The generalized-α \alpha α
Syntax
Two forms are available.
Copy 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 d d v v v
d n + 1 = d n + Δ t v n + Δ t 2 ( ( 1 2 − β ) a n + β a n + 1 ) , 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) , d n + 1 = d n + Δ t v n + Δ t 2 ( ( 2 1 − β ) a n + β a n + 1 ) , v n + 1 = v n + Δ t ( 1 − γ ) a n + Δ t γ a n + 1 . v_{n+1}=v_n+\Delta{}t\left(1-\gamma\right)a_n+\Delta{}t\gamma{}a_{n+1}. v n + 1 = v n + Δ t ( 1 − γ ) a n + Δ t γ a n + 1 . The equation of motion is expressed at somewhere between t n t_n t n t n + 1 t_{n+1} t n + 1
M a n + 1 − α m + C v n + 1 − α f + K d n + 1 − α f = F n + 1 − α f , Ma_{n+1-\alpha_m}+Cv_{n+1-\alpha_f}+Kd_{n+1-\alpha_f}=F_{n+1-\alpha_f}, M a n + 1 − α m + C v n + 1 − α f + K d n + 1 − α f = F n + 1 − α f , which can also be explicitly shown as
M ( ( 1 − α m ) a n + 1 + α m a n ) + C ( ( 1 − α f ) v n + 1 + α f v n ) + K ( ( 1 − α f ) d n + 1 + α f d n ) = ( 1 − α f ) F n + 1 + α f F n , 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, M ( ( 1 − α m ) a n + 1 + α m a n ) + C ( ( 1 − α f ) v n + 1 + α f v n ) + K ( ( 1 − α f ) d n + 1 + α f d n ) = ( 1 − α f ) F n + 1 + α f F n , where α m \alpha_m α m α f \alpha_f α f
Default Parameters
To obtain an unconditionally stable algorithm, the following conditions shall be satisfied.
α m ≤ α f ≤ 1 2 , β ≥ 1 4 + 1 2 ( α f − α m ) . \alpha_m\le\alpha_f\le\dfrac{1}{2},\quad\beta\ge\dfrac{1}{4}+\dfrac{1}{2}\left(\alpha_f-\alpha_m\right). α m ≤ α f ≤ 2 1 , β ≥ 4 1 + 2 1 ( α f − α m ) . Only one parameter is required, the spectral radius h o ∞ ho_\infty h o ∞
The following expressions are used to determine the values of all constants used.
α f = ρ ∞ ρ ∞ + 1 , α m = 2 ρ ∞ − 1 ρ ∞ + 1 , γ = 1 2 − ρ ∞ − 1 ρ ∞ + 1 , β = 1 ( ρ ∞ + 1 ) 2 . \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}. α f = ρ ∞ + 1 ρ ∞ , α m = ρ ∞ + 1 2 ρ ∞ − 1 , γ = 2 1 − ρ ∞ + 1 ρ ∞ − 1 , β = ( ρ ∞ + 1 ) 2 1 . 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 h o ∞ ho_\infty h o ∞ 0.5 0.5 0.5 0.8 0.8 0.8
Some special parameters can be chosen.
