A dual family of dissipative structure-dependent integration methods for structural nonlinear dynamics

A dual family of dissipative structure-dependent integration methods is proposed for structural nonlinear dynamics. It not only can be a family of two-step integration methods but also can be a family of one-step integration methods correspondingly. This family of methods is derived from an ingenious arrangement of the displacement difference equation, where the previous step data are applied to replace the previous two-step data by means of the asymptotic equation of motion. It has desirable properties, such as unconditional stability, explicitness of each time step, second-order accuracy and high-frequency numerical damping. In addition, it has no adverse properties that have been found in some structure-dependent integration methods, such as weak instability, conditional stability for stiffness hardening systems, high-frequency overshooting in steady-state responses and poor capability of seizing high nonlinearity. This family of methods contains most current semi-explicit, structure-dependent integration methods as it is a family of two-step integration methods although it also covers many brand-new members of two-step integration methods, and it is a brand-new family of one-step integration methods. It has the same properties as those of the generalized-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} method for linear elastic systems. However, it saves many computational efforts for solving inertial problems due to no involvement of nonlinear iterations per time step.