\section{Introduction}
-This example composes an advanced version of step-41 since it considers an
+This example is an extension of step-41, considering a contact problem with an
elasto-plastic material behavior with isotropic hardening in three dimensions.
-That means that we have to take care of an additional nonlinearity by the
-material behavior. An other difference compared to step-41 is that
-the contact area is arranged at the boundary of the deformable body now.\\
-Since you can slightly reach a few million degrees of freedom in three dimensions,
-even with adaptive mesh refinement, we decided to use trilinos and p4est to run
-our code in parallel. On the other hand we have to deal with hanging nodes because
-of the adaptive mesh, which is an other advance of step-41.
+That means that we have to take care of an additional nonlinearity: the
+material behavior. Since we consider a three dimensional problem here, a
+separate difference to step-41 is that the contact area is at the boundary of
+the deformable body now, rather than in the interior. Finally, compared to
+step-41, we also have to deal with
+hanging nodes because of the adaptive mesh in both the handling of the linear
+system as well as of the inequality constraints; in the latter case, we will
+have to deal with prioritizing whether the constraints from the hanging nodes
+or from the inequalities are more important.
+
+Since you can very easily reach a few million degrees of freedom in three
+dimensions, even with adaptive mesh refinement, we decided to use Trilinos and
+p4est to run our code in parallel, building on the framework of step-40 for
+the parallelization.
\section{Classical formulation}
\end{align*}
with $u\in H^2(\Omega)$. The vector valued function $u$ denotes the
displacement in the deformable body. The first two lines describe the elast-plastic
-material behaviour. Therein the equation shows the deformation $\varepsilon (u)$ as the additive
+material behavior. Therein the equation shows the deformation $\varepsilon (u)$ as the additive
decomposition of the elastic part $A\sigma$ and the plastic part $\lambda$. $A$ is defined as
the compliance tensor of fourth order which contains some material constants and $\sigma$ as the
symmetric stress tensor of second order. So we have to consider the inequality in the second
row component-by-component and furthermore we have to distinguish two cases.\\
-The continuous and convex function $\mathcal{F}$ denotes the von mises flow function
+The continuous and convex function $\mathcal{F}$ denotes the von Mises flow function
$$\mathcal{F}(\tau) = \vert\tau^D\vert - \sigma_0$$
with $\sigma_0$ as yield stress. If there is no plastic deformation - that is $\lambda=0$ - this yields $\vert\sigma^D\vert < \sigma_0$
and otherwise if $\lambda > 0$ it follows that $\vert\sigma^D\vert = \sigma_0$. That means if the stress is smaller as the yield stress
$$\left(\sigma,\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+.$$
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 behaviour. The most simple one is called linear isotropic hardening
+There are different constitutive laws to describe those material behavior. The most simple one is called linear isotropic hardening
with the flow function $\mathcal{F}(\tau,\eta) = \vert\tau^D\vert - (\sigma_0 + \gamma\eta)$.
It can be considered by establishing an additional term in our primal-mixed formulation:\\
Find a pair $\lbrace(\sigma,\xi),u\rbrace\in \Pi (W\times L^2(\Omega,\mathbb{R}))\times V^+$ with
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.\\
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 shows the theorem of Lions and Stampaccia
+It consists of a nonlinear variational inequality and has a unique solution as it shows the theorem of Lions and Stampaccia
(A proof can be found in Rodrigues: Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987).\\
-To handle the nonlinearity of the constitutive law we use a newton method and to deal with the contact we apply an
-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.
+To handle the nonlinearity of the constitutive law we use a Newton method and to deal with the contact we apply an
+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{Linearisation of the constitutive law for the Newton method}
-For the newton method we have to linearise the following semi-linearform
+For the Newton method we have to linearise the following semi-linearform
$$a(\psi;\varphi) := \left(P_{\Pi}(C\varepsilon(\varphi)),\varepsilon(\varphi)\right).$$
Becaus 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 to solve a minimisation problem for a known $u^i$ in every SQP-step of the form
time.
\section{The primal-dual active set algorithm combined with the inexact semi smooth
-newton method}
+Newton method}
-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
+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
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)] Assembel the newton matrix $a'(U^k;\varphi_i,\varphi_j)$ and the right-hand-side $F(U^k)$.
+ \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
\begin{align*}
AU^k + B\Lambda^k & = F, &\\
\section{Implementation}
-\end{document}
\ No newline at end of file
+\end{document}