# A new algorithm to increase speed of solving contact/friction problems: the Generalized Newton method

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

Modelling contact / friction is very important in nonlinear mechanics. Code_Aster has several methods to solve this problem. The most modern and most complete is called "continuous formulation" of the contact friction. It is based on the work the most advanced in the scientific community in this area, the continuous formualtion of the law of Signorini-Coulomb (augmented Lagrangian).

Currently, the resolution of this problem uses five levels of nested loops:

**–** 1) Loop over time steps
**–** 2) Loop geometry (matching and local database)
**–** 3) Loop over the threshold of friction (transformation of Coulomb’s probelm in Tresca’s one)
**–** 4) Loop over the status of contact
**–** 5) Loop of Newton (linearization of the problem)

Levels 2, 3 and 4 are specific to the problems of contact / friction and greatly penalize performance.

In partnership with the Laboratory of Mechanics and Acoustics of Marseille (Research Unit of CNRS), we propose the replacement of the algorithm by an algorithm called "generalized Newton method", resulting from the work of Alart, Curnier and Pietrzak .

The principle of this algorithm is to derive explicitly all terms non-linear problems contact / friction, taking into account such dependence on the direction of friction from the contact pressure. This modification of the terms of the Jacobian matrix makes non-symmetric but eliminates the loops 3 and 4 and thus save time.

To activate the generalized Newton, use ALGO_RESO_CONT = ’NEWTON’ and ALGO_RESO_FROT = ’NEWTON’ in command DEFI_CONTACT.

In the case of Shallow-test Ironing (ssnp153, see also this news), we divide the computation time by factor FOUR

Total number of Newton iterations | Relative CPU time | |
---|---|---|

Nested loops (old method) |
35087 | 1.0 |

Generalized Newton (new method) | 9056 | 0.25 |

This work has resulted in two scientific papers in international conferences.

*"Study of the robustness of an algorithm based on the Lagrangian stabilized for solving problems of contact friction"*, AD Kudawoo, F. Lebon, Abbas, T. for Soza, I. Rosu, National Symposium on Computational Structural, CSMA 2011, Giens, 9-13 May 2011

*"Two frictional contact algorithms based on stabilized Lagrangian formulation, and application"*, AD Kudawoo, F. Lebon, Abbas, T. for Soza, I. Rosu, International Conference on Computational Mechanics Contact ICCCM, Hannover, 15-17 June 2011