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J.E. Garay Labra 2020-08-18 15:49:00 +02:00
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presentation/pres03.tex
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@ -184,7 +184,7 @@ University of Groningen\\[0.5cm]
\onslide<3-> And a corrector field $\vec{w}$ which satisfies: \onslide<3-> And a corrector field $\vec{w}$ which satisfies:
\onslide<4-> \begin{align} \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} \\ \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} \vec w & = \vec 0 \quad \text{on} \quad \partial \Omega \label{eq:correctorBC}
\end{align} \end{align}
@ -198,7 +198,8 @@ University of Groningen\\[0.5cm]
\frametitle{The corrector field: Continuum problem} \frametitle{The corrector field: Continuum problem}
\footnotesize \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*} \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 \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*} \end{equation*}
@ -256,12 +257,12 @@ $
\footnotesize \footnotesize
\onslide<1-> \onslide<1->
\begin{theorem} \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} \end{theorem}
\onslide<2-> \onslide<2->
We can furthermore prove the following energy balance: We can furthermore prove the following energy balance:
\onslide<3-> \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} \begin{equation*}\label{eq:energy}
\| \vec w^k \|^2_{L_2(\Omega)} \leq \| \vec w^{k-1} \|^2_{L_2(\Omega)} \| \vec w^k \|^2_{L_2(\Omega)} \leq \| \vec w^{k-1} \|^2_{L_2(\Omega)}
\end{equation*} \end{equation*}
@ -359,7 +360,7 @@ Also perturbations were added into the measurements:
\begin{figure}[!hbtp] \begin{figure}[!hbtp]
\begin{center} \begin{center}
\includegraphics[height=0.5\textwidth]{images/perturbation_pres.png} \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{center}
\end{figure} \end{figure}
@ -557,6 +558,13 @@ Experiments using real 4D flow data
\section{Conclusions} \section{Conclusions}
\begin{frame}
\frametitle{Experiments}
\begin{center}
Conclusions
\end{center}
\end{frame}
\begin{frame} \begin{frame}
\frametitle{Conclusions and future work} \frametitle{Conclusions and future work}