3.6 Weak Formulation for FSI

General Formulation

The strong form of the virtual work statement is \(\delta W=0\) where

\[ \begin{equation} \begin{aligned}\delta W & =\int_{b}\delta\mathbf{v}^{s}\cdot\divg\boldsymbol{\sigma}^{s}\,dv\\ & +\int_{b}\delta\mathbf{w}\cdot\left[\divg\boldsymbol{\sigma}^{f}+\rho^{f}\left(\mathbf{b}-\mathbf{a}^{f}\right)\right]\,dv\\ & -\int_{b}\delta J^{f}\left[\frac{1}{J^{f}}\left(\dot{J}^{f}+\grad J^{f}\cdot\mathbf{w}\right)-\divg\mathbf{w}-\frac{\dot{J}^{s}}{J^{s}}\right]\,dv \end{aligned} \,.\label{eq:fsi-virtual-work-strong} \end{equation} \]

Here, \(b\) is the mixture domain (whose boundaries are defined on the solid), \(\delta\mathbf{v}^{s}\) is the virtual solid velocity, \(\delta\mathbf{w}\) is the virtual relative fluid velocity, and \(\delta J^{f}\) is the virtual energy density. The weak form of this statement may be written as the difference \(\delta W=\delta W_{ext}-\delta W_{int}\) between external virtual work \(\delta W_{ext}\) and internal virtual work \(\delta W_{int}\), where

\[ \begin{equation} \begin{aligned}\delta W_{int} & =\int_{b}\boldsymbol{\sigma}^{s}:\grad\delta\mathbf{v}^{s}\,dv\\ & +\int_{b}\boldsymbol{\tau}:\grad\delta\mathbf{w}\,dv+\int_{b}\delta\mathbf{w}\cdot\left(\grad p+\rho^{f}\mathbf{a}^{f}\right)\,dv\\ & +\int_{b}\delta J^{f}\left[\frac{1}{J^{f}}\left(\dot{J}^{f}+\grad J^{f}\cdot\mathbf{w}\right)-\frac{\dot{J}^{s}}{J^{s}}\right]\,dv\\ & +\int_{b}\mathbf{w}\cdot\grad\delta J^{f}\,dv \end{aligned} \,,\label{eq:fsi-internal-virtual-work} \end{equation} \]

and

\[ \begin{equation} \begin{aligned}\delta W_{ext} & =\int_{\partial b}\delta\mathbf{v}^{s}\cdot\mathbf{t}^{s}\,da\\ & +\int_{\partial b}\delta\mathbf{w}\cdot\mathbf{t}^{\tau}\,da+\int_{b}\delta\mathbf{w}\cdot\rho^{f}\mathbf{b}\,dv\\ & +\int_{\partial b}\delta J^{f}w_{n}\,da \end{aligned} \,.\label{eq:fsi-external-virtual-work} \end{equation} \]

Here, \(\mathbf{t}^{s}=\boldsymbol{\sigma}^{s}\cdot\mathbf{n}\) is the solid traction, \(\mathbf{t}^{\tau}=\boldsymbol{\tau}\cdot\mathbf{n}\) is the fluid viscous traction, and \(w_{n}=\mathbf{w}_{n}\cdot\mathbf{n}\) is the normal component of the relative fluid velocity on the boundary \(\partial b\), whose outward unit normal is \(\mathbf{n}\). In practice, both \(\boldsymbol{\sigma}^{s}\) and \(\mathbf{t}^{s}\) should contribute negligibly to \(\delta W\), but they cannot be set exactly to zero since we need some small elasticity to regularize the deforming mixture mesh.

Time Integration

In the generalized \(\alpha-\)method we evaluate displacements and velocities at the intermediate time \(t_{n+\alpha_{f}}=t_{n}+\alpha_{f}\left(t_{n+1}-t_{n}\right)\), such that

\[ \begin{aligned}\boldsymbol{\chi}_{n+\alpha_{f}}^{s} & =\left(1-\alpha_{f}\right)\boldsymbol{\chi}_{n}^{s}+\alpha_{f}\boldsymbol{\chi}_{n+1}^{s}\\ \mathbf{u}_{n+\alpha_{f}}^{s} & =\left(1-\alpha_{f}\right)\mathbf{u}_{n}^{s}+\alpha_{f}\mathbf{u}_{n+1}^{s}\\ \mathbf{v}_{n+\alpha_{f}}^{s} & =\left(1-\alpha_{f}\right)\mathbf{v}_{n}^{s}+\alpha_{f}\mathbf{v}_{n+1}^{s}\\ \mathbf{w}_{n+\alpha_{f}} & =\left(1-\alpha_{f}\right)\mathbf{w}_{n}+\alpha_{f}\mathbf{w}_{n+1}\\ J_{n+\alpha_{f}}^{f} & =\left(1-\alpha_{f}\right)J_{n}^{f}+\alpha_{f}J_{n+1}^{f} \end{aligned} \]

In particular, it follows that

\[ \mathbf{F}_{n+\alpha_{f}}^{s}=\frac{\partial\boldsymbol{\chi}_{n+\alpha_{f}}^{s}}{\partial\mathbf{X}}=\left(1-\alpha_{f}\right)\mathbf{F}_{n}^{s}+\alpha_{f}\mathbf{F}_{n+1}^{s} \]

\[ J_{n+\alpha_{f}}^{s}=\det\mathbf{F}_{n+\alpha_{f}}^{s} \]

\[ \dot{J}_{n+\alpha_{f}}^{s}=J_{n+\alpha_{f}}^{s}\mathbf{F}_{n+\alpha_{f}}^{-T}:\Grad\mathbf{v}_{n+\alpha_{f}}^{s} \]

\[ \mathbf{L}_{n+\alpha_{f}}^{s}=\Grad\mathbf{v}_{n+\alpha_{f}}^{s}\cdot\mathbf{F}_{n+\alpha_{f}}^{-1} \]

In practice however, we get better numerical results when using

\[ \dot{J}_{n+\alpha_{f}}^{s}=\frac{J_{n+1}^{s}-J_{n}^{s}}{\Delta t} \]

\[ \mathbf{L}_{n+\alpha_{f}}^{s}=\frac{\mathbf{F}_{n+1}-\mathbf{F}_{n}}{\Delta t}\cdot\mathbf{F}_{n+\alpha_{f}}^{-1} \]

Similarly, we evaluate velocity derivatives at the intermediate time \(t_{n+\alpha_{m}}=t_{n}+\alpha_{m}\left(t_{n+1}-t_{n}\right)\), such that

\[ \begin{aligned}\dot{\mathbf{v}}_{n+\alpha_{m}}^{s} & =\left(1-\alpha_{m}\right)\dot{\mathbf{v}}_{n}^{s}+\alpha_{m}\dot{\mathbf{v}}_{n+1}^{s}\\ \dot{\mathbf{w}}_{n+\alpha_{m}} & =\left(1-\alpha_{m}\right)\dot{\mathbf{w}}_{n}+\alpha_{m}\dot{\mathbf{w}}_{n+1}\\ \dot{J}_{n+\alpha_{m}}^{f} & =\left(1-\alpha_{m}\right)\dot{J}_{n}^{f}+\alpha_{m}\dot{J}_{n+1}^{f} \end{aligned} \]

The parameters \(\alpha_{f}\) and \(\alpha_{m}\) are evaluated from a single parameter \(\rho_{\infty}\) using

\[ \begin{equation} \alpha_{f}=\frac{1}{1+\rho_{\infty}}\,,\quad\alpha_{m}=\frac{1}{2}\frac{3-\rho_{\infty}}{1+\rho_{\infty}}\,,\label{eq:fsi-ga-alphas-1st} \end{equation} \]

for first-order systems, or

\[ \begin{equation} \alpha_{f}=\frac{1}{1+\rho_{\infty}}\,,\quad\alpha_{m}=\frac{2-\rho_{\infty}}{1+\rho_{\infty}}\,,\label{eq:fsi-ga-alphas-2nd} \end{equation} \]

for second-order systems, where \(0\le\rho_{\infty}\le1\). This parameter is the spectral radius for an infinite time step, which controls the amount of damping of high frequencies; a value of zero produces the greatest amount of damping, anihilating the highest frequency in one step, whereas a value of one preserves the highest frequency. Since the solid motion is governed by a second-order differential equation in time, we adopt the formulas for second-order systems.

