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.

Vibration Analysis of Thin Plates Resting on Pasternak Foundations by Element free Galerkin Method

“…Meshless method is an approximate solution to many boundary value problems which approximations are entirely based on nodal parameters. Varity of meshless methods have been introduced, namely the smoothed particle hydrodynamics (SPH) [8], the meshless local Petrov Galerkin method (MLPG) [9], the reproducing kernel particle method (RKPM) [10], the element free Galerkin method (EFG) [11] and the radial basis function approach [12][13][14][15]. Element free Galerkin method (EFG) is considered as one of meshless methods which is used for solution of bending, buckling and vibration of plates and shells.…”

Investigation on the spurious eigenvalues of vibration plates by non-dimensional dynamic influence function method

“…On the other hand, it has been mathematically proved that the NDIF [7] and the MFS [8,9] yields spurious eigenvalues in the solution of circular plate vibration. However, to our best knowledge, no theoretical proof is available for non-circular plates nowadays.…”

Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs

“…Although the results for circular or annular plates are available in the literature [1,2,3,4,5,6,7,8]; however, all of these publications except [1,6,8] are confined to plates with concentric hole. Although a large amount of papers on the circular membrane vibration with circular holes were published [9], very a few papers on the free vibration of plate with several holes can be found.…”