A New Meshless Method — Finite-Cover Based Element Free Method

Abstract

Presented in the paper is a so-called finite-cover-based element-free method that is aimed to solve both continuous and discontinuous deformation problems in a mathematically consistent framework as manifold method, but no requiring mesh generation. The method is mathematically based on finite circular cover numerical technique and multiple weighted moving least square approximation. In this method, overall volume of materials is overlaid by a series of overlapped circular mathematical covers. While cut by joints, interfaces of different media and physical boundaries, a mathematical cover may be divided into two or more completely disconnected parts that are defined as physical covers. Discontinuity of materials is characterized by discontinuity of physical cover instead of disconnection of influence support. Hence, influence domain, i.e. mathematical cover can be kept regular even in a discontinuous problem. On a set of physical covers containing unknown point under consideration, the multiple weighted moving least square approximation in conjunction with cover weighting functions defined on each mathematical cover is used to determine shape functions of the unknown point for variational principle. Afterwards, discrete equations of the boundary-value problem with discontinuity can be established using variational principle. Through numerical analyses, it is shown that the proposed method that shares successfully advantages of both the manifold method and mesh-free methods is theoretically rational and practically applicable.