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A new method for computation of eigenvector derivatives with distinct and repeated eigenvalues in structural dynamic analysis

机译：结构动态分析中具有重复特征值的特征向量导数计算的新方法

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摘要

Determining eigenvector derivatives is a challenging task due to the singularity of the coefficient matrices of the governing equations, especially for those structural dynamic systems with repeated eigenvalues. An effective strategy is proposed to construct a non-singular coefficient matrix, which can be directly used to obtain the eigenvector derivatives with distinct and repeated eigenvalues. This approach also has an advantage that only requires eigenvalues and eigenvectors of interest, without solving the particular solutions of eigenvector derivatives. The Symmetric Quasi-Minimal Residual (SQMR) method is then adopted to solve the governing equations, only the existing factored (shifted) stiffness matrix from an iterative eigensolution such as the subspace iteration method or the Lanczos algorithm is utilized. The present method can deal with both cases of simple and repeated eigenvalues in a unified manner. Three numerical examples are given to illustrate the accuracy and validity of the proposed algorithm. Highly accurate approximations to the eigenvector derivatives are obtained within a few iteration steps, making a significant reduction of the computational effort. This method can be incorporated into a coupled eigensolver/derivative software module. In particular, it is applicable for finite element models with large sparse matrices.
机译：由于控制方程的系数矩阵奇异，因此确定特征向量导数是一项艰巨的任务，尤其是对于那些具有重复特征值的结构动力学系统而言。提出了一种构造非奇异系数矩阵的有效策略，该矩阵可以直接用于获得特征值重复且重复的特征向量导数。该方法还具有仅需要感兴趣的特征值和特征向量的优点，而无需解决特征向量导数的特定解。然后采用对称拟最小残差（SQMR）方法求解控制方程，仅利用迭代本征解中现有的因式（位移）刚度矩阵，例如子空间迭代法或Lanczos算法。本方法可以统一方式处理简单和重复特征值的情况。给出了三个数值例子，说明了该算法的准确性和有效性。在几个迭代步骤中即可获得特征向量导数的高精度近似值，从而显着减少了计算量。该方法可以合并到耦合的本征求解器/派生软件模块中。特别适用于稀疏矩阵较大的有限元模型。

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