Numerical solution to the deflection of thin plates using the two-dimensional Berger equation with a meshless method based on multiple-scale Pascal polynomials

Abstract

• Two-dimensional linear and nonlinear Berger equation investigated. • Meshless multiple-scale Pascal polynomials method applied to fourth order equations. • Accurate results on complexly shaped domains with low number of collocation points . • Efficiency of proposed method demonstrated based on six problems. In this study, we employ Pascal polynomial basis in the two-dimensional Berger equation, which is a fourth order partial differential equation with applications to thin elastic plates. The polynomial approximation method based on Pascal polynomial basis can be readily adapted to obtain the numerical solutions of partial differential equations. However, a drawback with the polynomial basis is that the resulting coefficient matrix for the problem considered may be ill-conditioned. Due to this ill-conditioned behavior, we use a multiple-scale Pascal polynomial method for the Berger equation. The ill-conditioned numbers can be mitigated using this approach. Multiple scales are established automatically by selecting the collocation points in the multiple-scale Pascal polynomial method. This method is also a meshless method because there is no requirement to establish complex grids or for numerical integration. We present the solutions of six linear and nonlinear benchmark problems obtained with the proposed method on complexly shaped domains. The results obtained demonstrate the accuracy and effectiveness of the proposed method, as well showing its stability against large noise effects.