Multiscale localized differential quadature in 2D partial differential equation for mechanics of shape memory alloys

Abstract

In this research, the applicability of the Multiscale Localized Differential Quadrature (MLDQ) method in two-dimensional shape memory alloy (SMA) model was explored. The MLDQ method was governed in solving several partial differential equations. Besides, the finite difference (FD) method was used to solve some examples of partial differential equations and the solutions obtained were compared with those obtained by MLDQ method in order to show the accuracy of the numerical method. The MLDQ method was developed by increasing the number of grid points in critical region, and approximating the derivatives at the certain selected grid points. This present method together with the fourth-order Runge-Kutta (RK) method has been applied in differential equations such as wave equation and high gradient problems,. The MLDQ method can achieves accurate numerical solutions compared with FD method which is a low order numerical method by using a few number of grid points. The multiscale method was employed at the critical region which can break down the region of interest from coarser into finer grid points. Furthermore, FORTRAN programs were developed based on MLDQ method in solving some problems as above. The shared memory architecture of parallel computing was done by using OpenMP in order to reduce the time taken in simulating the numerical results. Consequently, the results show that the MLDQ method was a good numerical technique in two-dimensional SMA.