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

The contact integral may be discretized as $$\begin{equation} \delta G_{c}=\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left[t_{n}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{n}^{\left(1\right)}+w_{n}\left(\delta\tilde{p}^{\left(1\right)}-\delta\tilde{p}^{\left(2\right)}\right)+j_{n}\left(\delta\tilde{c}^{\left(1\right)}-\delta\tilde{c}^{\left(2\right)}\right)\right]\,.\label{eq640} \end{equation}$$ The variables may be interpolated over each element face according to $$\begin{equation} \begin{aligned}\delta\mathbf{v}^{\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\mathbf{v}_{a}^{\left(1\right)} & \delta\mathbf{v}^{\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\mathbf{v}_{b}^{\left(2\right)}\\ \Delta\mathbf{u}^{\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\mathbf{u}_{c}^{\left(1\right)} & \Delta\mathbf{u}^{\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\mathbf{u}_{d}^{\left(2\right)}\\ \delta\tilde{p}^{\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\tilde{p}_{a}^{\left(1\right)} & \delta\tilde{p}^{\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\tilde{p}_{b}^{\left(2\right)}\\ \Delta\tilde{p}^{\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\tilde{p}_{c}^{\left(1\right)} & \Delta\tilde{p}^{\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\tilde{p}_{d}^{\left(2\right)}\\ \delta\tilde{c}^{\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\tilde{c}_{a}^{\left(1\right)} & \delta\tilde{c}^{\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\tilde{c}_{b}^{\left(2\right)}\\ \Delta\tilde{c}^{\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\tilde{c}_{c}^{\left(1\right)} & \Delta\tilde{c}^{\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\tilde{c}_{d}^{\left(2\right)} \end{aligned} \,.\label{eq641} \end{equation}$$ Then, $$\begin{equation} \begin{aligned}\delta G_{c} & =\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left(\sum\limits _{a=1}^{m^{\left(1\right)}}\left[\begin{array}{ccc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta\tilde{p}_{a}^{\left(1\right)} & \delta\tilde{c}_{a}^{\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{a}^{\left(1\right)}\\ w_{a}^{\left(1\right)}\\ j_{a}^{\left(1\right)} \end{array}\right]\right.\\ & \left.+\sum\limits _{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{ccc} \delta\mathbf{v}_{b,k}^{\left(1\right)} & \delta\tilde{p}_{b,k}^{\left(1\right)} & \delta\tilde{c}_{b,k}^{\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{b,k}^{\left(1\right)}\\ w_{b,k}^{\left(1\right)}\\ j_{b,k}^{\left(1\right)} \end{array}\right]\right)\,, \end{aligned} \label{eq642} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{f}_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}t_{n}\mathbf{n}^{\left(1\right)} & \mathbf{f}_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}t_{n}\mathbf{n}^{\left(1\right)}\\ w_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}w_{n} & w_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}w_{n}\\ j_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}j_{n} & j_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}j_{n} \end{aligned} \,.