Table 1.

Augmented Lagrangian algorithm for frictionless contact of porous media

Initialization of contact iterations:

Set r = 0

$\text{Set}{\lambda }_n^{\left(r\right)}={\lambda }_n\text{from last time step}$

$\text{Set}{\lambda }_p^{\left(r\right)}={\lambda }_p\text{from last time step}$

Solution Step:

Do while contact iterations have not converged

Do while nonlinear iterations have not converged

Evaluate g(X(1)) and π(X(1)) for all X(1) ∊ ∂v(1)

$\text{Set}{t}_n=\{\begin{array}{cc}{\lambda }_n^{\left(r\right)}+{\epsilon }_{n}g\hfill & {\lambda }_n^{\left(r\right)}+{\epsilon }_{n}g<0\hfill \\ 0\hfill & {\lambda }_n^{\left(r\right)}+{\epsilon }_{n}g\ge 0\hfill \end{array}\phantom{\}}$

$\text{Set}\{\begin{array}{cc}{w}_n={\lambda }_p^{\left(r\right)}+{\epsilon }_p\pi \hfill & t_n<0\hfill \\ p^{\left(i\right)}=0\hfill & t_n=0\hfill \end{array}\phantom{\}}$

Solve for Δu and Δp

Check convergence of nonlinear iterations

End Do

If |g(X(1))|>GTOL or |π(X(1))|>PTOL for any X(1) ∊ γ(1)

Augment:

$\text{Set}{\lambda }_n^{(r+1)}=\{\begin{array}{cc}{\lambda }_n^{\left(r\right)}+{\epsilon }_{n}g\hfill & {\lambda }_n^{\left(r\right)}+{\epsilon }_{n}g<0\hfill \\ 0\hfill & {\lambda }_n^{\left(r\right)}+{\epsilon }_{n}g\ge 0\hfill \end{array}\phantom{\}}$

$\text{Set}{\lambda }_p^{(r+1)}=\{\begin{array}{cc}{\lambda }_p^{\left(r\right)}+{\epsilon }_p\pi \hfill & {\lambda }_n^{(r+1)}<0\hfill \\ 0\hfill & {\lambda }_n^{(r+1)}=0\hfill \end{array}\phantom{\}}$

r ← r + 1

Else contact has converged

End Do