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 $ho_\infty$ 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 $ho_\infty$ 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-$\alpha$