Lyapunov-Based Nonlinear Solution Algorithm for Structural Analysis

Abstract

A solution algorithm is proposed 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 demonstrated theoretically 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. The validation of implementing the algorithm is demonstrated using a geometrically nonlinear pendulum problem with a closed-form exact solution. Moreover, comparisons between the proposed algorithm and Newton-Raphson type algorithms are presented using several numerical examples from structural statics and dynamics. Finally, the scalability of the proposed Lyapunov-based algorithm is discussed via adaptive switching of nonlinear solution algorithms at the problematic time steps.