Adaptive Mesh Methods Based On Natural Boundary Reduction For Concave Exterior Problems

Based on the principle of natural boundary reduction,taking anisotropic and quasilinear exterior problems as examples,this thesis studies the feasibility of combining adaptive mesh method with natural boundary element method and coupling method in the elliptic concave angle region.The content is divided into the following four chapters:The first chapter introduces the background and significance of the topic,the summary of research status,and some definitions and theorems used in the thesis.In the second chapter,for the first time,the adaptive mesh method is applied to solve the anisotropic exterior problem on the boundary of elliptical arc.Firstly,the natural boundary element method of anisotropic problem on adaptive mesh is studied.By introducing elliptic coordinates,the equivalent variational problem is derived,and then the boundary is adaptively divided to obtain discrete variational problem.Then the error and convergence of the natural boundary solution are analyzed.Secondly,the coupling method of anisotropic problem on adaptive grid is studied.The elliptic arc artificial boundary is introduced into the unbounded region,and the region is divided into a bounded region and an unbounded region.After obtaining the equivalent variational problem,the adaptive mesh is divided in the bounded region to obtain the discrete variational problem.Then the error and convergence of the coupling solution are analyzed.Finally,the feasibility and effectiveness of the method are verified by several numerical examples.In the third chapter,for the first time,the adaptive mesh method is applied to solve the quasilinear exterior problem on the boundary of elliptical arc.Firstly,the natural boundary element method of quasilinear problem on adaptive grid is studied.By introducing Kirichhoff transformation,the equivalent variational problem is deduced,the boundary is adaptively divided,and the discrete variational problem is obtained.Then the error and convergence of the natural boundary solution are analyzed.Secondly,the coupling method of quasilinear problem on adaptive grid is studied.The elliptic arc artificial boundary is also introduced into the unbounded region,and the region is divided into a bounded region and a unbounded region.After obtaining the equivalent variational problem,the adaptive mesh is divided in the bounded region to obtain the discrete variational problem,and the error and convergence of the coupling solution are analyzed.Finally,the feasibility and effectiveness of the method are verified by several numerical examples.The fourth chapter summarizes the conclusion of the thesis and puts forward some problems worthy of further study.