7.8 Tied Fluid Interface
Under isothermal conditions which preclude any heat flux and for which the entropy is zero, the interface jump conditions derived from mass, linear momentum and energy balance across an interface \(\Gamma\) which is permeable to the fluid are
\[ \begin{equation} \left[\left[\rho\mathbf{u}^{\Gamma}\right]\right]\cdot\mathbf{n}^{\Gamma}=0\,,\label{eq:jump-mass} \end{equation} \]
\[ \begin{equation} \left[\left[\boldsymbol{\sigma}-\rho\mathbf{u}^{\Gamma}\otimes\mathbf{u}^{\Gamma}\right]\right]\cdot\mathbf{n}^{\Gamma}=\mathbf{0}\,,\label{eq:jump-linear-momentum} \end{equation} \]
and
\[ \begin{equation} \left[\left[\rho\left(\psi+\frac{1}{2}\mathbf{u}^{\Gamma}\cdot\mathbf{u}^{\Gamma}\right)\mathbf{u}^{\Gamma}-\boldsymbol{\sigma}^{T}\cdot\mathbf{u}^{\Gamma}\right]\right]\cdot\mathbf{n}^{\Gamma}=0\label{eq:jump-energy} \end{equation} \]
where
\[ \begin{equation} \mathbf{u}^{\Gamma}\equiv\mathbf{v}-\mathbf{v}^{\Gamma}\label{eq:Gamma-diffusive-velocity} \end{equation} \]
\(\rho\), \(\mathbf{v}\), \(\boldsymbol{\sigma}\) and \(\psi\) are the mass density, velocity, stress tensor and specific free energy of the fluid, respectively, and \(\mathbf{v}^{\Gamma}\) and \(\mathbf{n}^{\Gamma}\) are the velocity of, and outward unit normal to \(\Gamma\), respectively. A tied fluid interface \(\Gamma\), which allows fluid to cross \(\Gamma\), must enforce these continuity requirements. For fluid analyses, we let
\[ \boldsymbol{\sigma}=-p\mathbf{I}+\boldsymbol{\tau} \]
where \(p\) is the fluid pressure and \(\boldsymbol{\tau}\) is the viscous shear stress. Susbtituting this relation into the energy jump of eq.\eqref{eq:jump-energy}, neglecting the contribution of \(\boldsymbol{\tau}\) relative to that of \(p\), and making use of the jump condition from mass balance in eq.\eqref{eq:jump-mass}, we find that
\[ \begin{equation} \left[\left[\psi+\frac{p}{\rho}+\frac{1}{2}\mathbf{u}^{\Gamma}\cdot\mathbf{u}^{\Gamma}\right]\right]=0\,.\label{eq:jump-energy-redux} \end{equation} \]
Since \(\psi\) and \(p\) are functions of state that only depend on the fluid volume ratio \(J\) (in an isothermal analysis, see eq.(2.13-9)), letting \(J\) be continuous across \(\Gamma\) implies from eq.(2.13-7) that \(\rho\) is continuous across \(\Gamma\); thus, from eq.(2.13-7) we conclude that \(\mathbf{u}^{\Gamma}\cdot\mathbf{n}\) is continuous, hence eq.\eqref{eq:jump-energy-redux} is satisfied. In other words, letting \(\left[\left[J\right]\right]=0\) is a sufficient (but not necessary) condition to satisfy the jump conditions from mass and energy balance, as long as we also enforce \(\left[\left[\mathbf{u}^{\Gamma}\right]\right]\cdot\mathbf{n}=0\). For a viscous fluid, we also assume that the no-slip condition holds, implying that our jump requirement for the fluid velocity is even more stringent, namely \(\left[\left[\mathbf{u}^{\Gamma}\right]\right]=\mathbf{0}\) (or equivalently, \(\left[\left[\mathbf{v}\right]\right]=\mathbf{0}\)). According to the above relations between the stress and pressure, and recognizing that \(\left[\left[J\right]\right]=0\) implies \(\left[\left[p\right]\right]=0\), we may also simplify the linear momentum jump condition of eq.\eqref{eq:jump-linear-momentum} to \(\left[\left[\boldsymbol{\tau}\right]\right]\cdot\mathbf{n}^{\Gamma}=\mathbf{0}\). Therefore, the jump conditions that we need to enforce for a tied fluid interface \(\Gamma\) are
\[ \begin{equation} \begin{aligned}\left[\left[J\right]\right] & =0\\ \left[\left[\mathbf{v}\right]\right] & =\mathbf{0}\\ \left[\left[\boldsymbol{\tau}\right]\right]\cdot\mathbf{n}^{\Gamma} & =\mathbf{0} \end{aligned} \,.\label{eq:jump-tied-fluid} \end{equation} \]
These jump conditions are consistent with the choice of nodal degrees of freedom for fluid analyses (namely, \(\mathbf{v}\) and \(J\)). On a tied (or sliding) fluid interface, we don't have continuity of the mesh, therefore we can only enforce continuity of \(J\) and \(\mathbf{v}\) by enforcing continuity of the fluid pressure \(p\) and viscous traction \(\mathbf{t}^{\tau}\equiv\boldsymbol{\tau}\cdot\mathbf{n}^{\Gamma}\). However, in our fluid finite element formulation, \(p\) is not a natural boundary condition since it is calculated directly from \(J\). Instead, as explained in Section Computational Fluid Dynamics, the natural boundary condition for the fluid dilatation governing equation is continuity of \(v_{n}\) across \(\Gamma\), thus \(\left[\left[\mathbf{v}\right]\right]\cdot\mathbf{n}^{\Gamma}=0\).
