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首页> 外文期刊>Journal of Engineering Mechanics >Improved Woodbury Solution Method for Nonlinear Analysis with High-Rank Modifications Based on a Sparse Approximation Approach
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Improved Woodbury Solution Method for Nonlinear Analysis with High-Rank Modifications Based on a Sparse Approximation Approach

机译：基于稀疏近似法的高级修改，改进了伍德伯里解决方法的非线性分析方法

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摘要

In mathematics, the Woodbury formula is an efficient solution method for low-rank modifications that has been utilized by many researchers for the implementation of structural analyses with local material nonlinearity. The advantages of this method in local nonlinearity include the ability to avoid updating of the global stiffness matrix and to limit factorization to a matrix with a small dimension, which is known as the Schur complement. However, this matrix is generally dense, and its dimension depends on the scale of the nonlinear domains. When the condition of local nonlinearity is not satisfied, the problem becomes high-rank modifications and the Woodbury formula becomes inefficient. To overcome the limitation of the Woodbury formula and extend its high-efficiency advantage to more-general situations, an improved Woodbury method is proposed in which the dense Schur complement matrix is approximated using a banded and sparse matrix based on Saint Venant's principle. To eliminate the error caused by this approximation and minimize its adverse effect on iterative calculations, a displacement modification process was developed in terms of the tangent response of the structure so that the iterative rate of the proposed method can be accelerated. Moreover, an adaptive iterative strategy was established to further improve the computational performance of the proposed scheme. A numerical example demonstrates that the proposed scheme can be implemented more efficiently than the classical approach for the nonlinear analysis of structures.
机译：在数学中，伍德伯里公式是许多研究人员已经利用的低秩修改的有效解决方法，用于实施具有局部材料非线性的结构分析。该方法在局部非线性中的优点包括避免更新全局刚度矩阵的能力，并利用具有小维度的矩阵的分解，这被称为SCUR补充。然而，该矩阵通常是致密的，其尺寸取决于非线性域的比例。当不满足局部非线性的条件时，问题变为高级修改，伍德伯里公式变得效率低下。为了克服伍德伯里公式的限制并扩展其高效率的优势对更多一般情况，提出了一种改进的伍德伯里方法，其中使用基于圣文鸣原理的带状和稀疏矩阵近似密集的梭谱矩阵。为了消除该近似引起的误差并最小化其对迭代计算的不利影响，就结构的切线响应而开发了位移修改过程，从而可以加速所提出的方法的迭代速率。此外，建立了自适应迭代策略，以进一步提高所提出的方案的计算性能。数值示例演示了所提出的方案可以比结构非线性分析的经典方法更有效地实现。

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