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In the generalized $\alpha-$ method we evaluate displacements and velocities at the intermediate time $t_{n+\alpha_{f}}=t_{n}+\alpha_{f}\left(t_{n+1}-t_{n}\right)$ , such that $$\begin{aligned}\boldsymbol{\chi}_{n+\alpha_{f}}^{s} & =\left(1-\alpha_{f}\right)\boldsymbol{\chi}_{n}^{s}+\alpha_{f}\boldsymbol{\chi}_{n+1}^{s}\\ \mathbf{u}_{n+\alpha_{f}}^{s} & =\left(1-\alpha_{f}\right)\mathbf{u}_{n}^{s}+\alpha_{f}\mathbf{u}_{n+1}^{s}\\ \mathbf{v}_{n+\alpha_{f}}^{s} & =\left(1-\alpha_{f}\right)\mathbf{v}_{n}^{s}+\alpha_{f}\mathbf{v}_{n+1}^{s}\\ \mathbf{w}_{n+\alpha_{f}} & =\left(1-\alpha_{f}\right)\mathbf{w}_{n}+\alpha_{f}\mathbf{w}_{n+1}\\ J_{n+\alpha_{f}}^{f} & =\left(1-\alpha_{f}\right)J_{n}^{f}+\alpha_{f}J_{n+1}^{f} \end{aligned}$$ In particular, it follows that $$\mathbf{F}_{n+\alpha_{f}}^{s}=\frac{\partial\boldsymbol{\chi}_{n+\alpha_{f}}^{s}}{\partial\mathbf{X}}=\left(1-\alpha_{f}\right)\mathbf{F}_{n}^{s}+\alpha_{f}\mathbf{F}_{n+1}^{s}$$ $$J_{n+\alpha_{f}}^{s}=\det\mathbf{F}_{n+\alpha_{f}}^{s}$$ $$\dot{J}_{n+\alpha_{f}}^{s}=J_{n+\alpha_{f}}^{s}\mathbf{F}_{n+\alpha_{f}}^{-T}:\Grad\mathbf{v}_{n+\alpha_{f}}^{s}$$ $$\mathbf{L}_{n+\alpha_{f}}^{s}=\Grad\mathbf{v}_{n+\alpha_{f}}^{s}\cdot\mathbf{F}_{n+\alpha_{f}}^{-1}$$ In practice however, we get better numerical results when using $$\dot{J}_{n+\alpha_{f}}^{s}=\frac{J_{n+1}^{s}-J_{n}^{s}}{\Delta t}$$ $$\mathbf{L}_{n+\alpha_{f}}^{s}=\frac{\mathbf{F}_{n+1}-\mathbf{F}_{n}}{\Delta t}\cdot\mathbf{F}_{n+\alpha_{f}}^{-1}$$ Similarly, we evaluate velocity derivatives at the intermediate time $t_{n+\alpha_{m}}=t_{n}+\alpha_{m}\left(t_{n+1}-t_{n}\right)$ , such that $$\begin{aligned}\dot{\mathbf{v}}_{n+\alpha_{m}}^{s} & =\left(1-\alpha_{m}\right)\dot{\mathbf{v}}_{n}^{s}+\alpha_{m}\dot{\mathbf{v}}_{n+1}^{s}\\ \dot{\mathbf{w}}_{n+\alpha_{m}} & =\left(1-\alpha_{m}\right)\dot{\mathbf{w}}_{n}+\alpha_{m}\dot{\mathbf{w}}_{n+1}\\ \dot{J}_{n+\alpha_{m}}^{f} & =\left(1-\alpha_{m}\right)\dot{J}_{n}^{f}+\alpha_{m}\dot{J}_{n+1}^{f} \end{aligned}$$ The parameters $\alpha_{f}$ and $\alpha_{m}$ are evaluated from a single parameter $\rho_{\infty}$ using $$\begin{equation} \alpha_{f}=\frac{1}{1+\rho_{\infty}}\,,\quad\alpha_{m}=\frac{1}{2}\frac{3-\rho_{\infty}}{1+\rho_{\infty}}\,,\label{eq:fsi-ga-alphas-1st} \end{equation}$$ for first-order systems, or $$\begin{equation} \alpha_{f}=\frac{1}{1+\rho_{\infty}}\,,\quad\alpha_{m}=\frac{2-\rho_{\infty}}{1+\rho_{\infty}}\,,\label{eq:fsi-ga-alphas-2nd} \end{equation}$$ for second-order systems, 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. Since the solid motion is governed by a second-order differential equation in time, we adopt the formulas for second-order systems.