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 $\Gamma_C$,
+\right]^{\textrm{dim}}$ of $V$ restricted to $\Gamma_C$,
@f{gather*}K:=\{\mu\in W':\mu_T = 0,\quad\langle\mu n,v\rangle_{\Gamma_C}\geq 0,\quad
\forall v\in H_0^{\frac{1}{2}}(\Gamma_C), v \ge 0\text{ on }\Gamma_C \}@f}
of Lagrange multipliers, where $\langle\cdot,\cdot\rangle$
@f{gather*}\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.@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}
+ @f{gather*}\hat U^K := P_{\mathcal{A}_{k+1}}(U^k).@f}
<li> If $\mathcal{A}_{k+1} = \mathcal{A}_k$ and $\vert
F\left(U^{k}\right) \vert < \delta$ then stop, else set $k=k+1$ and go to
step (1).
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
+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, Schäfer,