> 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/generalizedalpha.md).

# GeneralizedAlpha

The generalized-$$\alpha$$ method provides second order accuracy with controllable algorithmic damping on high frequency response.

## Syntax

Two forms are available.

```
integrator GeneralisedAlpha (1) [2]
integrator GeneralizedAlpha (1) [2]
# (1) int, unique tag
# [2] double, spectral radius at infinite frequency, default: 0.5

integrator GeneralisedAlpha (1) (2) (3)
integrator GeneralizedAlpha (1) (2) (3)
# (1) int, unique tag
# (2) double, \alpha_f
# (3) double, \alpha_m
```

## Governing Equation

The generalized alpha method assumes that the displacement $$d$$ and the velocity $$v$$ are integrated as such,

$$
d\_{n+1}=d\_n+\Delta{}tv\_n+\Delta{}t^2\left(\left(\dfrac{1}{2}-\beta\right)a\_n+\beta{}a\_{n+1}\right)
,
$$

$$
v\_{n+1}=v\_n+\Delta{}t\left(1-\gamma\right)a\_n+\Delta{}t\gamma{}a\_{n+1}.
$$

The equation of motion is expressed at somewhere between $$t\_n$$ and $$t\_{n+1}$$.

$$
Ma\_{n+1-\alpha\_m}+Cv\_{n+1-\alpha\_f}+Kd\_{n+1-\alpha\_f}=F\_{n+1-\alpha\_f},
$$

which can also be explicitly shown as

$$
M\left(\left(1-\alpha\_m\right)a\_{n+1}+\alpha\_ma\_n\right)+C\left(\left(1-\alpha\_f\right)v\_{n+1}+\alpha\_fv\_n\right)
+K\left(\left(1-\alpha\_f\right)d\_{n+1}+\alpha\_fd\_n\right)=\left(1-\alpha\_f\right)F\_{n+1}+\alpha\_fF\_n,
$$

where $$\alpha\_m$$ and $$\alpha\_f$$ are two additional parameters.

## Default Parameters

To obtain an unconditionally stable algorithm, the following conditions shall be satisfied.

$$
\alpha\_m\le\alpha\_f\le\dfrac{1}{2},\quad\beta\ge\dfrac{1}{4}+\dfrac{1}{2}\left(\alpha\_f-\alpha\_m\right).
$$

Only one parameter is required, the spectral radius $$\rho\_\infty$$ that ranges from zero to one.

The following expressions are used to determine the values of all constants used.

$$
\alpha\_f=\dfrac{\rho\_\infty}{\rho\_\infty+1},\quad \alpha\_m=\dfrac{2\rho\_\infty-1}{\rho\_\infty+1},\quad
\gamma=\dfrac{1}{2}-\dfrac{\rho\_\infty-1}{\rho\_\infty+1},\quad \beta=\dfrac{1}{\left(\rho\_\infty+1\right)^2}.
$$

So that the resulting algorithm is unconditionally stable and has a second order accuracy. The target numerical damping for high frequencies is achieved while that of low frequencies is minimized.

The recommended values of the spectral radius $$\rho\_\infty$$ range from $$0.5$$ to $$0.8$$.

Some special parameters can be chosen.

| $$\alpha\_f$$ | $$\alpha\_M$$ | method         |
| ------------- | ------------- | -------------- |
| $$0.0$$       | $$0.0$$       | Newmark        |
| -             | $$0.0$$       | HHT-$$\alpha$$ |
| $$0.0$$       | -             | WBZ-$$\alpha$$ |

## Accuracy Analysis

![generalizedalpha-0.0](/files/lQbEiLKYJnEKyJnEbPi9) ![generalizedalpha-0.5](/files/ClTBfFx7Pzx4BxSqamRR) ![generalizedalpha-1.0](/files/tNrTyjUtK7i04VpPqF0F)


---

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