From 2a7a246219e2a595b235d5c69f3394288a84e512 Mon Sep 17 00:00:00 2001 From: frohne Date: Fri, 22 Feb 2013 03:58:08 +0000 Subject: [PATCH] extend the algorithm by a damping step - section 7 git-svn-id: https://svn.dealii.org/trunk@28521 0785d39b-7218-0410-832d-ea1e28bc413d --- .../examples/step-42/doc/intro-step-42.tex | 52 +++++++++++++------ 1 file changed, 36 insertions(+), 16 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 b92a1d6b61..9a11fbdf51 100644 --- a/deal.II/examples/step-42/doc/intro-step-42.tex +++ b/deal.II/examples/step-42/doc/intro-step-42.tex @@ -278,21 +278,30 @@ Now we describe an algorithm that combines the Newton-method, which we use for the nonlinear constitutive law, with the semismooth Newton method for the contact. It sums up the results of the sections before and works as follows: \begin{itemize} - \item[(0)] Initialize $\mathcal{A}_k$ and $\mathcal{F}_k$, such that $\mathcal{S} = \mathcal{A}_k \cup \mathcal{F}_k$ and $\mathcal{A}_k \cap \mathcal{F}_k = \emptyset$ and set $k = 1$. - \item[(1)] Assemble the Newton matrix $a'(U^k;\varphi_i,\varphi_j)$ and the right-hand-side $F(U^k)$. - \item[(2)] Find the primal-dual pair $(U^k,\Lambda^k)$ that satisfies + \item[(0)] Initialize $\mathcal{A}_k$ and $\mathcal{F}_k$, such that + $\mathcal{S} = \mathcal{A}_k \cup \mathcal{F}_k$ and $\mathcal{A}_k \cap + \mathcal{F}_k = \emptyset$ and set $k = 1$. The start value $U^0$ fullfills our + obstacle condition. + \item[(1)] Assemble the Newton matrix $a'(U^{k-1};\varphi_i,\varphi_j)$ and the + right-hand-side $F(U^{k-1})$. + \item[(2)] Find the primal-dual pair $(\tilde U^k,\Lambda^k)$ that satisfies \begin{align*} - AU^k + B\Lambda^k & = F, &\\ - \left[B^TU^k\right]_i & = G_i & \forall i\in\mathcal{A}_k,\\ + A\tilde U^k + B\Lambda^k & = F, &\\ + \left[B^T\tilde U^k\right]_i & = G_i & \forall i\in\mathcal{A}_k,\\ \Lambda^k_i & = 0 & \forall i\in\mathcal{F}_k. \end{align*} - \item[(3)] Define the new active and inactive sets by + \item[(3)] Damping for $k>2$ by applying a line search and calculating a linear + combination of $U^{k-1}$ and $\tilde U^k$. Find an + $\alpha_i:=2^{-i},(i=0,\ldots,10)$ so that $$U^k := \alpha_i\tilde U^k + + (1-\alpha_i)U^{k-1}$$ yields $$\vert + F\left(U^{k}\right) \vert < \vert F\left(U^{k-1}\right) \vert.$$ + \item[(4)] Define the new active and inactive sets by $$\mathcal{A}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + c\left(\left[B^TU^k\right]_i - G_i\right) > 0\rbrace,$$ $$\mathcal{F}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + c\left(\left[B^TU^k\right]_i - G_i\right) \leq 0\rbrace.$$ - \item[(4)] If $\mathcal{A}_{k+1} = \mathcal{A}_k$ and $\vert - F\left(U^{k+1}\right) \vert < \delta$ then stop, else set $k=k+1$ and go to + \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). \end{itemize} \noindent @@ -311,14 +320,25 @@ $$G_i = \begin{cases} 0, & \text{if}\quad i>m. \end{cases}$$\\ Compared to step-41, step (1) is added but it should be clear -from the sections above that we only linearize the problem. In step (2) we have to solve a linear -system of equations again. And now the solution has to fulfill two stopping -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.\\ -The idea of this method can also be found in Brunssen, Schmid, Schaefer, -Wohlmuth: 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. +from the sections above that we only linearize the problem. In step (2) we have +to solve a linear system of equations again. And now the solution has to fulfill two stopping +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 +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 +problem in the very first step ($k=0$) we get a reasonable start value but as you +can see in the results damping is important if we choose for example a ball as +obstacle.\\ +Damping our Newton method is more important for the nonlinearity cause by the +constitutive law as for the contact. For this reason we start to damp our method +for $k>2$ when we have two plastic iterations. Note that $U^1$ is a solution of +an elastic problem. + +\large{Umformulierung des Algorithmus, so dass $U^0$ Lösung des elastischen +Startproblems ist.} \section{Adaptive mesh refinement} -- 2.39.5