Franz Chouly

Robin Based Semi-Implicit Coupling in Fluid-Structure Interaction: Stability Analysis and Numerics

In this paper, we propose a semi-implicit coupling scheme for the numerical simulation of fluid-structure interaction systems involving a viscous incompressible fluid. The scheme is stable irrespective of the so-called added-mass effect and allows for conservative time-stepping within the structure. The efficiency of the scheme is based on the explicit splitting of the viscous effects and geometrical/convective nonlinearities through the use of the Chorin-Temam projection scheme within the fluid. Stability comes from the implicit pressure-solid coupling and a specific Robin treatment of the explicit viscous-solid coupling, derived from Nitsches method.

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

We 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.

Symmetric and non-symmetric variants of Nitsche's method for contact problems in elasticity: theory and numerical experiments

A general Nitsche method, which encompasses symmetric and non-symmetric variants, is proposed for frictionless unilateral contact problems in elasticity. The optimal convergence of the method is established both for two and three-dimensional problems and Lagrange affine and quadratic finite element methods. Two and three-dimensional numerical experiments illustrate the theory.

Simulation of the retroglossal fluid-structure interaction during obstructive sleep apnea

A method for computing the interaction between the airflow and the soft tissue during an Obstructive Apnea is presented. It is based on simplifications of the full continuum formulation (Navier-Stokes and finite elasticity) to ensure computation time compatible with clinical applications. Linear elasticity combined with a precomputation method allows fast prediction of the tissue deformation, while an asymptotic formulation of the full Navier-Stokes equations (Reduced Navier-Stokes/Prandtl equations) has been chosen for the flow. The accuracy of the method has already been assessed experimentally. Then, simulations of the complete collapsus at the retroglossal level in the upper airway have been carried out, on geometries extracted from pre-operative radiographies of two apneic patients. Post-operative geometries have been also used to check qualitatively if the predictions from the simulations are in agreement with the effects of the surgery

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 @e 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 @e=h. We finally extend the results to the case where given (Tresca) friction is taken into account.

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.

A local projection stabilized method for fictitious domains

In this work a local projection stabilization method is proposed for solving a fictitious domain problem. The method adds a suitable fluctuation term to the formulation, thus yielding the natural space for the Lagrange multiplier stable. Stability and convergence are proved and these results are illustrated with a numerical experiment.

Modelling the human pharyngeal airway: validation of numerical simulations using in vitro experiments

In the presented study, a numerical model which predicts the flow-induced collapse within the pharyngeal airway is validated using in vitro measurements. Theoretical simplifications were considered to limit the computation time. Systematic comparisons between simulations and measurements were performed on an in vitro replica, which reflects asymmetries of the geometry and of the tissue properties at the base of the tongue and in pathological conditions (strong initial obstruction). First, partial obstruction is observed and predicted. Moreover, the prediction accuracy of the numerical model is of 4.2% concerning the deformation (mean quadratic error on the constriction area). It shows the ability of the assumptions and method to predict accurately and quickly a fluid–structure interaction.

An Enhanced Parareal Algorithm for Partitioned Parabolic‐Hyperbolic Coupling

We present a parallel time‐marching scheme for coupled parabolic‐hyperbolic problems, as a prototype of fluid‐structure interaction problems involving a linear structure and a viscous fluid. No linearity assumption is made on the parabolic side. The classical Parareal scheme is applied to the parabolic part, while the modified algorithm proposed by Farhat et al. [1] is applied to the hyperbolic part. This hybrid Parareal treatment relies on the partitioned formulation of the coupled propagator. Numerical evidence shows that the resulting scheme is stable for a wide range of physical and discretization parameters.

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

Most 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, as in Chouly et al. (Math Comput 84:1089–1112, 2015). We carry out the numerical analysis of the method, and prove its well-posedness and optimal convergence in the