Research On Meshless Methods For Computational Electromagnetics

As a new type of numerical methods, meshless methods are independent of the concept of element which traditional element-type methods depends on. This type of methods avoid the onerous mesh generation and adaptive updating, thereby resulting in continuous differentiable approximations that are smooth function and require no post-processing, and therefore, they are paid very much attention by scientists and engineers in various computational researches because of their greatly theoretical and applicable value.Meshless methods, with their basic system, originate from computational mechanics and computational dynamics. The main purpose of this paper is to induces the basic system of meshless methods into computational electromagnetics researches and develop new type of meshless methods which are suitable for the investigated researches themselves. Therefore, two parts of important contents are studied in details in this paper, including researches on the common theoretical problems of meshless methods themselves and the special applicable background of computational electromagnetics, especially for the analysis of electromagnetic field of MicroElectroMechanicalSystem (MEMS) apparatus.With detailed analysis on theoretical problems and their implementation procedures, an important phenomenon on Element-Free Galerkin Method (EFGM) is discovered in this paper. The shape matrix of EFGM becomes ill conditioned, thereby resulting in numerical oscillations, when it uses high order polynomial basis. It is the opposition to common cases that high order polynomial basis results to high accuracy. Mathematical analysis on shape matrix and its inverse gives the cause of numerical oscillation. In order to solve this problem, this paper puts forward the theory of Meshless Method Based on Orthogonal Basis (MLMBOB) and analyzes its qualities on improving conditions of its shape matrix when compared with EFGM. Examples are given to prove all its merits.In this paper, a boundary meshless method (BMLM) for electromagnetic problems is presented. With difference to traditional boundary element method, BMLM combines a point interpolation method for construction of spatial shape functions for the governing equations, thus the spatial shape functions satisfy the Kronecker delta function and the essential boundary condition can be directly imposed on the boundary. Moreover, boundary polynomial point interpolation meshless method (BPPIM) and boundary radial point interpolation meshless method (BRPIM) are presented respectively for transient eddy current analysis.For comparing BPPIM and BRPIM, a representational example is illustrated, and accuracy analyses between them are expounded as well.Radial basis function methods are truly meshless method for approximating the solutions of PDEs. The shape parameter of RBF reflects its resolving power, and then decides the accuracy of approximations. General rules on solving PDEs with high accuracy and choosing proper shape parameters of RBFs are investigated. Taking the advantages of RBF method, a time matching RBF method for analyzing transient electromagnetic problems is presented and, in order to validate the stability and effectivity of the proposed method, implicit scheme and Crank-Nicolson scheme for discretization of time are analyzed for transient eddy current problem. For solving PDEs with complex domains and multiple mediums, sub-domain RBF collocation method and compactly supported RBF method are developed in this paper. They both can get a banding or block banding system matrix with low conditional number and high effectivity. In this paper, RBFs methods are also used for analyzing MEMS force-field coupling problem.Based on the balance ideology, this paper analyses the balance consisted in numerical techniques, including element type methods and meshless methods, and the essence of the balance as a universal law.