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Serge-Étienne Parent edited section_Constitutive_relationships_Retardation_factor__.tex
over 8 years ago
Commit id: a09039ead84be8c63da06acd3215dfa267de51bd
deletions | additions
diff --git a/section_Constitutive_relationships_Retardation_factor__.tex b/section_Constitutive_relationships_Retardation_factor__.tex
index 3073495..fef6b84 100644
--- a/section_Constitutive_relationships_Retardation_factor__.tex
+++ b/section_Constitutive_relationships_Retardation_factor__.tex
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Apply Gauss divergence theorem on the first part of the left-hand side.
\begin{align}
\int_{\Omega} \nabla \cdot \left( s D \nabla u \right) d\Omega = \int_{\Gamma_{N} \cup \Gamma_{D}} s
D \nabla u \cdot n d\Gamma
\end{align}
Because $s = 0$ on $\Gamma_{D}$.
\begin{align}
\int_{\Omega} \nabla \cdot \left( s D \nabla u \right) d\Omega = \int_{\Gamma_{N}} s
D \nabla u \cdot n d\Gamma
\end{align}
Put the pieces together.
\begin{align}
\int_{\Omega}\frac{\partial u}{\partial t} s d\Omega + \int_{\Omega}\nabla \cdot \left(\vec{v} u \right) s d\Omega - \int_{\Gamma_{N}} s
D \nabla u \cdot n d\Gamma + \int_{\Omega} D \nabla u \nabla s d\Omega = 0
\end{align}