A novel family of controllably dissipative composite integration algorithms for structural dynamic analysis

Abstract

In this paper, a new family of controllably dissipative composite algorithms is developed to obtain reliable numerical response of structural dynamic problems. The proposed algorithm is a self-starting, unconditionally stable and second-order accurate three sub-step composite algorithm. The new method includes two optimal sub-families of algorithms, both of which can control numerical dissipations in the high-frequency range by an intuitive way, and their numerical dissipations can range from the non-dissipative case to the asymptotic annihilating case. Besides, they actually involve only one free parameter and always share the identical effective stiffness matrices inside three sub-step to save the computational cost, which does not hold in some existing sub-step algorithms. Some numerical examples are given to show the superiority of the new algorithm with respect to controllable numerical dissipations and the ability of capturing the free-play nonlinearity.

Appendix A

Derivation of the amplification matrix A for the three sub-step composite algorithm can be divided into the following steps:

First, applying the trapezoidal rule in the first sub-step to the SDOF system yields

$$\begin{aligned} \begin{bmatrix} x_{t+\gamma _1 h}\\ h{\dot{x}}_{t+\gamma _1 h}\\ h^2\ddot{x}_{t+\gamma _1 h} \end{bmatrix}=A_{11}\begin{bmatrix} x_{t}\\ h{\dot{x}}_{t}\\ h^2\ddot{x}_{t} \end{bmatrix} \end{aligned}$$

(A1)

where the amplification matrix $A_{11}$ in the first sub-step can be written explicitly as

$$\begin{aligned}&A_{11}\nonumber \\&\quad =\frac{1}{\beta _1}\begin{bmatrix} 4\gamma _1\xi \Omega +4&2\gamma _1(2+\gamma \xi \Omega )&\gamma _1^2\\ -2\gamma _1\Omega ^2&4-\gamma _1^2\Omega ^2&2\gamma _1\\ -4\Omega ^2&-4\Omega (\gamma _1\Omega +2\xi )&-\gamma _1^2\Omega ^2-4\gamma _1\xi \Omega \\ \end{bmatrix} \end{aligned}$$

(A2)

where $\beta _1=\gamma _1^2\Omega ^2+4\gamma _1\xi \Omega +4$. $\omega =\Omega /h$ and $\xi$ are the undamped natural frequency and viscous damping ratio of the SDOF system, respectively.

Second, the similar calculation can be done in the second sub-step. And when $\gamma _2=2\gamma _1$, the matrix–vector form can be given as

$$\begin{aligned} \begin{bmatrix} x_{t+\gamma _2 h}\\ h{\dot{x}}_{t+\gamma _2 h}\\ h^2\ddot{x}_{t+\gamma _2 h} \end{bmatrix}=A_{22}\begin{bmatrix} x_{t+\gamma _1h}\\ h{\dot{x}}_{t+\gamma _1h}\\ h^2\ddot{x}_{t+\gamma _1h} \end{bmatrix} \end{aligned}$$

(A3)

And the amplification matrix $A_{22}$ is exactly identical to the amplification matrix $A_{11}$ in the first sub-step, namely $A_{22}=A_{11}$.

Then, the calculation in the third sub-step yields

$$\begin{aligned} \begin{bmatrix} x_{t+h}\\ h{\dot{x}}_{t+h}\\ h^2\ddot{x}_{t+h} \end{bmatrix}= & {} A_{2s}\begin{bmatrix} x_{t+\gamma _2 h}\\ h{\dot{x}}_{t+\gamma _2 h}\\ h^2\ddot{x}_{t+\gamma _2 h} \end{bmatrix}+A_{1s}\begin{bmatrix} x_{t+\gamma _1 h}\\ h{\dot{x}}_{t+\gamma _1 h}\\ h^2\ddot{x}_{t+\gamma _1 h} \end{bmatrix}\nonumber \\&\quad +A_{0s}\begin{bmatrix} x_{t}\\ h{\dot{x}}_{t}\\ h^2\ddot{x}_{t} \end{bmatrix} \end{aligned}$$

(A4)

where three iteration matrices $A_{2s}$, $A_{1s}$ and $A_{0s}$ can be expressed as

$$\begin{aligned} A_{2s}&=\frac{c_3}{\beta _2} \begin{bmatrix} 0&2c_4\xi \Omega +1&c_4\\ 0&-c_4\Omega ^2&1\\ 0&-\Omega ^2&-c_4\Omega ^2-2\xi \Omega \\ \end{bmatrix} \end{aligned}$$

(A5)

$$\begin{aligned} A_{1s}&=\frac{c_2}{c_3}A_{2s} \end{aligned}$$

(A6)

$$\begin{aligned} A_{0s}&=\frac{1}{\beta _2} \begin{bmatrix} 1+2c_4\xi \Omega&2c_1c_4\xi \Omega +c_1+c_4&c_1c_4\\ -c_4\Omega ^2&-c_1c_4\Omega ^2+1&c_1\\ -\Omega ^2&-c_1\Omega ^2-c_4\Omega ^2-2\xi \Omega&-c_1c_4\Omega ^2-2c_1\xi \Omega \\ \end{bmatrix} \end{aligned}$$

(A7)

where $\beta _2=c_4^2\Omega ^2+2c_4\xi \Omega +1$. The coefficients $c_1, c_2,c_3$ and $c_4$ are determined by Eqs. (12).

In the end, substituting Eqs. (A1) and (A3) into Eq. (A4) yields

$$\begin{aligned} \begin{bmatrix} x_{t+h}\\ h{\dot{x}}_{t+h}\\ h^2\ddot{x}_{t+h} \end{bmatrix}=A\begin{bmatrix} x_{t}\\ h{\dot{x}}_{t}\\ h^2\ddot{x}_{t} \end{bmatrix} \end{aligned}$$

(A8)

where the amplification matrix A of the new sub-step algorithm is given as follows

$$\begin{aligned} A=A_{2s}\cdot A_{22}\cdot A_{11}+A_{1s}\cdot A_{11} +A_{0s} \end{aligned}$$

(A9)

Hence, substituting Eqs. (A2), (A5)–(A7) into Eq. (A9) gives the explicit expression of amplification matrix A.