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7.1.6 Augmented Lagrangian Method
The augmented Lagrangian method is used in FEBio to enforce the contact constraints to a user-specified tolerance. This implies that the normal contact tractions are given by, Note that this assumption is consistent with the approach that was used in establishing the discretization of the linearization of the contact integral (7.1.5-3). In (7.1.6-1) is a penalty factor that is chosen arbitrarily.
The Newton-Raphson iterative method is now used to solve the nonlinear contact problem where Uzawa's method (REF) is employed to calculate the Lagrange multipliers . This implies that the Lagrange multipliers are kept fixed during the Newton-Raphson iterations. After convergence the multipliers are updated and a new NR procedure is started. This procedure can be summarized by the following four steps.
- Initialize the augmented Lagrangian iteration counter , and the initial guesses for the multipliers:
- Solve for , the solution vector corresponding to the fixed th iterate for the multipliers, where the contact tractions used to compute , the contact force, are governed by
- Update the Lagrange multipliers and iteration counters:
- Return to the solution phase.
Steps 2-4 of the above algorithm are generally repeated until all contact constraints are satisfied to a user-specified tolerance or little change in the solution vector from augmentation to augmentation is noted.