From 52c064fa7b70d779968e491e58d0039ca03c3659 Mon Sep 17 00:00:00 2001 From: bangerth Date: Sat, 27 Jul 2013 15:35:03 +0000 Subject: [PATCH] Minor adjustments. git-svn-id: https://svn.dealii.org/trunk@30175 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-42/doc/intro-step-42.tex | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 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 a6c6b1f752..7f2e577d33 100644 --- a/deal.II/examples/step-42/doc/intro-step-42.tex +++ b/deal.II/examples/step-42/doc/intro-step-42.tex @@ -83,7 +83,7 @@ is zero with the outward normal $n$. If there is contact ($u_n = g$) the tangent vanishes, because we consider a frictionless situation and the normal stress is negative. The gap $g$ comes with the start configuration of the obstacle and the deformable body. We refer that you have to ensure that the obstacle does not hit -the boundary of $\Gamma_C$ +the boundary of $\Gamma_C$ \section{Derivation of the variational inequality} @@ -227,7 +227,8 @@ Again the first case is for elastic and the second for plastic deformation. Just as in step-41 we compose a saddle point problem out of the minimisation problem. Again we do so to gain a formulation that allows us to solve a linear -system of equations finally.\\ +system of equations finally. + We introduce a Lagrange multiplier $\lambda$ and the convex cone $K\subset W'$, $W'$ dual space of the trace space $W:=\left[ H_0^{\frac{1}{2}}(\Gamma_C) \right]^{\textrm{dim}}$ of $V$ restricted to $\Gamma_C$, $$K:=\{\mu\in W':\mu_T = 0,\quad\langle\mu n,v\rangle_{\Gamma_C}\geq 0,\quad @@ -236,9 +237,8 @@ of Lagrange multipliers, where $\langle\cdot,\cdot\rangle$ denotes the duality pairing, i.e. a boundary integral, between $W'$ and $W$. Intuitively, $K$ is the cone of all "non-positive functions", except that $ K\subset \left( \left[ H_0^{\frac{1}{2}}(\Gamma_C) \right]^{\textrm{dim}} \right)' $ and -so contains other objects besides regular functions as well. This yields:\\ +so contains other objects besides regular functions as well. This yields: -\noindent \textit{Find $u\in V$ and $\lambda\in K$ such that} \begin{align*} \hat{a}(u,v) + b(v,\lambda) &= f(v),\quad &&v\in V\\ @@ -304,7 +304,7 @@ method for the contact. It sums up the results of the sections before and works $$\mathcal{F}_{k+1}:=\lbrace p\in\mathcal{S}:\Lambda^k_p + c\left(\left[B^TU^k\right]_p - G_p\right) \leq 0\rbrace.$$ Projection $U^k$ so that it holds the second equation in (2) - $$\hat U^K := P_{\mathcal{A}_{k+1}}(U^k).$$ + $$\hat U^K := P_{\mathcal{A}_{k+1}}(U^k).$$ \item[(5)] If $\mathcal{A}_{k+1} = \mathcal{A}_k$ and $\vert F\left(U^{k}\right) \vert < \delta$ then stop, else set $k=k+1$ and go to step (1). @@ -353,7 +353,7 @@ above that we only linearize the problem. In step (2) we have to solve a linear criteria. $\mathcal{A}_{k+1} = \mathcal{A}_k$ makes sure that the contact zones are iterated out and the second ensures an accurate enough residual which means that the plastic zones are also iterated out.\\ A similar method can also be found in Brunssen, Schmid, Schaefer, Wohlmuth: A -fast and robust iterative solver for nonlinear contact problems using a +fast and robust iterative solver for nonlinear contact problems using a primal-dual active set strategy and algebraic multigrid, Int. J. Numer. Meth. Engng, 2007, 69, pp. 524-543. But in advance we apply a line search to obtain a more robust method regarding the start value. Solving an elastic @@ -379,12 +379,12 @@ results it yields a quite reasonable adaptive mesh for the contact zone. This tutorial is essentailly a mixture of step-40 and step-41 but instead of PETSc we let the Trilinos library deal with parallelizing the linear algebra -(like in step-32). Since we are trying to solve a similar problem like in +(like in step-32). Since we are trying to solve a similar problem like in step-41 we will use the same methods but now in parallel. Another difficulty is the handling of the different constraints from -(the dirichlet conditons), the hanging nodes and the inequality condition that -arises from the contact. For this purpose we create three objects of type +(the dirichlet conditons), the hanging nodes and the inequality condition that +arises from the contact. For this purpose we create three objects of type ConstraintMatrix. Beside the ConstitutiveLaw class there is another new class called Input. This -- 2.39.5