Parallel Solution Of Large Finite Elemente Quations In Electromagnetic Problems

Finite element method(FEM)is one of the widely used computational electromagnetics method,its core idea is "numerical approximation" and "discretization".It discretizes the continuous problem and uses the assumed approximate function in each element to piecewise express the unknown field function to be solved in the solution domain,thus generating a finite element matrix.FEM can describe well about the problems of complex structure and heterogeneous medium which are widely used in solving continuity problems such as heat transfer,fluid mechanics,electromagnetic field.The FEM is used to analyze the electromagnetic problem,which can be concluded as solving partial finite element equations.Therefore,the fast solution of the equations becomes one of the key techniques of finite element calculation.At present,the complexity of the electromagnetic problem and the computing scale are increasing.So how to reduce storage demand and improve computational efficiency is of great theoretical and practical significance.First of all,for solving electromagnetic problems,we divide the solving areas into open area and closed area.The main differences between the former and the latter are as follows: the former need the virtual boundary to truncate with infinite analysis area and is changed into the finite solution area for analysis.In this paper,the Perfectly Matched Layer(PML)is used as the boundary condition to eliminate boundary reflection and then its basic principle is introduced to us.On this basis,we got the general finite element calculation formula,and then the large-scale linear equations are obtained along with the solving rate on GPU is optimized.Secondly,in the analysis of 3d electromagnetic problems,take a vectorial tetrahedron element in finite element analysis of vectorial field to approximately express functions of every cell.As a result,it solved the problems well when using interpolation node base unit to replace the vector electromagnetic field.The problems may just like a pseudo solution,not convenient to impose boundary conditions on the conductor surface and difficult to deal with conductor and dielectric edge conditions.And then,we study how to solve large-scale sparse linear equations based on GPU.On the base of storage format HYB which is widely used in lots of areas now,we replace the COO with CSR and combine it with ELL format,thus we put forward the HEC(Hybrid ELL and CSR)storage format,which is apply to the dimension of the coefficient matrix is not big and nonzero structure is simple.In the end,we proposed a storage format HMEC(Hybrid Multiple ELL and CSR)of sparse matrix to solve large sparse linear equations on GPU.First,we optimized the storage structure of the coefficient matrix by reordering and improved the irregularity of non-zero element distribution in sparse matrix.Secondly,we stored the coefficient matrix in a certain scale block.Then we adopt an approach by combining ELL and CSR storage format to adapt to different characteristics of blocks.At last,we took Bi-Conjugate Gradient Stabilized(BICGSTAB)and Conjugate Gradient(CG)iterative methods to solve large sparse linear systems,they are respectively preconditioned by incomplete-LU and incomplete-Cholesky factorization for asymmetric and symmetric positive definite linear matrices.The former is applicable to the condition number of coefficient matrix is much larger than 1 and the structure is more complex.For the partially sparse large-scale linear system,it has the advantages of fast speed,high precision and good stability.The latter makes full use of the diagonal advantage of matrix in the decomposition process and greatly improves the solution rate of large-scale sparse linear equations.