基于子结构的Woodbury非线性分析方法

A WOODBURY NONLINEAR ANALYSIS APPROACH BASED ON THE SUBSTRUCTURING METHOD

非线性分析是研究结构性能的重要手段,准确并高效的模拟非线性行为,对评估结构安全性具有重要意义。工程结构材料非线性行为一般仅发生在局部区域,考虑结构的局部非线性特征,往往能有效提高计算效率。Woodbury公式被应用于多种数值算法中来高效地求解局部材料非线性问题,该公式仅需对小规模的Schur补矩阵进行分解,避免了对整体刚度矩阵的分解运算。然而,Schur补矩阵通常不具备稀疏特征,且其阶数与非线性规模相关,因此,当非线性区域较大时,Woodbury公式的高效性受到限制。为此,该文提出了基于子结构的Woodbury非线性分析方法,该方法将Schur补矩阵分解为若干个子矩阵,大幅降低了非线性分析过程中Schur补矩阵的规模。最后将该方法应用于某钢框架结构的动力非线性分析,并从精度和效率两方面与传统Woodbury法做了对比;结果表明:该文方法在保证计算精度的前提下改善了Woodbury公式的计算性能,进一步拓宽了其适用范围。

Nonlinear analysis is an important means to study structural performance. Accurate and efficient simulation of nonlinear behavior is of great significance for evaluating structural safety. Material nonlinear behavior generally occurs in some local regions for most engineering structures. The computational efficiency of nonlinear analyses can be greatly improved by taking advantage of the characteristics of local nonlinearity. The Woodbury formula has been employed by multiple numerical algorithms to efficiently solve structural analysis problems with local nonlinearity. The use of the Woodbury formula only needs to factorize a small-scale Schur complement matrix and the corresponding operation of the global stiffness can be avoided. However, because the Schur complement matrix is dense and its dimension depends on the scale of the nonlinear domains, the achievement of high efficiency of the Woodbury formula requires the condition of local nonlinearity to be satisfied. To overcome the limitation of the Woodbury formula, a Woodbury nonlinear analysis approach based on the substructuring method is proposed, in which the order of the Schur complement matrix is considerably reduced by partitioning the matrix into several submatrices. The proposed method is applied to the dynamic nonlinear analysis of a steel frame structure, and a comparison is made with conventional structural analysis methods based on the Woodbury for mulain terms of accuracy and efficiency. The results show that the proposed method is sufficiently accurate and improves the computational performance of the Woodbury formula so that its scope of application is broadened.