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The contact integral (7.5.2-1) can be discretized as follows. First, we split the integration over all the slave surface elements. $$\begin{equation} W_{t}=\sum\limits _{e=1}^{nel}\int\limits _{\Gamma_{c}^{\left(e\right)}}\mathbf{T}\cdot\delta\mathbf{g}\,d\Gamma^{\left(e\right)}\,.\label{eq686} \end{equation}$$ The integration can be approximated by a quadrature rule, $$\begin{equation} W_{t}=\sum\limits _{e=1}^{nel}\sum\limits _{i=1}^{N_{int}^{\left(e\right)}}w^{i}j\left(\xi_{i}\right)\mathbf{T}\left(\xi_{i}\right)\cdot\delta\mathbf{g}\left(\xi_{i}\right)\,.\label{eq687} \end{equation}$$ If we use a nodally integrated elements, we have $$\begin{equation} \begin{aligned}\mathbf{w}^{\left(1\right)}\left(\xi_{i}\right) & =\mathbf{c}_{i}^{\left(1\right)}\,,\\ \mathbf{w}^{\left(2\right)}\left(\xi_{i}\right) & =\sum\limits _{j}N_{j}\left(\bar{\xi}_{i}\right)\mathbf{c}_{j}^{\left(2\right)}\,, \end{aligned} \label{eq688} \end{equation}$$ so that, $$\begin{equation} \delta\mathbf{g}\left(\xi_{i}\right)=\mathbf{c}_{i}^{\left(1\right)}-\sum\limits _{j}N_{j}\left(\bar{\xi}_{i}\right)\mathbf{c}_{j}^{\left(2\right)}\,.\label{eq689} \end{equation}$$ We can now write the contact integral (7.5.4-1) in its final form, $$\begin{equation} W_{t}=\sum\limits _{e=1}^{nel}\sum\limits _{i=1}^{N_{int}^{\left(e\right)}}w^{i}j\left(\xi_{i}\right)\left(\mathbf{N}\left(\xi_{i}\right)\mathbf{T}\left(\xi_{i}\right)\right)\cdot\delta\boldsymbol{\Phi}\,,\label{eq690} \end{equation}$$ where $$\begin{equation} \delta\boldsymbol{\Phi}^{T}\left(\xi_{i}\right)=\left[\begin{array}{ccccc} \mathbf{c}_{i}^{\left(1\right)} & \mathbf{c}_{1}^{\left(2\right)} & \mathbf{c}_{2}^{\left(2\right)} & \cdots & \mathbf{c}_{n}^{\left(2\right)}\end{array}\right]\,,\label{eq691} \end{equation}$$ $$\begin{equation} \mathbf{N}\left(\xi_{i}\right)=\left[\begin{array}{cccc} \mathbf{I} & -\mathbf{N}_{1} & \cdots & -\mathbf{N}_{n}\end{array}\right]\,,\label{eq692} \end{equation}$$ and $$\begin{equation} \mathbf{N}_{i}=\left[\begin{array}{ccc} N_{i} & 0 & 0\\ 0 & N_{i} & 0\\ 0 & 0 & N_{i} \end{array}\right]\,.\label{eq693} \end{equation}$$ For the linearized tied contact integral (7.5.3-1), a similar discretization procedure leads to, $$\begin{equation} \Delta W_{t}=\sum\limits _{e=1}^{nel}\sum\limits _{i=1}^{N_{int}^{\left(e\right)}}w^{i}j\left(\xi_{i}\right)\Delta\boldsymbol{\Phi}\cdot\mathbf{K}_{c}\delta\boldsymbol{\Phi}\,,\label{eq694} \end{equation}$$ where $$\begin{equation} \mathbf{K}_{c}=\varepsilon\mathbf{N}^{T}\mathbf{N}\,.\label{eq695} \end{equation}$$