Improved Woodbury approximation approach for inelasticity-separated solid model analysis
Improved Woodbury approximation approach for inelasticity-separated solid model analysis
复制标题
用于非弹性分离实体模型分析的改进伍德伯里近似方法
DOI：
10.1016/j.soildyn.2019.105926
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发表时间：
2020
影响因子：
4
通讯作者：
Z. Dong
中科院分区：
文献类型：
--
作者：
Gang Li;J. Li;L. Yu;D. Yu;Z. Dong
实体单元在传统有限元法中得到广泛应用，但由于需要重新计算和因式分解高维切线刚度矩阵，相对于杆、梁和壳单元，实体单元消耗更多的计算资源。针对三维实体结构的材料非线性分析，提出了一种非弹性分离实体单元模型和一种有效的数值求解方法。该实体单元模型是在文献[1]提出的非弹性分离有限元法（IS-FEM）的框架内发展起来的，当采用伍德伯里公式时，IS-FEM的计算效率优于传统的有限元法。将结构的切线刚度矩阵表示为不变整体刚度与另一个低秩的变化Schur补修正之和。然而，固体单元的非线性问题可能有更多的非弹性自由度（IDOFs），因此，IS-FEM的效率是低的，由于高阶Schur补修改。本文改进了伍德伯里近似法的求解过程，在伍德伯里公式的框架内应用组合近似法。新的求解格式在一定的迭代步数内只包含有限的全局初始刚度矩阵的回代和稀疏矩阵与向量的乘积，完全避免了Schur补矩阵分解和常数矩阵的预计算和存储过程。时间复杂度分析表明，与传统有限元法和精确伍德伯里公式的IS-FEM法相比，改进方法的效率有了很大提高，且效率提高的比例随着自由度的增加而增加.最后，数值算例验证了所提出的实体单元模型和改进的求解方法的正确性和有效性。
Solid elements are widely implemented in conventional finite element method (FEM), but they also consume more computing resources relative to the bar, beam and shell elements because of the high dimensional tangent stiffness matrix recalculation and factorization. In this work, an inelasticity-separated solid element model and an efficient numerical solution procedure are proposed for the material nonlinear analysis of three-dimensional (3D) entity structures. This solid element model was developed within the framework of the inelasticity-separated finite element method (IS-FEM) presented in prior studies [1], the computational efficiency of IS-FEM is better than that of conventional FEM for the local material nonlinearity problem when using the Woodbury formula. The tangent stiffness matrix of a structure is expressed by the sum of the invariant global stiffness and another changing Schur complement modification with a low-rank. However, a nonlinearity problem with solid elements may have more inelastic degrees of freedom (IDOFs), thus, the efficiency of IS-FEM is low due to the high-rank Schur complement modification. This study improved the solution procedure of the Woodbury approximation Method (WAM), which applies the combined approximation (CA) approach within the framework of the Woodbury formula. The new derived solution scheme only contains limited back-substitutions of the global initial stiffness matrix and product of the sparse matrix and vector during a certain iteration step, which completely avoids the Schur complement matrix factorization and the constant matrix precalculation and storage process. Time complexity analysis indicates that the efficiency of the improved methodology is greatly enhanced relative to both the conventional FEM and the IS-FEM with the exact Woodbury formula, and the ratio of the efficiency improvement increases with an increase in the degrees of freedom (DOFs). Finally, the numerical examples demonstrate the validity and efficiency of the presented solid element model and the improved solution procedure.