# Generalized Newton method for contact and friction

*by A.D. Kudawoo, EDF R&D PhD / AMA and LMA ; M. Abbas and T. De Soza, EDF R&D / AMA*

Solving contact and friction problems is an important issue for non-linear mechanics. Code_Aster offers a precise and robust formulation called "continuous formulation" of contact friction. It is a continuous formulation of the Signorini-Coulomb law in the form of augmented Lagrangian (DEFI_CONTACT / FORMULATION = ’CONTINUE’).

Historically (version 10 and above), solving this problem was using five levels of nested loops:
**–** 1) Loop over time steps
**–** 2) Loop on geometry (matching and local base)
**–** 3) Loop on the threshold of friction (transformation of the Coulomb problem into a series of Tresca problems)
**–** 4) Loop on the status of contact
**–** 5) Newton loop (linearization of the problem)

Levels 2, 3 and 4 are specific to the contact / friction solving.

Since version 11.0.28 (see this news), loops 3 and 4 were removed by applying a *partial* generalized Newton’s method , that is to derive an explicit part of the non-linear terms of the problem of contact / friction, taking into account for example the dependence of the rubbing direction relative to the contact pressure.

Now, since version 11.2.4, it is possible to use the *total* Newton generalized method, which removes the last loop for the contact / friction: the fixed point on the geometric configuration (loop 2)

To activate the generalized Newton, just specify in DEFI_CONTACT:

- ALGO_RESO_GEOM=’NEWTON’,
- ALGO_RESO_FROT=’NEWTON’
- ALGO_RESO_CONT=’NEWTON’

In the v11 version of Code_Aster, it is now the default.

Generalized Newton’s method has three effects on difficult problems (large strains, friction):

- Get a solution with a more severe geometric criterion (reliability)
- Get a solution impossible to obtain with the fixed point method or partial Newton’s method (robustness)
- Get a solution more quickly in some cases (performance)

In the test case "ring on block" (new test case to be published soon), gain up to 20% if one take friction into account.

Ring-on-block without friction |
Number of Newton iterations | ||

Fixed point loops (V10) | 2603 | ||

Generalized Newton method (V11) | 2216 |

Ring-on-block with friction |
Number of Newton iterations | ||

Fixed point loops (V10) | 4889 | ||

Generalized Newton method (V11) | 4054 |

On the test case ssnv128p (frictionnal block), the gain is greater:

Frictionnal block with friction |
Number of Newton iterations | ||

Fixed point loops (V10) | 1821 | ||

Generalized Newton method (V11) | 11 |

This work is a deliverable of the EDF project Advanced Numerical Methods in Mechanics, in partnership with the Laboratory of Mechanics and Acoustics Marseille (CNRS Unit Research ).