The Regularization Method Of Fundamental Solution For The Coating Structures And Inverse Problems

The ill-posed inverse problems are a widespread problem existing in modern engineering and technology.There is wide application background in the future to study the computation method about this problem.Since the unique characterization,such as heat insulation,anticorrosion,insulation,the application of thin body structure and coating structure is becoming more and more common.Hence,there also is important theory meaning and application value to study the effective numerical method about it.However,owing to the special structure of the thin body,the numerical computation is becoming a biggest challenge.The finite element method,finite difference method,boundary elements method are the most commonly used methods for solving the inverse problems and thin-body problems.While both the finite element method and finite difference method belong to the domain calculation methods,which means the whole computed domain need to be subdivided.It is well-known that complete the domain partition is not easy,especially,for the complex geometric structure.Furthermore,the implementation of boundary element method involves singular and nearly singular integrals and the calculation is tedious and time-consuming.The method of fundamental solution(MFS)is a simple,highly accurate meshless method,without dividing boundary element and computing integration,so it has potential advantages for development in the future.However,the efficiency of the MFS lies in the distribution of the source point,especially,the selection of distance between the fictitious boundary and the real boundary.If the distance is too small,the singularity can't be avoided caused by the coinciding of the source points and the collocation point.In contrast,if the distance is too large,high condition number of the linear systems will occur.To solve this problem,the regularization algorithm is introduced to the MFS which developed the regularization MFS.A large number of numerical examples show that the regularization MFS can effectively solve the ill posed system caused by inverse problem and thin body.This paper applies the mentioned regularization MFS method to lots of practical problems.Firstly,we will study the regularization MFS for the 2D boundary conditions identification potential and anisotropic potential problems.The equation system generated by the MFS is solved by the regularization method.From the results,it is easy to note that satisfactory results can be achieved by using TSVD method and Tikhonov regular method to solve linear systems with a large number of conditions and using the L-curve and GCV method to determine the regularization parameters.Then,the regularization MFS is applied to other problems such as potential thin body,potential anisotropic thin body,coating structure.Generally,since the thin body structure is in the micrometer or nanometer level,the numerical analysis of the physical parameters for thin structure has been a big challenge.And In many cases,some boundary quantities are unknown which increases the difficulty of the computation.To the authors' knowledge,the study on such problems rarely has been involved.However,in this paper we study the problem and present an effective numerical method which can accurately calculate the thinner coating problems and their inverse problems.Results indicate that the proposed method is simple,accurate and stable.In a word,our work broaden the scope of application of the MFS and provide a new and effective numerical method for the study of a variety of inverse problems of mathematical physics including the inversion problems of thin body and coating structure.