K.H. Chen

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.

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.

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.

A new concept of modal participation factor for numerical instability in the dual BEM for exterior acoustics

Abstract This paper presents the occurring mechanism why irregular frequencies are imbedded in the exterior acoustics using the dual boundary element method (BEM). The modal participation factor which dominates the numerical instability is derived for continuous and discrete systems. In addition, the irregular (fictitious) frequencies embedded in the singular or hypersingular integral equations are discussed, respectively. It is found that the irregular values depend on the kernels in the integral representation for the solution. A two-dimensional dual BEM program for the exterior acoustics was developed. Numerical experiments are conducted to verify the concept of modal participation factor.

Analytical study and numerical experiments for Laplace equation with overspecified boundary conditions

Abstract In this paper, the window function, e − αk 2 , is applied to regularize the divergent problem which occurs in the Laplace equation with overspecified boundary conditions in an infinite strip region. To deal with this ill-posed problem, the corner of the L-curve is chosen as the compromise point to determine the optimal α of the Gaussian window, e − αk 2 , so that the high wave-number ( k ) content can be suppressed instead of engineering judgement using the concept of a cutoff wave-number. From the examples shown, it is found that a reasonable solution of the unknown boundary potential can be reconstructed, and that both high wave-number content and divergent results can be avoided by using the proposed regularization technique.

Dual boundary element analysis of oblique incident wave passing a thin submerged breakwater

In this paper, the dual integral formulation for the modified Helmholtz equation in solving the propagation of oblique incident wave passing a thin barrier (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 transmission and reflection coefficients of oblique incident wave passing a thin rigid barrier are determined by the developed dual boundary element method program. Also, the results are obtained for the cases of wave scattering by a rigid barrier with a finite or zero thickness in a constant water depth and compared with those of experiment and analytical solution using eigenfunction expansion method. Good agreement is observed.

Numerical experiments for acoustic modes of a square cavity using the dual boundary element method

Abstract The dual integral formulation for the Helmholtz equation for use in solving the acoustic modes of a two-dimensional square cavity is derived, and a general dual boundary element method (BEM) program is developed. Numerical experiments for the degenerate acoustic modes of a square cavity are performed. It is found that the degenerate modes can be distinguished by specifying the normalized boundary data at different boundary points using either the singular integral equation (UT method) or the hypersingular integral equation (LM method). This technique can be employed to determine the multiplicity of the eigenvalue. Two examples with Dirichlet and Neumann boundary conditions are given to show the validity of the proposed technique. Sensitivity and failure in determining the acoustic modes by specifying the normalized data at the boundary locations near and on the node are examined, respectively. Also, numerical results are obtained using finite element method (FEM) and analytical solutions for comparison. Good agreement between them is obtained.

Dual boundary element analysis for cracked bars under torsion

A dual integral formulation for a cracked bar under torsion is derived, and a dual boundary element method is implemented. It is shown that as the thickness of the crack becomes thinner, the ill‐posedness for the linear algebraic matrix becomes more serious if the conventional BEM is used. Numerical experiments for solution instability due to ill‐posedness are shown. To deal with this difficulty, the hypersingular equation of the dual boundary integral formulation is employed to obtain an independent constraint equation for the boundary unknowns. For the sake of computational efficiency, the area integral for the torsion rigidity is transformed into two boundary integrals by using Green’s second identity and divergence theorem. Finally, the torsion rigidities for cracks with different lengths and orientations are solved by using the dual BEM, and the results compare well with the analytical solutions and FEM results.

Dual boundary element analysis of wave scattering from singularities

Abstract The dual boundary element method is used to obtain an efficient solution of the Helmholtz equation in the presence of geometric singularities. In particular, time-harmonic waves in a membrane which contains one or more fixed edge stringers (or cracks) are investigated. The hypersingular integral equation is used in the procedure to ensure a unique solution for the problem with a degenerate boundary. The method yields a solution for the entire membrane as well as the dynamic stress intensity factor. Numerical results are presented for a circular membrane containing a single edge stringer, two edge stringers and an internal stringer. Also, the first three critical wave numbers of the membrane with the homogeneous boundary condition are determined, and the dynamic stress intensity factors are found for problems with the nonhomogeneous boundary condition. Good agreement is found after comparing the results with exact solutions, and with results obtained using DtN-FEM and ABAQUS.