7.1.6 Augmented Lagrangian Method

1. Initialize the augmented Lagrangian iteration counter $k$ , and the initial guesses for the multipliers: $$\begin{equation} \begin{aligned}\lambda_{N_{n+1}}^{\left(0\right)} & =\lambda_{N_{n}}\,,\\ k & =0\,. \end{aligned} \label{eq591} \end{equation}$$
2. Solve for $\mathbf{d}_{n+1}^{\left(k\right)}$ , the solution vector corresponding to the fixed $k$ th iterate for the multipliers, $$\begin{equation} \mathbf{F}^{int}\left(\mathbf{d}_{n+1}^{\left(k\right)}\right)+\mathbf{F}^{c}\left(\mathbf{d}_{n+1}^{\left(k\right)}\right)=\mathbf{F}_{n+1}^{ext}\,,\label{eq592} \end{equation}$$ where the contact tractions used to compute $\mathbf{F}^{c}$ , the contact force, are governed by $$\begin{equation} t_{N_{n+1}}^{\left(k\right)}=\left\langle \lambda_{N_{n+1}}^{\left(k\right)}+\varepsilon_{N}g_{n+1}^{k}\right\rangle \,.\label{eq593} \end{equation}$$
3. Update the Lagrange multipliers and iteration counters: $$\begin{equation} \begin{aligned}\lambda_{N_{n+1}}^{\left(k+1\right)} & =\left\langle \lambda_{N_{n+1}}^{\left(k\right)}+\varepsilon_{N}g_{n+1}^{\left(k\right)}\right\rangle \,,\\ k & =k+1\,. \end{aligned} \label{eq594} \end{equation}$$
4. Return to the solution phase.