Search results for "Nitsche"
showing 6 items of 6 documents
Mixed and Nitsche's discretizations of frictional contact-mechanics in fractured porous media
2023
This work deals with the discretization of single-phase Darcy flows in fractured and deformable porous media, including frictional contact at the matrix-fracture interfaces. Fractures are described as a network of planar surfaces leading to so-called mixed dimensional models. Small displacements and a linear poro-elastic behavior are considered in the matrix. One key difficulty to simulate such coupled poro-mechanical models is related to the formulation and discretization of the contact mechanical sub-problem. Our starting point is based on the mixed formulation using facewise constant Lagrange multipliers along the fractures representing normal and tangential stresses. This is a natural c…
HHT-α and TR-BDF2 schemes for dynamic contact problems
2023
This work focuses on the numerical performance of HHT-α and TR-BDF2 schemes for dynamic frictionless unilateral contact problems between an elastic body and a rigid obstacle. Nitsche's method, the penalty method, and the augmented Lagrangian method are considered to handle unilateral contact conditions. Analysis of the convergence of an opposed value of the parameter α for the HHT-α method is achieved. The mass redistribution method has also been tested and compared with the standard mass matrix. Numerical results for 1D and 3D benchmarks show the functionality of the combinations of schemes and methods used.
A review on some discrete variational techniques for the approximation of essential boundary conditions
2022
We review different techniques to enforce essential boundary conditions, such as the (nonhomogeneous) Dirichlet boundary condition, within a discrete variational framework, and especially techniques that allow to account for them in a weak sense. Those are of special interest for discretizations such as geometrically unfitted finite elements or high order methods, for instance. Some of them remain primal, and add extra terms in the discrete weak form without adding a new unknown: this is the case of the boundary penalty and Nitsche techniques. Others are mixed, and involve a Lagrange multiplier with or without stabilization terms. For a simple setting, we detail the different associated for…
Sur la prise en compte de quelques conditions aux limites avec la méthode des éléments finis
2017
Ce cours a pour principal sujet la discrétisation de certaines conditions limites sur le bord d'un domaine à l'aide de la méthode des éléments finis. Nous étudions en particulier des problèmes elliptiques avec conditions de Dirichlet non-homogènes, puis avec conditions de Signorini. Les conditions de Signorini sont des conditions de contact unilatéral entre un solide et un support rigide. Différentes méthodes sont passées en revue, et nous détaillons l'analyse pour les méthodes classiques et celle de Nitsche.
HHT-alpha and predictor-corrector scheme for dynamic contact problem by Nitsche's method
2022
Dans ce travail nous sommes intéressés par le problème de contact unilatéral en dynamique et sans frottement. Nous nous concentrons sur l'évolution en temps de l'impact d'un corps élastique linéaire sur un obstacle rigide, et avons souhaité en particulier étudier comment combiner des schémas en temps comme HHT et prédicteur-correcteur avec un traitement du contact via Nitsche, ce qui n'a jamais été effectué auparavant. Nous présenterons aussi quelques résultats numériques en faisant attention au comportement numérique : stabilité du schéma, conservation ou non de l'énergie et oscillations parasites. Mots clés-problème de contact, méthode de Nitsche, éléments finis, dynamique.
Residual a posteriori error estimation for frictional contact with Nitsche method
2023
We consider frictional contact problems in small strain elasticity discretized with finite elements and Nitsche method. Both bilateral and unilateral contact problems are taken into account, as well as both Tresca and Coulomb models for the friction. We derive residual a posteriori error estimates for each friction model, following [Chouly et al, IMA J. Numer. Anal. (38) 2018, pp. 921-954]. For the incomplete variant of Nitsche, we prove an upper bound for the dual norm of the residual, for Tresca and Coulomb friction, without any extra regularity and without a saturation assumption. Numerical experiments allow to assess the accuracy of the estimates and their interest for adaptive meshing …