On the analysis of dispersion property and stable time step in meshfree method using the generalized meshfree approximation

• Published in: Finite elements in analysis and design 2011-07, Vol.47 (7), p.683-697
• Main Authors: Park, C.K., Wu, C.T., Kan, C.D.
• Format: Article
• Language: English
• Subjects: Approximation; Convex approximation; Dispersion error; Dispersions; Dynamic tests; Exact sciences and technology; Generalized meshfree approximation; Mathematical analysis; Mathematical models; Mathematics; Meshfree method; Meshless methods; Methods of scientific computing (including symbolic computation, algebraic computation); Non-convex approximation; Numerical analysis. Scientific computation; Partial differential equations; Sciences and techniques of general use; Stable time step; Transformations
• ISSN: 0168-874X; 1872-6925
• DOI: 10.1016/j.finel.2011.02.001

This paper studies the dispersion characteristic and stable time step of the meshfree method in explicit dynamic problems using the generalized meshfree (GMF) approximation. In the dispersion analysis, the von Neumann method is applied to analyze the numerical dispersion errors for the spatial semi-discretization of a partial differential equation (PDE) in meshfree method using various approximations. The study emphasizes on the influence from the Kronecker-delta property in the approximation that is constructed using the GMF approximation and the full transformation method. In the stable time step analysis, we derive the analytical value of the critical time step based on the eigenvalue analysis. In addition, special attention is paid on the study of the boundary condition effect in suggesting a stable time step used for the practical analysis. Three full-discretization examples are presented to verify the study in the semi-discretization analysis.