Efficient Numerical Solution Method for Large Deformation Analyses of Structures Based on the Updated Lagrangian Formulation

Abstract

Large deformation analyses of structures are of great importance to the evaluation of structural performance under extreme environmental loads, but currently available methods are time-consuming because of the requirement of factorizing large-scale matrices. The inelasticity-separated finite-element method (IS FEM), which can keep the global stiffness matrix unchanged and uses the Woodbury formula as the solver, was presented recently to provide a highly efficient tool for local material nonlinear analysis. To extend the high efficiency advantage of the IS FEM to large deformation analyses, in which the material nonlinearity may be nonlocal and the geometric nonlinearity should be considered, this paper proposes a novel numerical solution scheme by incorporating the updated Lagrangian (UL) formulation into the IS FEM framework. Within this scheme, a Woodbury approximation method (WAM) is introduced as an efficient solver, in which 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 induced by the approximation, an adaptive iteration strategy (AIS) is presented, in which the approximation error involved in WAM solution is evaluated based on energy norm concept, and the global stiffness matrix is required to be updated adaptively according to the calculated error. The high efficiency and accuracy of the proposed method are finally demonstrated by the time complexity analysis and numerical examples.