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The vector gap function $\mathbf{g}$ , representing the distance between the contact surfaces, is defined by $$\begin{equation} \mathbf{g}=\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}\,.\label{eq701} \end{equation}$$ The premise of a tied interface is that the parametric coordinates of $\mathbf{x}^{\left(1\right)}$ and $\mathbf{x}^{\left(2\right)}$ are both invariants (i.e., they are determined in the reference configuration and remain unchanged over time). The parametric coordinates of $\mathbf{x}^{\left(1\right)}$ correspond to the integration points on $\gamma^{\left(1\right)}$ , and those of $\mathbf{x}^{\left(2\right)}$ are evaluated once, in the reference configuration, by shooting a ray from the integration point on $\gamma^{\left(1\right)}$ to intersect $\gamma^{\left(2\right)}$ . It follows from this premise that $$\begin{equation} \begin{aligned}D\mathbf{x}^{\left(1\right)} & =\Delta\mathbf{u}^{\left(1\right)} & D\mathbf{x}^{\left(2\right)} & =\Delta\mathbf{u}^{\left(2\right)}\\ Dp^{\left(1\right)} & =\Delta p^{\left(1\right)} & Dp^{\left(2\right)} & =\Delta p^{\left(2\right)}\\ D\delta\mathbf{v}^{\left(1\right)} & =\mathbf{0} & D\delta\mathbf{v}^{\left(2\right)} & =\mathbf{0}\\ D\delta p^{\left(1\right)} & =0 & D\delta p^{\left(2\right)} & =0 \end{aligned} \,.\label{eq702} \end{equation}$$ If $\gamma^{\left(1\right)}$ and $\gamma^{\left(2\right)}$ are not initially conforming, continuity of fluid pressure and normal flux will only be enforced within the contact interface; unlike the sliding biphasic contact interface (Section 7.2↑), free-draining conditions are not set automatically on regions of $\gamma^{\left(1\right)}$ and $\gamma^{\left(2\right)}$ where a solution for $\mathbf{g}$ was not found. Therefore, these regions naturally enforce zero fluid flux (impermeable boundary), unless an explicit boundary condition on the pressure $p$ is prescribed over those regions.