A meshless method using local exponential basis functions with weak continuity up to a desired order

Abstract

This paper introduces a novel meshless method based on the local use of exponential basis functions (EBFs). The EBFs are found so that they satisfy the governing equations within a series of subdomains. The compatibility between the subdomains is weakly satisfied through the minimization of a suitable norm written for the residuals of the continuity conditions. The residual norm may contain any desirable order of continuity. This allows increasing the continuity of the solution without increasing the type of point-wise variables at each node. The solution procedure begins with the discretization of the solution domain into a set of nodal points and cloud construction on each nodal point. The approximate solution in the local coordinates of each cloud is constructed by a series of EBFs. A set of intermediate points are distributed throughout the domain and its boundary to apply the continuity between the local solutions of the adjacent clouds up to a desired order, and also to impose the boundary conditions. The main nodes may play the role of the intermediate points as well. The validity of the results is shown through some patch tests. Also some numerical examples are solved to illustrate the capabilities of the method. High convergence rate of the numerical results is one of the salient features of the proposed meshless method.