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

Liu

Hong

2003

Proc. R. Soc. Lond. A

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.

Section: Introductionmentioning

confidence: 99%

Section: Chief Treatment For Spurious Eigensolutionsmentioning

confidence: 99%

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

Liu

Hong

2003

Proc. R. Soc. Lond. A

“…Burton and Miller [3] solved the problem by combining singular and hypersingular equations with an imaginary constant. Chen et al [4] 235 methods to eliminate degenerate scale, scaling method and restriction method were discussed by Christiansen [33]. He also investigated the condition number of the influence matrix of the fictitious BEM and null-field approach [34].…”

Section: Introductionmentioning

confidence: 99%

Degenerate scale problem when solving Laplace's equation by BEM and its treatment

Lin

Numerical Meth Engineering

2004

Self Cite

SUMMARYIn this paper, Laplace problems are solved by using the dual boundary element method (BEM). It is found that a degenerate scale problem occurs if the conventional BEM is used. In this case, the influence matrix is rank deficient and numerical results become unstable. Both the circular and elliptical bars are studied analytically in the continuous system. In the discrete system, the Fredholm alternative theorem in conjunction with the SVD (Singular Value Decomposition) updating documents is employed to sort out the spurious mode which causes the numerical instability. Three regularization techniques, method of adding a rigid body mode, hypersingular formulation and CHEEF (Combined Helmholtz Exterior integral Equation Formulation) concept, are employed to deal with the rankdeficiency problem. The addition of a rigid body term, c, in the fundamental solution is proved to shift the original degenerate scale to a new degenerate scale by a factor e −c . The torsion rigidities are obtained and compared with analytical solutions. Numerical examples including elliptical, square and triangular bars were demonstrated to show the failure of conventional BEM in case of the degenerate scale. After employing the three regularization techniques, the accuracy of the proposed approaches is achieved.

“…Fundamental solutions are expanded into degenerate kernels in constructing the Green's function. Since a degenerate kernel separates the source and field points for the closed-form fundamental solution, it plays an important role in studying the rank-deficiency problems analytically in BEM, e.g., degenerate scale [4,6], spurious eigenvalues [5,8,10] and fictitious frequencies [2,3]. Also, it can study the order of pseudo-differential operator [7].…”

Section: Introductionmentioning

confidence: 99%

Alternative derivations for the Poisson integral formula

International Journal of Mathematical Education in Science and

2006

Self Cite

Poisson integral formula is revisited. The kernel in the Poisson integral formula can be derived in a series form through the direct BEM free of the concept of image point by using the null-field integral equation in conjunction with the degenerate kernels. The degenerate kernels for the closed-form Green's function and the series form of Poisson integral formula are also derived. Two and threedimensional cases are considered. Also, interior and exterior problems are both solved. Even though the image concept is required, the location of image point can be determined straightforward through the degenerate kernels instead of the method of reciprocal radii.