Research On Efficient Preconditioning Method For Higher Order Method Of Moments

With the development of science,the traditional analytical solutions are no longer applicable to solve the electromagnetic scattering or electromagnetic radiation problems of complex physical models.In recent decades,numerical methods have made great progress.Therefore,in the field of computational electromagnetics,many numerical algorithms have been developed,such as the finite element method and the finite difference time domain method for solving the differential form of Maxwell's equations;the method of moments for solving the integral form of Maxwell's equations,etc.When using numerical algorithms to solve many problems of electromagnetic calculation,it is usually to convert the solution of complex electromagnetic field problems to the solution of matrix equations.The method of moments is one of the numerical algorithms with higher theoretical accuracy.The low-order moment method uses the RWG basis function to discretely approximate the electromagnetic flow on the surface of the physical model,and the high-order moment method uses the highorder polynomial basis function to discretely approximate the electromagnetic flow on the surface of the physical model.but whether the low-order moment method or the high-order method of moments,a dense matrix equation will eventually be generated.When the direct solution method is used to solve the dense matrix equation,the calculation time complexity is high.And when the iterative solution method is used to solve the dense matrix equation,although the calculation time complexity can be reduced.The matrix equation generated by some problems may encounter the situation that the convergence rate is too slow or even does not converge when solving,because of the condition number of the generated dense matrix maybe very large.Therefore,scholars at domestic and abroad have done a lot of research on the process of adding preprocessing in the process of iterative solution methods.But for the high-order method of moments,when using high-order polynomial basis functions to discretely approximate the electromagnetic flow on the surface of the physical model,the condition number of the final dense matrix is not ideal,and the matrix is too illconditioned.So when using iterative solution solving the matrix equation,it is necessary to construct efficient preconditions,reduce the condition number of the matrix,improve the illconditioned characteristics of the matrix,and speed up the convergence rate of iterative solution.This article first briefly introduces the basic theory of the higher-order moment method and iterative solution method,and then constructs two efficient preconditions for the dense matrix equation generated by the higher-order moment method.And The validity of these two preconditions is verified by corresponding numerical examples.One is the matrix balance method that scales the basis function and the test function,and introduces three construction methods of left and right balance matrices.By adding the left and right balance matrices as bilateral preconditions to the preconditioning process of the iterative solution,speed up the convergence rate of the iterative solution method.And the efficiency of the matrix balancing method is verified by numerical examples.The other preconditions is based on the near-region matrix.Through the studying of the characteristics of the matrix generated by the high-order moment method,an effective method of sparseness of dense matrix is introduced,which extracts the elements that the interaction between the basis function and the test function is strong in the dense matrix as the near-region matrix,and guarantees the sparsity and symmetry of the near-region matrix.On the basis of the near-area matrix,using the SMILU decomposition technology provided in the Super LU library,a SMILU precondition based on the near-area matrix is constructed.Numerical examples show that the SMILU preconditioning based on the near-region matrix can greatly accelerate the convergence rate of iterative solution,which proves the efficiency of the SMILU preconditioning based on the near-region matrix.