To complete the integration scheme, we evaluate

\[ \begin{aligned}\beta & =\frac{1}{4}\left(1+\alpha_{m}-\alpha_{f}\right)^{2}\\ \gamma & =\frac{1}{2}+\alpha_{m}-\alpha_{f} \end{aligned} \]

then

\[ \begin{aligned}\mathbf{v}_{n+1}^{s} & =\mathbf{v}_{n}^{s}+\Delta t\left[\left(1-\gamma\right)\dot{\mathbf{v}}_{n}^{s}+\gamma\dot{\mathbf{v}}_{n+1}^{s}\right]\\ \mathbf{u}_{n+1}^{s} & =\mathbf{u}_{n}^{s}+\Delta t\mathbf{v}_{n}^{s}+\frac{\Delta t^{2}}{2}\left[\left(1-2\beta\right)\dot{\mathbf{v}}_{n}^{s}+2\beta\dot{\mathbf{v}}_{n+1}^{s}\right]\\ \mathbf{w}_{n+1} & =\mathbf{w}_{n}+\Delta t\left[\left(1-\gamma\right)\dot{\mathbf{w}}_{n}+\gamma\dot{\mathbf{w}}_{n+1}\right]\\ J_{n+1}^{f} & =J_{n}^{f}+\Delta t\left[\left(1-\gamma\right)\dot{J}_{n}^{f}+\gamma\dot{J}_{n+1}^{f}\right] \end{aligned} \]

Thus,

\[ \begin{aligned}\dot{\mathbf{v}}_{n+1}^{s} & =\frac{1}{\beta\Delta t}\left(\frac{\mathbf{u}_{n+1}^{s}-\mathbf{u}_{n}^{s}}{\Delta t}-\mathbf{v}_{n}^{s}\right)+\left(1-\frac{1}{2\beta}\right)\dot{\mathbf{v}}_{n}^{s}\\ \dot{\mathbf{w}}_{n+1} & =\frac{\mathbf{w}_{n+1}-\mathbf{w}_{n}}{\gamma\Delta t}+\left(1-\frac{1}{\gamma}\right)\dot{\mathbf{w}}_{n}=\frac{\mathbf{w}_{n+\alpha_{f}}-\mathbf{w}_{n}}{\alpha_{f}\gamma\Delta t}-\left(\frac{1}{\gamma}-1\right)\dot{\mathbf{w}}_{n}\\ \dot{J}_{n+1}^{f} & =\frac{J_{n+1}^{f}-J_{n}^{f}}{\gamma\Delta t}+\left(1-\frac{1}{\gamma}\right)\dot{J}_{n}^{f}=\frac{J_{n+\alpha_{f}}^{f}-J_{n}^{f}}{\alpha_{f}\gamma\Delta t}-\left(\frac{1}{\gamma}-1\right)\dot{J}_{n}^{f} \end{aligned} \]

According to this method ¹, the virtual work is evaluated using intermediate time step values, at \(t_{n+\alpha_{f}}\) for all parameters except \(\dot{\mathbf{v}}^{s}\), \(\dot{\mathbf{w}}\), and \(\dot{J}^{f}\), which are evaluated at \(t_{n+\alpha_{m}}\).

Discretization

The discretization of the internal work produces

\[ \begin{equation} \delta W_{int}=\sum_{a}\delta\mathbf{v}_{a}^{s}\cdot\int_{b}\mathbf{f}_{a}^{u}\,dv+\delta\mathbf{w}_{a}\cdot\int_{b}\mathbf{f}_{a}^{w}\,dv+\delta J_{a}^{f}\int_{b}f_{a}^{J}\,dv\label{eq:fsi-discretized-internal-work} \end{equation} \]

\[ \begin{aligned}\delta W_{int} & =\sum_{a}\delta\mathbf{v}_{a}^{s}\cdot\int_{b}\boldsymbol{\sigma}^{s}\cdot\grad N_{a}\,dv\\ & +\sum_{a}\delta\mathbf{w}_{a}\cdot\int_{b}\boldsymbol{\tau}\cdot\grad N_{a}\,dv+\sum_{a}\delta\mathbf{w}_{a}\cdot\int_{b}N_{a}\left(\grad p+\rho^{f}\mathbf{a}^{f}\right)\,dv\\ & +\sum_{a}\delta J_{a}^{f}\int_{b}N_{a}\left[\frac{1}{J^{f}}\left(\dot{J}^{f}+\grad J^{f}\cdot\mathbf{w}\right)-\frac{\dot{J}^{s}}{J^{s}}\right]\,dv\\ & +\sum_{a}\delta J_{a}^{f}\int_{b}\mathbf{w}\cdot\grad N_{a}\,dv \end{aligned} \]

where

\[ \begin{equation} \begin{aligned}\mathbf{f}_{a}^{u} & =\boldsymbol{\sigma}^{s}\cdot\grad N_{a}\\ \mathbf{f}_{a}^{w} & =\boldsymbol{\tau}\cdot\grad N_{a}+N_{a}\left(\grad p+\rho^{f}\mathbf{a}^{f}\right)\\ f_{a}^{J} & =N_{a}\left[\frac{1}{J^{f}}\left(\dot{J}^{f}+\grad J^{f}\cdot\mathbf{w}\right)-\frac{\dot{J}^{s}}{J^{s}}\right]+\mathbf{w}\cdot\grad N_{a} \end{aligned} \label{eq:fsi-discretized-residuals} \end{equation} \]

We used the following interpolations:

\[ \begin{equation} \begin{aligned}\delta\mathbf{v}^{s} & =\sum_{a}N_{a}\delta\mathbf{v}_{a}^{s} & \Delta\mathbf{u} & =\sum_{b}N_{b}\Delta\mathbf{u}_{b}\\ \grad\delta\mathbf{v}^{s} & =\sum_{a}\delta\mathbf{v}_{a}^{s}\otimes\grad N_{a} & \grad\Delta\mathbf{u} & =\sum_{b}\Delta\mathbf{u}_{b}\otimes\grad N_{b}\\ \divg\delta\mathbf{v}^{s} & =\sum_{a}\delta\mathbf{v}_{a}^{s}\cdot\grad N_{a} & \divg\Delta\mathbf{u} & =\sum_{b}\Delta\mathbf{u}_{b}\cdot\grad N_{b}\\ \delta\mathbf{w} & =\sum_{a}N_{a}\delta\mathbf{w}_{a} & \Delta\mathbf{w} & =\sum_{b}N_{b}\Delta\mathbf{w}_{b}\\ \grad\delta\mathbf{w} & =\sum_{a}\delta\mathbf{w}_{a}\otimes\grad N_{a} & \grad\Delta\mathbf{w} & =\sum_{b}\Delta\mathbf{w}_{b}\otimes\grad N_{b}\\ \delta J^{f} & =\sum_{a}N_{a}\delta J_{a}^{f} & \Delta J^{f} & =\sum_{b}N_{b}\Delta J_{b}^{f}\\ \grad\delta J^{f} & =\sum_{a}\delta J_{a}^{f}\grad N_{a} & \grad\Delta J^{f} & =\sum_{b}\Delta J_{b}^{f}\grad N_{b} \end{aligned} \label{eq:fsi-interpolations} \end{equation} \]

Note that the \(\grad\equiv\frac{\partial}{\partial\mathbf{x}}\) operator should be evaluated using \(\mathbf{x}_{n+\alpha_{f}}\).

The solution of the nonlinear equation \(\delta W=0\) is obtained by linearizing this relation as

\[ \begin{equation} \delta W+D\delta W\left[\Delta\mathbf{u}\right]+D\delta W\left[\Delta\mathbf{w}\right]+D\delta W\left[\Delta J^{f}\right]\approx0\,,\label{eq:fsi-linearized-virtual-work} \end{equation} \]

where the operator \(D\delta W\left[\cdot\right]\) represents the directional derivative of \(\delta W\) at \(\left(\mathbf{u}^{s},\mathbf{w},J^{f}\right)\) along an increment \(\Delta\mathbf{u}\) of \(\mathbf{u}^{s}\), \(\Delta\mathbf{w}\) of \(\mathbf{w}\), or \(\Delta J^{f}\) of \(J^{f}\).

