the real-valued u and q. Equations (9) mid (11) relating the imaginary parts of u and q defined on the boundary nodes produce the following discretized equations liunlliin = (26),(27) As a result, we obtained four discretized equations (22), (24), (26) and (27) for the original Helmhohz differential equation. Considering the homogeneous boundary condition, Eq. (2), the above-derived equations reduce to

5 Computational Acoustics 233 ArvXre = 0 AimXrc = 0 A,*,, = 0 AiiX = 0 where xre and xim are nonvanisliing real and imaginary terms of u and q on the boundary and the corresponding coefficients are denoted as A^ and A^. Equation (28) is rewritten as Are Aim - Aim Are Xrc\ 0 Xknl \0 (29) For the coefficient matrix on the left-hand-side Are - An] Aim Are (30) the proper equation is as usual det A = 0 (31) Since deta for the complex-valued matrix A is Gramian and is positive definite det A st 0 (32) which suggests that the zero-points of deta are local minima. The formulation indicated in Eqs. (28) and (32) are formally identical to the conventional one using the complex-valued fundamental solution of the Helmholtz equation. However, the expected difference is that the coefficient matrix A includes A explicitly as polynomials and that we do not need any recomputation on the boundary elements for each different magnitude of A. EIGENVALUE EXTRACTION Figures 1 and 2 are illustrative distributions of data and d(deta)/da with respect to A for the example problem shown later (Example 1). Although the map in Fig. 1 is employed conventionally for searching the zero-points of deta, it seems not easy to do since exact zero-points are not found due to numerical approximation. Therefore, even in the conventional method, local minima in place of zero-points are extracted on the map in Fig. 1. Further, Irom the last-indicated viewpoint, we mention that it is still different depending on magnification scale of the ordinate. The present scheme employs Fig. 2 instead of Fig. 1 because the curves of d(deta )/da vary across the abscissa (A-axis) and the zero-points are clearly detectable. For the purpose, the derivative of deta must be computable with ease, which fortunately was already certificated in (lie process of the formulation. Indicating d_ ^eta(a) (33)

6 234 Computational Acoustics we find k satisfying f (A) = 0 (34) by the Newton iteration, seems applicable for the map in Fig. 2. The first and second derivatives of del A are del A = del A* tri \ A. i da dk I dk del A = det A dk*,, _, dal 1' JcM-' da._,<** A tr\ A r + lr\ + A da- dk dk dt (35),(36) where (37) Thus the recurrence formula for the Newton iteration comes to dk (38) The following condition is supplemented for the local minima d' -deta^o (39) da' EXAMPLES Example computations are performed using piecevvise constant line boundary elements (constant elements) and the terms of the coefficient matrices PI, G are taken up to their magnitudes less than 10 ~. The first example is for a rectangular domain of L^ = 1, L with the boundary condition q - 0 along the entire boundary. Table 1 is the results obtained by 18 and 36 elements, compared with the analytical solution and the earlier results by the MRM. Good agreement is found between the analytical and present 36 element solutions. Two empty spaces in the column MRM indicate that the solutions were not obtained. The second example is for the case of mixed boundary condition shown in Fig. 3, which was treated previously by the adaptive boundary elements [16]. Table 2 compares a few results indicating the present one and proves its effectiveness. CONCLUSION The new complex-valued formulation for the eigenvalue problem of the scalar-valued

7 Computational Acoustics 235 Helmholtz equation was shown and the scheme for eigenvalue extraction by direct computation of the zero-points of the first derivative of the determinant of coefficient matrix. 1 he interrelationship between the present and the MRM formulations made it possible to derive the recurrence expression for the Newton iteration. Simple examples showed more effectiveness over the conventional and/or early scheme using the MRM in the real-valued formulation. REFERENCES 1. Niwa Y., Kobayashi S. and Kitahara M., Determination of Eigenvalues by Boundary Element Methods, Chapter 7, Developments in Boundary Element Methods, Vol. 2, Baneijee P. K. and Shaw R. P., eds., Appl. Sci. Pub., Beskos D., ed., Boundary Element Methods in Mechanics, North Holland, Partridge P. W. and Brebbia C. A., The dual reciprocity boundary element method for the Helmholtz equation, Proc. Int. Bound. Elms. Symp., Brebbia C. A. and Choudouet-Miranda A., eds., Comp. Mech. Pub./Springer Ver., , Baiierjee P. K., Ahmad S. and Waiig H. C., A new BEM formulation for acoustic eigenfrequency analysis, Int. J. Num. Meth. Eng., 26, , Coyette J. P. and Fyfe K. R., An improved formulation for acoustic eigemnode extraction from boundary element models, Trans. ASklE, J. Vib. Acoust., 112, , Ali A., Rajakumar C. and Yunus S. M., On the formulation of the acoustic boundary element eigenvalue problems, Int. J. Num. Meth. Eng., 31, , Kamiya N. and Andoh E., Eigenvalue analysis by boundary element method, J. Sound Vib., 160, , Kami y a N., Andoh E. and Nogae K., Three-dimensional eigenvalue analysis of the Helmholtz equation by multiple reciprocity boundary element method, Adv. Eng. Software, 16, , Kami y a N. and Andoh E., Standard eigenvalue analysis by boundary element method, Comm. Num. Meth. Eng., 9, , Itagaki M. and Brebbia C. A., Multiple reciprocity boundary element formulation for one-group fission neutron iteration problems, Eng. Anal. Bound. Elms., 11, 39-46, Itagaki M. and Brebbia C. A., Source iterative multiple reciprocity techniques for Helmholtz eigenvalue problems with boundary elements, Proc. Fifth Japan-China Symp. BEM, Tanaka M., Du Q. and Honma T., eds., Elsevier Sci. Pub., 79-88, Kirkup S. M. and A mini S., Solution of the Helmholtz problem via the boundary element method, Int. J. Niun. Meth. Eng., 36, , Kami y a N. and Wu S. T., Generalized eigenvalue formulation of the Ilelmholtz equation by Trefftz method, Int. J. Eng. Comp., Kamiya N. and Andoh E., A note on multiple reciprocity boundary element method: Neglect of domain term, Eng. Anal. Bound. Elms., 10, , Kamiya N. and Andoh E., A note on multiple recipocity integral formulation for Helmholtz equation, Comm. Num. Meth. Eng., 9, 9-13, Kamiya N., Nogae K. and Andoh E., Adaptive boundary element for eigenvalue analysis of the Helmholtz equation, Proc. 16th Int. Conf. BEM, Brebbia C. A. ed., Comp Mech. Pub./Elsevier Sci. Pub., 1994.

8 236 Computational Acoustics IUXIU T o 1 r f3 U wavenumbcr k Figure 1. Distribution of deta (Ex. 1) u=0 q= wav en umber k figure 2. Distribution of d(deta)/da'(ex. 1). Figure 3. Example 2. mode 1,0 0,1 1,1 2,0 2,1 0,2 1,2 3.0 Table 1. Results of Example 1. Anal. Sol MRM elms Present MRM elms Present Table 2. Results of Example 2. Present 72elms elms MRM adaptive