diff --git a/presentation/pres03.tex b/presentation/pres03.tex index 2cb30b6..657594d 100755 --- a/presentation/pres03.tex +++ b/presentation/pres03.tex @@ -184,7 +184,7 @@ University of Groningen\\[0.5cm] \onslide<3-> And a corrector field $\vec{w}$ which satisfies: \onslide<4-> \begin{align} -\vec{u} & \approx \vec{u}_{meas} + \vec{w} \quad \text{in} \quad \Omega \label{eq:corrector} \\ +\vec{u} & = \vec{u}_{meas} + \vec{w} \quad \text{in} \quad \Omega \label{eq:corrector} \\ \nabla \cdot \vec w & = 0 \quad \text{in} \quad \Omega \label{eq:correctorDiv} \\ \vec w & = \vec 0 \quad \text{on} \quad \partial \Omega \label{eq:correctorBC} \end{align} @@ -198,7 +198,8 @@ University of Groningen\\[0.5cm] \frametitle{The corrector field: Continuum problem} \footnotesize -\onslide<1-> Applying the decomposition $\vec{u} \approx \vec{u}_{meas} + \vec{w}$ into the original equation and writing a variational problem for $\vec w$ we have the following: Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that +\onslide<1-> Applying the decomposition $\vec{u} = \vec{u}_{meas} + \vec{w}$ into the original equation and writing a variational problem for $\vec w$ we have the following:\\ +Find $(\vec w(t) ,p(t)) \in H^1_0(\Omega)\times L^2(\Omega)$ such that \onslide<2-> \begin{equation*} \int_{\Omega} \rho \frac{\partial \vec{w}}{\partial t} \cdot \vec{v} + \rho \big ( ( \vec{u}_{meas} + \vec w) \cdot \nabla \big) \vec{w} \cdot \vec{v} + \rho \big ( \vec{w} \cdot \nabla \big) \vec{u}_{meas} \cdot \vec{v} + \mu \nabla \vec{w} : \nabla \vec{v} - p \nabla \cdot \vec{v} + q \nabla \cdot \vec{w} \notag \end{equation*} @@ -256,12 +257,12 @@ $\footnotesize \onslide<1-> \begin{theorem} -There exists a unique solution of Problem \ref{eq:Corrector_discrete} under condition: $$\rho/\tau + C_\Omega^{-2} \mu/2 - \rho 3 \| \nabla\vec u_{meas}^k\|_\infty > 0$$ for all$k>0$. +There exists a unique solution of Problem (\ref{eq:Corrector_discrete}) under the condition: $$\rho/\tau + C_\Omega^{-2} \mu/2 - \rho 3 \| \nabla\vec u_{meas}^k\|_\infty > 0$$ for all$k>0$. \end{theorem} \onslide<2-> We can furthermore prove the following energy balance: \onslide<3-> -\begin{theorem} For$(\vec w^k ,p^k)$solution of Problem \ref{eq:Corrector_discrete}, with$\ell_j(\vec v,q)=0$it holds +\begin{theorem} For$(\vec w^k ,p^k)$solution of Problem (\ref{eq:Corrector_discrete}), with$\ell_j(\vec v,q)=0$it holds \begin{equation*}\label{eq:energy} \| \vec w^k \|^2_{L_2(\Omega)} \leq \| \vec w^{k-1} \|^2_{L_2(\Omega)} \end{equation*} @@ -359,7 +360,7 @@ Also perturbations were added into the measurements: \begin{figure}[!hbtp] \begin{center} \includegraphics[height=0.5\textwidth]{images/perturbation_pres.png} -\caption{Different perturbation scenarios} +\caption{\small Different perturbation scenarios.$(\infty , 120 \%)$:$\vec{w} \times 200$,$(10 \ dB , 120 \%)$:$\delta \vec{u}, \vec{w} \times 4$, rest:$\vec{w} \times 4\$ } \end{center} \end{figure} @@ -557,6 +558,13 @@ Experiments using real 4D flow data \section{Conclusions} +\begin{frame} + \frametitle{Experiments} +\begin{center} +Conclusions +\end{center} +\end{frame} + \begin{frame} \frametitle{Conclusions and future work}