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,

$$d_{n+1}=d_n+\mathrm{\Delta }t{v}_n+\mathrm{\Delta }{t}^2(({\displaystyle \frac{1}{2}}-\beta )a_n+\beta a_{n+1}),$$

$$v_{n+1}=v_n+\mathrm{\Delta }t(1-\gamma )a_n+\mathrm{\Delta }t\gamma a_{n+1}.$$

The equation of motion is expressed at somewhere between $t_n$ and $t_{n+1}$.

$$M{a}_{n+1-{\alpha }_m}+C{v}_{n+1-{\alpha }_f}+K{d}_{n+1-{\alpha }_f}=F_{n+1-{\alpha }_f},$$

which can also be explicitly shown as

$$M((1-{\alpha }_m)a_{n+1}+{\alpha }_m{a}_n)+C((1-{\alpha }_f)v_{n+1}+{\alpha }_f{v}_n)+K((1-{\alpha }_f)d_{n+1}+{\alpha }_f{d}_n)=(1-{\alpha }_f)F_{n+1}+{\alpha }_f{F}_n,$$

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

Default Parameters

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

$${\alpha }_m\le {\alpha }_f\le {\displaystyle \frac{1}{2}},\beta \ge {\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2}}({\alpha }_f-{\alpha }_m).$$

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

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

$${\alpha }_f={\displaystyle \frac{{\rho }_{\mathrm{\infty }}}{{\rho }_{\mathrm{\infty }}+1}},{\alpha }_m={\displaystyle \frac{2{\rho }_{\mathrm{\infty }}-1}{{\rho }_{\mathrm{\infty }}+1}},\gamma ={\displaystyle \frac{1}{2}}-{\displaystyle \frac{{\rho }_{\mathrm{\infty }}-1}{{\rho }_{\mathrm{\infty }}+1}},\beta ={\displaystyle \frac{1}{{({\rho }_{\mathrm{\infty }}+1)}^2}}.$$

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 ${\rho }_{\mathrm{\infty }}$ 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-$\alpha$