3.10 Generalized \(\alpha-\)Method

The generalized \(\alpha-\)method is used for temporal discretization of governing equations in fluid mechanics. For this method we combine the degrees of freedom into \(\mathbf{Y}_{n}=\left\{ \mathbf{v},J\right\} _{n}\), where the subscript \(n\) denotes time \(t_{n}\); similarly, we let \(\dot{\mathbf{Y}}_{n}=\left\{ \frac{\partial\mathbf{v}}{\partial t},\frac{\partial J}{\partial t}\right\} _{n}\). According to this method ¹, the virtual work is evaluated at \(\delta W\left(\dot{\mathbf{Y}}_{n+\alpha_{m}},\mathbf{Y}_{n+\alpha_{f}}\right)\), where \(t_{n+\alpha}=t_{n}+\alpha\Delta t\) and \(\Delta t=t_{n+1}-t_{n}\). Here,

\[ \begin{equation} \begin{aligned}\mathbf{Y}_{n+\alpha_{f}} & =\alpha_{f}\mathbf{Y}_{n+1}+\left(1-\alpha_{f}\right)\mathbf{Y}_{n}\\ \dot{\mathbf{Y}}_{n+\alpha_{m}} & =\alpha_{m}\dot{\mathbf{Y}}_{n+1}+\left(1-\alpha_{m}\right)\dot{\mathbf{Y}}_{n} \end{aligned} \,.\label{eq:ga-Y-Ydot} \end{equation} \]

The parameters \(\alpha_{f}\) and \(\alpha_{m}\) are evaluated from a single parameter \(\rho_{\infty}\) using

\[ \begin{equation} \alpha_{f}=\frac{1}{1+\rho_{\infty}}\,,\quad\alpha_{m}=\frac{1}{2}\frac{3-\rho_{\infty}}{1+\rho_{\infty}}\,,\label{eq:ga-alphas} \end{equation} \]

where \(0\le\rho_{\infty}\le1\). This parameter is the spectral radius for an infinite time step, which controls the amount of damping of high frequencies; a value of zero produces the greatest amount of damping, anihilating the highest frequency in one step, whereas a value of one preserves the highest frequency.

The linearization of \(\delta W\left(\dot{\mathbf{Y}}_{n+\alpha_{m}},\mathbf{Y}_{n+\alpha_{f}}\right)\) reported in Section Temporal Discretization and Linearization is effectively performed along an increment \(\Delta\mathbf{Y}\) of \(\mathbf{Y}_{n+1}\) so that the solution to \(\delta W=0\) produces \(\mathbf{Y}_{n+1}\). Based on Newmark integration, we have

\[ \begin{equation} \dot{\mathbf{Y}}_{n+1}=\frac{\mathbf{Y}_{n+1}-\mathbf{Y}_{n}}{\gamma\Delta t}-\left(\frac{1}{\gamma}-1\right)\dot{\mathbf{Y}}_{n}\,.\label{eq:ga-Newmark-integration} \end{equation} \]

where, according to the generalized \(\alpha-\)method,

\[ \begin{equation} \gamma=\frac{1}{2}+\alpha_{m}-\alpha_{f}\,.\label{eq:ga-ksi-gamma} \end{equation} \]

Therefore, in this scheme, \(\dot{\mathbf{Y}}_{n+\alpha_{m}}\) is evaluated from

\[ \begin{equation} \dot{\mathbf{Y}}_{n+\alpha_{m}}=\left(1-\frac{\alpha_{m}}{\gamma}\right)\dot{\mathbf{Y}}_{n}+\frac{\xi}{\Delta t}\left(\mathbf{Y}_{n+\alpha_{f}}-\mathbf{Y}_{n}\right)\,,\quad\xi\equiv\frac{\alpha_{m}}{\alpha_{f}\gamma}.\label{eq:ga-Ydot-evaluation} \end{equation} \]

Using \eqref{eq:ga-Y-Ydot} and \eqref{eq:ga-Ydot-evaluation}, we find that

\[ \begin{equation} \begin{aligned}D\mathbf{Y}_{n+\alpha_{f}}\left[\Delta\mathbf{Y}\right] & =\alpha_{f}\Delta\mathbf{Y}\\ D\dot{\mathbf{Y}}_{n+\alpha_{m}}\left[\Delta\mathbf{Y}\right] & =\frac{\alpha_{m}}{\gamma}\frac{\Delta\mathbf{Y}}{\Delta t} \end{aligned} \label{eq:ga-Y-Ydot-linearization} \end{equation} \]

Given the solution \(\left(\dot{\mathbf{Y}}_{n+\alpha_{m}},\mathbf{Y}_{n+\alpha_{f}}\right)\), the solution at \(t_{n+1}\) is evaluated from

\[ \begin{equation} \begin{aligned}\mathbf{Y}_{n+1} & =\mathbf{Y}_{n}+\frac{\mathbf{Y}_{n+\alpha_{f}}-\mathbf{Y}_{n}}{\alpha_{f}}\,,\\ \dot{\mathbf{Y}}_{n+1} & =\dot{\mathbf{Y}}_{n}+\frac{\dot{\mathbf{Y}}_{n+\alpha_{m}}-\dot{\mathbf{Y}}_{n}}{\alpha_{m}}\,. \end{aligned} \label{eq:ga-Y-updates} \end{equation} \]

Four different options are presented in ¹ for initializing \(\mathbf{Y}_{n+1}\) and \(\dot{\mathbf{Y}}_{n+1}\) at the beginning of time step \(t_{n+1}\); the first three of these have been implemented in FEBio. For steady flows these authors recommend disregarding \(\rho_{\infty}\) and setting \(\alpha_{f}=\alpha_{m}=\gamma=1\) to recover the backward Euler scheme.

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1. Jansen, Kenneth E; Whiting, Christian H; Hulbert, Gregory M. "A generalized-\(\alpha\) method for integrating the filtered {Navier}--{Stokes} equations with a stabilized finite element method." Comput. Methods Appl. Mech. Engrg., vol. 190, pp. 305--319 (2000). ↩↩