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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 \[ (I+\eta ^{t} \hskip-1.42263779528ptD\theta )^{-1}=I-\eta ^{t}\hskip-1.42263779528ptD\theta \ldots \] |
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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: \[ D\theta _{ij}=\displaystyle \frac{\partial \theta _ i}{\partial x_ j}. \] |
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