From 82fa79f861a379cc759ebb8b4d5b2071f2db3ade Mon Sep 17 00:00:00 2001 From: frohne Date: Wed, 27 Feb 2013 22:50:22 +0000 Subject: [PATCH] fixing the right-hand-side in the linearization; removeing some backslashes git-svn-id: https://svn.dealii.org/trunk@28624 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-42/doc/intro-step-42.tex | 9 +++++---- deal.II/examples/step-42/doc/intro.dox | 9 ++++++--- 2 files changed, 11 insertions(+), 7 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 811781ad02..0a8e2d649e 100644 --- a/deal.II/examples/step-42/doc/intro-step-42.tex +++ b/deal.II/examples/step-42/doc/intro-step-42.tex @@ -182,9 +182,9 @@ To handle the nonlinearity of the constitutive law we use a Newton method and to active set method like in step-41. To be more concrete we combine both methods to an inexact semi smooth Newton method - inexact since we use an iterative solver for the linearised problems in each Newton step. -\section{Linearisation of the constitutive law for the Newton method} +\section{Linearization of the constitutive law for the Newton method} -For the Newton method we have to linearise the following semi-linearform +To apply Newton method we have to linearise the following semi-linearform $$a(\psi;\varphi) := \left(P_{\Pi}(C\varepsilon(\varphi)),\varepsilon(\varphi)\right).$$ Because we have to find the solution $u$ in the convex set $V^+$, we have to apply an SQP-method (SQP: sequential quadratic programming). That means we have @@ -196,8 +196,9 @@ to solve a minimisation problem for a known $u^i$ in every SQP-step of the form &\rightarrow& \textrm{min},\quad u^{i+1}\in V^+. \end{eqnarray*} Neglecting the constant terms $ a(u^i;u^i)$ and $ a'(u^i;u^i,u^i)$ we obtain the -following minimisation problem $$\dfrac{1}{2} a'(u^i;u^{i+1},u^{i+1}) - F(u^i)\rightarrow \textrm{min},\quad u^{i+1}\in V^+$$ with -$$F(\varphi) := \left(a'(\varphi;\varphi,u^{i+1}) - a(\varphi;u^{i+1}) \right).$$ +following minimisation problem $$\dfrac{1}{2} a'(u^i;u^{i+1},u^{i+1}) - +F(u^{i+1})\rightarrow \textrm{min},\quad u^{i+1}\in V^+$$ with $$F(\varphi) := +\left(a'(u^{i};u^{i},\varphi) - a(u^{i},\varphi) \right).$$ In the case of our constitutive law the Fr\'echet derivative of the semi-linearform $a(.;.)$ at the point $u^i$ is diff --git a/deal.II/examples/step-42/doc/intro.dox b/deal.II/examples/step-42/doc/intro.dox index 4d5be71f76..e90f986369 100644 --- a/deal.II/examples/step-42/doc/intro.dox +++ b/deal.II/examples/step-42/doc/intro.dox @@ -78,7 +78,8 @@ plastic deformations. The third equation is called equilibrium condition with a force of volume density $f$ which we will neglect in our example. The boundary of $\Omega$ separates as follows $\Gamma=\Gamma_D\bigcup\Gamma_C$ and $\Gamma_D\bigcap\Gamma_C=\emptyset$. -At the boundary $\Gamma_D$ we have zero Dirichlet conditions. $\Gamma_C$ denotes the potential contact boundary.\\ +At the boundary $\Gamma_D$ we have zero Dirichlet conditions. $\Gamma_C$ denotes the potential contact boundary. + The last two lines decribe the so-called Signorini contact conditions. If there is no contact the normal stress @f{gather*} \sigma_n = \sigma n\cdot n@f} is zero with the outward normal $n$. If there is contact ($u_n = g$) the tangential stress $\sigma_t = \sigma\cdot n - \sigma_n n$ @@ -116,7 +117,8 @@ tensor $C=A^{-1}$ we achieve the primal-mixed version of our problem: Find a pair $\lbrace\sigma,u\rbrace\in \Pi W\times V^+$ with @f{gather*}\left(\sigma,\tau - \sigma\right) - \left(C\varepsilon(u), \tau - \sigma\right) \geq 0,\quad \forall \tau\in \Pi W@f} @f{gather*}\left(\sigma,\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+.@f} -Therein $\varepsilon$ denotes the linearised deformation tensor with $\varepsilon(u) := \dfrac{1}{2}\left(\nabla u + \nabla u^T\right)$ for small deformations.\\ +Therein $\varepsilon$ denotes the linearised deformation tensor with $\varepsilon(u) := \dfrac{1}{2}\left(\nabla u + \nabla u^T\right)$ for small deformations. + Most materials - especially metals - have the property that they show some hardening effects during the forming process. There are different constitutive laws to describe those material behaviors. The simplest one is called linear isotropic hardening described by the flow function @@ -180,7 +182,8 @@ tr(\varepsilon(u))$ the volumetric part of the stress tensor. In the next step we test in a pointwise sense where the deviator part of the stress in a norm is bigger than the yield stress. If there are such points we -project the deviator stress in those points back to the yield surface. Methods of this kind are called projections algorithm or radial-return-algorithm.\\ +project the deviator stress in those points back to the yield surface. Methods of this kind are called projections algorithm or radial-return-algorithm. + Now we have a primal formulation of our elasto-plastic contact problem which only depends on the displacement $u$. It consists of a nonlinear variational inequality and has a unique solution as it satisfies the theorem of Lions and Stampaccia. A proof can be found in -- 2.39.5