M.S. Matbuly

Natural frequencies of a functionally graded cracked beam using the differential quadrature method

The present work is concerned with the free vibration analysis of an elastically supported cracked beam. The beam is made of a functionally graded material and rested on a Winkler–Pasternak foundation. The line spring model is employed to formulate the problem. The method of differential quadrature is applied to solve it. The obtained results agreed with the previous similar ones. Further, a parametric study is introduced to investigate the effects of the geometric and elastic characteristics of the problem on the values of natural frequencies and mode shape functions.

Vibration analysis of structural elements using differential quadrature method

The method of differential quadrature is employed to analyze the free vibration of a cracked cantilever beam resting on elastic foundation. The beam is made of a functionally graded material and rests on a Winkler–Pasternak foundation. The crack action is simulated by a line spring model. Also, the differential quadrature method with a geometric mapping are applied to study the free vibration of irregular plates. The obtained results agreed with the previous studies in the literature. Further, a parametric study is introduced to investigate the effects of geometric and elastic characteristics of the problem on the natural frequencies.

Analysis of composite plates using moving least squares differential quadrature method

Abstract In this work, the moving least squares differential quadrature method (MLSDQM) is employed to analyze bending problems of composite plates. Based on a transverse shear theory, the governing equations of the problem are derived. The transverse deflection and two rotations of the plate are independently approximated with MLS approximations. The weighting coefficients used in the MLSDQ approximation are obtained through the fast computation of the MLS shape functions and their partial derivatives. The obtained results are compared with the previous analytical and numerical ones. Further a parametric study is introduced to investigate the effects of elastic and geometric characteristics on the values of transverse deflection of the plate.

A modified diffusion coefficient technique for the convection diffusion equation

A new modified diffusion coefficient (MDC) technique for solving convection diffusion equation is proposed. The Galerkin finite-element discretization process is applied on the modified equation rather than the original one. For a class of one-dimensional convection-diffusion equations, we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size, then, prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and/or equation coefficients. The application of the derived analytic formula of MDC is extended for other classes of convection-diffusion equations, where the analytic formula is computed locally within each element according to the mesh size and the values of associated coefficients in each direction. The numerical results of the proposed approach for two-dimensional, variable coefficients, with boundary layers, convection-dominated problems show stable and accurate solutions even on coarse grids. Accordingly, multigrid based solvers retain their efficient convergence rates.

Iterative differential quadrature solutions for Bratu problem

Abstract A numerical scheme based on differential quadrature methods, is introduced for solving Bratu problem. The problem is firstly reduced to an iterative one. Then, both of differential quadrature method (DQM) and moving least squares differential quadrature method (MLSDQM) are applied to solve iteratively the nonlinear problem. The proposed scheme successfully computes multiple solutions to Bratu’s problem. The obtained results agree with the 1D and 2D closed forms. Further a parametric study is introduced to investigate the computational characteristics of the proposed scheme.

Buckling analysis of composite plates using moving least squares differential quadrature method

ABSTRACT This work concerns with buckling and vibration analysis of composite plates based on a transverse shear theory. A numerical scheme is introduced to determine the angular frequencies and critical buckling loads of such plates. Moving least square differential quadrature method is employed to reduce the problem to that of eigen value problem. The accuracy and efficiency of the proposed scheme is examined with different computational characteristics, (radius of support domain, basis completeness order, and scaling factors). The obtained results agreed, at less execution time, with the previous ones. Further, a parametric study is introduced to investigate the influence of elastic and geometric characteristics, (Youngs modulus gradation ratio, shear modulus gradation ratio, Poissons ratio, loading parameter, and aspect ratio), of the composite on the values of critical buckling load, natural frequencies, and behavior of mode shape functions.