### 29.2.3 Constraint enforcement methods for ABAQUS/Standard contact pairs

Products: ABAQUS/Standard  ABAQUS/CAE

### Overview

Contact constraint enforcement methods in ABAQUS/Standard:

• are specified as part of the surface interaction definition;

• determine how contact constraints imposed by a contact pair's physical pressure-overclosure relationship (see Contact pressure-overclosure relationships, Section 30.1.2) are resolved numerically in an analysis;

• can either strictly enforce or approximate the physical pressure-overclosure relationships;

• can be modified to resolve convergence difficulties due to overconstraints; and

• sometimes utilize Lagrange multiplier degrees of freedom.

The available constraint enforcement methods for normal contact in ABAQUS/Standard are discussed in detail in this section. The frictional constraint enforcement methods in ABAQUS/Standard are assigned independently of those for the normal contact constraints and are discussed in Frictional behavior, Section 30.1.5. The use of Lagrange multipliers in contact calculations is also covered in this section.

### Available 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.

In many cases the contact constraints can be enforced with or without Lagrange multipliers. Lagrange multipliers can add significantly to solution cost, but they also protect against numerical errors related to ill-conditioning that can occur if a high contact stiffness is in effect. Any Lagrange multipliers associated with contact are present only for active contact constraints, so the number of equations will change as the contact status changes. As will be discussed in more detail, ABAQUS/Standard will choose whether or not to use Lagrange multipliers automatically, based on the contact stiffness.

The constraint enforcement method depends in part on your choice of pressure-overclosure relationship. “Hard” contact (the default pressure-overclosure relationship) can be enforced using any of the three constraint enforcement methods. However, a “softened” contact relationship can be enforced only using the direct method.

The default constraint enforcement method is usually the direct method with Lagrange multipliers. There are two exceptions to this rule:

• the finite-sliding, surface-to-surface contact formulation uses the penalty method without Lagrange multipliers by default; and

• three-dimensional self-contact with node-to-surface discretization uses the augmented Lagrange method without Lagrange multipliers by default.

### Direct method

The direct method strictly enforces a given pressure-overclosure behavior for each constraint, without approximation or use of augmentation iterations. The direct method is used by default to strictly enforce contact constraints for all types of contact except finite-sliding, surface-to-surface contact and three-dimensional, node-to-surface self-contact. For “hard” pressure-overclosure relationships you can specify an alternative constraint enforcement method if the direct method is used by default; you should consider the following factors in this decision:

• Direct enforcement of hard contact provides accurate solutions with reasonable efficiency in many cases.

• Alternative solution 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.

• Lagrange multipliers are always used for active contact constraints with directly enforced hard contact.

• Overconstraints due to overlapping contact definitions or the combination of contact and other constraint types (see Overconstraint checks, Section 28.6.1) should be avoided for directly enforced hard contact.

#### Softened pressure-overclosure relationships

The direct method is the only method that can be used to enforce “softened” pressure-overclosure relationships. The direct method can be used to model softened contact behavior regardless of the type of contact formulation; however, modeling stiff interface behavior with a contact formulation that is prone to overconstraints can be difficult. By default, Lagrange multipliers are always used for exponential pressure-overclosure relationships. Whether or not Lagrange multiplier degrees of freedom are used by default for linear or tabular pressure-overclosure relationships depends on the maximum slope of the pressure-overclosure relationship: Lagrange multipliers are used if the maximum slope exceeds 1000 times the underlying element stiffness (as computed by ABAQUS/Standard); otherwise, the constraints are enforced without Lagrange multipliers. Softened pressure-overclosure relationships are discussed in more detail in Contact pressure-overclosure relationships, Section 30.1.2.

#### Limitations of the direct method

Because of its strict interpretation of contact constraints, hard contact simulations utilizing the direct enforcement method are susceptible to overconstraint issues. As a result, directly enforced hard contact is not available for contact pairs in the following situations:

• finite-sliding, surface-to-surface formulations; and

In both of these instances you can use an alternate enforcement method or the direct method with a softened pressure-overclosure relationship.

You may experience similar overconstraint problems with symmetric master-slave contact pairs (see Using symmetric master-slave contact pairs to improve contact modeling” in “Defining contact pairs in ABAQUS/Standard, Section 29.2.1). Although directly enforced hard contact is the default for these contact pairs, it is recommended that you use an alternate enforcement method or a softened contact relationship.

