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中文核心期刊

基于本征正交分解的多重多级子结构方法

The Multi-Level Substructuring Method Based on Proper Orthogonal Decomposition1)

  • 摘要: 本文提出了一种新的基于本征正交分解(Proper Orthogonal Decomposition, POD)的多重多级子结构方法—。该方法通过引入主从自由度的概念,在传统静凝聚的基础上进行了进一步的降阶计算。具体而言是利用模态分解和奇异值分解技术,构建了低阶模态和高阶近似模态作为降阶基底,分别用来对静凝聚中的数值基函数和内部刚度矩阵进行降阶操作,显著减少了计算过程中所需的存储资源。同时为了实现子结构之间的高效连接,通过对子结构边界部分的低阶模态进行了奇异值分解,可获得能够在同一线性空间内表示所有子结构变形的正交基底。这一基底能够对子结构出口矩阵进一步降阶,极大地提高了多重多级子结构方法的计算速度。此外,本文还详细阐述了如何将多个子结构进行拼接,并提出了一个巧妙的方法来解决刚体模态对应零特征值的问题。通过定量地分析时间复杂度和空间复杂度,证明了所提出方法不仅在空间复杂度上有了显著改善,而且在自振模态特征值问题上的时间和空间复杂度也得到了有效控制。最后,通过数值算例验证了该方法的有效性和可靠性,且随着正交基数量的增加,算法表现出良好的数值精度和收敛性以及计算效率。

     

    Abstract: This paper proposes a novel multi-level substructuring method based on Proper Orthogonal Decomposition (POD). The method introduces the concept of master-slave degrees of freedom and extends traditional static condensation through further order reduction. Specifically, modal decomposition and Singular Value Decomposition (SVD) techniques are employed to construct low-order modal bases and high-order approximate modal bases. These bases are utilized to reduce the numerical basis functions in static condensation and approximate the internal stiffness matrix, significantly decreasing storage requirements during computation. To achieve efficient connectivity between substructures, orthogonal bases capable of representing all substructural deformations within a unified linear space are derived by applying SVD to the low-order modal components of substructural boundaries. These orthogonal bases enable additional dimensionality reduction of substructural interface matrices, substantially improving computational speed in multi-level substructuring. Furthermore, the paper elaborates on substructural assembly techniques and presents an innovative approach to address the issue of rigid body modes corresponding to zero eigenvalues. Quantitative analysis of temporal and spatial complexity demonstrates that the proposed method not only achieves remarkable improvements in spatial complexity but also effectively controls computational costs for eigenvalue problems in vibration mode analysis. Numerical examples validate the method's effectiveness and reliability, showing enhanced numerical accuracy, convergence performance, and computational efficiency with increasing numbers of orthogonal bases.

     

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