| The meshless method is a tool for numerical analysis,developed on the basis of the grid method.It avoids mesh division,has simple calculation principle,the governing equation directly acts on discrete points,has fast convergence speed and relatively high precision.This method can be applied to the problems that are difficult to solve due to mesh distortion.In this paper,we review the current research status of Helmholtz equation on the characteristic frequency problems of waveguide structure and thin film vibration,discuss the feasibility of the Localized method of Fundamental Solutions(LMFS)in the theoretical calculation of electromagnetic waveguide structure transmission,and analyze the propagation performance of waveguide structures in different cross-sectional forms.At the same time,different degenerative boundary conditions are applied to the thin film structure,and LMFS is used to study the characteristic frequency problem of free vibration of the film.The development and classification of meshless methods are reviewed,and the computational characteristics of such methods are introduced in this paper.LMFS is a regional grid-free numerical analysis method that does not rely on grid generation and has no complex numerical integration process.It belongs to the strong format meshless method.Not only does it retain the high precision of the method of fundamental solutions(MFS),but the coefficient matrix generated by interpolation is sparse,which greatly improves the computational efficiency and avoids the ill-conditioned matrix phenomenon.One of the innovations of this paper is that the technique of Method of External Source(MES)is combined with the LMFS numerical analysis Method to numerically simulate the waveguide structures in simply and multi-connected domains.The second innovation is to use the method of regional point selection to solve the problem of crossing degenerate boundary when selecting local points,which can effectively avoid the generation of degenerate boundary singularity,and study the natural vibration characteristics of thin films with different numbers of degenerate boundary.The program developed in this paper is of great significance to the meshless method for solving the characteristic frequencies of electromagnetic waveguide structures and thin film structures.The main work and results are as follows:(1)This paper reviews the development of electromagnetic waveguide structure and the research status of its characteristic frequency,analyzes the transmission characteristics of the waveguide structure,and looks for an effective numerical simulation method to solve the eigenfrequency.The paper summarizes the development and research status of the free vibration theory of thin film structure,and explores the characteristic frequency problem of thin film vibration with degenerative boundary.(2)The Helmholtz equation is introduced,the basic principle and derivation of LMFS and MES are described in detail,and the method is applied to the characteristic frequency simulation of electromagnetic waveguide and thin film vibration,the regional selection method is used to select points in local domain.Comparing the results of numerical case calculation with those of previous studies,it is found that LMFS has high accuracy and good efficiency,thus verifying the feasibility of the calculation method.(3)By analyzing the numerical results,it can be found that there is a unique characteristic mode corresponding to the characteristic frequency of the solution.The mode at low order frequency is simple,while the mode at high order frequency is relatively complex.In this paper,a program for solving the characteristic frequency problem under the Helmholtz equation is developed based on MATLAB,and the developed program is summarized and shared,which provides a certain research basis for the following work. |
