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
&\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
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 $\sigma_t = \sigma\cdot n - \sigma_n n$
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
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
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