Applications Of Localized Method Of Fundamental Solution For Inhomogeneous Problems Based On Recursive Composite Multiple Reciprocity Method

Numerical simulation methods are developing very fast in recent decades and provide important access to mathematical and physical problems.In many engineering problems,some mesh based numerical methods such as the Finite Element Method(FEM)and Boundary Element Method(BEM)are very common in various fields.At the same time,in order to deal with problems which can hardly be described by mesh,the meshless methods are proposed.Among them,the fundamental solutions of method(MFS)is a typical boundary type meshless method,which has the advantages of no need of interior nodes,simple calculation process,high accuracy of numerical results,and simple programming.However,there are various shortcomings in the MFS.First,the discretized matrix is a full matrix,which requires a large amount of computation and is not easily be applied to large-scale engineering problems;Second,due to the fundamental solution,inhomogeneous terms of the governing equations cannot be directly solved,and other auxiliary techniques are needed to deal with the inhomogeneous partial differential equations.In order to address the above shortcomings,this article adopts the recursive composite multiple reciprocity technique(RC-MRM),combined with the LMFS to solve inhomogeneous problems.Among them,the LMFS incorporates the characteristics of both the domain type method and the boundary type method,which can not only obtain high-precision calculation results,but also improve computational efficiency and stability during the localization process.The recursive composite multiple reciprocity method has the computational advantage of no need of seeking specific solutions.This article combines the theories of the two existing algorithms and applies the combined algorithm to various inhomogeneous problems for the first time.The main work of this article can be divided into three parts:(1)theoretical introduction of the proposed the LMFS based on RC-MRM;(2)Analysis and verification of the proposed method;(3)Improvement of the proposed methods.Firstly,in Chapter 2,this article mainly elaborates on how to combine the two existing methods,RC-MRM and LMFS in theory,and extends the theory to inhomogeneous problems that were previously difficult to solve directly by using the RC-MRM through Taylor expansion;Secondly,in terms of analysis and validation of the proposed method,this paper verifies the computational accuracy and efficiency of the method in Chapters 3 and 6for several typical inhomogeneous problems and Cauchy problems.At the same time,in order to discuss the numerical characteristics of the LMFS based on RC-MRM,a comparison is made with several other typical mesh based and meshless methods,and suggestions for selecting algorithm parameters in this paper are given;Finally,in terms of algorithm improvement,theoretical improvements and extensions are made to the RC-MRM.For the Winkler plate bending problem,the computational nodes selection mode has been changed,making the solution results that could have been solved by the improved method in the past more stable.Moreover,the special boundary Winkler plate bending problem that could not have been solved by this method in the past can be solved by the improved method in this paper.For inhomogeneous linear elasticity problems,the Hormander operator decomposition technique is used to apply the RC-MRM for the first time to governing equation group,expanding the theoretical content of this method in solving inhomogeneous governing equation group.At the same time,it enables the solution of inhomogeneous linear elastic problems that have not yet been solved by the RC-MRM to be solved in combination with LMFS.The above work not only provides theoretical expansion and numerical verification of the LMFS based on RC-MRM,but also provides new ideas for solving different inhomogeneous problems.