Contact constraint enforcement methods in Abaqus/Standard

There are three contact constraint enforcement methods available in Abaqus/Standard:

• The direct method attempts to strictly enforce a given pressure-overclosure behavior per constraint, without approximation or use of augmentation iterations.

• The penalty method is a stiff approximation of hard contact.

• The augmented Lagrange method uses the same kind of stiff approximation as the penalty method, but also uses augmentation iterations to improve the accuracy of the approximation.

The default constraint enforcement method depends on interaction characteristics, as follows:

• The penalty method is used by default for finite-sliding, surface-to-surface contact (including general contact) if a hard pressure-overclosure relationship is in effect.

• The augmented Lagrange method is used by default for three-dimensional self-contact with node-to-surface discretization if a hard pressure-overclosure relationship is in effect.

• The direct method is the default in all other cases.

You should consider the following factors when choosing the contact enforcement method:

• The direct method must be used for contact pairs with a softened pressure-overclosure relationship (see Contact pressure-overclosure relationships).

• The direct method strictly enforces the specified pressure-overclosure behavior consistent with the constraint formulation

• The penalty or augmented Lagrange constraint enforcement methods sometimes provide more efficient solutions (generally due to reduced calculation costs per iteration and a lower number of overall iterations per analysis) at some (typically small) sacrifice in solution accuracy. See the discussions of the penalty and augmented Lagrange methods below.

• Overconstraints due to overlapping contact definitions or the combination of contact and other constraint types (see Overconstraint Checks) should be avoided for directly enforced hard contact.

The direct method strictly enforces a given pressure-overclosure behavior for each constraint, without approximation or use of augmentation iterations.

SURFACE INTERACTION, NAME=interaction_property_name
SURFACE BEHAVIOR, DIRECT

Interaction module: contact property editor: MechanicalNormal Behavior: Constraint enforcement method: Direct (Standard)

The penalty method approximates hard pressure-overclosure behavior. With this method the contact force is proportional to the penetration distance, so some degree of penetration will occur. Advantages of the penalty method include:

• Numerical softening associated with the penalty method can mitigate overconstraint issues and reduce the number of iterations required in an analysis.

• The penalty method can be implemented such that no Lagrange multipliers are used, which allows for improved solver efficiency.

The linear penalty method can be used within an augmentation iteration scheme that drives down the penetration distance. This so-called augmented Lagrange method applies only to hard pressure-overclosure relationships. The following describes the sequence that occurs in each increment with this approach:

1. Abaqus/Standard finds a converged solution with the penalty method.

2. If a slave node penetrates the master surface by more than a specified penetration tolerance, the contact pressure is “augmented” and another series of iterations is executed until convergence is once again achieved.

3. Abaqus/Standard continues to augment the contact pressure and find the corresponding converged solution until the actual penetration is less than the penetration tolerance.

The augmented Lagrange method may require additional iterations in some cases; however, this approach can make the resolution of contact conditions easier and avoid problems with overconstraints, while keeping penetrations small. The augmented Lagrange method is used by default for three-dimensional self-contact using node-to-surface discretization.

The default penetration tolerance is one-tenth of a percent of the characteristic interface length except in the following cases:

• if you specify a penalty stiffness scaling factor, $f_k$ , of less than 1.0 (using the interface discussed below), Abaqus/Standard will automatically scale the default penetration tolerance by a factor of $\frac{1}{\sqrt{f_k}}$ (which will be greater than or equal to 1.0);

• the default penetration tolerance for finite-sliding, surface-to-surface contact is five percent of the characteristic interface length, subject to the scaling discussed in the previous bullet point.

The default penalty stiffness for the augmented Lagrange method is 1000 times the representative underlying element stiffness. Lagrange multipliers are used for the augmented Lagrange method if the penalty stiffness exceeds 1000 times the representative underlying element stiffness computed by Abaqus/Standard; otherwise, no Lagrange multipliers are used. Therefore, Lagrange multipliers are not used for the augmented Lagrange method with the default penalty stiffness.

SURFACE INTERACTION, NAME=interaction_property_name
SURFACE BEHAVIOR, AUGMENTED LAGRANGE

Interaction module: contact property editor: MechanicalNormal Behavior: Constraint enforcement method: Augmented Lagrange (Standard) 

Using Lagrange multipliers to enforce contact constraints can add significantly to the solution cost, but they also protect against numerical errors related to ill-conditioning that can occur if a high contact stiffness is in effect. Abaqus/Standard automatically chooses whether the constraint method makes use of Lagrange multipliers, based on a comparison of the contact stiffness to the underlying element stiffness. Table 1 summarizes the use of Lagrange multipliers. Lagrange multipliers are not used for the default contact stiffnesses associated with the penalty and augmented Lagrange approximations of hard contact. Any Lagrange multipliers associated with contact are present only for active contact constraints, so the number of equations may change as the contact status changes.