Fast Electromagnetic Mortar Finite Element Analyses with High Accuracy Based on Algebraic Multigrid Methods

(1)砂浆有限元分析迭代方法研究在砂浆有限元分析中，有两种格式可供选择：一种为拉格朗日乘子格式，另一种为消元格式。对于这两种形式的线性方程，我们提出了几种新的基于不完全Cholesky分解的预条件技术。其中一个提出的预条件将不完全Cholesky分解应用于近似系数矩阵。(2)基于代数多重网格法的砂浆有限元分析在一个初步的研究中，我们发展了代数多重网格法，它是由有限边单元和节点单元分析产生的稀疏矩阵的有效预条件。对于砂浆有限元分析，w…在此基础上提出了一种基于AMG方法的有效迭代求解器。由于变量消元公式导致了稠密的系数矩阵，而AMG预条件算子被有效地应用于稀疏矩阵，该稀疏矩阵是原始稠密矩阵的近似。数值试验表明，与传统的求解器相比，所提出的求解器具有良好的性能。基于代数多重网格法的快速非线性有限元分析我们提出了一种以磁场标量势为未知量的快速非线性有限元分析方法。在牛顿-拉夫森(NR)迭代中，线性方程组由AMG预条件共轭梯度(CG)求解器求解。采用线搜索法提高了NR迭代的收敛速度。数值结果表明，AMG-CG求解器对一个基准模型的非线性分析是有效的。此外，通过放宽CG迭代的收敛准则，获得了相当大的加速。较少

(1)Research on iterative methods for mortar finite element analysesTwo type formulations are selectable in mortar finite element analyses: one with Lagrange multipliers and the other with elimination of variables on the slave-side boundaries. We propose several new preconditioning techniques, which are based on the incomplete Cholesky decomposition, for linear equations arising from the both formulations. One of the proposed preconditioners applies the incomplete Cholesky decomposition to an approximate coefficient matrix. Numerical tests demonstrate that a combination of the variable elimination and approximation of the matrix is promising from a viewpoint of the elapsed time to solve the equations.(2)Mortar finite element analyses using algebraic multigrid methods In a preliminary study, we develop algebraic multigrid (AMG) methods, which are efficient preconditioners for sparse matrices arising from the finite edge-and nodal element analyses.For the mortar finite element analyses, w … More e propose an effective iterative solver preconditioned by the AMG method. Whereas the variable elimination formulation leads to a dense coefficient matrix, the AMG preconditioner is efficiently applied to a sparse matrix, which is an approximation of the original dense matrix. Numerical tests demonstrate an excellent performance of the proposed solver, compared with conventional solvers. The proposed solver can be applied to problems with anti-periodic boundary conditions.(3)Fast nonlinear magnetic analyses using algebraic multigrid methodsWe present a fast nonlinear finite element analysis with the magnetic scalar potential as unknowns. In the Newton-Raphson (NR) iterations, the linear equations are solved by an AMG preconditioned conjugate gradient (CG) solver. The line-search method is applied to improve the convergence of NR iterations. Numerical results show that the AMG-CG solver is efficient for the nonlinear analysis of a benchmark model. Moreover, a considerable speed-up is achieved by relaxing the convergence criterion of the CG iterations. Less