Performance Analysis And Improvement Of The Woodbury Method In Solving Structural Nonlinear Problems

Nonlinear problems generally exist in various engineering fields.Structural nonlinear analysis,which provides an effective means for structural performance evaluation,can effectively simulate the entire response process of structure under different loads and further grasp the mechanical characteristics of the structure.At present,finite element method(FEM)is a commonly used method of structural nonlinear analysis.However,as the increase of the structural scale and refinement of the analysis model,the computational cost of finite element analysis will substantially increase.Although the continuous development of the computer technology has alleviated this problem to some extent,the development of efficient nonlinear analysis method is still the fundamental way to solve this problem due to the complexity of the nonlinear analysis.Many researchers have put forward efficient methods based on the characteristics of structural nonlinear analysis,but each method has its own applicability and limitations.Therefore,to fully understand the computational performance of nonlinear analysis methods and develop efficient algorithms for different problems are still the research focus of the structural nonlinear analysis.The elements of the stiffness matrix are often partially changed for local nonlinear problems,in which the tangent stiffness matrix in each incremental step can be written as the sum of the initial stiffness matrix and its low rank perturbation matrix so that the structural response can be efficiently solved by the Woodbury formula.The above mentioned nonlinear analysis method can be called the nonlinear analysis method based on the Woodbury formula(called Woodbury method for simplicity).At present,only preliminary theoretical understanding of the computational performance of the Woodbury method is obtained,and there is still lack of systematic research on it.In this paper,the Woodbury method is chosen as the research object,and the structural nonlinear analysis is taken as the starting point.The computational performance of Woodbury method is comprehensively studied from both the solution of the displacement response in each incremental step(by the Woodbury formula)and the iterative calculation,including quantitative analysis of computational efficiency and accuracy verification.Meanwhile,the improved method is proposed according to the limitation of Woodbury method,and the verification of its efficiency and accuracy are conducted.The main research contents are shown as follows:(1)Theoretical research on the efficiency analysis of the Woodbury method is carried out.Firstly,the theory of Woodbury formula is introduced,and the process of solving linear equations by the Woodbury formula and the direct decomposition method is compared.The fundamental reason that the Woodbury formula can efficiently solve the inverse of low-rank modified matrix is theoretically explained.Secondly,the basic theory of Woodbury method for structural nonlinear analysis is introduced,and its general solution flow of nonlinear analysis is summarized.The factors affecting the efficiency of the Woodbury method and their relationship are analyzed based on the nonlinear analysis characteristics.Finally,the time complexity theory which is the quantitative efficiency evaluation method is introduced,and the measures to reduce the time complexity are given with examples.(2)The quantitative efficiency evaluation of Woodbury method is carried out,and the application scope of this method is obtained.The time complexity models of Woodbury method and FEM(by LDLT factorization method)for solving linear equations in each iteration step are established by using the time complexity theory,and the quantitative comparison analysis is carried out.The results show that the Woodbury method has significant efficiency advantages over the FEM for local nonlinear problems.However,the efficiency of Woodbury method significantly decreases with the increase of the nonlinear range,which is even lower than that of the FEM.Therefore,the Woodbury method is no longer efficient in this case.(3)The research on the iterative performance of Woodbury method is carried out.According to its iterative characteristics,the two-point method with three convergence order,the two-point method with four convergence order and the three-point method are selected to improve the iterative solution,and the comparison with the traditional Newton-Raphson(N-R)method and modified Newton method is carried out.The time complexity models of the above five iterative algorithms solving the equilibrium equations of the Woodbury method are obtained,and the efficiency comparison of the other four iterative algorithms with N-R method is quantitatively conducted to obtain the applicable conditions of each algorithm.The static and dynamic simulation examples validate the accuracy of the improved iterative algorithms.The calculation performance of the five iterative algorithms is comprehensively compared from the perspective of accuracy,convergence,convergence rate and efficiency.Then,the theoretical basis and suggestions are provided for the selection of the iterative algorithms in practical problems,and the iterative performance optimization of Woodbury method is realized.(4)The improved method called Woodbury Approximation Method(WAM)is proposed according to the limitation of the Woodbury method.Approximate Woodbury Formula based on the idea of approximate method is proposed to solve the structural response in the linear iteration step.Meanwhile,a new forcing term which ensures the WAM with superlinear convergence rate is proposed to prevent the convergence rate of iterative calculation decreasing significantly because of approximate solution.The time complexity model of the WAM is obtained,and the quantitative comparisons with Woodbury method and FEM are carried out.The theoretical and numerical results show that the WAM can obtain both high accuracy with only few basis vectors and high convergence rate.The WAM is more efficient than the Woodbury method and FEM for problems with large part of nonlinear regions,indicating that the WAM enlarges the application scope of the Woodbury method.