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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 DECOMPOSITION

  • 摘要: 提出了一种新颖的基于本征正交分解(proper orthogonal decomposition, POD)的多重多级子结构方法. 该方法在传统静凝聚(将内部自由度降阶至边界主自由度)的基础上, 引入了两级独立的POD降阶: 首先, 构建低阶振动模态和基于POD的高阶近似模态共同作为降阶基底, 分别用于近似静凝聚中的数值基函数(约束模态)和缩减后的内部动力学行为, 以显著降低存储需求. 其次, 也是本方法实现子结构高效拼接的关键, 即对所有子结构的边界降阶模态施加奇异值分解(singular value decomposition, SVD), 从而生成一组公共的正交界面基底. 该基底确保了所有子结构的边界变形能在同一线性空间内表达, 极大简化了组装过程并提升了计算速度. 此外, 还探讨了针对复杂拓扑边界的降阶处理办法, 以及如何消除刚体模态对应的零特征值对计算稳定性的影响. 通过对算法复杂度的定量分析表明, 本方法在空间和时间复杂度上均优于传统子结构法. 最后的数值算例证实, 方法的计算精度和效率随着所采用正交基数量的增加而稳定提升, 展现了其良好的收敛性与可靠性.

     

    Abstract: This paper proposes a novel multi-level substructuring method based on proper orthogonal decomposition (POD) to address the high computational cost associated with the dynamic analysis of large-scale structures. The method introduces a two-level independent POD reduction scheme, building upon the framework of traditional static condensation, which reduces internal degrees of freedom to boundary master degrees of freedom. The first level of reduction involves constructing a reduced-order basis by combining low-order vibration modes and POD-generated high-order approximation modes. This composite basis is employed to approximate the numerical basis functions (e.g., constraint modes) within the static condensation process and to capture the essential characteristics of the reduced internal dynamic behavior, thereby significantly decreasing storage requirements. The second level, which is crucial for enabling the efficient assembly of multiple substructures, applies singular value decomposition (SVD) to the reduced boundary modes obtained from all individual substructures. This critical step generates a common set of orthogonal interface bases, ensuring that the boundary deformations of all substructures can be consistently represented within the same linear space. This universal representation greatly simplifies the substructure assembly process and substantially increases the overall computational speed. Furthermore, the paper elaborates on specific strategies for the reduced-order treatment of complex topological boundaries and discusses methods to eliminate the adverse impact of zero eigenvalues associated with rigid body modes on computational stability. A rigorous quantitative analysis of the algorithm's complexity demonstrates its superiority over traditional substructuring methods in terms of both time and space complexity. Finally, the effectiveness and reliability of the proposed method are confirmed through numerical examples. The results indicate that the computational accuracy and efficiency improve steadily as the number of orthogonal bases increases, demonstrating good numerical precision, convergence behavior, and computational reliability. The method presents a robust and efficient framework for the model order reduction of complex structural systems.

     

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