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184.108.40.206 Augmented Lagrangian Scheme
The augmented Lagrangian scheme employed in this study is first order and utilizes Uzawa's algorithm , where multipliers are updated outside of the Newton step, producing a double loop algorithm (see the texts by Laursen  and Wriggers  for a discussion of Uzawa's algorithm applied to frictional contact problems). Such an approach preserves the quadratic convergence of Newton's method near solution points. The presented scheme is a modification of the approach suggested by Simo and Laursen .
During stick, the traction is calculated by augmenting the vector gap , where is the vectorial Lagrange multiplier in stick. In slip, we first calculate the normal component of the contact traction by augmenting the normal gap , where is the normal Lagrange multiplier. The total traction vector in slip is then defined to be where is given by eq.(220.127.116.11-2). In this approach the Lagrange multiplier augments the traction in stick, but in slip only the normal component of traction is augmented by and the tangential traction is directly prescribed from the augmented normal component. This approach has the advantage of preserving 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 controlled by the slip criterion is employed to differentiate between stick and slip, presented in Section 18.104.22.168↓.
The Lagrange multipliers and are held constant during each Newton step. Outside of the Newton loop, in this study we propose a novel update scheme where one of these multipliers is considered active and is updated from the kinematic data ( or ), and the other is considered passive and is derived from the active multiplier. The contact status is determined via eq.(22.214.171.124-1). If the current status is stick ( ), we update (active) and derive (passive), Alternatively, if the current contact status is slip ( ), we update (active) and derive (passive), An active-passive strategy for the multipliers ensures consistency when the contact status switches between stick and slip and is made possible due to this formulation's use of a single penalty parameter , in conjunction with an exact return mapping for slip.
Augmentations proceed until a tolerance related to a convergence criterion is met. In this formulation, two separate convergence criteria and their associated tolerances are defined. The first criterion considers the relative change of the norms of the active Lagrange multipliers between successive iterations, where the associated unitless tolerance specifies the largest allowable change. Convergence is achieved when where represents the total norm of the active multipliers across the contact surface at augmentation step , calculated by summing all the individual norms, In this expression, is the number of element faces on , is the number of integration points on the element face, and is the norm of the active Lagrange multiplier at the integration point on the element of at augmentation step , defined by The second criterion is a gap tolerance, where augmentations will continue until the magnitude of the gap is lower than the specified tolerance at every location. Convergence of the augmentations requires where is associated with points currently sticking and with those in slip. The tolerance has units of length, allowing enforcement of the non-penetration and stick constraints to arbitrarily small precision.