Certain second-order element faces do not perform well in directly enforced hard contact relationships. See Three-dimensional surfaces with second-order faces” in “Common difficulties associated with contact modeling in ABAQUS/Standard, Section 29.2.11, for details on this issue.

### Penalty method

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.

When the penalty method is used, ABAQUS/Standard will, by default, set the so-called penalty stiffness to 10 times a representative underlying element stiffness. You may scale or reassign the penalty stiffness, as discussed below. Contact penetrations resulting from the default penalty stiffness will not significantly affect results in most cases; however, these penetrations can sometimes contribute to some degree of stress inaccuracy (for example, with displacement-controlled loading and a coarse mesh). The penalty method is used by default for the finite-sliding, surface-to-surface contact formulation.

The penalty method typically does not use Lagrange multiplier degrees of freedom. A variation of the penalty method that makes use of Lagrange multipliers to avoid ill-conditioning issues for high penalty stiffness (at some computational expense) is also provided in ABAQUS/Standard. Lagrange multipliers are used if the penalty stiffness exceeds 1000 times the representative underlying element stiffness computed by ABAQUS/Standard. Therefore, Lagrange multipliers are not used with the default penalty stiffness.

 Input File Usage: Use both of the following options: ```*SURFACE INTERACTION, NAME=interaction_property_name *SURFACE BEHAVIOR, PENALTY```

 ABAQUS/CAE Usage: Interaction module: contact property editor: MechanicalNormal Behavior: Constraint enforcement method: Penalty (Standard)

#### Modifying the penalty stiffness

If you are interested in investigating the effects of modifying the penalty stiffness, it is generally recommended that you consider order-of-magnitude changes. Increasing the penalty stiffness above the threshold value discussed above will, by default, introduce Lagrange multipliers. As part of the surface behavior definition, you can specify the penalty stiffness, shift the pressure-overclosure relationship by specifying the clearance at which the contact pressure is zero, or scale the default or specified penalty stiffness by a factor. You can also scale the penalty stiffness on a step-by-step basis, which will act as an additional multiplier on any scale factor specified as part of the surface behavior definition.

 Input File Usage: To modify the penalty behavior in the surface behavior definition: ```*SURFACE BEHAVIOR, PENALTY penalty stiffness, clearance at zero pressure, factor```To scale the penalty stiffness on a step-by-step basis:`*CONTACT CONTROLS, STIFFNESS SCALE FACTOR=factor`

 ABAQUS/CAE Usage: To modify the penalty behavior in the surface behavior definition: Interaction module: contact property editor: MechanicalNormal Behavior: Constraint enforcement method: Penalty (Standard): Contact stiffness: Specify: penalty stiffness, Contact stiffness scale factor: factor, Clearance at which contact pressure is zero: clearance at zero pressureTo scale the penalty stiffness on a step-by-step basis:Interaction module: ABAQUS/Standard contact controls editor: Augmented Lagrange: Stiffness scale factor: factor

#### Limitations of the penalty method

The penalty method cannot be used for debonded surfaces.

If the penalty method is specified, Lagrange multipliers are always used during analysis steps with the following procedures:

If surface elements have been used to define a contact surface on the exterior of a substructure (see Contact modeling if substructures are present, Section 29.2.9), ABAQUS/Standard interprets the underlying element stiffness to be zero. This can lead to difficulty in determining the default penalty stiffness and may cause numerical problems during the analysis.

### Augmented Lagrange method

The 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, , of less than 1.0 (using the interface discussed below), ABAQUS/Standard will automatically scale the default penetration tolerance by a factor of (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 depends on the contact formulation used. For the finite-sliding, surface-to-surface formulation the default penalty stiffness is the same as for the penalty method without augmentation (i.e., equal to 10 times the representative underlying element stiffness computed by ABAQUS/Standard). For self-contact using node-to-surface discretization, the default penalty stiffness is 100 times the representative underlying element stiffness. For all other contact formulations the default penalty stiffness 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.

 Input File Usage: Use both of the following options: ```*SURFACE INTERACTION, NAME=interaction_property_name *SURFACE BEHAVIOR, AUGMENTED LAGRANGE```

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

#### Modifying the penetration tolerance for the augmented Lagrange method

You can modify the penetration tolerance for the augmented Lagrange method on a step-by-step basis by specifying an absolute or relative penetration tolerance. The relative penetration tolerance is specified with respect to a characteristic length computed by ABAQUS/Standard, which represents a typical facet dimension. The default penetration tolerance was discussed above. The default penetration tolerance is increased automatically if you set the penalty stiffness scale factor to a value less than 1.0 (also discussed above); however, ABAQUS/Standard will not adjust any directly specified penetration tolerance. Choosing a very small penetration tolerance may result in an excessive number of augmentation iterations.

