When choosing a formulation to be used in a contact region, you basically decide the mathematical method that the code will use to enforce the contact compatibility condition (for example, no penetration allowed, no separation, no sliding, or sliding with some resistance).
There are four basic formulations:
- Penalty method
- Augmented Lagrange
- Pure Lagrange
- Multipoint constraint (or MPC)
This post focuses on the first two formulations.
The penalty method introduces a force at the contact detection point(s) that has penetrated across the target surface with the express purpose of eliminating the penetration. This method uses very simple formulas:
Fc = kc*Dp
for contact detection points that penetrate across the target, and
Fc = zero
for open contact detection points. kc is the contact stiffness (also called penalty stiffness); it is a predetermined property of the contact element. Dp is the penetration at the contact element. Hence, the larger the penetration, the greater the calculated force. The challenge here is that the magnitude of the force necessary to prevent penetration is completely unknown beforehand. Obviously, the force needs to be large enough to push the contact surface back to the target surface and, thereby, eliminate unwanted penetration — but not so large that it pushes the contact completely off the target surface, causing error and instability. A positive aspect of the penalty method is that it is elegantly simple. The negative is that you end up with a finite amount of penetration at the end of the load step. Of course, this penetration is necessary for a contact force to be generated. It is important, therefore, that the contact stiffness be large enough so that the resulting penetration is negligibly small; but the contact stiffness cannot be so large as to cause instability and nonconvergence. This same strategy is used both in the opposite direction to prevent separation (with bonded or no-separation behaviors) and in the tangential direction to enforce frictional resistance and no-sliding behavior.
The augmented Lagrange method is very similar to the penalty method. The calculated force at the contact detection point(s) is
Fc=kc*Dp+ l
in which l is an internally calculated term that augments the penalty-based force calculation. The purpose of the augmentation is to reduce sensitivity to contact stiffness. All things being equal, the augmented Lagrange method should produce less penetration than the pure penalty method, but it might take more iterations to converge. The program-controlled default formulation for contact between flexible bodies is augmented Lagrange.
The optimal value for contact stiffness in these methods is one that generates a converged result in a reasonable number of iterations with a resulting penetration (or elastic slip in the tangential direction) that is inside acceptable tolerance. Often, such an optimal value will vary as you progress through the load path. To enhance convergence, the program automatically adjusts the stiffness based on the current mean stress of the underlying elements and allowable penetration. You can influence the code-calculated stiffness value by manually defining a multiply factor on the stiffness.
To read more about these formulations, refer to the documentation for the CONTA174 element as well as // Contact Technology Guide // 3. Surface-to-Surface Contact // 3.9.3 Selecting an Algorithm.
My next post will discuss the normal Lagrange and MPC algorithms. Stay tuned!
