Jeng-Tzong Chen

On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations

In this paper, it is proved that the two approaches, known in the literature as the method of fundamental solutions (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in formulation. In deriving the equivalence of the Trefftz method and the MFS for the Laplace and biharmonic problems, it is interesting to find that the complete set in the Trefftz method for the Laplace and biharmonic problems are embedded in the degenerate kernels of the MFS. The degenerate scale appears using the MFS when the geometrical matrix is singular. The occurring mechanism of the degenerate scale in the MFS is also studied by using circulant. The comparison of accuracy and efficiency of the two methods was addressed.

A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function

In this paper, a meshless method for solving the eigenfrequencies of clamped plates using the radial basis function (RBF) is proposed. The coefficients of influence matrices are easily determined by the two-point function. By employing the RBF in the imaginary-part fundamental solution, true eigensolutions instead of spurious one are obtained for plate vibration. In order to obtain the eigenvalues and boundary modes at the same time, singular value decomposition technique is utilized. Two examples, circular and rectangular clamped plates, are demonstrated to see the validity of the present method.

A study on the multiple reciprocity method and complex-valued formulation for the Helmholtz equation

Abstract The relation between the multiple reciprocity method and the complex-valued formulation for the Helmholtz equation is re-examined in this paper. Both the singular and hypersingular integral equations derived from the conventional multiple reciprocity method are identical to the real parts of the complex-valued singular and hypersingular integral equations, provided that the fundamental solution chosen in the multiple reciprocity method is proper. The problem of spurious eigenvalues occurs when we use either a singular or hypersingular equation only in the multiple reciprocity method because information contributed by the imaginary part of the complex-valued formulation is lost. To filter out the spurious eigenvalues in the conventional multiple reciprocity method, singular and hypersingular equations are combined together to provide sufficient constraint equations. Several one-dimensional examples are used to examine the relation between the conventional multiple reciprocity method and the complex-valued formulation. Also, a new complete multiple reciprocity method in one-dimensional cases, which involves real and imaginary parts, is proposed by introducing the imaginary part in the undetermined coefficient in the zeroth-order fundamental solution. Based on this complete multiple reciprocity method, it is shown that the kernels derived from the multiple reciprocity method are exactly the same as those obtained in the complex-valued formulation.

Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants

For a potential problem, the boundary integral equation approach has been shown to yield a nonunique solution when the geometry is equal to a degenerate scale. In this paper, the degenerate scale problem in boundary element method (BEM) is analytically studied using the degenerate kernels and circulants. For the circular domain problem, the singular problem of the degenerate scale with radius one can be overcome by using the hypersingular formulation instead of the singular formulation. A simple example is shown to demonstrate the failure using the singular integral equations. To deal with the problem with a degenerate scale, a constant term is added to the fundamental solution to obtain the unique solution and another numerical example with an annular region is also considered.

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.

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

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 modified 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.

An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation

The boundary integral equation approach has been shown to suffer a nonunique solution when the geometry is equal to a degenerate scale for a potential problem. In this paper, the degenerate scale problem in boundary element method for the two-dimensional Laplace equation is analytically studied in the continuous system by using degenerate kernels and Fourier series instead of using discrete system using circulants [Engng Anal. Bound. Elem. 25 (2001) 819]. For circular and multiply-connected domain problems, the rank-deficiency problem of the degenerate scale is solved by using the combined Helmholtz exterior integral equation formulation (CHEEF) concept. An additional constraint by collocating a point outside the domain is added to promote the rank of influence matrix. Two examples are shown to demonstrate the numerical instability using the singular integral equation for circular and annular domain problems. The CHEEF concept is successfully applied to overcome the degenerate scale and the error is suppressed in the numerical experiment.

