Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply connected problem

The spurious eigenvalues of an annular domain have been verified for the singular and hypersingular boundary‐element methods (BEMs) and circumvented by using the Burton‐Miller approach. Do they also occur in other formulations: continuous formulations such as the singular and hypersingular boundary integral equations (BIEs), the null‐field BIEs and the fictitious BIEs, or such discrete formulations as the null‐field BEMs and the fictitious BEMs? For the ten formulations of the multiply connected problem the study of otherwise the same issues is continued in the present paper. By using the degenerate kernels and the Fourier series, it is demonstrated analytically for the six continuous formulations of BIEs that spurious eigensolutions depend on the geometry of the inner boundary but not on that of the outer boundary. This conclusion can be extended to the six discrete formulations of BEMs. To filter out the spurious eigenvalues, the CHIEF (combined Helmholtz integral equation formulation) method is used here instead of the Burton‐Miller approach. The optimum number and appropriate positions of the CHIEF points are also addressed. It is then shown that, in the null‐field and fictitious BEMs, the spurious and true eigenvalues can be detected and distinguished by using the singular‐value‐decomposition‐updating techniques in conjunction with the Fredholm alternative theorem. Illustrative examples show the validity of the proposed methodologies.

[1]  J. R. Hutchinson.  Analysis of Plates and Shells by Boundary Collocation , 1991 .

[2]  Wolfgang L. Wendland,et al.  Boundary integral methods in low frequency acoustics , 2000 .

[3]  H. Hong,et al.  Review of Dual Boundary Element Methods With Emphasis on Hypersingular Integrals and Divergent Series , 1999 .

[4]  Ming-Te Liang,et al.  ANALYTICAL STUDY AND NUMERICAL EXPERIMENTS FOR RADIATION AND SCATTERING PROBLEMS USING THE CHIEF METHOD , 2001 .

[5]  G. Mey.  Calculation of eigenvalues of the Helmholtz equation by an integral equation , 1976 .

[6]  Elizabeth R. Jessup,et al.  Matrices, Vector Spaces, and Information Retrieval , 1999, SIAM Rev..

[7]  H. Saunders,et al.  Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates , 1985 .

[8]  Paul A. Martin,et al.  ON THE NULL-FIELD EQUATIONS FOR THE EXTERIOR PROBLEMS OF ACOUSTICS , 1980 .

[9]  Norio Kamiya,et al.  A new complex-valued formulation and eigenvalue analysis of the Helmholtz equation by boundary element method , 1996 .

[10]  Arthur W. Leissa,et al.  Vibration of Plates , 2021, Solid Acoustic Waves and Vibration.

[11]  A. Nowak,et al.  The Multiple reciprocity boundary element method , 1994 .

[12]  S. Chyuan,et al.  Boundary element analysis for the Helmholtz eigenvalue problems with a multiply connected domain , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[13]  G. de Mey,et al.  A simplified integral equation method for the calculation of the eigenvalues of Helmholtz equation , 1977 .

[14]  Jeng-Tzong Chen,et al.  On fictitious frequencies using circulants for radiation problems of a cylinder , 2000 .

[15]  N. Kamiya,et al.  A note on multiple reciprocity integral formulation for the Helmholtz equation , 1993 .

[16]  T. K. Rengarajan,et al.  The use of CHIEF to obtain unique solutions for acoustic radiation using boundary‐integral equations , 1987 .

[17]  J. Achenbach,et al.  Off-boundary approach to the boundary element method , 1988 .

[18]  Weichung Yeih,et al.  APPLICATIONS OF THE GENERALIZED SINGULAR-VALUE DECOMPOSITION METHOD ON THE EIGENPROBLEM USING THE INCOMPLETE BOUNDARY ELEMENT FORMULATION , 2000 .

[19]  Shyh-Rong Kuo,et al.  Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real‐part dual BEM , 2000 .

[20]  Jeng-Tzong Chen,et al.  Dual integral formulation for determining the acoustic modes of a two-dimensional cavity with a degenerate boundary , 1998 .

[21]  Carlos Alberto Brebbia,et al.  The multiple-reciprocity method. A new approach for transforming BEM domain integrals to the boundary , 1989 .

[22]  Jianxin Zhou,et al.  Vibration and damping in distributed systems , 1993 .

[23]  Jeng-Tzong Chen,et al.  On fictitious frequencies using dual series representation , 1998 .

[24]  J. Hwang,et al.  A retracted boundary integral equation for exterior acoustic problem with unique solution for all wave numbers , 1991 .

[25]  Yinglong Zhang,et al.  On the dual reciprocity boundary element method , 1993 .

[26]  G. F. Miller,et al.  The application of integral equation methods to the numerical solution of some exterior boundary-value problems , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[27]  P. W. Partridge,et al.  The dual reciprocity boundary element method , 1991 .

[28]  Richard Paul Shaw,et al.  Helmholtz‐equation eigenvalues and eigenmodes for arbitrary domains , 1974 .

[29]  F. C. Wong,et al.  Determination of Spurious Eigenvalues and Multiplicities of True Eigenvalues in the Dual Multiple Reciprocity Method Using the Singular-Value Decomposition Technique , 2000 .

[30]  H. A. Schenck.  Improved Integral Formulation for Acoustic Radiation Problems , 1968 .

[31]  H. Hong,et al.  Derivations of Integral Equations of Elasticity , 1988 .

[32]  I L Chen,et al.  A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method. , 2001, The Journal of the Acoustical Society of America.

[33]  Satoshi Tomioka,et al.  Power iterative multiple reciprocity boundary element method for solving three-dimensional Helmholtz eigenvalue problems , 1997 .