To linearize the virtual work, we need to express the integrals appearing in \(\delta W_{int}\) and \(\delta W_{ext}\) over the material frame of the finite element solid domain. For notational convenience, we let \(J\equiv J^{s}\) and \(\mathbf{F}\equiv\mathbf{F}^{s}\). Thus,

\[ \begin{equation} \begin{aligned}\int_{b}\boldsymbol{\sigma}^{s}:\grad\delta\mathbf{v}^{s}\,dv & =\int_{B}\mathbf{F}\cdot\mathbf{S}^{s}:\Grad\delta\mathbf{v}^{s}\,dV\end{aligned} \,,\label{eq:fsi-work-material-1} \end{equation} \]

where \(\mathbf{S}^{s}=J\cdot\mathbf{F}^{-1}\cdot\boldsymbol{\sigma}^{s}\cdot\mathbf{F}^{-T}\) is the second Piola-Kirchhoff stress for the solid material. Similarly,

\[ \begin{equation} \int_{b}\boldsymbol{\tau}:\grad\delta\mathbf{w}\,dv=\int_{B}\boldsymbol{\tau}\cdot J\mathbf{F}^{-T}:\Grad\delta\mathbf{w}\,dV\,.\label{eq:fsi-work-material-2} \end{equation} \]

Next,

\[ \begin{equation} \begin{aligned} & \int_{b}\delta\mathbf{w}\cdot\left(\grad p+\rho^{f}\mathbf{a}^{f}\right)\,dv\\ & =\int_{B}\delta\mathbf{w}\cdot\left(J\mathbf{F}^{-T}\cdot\Grad p+J\rho^{f}\left(\dot{\mathbf{w}}+\dot{\mathbf{v}}^{s}\right)+\rho^{f}\left(\Grad\mathbf{w}+\Grad\mathbf{v}^{s}\right)\cdot\mathbf{W}\right)\,dV \end{aligned} \label{eq:fsi-work-material-3-4} \end{equation} \]

\[ \begin{equation} \int_{b}\delta J^{f}\left[\frac{1}{J^{f}}\left(\dot{J}^{f}+\grad J^{f}\cdot\mathbf{w}\right)-\frac{\dot{J}}{J}\right]\,dv=\int_{B}\delta J^{f}\left[\frac{1}{J^{f}}\left(J\dot{J}^{f}+\Grad J^{f}\cdot\mathbf{W}\right)-\dot{J}\right]\,dV\label{eq:fsi-work-material-5} \end{equation} \]

\[ \begin{equation} \int_{b}\mathbf{w}\cdot\grad\delta J^{f}\,dv=\int_{B}\Grad\delta J^{f}\cdot\mathbf{W}\,dV\label{eq:fsi-work-material-6} \end{equation} \]

where \(\mathbf{W}=J\mathbf{F}^{-1}\cdot\mathbf{w}\).

Keep in mind that we are solving for the motions \(\boldsymbol{\chi}_{n+1}^{\iota}\) at \(t_{n+1}\). Therefore,

\[ \begin{aligned}D\mathbf{u}^{s}\left[\Delta\mathbf{u}\right] & =\alpha_{f}\Delta\mathbf{u}\\ D\mathbf{F}\left[\Delta\mathbf{u}\right] & =\alpha_{f}\Grad\Delta\mathbf{u}\\ DJ\left[\Delta\mathbf{u}\right] & =\alpha_{f}J\left(\divg\Delta\mathbf{u}\right)\\ D\dot{J}\left[\Delta\mathbf{u}\right] & =D\left(J\mathbf{F}^{-T}:\Grad\mathbf{v}^{s}\right)\left[\Delta\mathbf{u}\right]\\ & =\alpha_{f}J\left[\left(\divg\mathbf{v}^{s}+\frac{\gamma}{\beta\Delta t}\right)\left(\divg\Delta\mathbf{u}\right)-\left(\grad\Delta\mathbf{u}\right)^{T}:\mathbf{L}^{s}\right]\\ D\mathbf{v}^{s}\left[\Delta\mathbf{u}\right] & =\frac{\alpha_{f}\gamma}{\beta\Delta t}\Delta\mathbf{u}\\ D\mathbf{w}\left[\Delta\mathbf{w}\right] & =\alpha_{f}\Delta\mathbf{w}\\ DJ^{f}\left[\Delta J^{f}\right] & =\alpha_{f}\Delta J^{f}\\ D\rho^{f}\left[\Delta J^{f}\right] & =-\alpha_{f}\frac{\rho^{f}}{J^{f}}\Delta J^{f} \end{aligned} \]

In general, \(\boldsymbol{\sigma}^{s}\) (and thus, \(\mathbf{S}^{s}\)) is only a function of the solid strain, such as the right Cauchy-Green tensor \(\mathbf{C}=\mathbf{F}^{T}\cdot\mathbf{F}\) or the Green-Lagrange strain \(\mathbf{E}=\left(\mathbf{C}-\mathbf{I}\right)/2\).

\[ D\mathbf{E}\left[\Delta\mathbf{u}\right]=\frac{\alpha_{f}}{2}\left(\Grad^{T}\Delta\mathbf{u}\cdot\mathbf{F}+\mathbf{F}^{T}\cdot\Grad\Delta\mathbf{u}\right) \]

Therefore, following the standard approach in solid mechanics, the linearization of \(\mathbf{S}^{s}\) is

\[ \begin{equation} \begin{aligned}D\mathbf{S}^{s} & =D\mathbf{S}^{s}\left[\Delta\mathbf{u}\right]=\frac{\partial\mathbf{S}^{s}}{\partial\mathbf{E}}:D\mathbf{E}\left[\Delta\mathbf{u}\right]\\ & =\alpha_{f}\boldsymbol{\mathbb{C}}^{s}:\frac{1}{2}\left(\Grad^{T}\Delta\mathbf{u}\cdot\mathbf{F}+\mathbf{F}^{T}\cdot\Grad\Delta\mathbf{u}\right)\\ & =\alpha_{f}\boldsymbol{\mathbb{C}}^{s}:\left(\mathbf{F}^{T}\,\underline{\otimes}\,\mathbf{F}^{T}\right):\Delta\boldsymbol{\varepsilon} \end{aligned} \label{eq:fsi-solid-stress-linearization} \end{equation} \]

In general, \(\boldsymbol{\tau}\) is a function of the fluid volume ratio \(J^{f}\) and the rate of deformation of the fluid, \(\mathbf{D}^{f}=\frac{1}{2}\left(\mathbf{L}^{f}+\left(\mathbf{L}^{f}\right)^{T}\right)\). However, since \(\mathbf{v}^{f}\) is not a degree of freedom, we need to let \(\mathbf{D}^{f}=\mathbf{D}^{w}+\mathbf{D}^{s}\), where \(\mathbf{D}^{w}=\frac{1}{2}\left(\mathbf{L}^{w}+\left(\mathbf{L}^{w}\right)^{T}\right)\) and \(\mathbf{D}^{s}=\frac{1}{2}\left(\mathbf{L}^{s}+\left(\mathbf{L}^{s}\right)^{T}\right)\). Thus,

\[ \begin{equation} D\boldsymbol{\tau}=\boldsymbol{\mathcal{C}}^{\tau}:\left(D\mathbf{D}^{f}\left[\Delta\mathbf{w}\right]+D\mathbf{D}^{f}\left[\Delta\mathbf{u}\right]\right)+\frac{\partial\boldsymbol{\tau}}{\partial J^{f}}DJ^{f}\left[\Delta J^{f}\right]\label{eq:fsi-fluid-stress-linearization} \end{equation} \]

where

\[ \begin{equation} \boldsymbol{\mathcal{C}}^{\tau}=\frac{\partial\boldsymbol{\tau}}{\partial\mathbf{D}^{f}}\label{eq:fsi-viscous-tangent-tensor} \end{equation} \]

Linearization of \(\int_{B}\mathbf{F}\cdot\mathbf{S}^{s}:\Grad\delta\mathbf{v}^{s}\,dV\)

\[ \begin{equation} \begin{aligned} & D\left(\int_{B}\mathbf{F}\cdot\mathbf{S}^{s}:\Grad\delta\mathbf{v}^{s}\,dV\right)\left[\Delta\mathbf{u}\right]\\ & =\int_{b}\alpha_{f}\left(\boldsymbol{\sigma}^{s}:\grad^{T}\Delta\mathbf{u}\cdot\grad\delta\mathbf{v}^{s}+\grad\delta\mathbf{v}^{s}:\boldsymbol{\mathcal{C}}^{s}:\Delta\boldsymbol{\varepsilon}\right)\,dv \end{aligned} \label{eq:fsi-term-1-Du} \end{equation} \]

where

\[ \boldsymbol{\mathcal{C}}^{s}=J^{-1}\left(\mathbf{F}\,\underline{\otimes}\,\mathbf{F}\right):\boldsymbol{\mathbb{C}}^{s}:\left(\mathbf{F}^{T}\,\underline{\otimes}\,\mathbf{F}^{T}\right) \]

