Power iterative multiple reciprocity boundary element method for solving three-dimensional Helmholtz eigenvalue problems

Itagaki, Masafumi; Nishiyama, Shusuke; Tomioka, Satoshi; Enoto, Takeaki; Sahashi, Naoki

doi:10.1016/s0955-7997(97)00055-6

Cited by 8 publications

(4 citation statements)

References 9 publications

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“…1992) and the multiple reciprocity method (MRM) (Kamiya & Andoh 1993;Nowak & Brebbia 1989;Nowak & Neves 1994) have been widely used. One advantage of the MRM, which uses the Laplace-type fundamental solution, is that only real-valued computation is needed (Itagaki & Brebbia 1993Itagaki et al 1997). Therefore, the MRM is indeed no more than the real part of the complex-valued formulation (Kamiya et al .…”

Section: Introductionmentioning

confidence: 99%

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

Chen

Liu

Hong

2003

Proc. R. Soc. Lond. A

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The spurious eigenvalues of an annular domain have been veri ed 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-eld BIEs and the ctitious BIEs, or such discrete formulations as the null-eld BEMs and the ctitious 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 lter 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-eld and ctitious BEMs, the spurious and true eigenvalues can be detected and distinguished by using the singular-value-decompositionupdating techniques in conjunction with the Fredholm alternative theorem. Illustrative examples show the validity of the proposed methodologies.

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Section: Introductionmentioning

confidence: 99%

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

Chen

Liu

Hong

2003

Proc. R. Soc. Lond. A

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“…For the Helmholtz eigenvalue problem with a simply connected domain, the dual reciprocity method (DRM) (Silva & Venturini 1988) and the multiple reciprocity method (MRM) (Kamiya & Andoh 1993;Nowak & Neves 1994;Nowak & Brebbia 1989) have been widely used recently. One advantage of the MRM using the Laplacetype fundamental solution is that only real-valued computation is used instead of the MRM using the Helmholtz-type fundamental solution (Itagaki & Brebbia 1993Itagaki et al . 1997).…”

Section: Introductionmentioning

confidence: 99%

Boundary element analysis for the Helmholtz eigenvalue problems with a multiply connected domain

Chen

Lin

Kuo

et al. 2001

Proc. R. Soc. Lond. A

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For a Helmholtz eigenvalue problem with a multiply connected domain, the boundary integral equation approach as well as the boundary-element method is shown to yield spurious eigenvalues even if the complex-valued kernel is used. In such a case, it is found that spurious eigenvalues depend on the geometry of the inner boundary. Demonstrated as an analytical case, the spurious eigenvalue for a multiply connected problem with its inner boundary as a circle is studied analytically. By using the degenerate kernels and circulants, an annular case can be studied analytically in a discrete system and can be treated as a special case. The proof for the general boundary instead of the circular boundary is also derived. The Burton{Miller method is employed to eliminate spurious eigenvalues in the multiply connected case. Moreover, a modi ed method considering only the real-part formulation is provided. Five examples are shown to demonstrate that the spurious eigenvalues depend on the shape of the inner boundary. Good agreement between analytical prediction and numerical results are found.

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“…Different to finite element approaches which require a discretization of the computational domain Ω, the use of boundary integral formulations and boundary element methods to solve the eigenvalue problem needs only a discretization of the boundary Γ. For the discretization of the boundary integral eigenvalue problem collocation schemes [4,5,14,16,17,22,27,29] and Galerkin methods [6,7,31,34] are considered. Both methods yield algebraic nonlinear eigenvalue problems where the matrix entries are transcendental functions with respect to the eigenparameter.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of a Galerkin Boundary Element Method for the Dirichlet Laplacian Eigenvalue Problem

Steinbach¹,

Unger²

2012

SIAM J. Numer. Anal.

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In this paper, a rigorous convergence and error analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem is presented. The formulation of the eigenvalue problem in terms of a boundary integral equation yields a nonlinear boundary integral operator eigenvalue problem. This nonlinear eigenvalue problem and its Galerkin approximation are analyzed in the framework of eigenvalue problems for holomorphic Fredholm operator-valued functions. The convergence of the approximation is shown and quasi-optimal error estimates are presented. Numerical experiments are given confirming the theoretical results.

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Power iterative multiple reciprocity boundary element method for solving three-dimensional Helmholtz eigenvalue problems

Cited by 8 publications

References 9 publications

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

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

Boundary element analysis for the Helmholtz eigenvalue problems with a multiply connected domain

Convergence Analysis of a Galerkin Boundary Element Method for the Dirichlet Laplacian Eigenvalue Problem

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