Research On Fast Algorithms For High-precision Finite Element Method In Fluid And Electromagnetic Fields

In recent decades,with the rapid development of computer technology,numerical simulation methods have been widely applied in various scientific fields,industrial design and manufacturing related numerical simulation software.Especially the finite element method and its related extended numerical methods have achieved considerable results in the application of fluid mechanics,electromagnetics and other fields.With the increasing demand for computational accuracy in engineering applications,many researchers have conducted research on high-precision finite element methods and have obtained significant achievements.However,the improvement in accuracy is often accompanied by a significant increase in the unknowns of the solving elements inevitably reducing the computational efficiency of the finite element method.Besides,when solving large matrix systems,these numerical simulations require a lot of computation and data processing,resulting in significant computational costs and memory consumption,greatly weakening computational performance.Therefore,studying efficient and fast algorithms to improve computational efficiency is of great significance for the application of high-precision finite element methods in practical engineering and related numerical simulation software.This dissertation focuses on high-precision finite element numerical simulation techniques in subsonic fluid calculation and electromagnetic scattering problems,and studies the improvement of computational efficiency of finite element method to reduce the computational time of large nonlinear/linear systems and significantly save computational resources.The dissertation proposes the POD-DG method based on proper orthogonal decomposition(POD)model reduction,DEIM-DG method based on discrete empirical interpolation method(DEIM)and dual P-type multigrid method for the discontinuous Galerkin(DG)finite element method in fluids,and extends the P-type multigrid preprocessing technique to higher-order finite element methods in electromagnetic calculations.Finally,a high-efficiency subspace iterative algorithm combining low rank approximation and GCRO-DR(GCRO with deflated restarting)cyclic subspace is proposed for the problem of multiple right hands sides in electromagnetic scattering.The main research work can be summarized as follows:1.A POD-DG method based on proper orthogonal decomposition(POD)model reduction method is proposed for large-scale nonlinear systems generated by the DG finite element method to solve the subsonic problem of Euler equations in fluids.This method aims to effectively improve the computational efficiency of the DG method by reducing the dimensionality of the computing system.Firstly select the instantaneous solution of the DG discretization and construct the instantaneous snapshot matrix to search for the optimal low dimensional approximation of the global system.Then the feature system is obtained through singular value decomposition(SVD)analysis,and based on this,a POD basis vector space is established.Finally,the Galerkin solution of the flow field is projected onto a low dimensional space composed of the POD basis vector space to obtain the POD-DG format for efficient solution.The numerical experimental results show that with the POD-DG method,the computational efficiency has been improved after POD model reduction,saving computational time and cost,while still maintaining high computational accuracy.2.A DEIM-DG method is proposed based on the POD model reduction to address the computational complexity caused by nonlinear terms in Euler equations.This method uses the discrete empirical interpolation method(DEIM)to reduce the dimensionality of nonlinear terms,further optimizing the computational efficiency of the DG algorithm and achieving effective model reduction of the POD algorithm.In the implementation process of the method,the idea of POD is first used to select appropriate time flow solutions and nonlinear residual in the full-order system,and construct instantaneous snapshot matrices to obtain POD orthogonal basis vectors in low dimensional space.Then,by using the POD basis vectors of nonlinear terms for interpolation and index,the optimal number of interpolation points and the Boolean matrix containing the position information of the interpolation points are determined.Finally,only a few selected nonlinear terms are iteratively calculated on the spatial grid points,while the information of the remaining points is estimated through interpolation.On the basis of the POD-DG method,the calculation format of DEIM-DG was constructed by combining the DEIM method,achieving corresponding complete model reduction.The numerical results show that the improved DEIM-DG format has significantly improved computational efficiency,especially in the case of mesh refinement,where the improvement in computational efficiency is particularly prominent.3.The dual P-type multigrid method is proposed for solving the large sparse linear system formed by implicit solution of the DG algorithm for solving Euler equations in fluid.The P-type multigrid method is used as a preprocessing technique and an iterative technique respectively to accelerate convergence,achieving double accelerated convergence of the solution system from two aspects: matrix preprocessing and time iteration.Specifically,this method first applies P-type multigrid preprocessing technology to transform the originally complex large pathological sparse matrix into a relatively small block matrix and perform inverse processing.Furthermore,the idea of the P-type multigrid method to perform cyclic iterations on approximate solutions of different orders is employed to form a P-type multigrid accelerated convergence iterative technique for time iteration.Finally,the P-type multigrid preprocessing technique in the dual P-type multigrid method is extended to the high-order finite element method for solving three-dimensional electromagnetic problems,and combined with multifrontal block ILU preconditioner to further improve the convergence speed of the iteration.The numerical results show that the dual P-type multigrid method and P-type multigrid preprocessing have achieved significant results in fluid and electromagnetic fields respectively,and the computational efficiency of the linear system after preprocessing has been greatly improved.4.A high-efficiency subspace iteration method combining singular value decomposition low rank approximation and GCRO-DR(GCRO with deflated restarting)cyclic Krylov subspace iteration method is proposed to address the efficiency issue of solving multiple right hand sides in electromagnetic scattering.Firstly,by using low rank approximation,the discretized large sparse linear matrix is decomposed into low rank matrices to reduce the number of coefficient vectors and only retain linearly independent coefficient vectors.Subsequently,using the idea of "recycling" in the GCRO-DR algorithm,the low rank linear equation is iteratively iterated,and the subspace information generated by the previous cycle(such as residual vector information)is retained to preprocess the next cycle,thereby achieving iterative convergence acceleration.The numerical experimental results have verified the fast convergence characteristics of the method and demonstrated its effectiveness in improving the efficiency of solving multiple right hand sides electromagnetic scattering problems.