Similarly,

\[ \begin{equation} D\left(\int_{B}\mathbf{F}\cdot\mathbf{S}^{s}:\Grad\delta\mathbf{v}^{s}\,dV\right)\left[\Delta\mathbf{w}\right]=0\label{eq:fsi-term-1-Dw} \end{equation} \]

and

\[ \begin{equation} D\left(\int_{B}\mathbf{F}\cdot\mathbf{S}^{s}:\Grad\delta\mathbf{v}^{s}\,dV\right)\left[\Delta J^{f}\right]=0\label{eq:fsi-term-1-DJ} \end{equation} \]

The discretized version produces

\[ \begin{aligned} & \int_{b}\alpha_{f}\left(\boldsymbol{\sigma}^{s}:\grad^{T}\Delta\mathbf{u}\cdot\grad\delta\mathbf{v}^{s}+\grad\delta\mathbf{v}^{s}:\boldsymbol{\mathcal{C}}^{s}:\Delta\boldsymbol{\varepsilon}\right)\,dv\\ & =\sum_{a}\delta\mathbf{v}_{a}^{s}\cdot\sum_{b}\int_{b}\mathbf{K}_{ab}^{uu}\,dv\cdot\Delta\mathbf{u}_{b} \end{aligned} \]

where

\[ \begin{equation} \mathbf{K}_{ab}^{uu}=\alpha_{f}\left(\left(\grad N_{a}\cdot\boldsymbol{\sigma}^{s}\cdot\grad N_{b}\right)\mathbf{I}+\left(\grad N_{a}\cdot\boldsymbol{\mathcal{C}}^{s}\cdot\grad N_{b}\right)\right)\label{eq:fsi-term-1-Kuu} \end{equation} \]

\[ \begin{equation} \mathbf{K}_{ab}^{uw}=\mathbf{0}\label{eq:term-1-Kuw} \end{equation} \]

\[ \begin{equation} \mathbf{k}_{ab}^{uJ}=\mathbf{0}\label{eq:term-1-kuJ} \end{equation} \]

Linearization of \(\int_{B}\boldsymbol{\tau}\cdot J\mathbf{F}^{-T}:\Grad\delta\mathbf{w}\,dV\)

\[ \begin{equation} \begin{aligned} & D\left(\int_{B}\boldsymbol{\tau}\cdot J\mathbf{F}^{-T}:\Grad\delta\mathbf{w}\,dV\right)\left[\Delta\mathbf{u}\right]\\ & =\int_{b}\alpha_{f}\left(\boldsymbol{\tau}:\grad\delta\mathbf{w}\cdot\Delta\mathbf{G}+\grad\delta\mathbf{w}:\boldsymbol{\mathcal{C}}^{\tau}:\mathbf{M}\cdot\grad\Delta\mathbf{u}\right)\,dv \end{aligned} \label{eq:fsi-term-2-Du} \end{equation} \]

\[ \begin{aligned}D\left(\int_{B}\boldsymbol{\tau}\cdot J\mathbf{F}^{-T}:\Grad\delta\mathbf{w}\,dV\right)\left[\Delta\mathbf{u}\right] & =\int_{B}D\left(\boldsymbol{\tau}\cdot J\mathbf{F}^{-T}\right)\left[\Delta\mathbf{u}\right]:\Grad\delta\mathbf{w}\,dV\\ & =\int_{B}\left(D\boldsymbol{\tau}\left[\Delta\mathbf{u}\right]\cdot J\mathbf{F}^{-T}+\boldsymbol{\tau}\cdot D\left(J\mathbf{F}^{-T}\right)\left[\Delta\mathbf{u}\right]\right):\Grad\delta\mathbf{w}\,dV \end{aligned} \]

where we used

\[ \boldsymbol{\mathcal{C}}^{\tau}=\frac{\partial\boldsymbol{\tau}}{\partial\mathbf{D}^{f}}\,. \]

Similarly,

\[ \begin{equation} \begin{aligned}D\left(\int_{B}\boldsymbol{\tau}\cdot J\mathbf{F}^{-T}:\Grad\delta\mathbf{w}\,dV\right)\left[\Delta\mathbf{w}\right] & =\int_{b}\alpha_{f}\grad\delta\mathbf{w}:\boldsymbol{\mathcal{C}}^{\tau}:\grad\Delta\mathbf{w}\,dv\end{aligned} \label{eq:fsi-term-2-Dw} \end{equation} \]

\[ \begin{equation} D\left(\int_{B}\boldsymbol{\tau}\cdot J\mathbf{F}^{-T}:\Grad\delta\mathbf{w}\,dV\right)\left[\Delta J^{f}\right]=\int_{b}\alpha_{f}\Delta J^{f}\boldsymbol{\tau}_{J}^{\prime}:\grad\delta\mathbf{w}\,dv\label{eq:fsi-term-2-DJ} \end{equation} \]

where

\[ \boldsymbol{\tau}_{J}^{\prime}=\frac{\partial\boldsymbol{\tau}}{\partial J^{f}} \]

The discretized version produces

\[ \begin{aligned} & \int_{b}\alpha_{f}\left(\boldsymbol{\tau}:\grad\delta\mathbf{w}\cdot\Delta\mathbf{G}+\grad\delta\mathbf{w}:\boldsymbol{\mathcal{C}}^{\tau}:\mathbf{M}\cdot\grad\Delta\mathbf{u}\right)\,dv\\ & =\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{K}_{ab}^{wu}\,dv\cdot\Delta\mathbf{u}_{b} \end{aligned} \]

where

\[ \begin{equation} \mathbf{K}_{ab}^{wu}=\alpha_{f}\boldsymbol{\tau}\cdot\left(\grad N_{a}\otimes\grad N_{b}-\grad N_{b}\otimes\grad N_{a}\right)+\left(\grad N_{a}\cdot\boldsymbol{\mathcal{C}}^{\tau}\cdot\grad N_{b}\right)\cdot\mathbf{M}\label{eq:fsi-term-2-Kwu} \end{equation} \]

\[ \int_{b}\alpha_{f}\grad\delta\mathbf{w}:\boldsymbol{\mathcal{C}}^{\tau}:\grad\Delta\mathbf{w}\,dv=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{K}_{ab}^{ww}\,dv\cdot\Delta\mathbf{w}_{b} \]

where

\[ \begin{equation} \mathbf{K}_{ab}^{ww}=\alpha_{f}\grad N_{a}\cdot\boldsymbol{\mathcal{C}}^{\tau}\cdot\grad N_{b}\label{eq:fsi-term-2-Kww} \end{equation} \]

\[ \int_{b}\alpha_{f}\Delta J^{f}\boldsymbol{\tau}_{J}^{\prime}:\grad\delta\mathbf{w}\,dv=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{k}_{ab}^{wJ}\,dv\,\Delta J_{b}^{f} \]

where

\[ \begin{equation} \mathbf{k}_{ab}^{wJ}=\alpha_{f}N_{b}\boldsymbol{\tau}_{J}^{\prime}\cdot\grad N_{a}\label{eq:fsi-term-2-kwJ} \end{equation} \]

Linearization of \(\int_{B}\delta\mathbf{w}\cdot J\mathbf{F}^{-T}\cdot\Grad p\,dV\)

\[ \begin{equation} \boxed{\begin{aligned} & D\left(\int_{B}\delta\mathbf{w}\cdot J\mathbf{F}^{-T}\cdot\Grad p\,dV\right)\left[\Delta\mathbf{u}\right]\\ & =\int_{b}\alpha_{f}\delta\mathbf{w}\cdot\Delta\mathbf{G}^{T}\cdot\grad p\,dv \end{aligned} }\label{eq:fsi-term-3-Du} \end{equation} \]

\[ \begin{equation} \boxed{D\left(\int_{B}\delta\mathbf{w}\cdot J\mathbf{F}^{-T}\cdot\Grad p\,dV\right)\left[\Delta\mathbf{w}\right]=0}\label{eq:fsi-term-3-Dw} \end{equation} \]

\[ \begin{equation} \boxed{\begin{aligned} & D\left(\int_{B}\delta\mathbf{w}\cdot J\mathbf{F}^{-T}\cdot\Grad p\,dV\right)\left[\Delta J^{f}\right]\\ & =\int_{b}\alpha_{f}\delta\mathbf{w}\cdot\left(\Delta J^{f}p^{\prime\prime}\left(J^{f}\right)\grad J^{f}+p^{\prime}\left(J^{f}\right)\grad\Delta J^{f}\right)\,dv \end{aligned} }\label{eq:fsi-term-3-DJ} \end{equation} \]

