A unified scheme for nonlinear dynamic direct time integration methods: a comparative study on the application of multi-point methods

Abstract

In this article, we first present a unified scheme to apply nonlinear dynamic time integration methods. The unified scheme covers many existing time integration methods as exceptional cases. This paper has investigated time integration methods, including the Newmark, Wilson, Houbolt, and $\rho _{\infty }$-Bathe method. We then implement the multi-point methods as the nonlinear solution schemes along with the direct time integration methods in nonlinear dynamic analysis. Also, a unified scheme for applying single-point and multi-point methods is presented. Finally, we demonstrate with numerical examples that the unified scheme provides a framework for comparing direct time integration methods. We also investigate the performance of multi-point methods as nonlinear solution methods in detail.

Appendix 1: Unifying \(a_{0}\) in the unified scheme

The coefficient $a_{0}$ in the unified scheme is as follows:

$$\begin{aligned} a_{0}=\frac{1}{4\beta (q_{2}\theta \Delta t)^{2}}(-\frac{H}{2}+1), \end{aligned}$$

(58)

where is used in Eq. (59):

$$\begin{aligned} {\varvec{\delta }} _{\text {ssi}}=a_{0}{{\mathbf {M}}}+a_{1}{{\mathbf {C}}}, \end{aligned}$$

(59)

The unified scheme includes the four time integration methods, so we obtain the coefficient $a_{0}$ according to the time integration methods:

1.1 Unifying $a_{0}$ in $\rho _{\infty }$-Bathe method

The $\rho _{\infty }$-Bathe method is a composite method. In this method, the values of dynamic components $\varvec{\delta } _{1}$ and $\varvec{\delta } _{2}$ are as follows:

$$\begin{aligned} {\varvec{\delta }} _{1}& = \frac{4}{(\gamma \Delta t)^{2}}{{\mathbf {M}}}+\frac{1}{\gamma \Delta t}{{\mathbf {C}}}, \end{aligned}$$

(60)

$$\begin{aligned} {\varvec{\delta }} _{2}& = \frac{1}{(q_{2} \Delta t)^{2}}{{\mathbf {M}}}+\frac{1}{q_{2} \Delta t}{{\mathbf {C}}}, \end{aligned}$$

(61)

where

$$\begin{aligned} q_{2}& = -\gamma q_{1}+\frac{1}{2}, \end{aligned}$$

(62)

$$\begin{aligned} q_{1}& = \frac{\rho _{\infty }+1}{2\gamma (\rho _{\infty }-1)+4}, \end{aligned}$$

(63)

By comparing Eqs. (60) and (61) with Eq. (59), the following can be expressed:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {If}:{\text {ssi}}=1\rightarrow a_{0}=\frac{4}{(\gamma \Delta t)^{2}}, \\ \text {If}:{\text {ssi}}=2\rightarrow a_{0}=\frac{1}{(q_{2} \Delta t)^{2}}, \end{array}\right. } \end{aligned}$$

(64)

Therefore, $a_{0}=\frac{1}{(q_{2} \Delta t)^{2}}$ can be considered as the comprehensive state of the coefficient $a_{0}$, when $q_{2}$ be expressed as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text { if }: {\text {ssi}}=1\rightarrow q_{2}=\frac{\gamma }{2}, \\ \text { if }: {\text {ssi}}=2\rightarrow q_{2}=-\gamma q_{1}+\frac{1}{2}, \end{array}\right. } \end{aligned}$$

(65)

Now we require to obtain a unified equation for $q_{2}$ in terms of ssi to satisfy Eq. (65):

$$\begin{aligned} q_{2}=\gamma q_{1}(-1)^{{\text {ssi}}-1}+\frac{{\text {ssi}}-1}{\text {ssi}}, \end{aligned}$$

(66)

We can express for $q_{1}$ a unified equation in terms of y by comparing Eq. (65) with Eq. (66). Therefore, Eq. (63) is rewritten as follows:

$$\begin{aligned} q_{1}=\frac{(\rho _{\infty }+1)^{ssi-1}}{2\gamma (\rho _{\infty }-1)({\text {ssi}}-1)+2{\text {ssi}}}, \end{aligned}$$

(67)

1.2 Unifying $a_{0}$ in Newmark method

The Newmark method is a simple method. Therefore, $\delta _{1}$ in the unified scheme for the Newmark method is as follows:

$$\begin{aligned} \delta _{1}=\frac{1}{\beta \Delta t^{2}}{\mathbf {M}}+\frac{\alpha }{\beta \Delta t}{\mathbf {C}}, \end{aligned}$$

(68)

The dynamic component of $\delta _{1}$ is compared between the $\rho _{\infty }$-Bathe and Newmark methods, and it is observed that the coefficient $a_{0}$ in the Newmark method has no $\gamma$. Therefore, we consider $\gamma$ as a constant and neutral parameter in the Newmark method ($\gamma$ = 1). The coefficient $a_{0}$ obtained in the previous step is written as follows to include both methods:

$$\begin{aligned} a_{0}=\frac{1}{4\beta (q_{2}\Delta t)^{2}}, \end{aligned}$$

(69)

1.3 Unifying $a_{0}$ in Wilson method

The Wilson method is a simple method and is similar to the linear acceleration method ($\alpha =\frac{1}{2},\,\beta =\frac{1}{6}$). $\delta _{1}$ in the unified scheme for the Wilson method is as follows:

$$\begin{aligned} \delta _{1}=\frac{6}{(\theta \Delta t)^{2}}{\mathbf {M}}+\frac{3}{\theta \Delta t}{\mathbf {C}}, \end{aligned}$$

(70)

Equation (69) can be rewritten by comparing Eqs. (59) and (69) with Eq. (70):

$$\begin{aligned} a_{0}=\frac{1}{4\beta (q_{2}\theta \Delta t)^{2}}, \end{aligned}$$

(71)

1.4 Unifying $a_{0}$ in Houbolt method

The Houbolt method is a simple method. $\delta _{1}$ in the unified scheme for the Houbolt method is as follows:

$$\begin{aligned} \delta _{1}=\frac{2}{\Delta t^{2}}{\mathbf {M}}+\frac{11}{6\Delta t}{\mathbf {C}}, \end{aligned}$$

(72)

By choosing the Houbolt method for analysis in the unified scheme, the variables of other methods should be considered as constant and neutral parameters ($\gamma =1,\,\theta =1,\,\alpha =\frac{1}{2},\,\beta =\frac{1}{4}$). Therefore, Eq. (71) can be rewritten by comparing Eqs. (59) and (71) with Eq. (72):

$$\begin{aligned} a_{0}=\frac{1}{4\beta (q_{2}\theta \Delta t)^{2}}\left(-\frac{H}{2}+1\right), \end{aligned}$$

(73)

In the unified scheme can be selected the Houbolt method by $H=1$, and the coefficients of the Houbolt method can be activated.