I.L. 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 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.

USING THE METHOD OF FUNDAMENTAL SOLUTIONS IN CONJUNCTION WITH THE DEGENERATE KERNEL IN CYLINDRICAL ACOUSTIC PROBLEMS

Abstract This paper proposes applications of the method of fundamental solutions (MFS) to exterior acoustic radiation and scattering problems. By using the two‐point function of fundamental solutions, the coefficients of influence matrices are easily determined. It is found that this method also pordoces irregular frequencies as well as the boundary element method does. The position of the irregular frequency depends on the source point location. To avoid this numerical instability, the mixed‐layer potential method is employed to deal with the problem. Based on the circulant properties and degenerate kernels, an analytical scheme in the discrete system of a cylinder is achieved to demonstrate the existence of irregular frequencies. Three numerical examples of uniform radiation, nonuniform radiation and scattering problems of a circular cylinder are examined and are compared with the results by using direct BEM.

Mathematical analysis and treatment for the true and spurious eigenequations of circular plates by the meshless method using radial basis function

Abstract In this paper, a meshless method for solving the eigenproblems of plate vibration using the radial basis function (RBF) is proposed. By employing the RBF in the imaginary‐part fundamental solution, spurious eigenequations in conjunction with the true ones are obtained at the same time. Mathematical analysis for the appearance of spurious eigenequations by using degenerate kernel and circulant is done through a circular plate for a discrete system. In order to obtain the true and spurious eigenequations, six (C 2 4) formulations (either two combinations from the four types of potentials, single, double, triple and quadruple) of meshless methods are employed in conjunction with the SVD technique. The spurious eigenequation in each formulation is found and is filtered out by using the SVD updating technique. Three cases, clamped, simply‐supported and free circular plates, are demonstrated to check the validity of the meshless methods.

Eigenanalysis for membranes with stringers using conventional BEM in conjunction with SVD technique

It is well known that either the multi-domain boundary element method (BEM) or the dual BEM can solve boundary value problems with degenerate boundaries where double values are defined, i.e., two portions of boundary meet together with a zero thickness. In this paper, the eigensolutions for membranes with stringers are obtained in a single domain by using the conventional BEM in conjunction with the SVD technique. By adopting the SVD technique for rank revealing, the nontrivial boundary mode can be detected by the successive zero singular values which are not due to the degeneracy of degenerate boundary. The boundary modes are obtained according to the right unitary vectors with respect to the zero singular values in the SVD. Three examples, a single-edge stringer, a double-edge stringer and a central stringer in a circular membrane, are considered. The results of the present method, are compared with those of the multi-domain BEM, the dual BEM, the DtN method (Dirichlet to Neumann), the FEM (ABAQUS) and analytical solutions if available. Good agreement is obtained. The goal to deal with the eigenproblem in a single domain without hypersingularity is achieved.

FREE VIBRATION ANALYSIS OF MULTIPLY CONNECTED PLATES USING THE METHOD OF FUNDAMENTAL SOLUTIONS

In this paper, the method of fundamental solutions (MFS) for solving the eigenfrequencies of multiply connected plates is proposed. The coefficients of influence matrices are easily determined when the fundamental solution is known. True and spurious eigensolutions appear at the same time. It is found that the spurious eigensolution using the MFS depends on the location of the inner boundary where the fictitious sources are distributed. To verify this finding, mathematical analysis for the appearance of spurious eigenequations using degenerate kernels and circulants is done by demonstrating an annular plate with a discrete model. In order to obtain the true eigensolution, the Burton & Miller method is utilized to filter out the spurious eigensolutions. One example is demonstrated analytically and numerically to see the validity of the present method.