The discretization of these terms produces

\[ \int_{b}\alpha_{f}\delta\mathbf{w}\cdot\Delta\mathbf{G}^{T}\cdot\grad p\,dv=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{K}_{ab}^{wu}\,dv\cdot\Delta\mathbf{u}_{b} \]

where

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{wu}=\alpha_{f}N_{a}\left(\grad p\otimes\grad N_{b}-\grad N_{b}\otimes\grad p\right)}\label{eq:fsi-term-3-Kwu} \end{equation} \]

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{ww}=\mathbf{0}}\label{eq:term-3-Kww} \end{equation} \]

\[ \int_{b}\alpha_{f}\delta\mathbf{w}\cdot\left(\Delta J^{f}p^{\prime\prime}\left(J^{f}\right)\grad J^{f}+p^{\prime}\left(J^{f}\right)\grad\Delta J^{f}\right)\,dv=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{k}_{ab}^{wJ}\,dv\Delta J_{b}^{f} \]

\[ \begin{equation} \boxed{\mathbf{k}_{ab}^{wJ}=\alpha_{f}N_{a}\left(N_{b}p^{\prime\prime}\left(J^{f}\right)\grad J^{f}+p^{\prime}\left(J^{f}\right)\grad N_{b}\right)}\label{eq:fsi-term-3-kwJ} \end{equation} \]

Linearization of \(\int_{B}\delta\mathbf{w}\cdot\rho^{f}\left[J\left(\dot{\mathbf{w}}+\dot{\mathbf{v}}^{s}\right)+\left(\Grad\mathbf{w}+\Grad\mathbf{v}^{s}\right)\cdot\mathbf{W}\right]\,dV\)

\[ \int_{B}\delta\mathbf{w}\cdot\rho^{f}\left[J\left(\dot{\mathbf{w}}+\dot{\mathbf{v}}^{s}\right)+\left(\Grad\mathbf{w}+\Grad\mathbf{v}^{s}\right)\cdot\mathbf{W}\right]\,dV \]

\[ \begin{equation} \boxed{\begin{aligned} & D\left(\int_{B}\delta\mathbf{w}\cdot\rho^{f}\left[J\left(\dot{\mathbf{w}}+\dot{\mathbf{v}}^{s}\right)+\left(\Grad\mathbf{w}+\Grad\mathbf{v}^{s}\right)\cdot\mathbf{W}\right]\,dV\right)\left[\Delta\mathbf{u}\right]\\ & =\int_{b}\delta\mathbf{w}\cdot\alpha_{f}\rho^{f}\left[\left(\divg\Delta\mathbf{u}\right)\left(\dot{\mathbf{w}}+\dot{\mathbf{v}}^{s}\right)+\frac{\gamma}{\beta\Delta t}\grad\Delta\mathbf{u}\cdot\mathbf{w}+\frac{\alpha_{m}}{\alpha_{f}\beta\Delta t^{2}}\Delta\mathbf{u}\right]\,dv\\ & +\int_{b}\delta\mathbf{w}\cdot\alpha_{f}\rho^{f}\mathbf{L}^{f}\cdot\Delta\mathbf{G}\cdot\mathbf{w}\,dv \end{aligned} }\label{eq:fsi-term-4-Du} \end{equation} \]

\[ \begin{equation} \boxed{\begin{aligned} & D\left(\int_{B}\delta\mathbf{w}\cdot\rho^{f}\left[J\left(\dot{\mathbf{w}}+\dot{\mathbf{v}}^{s}\right)+\left(\Grad\mathbf{w}+\Grad\mathbf{v}^{s}\right)\cdot\mathbf{W}\right]\,dV\right)\left[\Delta\mathbf{w}\right]\\ & =\int_{b}\delta\mathbf{w}\cdot\alpha_{f}\rho^{f}\left[\left(\frac{\alpha_{m}}{\alpha_{f}\gamma\Delta t}\mathbf{I}+\mathbf{L}^{f}\right)\cdot\Delta\mathbf{w}+\grad\Delta\mathbf{w}\cdot\mathbf{w}\right]\,dv \end{aligned} }\label{eq:fsi-term-4-Dw} \end{equation} \]

\[ \begin{equation} \boxed{\begin{aligned} & D\left(\int_{B}\delta\mathbf{w}\cdot\rho^{f}\left[J\left(\dot{\mathbf{w}}+\dot{\mathbf{v}}^{s}\right)+\left(\Grad\mathbf{w}+\Grad\mathbf{v}^{s}\right)\cdot\mathbf{W}\right]\,dV\right)\left[\Delta J^{f}\right]\\ & =-\int_{b}\delta\mathbf{w}\cdot\alpha_{f}\frac{\rho^{f}}{J^{f}}\Delta J^{f}\mathbf{a}^{f}\,dv \end{aligned} }\label{eq:fsi-term-4-DJ} \end{equation} \]

The discretized version is given by

\[ \begin{aligned} & \int_{b}\delta\mathbf{w}\cdot\alpha_{f}\rho^{f}\left[\left(\divg\Delta\mathbf{u}\right)\left(\dot{\mathbf{w}}+\dot{\mathbf{v}}^{s}\right)+\frac{\gamma}{\beta\Delta t}\grad\Delta\mathbf{u}\cdot\mathbf{w}+\frac{\alpha_{m}}{\alpha_{f}\beta\Delta t^{2}}\Delta\mathbf{u}\right]\,dv\\ & +\int_{b}\delta\mathbf{w}\cdot\alpha_{f}\rho^{f}\mathbf{L}^{f}\cdot\Delta\mathbf{G}\cdot\mathbf{w}\,dv\\ & =\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{K}_{ab}^{wu}\,dv\cdot\Delta\mathbf{u}_{b} \end{aligned} \]

where

\[ \begin{equation} \boxed{\begin{aligned}\mathbf{K}_{ab}^{wu} & =\alpha_{f}N_{a}\rho^{f}\left[\mathbf{a}^{f}\otimes\grad N_{b}-\left(\grad N_{b}\cdot\mathbf{w}\right)\mathbf{L}^{f}+\left(\frac{\gamma}{\beta\Delta t}\left(\grad N_{b}\cdot\mathbf{w}\right)+\frac{\alpha_{m}}{\alpha_{f}\beta\Delta t^{2}}N_{b}\right)\mathbf{I}\right]\end{aligned} }\label{eq:fsi-term-4-Kwu} \end{equation} \]

\[ \begin{aligned} & \int_{b}\delta\mathbf{w}\cdot\alpha_{f}\rho^{f}\left[\left(\frac{\alpha_{m}}{\alpha_{f}\gamma\Delta t}\mathbf{I}+\mathbf{L}^{f}\right)\cdot\Delta\mathbf{w}+\grad\Delta\mathbf{w}\cdot\mathbf{w}\right]\,dv\\ & =\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{K}_{ab}^{ww}\,dv\cdot\Delta\mathbf{w}_{b} \end{aligned} \]

where

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{ww}=\alpha_{f}N_{a}\rho^{f}\left[N_{b}\left(\frac{\alpha_{m}}{\alpha_{f}\gamma\Delta t}\mathbf{I}+\mathbf{L}^{f}\right)+\left(\grad N_{b}\cdot\mathbf{w}\right)\mathbf{I}\right]}\label{eq:fsi-term-4-Kww} \end{equation} \]

\[ -\int_{B}\delta\mathbf{w}\cdot\alpha_{f}\frac{\rho^{f}}{J^{f}}\Delta J^{f}\mathbf{a}^{f}\,dv=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{k}_{ab}^{wJ}\,dv\Delta J_{b}^{f} \]

where

\[ \begin{equation} \boxed{\mathbf{k}_{ab}^{wJ}=-\alpha_{f}N_{a}N_{b}\frac{\rho^{f}}{J^{f}}\mathbf{a}^{f}}\label{eq:fsi-term-4-kwJ} \end{equation} \]

