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

The contact integral may be discretized as $$\begin{equation} \delta G_{c}=\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left[\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}+w_{n}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\right]\,.\label{eq708} \end{equation}$$ The variables may be interpolated over each element face according to $$\begin{equation} \begin{aligned}\delta\mathbf{v}^{\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\mathbf{v}_{a}^{\left(1\right)} & \delta\mathbf{v}^{\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\mathbf{v}_{b}^{\left(2\right)}\\ \Delta\mathbf{u}^{\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\mathbf{u}_{c}^{\left(1\right)} & \Delta\mathbf{u}^{\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\mathbf{u}_{d}^{\left(2\right)}\\ \delta p^{\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta p_{a}^{\left(1\right)} & \delta p^{\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta p_{b}^{\left(2\right)}\\ \Delta p^{\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta p_{c}^{\left(1\right)} & \Delta p^{\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta p_{d}^{\left(2\right)} \end{aligned} \,.\label{eq709} \end{equation}$$ Then, $$\begin{equation} \begin{aligned}\delta G_{c} & =\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left(\sum\limits _{a=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta p_{a}^{\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{a}^{\left(1\right)}\\ w_{a}^{\left(1\right)} \end{array}\right]\right.\\ & \left.+\sum\limits _{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{b,k}^{\left(1\right)} & \delta p_{b,k}^{\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{b,k}^{\left(1\right)}\\ w_{b,k}^{\left(1\right)} \end{array}\right]\right) \end{aligned} \,,\label{eq710} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{f}_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}\mathbf{t} & \mathbf{f}_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}\mathbf{t}\\ w_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}w_{n} & w_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}w_{n} \end{aligned} \,.\label{eq711} \end{equation}$$ Similarly, $$\begin{equation} \begin{aligned}-D\delta G_{c} & =\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\\ & \times\left(\sum\limits _{a=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta p_{a}^{\left(1\right)}\end{array}\right]\cdot\left(\sum\limits _{c=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \mathbf{K}_{ac}^{\left(1,1\right)} & \mathbf{0}\\ \mathbf{k}_{ac}^{\left(1,1\right)} & k_{ac}^{\left(1,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{c}^{\left(1\right)}\\ \Delta p_{c}^{\left(1\right)} \end{array}\right]\right.\right.\\ & +\left.\sum\limits _{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \mathbf{K}_{ad,k}^{\left(1,2\right)} & \mathbf{0}\\ \mathbf{0} & k_{ad,k}^{\left(1,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{d}^{\left(2\right)}\\ \Delta p_{d}^{\left(2\right)} \end{array}\right]\right)\\ & +\sum\limits _{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{b,k}^{\left(2\right)} & \delta p_{b,k}^{\left(2\right)}\end{array}\right]\cdot\left(\sum\limits _{c=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \mathbf{K}_{bc,k}^{\left(2,1\right)} & \mathbf{0}\\ \mathbf{k}_{bc,k}^{\left(2,1\right)} & k_{bc,k}^{\left(2,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{c}^{\left(1\right)}\\ \Delta p_{c}^{\left(1\right)} \end{array}\right]\right.\\ & +\left.\left.\sum\limits _{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \mathbf{K}_{bd,k}^{\left(2,2\right)} & \mathbf{0}\\ \mathbf{0} & k_{bd,k}^{\left(2,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{d}^{\left(2\right)}\\ \Delta p_{d}^{\left(2\right)} \end{array}\right]\right)\right)\,, \end{aligned} \label{eq712} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{K}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}\left(\varepsilon_{n}N_{c}^{\left(1\right)}\mathbf{I}+\mathbf{t}\otimes\mathbf{a}_{c}^{\left(1\right)}\right)\\ \mathbf{K}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{n}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\mathbf{I}\\ \mathbf{K}_{bc,k}^{\left(2,1\right)} & =-N_{b}^{\left(2\right)}\left(\varepsilon_{n}N_{c}^{\left(1\right)}\mathbf{I}+\mathbf{t}\otimes\mathbf{a}_{c}^{\left(1\right)}\right)\\ \mathbf{K}_{bd,k}^{\left(2,2\right)} & =\varepsilon_{n}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)}\mathbf{I} \end{aligned} \,,\label{eq713} \end{equation}$$ $$\begin{equation} \begin{aligned}\mathbf{k}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}w_{n}\mathbf{a}_{c}^{\left(1\right)}\\ \mathbf{k}_{bc,k}^{\left(2,1\right)} & =-N_{b}^{\left(2\right)}w_{n}\mathbf{a}_{c}^{\left(1\right)} \end{aligned} \,,\label{eq714} \end{equation}$$ $$\begin{equation} \begin{aligned}k_{ac}^{\left(1,1\right)} & =-\varepsilon_{p}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\\ k_{ad,k}^{\left(1,2\right)} & =\varepsilon_{p}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ k_{bc,k}^{\left(2,1\right)} & =\varepsilon_{p}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\\ k_{bd,k}^{\left(2,2\right)} & =-\varepsilon_{p}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \,,\label{eq715} \end{equation}$$ and $$\begin{equation} \mathbf{a}_{c}^{\left(1\right)}=\frac{1}{J_{\eta}^{\left(1\right)}}\mathbf{n}^{\left(1\right)}\times\left(\mathbf{g}_{2}^{\left(1\right)}\frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}-\mathbf{g}_{1}^{\left(1\right)}\frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right)\,.\label{eq716} \end{equation}$$