BatheTwoStep

Starting from version 2.7, customisation of spectral radius ($\rho_\infty$) and sub-step size ($\gamma$) is supported.

The algorithm is known to be able to conserve energy and momentum.

References:

1. 10.1016/j.compstruc.2006.09.004

2. 10.1016/j.compstruc.2018.11.001

The implementation follows the original algorithm. The only difference is the step size defined in suanPan is the mean step size of two sub-step sizes, that is $\Delta{}t/2$ in the references.

For further discussions on this type of algorithm, one can check the following references.

1. 10.1007/s11071-023-09065-7

Syntax

integrator BatheTwoStep (1) [2] [3]
# (1) int, unique integrator tag
# [2] double, spectral radius, \rho_\infty, default: 0
# [2] double, sub-step size, gamma, default: 0.5

Using integrator BatheTwoStep (1) with two optional parameters omitted gives the same results as in versions prior to version 2.7.

The spectral radius $\rho_\infty$ ranges from 0 to 1, both ends inclusive.

The sub-step size $\gamma$ ranges from 0 to 1, both ends exclusive.

Theory

The First Sub-step

For trapezoidal rule,

$$v_{n+1}=v_n+\dfrac{\gamma\Delta{}t}{2}\left(a_n+a_{n+1}\right),$$

$$u_{n+1}=u_n+\dfrac{\gamma\Delta{}t}{2}\left(v_n+v_{n+1}\right).$$

Then,

$$u_{n+1}=u_n+\dfrac{\gamma\Delta{}t}{2}\left(v_n+v_n+\dfrac{\gamma\Delta{}t}{2}\left(a_n+a_{n+1}\right)\right),$$

$$\Delta{}u=\gamma\Delta{}tv_n+\dfrac{\gamma^2\Delta{}t^2}{4}a_n+\dfrac{\gamma^2\Delta{}t^2}{4}a_{n+1}.$$

One could obtain

$$a_{n+1}=\dfrac{4}{\gamma^2\Delta{}t^2}\Delta{}u-\dfrac{4}{\gamma\Delta{}t}v_n-a_n,\qquad \Delta{}a=\dfrac{4}{\gamma^2\Delta{}t^2}\Delta{}u-\dfrac{4}{\gamma\Delta{}t}v_n-2a_n,$$

$$v_{n+1}=\dfrac{2}{\gamma\Delta{}t}\Delta{}u-v_n,\qquad \Delta{}v=\dfrac{2}{\gamma\Delta{}t}\Delta{}u-2v_n.$$

The effective stiffness is then

$$\bar{K}=K+\dfrac{2}{\gamma\Delta{}t}C+\dfrac{4}{\gamma^2\Delta{}t^2}M.$$

The Second Sub-step

The second step is computed by

$$v_{n+2}=v_n+\Delta{}t\left(q_0a_n+q_1a_{n+1}+q_2a_{n+2}\right),$$

$$u_{n+2}=u_n+\Delta{}t\left(q_0v_n+q_1v_{n+1}+q_2v_{n+2}\right).$$

The parameters satisfy $q_0+q_1+q_2=1$, and

$$q_1=\dfrac{\rho_\infty+1}{2\gamma\left(\rho_\infty-1\right)+4},\qquad q_2=0.5-\gamma{}q_1.$$

Hence,

$$a_{n+2}=\dfrac{v_{n+2}-v_n}{q_2\Delta{}t}-\dfrac{q_0}{q_2}a_n-\dfrac{q_1}{q_2}a_{n+1},$$

$$v_{n+2}=\dfrac{u_{n+2}-u_n}{q_2\Delta{}t}-\dfrac{q_0}{q_2}v_n-\dfrac{q_1}{q_2}v_{n+1}.$$

The effective stiffness is then

$$\bar{K}=K+\dfrac{1}{q_2\Delta{}t}C+\dfrac{1}{q_2^2\Delta{}t^2}M.$$

Accuracy Analysis

bathetwostep-0.0

bathetwostep-0.5

bathetwostep-1.0