Finite difference method

Abstract: Four methods for the calculation of derivatives of vibration mode shapes (eigenvectors) with respect to design parameters are described. These are finite-difference method, modal method, a modified modal method and Nelson's method. The methods are implemented in a general-purpose commercial finite-element program and applied to the following test problems: a cantilever beam and a stiffened cylinder with a cutout. Design variables are a beam tip mass, a beam root height, and specific dimensions of the cylinder model. The methods are compared on the basis of central processor (CP) seconds required to obtain the derivatives, and two of the methods are also evaluated for the rapidity of convergence. Data is presented showing the amount of CP time used to compute the first four eigenvector derivatives for each example problem. A scalar measure of the error in the mode shape derivative is defined, and numerical results illustrating the rapidity of convergence of the approximate derivative to the exact derivative are presented. Results indicate an advantage in using Nelson's method because this method is exact and requires less CP time, especially when derivatives with respect to several design variables are computed.

Abstract: This paper describes the application of radial basis function (RBF) based finite difference type scheme (RBF-FD) for solving steady convection–diffusion equations. Numerical studies are made using multiquadric (MQ) RBF. By varying the shape parameter in MQ, the accuracy of the solution is seen to be highly improved for large values of Reynolds' numbers. The developed scheme has been compared with the corresponding finite difference scheme and found that the solutions obtained using the former are non-oscillatory. Copyright © 2007 John Wiley & Sons, Ltd.

Abstract: A novel meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to accurately analyze the two–dimensional shallow water equations (SWEs). The SWEs are a hyperbolic system of first-order nonlinear partial differential equations and can be used to describe various problems in hydraulic and ocean engineering, so it is of great importance to develop an efficient and accurate numerical model to analyze the SWEs. According to split-coefficient matrix methods, the SWEs can be transformed to a characteristic form, which can easily present information of characteristic in the correct directions. The GFDM and the second-order Runge-Kutta method are adopted for spatial and temporal discretization of the characteristic form of the SWEs, respectively. The GFDM is one of the newly-developed domain-type meshless methods, so the time-consuming tasks of mesh generation and numerical quadrature can be truly avoided. To use the moving-least squares method of the GFDM, the spatial derivatives at every node can be expressed as linear combinations of nearby function values with different weighting coefficients. In order to properly cooperate with the split-coefficient matrix methods and the characteristic of the SWEs, a new way to determine the shape of star in the GFDM is proposed in this paper to capture the wave transmission. Numerical results and comparisons from several examples are provided to verify the merits of the proposed meshless scheme. Besides, the numerical results are compared with other solutions to validate the accuracy and the consistency of the proposed meshless numerical scheme.