Dual integral formulation for determining the acoustic modes of a two-dimensional cavity with a degenerate boundary

Abstract In this paper, the dual integral formulation for the Helmholtz equation used in solving the acoustic modes of a two-dimensional cavity with a degenerate boundary is derived. All the improper integrals for the kernel functions in the dual integral equations are reformulated into regular integrals by integrating by parts and are calculated by means of the Gaussian quadrature rule. The jump properties for the single layer potential, double layer potential and their directional derivatives are examined and the potential distributions are shown. To demonstrate the validity of the present formulation, the acoustic frequencies and acoustic modes of the two-dimensional cavity with an incomplete partition are determined by the developed dual BEM program. Also, the numerical results are compared with those of the ABAQUS program, FEM by Petyt and the dual multiple reciprocity method. Good agreement between the present formulation and measurements by Petyt is also shown.

Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real-part dual BEM

It has been found recently that the multiple reciprocity method (MRM) (Chen and Wong. Engng. Anal. Boundary Elements 1997; 20(1):25–33; Chen. Processings of the Fourth World Congress on Computational Mechanics, Onate E, Idelsohn SR (eds). Argentina, 1998; 106; Chen and Wong. J. Sound Vibration 1998; 217(1): 75–95.) or real-part BEM (Liou, Chen and Chen. J. Chinese Inst. Civil Hydraulics 1999; 11(2):299–310 (in Chinese)). results in spurious eigenvalues for eigenproblems if only the singular (UT) or hypersingular (LM) integral equation is used. In this paper, a circular cavity is considered as a demonstrative example for an analytical study. Based on the framework of the real-part dual BEM, the true and spurious eigenvalues can be separated by using singular value decomposition (SVD). To understand why spurious eigenvalues occur, analytical derivation by discretizing the circular boundary into a finite degree-of-freedom system is employed, resulting in circulants for influence matrices. Based on the properties of the circulants, we find that the singular integral equation of the real-part BEM for a circular domain results in spurious eigenvalues which are the zeros of the Bessel functions of the second kind, Y (kρ), while the hypersingular integral equation of the real-part BEM results in spurious eigenvalues which are the zeros of the derivative of the Bessel functions of the second kind, Yn′(kρ). It is found that spurious eigenvalues exist in the real-part BEM, and that they depend on the integral representation one uses (singular or hypersingular; single layer or double layer) no matter what the given types of boundary conditions for the interior problem are. Furthermore, spurious modes are proved to be trivial in the circular cavity through analytical derivations. Numerically, they appear to have the same nodal lines of the true modes after normalization with respect to a very small nonzero value. Two examples with a circular domain, including the Neumann and Dirichlet problems, are presented. The numerical results for true and spurious eigensolutions match very well with the theoretical prediction. Copyright

Applications of the dual integral formulation in conjunction with fast multipole method in large-scale problems for 2D exterior acoustics

In this paper, we solve the large-scale problem for exterior acoustics by employing the concept of fast multipole method (FMM) to accelerate the construction of influence matrix in the dual boundary element method (DBEM). By adopting the addition theorem, the four kernels in the dual formulation are expanded into degenerate kernels, which separate the field point and source point. The separable technique can promote the efficiency in determining the coefficients in a similar way of the fast Fourier transform over the Fourier transform. The source point matrices decomposed in the four influence matrices are similar to each other or only some combinations. There are many zeros or the same influence coefficients in the field point matrices decomposed in the four influence matrices, which can avoid calculating repeatedly the same terms. The separable technique reduces the number of floating-point operations from OðN 2 Þ to OðN log a ðNÞÞ; where N is number of elements and a is a small constant independent of N: To speed up the convergence in constructing the influence matrix, the center of multipole is designed to locate on the center of local coordinate for each boundary element. This approach enhances convergence by collocating multipoles on each center of the source element. The singular and hypersingular integrals are transformed into the summability of divergent series and regular integrals. Finally, the FMM is shown to reduce CPU time and memory requirement thus enabling us apply BEM to solve for large-scale problems. Five moment FMM formulation was found to be sufficient for convergence. The results are compared well with those of FEM, conventional BEM and analytical solutions and it shows the accuracy and efficiency of the FMM when compared with the conventional BEM. q 2003 Elsevier Ltd. All rights reserved.