Improved Woodbury approximation approach for inelasticity-separated solid model analysis

Abstract

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.