3.2 Changement d’ouvert

La transformation géométrique

 

 

Notations: $\Omega ^\eta =F^\eta (\Omega )$ :

Soit $\eta \in {\mathbb R}$ destiné à être petit.

\[ \boxed {x^\eta =x+\eta \theta (x),\Omega ^\eta =F^\eta (\Omega ).} \]

On introduit ($x^\eta =F^\eta (x)$):

\[ v\in H^1_0(\Omega ^\eta ),\rightarrow v^\eta (x)=v\circ F^\eta (x). \]

On notera que:

\[ \det (I+\eta D\theta )=1+\eta \hbox{div}(\theta )+\ldots , \]

et

\[ (I+\eta ^{t} \hskip-1.42263779528ptD\theta )^{-1}=I-\eta ^{t}\hskip-1.42263779528ptD\theta \ldots \]
 

 

Règles de calcul :

\[ \displaystyle \int _{\Omega ^\eta }v(x^\eta )=\displaystyle \int _\Omega v^\eta (x)\det (I+\eta D\theta ), \]\[ \nabla ^\eta v(x^\eta )=(I+\eta ^{t} \hskip-0.853582677165ptD\theta )^{-1}\nabla v^\eta (x) \]

où:

\[ \nabla ^\eta = ^{t} \hskip-0.853582677165pt[\displaystyle \frac{\partial .}{\partial x^\eta }] \]

et

\[ \nabla =^{t} \hskip-0.853582677165pt[\displaystyle \frac{\partial .}{\partial x}]. \]

Enfin:

\[ D\theta _{ij}=\displaystyle \frac{\partial \theta _ i}{\partial x_ j}. \]