Abstract. The purpose of this paper is to provide a priori error estimates on the approximation of contact conditions in the framework of the eXtended Finite-Element Method (XFEM) for two dimensional elastic bodies. This method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. We consider a stabilized Lagrange multiplier method whose particularity is that no discrete inf-sup condition is needed in the convergence analysis. The contact condition is prescribed on the crack with a discrete multiplier which is the trace on the crack of a finiteelement method on the non-cracked domain, avoiding the definition of a specific mesh of the crack. Additionally, we present numerical experiments which confirm the efficiency of the proposed method.Mathematics Subject Classification. 74M15, 74S05, 35M85.
We present a new finite element method, called φ-FEM, to solve numerically elliptic partial differential equations using simple computational grids, not fitted to the boundary of the physical domain. The geometry of the domain is taken into account using a level-set function, which is a popular tool to deal with complicated or evolving domains. Our approach belongs to the family of fictitious domain methods (or immersed boundary methods) and is close to recent methods of cutFEM/XFEM type. Contrary to the latter, φ-FEM does not need any non-standard numerical integration on cut mesh elements or on the actual boundary, while assuring the optimal convergence orders with finite elements of any degree and providing reasonably well conditioned discrete problems. The first version of φ-FEM was introduced in [1] for the case of essential (Dirichlet) boundary conditions. The approximation to the solution is obtained there as a product of a finite element function with the given level-set function, also approximated by finite elements. The case of natural (Neumann or Robin) boundary conditions is less straightforward. A variant of φ-FEM for this case is proposed in [2]. We introduce there the gradient of the primary solution as an auxiliary variable (only on the mesh cells cut by the boundary, so that the size of the numerical system is only slightly increased).
This paper is concerned with residual error estimators for finite element approximations of Coulomb frictional contact problems. A recent uniqueness result by Renard in [72] for the continuous problem allows us to perform an a posteriori error analysis. We propose, study and implement numerically two residual error estimators associated with two finite element discretizations. In both cases the estimators permit to obtain upper and lower bounds of the discretization error.
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