Woodbury Approximation Method for Structural Nonlinear Analysis

Abstract

As an exact method, the Woodbury formula is used to solve local material nonlinearity problems in structural analysis, in which the calculation and factorization of the global stiffness matrix are avoidable compared to the direct method used in the conventional finite-element method (FEM). This study uses the time-complexity theory to evaluate the efficiency of the Woodbury formula. The results show that the time complexity increases as the number of the inelastic degrees of freedom (IDOF) increases, and the Woodbury formula is more efficient than the LDLT factorization method only when nonlinearity appears within local, small regions. For example, for a structure with a total of 1, degrees of freedom (DOF), the efficiency of the Woodbury formula is higher than the LDLT method when the number of IDOF is less than 1% of the total DOF (1,). Although the nonlinearity only occurring in small partial domains is common for most engineering structures, this low efficiency threshold still limits the Woodbury formula application for some structures with large part of nonlinear regions. To extend this efficiency threshold value, a Woodbury approximation method (WAM) is proposed that incorporates the idea of a combined approximations (CA) approach into the framework of the Woodbury formula, in which the reduced-basis method and binomial series expansion are used to solve the system of linear equations whose scale depends on the number of IDOF. The accuracy considerations for the approximate solution are discussed and the convergence criterion for error evaluation is presented. Moreover, the time-complexity analysis indicates that the limit on the efficiency is enhanced greatly by the proposed WAM under the comparable computational accuracy to the Woodbury formula, such as expanding above percentage of 1 to 7%. Finally, a numerical example is presented to prove that the WAM can provide accurate results with high efficiency, and thus, has greater potential for solving nonlinear problems.