Study Of Fundamental Solution-type Meshless Methods For Electromagnetic Scattering Problems

For two-dimensional and three-dimensional electromagnetic scattering problems,traditional numerical solution methods require a mesh dissection of the target,which means that a time-consuming meshing process needs to be undergone,while the complex geometry needs to be simplified.In contrast to traditional methods,meshless methods can replace the mesh with discrete nodes in the geometric description,completely or partially eliminating the mesh and reducing the cost of meshing.In recent years,meshless methods have become a popular numerical method,providing an effective numerical solution for solving various engineering computational problems.The method of fundamental solution(MFS),as a meshless method,has been used to solve electromagnetic problems.However,MFS all have certain limitations and open problems,including its accuracy and efficiency strongly depend on the location of the auxiliary boundary and the distribution of the source points,as well as the pathological state of the system matrix.Also,the existence of controversial auxiliary boundaries outside the physical domain has prevented MFS from becoming a popular numerical method.For this reason,a regularized method of fundamental solution(RMFS)is proposed for electromagnetic scattering problems to avoid the headache of auxiliary boundary selection problem in MFS.The main contents are:(1)Some concepts of fundamental solutions of partial differential equations and the fundamental theory of MFS are described,mainly differential operators,self-adjoint operators,fundamental solutions and the principles of numerical implementation of MFS.The principles of the method of moments are briefly described.(2)An RMFS is proposed for the electromagnetic scattering problem of two-dimensional PEC cylinders under TMz and TEz wave incidence,which is based on the null-field integral equation and the construction of solutions,using the de-singularized subtraction and add-back(SAB)techniques to regularize the singularities of the fundamental solution,and is used to solve the singularities of the fundamental solution when the points are arranged on the physical boundary.Numerical examples are given to verify the correctness,convergence and stability of the proposed method.A method for dealing with fluting targets is also given to improve the accuracy of RMFS.(3)For the electromagnetic scattering problem of a three-dimensional target,it is decomposed into a system of independent Helmholtz equations with mutually coupled boundary conditions.The singularity of the fundamental solution when the points are arranged on the physical boundary is solved using the null-field integral equation,the boundary integral equation and the constructed solution.Numerical examples are given to verify the correctness,convergence and stability of the proposed method,and to further verify the superiority of RMFS over MFS by comparison.The effectiveness of the proposed method is verified by numerical simulations.The results show that RMFS can avoid the problem of auxiliary boundary selection in MFS and has great potential in solving the electromagnetic scattering problem of PEC targets.