A stable 3D contact formulation using X-FEM

Abstract

This paper presents a 3D non-locking contact approach, within the eXtended Finite Element Method (X-FEM) framework. X-FEM allows one to introduce interface independently of the mesh. The contact problem on the interface leads to an Augmented Lagrangian formulation derived from the discretization of its continuous formulation. It is shown that a simple choice of the Lagrange multiplier space is not suitable and leads to contact pressure oscillations. An algorithm for the restriction of the Lagrange multiplier approximation space is proposed to stabilize the formulation. The stability of the mixed displacement-contact pressure formulation is discussed in terms of convergence of the energy error. Numerical examples performed with the Finite Element software Code_Aster illustrate this approach while solving three-dimensional problems with contact. RÉSUMÉ. Cet article présente une formulation stabilisée pour les problèmes de contact en 3D, dans le cadre de la méthode des éléments finis étendue (X-FEM), méthode qui autorise des interfaces indépendantes du maillage. Pour la formulation du problème de contact sur l'interface, nous utilisons un Lagrangien Augmenté, qui dérive de la discrétisation du problème de contact écrit sous sa forme continue. Nous montrons qu'un choix simple de l'espace des multiplicateurs de Lagrange n'est pas satisfaisant car cela conduit à des oscillations des pressions de contact. Un algorithme de restriction de l'espace d'approximation des multiplicateurs de Lagrange est proposé afin de stabiliser la formulation. La stabilité de la formulation mixte (déplacement-pression de contact) est démontrée à l'aide des taux de convergence des erreurs. Des exemples numériques réalisés avec le logiciel éléments finis Code_Aster illustrent cette approche pour la résolution de problèmes tridimensionnels avec contact.

A contact algorithm for frictional crack propagation with the extended finite element method

“…We realize that the penalty method may not be optimal for some problems since it could lead to poor conditioning of the system of equations in some cases. To address this concern, we also consider an augmented Lagrangian technique [43][44][45] and monitor the efficacy and adequacy of the standard penalty approach. For both the standard penalty and augmented Lagrangian procedures the FE equations allow the balance of linear momentum to be written in a standard residual form and linearized consistently for a full Newton-Raphson iteration.…”

Embedded solids of any dimension in the X-FEM. Part II – Imposing Dirichlet boundary conditions

“…A new algorithm is introduced to reduce the Lagrange multiplier space for any codimension of the boundary embedded in a mismatching mesh. One improvement with respect to existing algorithms [18,66,48,44] is its ability to build stable Lagrange multiplier spaces along embedded lines in 3D (codimension two). To our best knowledge, the present work is the first study investigating Dirichlet boundary conditions on this setting within non-conforming meshes.…”

“…They allow to extend stable Lagrange multiplier methods [18,66,48,44] to 1D boundaries embedded in 3D (codimension two) in a single framework.…”

“…This selection must satisfy the following conditions to be consistent with FEM. Similar concepts have been introduced in [18,66,48], but only edges of the mesh were used to identify the shape functions. Furthermore, the non-uniqueness in the choice of the resulting Lagrange multiplier space was reported.…”

“…The parameter is computed with analytical standard hat functions that is too rich, a first algorithm has been proposed in [18] to deplete the Lagrange multiplier space. A second algorithm has been introduced in [66] and extended for large sliding [67], improving the accuracy of the computed fields at the expense of the resolution of a global problem. Finally, a third algorithm [48] combines both a local construction of the Lagrange multiplier space and good accuracy.…”