Inexact Newton Woodbury method for efficient structural nonlinear analysis

Abstract

Nonlinear analysis has become an indispensable means for structural performance evaluation, but is still an expensive process especially for large and complex structures. Although the computer technology has largely improve the performance of nonlinear analysis, more efficient and accurate methods still attract the attention of the researchers. The Woodbury nonlinear method which uses the Woodbury formula to solve the incremental displacement response is an efficient kind of method for local nonlinear analysis, but it is no longer applicable for problems with large part of nonlinearity. Although some approximate methods have been proposed to improve the solution of linear equations in Woodbury nonlinear method, the induced error cannot be effectively controlled which leads to decreasing the convergence rate of the iterative computation and offsetting the improved computational efficiency. To address this problem, this paper proposes the Inexact Newton Woodbury (INW) method for efficient nonlinear analysis, which comprehensively considers the efficiency of both the solution of the linear equations and iterative computation. In terms of the linear equations, the improved Woodbury method is presented, where the approximation method is combined with the Woodbury formula, considerably enhancing the solution efficiency of the linear equations and largely extending the application range of the Woodbury formula. Moreover, to ensure the convergence rate which is another important influence factor of the overall nonlinear analysis efficiency, the inexact Newton condition with a new proposed forcing term applicable for the INW method is employed to make the iterative computation with superlinear convergence rate. Then, the time complexity analysis is conducted to make a comprehensive evaluation of the proposed INW method and the traditional Finite Element Method (FEM) from the aspects of the linear equations solution and iterative computation. The results show that the INW method is more efficient than the FEM especially for problems with large part of nonlinearity. Finally, a simulation example is presented to validate the efficiency and accuracy of the INW method. The results demonstrate that the INW method has higher efficiency under comparable accuracy with the Woodbury nonlinear method so that it has significantly potential for large part of nonlinearity problems.

CE Database Subject Headings: Engineering structures; finite element method; material nonlinearity; time complexity

Keywords:

• Nonlinear analysis
• Woodbury formula
• computational efficiency
• accuracy

Additional information

Funding

Funding for authors was provided by the National Natural Science Foundation of China (grant No. 52038002, 52008075), Liaoning Revitalization Talents Program (grant No. XLYC1902043), and “Northeast Electric Power University Ph.D. Startup Fund” (Grant No. BSJXM-2020217).