Efficient Woodbury-CA Hybrid Method for Structures with Material and Geometric Nonlinearities

Nonlinearity problems in engineering structures primarily involve material nonlinearity and geometric nonlinearity. Significant effort has gone toward developing accurate and efficient models and methods to simulate these nonlinear behaviors of structures. Although advanced hardware technology has greatly enhanced the computational performance of nonlinear analyses, many researchers still pay extra attention to developing more efficient numerical solution methods with the emergence of complex large-scale structures. The inelasticity-separated finite element method (IS FEM), as an efficient algorithm, is suitable for solving local material nonlinearity problems by keeping the global stiffness matrix unchanged throughout the whole computational process such that the Woodbury formula can be used as an effective tool. However, this procedure does not obviously improve the computational efficiency for structures with large deformation because widely distributed geometric nonlinearity commonly occurs throughout entire structures rather than in local domains. Furthermore, the global stiffness matrix, which is equal to the sum of the initial stiffness matrix and geometric stiffness matrix, changes in real time. This study proposes an efficient Woodbury-CA hybrid (WCH) method by incorporating the combined approximations (CA) method into the framework of the IS FEM to obtain the response of engineering structures with hybrid nonlinear behaviors (both material and geometric nonlinearities) under external loads. Within this framework, the solution of linear equations in the Woodbury formula, which is related only to the geometric nonlinear behaviors of structures, can be obtained by employing the CA method; two other global stiffness matrices, which are used to formulate the Schur complement matrices, are approximated as constant matrices for small periods of time. Additional error induced by these approximations can be eliminated by updating the global stiffness matrix when the optimal adaptive criterion (AC) used for evaluating the difference between the exact solution and approximate solution is not satisfied. Additionally, the time complexity theory is used to evaluate the computational efficiency of the proposed method; the results show that the WCH method has outstanding advantages over the conventional finite element method (FEM). The proposed method is validated against the FEM results via two different numerical examples and has greater potential for solving local material and global geometric nonlinearity problems.