> For the complete documentation index, see [llms.txt](https://tlcfem.gitbook.io/suanpan-manual/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tlcfem.gitbook.io/suanpan-manual/integrator/implicit/bathetwostep.md).

# 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](https://doi.org/10.1016/j.compstruc.2006.09.004)
2. [10.1016/j.compstruc.2018.11.001](https://doi.org/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](https://doi.org/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](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-a7a2353ba7b5f8612538c84275b6939a2f6029de%2Fbathetwostep-0.0.svg?alt=media) ![bathetwostep-0.5](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-3ea8997ecdcdcf234e520532f0143b6c5f7007f6%2Fbathetwostep-0.5.svg?alt=media) ![bathetwostep-1.0](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-9859c7c280019d889b9dbab4f37ac5bd80b92602%2Fbathetwostep-1.0.svg?alt=media)


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