Research And Implementation Of Multilevel Green's Function Interpolation Algorithm For Electromagnetic Problems Of Low And Intermediate Frequencies

It is well known that the use of computational electromagnetic methods for the fast solution of the radiation and scattering problems of targets in complex electromagnetic environments has always been a hot topic in the field of electromagnetic fields.Whether in the actual engineering applications,or in the field of navigation industry,computational electromagnetic methods are of concern,and thus more and more researchers are engaged in this area.However,with the comprehensive investigation of this field,it is easy to know that electromgentic problems of high and intermediate frequencies have been widely solved.By comparsion,solutions of low-frequency electromgentic problems are realtively rare.Hence it is worthwhile studing this area in depth.Method of Moments(MoM)has been regarded as an effective numerical method for solving electromagnetic scattering and radiation.After several decades,many techniques have been designed to use limited computational resources to solve electrically large problems.Various fast algorithms have been developed to improve the efficiency and accuracy of the computational methods,including multilevel fast multipole algorithm(MLFMA),adaptive integration method(AIM),sparse matrix standard grid method(SMCG),pre-corrected fast Fourier transform method(PFFT).These methods can greatly reduce the computational complexities.In this thesis,a newly developed kernel-independent fast algorithm,e.g.,multilevel Green's function interpolation method(MLGFIM)is employed.Based on the MoM,the MLGFIM has been studied in the thesis.A detailed study of the interpolation scheme of the Green's function has been conducted.By comparing the different interpolated points and interpolated functions,the optimal interpolation scheme for low-frequency problems has been proposed.And then this interpolation method is introduced into the MLGFIM to solve the low-frequency electromagnetic problems.In the low-frequency problems,the eigenvalues of the electric field integral equation(EFIE)mainly distribute at zero and infinity,and thus the "break-down" phenomenon occurs,which results in the huge conditioning number of the impedance matrix in the MoM.Hence the accuracy of the algorithm greatly degrades,and even the wrong results are obtained.In order to overcome this phenomenon,a lot of researches have been carried out.The loop-tree and loop-star basis function decompositions have been introduced to improve the problem.But how to efficiently obtain these basis functions has still been a problem to be solved.Preconditioning techniques have been widely used because they can improve conditioning number of the impedance matrix and hence speed up the convergence of iterative algorithms.The intrinsic reason for the "break-dowm" phenomenon at the low frequency is that the spectrum of the EFIE operator accumulates at zero and infinity.Hence the Calderon identity and the Calderon preconditioner is introduced into the EFIE.With the use of the the Calderon preconditioner,the operator with the first kind can be converted into the operator with the second kind,and thus the "break-down" phenomenon can be avoided.In addition,for the solution of linear equations,two kinds of soution methods are generally used.The first one is the direct method,for example LU decomposition method,Gaussian elimination method.Theses methods are employed in the case of the problems with a small number of the unknowns.When the unknowns are large,the iterative method is preferred.Various iterative solvers have been applied to solve the electromangetic problems.In this thesis,several kinds of iterative solvers are compared according to their accuracy and convergent rates.Finally,we use the MLGFIM to speed up the solution of the electromagnetic problem.The Calderon preconditioning technique is used to overcome the "break-down" phenomenon of the EFIE operator at low frequencies.With the appropriate iterative solvers,some low-frequency problems are solved.The numerical examples are given to demonstrate the correctness and efficiency of the proposed algorithm.