A Simple, Fast, and Accurate Time-Integrator for Strongly Nonlinear Dynamical Systems

• Published in: Computer modeling in engineering & sciences Vol. 100; no. 3; pp. 249 - 275
• Main Authors: Elgohary, T A, Dong, L, Junkins, J L, Atluri, S N
• Format: Journal Article
• Language: English
• Published: Henderson Tech Science Press 2014
• Subjects: Algorithms; Boundary value problems; Collocation methods; Computational efficiency; Computer simulation; Computing costs; Differential equations; Duffing oscillators; Dynamical systems; Exact solutions; Implicit methods; Initial value problems; Integrators; Mathematical models; Methods; Nonlinear analysis; Nonlinear differential equations; Nonlinear dynamics; Nonlinear equations; Nonlinear systems; Nonlinearity; Numerical methods; Numerical stability; Ordinary differential equations; Radial basis function; Runge-Kutta method; Time domain analysis
• ISSN: 1526-1492; 1526-1506
• DOI: 10.3970/cmes.2014.100.249

In this study, we consider Initial Value Problems (IVPs) for strongly nonlinear dynamical systems, and study numerical methods to analyze short as well as long-term responses. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables of positions as well as velocities. For each discrete-time interval Radial Basis Functions (RBFs) are assumed as trial functions for the mixed variables in the time domain. A simple collocation method is developed in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points as well as collocation points. Three numerical examples are provided to compare the present algorithm with explicit as well implicit methods in terms of accuracy, required size of time-interval (or step) and computational cost. The present algorithm is compared against, the second order central difference method, the classical Runge-Kutta method, the adaptive Runge-Kutta- Fehlberg method, the Newmark-β and the Hilber-Hughes-Taylor methods. First the highly nonlinear Duffing oscillator is analyzed and the solutions obtained from all algorithms are compared against the analytical solution for free oscillation at long times. A Duffing oscillator with impact forcing function is next solved. Solutions are compared against numerical solutions from state of the art ODE45 numerical integrator for long times. Finally, a nonlinear 3-DOF system is presented and results from all algorithms are compared against ODE45. It is shown that the present RBF-Coll algorithm is very simple, efficient and very accurate in obtaining the solution for the nonlinear IVP. Since other presented methods require a much smaller step size and higher computational cost, the proposed algorithm is advantageous and has promising applications in solving nonlinear dynamical systems. The extension of the present algorithm to orbit propagation problems with perturbations, will be pursued in our future studies. Issues of numerical stability for various time-integrators will also be explored in future studies.