# Shape Derivatives

The goal of our project aims to perform an optimization problem over the geometry of our domain:

\begin{align}
\boldsymbol{x} = \mathrm{argmin}_{\boldsymbol{x}} E( \boldsymbol{u}(\boldsymbol{x})) \quad s.t. \quad \boldsymbol{x}_b = \boldsymbol{y}
\end{align}

Where $\boldsymbol{u}$ is given by its own minimization problem:

\begin{align}
\boldsymbol{u} = \mathrm{argmin}_{\boldsymbol{u}} E(\boldsymbol{u}) \quad s.t. \quad \boldsymbol{u}_b = \boldsymbol{f} \quad \forall \boldsymbol{u} \in \Omega(\boldsymbol{x})
\end{align} hi

## Open Questions
1. What optimization algorithm can we use for the first equation? Gradient descent is a start, can we do a newton method? Why/Why not? What about the Sobolov-preconditioning solver of the Repulsive Curves paper ?

2. Where does this fit in existing shape derivatives work?