Linearization of \(\int_{B}\delta J^{f}\left[\frac{1}{J^{f}}\left(J\dot{J}^{f}+\Grad J^{f}\cdot\mathbf{W}\right)-\dot{J}\right]\,dV\)

\[ \int_{B}\delta J^{f}\left[\frac{1}{J^{f}}\left(J\dot{J}^{f}+\Grad J^{f}\cdot\mathbf{W}\right)-\dot{J}\right]\,dV \]

\[ \begin{aligned} & D\left(\int_{B}\delta J^{f}\left[\frac{1}{J^{f}}\left(J\dot{J}^{f}+\Grad J^{f}\cdot\mathbf{W}\right)-\dot{J}\right]\,dV\right)\left[\Delta\mathbf{u}\right]\\ & =\int_{B}\delta J^{f}\left[\frac{1}{J^{f}}\left(DJ\left[\Delta\mathbf{u}\right]\dot{J}^{f}+\Grad J^{f}\cdot D\mathbf{W}\left[\Delta\mathbf{u}\right]\right)-D\dot{J}\left[\Delta\mathbf{u}\right]\right]\,dV\\ & =\int_{b}\delta J^{f}\alpha_{f}\left[\left(\frac{\dot{J}^{f}}{J^{f}}-\divg\mathbf{v}^{s}-\frac{\gamma}{\beta\Delta t}\right)\left(\divg\Delta\mathbf{u}\right)+\mathbf{w}\cdot\left[\left(\divg\Delta\mathbf{u}\right)\mathbf{I}-\grad^{T}\Delta\mathbf{u}\right]\cdot\frac{\grad J^{f}}{J^{f}}+\left(\grad\Delta\mathbf{u}\right)^{T}:\mathbf{L}^{s}\right]\,dv \end{aligned} \]

\[ \begin{equation} \boxed{\begin{aligned} & D\left(\int_{B}\delta J^{f}\left[\frac{1}{J^{f}}\left(J\dot{J}^{f}+\Grad J^{f}\cdot\mathbf{W}\right)-\dot{J}\right]\,dV\right)\left[\Delta\mathbf{u}\right]\\ & =\int_{b}\delta J^{f}\alpha_{f}\left[\left(\frac{\dot{J}^{f}}{J^{f}}-\divg\mathbf{v}^{s}-\frac{\gamma}{\beta\Delta t}\right)\left(\divg\Delta\mathbf{u}\right)+\mathbf{w}\cdot\Delta\mathbf{G}^{T}\cdot\frac{\grad J^{f}}{J^{f}}+\left(\grad\Delta\mathbf{u}\right)^{T}:\mathbf{L}^{s}\right]\,dv \end{aligned} }\label{eq:fsi-term-5-Du} \end{equation} \]

\[ \begin{equation} \boxed{\begin{aligned} & D\left(\int_{B}\delta J^{f}\left[\frac{1}{J^{f}}\left(J\dot{J}^{f}+\Grad J^{f}\cdot\mathbf{W}\right)-\dot{J}\right]\,dV\right)\left[\Delta\mathbf{w}\right]\\ & =\int_{b}\delta J^{f}\alpha_{f}\frac{1}{J^{f}}\Delta\mathbf{w}\cdot\grad J^{f}\,dv \end{aligned} }\label{eq:fsi-term-5-Dw} \end{equation} \]

\[ \begin{equation} \boxed{\begin{aligned} & D\left(\int_{B}\delta J^{f}\left[\frac{1}{J^{f}}\left(J\dot{J}^{f}+\Grad J^{f}\cdot\mathbf{W}\right)-\dot{J}\right]\,dV\right)\left[\Delta J^{f}\right]\\ & =\int_{b}\delta J^{f}\alpha_{f}\frac{1}{J^{f}}\left(\left[\frac{\alpha_{m}}{\alpha_{f}\gamma\Delta t}-\frac{1}{J^{f}}\left(\dot{J}^{f}+\mathbf{w}\cdot\grad J^{f}\right)\right]\Delta J^{f}+\mathbf{w}\cdot\grad\Delta J^{f}\right)\,dv \end{aligned} }\label{eq:fsi-term-5-DJ} \end{equation} \]

The discretization of these terms produces

\[ \begin{aligned} & \int_{b}\delta J^{f}\alpha_{f}\left[\left(\frac{\dot{J}^{f}}{J^{f}}-\divg\mathbf{v}^{s}-\frac{\gamma}{\beta\Delta t}\right)\left(\divg\Delta\mathbf{u}\right)+\mathbf{w}\cdot\Delta\mathbf{G}^{T}\cdot\frac{\grad J^{f}}{J^{f}}+\left(\grad\Delta\mathbf{u}\right)^{T}:\mathbf{L}^{s}\right]\,dv\\ & =\sum_{a}\delta J_{a}^{f}\sum_{b}\int_{b}\mathbf{k}_{ab}^{Ju}\,dv\cdot\Delta\mathbf{u}_{b} \end{aligned} \]

where

\[ \begin{equation} \boxed{\begin{aligned}\mathbf{k}_{ab}^{Ju} & =\alpha_{f}N_{a}\left[\left(\frac{\dot{J}^{f}}{J^{f}}-\divg\mathbf{v}^{s}-\frac{\gamma}{\beta\Delta t}\right)\grad N_{b}+\left(\grad N_{b}\otimes\frac{\grad J^{f}}{J^{f}}-\frac{\grad J^{f}}{J^{f}}\otimes\grad N_{b}\right)\cdot\mathbf{w}+\left(\mathbf{L}^{s}\right)^{T}\cdot\grad N_{b}\right]\end{aligned} }\label{eq:fsi-term-5-kJu} \end{equation} \]

\[ \int_{b}\delta J^{f}\alpha_{f}\frac{1}{J^{f}}\Delta\mathbf{w}\cdot\grad J^{f}\,dv=\sum_{a}\delta J_{a}^{f}\sum_{b}\int_{b}\mathbf{k}_{ab}^{Jw}\,dv\cdot\Delta\mathbf{w}_{b} \]

where

\[ \begin{equation} \boxed{\mathbf{k}_{ab}^{Jw}=\alpha_{f}\frac{N_{a}N_{b}}{J^{f}}\grad J^{f}}\label{eq:fsi-term-5-kJw} \end{equation} \]

\[ \begin{aligned} & \int_{b}\delta J^{f}\alpha_{f}\frac{1}{J^{f}}\left(\left[\frac{\alpha_{m}}{\alpha_{f}\gamma\Delta t}-\frac{1}{J^{f}}\left(\dot{J}^{f}+\mathbf{w}\cdot\grad J^{f}\right)\right]\Delta J^{f}+\mathbf{w}\cdot\grad\Delta J^{f}\right)\,dv\\ & =\sum_{a}\delta J_{a}^{f}\sum_{b}\int_{b}k_{ab}^{JJ}\,dv\Delta J_{b}^{f} \end{aligned} \]

where

\[ \begin{equation} \boxed{k_{ab}^{JJ}=\alpha_{f}N_{a}\frac{1}{J^{f}}\left(N_{b}\left[\frac{\alpha_{m}}{\alpha_{f}\gamma\Delta t}-\frac{1}{J^{f}}\left(\dot{J}^{f}+\mathbf{w}\cdot\grad J^{f}\right)\right]+\mathbf{w}\cdot\grad N_{b}\right)}\label{eq:fsi-term-5-kJJ} \end{equation} \]

Linearization of \(-\int_{B}\Grad\delta J^{f}\cdot\mathbf{W}\,dV\)

\[ -\int_{B}\Grad\delta J^{f}\cdot\mathbf{W}\,dV \]

\[ \begin{equation} \boxed{D\left(\int_{B}\Grad\delta J^{f}\cdot\mathbf{W}\,dV\right)\left[\Delta\mathbf{u}\right]=\int_{b}\alpha_{f}\mathbf{w}\cdot\Delta\mathbf{G}^{T}\cdot\grad\delta J^{f}\,dv}\label{eq:fsi-term-6-Du} \end{equation} \]

\[ \begin{aligned} & D\left(\int_{B}\Grad\delta J^{f}\cdot\mathbf{W}\,dV\right)\left[\Delta\mathbf{u}\right]\\ & =\int_{B}\Grad\delta J^{f}\cdot D\mathbf{W}\left[\Delta\mathbf{w}\right]\,dV\\ & =\int_{B}\alpha_{f}\Delta\mathbf{w}\cdot\grad\delta J^{f}\,dv \end{aligned} \]

\[ \begin{equation} \boxed{D\left(\int_{B}\Grad\delta J^{f}\cdot\mathbf{W}\,dV\right)\left[\Delta\mathbf{w}\right]=\int_{B}\alpha_{f}\Delta\mathbf{w}\cdot\grad\delta J^{f}\,dv}\label{eq:fsi-term-6-Dw} \end{equation} \]

\[ \begin{equation} \boxed{D\left(\int_{B}\Grad\delta J^{f}\cdot\mathbf{W}\,dV\right)\left[\Delta J^{f}\right]=0}\label{eq:fsi-term-6-DJ} \end{equation} \]

