Improved Woodbury Solution Method for Nonlinear Analysis with High-Rank Modifications Based on a Sparse Approximation Approach

In mathematics, the Woodbury formula is an efficient solution method for low-rank modifications that has been utilized by many researchers for the implementation of structural analyses with local material nonlinearity. The advantages of this method in local nonlinearity include the ability to avoid updating of the global stiffness matrix and to limit factorization to a matrix with a small dimension, which is known as the Schur complement. However, this matrix is generally dense, and its dimension depends on the scale of the nonlinear domains. When the condition of local nonlinearity is not satisfied, the problem becomes high-rank modifications and the Woodbury formula becomes inefficient. To overcome the limitation of the Woodbury formula and extend its high-efficiency advantage to more-general situations, an improved Woodbury method is proposed in which the dense Schur complement matrix is approximated using a banded and sparse matrix based on Saint Venant's principle. To eliminate the error caused by this approximation and minimize its adverse effect on iterative calculations, a displacement modification process was developed in terms of the tangent response of the structure so that the iterative rate of the proposed method can be accelerated. Moreover, an adaptive iterative strategy was established to further improve the computational performance of the proposed scheme. A numerical example demonstrates that the proposed scheme can be implemented more efficiently than the classical approach for the nonlinear analysis of structures.