Application Of Complex Variables Meshless Local Petrov-Galerkin Method

During the last three decades, a new class of computational methods, which are called the meshless method, has been developed. The meshless method enables solving PDE’s numerically only based on a set of points without the need of an additional mesh. This method is different from the conventional finite element method which is based on the meshes or elements. The shape functions of the meshless methods are constructed by using the nodes in a moving local domain. These features make the meshless method a hot point in the field of computational mechanics.The meshless local Petrov‐Galerkin method based on the moving least squares approximation is one of the resent meshless approaches. The main advantage of this method over other meshless methods is that this method is a “truly meshless” approach. In other words, no background mesh is required for the interpolation of the solution variables or for the evaluation of the various integrals appearing in the local weak formulation of the problem. But there exists an inconvenience that the trial function constructed by the moving least square method has great computation cost, furthermore, the final equations is sometimes ill‐conditioned. The main objective of this paper is to develop the complex variable meshless local Petrov‐Galerkin method. In this method, the complex variable moving least squares approximation is used to construct the trial function and the Heaviside step function is chosen for representing the test function. Furthermore, the present method is presented to study transient heat conduction problems and elastodynamic problems. The main researches are as follows:The complex variable meshless local Petrov‐Galerkin method is applied to solve the transient heat conduction problems, and the complex variable meshless local Petrov‐Galerkin method for two‐dimensional transient heat conduction problems is presented. And the corresponding formulae are deduced.The complex variable meshless local Petrov‐Galerkin method is applied to the two‐dimensional elastodynamic problems, and the complex variable meshless local Petrov‐Galerkin method for two‐dimensional elastodynamics problems is presented. And the corresponding formulae are obtained in detail.In order to demonstrate the validity of the complex variable meshless local Petrov‐Galerkin method, the corresponding MATLAB codes have been written, and some numerical examples are provided.