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GeneralizedAlpha

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

Syntax

Two forms are available.

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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-\(\alpha\)