1. Introduction

The finite element method is currently massively used in numerical mechanics to simulate the problems of unilateral contact between two deformable objects. These are problems with a non-linear boundary condition, namely respecting a non-interpenetration condition for the displacement field, a sign condition for the normal pressure on the contact zone and a condition of complementarity between these two fields. In the general case, such problems lead to the management of incompatible meshes (i.e. whose nodes in the contact areas of the objects in question are not coincident).

The most used methods are based on node-segment algorithms in \(2D\) and node-facet algorithms in \(3D\) (in code_aster, these are the formulations DISCRETE and CONTINUE in mode INTEGRATION =” NOEUD “). These methods are known to fail patch tests in the general case of incompatible meshes and to lead to sub-optimal mathematical analyses. On the other hand, the academic community has been very productive in this field over the past fifteen years. The most successful methods are mortar methods adapted to contact.

These methods consist in using a simple projection \({L}^{2}\) onto a joint space, i.e., a space of finite elements defined between the two objects. The different choices of joint spaces generate a whole family of mortar-type methods. However, these methods have significant disadvantages when considering their generic implementation in an industrial code.

The method proposed here gives numerical and mathematical results of as good quality as those obtained by the mortar methods while maintaining the aspects (locality, analytical definition of joint spaces, etc…) making the implementation in code_aster easier by maintaining the current architecture of the code

To do this, we will define a simple contact condition which consists in satisfying the contact conditions on average on each macro-element of an appropriately defined macro-mesh (which will have to be prepared using the option “DECOUPE_LAC” of the operator CREA_MAILLAGE).