Theory Manual Version 3.6
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Subsection 7.2.4: Penalty Scheme Up Section 7.2: Biphasic Contact Subsection 7.2.6: Stick-Slip Algorithm

### 7.2.5 Augmented Lagrangian Scheme

The augmented Lagrangian scheme presented herein was developed in Section 7.1.9.6↑ [114] as a modification of the approach proposed by Simo and Laursen [93]. Briefly, this is a first-order augmentation scheme that utilizes Uzawa's algorithm [23], where the multipliers are updated outside of the Newton step, producing a double loop algorithm [110] and preserving quadratic convergence of Newton's method near solution points.
During stick, the traction is calculated by augmenting the vector gap , where is the vectorial Lagrange multiplier in stick. In slip, the normal component of the contact traction is first calculated by augmenting the normal gap , where is the normal Lagrange multiplier. The total traction vector in slip is then directly prescribed as where eq.(7.2.5-2) has been used. The update formulas for the Lagrange multipliers can be found in Section 7.1.9.6↑ [114]. Here it suffices to note that by augmenting only the normal gap and employing eq.(7.2.5-3), we have ensured an exact mapping to the proper tangential traction in slip, which is consistent with the augmented normal traction. As in the penalty case, a trial state and return map, presented in Section 7.2.6↓ and controlled by the slip criterion, is used to differentiate between stick and slip.
In this augmentation scheme, the normal fluid flux is given by where is the fluid pressure Lagrange multiplier, and the update formula for has been detailed in our prior work on frictionless biphasic contact [8] and is given by Unlike and [114], is not dependent on the stick-slip status. However, as noted before, eq.(7.2.5-4) must be evaluated after the stick-slip status has been resolved.
Subsection 7.2.4: Penalty Scheme Up Section 7.2: Biphasic Contact Subsection 7.2.6: Stick-Slip Algorithm