Development of A Robust Nonlinear Solution Algorithm for Structural Analysis

This paper proposes a solution algorithm for nonlinear structural analysis problems involving static and/or dynamic loads based on the Lyapunov stability theory. The main idea is to reformulate the equations of motion into a hypothetical dynamical system characterized by a set of ordinary differential equations, whose equilibrium points represent the solutions of the nonlinear structural problems. Starting from the Lyapunov stability theory, it is theoretically demonstrated that this hypothetical dynamical system is characterized by a global convergence to the equilibrium points for structural dynamics, i.e., the convergence is guaranteed independently of the selection of the initial guess. This feature overcomes the inherent limitations of the traditional iterative minimization algorithms and relaxes the restriction on the selection of the initial guess for various structural nonlinear behaviors. Moreover, comparisons between the proposed algorithm and regular Newton-Raphson algorithm are presented using several numerical examples from structural dynamics. Finally, the scalability of the proposed Lyapunov-based algorithm is discussed via adaptive switching of nonlinear solution algorithms at the problematic time steps. Postdoctoral Scholar, Dept. of CEE, UC Berkeley, CA 94720 (email: benliangxiao@berkeley.edu) Taisei Professor of Civil Engineering and Director of PEER, Dept. of CEE, UC Berkeley, CA 94720 Liang X, Mosalam KM. Development of a Robust Nonlinear Solution Algorithm for Structural Analysis. Proceedings of the 11 National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Los Angeles, CA. 2018. Eleventh U.S. National Conference on Earthquake Engineering Integrating Science, Engineering & Policy June 25-29, 2018 Los Angeles, California Development of a Robust Nonlinear Solution Algorithm for Structural Analysis X. Liang and K. Mosalam