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A preconditioned conjugate gradient method for computing eigenvector derivatives with distinct and repeated eigenvalues

机译：计算具有不同和重复特征值的特征向量导数的预处理共轭梯度方法

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摘要

A preconditioned conjugate gradient method is proposed for computing eigenvector derivatives with distinct and repeated eigenvalues in the real symmetric eigensystems. In view of singular character of the coefficient matrices of the governing equations for particular solutions of eigenvector derivatives, a modified governing equation for the complementary part of the computed modal contribution excluding those of the repeated modes is introduced, and its coefficient matrix is symmetric and positive definite. The existing factored (shifted) stiffness matrix from an iterative eigensolution such as Lanczos or Subspace Iteration is then utilized as preconditioner. High accurate approximations to particular solutions of eigenvector derivatives can be provided with a few iterations. The present method can deal with both cases of simple and repeated eigenvalues in a unified manner, and can be integrated into a coupled eigensolver/derivative software module. It is especially suitable for the large sparse matrices that arise in industrial-size finite element models. Finally, two numerical examples are used to demonstrate the superior efficiency and fast convergence of the present method.
机译：提出了一种预处理共轭梯度方法，用于在实对称特征系统中计算具有不同和重复特征值的特征向量导数。鉴于特征向量导数特定解的控制方程系数矩阵的奇异性，针对计算出的模态贡献的互补部分（除重复模式之外）引入了修正的控制方程，其系数矩阵对称且为正定。然后将来自迭代本征解（例如Lanczos或子空间迭代）的现有分解（位移）刚度矩阵用作前置条件。特征向量导数的特定解的高精度近似可以通过几次迭代来提供。本方法可以统一的方式处理简单和重复特征值的情况，并且可以集成到耦合的特征求解器/导数软件模块中。它特别适合于工业规模的有限元模型中出现的大型稀疏矩阵。最后，使用两个数值示例来证明本方法的优越效率和快速收敛性。

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