 Input File Usage: To specify an absolute penetration tolerance: `*CONTACT CONTROLS, ABSOLUTE PENETRATION TOLERANCE=tolerance`To specify a relative penetration tolerance:`*CONTACT CONTROLS, RELATIVE PENETRATION TOLERANCE=tolerance`

 ABAQUS/CAE Usage: Interaction module: ABAQUS/Standard contact controls editor: Augmented Lagrange: Penetration tolerance: Absolute: tolerance or Relative: tolerance

#### Modifying the penalty stiffness for the augmented Lagrange method

As with the penalty method, you can specify the penalty stiffness, shift the pressure-overclosure relationship by specifying the clearance at which the contact pressure is zero, or scale the default or specified penalty stiffness by a factor as part of the surface behavior definition. You can also scale the penalty stiffness on a step-by-step basis, which will act as an additional multiplier on any scale factor specified as part of the surface behavior definition. Choosing a very low penalty stiffness may result in an excessive number of augmentation iterations.

 Input File Usage: To modify the penalty behavior in the surface behavior definition: ```*SURFACE BEHAVIOR, AUGMENTED LAGRANGE penalty stiffness, clearance at zero pressure, factor```To scale the penalty stiffness on a step-by-step basis:`*CONTACT CONTROLS, STIFFNESS SCALE FACTOR=factor`

 ABAQUS/CAE Usage: To modify the penalty behavior in the surface behavior definition: Interaction module: contact property editor: MechanicalNormal Behavior: Constraint enforcement method: Augmented Lagrange (Standard): Contact stiffness: Specify: penalty stiffness, Contact stiffness scale factor: factor, Clearance at which contact pressure is zero: clearance at zero pressureTo scale the penalty stiffness on a step-by-step basis:Interaction module: ABAQUS/Standard contact controls editor: Augmented Lagrange: Stiffness scale factor: factor

#### Limitations of the augmented Lagrange method

The augmented Lagrange method cannot be used for debonded surfaces.

If the augmented Lagrange method is specified, Lagrange multipliers are always used during analysis steps with the following procedures:

If surface elements have been used to define a contact surface on the exterior of a substructure (see Contact modeling if substructures are present, Section 29.2.9), ABAQUS/Standard interprets the underlying element stiffness to be zero. This can lead to difficulty in determining the default penalty stiffness and may cause numerical problems during the analysis.

### Specifying directly whether or not the contact constraint method should use Lagrange multipliers

ABAQUS/Standard will automatically choose whether the constraint method makes use of Lagrange multipliers according to the criteria discussed above for the various constraint methods. Table 29.2.3–1 summarizes the default use of Lagrange multipliers.

Table 29.2.3–1 Default use of Lagrange multipliers in constraint enforcement methods.

Constraint MethodUse Lagrange Multipliers by Default
YesNo1
Direct, hard contactAlwaysNever
Direct, exponential softened contactAlwaysNever
Direct, linear softened contactIf If
Direct, tabular softened contactIf If
Penalty, hard contactIf If
Augmented Lagrange, hard contactIf If
= maximum slope of pressure-overclosure relationship
= penalty stiffness
= underlying element stiffness
1Lagrange multipliers are always used, regardless of the constraint enforcement method or stiffness, in the following cases: design sensitivity analyses, direct steady-state dynamics analyses, analyses using the quasi-Newton method, analyses using the contact iterations solution technique.

You can override the default Lagrange multiplier behavior except in the following cases:
However, it is generally recommended that you do not override the default choice, because:
• Using Lagrange multipliers for cases with relatively small to moderate penalty stiffness generally reduces solver efficiency without significantly improving results.

• Not using Lagrange multipliers for cases with large values of penalty stiffness can lead to numerical ill-conditioning in the equation solver.

 Input File Usage: To specify that Lagrange multipliers should not be used by the constraint enforcement method: `*CONTACT CONTROLS, LAGRANGE MULTIPLIER=NO`Use either of the following options to specify that Lagrange multipliers must be used by the constraint enforcement method:`*CONTACT CONTROLS, LAGRANGE MULTIPLIER=YES ``*CONTACT CONTROLS, LAGRANGE MULTIPLIER`

 ABAQUS/CAE Usage: Interaction module: ABAQUS/Standard contact controls editor: Enforce using Lagrange multipliers: Off or On