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## Quasi-symmetrical formulations for contact and friction between deformable bodies: application to 3D forging

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Lionel Fourment
Sorin Popa
• Fonction : Auteur

#### Résumé

To properly handle the contact conditions without introducing spurious numerical constraints, the master / slave approach is inescapable but it results into a non-symmetric formulation for non-coinciding meshes [1]. From a theoretical standpoint, this unsatisfactory treatment of the contact area results into a decrease of the convergence rate of the finite element method [2]. From a more practical standpoint, severe problems arise when the discretization of the master surface is much finer than the slave surface. In metal forming, the workpiece is always the slave while the tools are the masters. So aiming at accurate tool stress calculations require masters meshes that are locally much finer than the corresponding mesh on the slave-workpiece. With a standard formulation, parts of the tool contact surface may result to be numerically unloaded, so providing very inaccurate finite element solution where high accuracy is required. A symmetric formulation has been proposed in [3], but it introduces spurious constraints. In [1], an accurate calculation of the contact conditions between the contacting bodies in proposed, while in [2], a L2 enhanced projection of the displacement field on the contact surface is developed. Both algorithms are written in 2D for an integrated formulation. Their extension in 3D seems quite uneasy. We then proposed a quasi-symmetric formulation [4]. It can be compared to [3] but the contact Lagrange multipliers are not duplicated on both contact surfaces. It so allows avoiding introducing spurious constraints, while keeping a simple and almost symmetric formulation. The implementation is not too complex. It is carried out into the FORGE3® finite element software, where a nodal (node to facet) contact formulation is utilized and the contact conditions are handled by a penalty method. A series of patch tests that have been proposed in [1] and [2], allows evaluating the convergence rate of this formulation and its robustness.

#### Domaines

Sciences de l'ingénieur [physics] Matériaux

### Dates et versions

hal-00509842 , version 1 (16-08-2010)

### Identifiants

• HAL Id : hal-00509842 , version 1

### Citer

Lionel Fourment, Sorin Popa. Quasi-symmetrical formulations for contact and friction between deformable bodies: application to 3D forging. 8th International Conference on Computational Plasticity CIMNE UPC, Sep 2005, Barcelona, Spain. ⟨hal-00509842⟩

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