A self-starting dissipative alternative to the central difference methods

Abstract

This paper focuses mainly on developing single-step explicit integration algorithms considering the implicit treatment of velocity. A novel explicit algorithm (GSSI) is proposed and recommended as a self-starting dissipative alternative to the central difference methods in transient analysis. GSSI not only shares the same advantages as the central difference methods, such as second-order accuracy and computational cost, but also achieves flexible dissipation control and self-starting property. Remarkably, GSSI provides a significantly larger stability bound than the central difference methods in the damped case. GSSI imposes two algorithmic parameters ($\rho _s$ and $\rho _b$) to flexibly adjust numerical dissipation at the bifurcation point. In general, the $\rho _b$ controls numerical dissipation at the bifurcation point, while the $\rho _s$ further adjusts the amount of dissipation in the low-frequency range. Spectral analysis and numerical examples are given solved to show the superiority of GSSI over the existing single-step explicit methods with respect to accuracy, stability, and dissipation control.

The identity among the classical CD, half-step, and Newmark explicit methods

First, when the time step $\varDelta t$ is assumed to be constant, the classical CD method (2) is algebraically identical to the Newmark explicit scheme (4). The classical CD method is rewritten herein as

$$\begin{aligned} \dot{{{\mathbf {U}}}}_{n}&=\frac{1}{2\varDelta t}({{\mathbf {U}}}_{n+1}-{{\mathbf {U}}}_{n-1}) \end{aligned}$$

(A1a)

$$\begin{aligned} \ddot{{{\mathbf {U}}}}_{n}&=\frac{1}{\varDelta t^2}({{\mathbf {U}}}_{n+1}-2{{\mathbf {U}}}_n+{{\mathbf {U}}}_{n-1}). \end{aligned}$$

(A1b)

The displacement vector ${{\mathbf {U}}}_{n-1}$ can be eliminated in Eq. (A1) to yield

$$\begin{aligned} {{\mathbf {U}}}_{n+1}={{\mathbf {U}}}_{n}+\varDelta t\dot{{{\mathbf {U}}}}_n+\dfrac{1}{2}\varDelta t^2\ddot{{{\mathbf {U}}}}_n, \end{aligned}$$

(A2)

whereas the displacement vector ${{\mathbf {U}}} _{n+1}$ can also be eliminated in Eq. (A1) to give

$$\begin{aligned} \dot{{{\mathbf {U}}}}_n=\dfrac{{{\mathbf {U}}}_{n}-{{\mathbf {U}}}_{n-1}}{\varDelta t}+\dfrac{1}{2}\varDelta t\ddot{{{\mathbf {U}}}}_n. \end{aligned}$$

(A3)

With the time step $\varDelta t$ unchanged, Eq. (A3) can go one time step further to give

$$\begin{aligned} \dot{{{\mathbf {U}}}}_{n+1}=\dfrac{{{\mathbf {U}}}_{n+1}-{{\mathbf {U}}}_{n}}{\varDelta t}+\dfrac{1}{2}\varDelta t\ddot{{{\mathbf {U}}}}_{n+1}. \end{aligned}$$

(A4)

Substituting Eq. (A2) into Eq. (A4) to eliminate ${{\mathbf {U}}} _{n+1}$ yields

$$\begin{aligned} \dot{{{\mathbf {U}}}}_{n+1}=\dot{{{\mathbf {U}}}}_n+\dfrac{1}{2}\varDelta t\left( \ddot{{{\mathbf {U}}}}_n+\ddot{{{\mathbf {U}}}}_{n+1}\right) . \end{aligned}$$

(A5)

As one can see, Eqs. (A2) and (A5) are integration formulas of the Newmark explicit method (4). Hence, the classical CD method is algebraically identical to the Newmark explicit scheme when the time step remains unchanged.

Next, when the time step $\varDelta t$ is assumed to be constant, the half-step scheme (2) is algebraically identical to the Newmark explicit method (4). The half-step scheme (3) is rewritten as follows.

$$\begin{aligned} \dot{{{\mathbf {U}}}}_{n+1/2}&=\dot{{{\mathbf {U}}}}_{n-1/2}+\varDelta t\ddot{{{\mathbf {U}}}}_n \end{aligned}$$

(A6a)

$$\begin{aligned} {{\mathbf {U}}}_{n+1}&={{\mathbf {U}}}_n+\varDelta t\dot{{{\mathbf {U}}}}_{n+1/2} \end{aligned}$$

(A6b)

$$\begin{aligned} \dot{{{\mathbf {U}}}}_{n+1}&=\dot{{{\mathbf {U}}}}_{n+1/2}+\frac{\varDelta t}{2}\ddot{{{\mathbf {U}}}}_{n+1} \end{aligned}$$

(A6c)

The schemes above can be re-organized as

$$\begin{aligned} {{\mathbf {U}}}_{n+1}&={{\mathbf {U}}}_n+\varDelta t\dot{{{\mathbf {U}}}}_{n+1}-\dfrac{1}{2}\varDelta t^2\ddot{{{\mathbf {U}}}}_{n+1} \end{aligned}$$

(A7a)

$$\begin{aligned} \dot{{{\mathbf {U}}}}_{n+1}&=\dot{{{\mathbf {U}}}}_{n-1/2}+\varDelta t\ddot{{{\mathbf {U}}}}_n+\frac{\varDelta t}{2}\ddot{{{\mathbf {U}}}}_{n+1}. \end{aligned}$$

(A7b)

With the time step $\varDelta t$ unchanged, Eq. (A6b) and Eq. (A7a) can go one integration step back to give

$$\begin{aligned} {{\mathbf {U}}}_n&={{\mathbf {U}}}_{n-1}+\varDelta t\dot{{{\mathbf {U}}}}_{n-1/2} \end{aligned}$$

(A8a)

$$\begin{aligned} {{\mathbf {U}}}_n&={{\mathbf {U}}}_{n-1}+\varDelta t\dot{{{\mathbf {U}}}}_n-\dfrac{1}{2}\varDelta t^2\ddot{{{\mathbf {U}}}}_n. \end{aligned}$$

(A8b)

Equation (A8) reveals that the velocity vector $\dot{{{\mathbf {U}}}}_{n-1/2}$ should be calculated as $\dot{{{\mathbf {U}}}}_{n-1/2}=\dot{{{\mathbf {U}}}}_n-\dfrac{1}{2}\varDelta t\ddot{{{\mathbf {U}}}}_n$, and thus Eq. (A7b) can be further simplified as

$$\begin{aligned} \dot{{{\mathbf {U}}}}_{n+1}=\dot{{{\mathbf {U}}}}_n+\dfrac{1}{2}\varDelta t\left( \ddot{{{\mathbf {U}}}}_n+\ddot{{{\mathbf {U}}}}_{n+1}\right) . \end{aligned}$$

(A9a)

Substituting the equation above into Eq. (A7a) to eliminate $\dot{{{\mathbf {U}}}} _{n+1}$ yields

$$\begin{aligned} {{\mathbf {U}}}_{n+1}={{\mathbf {U}}}_{n}+\varDelta t\dot{{{\mathbf {U}}}}_n+\dfrac{1}{2}\varDelta t^2\ddot{{{\mathbf {U}}}}_n. \end{aligned}$$

(A9b)

Obviously, Eqs. (A9a) and (A9b) are integration formulas of the Newmark explicit method. Hence, the half-step scheme (3) is algebraically identical to the Newmark explicit method when the time step remains unchanged.

Finally, it is concluded that the classical CD, half-step, and Newmark explicit methods are algebraically identical to each other on the condition that the time step $\varDelta t$ remains unchanged.