From d1835792c96e24f7660bcd7abb3788fa1a7a13f5 Mon Sep 17 00:00:00 2001 From: frohne Date: Tue, 5 Feb 2013 04:10:36 +0000 Subject: [PATCH] minor edits git-svn-id: https://svn.dealii.org/trunk@28229 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-42/doc/intro-step-42.tex | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/deal.II/examples/step-42/doc/intro-step-42.tex b/deal.II/examples/step-42/doc/intro-step-42.tex index ba71b4afbd..c4b7b0cc26 100644 --- a/deal.II/examples/step-42/doc/intro-step-42.tex +++ b/deal.II/examples/step-42/doc/intro-step-42.tex @@ -79,7 +79,8 @@ deformable body. \section{Derivation of the variational inequality} -As a starting point we want to minimise an energy functional: +As a starting point to derive the equations above, let us imagine that we want +to minimise an energy functional: $$E(\tau) := \dfrac{1}{2}\int\limits_{\Omega}\tau A \tau d\tau,\quad \tau\in \Pi W^{div}$$ with $$W^{div}:=\lbrace \tau\in @@ -88,14 +89,15 @@ $$\Pi \Sigma:=\lbrace \tau\in \Sigma, \mathcal{F}(\tau)\leq 0\rbrace$$ as the set of admissible stresses which is defined by a continious, convex flow function $\mathcal{F}$. -With the goal to derive the dual formulation of the minimisation problem, we define a lagrange function: +With the goal of deriving the dual formulation of the minimisation +problem, we define a lagrange function: $$L(\tau,\varphi) := E(\tau) + (\varphi, div(\tau)),\quad \lbrace\tau,\varphi\rbrace\in\Pi W^{div}\times V^+$$ with $$V^+ := \lbrace u\in V: u_n\leq g \text{ on } \Gamma_C \rbrace$$ $$V:=\left[ H_0^1 \right]^{dim}:=\lbrace u\in \left[H^1(\Omega)\right]^{dim}: u = 0 \text{ on } \Gamma_D\rbrace$$ -By building the fr\'echet derivatives of $L$ for both components we obtain the dual formulation for the stationary case -which is known as \textbf{Hencky-Type-Model}:\\ +By building the Fr\'echet derivatives of $L$ for both components we obtain the +dual formulation for the stationary case which is known as \textbf{Hencky-Type-Model}:\\ Find a pair $\lbrace\sigma,u\rbrace\in \Pi W\times V^+$ with $$\left(A\sigma,\tau - \sigma\right) + \left(u, div(\tau) - div(\sigma)\right) \geq 0,\quad \forall \tau\in \Pi W^{div}$$ $$-\left(div(\sigma),\varphi - u\right) \geq 0,\quad \forall \varphi\in V^+.$$ -- 2.39.5