Augmented Lagrange 2

[krishp]

Hi Folks,

What is exactly Augmented Lagrange formulation. I only know that it uses iterative solver in an effective way. But based on what it works in this way when compare to the Pure penalty or Normal lagrange formulations. If anybody find any helpful information, please copy the links too. Thanks in advance.

Tobias.

 

[xerf]

The augmented Lagrange formulation is a method for imposing constraints. It is actually a combination/compromise of Lagrange multiplier approach and penalty approach.

The augmented Lagrange approach is used for contact modeling based on monitoring the gaps between pairs of nodes. A constraint is inserted when the gap between 2 nodes (on opposite surfaces) becomes zero or negative.

For the general theoretical background you might be interested in the text: "Linear and nonlinear programming" by D.Luenberger.

An interesting paper:
Zavarise, Wriggers. A superlinear convergent augmented Lagrangian procedure for contact problems, Engineering Computations, 16, 88-119, 1999.

 

[torpen]

Hello,

The aim is to introduce kinematic constraints between dofs (contact, rigid body, hing joint, equality, ...).

U* = U + 1/2*p*phi^2 + k*lambda*phi

where,
k is a scale factor and phi is the kinematic constraint.


We consider 2 one-dimensional rods.

1 ____ 2 3 ____ 4

Node 4 is fixed and node 1 is loaded.
The stiffness of the rods is equal to unity and we apply an equality between the displacements of the nodes 2 and 3.

Penalty method
A large stiffness is defined between dofs. We minimize:
U* = U + 1/2*p*(u3-u2)^2
If k is too large according to the stiffness of the structure, the both displacements are about equal but numerical problems occur.
If k is not enough large, the equality is not respected.

The system is:

| 1 -1 0 | | u1 | | F |
| 0 1+p -p | | u2 | = | 0 |
| 0 -p 1+p | | u3 | | 0 |

We find:
u1 = 2*F + F/p; u2 = F + F/p and u3 = F
If p tends to infinity we have the exact solution but the systme is ill-conditioned.

Lagrangian method
We minimize:
U* = U + lambda*(u3-u2)

The equality is exactly respected but this method is sensitive to the numbering of the dofs.
If the resolution is performed by the Gauss method, the second pivot is equal to zero and pivoting is essential.

The system is:

| 1 -1 0 0 | | u1 | | F |
| -1 1 0 1 | | u2 | = | 0 |
| 0 0 1 1 | | u3 | = | 0 |
| 0 1 -1 0 | | lambda | | 0 |

We find:
u1 = 2*F; u2 = u3 = F and lambda = F

Augmented Lagrangian method
We minimize:
U* = U + 1/2*p*(u3-u2)^2 + lambda*(u3-u2)

The results do not depend on the parameter p.

The system is:

| 1 -1 0 0 | | u1 | | F |
| -1 1+p -p 1 | | u2 | = | 0 |
| 0 -p 1+p 1 | | u3 | = | 0 |
| 0 1 -1 0 | | lambda | | 0 |

We find:
u1 = 2*F; u2 = u3 = F and lambda = F

The problem of pivoting is solved if lambda is after the dofs of the constraint.


Regards,

Torpen.