Research Of Fast Direct Solver Based On Higher Order Hierarchical Vector Basis Functions

The fast direct method to the integral equation can explicitly construct the inverse of the system matrix.Compared with iterative solvers,direct solvers don't suffer from convergence problem.At the same time,the direct solution method only needs to perform one matrix-vector multiplication of the inverse matrix and different excitation vectors in a multiple right-hand sides problem,so it has high computational efficiency.Despite the high efficiency of the fast direct solver,it is still very challenging to solve electrically large problems,since the number of unknowns of the system are prohibitively large when the popular Rao-Wilton-Glisson(RWG)basis functions are used for discretization.To mitigate the computational burden,this thesis studies a fast and direct solution method based on higher order hierarchical vector(HOHV)basis functions.Due to the higher order modeling,the number of unknowns is significantly reduced,and it is possible to solve electrically large targets with much less computational cost.Firstly,this thesis studies a hierarchical matrix LU fast direct solver based on higher order approachs(H-LU-HO)for the fast analysis of electromagnetic scattering problems.In higher order modeling,the surface of the object is discretized with HO curved elements and the HOHV basis functions based on curved surfaces are employed,greatly reducing the number of unknowns.Meanwhile,an efficient assembly scheme based on patch coupling is developed to accelerate the low rank matrices decompossion in H-LU-HO.Numerical examples show that the computation time and memory cost is reduced significantly and the solution efficiency is improved by the H-LU-HO direct solver.Then,this thesis proposes a modified hierarchically off-diagonal low-rank matrices fast direct solver based on HOHV basis functions(M-HODLR-HO).The M-HODLR direct solver uses modified compression method to compress off-diagonal matrix blocks,and combines the adaptive SVD strategy to improve the solution efficiency and solution accuracy.In addition,by combining higher order approachs,M-HODLR-HO greatly reduces the number of patch in geometric modeling and number of unknowns,as well as accelerates the solution process of the system matrix equation.In a numerical example,M-HODLR-HO is applied in calculating the monostatic RCS of an electrically large and complex plane with more than 100 wavelengths,which shows the ability of the proposed method in large-and multi-scale simulations.Finally,the M-HODLR fast direct solver for solving the dielectric integral equation PMCHWT(M-HODLR-PMCHWT)is proposed in this thesis.The M-HODLRPMCHWT fast direct solver processes the equivalent current and magnetic current at the same time when constructing the M-HODLR matrix,so the form of the constructed system matrix is simpler.Consequantly,it is easier to implement a fast direct solver,and hence the inverse matrix can be quickly calculated.Numerical example results demonstrate the effectiveness of the method.