A Nitsche-based domain decomposition method for hypersingular integral equations

We introduce and analyze a Nitsche-based domain decomposition method for the solution of hypersingular integral equations. This method allows for discretizations with non-matching grids without the necessity of a Lagrangian multiplier, as opposed to the traditional mortar method. We prove its almost quasi-optimal convergence and underline the theory by a numerical experiment.Comment: 21 pages, 5 figure

A local projection stabilized method for fictitious domains

In this work a local projection stabilization method is proposed to solve a fictitious domain problem. The method adds a suitable fluctuation term to the formulation thus rendering the natural space for the Lagrange multiplier stable. Stability and convergence are proved and these results are illustrated by a numerical experiment.Comment: Submitted Preprin

An adaptation of Nitsche's method to the Tresca friction problem

International audienceWe propose a simple adaptation to the Tresca friction case of the Nitsche-based finite element method given in [Chouly-Hild, 2012, Chouly-Hild-Renard, 2013] for frictionless unilateral contact. Both cases of unilateral and bilateral contact with friction are taken into account, with emphasis on frictional unilateral contact for the numerical analysis. We manage to prove theoretically the fully optimal convergence rate of the method in the H^1(Ω)-norm which is O(h^(1/2+nu)) when the solution lies in H^{3/2+nu}(Ω), 0 < ν ≤ 1/2, in two dimensions and three dimensions, for Lagrange piecewise linear and quadratic finite elements. No additional assumption on the friction set is needed to obtain this proof

A Nitsche-based method for unilateral contact problems: numerical analysis

International audienceWe introduce a Nitsche-based formulation for the finite element discretization of the unilateral contact problem in linear elasticity. It features a weak treatment of the non-linear contact conditions through a consistent penalty term. Without any additional assumption on the contact set, we can prove theoretically its fully optimal convergence rate in the H1(Ω)-norm for linear finite elements in two dimensions, which is O(h^(1/2+ν)) when the solution lies in H^(3/2+ν)(Ω), 0 < ν ≤ 1/2. An interest of the formulation is that, conversely to Lagrange multiplier-based methods, no other unknown is introduced and no discrete inf-sup condition needs to be satisfied

On convergence of the penalty method for unilateral contact problems

We present a convergence analysis of the penalty method applied to unilateral contact problems in two and three space dimensions. We first consider, under various regularity assumptions on the exact solution to the unilateral contact problem, the convergence of the continuous penalty solution as the penalty parameter $\varepsilon$ vanishes. Then, the analysis of the finite element discretized penalty method is carried out. Denoting by $h$ the discretization parameter, we show that the error terms we consider give the same estimates as in the case of the constrained problem when the penalty parameter is such that $\varepsilon = h$

Comparison of computations of asymptotic flow models in a constricted channel

International audienceWe aim at comparing computations with asymptotic models issued from incom- pressible Navier-Stokes at high Reynolds number: the Reduced Navier-Stokes/Prandtl (RNS/P) equations and the Double Deck (DD) equations. We treat the case of the steady two dimensional flow in a constricted pipe. In particular, finite differences and finite element solvers are compared for the RNS/P equations. It results from this study that the two codes compare well. Numerical examples also illustrate the interest of these asymptotic models as well as the flexibility of the finite element solver

A time-parallel framework for coupling finite element and lattice Boltzmann methods

International audienceIn this work we propose a new numerical procedure for the simulation of time-dependent problems based on the coupling between the finite element method and the lattice Boltzmann method. The procedure is based on the Parareal paradigm and allows to couple efficiently the two numerical methods, each one working with its own grid size and time-step size. The motivations behind this approach are manifold. Among others, we have that one technique may be more efficient, or physically more appropriate or less memory consuming than the other depending on the target of the simulation and/or on the sub-region of the computational domain. Furthermore, the coupling with finite element method may circumvent some difficulties inherent to lattice Boltzmann discretization, for some domains with complex boundaries, or for some boundary conditions. The theoretical and numerical framework is presented for parabolic equations, in order to describe and validate numerically the methodology in a simple situation

A Nitsche finite element method for dynamic contact

International audienceWe present a new approximation of elastodynamic frictionless contact problems based both on the finite element method and on an adaptation of Nitsche's method which was initially designed for Dirichlet's condition

An unbiased Nitsche's approximation of the frictional contact between two elastic structures

International audienceMost of the numerical methods dedicated to the contact problem involving two elastic bodies are based on the master/slave paradigm. It results in important detection difficulties in the case of self-contact and multi-body contact, where it may be impractical, if not impossible , to a priori nominate a master surface and a slave one. In this work we introduce an unbiased finite element method for the finite element approximation of frictional contact between two elastic bodies in the small deformation framework. In the proposed method the two bodies expected to come into contact are treated in the same way (no master and slave surfaces). The key ingredient is a Nitsche-based formulation of contact conditions. We carry out the numerical analysis of the method, and prove its well-posedness and optimal convergence in the H 1-norm. Numerical experiments are performed to illustrate the theoretical results and the performance of the method

A Nitsche finite element method for dynamic contact : 1. Semi-discrete problem analysis and time-marching schemes

International audienceThis paper presents a new approximation of elastodynamic frictionless contact problems based both on the finite element method and on an adaptation of Nitsche's method which was initially designed for Dirichlet's condition. A main interesting characteristic is that this approximation produces well-posed space semi-discretizations contrary to standard finite element discretizations. This paper is then mainly devoted to present an analysis of the semi-discrete problem in terms of consistency, well-posedness and energy conservation, and also to study the well-posedness of some time-marching schemes (theta-scheme, Newmark and a new hybrid scheme). The stability properties of the schemes and the corresponding numerical experiments can be found in a second paper