Study Of Domain Decomposition Methods Based On Surface Integral Equations

Electromagnetic(EM)fields are invisible and intangible,but they exist everywhere in our lives.Whether in the military or civilian fields,the EM environments are becoming more and more complex.The EM problems are becoming more and more meticulous and larger.Among so many EM numerical methods,the surface integral equation method has been widely concerned by scholars in the field of computational electromagnetics,due to its high theoretical accuracy and few discrete elements.However,in the face of the ever-increasing EM simulation needs,even the fast algorithm of the integral equation method is difficult to solve the electrically super-large problems and system-level problems in the real-world EM environment within the limited computing resources.For example,system-level EM simulation problems such as the large airborne antenna arrays and electrically super-large problems of the analysis of stealth characteristics of ships.In order to solve large and complex EM simulation problems within acceptable research time,with existing limited computing resources,without loss of accuracy,the surface integral equation based domain decomposition method(SIE-DDM)combined with the parallel computing strategy and the out-of-core computing strategy is studied in this dissertation.The method can be used to efficiently and accurately solve the EM simulation problems of the scattering characteristics analysis for low radar cross section target,ship stealth characteristics analysis and the disturbance analysis for airborne large antenna arrays by rotor modulation effect.The main contributions of this dissertation can be summarized as follows.1.The singularity in the integral kernel when computing the EM field(integral singularity)on the surface or the EM field near the surface(numerical integral singularity)of PEC and medium objects using the method of moments is studied in depth.Discussed the singularity of the integral when using the Green function method to solve the EM field.According to the singular value expansion method,the analytical expression of the surface EM field integral formula is derived and is extended to the integral formula of near-surface EM field.This provides the basis for the realization of accurate computations for the three SIE-DDM proposed later in this dissertation,which use the mutual coupling EM field instead of the mutual coupling impedance to synthesize the interaction effects between subdomains.2.The non-overlapping and non-conformal domain decomposition method(NNDDM)for multiscale PEC targets is studied in detail.The NNDDM can independently mesh each subdomain according to the electrical size structure characteristics of the model.A direct solver based on LU decomposition is used inside the subdomains to ensure the accuracy of the subdomains' solution.The outside of the subdomains(the entire domain decomposition system)adopt a steady iterative solver,which makes it possible to quickly obtain the solution of the entire system problem through only a few simple iterations,and speeds up the computation.When calculating the mutual coupling between subdomains,the mutual coupling EM field excitation source from the other subdomains is superimposed on the original excitation source of plane wave for each subdomain.In this way,the continuity of the tangential field and the normal field constrained by the second-order transmission condition is implicitly realized.Among them,the surface current of the self-subdomain is used to compute the mutual coupling excitation EM field on the artificial virtual interface,and the surface current of the other subdomains is used to compute the mutual coupling excitation EM field on other non-interfaces.The advantage of this method is that no additional constraints need to be added,which simplifies the filling of the system matrix,optimizes the complexity of program implementation,and reduces memory consumption and computational complexity.3.The matrix-partitioned domain decomposition method(MP-DDM)for large-scale PEC and dielectric targets is studied in depth.Compared with the NNDDM,the meshing strategy of MP-DDM is more convenient,eliminating the tedious manual model preprocessing process,and the existing mesh partitioning algorithm(such as METIS software package,etc.)can be used for adaptive domain partitioning.A direct solver based on LU decomposition is used inside the subdomains to ensure the accuracy of the subdomains' solution.Outside the subdomain(in the entire domain decomposition system),an iterative solver based on the Krylov subspace and a left-hand preconditioning strategy are adopted to ensure the stability and rapid convergence of the entire system.The outside unsteady iterative solver has high universality for solving various EM models.When calculating the mutual coupling between subdomains,a singular integral processing strategy at the boundary lines of the subdomains is proposed.That is,the 1/4 impedance elements are used between adjacent subdomains to participate in the mutual coupling calculation to improve the accuracy of the solution.The advantage of this method is that it can simplify modeling complexity,reduce memory consumption,and improve solution efficiency while ensuring the accuracy of results.4.The higher order basis function based domain decomposition method(HOB-DDM)for objects with variable parts is studied in detail.The HOB-DDM can partition the electrically large object model roughly into subdomains according to structurally variable and structurally invariant,and partition variable parts into independent subdomains.A direct solver based on LU decomposition is used inside the subdomains.Store the decomposed submatrices out of core,and use the decomposed submatrices repeatedly in the subsequent outside iteration process to speed up the solution of the matrix equation of the entire domain decomposition system.The outside of the subdomains(the entire domain decomposition system)adopt a steady iterative solver,so that the solution of the entire system problem can be quickly obtained by only a few simple iterations.The advantage of this method is that for complex targets containing variable parts,only the subdomains corresponding to the variable parts(usually small subdomains)need to be filled and decomposed repeatedly during the design process.For the subdomains corresponding to the invariant parts(usually larger main parts),the submatrices only can be filled and decomposed once,and be reused in the outer iteration.In the outside iteration process,the subdomains corresponding to the variable model parts is mutually coupled with the other unchanging subdomains,which can significantly reduce the computing time and speed up the design cycle.5.For the three SIE-DDMs proposed in this dissertation,the parallel strategy corresponding to each method is carefully studied.For the NNDDM aimed at solving multiscale problems and the HOB-DDM aimed at solving objects with variable parts,since they all partition subdomains according to the structural characteristics of the target to be solved,it is difficult to ensure the balance of the subdomains' scale.Therefore,a parallel strategy of "parallel in subdomains and serial among subdomains" is designed to ensure load balance among processes,and improve the computational efficiency of parallel SIE-DDM programs.For the MP-DDM aimed at solving large-scale problems,since its subdomain partitioning strategy is not limited to the structural characteristics of the models,it can achieve as balanced subdomain partitioning as possible.Therefore,in addition to the parallel strategy of "parallel in subdomains and serial among subdomains",a parallel strategy of "parallel among subdomains and serial in subdomains" is also designed.This parallel strategy is more in line with the natural parallel state of the domain decomposition method itself,and is more conducive to the scalability of parallel programs.By using the three proposed method,an electrically large target can be analyzed in a common workstation,which provides a great convenience for the users.If the program is transplanted to a high-performance computing platform,the scale of the problems that can be solved will double.