\subsection{Parametrization and constraints}

\begin{frame}
\frametitle{Problem parametrization}

Elliptical cracks are considered:

\vspace{\baselineskip}

\begin{columns}
	\begin{column}{0.60\textwidth}
		\begin{figure}
			\centering
			\includegraphics[width=0.90\textwidth]{../Figures/ellipse_parametrization.eps}
		\end{figure}
	\end{column}
	
	\begin{column}{0.40\textwidth}
		Parameters:
		\vspace{\baselineskip}
		\begin{itemize}
		\setlength\itemsep{2em}
			\item Coordinates of center point $\mathbf{x}_0$ ($\left\{x_0,y_0,z_0\right\}$)
			\item Rotation about the three axes $\theta_x, \theta_y$ and $\theta_z$
			\item Lengths $a$ and $b$
		\end{itemize}
	\end{column}
\end{columns}

\end{frame}

\begin{frame}
\frametitle{Problem parametrization}

Scaling of parameters:

\vspace{\baselineskip}

\begin{equation*}\label{eq:param_encoding}
	p_i=\frac{p_{i_1}+p_{i_2}}{2}+\frac{p_{i_2}-p_{i_1}}{2} \sin\left(\frac{\beta_i}{10} \cdot \frac{\pi}{2}\right)
\end{equation*}

where:

\begin{description}
\setlength\itemsep{1em}
\item[$\beta_i$] are design variables
\item[$p_i$] are geometrical parameters of the crack
\item[$p_{i_1},p_{i_2}$] are lower and upper values for the parameters
\end{description}

\end{frame}