See Section Fluid Mechanics for a review of fluid materials, and for additional details on the FEBio fluid solver. The tied fluid interface is defined between surfaces \(\gamma^{\left(1\right)}\) and \(\gamma^{\left(2\right)}\). Due to continuity requirements on the viscous traction and normal fluid velocity, the external virtual work resulting from tractions \(\mathbf{t}^{\tau\left(i\right)}\) and normal velocities \(v_{n}^{\left(i\right)}\) (\(i=1,2)\) may be combined into the tied interface integral
\[ \begin{equation} \begin{aligned}\delta G_{c} & =\int_{\gamma^{\left(1\right)}}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}^{\tau\left(1\right)}da^{\left(1\right)}\\ & -\int_{\gamma^{\left(1\right)}}\left(\delta J^{\left(1\right)}-\delta J^{\left(2\right)}\right)v_{n}^{\left(1\right)}da^{\left(1\right)}\,. \end{aligned} \label{eq696-2} \end{equation} \]
To evaluate and linearize \(\delta G_{c}\), define the covariant basis vectors on each surface as
\[ \begin{equation} \mathbf{g}_{\alpha}^{\left(i\right)}=\frac{\partial\mathbf{x}^{\left(i\right)}}{\partial\eta_{\left(i\right)}^{\alpha}},\quad\alpha=1,2,\label{eq697-2} \end{equation} \]
where \(\mathbf{x}^{\left(i\right)}\) represents the spatial position of points on \(\gamma^{\left(i\right)}\), and \(\eta_{\left(i\right)}^{\alpha}\) represent the parametric coordinates of that point. The unit outward normal on each surface is then given by
\[ \begin{equation} \mathbf{n}^{\left(i\right)}=\frac{\mathbf{g}_{1}^{\left(i\right)}\times\mathbf{g}_{2}^{\left(i\right)}}{\left|\mathbf{g}_{1}^{\left(i\right)}\times\mathbf{g}_{2}^{\left(i\right)}\right|}\,.\label{eq698-2} \end{equation} \]
Now the contact integral may be rewritten as
\[ \begin{equation} \begin{aligned}\delta G_{c} & =\int_{\gamma^{\left(1\right)}}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}^{\tau}\,\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\,d\eta_{\left(1\right)}^{1}d\eta_{\left(1\right)}^{2}\\ & +\int_{\gamma^{\left(1\right)}}\left(\delta J^{\left(1\right)}-\delta J^{\left(2\right)}\right)v_{n}\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\,d\eta_{\left(1\right)}^{1}d\eta_{\left(1\right)}^{2} \end{aligned} \,,\label{eq699-2} \end{equation} \]
where \(\mathbf{t}^{\tau}\equiv\mathbf{t}^{\tau\left(1\right)}\) and \(v_{n}\equiv v_{n}^{\left(1\right)}\). The linearization \(D\delta G_{c}\) of \(\delta G_{c}\) has the form
\[ \begin{equation} D\delta G_{c}=\sum\limits_{i=1}^{2}D\delta G_{c}\left[\Delta\mathbf{v}^{\left(i\right)}\right]+D\delta G_{c}\left[\Delta J^{\left(i\right)}\right]\,.\label{eq700-2} \end{equation} \]
Gap Functions
The premise of a tied interface is that the parametric coordinates of coincident points \(\mathbf{x}^{\left(1\right)}\) and \(\mathbf{x}^{\left(2\right)}\) on \(\gamma^{\left(1\right)}\) and \(\gamma^{\left(2\right)}\) 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 by shooting a ray from the integration point on \(\gamma^{\left(1\right)}\) to intersect \(\gamma^{\left(2\right)}\).
