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

The contact integral may be discretized as $$\begin{equation} \delta G_{c}=\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left[\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}^{\tau}-\left(\delta J^{\left(1\right)}-\delta J^{\left(2\right)}\right)v_{n}\right]\,.\label{eq:TFI-discrete-1} \end{equation}$$ The variables may be interpolated over each element face according to $$\begin{equation} \begin{aligned}\delta\mathbf{v}^{\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\mathbf{v}_{a}^{\left(1\right)} & \delta\mathbf{v}^{\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\mathbf{v}_{b}^{\left(2\right)}\\ \Delta\mathbf{v}^{\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\mathbf{v}_{c}^{\left(1\right)} & \Delta\mathbf{v}^{\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\mathbf{v}_{d}^{\left(2\right)}\\ \delta J^{\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta J_{a}^{\left(1\right)} & \delta J^{\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta J_{b}^{\left(2\right)}\\ \Delta J^{\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta J_{c}^{\left(1\right)} & \Delta J^{\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta J_{d}^{\left(2\right)} \end{aligned} \,.\label{eq:TFI-discrete-2} \end{equation}$$ Then, $$\begin{equation} \begin{aligned}\delta G_{c} & =\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left(\sum\limits _{a=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta J_{a}^{\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{a}^{\left(1\right)}\\ v_{a}^{\left(1\right)} \end{array}\right]\right.\\ & \left.+\sum\limits _{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{b,k}^{\left(1\right)} & \delta J_{b,k}^{\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{b,k}^{\left(1\right)}\\ v_{b,k}^{\left(1\right)} \end{array}\right]\right) \end{aligned} \,,\label{eq:TFI-discrete-3} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{f}_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}\mathbf{t}^{\tau} & \mathbf{f}_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}\mathbf{t}^{\tau}\\ w_{a}^{\left(1\right)} & =-N_{a}^{\left(1\right)}v_{n} & w_{b,k}^{\left(2\right)} & =N_{b}^{\left(2\right)}v_{n} \end{aligned} \,.\label{eq:TFI-discrete-4} \end{equation}$$ Similarly, $$\begin{equation} \begin{aligned}-D\delta G_{c} & =\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\\ & \times\left(\sum\limits _{a=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta J_{a}^{\left(1\right)}\end{array}\right]\cdot\left(\sum\limits _{c=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \mathbf{K}_{ac}^{\left(1,1\right)} & \mathbf{0}\\ \mathbf{0} & k_{ac}^{\left(1,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{v}_{c}^{\left(1\right)}\\ \Delta J_{c}^{\left(1\right)} \end{array}\right]\right.\right.\\ & +\left.\sum\limits _{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \mathbf{K}_{ad,k}^{\left(1,2\right)} & \mathbf{0}\\ \mathbf{0} & k_{ad,k}^{\left(1,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{v}_{d}^{\left(2\right)}\\ \Delta J_{d}^{\left(2\right)} \end{array}\right]\right)\\ & +\sum\limits _{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{b,k}^{\left(2\right)} & \delta J_{b,k}^{\left(2\right)}\end{array}\right]\cdot\left(\sum\limits _{c=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \mathbf{K}_{bc,k}^{\left(2,1\right)} & \mathbf{0}\\ \mathbf{0} & k_{bc,k}^{\left(2,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{v}_{c}^{\left(1\right)}\\ \Delta J_{c}^{\left(1\right)} \end{array}\right]\right.\\ & +\left.\left.\sum\limits _{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \mathbf{K}_{bd,k}^{\left(2,2\right)} & \mathbf{0}\\ \mathbf{0} & k_{bd,k}^{\left(2,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{v}_{d}^{\left(2\right)}\\ \Delta J_{d}^{\left(2\right)} \end{array}\right]\right)\right)\,, \end{aligned} \label{eq:TFI-discrete-5} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{K}_{ac}^{\left(1,1\right)} & =\varepsilon_{t}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\left(\mathbf{I}-\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\right)\\ \mathbf{K}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{t}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\left(\mathbf{I}-\mathbf{n}^{\left(2\right)}\otimes\mathbf{n}^{\left(2\right)}\right)\\ \mathbf{K}_{bc,k}^{\left(2,1\right)} & =-\varepsilon_{t}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\left(\mathbf{I}-\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\right)\\ \mathbf{K}_{bd,k}^{\left(2,2\right)} & =\varepsilon_{t}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)}\left(\mathbf{I}-\mathbf{n}^{\left(2\right)}\otimes\mathbf{n}^{\left(2\right)}\right) \end{aligned} \,,\label{eq:TFI-discrete-6} \end{equation}$$ $$\begin{equation} \begin{aligned}k_{ac}^{\left(1,1\right)} & =-K\varepsilon_{n}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\\ k_{ad,k}^{\left(1,2\right)} & =K\varepsilon_{n}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ k_{bc,k}^{\left(2,1\right)} & =K\varepsilon_{n}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\\ k_{bd,k}^{\left(2,2\right)} & =-K\varepsilon_{n}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \,.\label{eq:TFI-discrete-7} \end{equation}$$