Strain energy density can then be calculated
\begin{equation} \label{eq:W} \label{eq:W}W=\frac{1}{2}\boldsymbol{\sigma\epsilon}\\ \end{equation}and the total potential energy found
\begin{equation} \label{eq:psi} \label{eq:psi}\boldsymbol{\prod}=\boldsymbol{\int_{\Omega}}W\mathit{d}\boldsymbol{\Omega}-\boldsymbol{\int_{\Omega}B_{s}.u}\mathit{d}\boldsymbol{\Omega}-\boldsymbol{\int_{\Omega_{1}}B_{c}.u}\mathit{d}\boldsymbol{\Omega_{1}}-\boldsymbol{\int_{\Gamma_{1}}T_{s}.u}\mathit{d}\boldsymbol{\Gamma_{1}}\\ \end{equation} \begin{equation} \boldsymbol{\Omega_{1}}=\{(x,y,z)\in\boldsymbol{\Omega}:z>S_{c}\}\nonumber \\ \end{equation} \begin{equation} \boldsymbol{\Gamma_{1}}=\{(x,y,z)\in\boldsymbol{\Gamma}:x>0,z>S_{c}\}\nonumber \\ \end{equation}By taking the directional derivative of \(\prod\) with respect to the change in \(u\) and setting it to zero the displacement field \(u\) can be calculated at the minimum potential energy.
\begin{equation} \label{eq:F_new} \label{eq:F_new}F=\boldsymbol{\bigtriangledown_{u}}\prod(\boldsymbol{u})=0\\ \end{equation}Subject to the Dirichlet boundary conditions, Equations \ref{eq:BC1} and \ref{eq:BC2}.
\begin{equation} \label{eq:BC1} \label{eq:BC1}\boldsymbol{u}|_{\boldsymbol{\Omega_{bc1}}}=\begin{bmatrix}u_{x}\\ u_{y}\\ 0\end{bmatrix}\hskip 5.690551pt\\ \end{equation} \begin{equation} \boldsymbol{\Omega_{bc1}}=\{(x,y,z)\in\boldsymbol{\Omega}:z>tol_{z}\}\nonumber \\ \end{equation}and
\begin{equation} \label{eq:BC2} \label{eq:BC2}\boldsymbol{u}|_{\boldsymbol{\Omega_{bc2}}}=\begin{bmatrix}0\\ 0\\ 0\end{bmatrix}\hskip 5.690551pt\\ \end{equation} \begin{equation} \boldsymbol{\Omega_{bc2}}=\{(x,y,z)\in\boldsymbol{\Omega_{bc1}}:(x,y)<tol,(x,y)>-tol\}\nonumber \\ \end{equation}