The vector gap function \(\mathbf{g}\), representing the difference between velocities across the interface, is defined by
\[ \begin{equation} \mathbf{g}=\mathbf{v}^{\left(2\right)}-\mathbf{v}^{\left(1\right)}\,,\label{TFI-vt-gap} \end{equation} \]
We may similarly define the scalar gap function \(\pi\), representing the difference between fluid dilatations (or volume ratios) across the interface,
\[ \begin{equation} \pi=J^{\left(2\right)}-J^{\left(1\right)}\,.\label{eq:TFI-e-gap} \end{equation} \]
Penalty Method
Let the tied interface viscous traction be described by the penalty function,
\[ \begin{equation} \mathbf{t}^{\tau}=\varepsilon_{t}\mathbf{g}\,,\label{eq:TFI-t-penalty} \end{equation} \]
where \(\varepsilon_{t}\) is a penalty factor associated with \(\mathbf{t}^{\tau}\). The penalty factor \(\varepsilon_{t}\) is expressed in units of viscosity per length. Therefore, in an auto-penalty scheme, it may be scaled to the ratio of fluid viscosity to element thickness. Similarly, let
\[ \begin{equation} v_{n}=\varepsilon_{n}\pi=\varepsilon_{n}\left(J^{\left(2\right)}-J^{\left(1\right)}\right)\,,\label{eq:TFI-v-penalty} \end{equation} \]
where \(\varepsilon_{n}\) is a penalty factor associated with \(v_{n}\). Note that \(\varepsilon_{n}\) has units of velocity. It follows that
\[ \begin{equation} \begin{aligned}D\mathbf{t}^{\tau} & =\varepsilon_{t}\left(\Delta\mathbf{v}^{\left(2\right)}-\Delta\mathbf{v}^{\left(1\right)}\right)\,,\\ Dv_{n} & =\varepsilon_{n}\left(\Delta J^{\left(2\right)}-\Delta J^{\left(1\right)}\right)\,. \end{aligned} \label{eq:TFI-penalty-linearization} \end{equation} \]
Given these relations, it can be shown that the directional derivatives of the various terms appearing in the integrand of \(\delta G_{c}\) are
\[ \begin{equation} \begin{aligned} & -D\left(J_{\eta}^{\left(1\right)}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}^{\tau}\right)=\\ & \varepsilon_{t}J_{\eta}^{\left(1\right)}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\left(\Delta\mathbf{v}^{\left(1\right)}-\Delta\mathbf{v}^{\left(2\right)}\right) \end{aligned} \,,\label{eq:TFI-G-linearization-v} \end{equation} \]
\[ \begin{equation} D\left(v_{n}\left(\delta J^{\left(1\right)}-\delta J^{\left(2\right)}\right)J_{\eta}^{\left(1\right)}\right)=-\varepsilon_{n}J_{\eta}^{\left(1\right)}\left(\delta J^{\left(1\right)}-\delta J^{\left(2\right)}\right)\left(\Delta J^{\left(1\right)}-\Delta J^{\left(2\right)}\right)\,,\label{eq:TFI-G-linearization-e} \end{equation} \]
where \(J_{\eta}^{\left(1\right)}=\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\).
Discretization
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
\[ \begin{equation} \begin{aligned}\mathbf{K}_{ac}^{\left(1,1\right)} & =\varepsilon_{t}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\left(\mathbf{I}-\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\right)\\ \mathbf{K}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{t}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\left(\mathbf{I}-\mathbf{n}^{\left(2\right)}\otimes\mathbf{n}^{\left(2\right)}\right)\\ \mathbf{K}_{bc,k}^{\left(2,1\right)} & =-\varepsilon_{t}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\left(\mathbf{I}-\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\right)\\ \mathbf{K}_{bd,k}^{\left(2,2\right)} & =\varepsilon_{t}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)}\left(\mathbf{I}-\mathbf{n}^{\left(2\right)}\otimes\mathbf{n}^{\left(2\right)}\right) \end{aligned} \,,\label{eq:TFI-discrete-6} \end{equation} \]
\[ \begin{equation} \begin{aligned}k_{ac}^{\left(1,1\right)} & =-K\varepsilon_{n}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\\ k_{ad,k}^{\left(1,2\right)} & =K\varepsilon_{n}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ k_{bc,k}^{\left(2,1\right)} & =K\varepsilon_{n}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\\ k_{bd,k}^{\left(2,2\right)} & =-K\varepsilon_{n}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \,.\label{eq:TFI-discrete-7} \end{equation} \]