The discretization of these terms produces

\[ \int_{b}\alpha_{f}\mathbf{w}\cdot\Delta\mathbf{G}^{T}\cdot\grad\delta J^{f}\,dv=\sum_{a}\delta J_{a}^{f}\sum_{b}\int_{v}\mathbf{k}_{ab}^{Ju}\,dv\cdot\Delta\mathbf{u}_{b} \]

where

\[ \begin{equation} \boxed{\mathbf{k}_{ab}^{Ju}=\alpha_{f}\left(\grad N_{b}\otimes\grad N_{a}-\grad N_{a}\otimes\grad N_{b}\right)\cdot\mathbf{w}}\label{eq:fsi-term-6-kJu} \end{equation} \]

\[ \int_{B}\alpha_{f}\Delta\mathbf{w}\cdot\grad\delta J^{f}\,dv=\sum_{a}\delta J_{a}^{f}\sum_{b}\int_{b}\mathbf{k}_{ab}^{Jw}\,dv\cdot\Delta\mathbf{w}_{b} \]

where

\[ \begin{equation} \boxed{\mathbf{k}_{ab}^{Jw}=\alpha_{f}N_{b}\grad N_{a}}\label{eq:fsi-term-6-kJw} \end{equation} \]

\[ \begin{equation} \boxed{k_{ab}^{JJ}=0}\label{eq:fsi-term-6-kJJ} \end{equation} \]

Body force term \(\int_{b}\delta\mathbf{w}\cdot\rho^{f}\mathbf{b}\,dv\)

The body force term may be discretized as

\[ \int_{b}\delta\mathbf{w}\cdot\rho^{f}\mathbf{b}\,dv=\sum_{a}\delta\mathbf{w}_{a}\cdot\int_{b}N_{a}\rho^{f}\mathbf{b}\,dv \]

To linearizes this expression, we first evaluate it in the material domain,

\[ \int_{b}\delta\mathbf{w}\cdot\rho^{f}\mathbf{b}\,dv=\int_{B}\delta\mathbf{w}\cdot\rho^{f}J\mathbf{b}\,dV \]

Now,

\[ \begin{equation} \boxed{D\left(\int_{B}\delta\mathbf{w}\cdot\rho^{f}J\mathbf{b}\,dV\right)\left[\Delta\mathbf{u}\right]=\int_{b}\alpha_{f}\rho^{f}\left(\divg\Delta\mathbf{u}\right)\delta\mathbf{w}\cdot\mathbf{b}\,dv}\label{eq:fsi-bf-1} \end{equation} \]

\[ \begin{equation} \boxed{D\left(\int_{B}\delta\mathbf{w}\cdot\rho^{f}J\mathbf{b}\,dV\right)\left[\Delta\mathbf{w}\right]=\int_{b}\delta\mathbf{w}\cdot\rho^{f}D\mathbf{b}\left[\Delta\mathbf{w}\right]\,dv}\label{eq:fsi-bf-2} \end{equation} \]

\[ \begin{equation} \boxed{D\left(\int_{B}\delta\mathbf{w}\cdot\rho^{f}J\mathbf{b}\,dV\right)\left[\Delta J^{f}\right]=-\int_{b}\Delta J^{f}\frac{\alpha_{f}\rho^{f}}{J^{f}}\delta\mathbf{w}\cdot\mathbf{b}\,dv}\label{eq:fsi-bf-3} \end{equation} \]

The discrete forms of these expressions are given by

\[ \int_{b}\alpha_{f}\rho^{f}\left(\divg\Delta\mathbf{u}\right)\delta\mathbf{w}\cdot\mathbf{b}\,dv=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{K}_{ab}^{wu}\,dv\cdot\Delta\mathbf{u}_{b} \]

where

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{wu}=\alpha_{f}\rho^{f}N_{a}\mathbf{b}\otimes\grad N_{b}}\label{eq:fsi-bf-4} \end{equation} \]

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{ww}=\alpha_{f}\rho^{f}N_{a}N_{b}\frac{\partial\mathbf{b}}{\partial\mathbf{w}}}\label{eq:fsi-bf-5} \end{equation} \]

\[ -\int_{b}\Delta J^{f}\frac{\alpha_{f}\rho^{f}}{J^{f}}\delta\mathbf{w}\cdot\mathbf{b}\,dv=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{b}\mathbf{k}_{ab}^{wJ}\,dv\,\Delta J_{b}^{f} \]

where

\[ \begin{equation} \boxed{\mathbf{k}_{ab}^{wJ}=-\alpha_{f}N_{a}N_{b}\frac{\rho^{f}}{J^{f}}\mathbf{b}}\label{eq:fsi-bf-6} \end{equation} \]

Fluid traction acting on solid interface

At a fluid-solid interface \(\Gamma\), the fluid traction \(\mathbf{t}^{f}=\boldsymbol{\sigma}^{f}\cdot\mathbf{n}\) acts on the solid surface, \(\mathbf{t}^{s}=-\mathbf{t}^{f}\), where \(\mathbf{n}\) is the outward normal to the fluid surface. The resulting virtual work on the solid domain is

\[ \delta G=\int_{\Gamma}\delta\mathbf{v}^{s}\cdot\mathbf{t}_{n+\alpha_{f}}^{s}\,da=\int_{\Gamma}-\delta\mathbf{v}^{s}\cdot\boldsymbol{\sigma}^{f}\cdot\mathbf{n}\,da=\int_{\Gamma_{\eta}}-\delta\mathbf{v}^{s}\cdot\boldsymbol{\sigma}^{f}\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\,d\eta^{1}d\eta^{2} \]

The linearization of this work is given by

\[ D\delta G\left[\Delta\mathbf{u}\right]=\int_{\Gamma_{\eta}}-\delta\mathbf{v}^{s}\cdot\alpha_{f}\left[\left(\boldsymbol{\mathcal{C}}^{v}:\mathbf{M}\cdot\grad\Delta\mathbf{u}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)+\boldsymbol{\sigma}^{f}\cdot\left(\frac{\partial\Delta\mathbf{u}}{\partial\eta^{1}}\times\mathbf{g}_{2}+\mathbf{g}_{1}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{2}}\right)\right]\,d\eta^{1}d\eta^{2} \]

where

\[ \mathbf{g}_{\alpha}=\frac{\partial\mathbf{x}}{\partial\eta^{\alpha}} \]

are covariant basis vectors on \(\Gamma\). Similarly,

\[ D\delta G\left[\Delta\mathbf{w}\right]=\int_{\Gamma_{\eta}}-\delta\mathbf{v}^{s}\cdot\alpha_{f}\left[\left(\boldsymbol{\mathcal{C}}^{v}:\grad\Delta\mathbf{w}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\right]\,d\eta^{1}d\eta^{2} \]

and

\[ D\delta G\left[\Delta J^{f}\right]=\int_{\Gamma_{\eta}}-\Delta J^{f}\delta\mathbf{v}^{s}\cdot\alpha_{f}\left[\left(-p^{\prime}\left(J^{f}\right)+\boldsymbol{\tau}_{J}^{\prime}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\right]\,d\eta^{1}d\eta^{2} \]

The discretized form of these equations is

\[ \delta G=\sum_{a}\delta\mathbf{v}_{a}^{s}\cdot\int_{\Gamma}\mathbf{f}_{a}\,d\eta^{1}d\eta^{2} \]

where

\[ \begin{equation} \boxed{\mathbf{f}_{a}=-N_{a}\boldsymbol{\sigma}^{f}\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)}\label{eq:fsi-ft-1} \end{equation} \]

\[ D\delta G\left[\Delta\mathbf{u}\right]=\sum_{a}\delta\mathbf{v}_{a}^{s}\cdot\sum_{b}\int_{\Gamma_{\eta}}\mathbf{K}_{ab}^{uu}\,d\eta^{1}d\eta^{2}\cdot\Delta\mathbf{u}_{b} \]

where

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{uu}=-\alpha_{f}N_{a}\left(\left[\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\cdot\boldsymbol{\mathcal{C}}^{v}\cdot\grad N_{b}\right]\cdot\mathbf{M}+\boldsymbol{\sigma}^{f}\cdot\boldsymbol{\mathcal{A}}\left\{ \frac{\partial N_{b}}{\partial\eta^{2}}\mathbf{g}_{1}-\frac{\partial N_{b}}{\partial\eta^{1}}\mathbf{g}_{2}\right\} \right)}\label{eq:fsi-ft-2} \end{equation} \]

and

\[ D\delta G\left[\Delta\mathbf{w}\right]=\sum_{a}\delta\mathbf{v}_{a}^{s}\cdot\sum_{b}\int_{\Gamma_{\eta}}\mathbf{K}_{ab}^{uw}\,d\eta^{1}d\eta^{2}\cdot\Delta\mathbf{w}_{b} \]

where

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{uw}=-\alpha_{f}N_{a}\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\cdot\boldsymbol{\mathcal{C}}^{v}\cdot\grad N_{b}}\label{eq:fsi-ft-3} \end{equation} \]

\[ D\delta G\left[\Delta J^{f}\right]=\sum_{a}\delta\mathbf{v}_{a}^{s}\cdot\sum_{b}\int_{\Gamma_{\eta}}\mathbf{k}_{ab}^{uJ}\,d\eta^{1}d\eta^{2}\,\Delta J_{b}^{f} \]

where

\[ \begin{equation} \boxed{\mathbf{k}_{ab}^{uJ}=-\alpha_{f}N_{a}N_{b}\left(-p^{\prime}\left(J^{f}\right)+\boldsymbol{\tau}_{J}^{\prime}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)}\label{eq:fsi-ft-4} \end{equation} \]

