基于近似Woodbury公式的梁单元几何大变形分析  

Geometric large deformation analysis for beam element based on the approximate Woodbury formula

作　　者：李钢[1] 靳永强 Li Gang;Jin Yongqiang(State Key laboratory of Costal and Offshore Engineering,Dalian University of Technology,Dalian 116024,China)

机构地区：[1]大连理工大学海岸和近海工程国家重点实验室,辽宁大连116024

出　　处：《土木工程学报》2021年第11期47-56,共10页China Civil Engineering Journal

基　　金：国家重点研发计划(2018YFC1504303);国家自然科学基金(52038002)。

摘　　要：工程结构发生材料非线性时,往往伴随着几何大变形行为。而在非线性分析过程中,刚度矩阵的分解和求逆是不可避免的。该文将隔离非线性有限元法嵌入到更新拉格朗日格式中,建立了考虑材料非线性和几何大变形行为的梁单元控制方程。为了实现控制方程的高效求解,提出了近似Woodbury公式,其核心思想是:与舒尔补矩阵相关的线性方程组采用组合近似法求解,实时变化的整体刚度矩阵被假定在一段时间内保持不变,从而避免矩阵的实时更新和分解,拓宽了Woodbury公式的应用范围。依据时间复杂度分析,对该文方法进行了效率评估。结果表明:对于结构出现大范围材料非线性时的几何大变形问题,近似Woodbury公式依然具有高效性;通过对某钢筋混凝土框架结构进行地震反应分析,论证了该文方法在满足工程精度要求的前提下,能够高效求解结构出现大范围材料非线性时的几何大变形问题。In general,material nonlinearity and geometric large deformation behaviors occur concurrently in engineering structures under external loads.Many numerical solution methods can successfully capture the response of structures,but they are time-consuming because of the requirement of repeatedly factorizing matrices.In this work,the governing equation of beam element involving the material nonlinearity and geometric large deformation behaviors is established by incorporating the inelasticity-separated finite element method(IS FEM)into the framework of the updated Lagrangian(UL)formulation.To efficiently solve the governing equation and overcome the limitations of Woodbury formula,a Woodbury approximation method(WAM)is presented as an efficient solver.The basic idea behind the developed WAM is that the changing global stiffness matrix is approximated as a constant matrix within a short time period,and a linear equation related to the Schur complement matrix is solved by the combined approximations(CA)method.To eliminate the additional error stemming from the approximation,an adaptive iteration strategy(AIS)based on the energy norm is adopted,in which the difference between the developed WAM solution and the exact solution is evaluated.Furthermore,time complexity analysis is used to verify the high-efficiency advantage of the proposed method.Finally,a reinforced concrete frame used as numerical example demonstrates that the proposed method can be implemented for nonlocal material nonlinearity and large deformation analyses of beam element.

关 键 词：梁单元 几何大变形 近似Woodbury公式 组合近似法 时间复杂度 

分 类 号：TU311.41[建筑科学—结构工程]