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Subsection 8.4.1: The penalty method Up Section 8.4: Guidelines for Contact Problems Subsection 8.4.3: Initial Separation

### 8.4.2 Augmented Lagrangian Method

In theory, in the presence of constraints, the corresponding Lagrange multipliers must be calculated which, in the case of contact, correspond to the contact forces that enforce the contact constraint exactly. Unfortunately, the exact evaluation of these Lagrange multiplies is numerically very challenging and therefore in FEBio it was decided to evaluate these Lagrange multipliers in an iterative method, named the Augmented Lagrangian method. Using this method, FEBio will solve a time step several times (these iterations are termed augmentations), where each time the approximate Lagrange mutlipliers are updated (or augmented).
The obvious drawback of this method is that now each time step has to be solved several times. However, the advantage of avoiding the numerical problems of obtaining the exact Lagrange multipliers and the fact that in most contact problems very little extra work is needed to solve these augmentations, makes this method very attractive. In addition, it often gives better enforcement of the contact constraint compared to the penalty method.
So why not just use the augmented Lagrangian method? Well, often the penalty method will give good results and since the penalty method is much faster, it is often the preferred choice of many analysists.
In many cases, users are only interested in the final time step. For these users it may be of interest that it is possible to use the augmented Lagrange method only in the final time step. This can be done by defining a loadcurve for the laugon contact parameter. Then, define the corresponding loadcurve as zero everywhere except for the final time step. FEBio will now only use the augmented Lagrangian method in the final time step (and the penalty method all other steps).
Subsection 8.4.1: The penalty method Up Section 8.4: Guidelines for Contact Problems Subsection 8.4.3: Initial Separation