Linearized Stability and Accuracy for Step-by-Step Solutions of Certain Nonlinear Systems

Abstract

In this work, stability and accuracy of the Newmark method for nonlinear systems are obtained from a linearized analysis. This analysis reveals that an unconditionally stable integration method for linear elastic systems is unconditionally stable for nonlinear systems and a conditionally stable integration method for linear elastic systems remains conditionally stable for nonlinear systems except that its upper stability limit might vary with the step degree of nonlinearity and step degree of convergence. A sufficient condition to have a stable computation for nonlinear systems in a whole step-by-step integration procedure is also developed in this study. Furthermore, it is also found that numerical accuracy in the solution of nonlinear systems is closely related to the step degree of nonlinearity and step degree of convergence although its characteristics are similar to those of the preceding works for linear elastic systems. Since these results are obtained from a linearized analysis, they can be applicable to the nonlinear systems that satisfied the simplifications for the analysis but may not be applicable to general nonlinear systems.