The radial point interpolation meshfree discretization is a very efficient numerical framework for the analysis of piezoelectricity, in which the fundamental electrostatic equations governing piezoelectric media are solved without mesh generation. Due to the mechanical-electrical coupling property and the piezoelectric constant, the discrete linear system is sparse, of generalized saddle point form and often very ill conditioned. In this work, we propose a technique for constructing a family of cell-by-cell approximate Schur complement matrices, to be used in preconditioning to accelerate the convergence of Krylov subspace iteration methods for such problems. The approximate Schur complement matrices are simply and cheaply constructed in the process of the meshfree discretization and have a sparse structure. It is proved that the so-constructed approximate Schur complement matrices are spectrally equivalent to the exact Schur complement matrix, which leads to very fast convergence when used in preconditioning. In addition, nondimensionalization of the piezoelectric equations is considered to make the computations more stable. The robustness and the efficiency of the proposed preconditioners is illustrated numerically on two test problems, arising from a piezoelectric strip shear deformation problem and a piezoelectric strip bending problem. Numerical results show that the number of iterations to achieve a given tolerance is independent of the number of degrees of freedom as well as of the various problem parameters.

中文翻译：

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无网格离散压电方程预处理中的逐单元近似 Schur 补充技术

径向点插值无网格离散化是用于分析压电性的非常有效的数值框架，其中无需网格生成即可求解控制压电介质的基本静电方程。由于机电耦合特性和压电常数，离散线性系统是稀疏的，具有广义鞍点形式，并且常常是病态的。在这项工作中，我们提出了一种用于构建一系列逐单元近似 Schur 补矩阵的技术，用于预处理以加速此类问题的 Krylov 子空间迭代方法的收敛。近似 Schur 补矩阵在无网格离散化过程中构造简单且成本低廉，并且具有稀疏结构。证明如此构造的近似 Schur 补矩阵在谱上等效于精确的 Schur 补矩阵，这在用于预处理时会导致非常快的收敛。此外，压电方程的无量纲化被认为使计算更加稳定。所提出的预处理器的鲁棒性和效率在两个测试问题上进行了数值说明，这两个测试问题来自压电条剪切变形问题和压电条弯曲问题。数值结果表明，达到给定容差的迭代次数与自由度数以及各种问题参数无关。当用于预处理时，这会导致非常快的收敛。此外，压电方程的无量纲化被认为使计算更加稳定。所提出的预处理器的鲁棒性和效率在两个测试问题上进行了数值说明，这两个测试问题来自压电条剪切变形问题和压电条弯曲问题。数值结果表明，达到给定容差的迭代次数与自由度数以及各种问题参数无关。当用于预处理时，这会导致非常快的收敛。此外，压电方程的无量纲化被认为使计算更加稳定。所提出的预处理器的鲁棒性和效率在两个测试问题上进行了数值说明，这两个测试问题来自压电条剪切变形问题和压电条弯曲问题。数值结果表明，达到给定容差的迭代次数与自由度数以及各种问题参数无关。由压电条剪切变形问题和压电条弯曲问题引起。数值结果表明，达到给定容差的迭代次数与自由度数以及各种问题参数无关。由压电条剪切变形问题和压电条弯曲问题引起。数值结果表明，达到给定容差的迭代次数与自由度数以及各种问题参数无关。