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的迭代方法有两种：一种是采用拉格朗日乘子的迭代方法，另一种是在从侧边界上消去变量的迭代方法。我们提出了几个新的预处理技术，这是基于不完全Cholesky分解，从这两个配方所产生的线性方程。其中一个预条件子将不完全Cholesky分解应用于近似系数矩阵。数值试验表明，从求解方程所需时间的角度来看，变量消去和矩阵逼近相结合是有希望的。（2）砂浆有限元分析的代数多重网格方法在初步研究中，我们发展了代数多重网格（AMG）方法，它是有限边元和节点元分析中产生的稀疏矩阵的有效预处理器。对于砂浆有限元分析，我们使用代数多重网格方法，并将其应用于砂浆有限元分析中。 ...更多信息 提出了一种有效的AMG方法预处理迭代求解器。而变量消除公式导致一个密集的系数矩阵，AMG预处理有效地应用于稀疏矩阵，这是一个原始的密集矩阵的近似。数值试验表明，与传统的求解器相比，所提出的求解器具有优异的性能。所提出的求解器可以应用于反周期边界条件的问题。（3）基于代数多重网格方法的快速非线性磁分析我们提出了一种以磁标位为未知数的快速非线性有限元分析方法。在Newton-Raphson（NR）迭代中，线性方程组由AMG预处理共轭梯度（CG）求解器求解。采用线搜索法提高了NR迭代的收敛性。数值结果表明，AMG-CG求解器是有效的非线性分析的基准模型。此外，通过放宽CG迭代的收敛准则，实现了相当大的速度提高。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