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

(1)砂浆有限元分析的迭代方法研究砂浆有限元分析有两种类型的公式可供选择：一种是拉格朗日乘子，另一种是消除从侧边界上的变量。我们提出了几种新的基于不完全Cholesky分解的预处理技术，用于由这两种公式产生的线性方程。所提出的预条件之一是对近似系数矩阵应用不完全Cholesky分解。数值试验表明，从消耗时间的角度出发，将变量消去法与矩阵逼近法结合起来求解方程是有希望的。在初步研究中，我们开发了代数多重网格（AMG）方法，它是有限边单元和节点单元分析产生的稀疏矩阵的有效预处理。针对砂浆有限元分析，本文提出了一种以AMG法为前提的有效迭代求解方法。变量消去公式得到密集系数矩阵，而AMG预调节器有效地应用于稀疏矩阵，它是原始密集矩阵的近似值。数值试验表明，与传统的求解器相比，该求解器具有良好的性能。该方法适用于具有反周期边界条件的问题。(3)基于代数多网格法的快速非线性磁场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Newton-Raphson （NR）迭代中，线性方程组采用AMG预条件共轭梯度（CG）求解器求解。为了提高NR迭代的收敛性，采用了直线搜索方法。数值结果表明，AMG-CG求解器对于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。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