A high precision direct integration scheme for nonlinear dynamic systems

Based on the high precision direct (HPD) integration scheme for linear systems, a high precision direct integration scheme for nonlinear (HPD-NL) dynamic systems is developed. The method retains all the advantages of the standard HPD scheme (high precision with large time-steps and computational efficiency) while allowing nonlinearities to be introduced with little additional computational effort. In addition, limitations on minimum time step resulting from the approximation that load varies linearly between timesteps are reduced by introducing a polynomial approximation of the load. This means that, in situations where a rapidly varying or transient dynamic load occurs, a larger time-step can still be used while maintaining a good approximation of the forcing function and, hence, the accuracy of the solution. Numerical examples of the HPD-NL scheme compared with Newmark's method and the fourth-order Runge-Kutta (Kutta 4) method are presented. The examples demonstrate the high accuracy and numerical efficiency of the proposed method.