An unconditionally stable time integration method with controllable dissipation for second-order nonlinear dynamics

Abstract

This paper proposes a two-sub-step time integration method with controllable dissipation to solve nonlinear dynamic problems. The proposed method has second-order accuracy, unconditional stability and zero-order overshoots. In addition, different from most existing time integration methods, the present method is self-starting, and initial acceleration vector is not required. Importantly, the well-known BN-stability theory for first-order nonlinear dynamics is employed to design algorithmic parameters; thus, the present method is BN-stable, or unconditionally stable for nonlinear dynamics. The present method can give stable and accurate predictions for nonlinear problems in which some excellent methods such as the trapezoidal rule and the ρ_∞-Bathe method fail. A few representative nonlinear numerical examples show that the proposed method enjoys advantages in accuracy, stability and energy conservation compared with the trapezoidal rule and the ρ_∞-Bathe method.

Appendix A: Effective stiffness matrices and load vectors

The time-stepping equations have the forms as

$$ \left\{ {\begin{array}{*{20}l} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\user2{K}} }_{1} {\user2{x}}_{{t + c_{1} \Delta t}} = {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\user2{R}} }_{1} } \hfill \\ {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\user2{K}} }_{2} {\user2{x}}_{{t + c_{2} \Delta t}} = {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\user2{R}} }_{2} } \hfill \\ \end{array} } \right. $$

(A.1)

where the effective stiffness matrices are

$$ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\user2{K}} }_{1} = \frac{1}{{c_{1}^{2} \Delta t^{2} }}{\user2{M}} + \frac{1}{{c_{1} \Delta t}}{\user2{C}} + {\user2{K}} $$

(A.2)

$$ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\user2{K}} }_{2} = \frac{1}{{c_{2}^{2} \alpha^{2} \Delta t^{2} }}{\user2{M}} + \frac{1}{{c_{2} \alpha \Delta t}}{\user2{C}} + {\user2{K}} $$

(A.3)

and the load vectors are

$$ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\user2{R}} }_{1} = {\user2{Q}}(t + c_{1} \Delta t) + {\user2{M}}\left( {\frac{1}{{c_{1}^{2} \Delta t^{2} }}{\user2{x}}_{t} + \frac{1}{{c_{1} \Delta t}}\dot{\user2{x}}_{t} } \right){ + }\frac{1}{{c_{1} \Delta t}}{\user2{Cx}}_{t} $$

(A.4)

$$ \begin{aligned} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\user2{R}} }_{2} & = {\user2{Q}}(t + c_{2} \Delta t) + {\user2{M}}\left[\frac{1}{{c_{2}^{2} \alpha^{2} \Delta t^{2} }}{\user2{x}}_{t} + \frac{1}{{c_{2} \alpha^{2} \Delta t}}\left( {1 - \alpha } \right)\dot{\user2{x}}_{{t + c_{1} \Delta t}} { + }\frac{1}{{c_{2} \alpha \Delta t}}\dot{\user2{x}}_{t} + \frac{1}{\alpha }\left( {1 - \alpha } \right)\ddot{\user2{x}}_{{t + c_{1} \Delta t}} \right]\\ & \quad + C\left[ {\frac{1}{{c_{2} \alpha \Delta t}}{\user2{x}}_{t} + \frac{1}{\alpha }\left( {1 - \alpha } \right)\ddot{\user2{x}}_{{t + c_{1} \Delta t}} } \right] \\ \end{aligned} $$

(A.5)