True and spurious eigensolutions for the problems with the mixed-type boundary conditions using BEMs

Boundary integral equations and boundary element methods were employed semi-analytically and numerically to study the occurrence of spurious eigenvalues for the interior Helmholtz equations with the mixed-type boundary conditions. The degenerate kernels of fundamental function and Fourier series were utilized in the null-field integral equation to derive the true and spurious eigenfunctions semi-analytically. The complex-valued boundary element method (BEM), the real-part BEM, the imaginary-part BEM and the multiple reciprocity method were utilized to solve the eigenproblem. The results are compared with those of FEM. A simple case of one-dimensional eigenproblem was also addressed. Moreover, the SVD updating techniques in conjunction with the Fredholm's alternative theorem were adopted to sort out the spurious eigenvalues and extract the true ones. It is emphasized that the occurrence of spurious eigenvalues depends on the adopted method (singular or hypersingular formulation) no matter what the given type of boundary conditions for the problem is. The illustrative examples were verified successfully and the numerical results matched well with those of FEM.

[1]  K. H. Chen,et al.  Determination of spurious eigenvalues and multiplicities of true eigenvalues using the real-part dual BEM , 1999 .

[2]  A. Laub,et al.  The singular value decomposition: Its computation and some applications , 1980 .

[3]  K. H. Chen,et al.  COMMENTS ON &&VIBRATION ANALYSIS OF ARBITRARY SHAPED MEMBRANES USING NON-DIMENSIONAL DYNAMIC INFLUENCE FUNCTION'' , 2000 .

[4]  Jeng-Tzong Chen,et al.  Recent development of dual BEM in acoustic problems , 2000 .

[5]  Gene H. Golub,et al.  Matrix computations , 1983 .

[6]  Lee,et al.  Eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point-matching , 2000, The Journal of the Acoustical Society of America.

[7]  Young-Seok Kang,et al.  VIBRATION ANALYSIS OF ARBITRARILY SHAPED MEMBRANES USING NON-DIMENSIONAL DYNAMIC INFLUENCE FUNCTION , 1999 .

[8]  F. Casadei,et al.  Containment of blast phenomena in underground electrical power plants , 1998 .

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

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

[12]  A nonsingular integral formulation for the Helmholtz eigenproblems of a circular domain , 1999 .

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

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

[15]  I. Wolff,et al.  The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems , 1994 .

[16]  Jeng-Tzong Chen,et al.  A study on the multiple reciprocity method and complex-valued formulation for the Helmholtz equation , 1998 .

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

[18]  F. C. Wong,et al.  Analytical derivations for one-dimensional eigenproblems using dual boundary element method and multiple reciprocity method , 1997 .

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

[20]  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 .