<h3>The primal-dual active set algorithm combined with the inexact semi smooth
Newton method</h3>
-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:
+Now we describe an algorithm that combines the damped semismooth 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:
<ol>
- <li> 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 $k = 1$.
- <li> Assemble the Newton matrix $a'(U^k;\varphi_i,\varphi_j)$ and the right-hand-side $F(U^k)$.
- <li> Find the primal-dual pair $(U^k,\Lambda^k)$ that satisfies
+ <li> 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 $\hat U^0 :=
+ P_{\mathcal{A}_k}(0)$ fullfills our obstacle condition.
+ <li> Assemble the Newton matrix $A := a'(\hat
+ U^{k-1};\varphi_i,\varphi_j)$ and the right-hand-side $F(\hat U^{k-1})$.
+ <li> Find the primal-dual pair $(\bar U^k,\Lambda^k)$ that satisfies
@f{align*}
- AU^k + B\Lambda^k & = F, &\\
- \left[B^TU^k\right]_i & = G_i & \forall i\in\mathcal{A}_k,\\
+ A\bar U^k + B\Lambda^k & = F, &\\
+ \left[B^T\bar U^k\right]_i & = G_i & \forall i\in\mathcal{A}_k,\\
\Lambda^k_i & = 0 & \forall i\in\mathcal{F}_k.
@f}
+ <li> Damping for $k>2$ by applying a line search and calculating a linear
+ combination of $U^{k-1}$ and $\bar U^k$. Find an
+ $\alpha_i:=2^{-i},(i=0,\ldots,10)$ so that
+ @f{gather*}U^k := \alpha_i\bar U^k +
+ (1-\alpha_i)U^{k-1}@f}
+ yields
+ @f{gather*}\vert F\left(U^{k}\right) \vert < \vert F\left(U^{k-1}\right) \vert.\f}
<li> Define the new active and inactive sets by
@f{gather*}\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,@f}
@f{gather*}\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.@f}
+ Projection $U^k$ so that it holds the second equation in (2)
+ @f{gather*}\hat U^K := P_{\mathcal{A}_{k+1}}(U^k).@f}
<li> 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
- step (2).
+ F\left(U^{k}\right) \vert < \delta$ then stop, else set $k=k+1$ and go to
+ step (1).
</ol>
-The mass matrix $B\in\mathbb{R}^{n\times m}$, $n>m$, is quadratic in our
-situation since $\Lambda^k$ is only defined on $\Gamma_C$:
+The meaning of the decorated and none decorated $U$ is as follows: $\bar U$
+denotes the solution of the linear system of equations in (2), $U$ is the
+damped solution and equals to $\bar U$ if the damping parameter $\alpha_0 =
+1$ and $\hat U := P_{\mathcal{A}}(U)$ is the projection of the active
+components in $\mathcal{A}$ to the gap
+
+@f{gather*}P_{\mathcal{A}}(U):=\begin{cases}
+U_i, & \textrm{if}\quad i\notin\mathcal{A}\\
+{}^{G_i}/_{B^T_i}, & \textrm{if}\quad
+i\in\mathcal{A}.
+\end{cases}@f}\\
+The mass matrix $B\in\mathbb{R}^{n\times m}$, $n>m$, is quadratic in our situation since $\Lambda^k$ is only defined on $\Gamma_C$:
@f{gather*}B_{ij} = \begin{cases}
\int\limits_{\Gamma_C}\varphi_i^2(x)dx, & \text{if}\quad i=j\\
0, & \text{if}\quad i\neq j.
\end{cases}@f}
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
+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, Schäfer,
+A similar method can also be found in Brunssen, Schmid, Schäfer,
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.
+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 ($k=1$) 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 and $U^2$ is the first plastic solution. A linear combination
+between these both results in stresses which are not in the convex set
+of the feasible stresses.
<h3>Adaptive mesh refinement</h3>
projects / dealii-svn.git / commitdiff