$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

We use the following interpolations: $$\begin{equation} \begin{aligned}\delta\mathbf{v} & =\sum_{a}N_{a}\delta\mathbf{v}_{a} & \Delta\mathbf{u} & =\sum_{b}N_{b}\Delta\mathbf{u}_{b}\\ \grad\delta\mathbf{v} & =\sum_{a}\delta\mathbf{v}_{a}\otimes\grad N_{a} & \grad\Delta\mathbf{u} & =\sum_{b}\Delta\mathbf{u}_{b}\otimes\grad N_{b}\\ \divg\delta\mathbf{v} & =\sum_{a}\delta\mathbf{v}_{a}\cdot\grad N_{a} & \divg\Delta\mathbf{u} & =\sum_{b}\Delta\mathbf{u}_{b}\cdot\grad N_{b} \end{aligned} \,,\label{eq:interpolations-1} \end{equation}$$ where $N_{a}\left(\eta^{1},\eta^{2},\eta^{3}\right)$ are shape functions of the element parametric coordinates $\left(\eta^{1},\eta^{2},\eta^{3}\right)$ . Note that the $\grad\equiv\frac{\partial}{\partial\mathbf{x}}$ operator should be evaluated at $t_{n+\alpha_{f}}$ , using $\mathbf{x}_{n+\alpha_{f}}$ . For example, in the case of a scalar function $f$ , $$\begin{aligned}\grad f & =\frac{\partial f}{\partial\mathbf{x}_{n+\alpha_{f}}}=\frac{\partial f}{\partial\eta^{i}}\mathbf{g}_{n+\alpha_{f}}^{i}\\ \mathbf{g}_{n+\alpha_{f}}^{i} & =\frac{\partial\eta^{i}}{\partial\mathbf{x}_{n+\alpha_{f}}} \end{aligned} \,,$$ where the contravariant basis vectors $\mathbf{g}_{n+\alpha_{f}}^{i}$ may be evaluated from the covariant basis vectors $$\mathbf{g}_{i}^{n+\alpha_{f}}=\frac{\partial\mathbf{x}_{n+\alpha_{f}}}{\partial\eta^{i}}=\left(1-\alpha_{f}\right)\frac{\partial\mathbf{x}_{n}}{\partial\eta^{i}}+\alpha_{f}\frac{\partial\mathbf{x}_{n+1}}{\partial\eta^{i}}$$ using $\mathbf{g}_{i}^{n+\alpha_{f}}\cdot\mathbf{g}_{n+\alpha_{f}}^{j}=\delta_{i}^{j}$ .