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# Generalized Newton method for contact and friction

2 October 2012

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 frictionNumber of Newton iterations
Fixed point loops (V10) 2603
Generalized Newton method (V11) 2216
Ring-on-block with frictionNumber 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 frictionNumber 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 ).