The Regularization Method Of Fundamental Solution For The Potential And Elasticity Cauchy Inverse Problems

Mathematical physics inverse problem is widespread problem in engineering and computational mathematics.And Cauchy inverse problem is a classical inverse problem.Anisotropic thin body structure has the characteristics that heat transfer coefficient is changing with directions,and the size of thickness is small.With the development of science and technology,this kind of new materials in the application of practical engineering will be more and more widely.Therefore it has the important theory significance and the application background to study effective numerical methods for this problem.Finite Element Method(FEM),Finite Difference Method(FDM)and Boundary Element Method(BEM)are commonly used numerical methods.The Method of Fundamental Solution(MFS)is a simple,highly accurate mesh-less method,without dividing its boundary and the region.For its excellent characteristics of high computing accuracy,fast convergence,simple programming,the MFS become a method which has great potential advantages.However,the MFS is inevitably involved to processing the ill-conditioned linear system of equations discrete from the inverse problem similarly.In addition,the effectiveness of the MFS is influenced by the position of the source point to a certain extent,especially the distance between the virtual boundary and true boundary.This paper present the regularization MFS.Examples of 3D conventional structure and 2D anisotropy thin body structure Cauchy inverse problem show that the regularization MFS can effectively solve the ill-conditioned linear system and extend the scope of "distance".In this work,we study the regularization MFS for 2D elasticity Cauchy inverse problems in chapter three.Then the 3D potential and elasticity Cauchy inverse problems are investigated in chapter four and five.The ill-conditioned linear system of equations discreted by the MFS is solved by TSVD method and the Tikhonov regular method.And the parameters used in TSVD and Tikhonov are determined by the L-curve and GCV method,good effect has been achieved.In chapter six we present an improved method of fundamental solutions and apply it to 2D anisotropy thin body structure Cauchy inverse problem.Numerical results indicate that the proposed method is accurate and stable.Effective results can be achieved even a part of boundary information are unknown,influence by noise or the thickness of the thin body structure down to 1E-09.In a word,our work expands the scope of application of the MFS and provides a new and effective numerical method for the study of kinds of inverse problems.