An extended forward-backwards implicit scheme for a nonlinear mass-spring finite element time-dependent system

Abstract

This paper reports the development of extended forward-backwards difference numerical solution procedures for a class of nonlinear systems. The theoretical accuracy analysis revealed a high-order accuracy, with an error of order O for displacement, and an error of order O for velocity. This is an improvement over simple second-order accurate implicit integration schemes. The Lyapunov stability analysis showed that the algorithm developed was unconditionally stable, and values of integration parameters, , can be determined from stability analyses. A linear system of two-degrees-of-freedom was initially solved to illustrate how to extend the methods to deal with multiple-degrees-of-freedom systems using matrices and vectors. The results of Thomson [16] were confirmed. Two nonlinear finite element problems were successfully solved. The hundred-degrees-of-freedom system confirmed the results of Wang [19]. The graphs of displacement versus time look similar. In addition, phase trajectory plots and a velocity versus displacement graph revealed the property of a closed path for the nonlinear mass-spring system. The accuracy of the results was not compared because such an exercise would require gaining access to the authors’ data.