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

The virtual work $\delta F$ on a biphasic-FSI interface was given in Eq. (3.7.1-10). The linearizations of $\delta F$ are given by $$\begin{equation} \begin{aligned}D\delta F\left[\Delta\mathbf{u}\right]= & \int_{\partial\Omega^{b}}\delta\mathbf{v}\cdot\alpha_{f}\frac{\varphi^{s}}{\varphi^{f}}\left(\divg\Delta\mathbf{u}\right)\mathbf{T}_{d}\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right) d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}-\delta\mathbf{v}\cdot\boldsymbol{\mathcal{C}}^{\tau}:\alpha_{f}\left(-\frac{\varphi^{s}}{\varphi^{f^{2}}}\left(\divg\Delta\mathbf{u}\right)\mathbf{D}^{w}+\mathbf{M}\cdot\grad\Delta\mathbf{u}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right) d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}-\delta\mathbf{v}\cdot\boldsymbol{\mathcal{C}}^{\tau}:\alpha_{f}\left(2\frac{\varphi^{s}}{\varphi^{f^{3}}}\left(\divg\Delta\mathbf{u}\right)\mathbf{w}\otimes\grad\varphi^{f}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right) d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}-\delta\mathbf{v}\cdot\boldsymbol{\mathcal{C}}^{\tau}:\alpha_{f}\left(-\frac{1}{\varphi^{f^{2}}}\mathbf{w}\otimes\left(-\grad^{T}\Delta\mathbf{u}\cdot\grad\varphi^{f}\right)\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right) d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}\delta\mathbf{v}\cdot\boldsymbol{\mathcal{C}}^{\tau}:\alpha_{f}\left(\frac{\varphi^{s}}{\varphi^{f^{2}}}\mathbf{w}\otimes\left(-\frac{1}{J}\left(\divg\Delta\mathbf{u}\right)\grad J+\grad\left(\divg\Delta\mathbf{u}\right)\right)\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right) d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}-\delta\mathbf{v}\cdot\alpha_{f}\boldsymbol{\tau}\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}\,,\\ D\delta F\left[\Delta\mathbf{w}\right] & =\int_{\partial\Omega^{b}}-\delta\mathbf{v}\cdot\alpha_{f}\boldsymbol{\mathcal{C}}^{\tau}:\left(\frac{1}{\varphi^{f}}\grad\Delta\mathbf{w}-\frac{1}{\varphi^{f^{2}}}\Delta\mathbf{w}\otimes\grad\varphi^{f}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right) d\eta^{1}d\eta^{2}\,,\\ D\delta F\left[\Delta J^{f}\right] & =\int_{\partial\Omega^{b}}-\delta\mathbf{v}\cdot\alpha_{f}\left(-p^{\prime}\mathbf{I}+\boldsymbol{\tau}_{J}^{\prime}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\Delta J^{f} d\eta^{1}d\eta^{2}\,. \end{aligned} \label{eq:BFSI-int-lin} \end{equation}$$ The discretized forms of this virtual work is $$\begin{equation} \delta F=\sum_{a}\delta\mathbf{v}_{a}\cdot\mathbf{f}_{a}\,,\label{eq:BSI-int-discret} \end{equation}$$ where $$\begin{equation} \mathbf{f}_{a}=\int_{\partial\Omega^{b}}-N_{a}\left(-p\mathbf{I}+\boldsymbol{\tau}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right) d\eta^{1}d\eta^{2}\,.\label{eq:BFSI-int-discret-parts} \end{equation}$$ The discretized forms of the linearizations are $$\begin{equation} \begin{aligned}D\delta F\left[\Delta\mathbf{u}\right] & =\sum_{a}\delta\mathbf{v}_{a}\cdot\sum_{b}\mathbf{K}_{ab}^{uu}\cdot\Delta\mathbf{u}_{b}\,,\\ D\delta F\left[\Delta\mathbf{w}\right] & =\sum_{a}\delta\mathbf{v}_{a}\cdot\sum_{b}\mathbf{K}_{ab}^{uw}\cdot\Delta\mathbf{w}_{b}\,,\\ D\delta F\left[\Delta J^{f}\right] & =\sum_{a}\delta\mathbf{v}_{a}\cdot\sum_{b}\mathbf{k}_{ab}^{uJ}\Delta J_{b}^{f}\,, \end{aligned} \label{eq:BFSI-int-lin-discret} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{K}_{ab}^{uu}= & \int_{\partial\Omega^{b}}\alpha_{f}N_{a}\frac{\varphi^{s}}{\varphi^{f}}\left(\boldsymbol{\tau}\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\right)\otimes\grad N_{b} d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}-\alpha_{f}N_{a}\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\cdot\boldsymbol{\mathcal{C}}_{d}\cdot\grad N_{b}\cdot\left(-\frac{\varphi^{s}}{\varphi^{f^{2}}}\mathbf{D}^{w}+\mathbf{M}\right) d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}-\alpha_{f}N_{a}\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\cdot\boldsymbol{\mathcal{C}}^{\tau}\cdot\mathbf{w}\cdot\left(2\frac{\varphi^{s}}{\varphi^{f^{3}}}\grad\varphi^{f}\otimes\grad N_{b}\right) d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}-\alpha_{f}N_{a}\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\cdot\boldsymbol{\mathcal{C}}^{\tau}\cdot\mathbf{w}\cdot\left(\frac{1}{\varphi^{f^{2}}}\grad N_{b}\otimes\grad\varphi^{f}\right) d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}\alpha_{f}N_{a}\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\cdot\boldsymbol{\mathcal{C}}^{\tau}\cdot\mathbf{w}\cdot\frac{\varphi^{s}}{\varphi^{f^{2}}}\left(-\frac{1}{J}\grad J\otimes\grad N_{b}+\grad\grad N_{b}\right) d\eta^{1}d\eta^{2}\\ & +\int_{\partial\Omega^{b}}-\alpha_{f}N_{a}\left(-p\mathbf{I}+\boldsymbol{\tau}\right)\cdot\boldsymbol{\mathcal{A}}\left\{ -\mathbf{g}_{2}\frac{\partial N_{b}}{\partial\eta^{1}}+\mathbf{g}_{1}\frac{\partial N_{b}}{\partial\eta^{2}}\right\} d\eta^{1}d\eta^{2}\,,\\ \mathbf{K}_{ab}^{uw}= & \int_{\partial\Omega^{b}}-\alpha_{f}N_{a}\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right)\cdot\boldsymbol{\mathcal{C}}_{d}\cdot\frac{1}{\varphi^{f}}\left(\grad N_{b}-\frac{1}{\varphi^{f}}N_{b}\grad\varphi^{f}\right) d\eta^{1}d\eta^{2}\,,\\ \mathbf{k}_{ab}^{uJ}= & \int_{\partial\Omega^{b}}-\alpha_{f}N_{a}N_{b}\left(-p^{\prime}\mathbf{I}+\boldsymbol{\tau}_{J}^{\prime}\right)\cdot\left(\mathbf{g}_{1}\times\mathbf{g}_{2}\right) d\eta^{1}d\eta^{2}\,. \end{aligned} \label{eq:BFSI-int-lin-discret-parts} \end{equation}$$ Note that $\boldsymbol{\mathcal{A}}\left\{ \cdot\right\}$ represents the skew-symmetric tensor form of the bracketed vector.