Special Boundary Conditions

Backflow Stabilization

Let the normal component of the viscous traction be given by

\[ \begin{equation} t_{n}^{\tau}=\begin{cases} \beta\rho_{r}w_{n}^{2} & w_{n}<0\\ 0 & w_{n}\ge0 \end{cases}\,.\label{eq:fsi-BFS-normal-traction} \end{equation} \]

Then, the contribution of this traction to the virtual external work \(\delta W_{ext}\) is

\[ \begin{equation} \delta G=\int_{\partial\Omega}\delta\mathbf{w}\cdot t_{n}^{\tau}\mathbf{n}\,da\,.\label{eq:fsi-BFSckflow-virtual-work} \end{equation} \]

The linearization of \(\delta G\) along an increment \(\Delta\mathbf{w}\) in the relative velocity is given by

\[ \begin{equation} D\delta G\left[\Delta\mathbf{w}\right]=\int_{\partial\Omega}\delta\mathbf{w}\cdot\alpha_{f}\mathbf{K}_{n}\cdot\Delta\mathbf{w}\,da\,,\label{eq:fsi-BFS-linearization} \end{equation} \]

where

\[ \begin{equation} \mathbf{K}_{n}=\begin{cases} 2\beta\rho_{r}w_{n}\left(\mathbf{n}\otimes\mathbf{n}\right) & v_{n}<0\\ \mathbf{0} & v_{n}\ge0 \end{cases}\,.\label{eq:fsi-BFS-stiffness} \end{equation} \]

Similarly,

\[ D\delta G\left[\Delta\mathbf{u}\right]=\int_{\partial\Omega_{\eta}}\delta\mathbf{w}\cdot\alpha_{f}t_{n}^{\tau}\left(\mathbf{g}_{1}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{2}}-\mathbf{g}_{2}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{1}}\right)\,d\eta^{1}d\eta^{2} \]

The discretized form of \(\delta G\) is

\[ \begin{equation} \delta G=\sum_{a}\delta\mathbf{w}_{a}\cdot\int_{\partial\Omega_{\eta}}\mathbf{f}_{a}^{n}\,d\eta^{1}d\eta^{2},\quad\boxed{\mathbf{f}_{a}^{n}=N_{a}t_{n}^{\tau}\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)}\,,\label{eq:fsi-BFS-discretized-work} \end{equation} \]

whereas the discretized form of \(D\delta G\left[\Delta\mathbf{v}\right]\) is

\[ \begin{equation} D\delta G\left[\Delta\mathbf{v}\right]=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\left(\int_{\partial\Omega_{\eta}}\mathbf{K}_{ab}^{wu}\,d\eta^{1}d\eta^{2}\cdot\Delta\mathbf{u}+\int_{\partial\Omega_{\eta}}\mathbf{K}_{ab}^{ww}\,d\eta^{1}d\eta^{2}\cdot\Delta\mathbf{w}_{b}\right)\,,\label{eq:BFS-discretized-tangent-1} \end{equation} \]

with

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{ww}=\alpha_{f}N_{a}N_{b}\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|\mathbf{K}_{n}}\,,\label{eq:fsi-BFS-Kww} \end{equation} \]

and

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{wu}=\alpha_{f}N_{a}t_{n}^{\tau}\boldsymbol{\mathcal{A}}\left\{ \frac{\partial N_{b}}{\partial\eta^{2}}\mathbf{g}_{1}-\frac{\partial N_{b}}{\partial\eta^{1}}\mathbf{g}_{2}\right\} }\label{eq:fsi-BFS-Kwu} \end{equation} \]

Tangential Stabilization

The tangential traction given by

\[ \begin{equation} \mathbf{t}_{t}^{\tau}=-\beta\rho_{r}\left|\mathbf{w}_{t}\right|\mathbf{w}_{t}\,.\label{eq:fsi-tangential-stabilization} \end{equation} \]

This form shows that \(\mathbf{t}_{t}^{\tau}\) opposes tangential flow. The external virtual work for this traction is

\[ \begin{equation} \delta G=\int_{\partial\Omega}\delta\mathbf{w}\cdot\mathbf{t}_{t}^{\tau}\,da\,.\label{eq:fsi-TS-external-work} \end{equation} \]

Its linearization along an increment \(\Delta\mathbf{w}\) is

\[ \begin{equation} D\delta G\left[\Delta\mathbf{w}\right]=\int_{\partial\Omega}\delta\mathbf{w}\cdot\alpha_{f}\mathbf{K}_{t}\cdot\Delta\mathbf{w}\,da\,,\label{eq:fsi-TS-linearization} \end{equation} \]

where it can be shown that

\[ \begin{equation} \mathbf{K}_{t}=-\beta\rho_{r}\left|\mathbf{w}_{t}\right|\left(\mathbf{I}-\mathbf{n}\otimes\mathbf{n}+\frac{\mathbf{w}_{t}}{\left|\mathbf{w}_{t}\right|}\otimes\frac{\mathbf{w}_{t}}{\left|\mathbf{w}_{t}\right|}\right)\,.\label{eq:fsi-TS-stiffness} \end{equation} \]

Similarly,

\[ D\delta G\left[\Delta\mathbf{u}\right]=\int_{\partial\Omega_{\eta}}\delta\mathbf{w}\cdot\alpha_{f}\left(\mathbf{t}_{t}^{\tau}\otimes\mathbf{n}\right)\cdot\left(\frac{\partial\Delta\mathbf{u}}{\partial\eta^{1}}\times\mathbf{g}_{2}+\mathbf{g}_{1}\times\frac{\partial\Delta\mathbf{u}}{\partial\eta^{2}}\right)\,d\eta^{1}d\eta^{2}\,. \]

The discretized form of \(\delta G\) is

\[ \begin{equation} \delta G=\sum_{a}\delta\mathbf{v}_{a}\cdot\int_{\partial\Omega}\mathbf{f}_{a}^{\tau}\,d\eta^{1}d\eta^{2}\,,\quad\boxed{\mathbf{f}_{a}^{\tau}=N_{a}\mathbf{t}_{t}^{\tau}\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|}\,.\label{eq:fsi-TS-discretized-work} \end{equation} \]

The discretized form of \(D\delta G\left[\Delta\mathbf{v}\right]\) is

\[ \begin{equation} D\delta G\left[\Delta\mathbf{v}\right]=\sum_{a}\delta\mathbf{w}_{a}\cdot\sum_{b}\int_{\partial\Omega_{\eta}}\left(\mathbf{K}_{ab}^{wu}\cdot\Delta\mathbf{u}_{b}+\mathbf{K}_{ab}^{ww}\cdot\Delta\mathbf{w}_{b}\right)\,d\eta^{1}d\eta^{2}\,,\label{eq:TS-discretized-tangent-1} \end{equation} \]

where

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{ww}=\alpha_{f}N_{a}N_{b}\left|\mathbf{g}_{1}\times\mathbf{g}_{2}\right|\mathbf{K}_{t}}\,,\label{eq:fsi-TS-Kww} \end{equation} \]

and

\[ \begin{equation} \boxed{\mathbf{K}_{ab}^{wu}=\alpha_{f}N_{a}\left(\mathbf{t}_{t}^{\tau}\otimes\mathbf{n}\right)\cdot\boldsymbol{\mathcal{A}}\left\{ \frac{\partial N_{b}}{\partial\eta^{2}}\mathbf{g}_{1}-\frac{\partial N_{b}}{\partial\eta^{1}}\mathbf{g}_{2}\right\} }\,.\label{eq:fsi-TS-Kwu} \end{equation} \]

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1. Jansen, Kenneth E; Whiting, Christian H; Hulbert, Gregory M. "A generalized-\(\alpha\) method for integrating the filtered {Navier}--{Stokes} equations with a stabilized finite element method." Comput. Methods Appl. Mech. Engrg., vol. 190, pp. 305--319 (2000). ↩