Study of the Fragile Points Method for solving two-dimensional linear and nonlinear wave equations on complex and cracked domains

Abstract

This paper presents the meshless Fragile Points Method (FPM) for the two-dimensional linear and nonlinear wave equations on irregular and complex domains. Time derivatives of the wave equation are discretized using finite difference schemes, and the theorems of stability and convergence of this semi-discrete scheme are given. A simple Galerkin was applied with the domain divided into subdomains by a Voronoi diagram for spatial discretization of problems. Numerical flux corrections are applied to avoid incompatibilities in this study. In case of encountering a crack or discontinuity in the common neighborhood of the selected points, the connection between the two points is cut off without making any particular change in the result. Hence in this method, the points are called “fragile” and it is suitable for solving problems with cracks and discontinuities. A predictor–corrector scheme is used to obtain an appropriate approximation for the nonlinear term in the wave equation. FPM yields symmetric and sparse point stiffness matrices, making it a computationally efficient method that reaches the appropriate accuracy in a short time according to the conditions of the problem. Test problems on complex and cracked domains are evaluated that show the acceptable accuracy and efficiency of the proposed method.