A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method

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%

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

Liu

Hong

2003

Proc. R. Soc. Lond. A

“…He proposed the combined Helmholtz interior integral equation formulation (CHIEF) method by collocating the point outside the domain as an auxiliary constraint to promote the rank of influence matrices. Chen et al [8,9] extended the CHIEF method to combined Helmholtz exterior integral equation formulation (CHEEF) method for overcoming the problem of spurious eigenvalues in the eigenproblem. However, both CHIEF and CHEEF ideas may cause failure in selecting the point.…”

Section: Introductionmentioning

confidence: 99%

“…Wu et al [10] proposed a CHIEF-block idea to enforce the constraints in a weighted residual sense over a small region. If the CHIEF point locates on or near the nodal line of its corresponding mode, it may not provide a valid constraint [8,9]. To overcome this problem, Chen et al [8,9] presented an analytical study to select the valid number and position of CHIEF points for the circular case by using circulants.…”

Section: Introductionmentioning

confidence: 99%

Fictitious frequency revisited

Engineering Analysis with Boundary Elements

Lin

Tsai

2009

a b s t r a c tThe nonexistence and nonuniqueness problems associated with integral equation methods for exterior acoustics are revisited. The Fredholm alternative theorem in conjunction with the singular value decomposition updating technique is used to simultaneously determine the fictitious frequencies and corresponding modes in exterior acoustics. After selecting the combined Helmholtz interior integral equation formulation (CHIEF) points, the influence row vectors are obtained. A criterion in selecting the minimum number of CHIEF points and their positions is proposed by testing the orthogonal condition between the influence row vector and the right unitary vector. It is proved in the discrete system for arbitrary-shape problems that the source of numerical instability of irregular frequency originates from the zero divided by zero using the generalized coordinates of unitary vectors. The mathematical structures of the four influence matrices in the dual boundary element method (BEM) are examined by using the left and right unitary matrices. Extracting the true eigenvalue and filtering out the fictitious frequency can be unified by using the updating term and updating document, respectively. Radiation problems of a cylinder and a square rod are demonstrated to see the validity of the present formulation.

“…If we need to look for the eigenmode as well as eigenvalue as usual, the sorting for the spurious eigenequations pays a small price by identifying the mode shapes. Chen et al [3] commented that the spurious modes can be reasonable which may mislead the judgement of the true and spurious ones, since the true and spurious modes may have the same nodal line in case of different eigenvalues. This is the reason why Chen et al have developed many systematic techniques [2,3,4,5,6,7], for sorting out the true and the spurious eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

“…Chen et al [3] commented that the spurious modes can be reasonable which may mislead the judgement of the true and spurious ones, since the true and spurious modes may have the same nodal line in case of different eigenvalues. This is the reason why Chen et al have developed many systematic techniques [2,3,4,5,6,7], for sorting out the true and the spurious eigenvalues. Niwa et al [21] also stated that ``One must take care to use the complete Green's function for outgoing waves, as attempts to use just the real (singular) or imaginary (regular) part separately will not provide the complete spectrum".…”

Section: Introductionmentioning

confidence: 99%

Mathematical analysis and numerical study of true and spurious eigenequations for free vibration of plates using real-part BEM

Lin

Computational Mechanics

et al. 2004

Self Cite

In this paper, an imaginary-part BEM for solving the eigenfrequencies of plates is proposed for avoiding singularity and saving half effort in computation instead of using the complex-valued BEM. By employing the imaginary-part fundamental solution, the spurious eigenequations in conjunction with the true eigenequation are obtained for free vibration of plate. To verify this finding, the circulant is adopted to analytically derive the true and spurious eigenequations in the discrete system of a circular plate. In order to obtain the eigenvalues and boundary modes at the same time, the singular value decomposition (SVD) technique is utilized. The analytical solutions are derived in the discrete system. Three cases, clamped, simply-supported and free circular plates, are demonstrated analytically and numerically to see the validity of the present method. SVD updating technique is adopted to suppress the ocurrence of the spurious eigenvalues, and a clamped plate is demonstrated analytically for the discrete system in this paper.