Numerical Characteristics of Explicit Method for Nonlinear Systems

Numerical properties of the Newmark explicit method in the solution of nonlinear systems are evaluated by introducing a parameter, which is named the instantaneous degree of nonlinearity, to monitor the variation of stiffness with time. Stability analysis reveals that the upper stability limit is inversely proportional to the square root of the instantaneous degree of nonlinearity and thus it is no longer equal to 2 for nonlinear systems. In fact, it is shrunk as instantaneous degree of nonlinearity is larger than 1 while it is enlarged as instantaneous degree of nonlinearity is less than 1. It is also proved that the satisfaction of stability limits for each time step implies a stable computation in the complete step-by-step integration procedure. Accuracy analysis shows that the relative period error is increased with the increase of the instantaneous degree of nonlinearity for a given product of the initial natural frequency and time step. Furthermore, a rough guideline is proposed for accurate integration of nonlinear systems and its appropriateness is confirmed with numerical examples.