\label{eq643} \end{equation}$$ 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}{ccc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta\tilde{p}_{a}^{\left(1\right)} & \delta\tilde{c}_{a}^{\left(1\right)}\end{array}\right]\cdot\left(\sum\limits _{c=1}^{m^{\left(1\right)}}\left[\begin{array}{ccc} \mathbf{K}_{ac}^{\left(1,1\right)} & \mathbf{0} & \mathbf{0}\\ \mathbf{g}_{ac}^{\left(1,1\right)} & g_{ac}^{\left(1,1\right)} & 0\\ \mathbf{h}_{ac}^{\left(1,1\right)} & 0 & h_{ac}^{\left(1,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{c}^{\left(1\right)}\\ \Delta\tilde{p}_{c}^{\left(1\right)}\\ \Delta\tilde{c}_{c}^{\left(1\right)} \end{array}\right]\right.\right.\\ & +\left.\sum\limits _{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{ccc} \mathbf{K}_{ad,k}^{\left(1,2\right)} & \mathbf{0} & \mathbf{0}\\ \mathbf{g}_{ad,k}^{\left(1,2\right)} & g_{ad,k}^{\left(1,2\right)} & 0\\ \mathbf{h}_{ad,k}^{\left(1,2\right)} & 0 & h_{ad,k}^{\left(1,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{d}^{\left(2\right)}\\ \Delta\tilde{p}_{d}^{\left(2\right)}\\ \Delta\tilde{c}_{d}^{\left(2\right)} \end{array}\right]\right)\\ & +\sum\limits _{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{ccc} \delta\mathbf{v}_{b,k}^{\left(2\right)} & \delta\tilde{p}_{b,k}^{\left(2\right)} & \delta\tilde{c}_{b,k}^{\left(2\right)}\end{array}\right]\cdot\left(\sum\limits _{c=1}^{m^{\left(1\right)}}\left[\begin{array}{ccc} \mathbf{K}_{bc,k}^{\left({2,1}\right)} & \mathbf{0} & \mathbf{0}\\ \mathbf{g}_{bc,k}^{\left(2,1\right)} & g_{bc,k}^{\left(2,1\right)} & 0\\ \mathbf{h}_{bc,k}^{\left(2,1\right)} & 0 & h_{bc,k}^{\left(2,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{c}^{\left(1\right)}\\ \Delta\tilde{p}_{c}^{\left(1\right)}\\ \Delta\tilde{c}_{c}^{\left(1\right)} \end{array}\right]\right.\\ & +\left.\left.\sum\limits _{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{ccc} \mathbf{K}_{bd,k}^{\left(2,2\right)} & \mathbf{0} & \mathbf{0}\\ \mathbf{g}_{bd,k}^{\left(2,2\right)} & g_{bd,k}^{\left(2,2\right)} & 0\\ \mathbf{h}_{bd,k}^{\left(2,2\right)} & 0 & h_{bd,k}^{\left(2,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{d}^{\left(2\right)}\\ \Delta\tilde{p}_{d}^{\left(2\right)}\\ \Delta\tilde{c}_{d}^{\left(2\right)} \end{array}\right]\right)\right)\,, \end{aligned} \label{eq644} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{K}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}\left(\varepsilon_{n}N_{c}^{\left(1\right)}\mathbf{N}^{\left(1\right)}+t_{n}\mathbf{A}_{c}^{\left(1\right)}\right)\\ \mathbf{K}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{n}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\mathbf{N}^{\left(1\right)}\\ \mathbf{K}_{bc,k}^{\left(2,1\right)} & =-N_{c}^{\left(1\right)}\left(\varepsilon_{n}N_{b}^{\left(2\right)}\mathbf{N}^{\left(1\right)}+t_{n}\mathbf{M}_{b}^{\left(2\right)}\right)-t_{n}N_{b}^{\left(2\right)}\mathbf{A}_{c}^{\left(1\right)}\\ \mathbf{K}_{bd,k}^{\left(2,2\right)} & =N_{d}^{\left(2\right)}\left(\varepsilon_{n}N_{b}^{\left(2\right)}\mathbf{N}^{\left(1\right)}+t_{n}\mathbf{M}_{b}^{\left(2\right)}\right) \end{aligned} \,,\label{eq645} \end{equation}$$ $$\begin{equation} \begin{aligned}\mathbf{g}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}\left(\varepsilon_{p}N_{c}^{\left(1\right)}\mathbf{p}^{\left(1\right)}-w_{n}\mathbf{A}_{c}^{\left(1\right)}\cdot\mathbf{n}^{\left(1\right)}\right)\\ \mathbf{g}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{p}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\mathbf{p}^{\left(1\right)}\\ \mathbf{g}_{bc,k}^{\left(2,1\right)} & =N_{c}^{\left(1\right)}\left(-\varepsilon_{p}N_{b}^{\left(2\right)}\mathbf{p}^{\left(1\right)}+w_{n}\mathbf{m}_{b}^{\left(2\right)}\right)+w_{n}N_{b}^{\left(2\right)}\mathbf{A}_{c}^{\left(1\right)}\cdot\mathbf{n}^{\left(1\right)}\\ \mathbf{g}_{bd,k}^{\left(2,2\right)} & =N_{d}^{\left(2\right)}\left(\varepsilon_{p}N_{b}^{\left(2\right)}\mathbf{p}^{\left(1\right)}-w_{n}\mathbf{m}_{b}^{\left(2\right)}\right) \end{aligned} \,,\label{eq646} \end{equation}$$ $$\begin{equation} \begin{aligned}g_{ac}^{\left(1,1\right)} & =-\varepsilon_{p}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\\ g_{ad,k}^{\left(1,2\right)} & =\varepsilon_{p}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ g_{bc,k}^{\left(2,1\right)} & =\varepsilon_{p}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\\ g_{bd,k}^{\left(2,2\right)} & =-\varepsilon_{p}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \,,\label{eq647} \end{equation}$$ $$\begin{equation} \begin{aligned}\mathbf{h}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}\left(\varepsilon_{c}N_{c}^{\left(1\right)}\mathbf{q}^{\left(1\right)}-j_{n}\mathbf{A}_{c}^{\left(1\right)}\cdot\mathbf{n}^{\left(1\right)}\right)\\ \mathbf{h}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{c}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\mathbf{q}^{\left(1\right)}\\ \mathbf{h}_{bc,k}^{\left(2,1\right)} & =N_{c}^{\left(1\right)}\left(-\varepsilon_{c}N_{b}^{\left(2\right)}\mathbf{q}^{\left(1\right)}+j_{n}\mathbf{m}_{b}^{\left(2\right)}\right)+j_{n}N_{b}^{\left(2\right)}\mathbf{A}_{c}^{\left(1\right)}\cdot\mathbf{n}^{\left(1\right)}\\ \mathbf{h}_{bd,k}^{\left(2,2\right)} & =N_{d}^{\left(2\right)}\left(\varepsilon_{c}N_{b}^{\left(2\right)}\mathbf{q}^{\left(1\right)}-j_{n}\mathbf{m}_{b}^{\left(2\right)}\right) \end{aligned} \,,\label{eq648} \end{equation}$$ $$\begin{equation} \begin{aligned}h_{ac}^{\left(1,1\right)} & =-\varepsilon_{c}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\\ h_{ad,k}^{\left(1,2\right)} & =\varepsilon_{c}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ h_{bc,k}^{\left(2,1\right)} & =\varepsilon_{c}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\\ h_{bd,k}^{\left(2,2\right)} & =-\varepsilon_{c}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \,,\label{eq649} \end{equation}$$ and $$\begin{equation} \begin{aligned}\mathbf{N}^{\left(1\right)} & =\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\\ \mathbf{A}_{c}^{\left(1\right)} & =\frac{1}{J_{\eta}^{\left(1\right)}}\boldsymbol{\mathcal{A}}\left\{ \frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}\mathbf{g}_{2}^{\left(1\right)}-\frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\mathbf{g}_{1}^{\left(1\right)}\right\} \\ \mathbf{M}_{b}^{\left(2\right)} & =\mathbf{n}^{\left(2\right)}\otimes\mathbf{m}_{b}^{\left(2\right)}\\ \mathbf{m}_{b}^{\left(2\right)} & =\frac{\partial N_{b}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\mathbf{g}_{\left(2\right)}^{\alpha}\\ \mathbf{p}^{\left(1\right)} & =\frac{\partial\tilde{p}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{\alpha}}\mathbf{g}_{\left(1\right)}^{\alpha}\\ \mathbf{q}^{\left(1\right)} & =\frac{\partial\tilde{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{\alpha}}\mathbf{g}_{\left(1\right)}^{\alpha} \end{aligned} \,.\label{eq650} \end{equation}$$