Michel Fournié - Academia.edu (original) (raw)

Papers by Michel Fournié

HAL (Le Centre pour la Communication Scientifique Directe), 2018

IFAC-PapersOnLine

We study the numerical approximation of the fluid structure interaction for stabilization of the ... more We study the numerical approximation of the fluid structure interaction for stabilization of the fluid flow around an unstable stationary solution in a two dimensional domain, in the presence of boundary perturbations. We use a feedback control law recently proposed in [Airiau et al. (2017)] which is able to stabilize the nonlinear semi-discrete controlled system and based on the fluid only. Using Dirichlet boundary feedback, we deduce a boundary structure displacement. The fluid structure closed loop feedback is tested numerically using a fictitious domain finite element method based on extended Finite Element.

Lecture Notes in Computational Science and Engineering, 2020

The present study deals with the numerical simulation of a fluid-structure interaction problem. T... more The present study deals with the numerical simulation of a fluid-structure interaction problem. The fluid is represented by the incompressible Navier-Stokes equations and the structure is described by an ODE depending on two degrees of freedom. A recent fictitious domain method on a fixed mesh is considered. For that choice, we provide several tricks to meet the difficulties arising from the fluidstructure interaction. All developed tools can be applied to very general geometries and deformations of the structure. Finally, numerical simulations are conducted in a realistic aeronautics configuration.

We study the numerical approximation of a 2d fluid–structure interaction problem stabilizing the ... more We study the numerical approximation of a 2d fluid–structure interaction problem stabilizing the fluid flow around an unstable stationary solution in presence of boundary perturbations. The structure is governed by a finite number of parameters and a feedback control law acts on their accelerations. The existence of strong solutions and the stabilization of this fluid–structure system were recently studied in [3]. The present work is dedicated to the numerical simulation of the problem using a fictitious domain method based on extended Finite Element [4]. The originality of the present work is to propose efficient numerical tools that can be extended in a simple manner to any fluid-structure control simulation. Numerical tests are given and the stabilization at an exponential decay rate is observed for small enough initial perturbations.

arXiv: Numerical Analysis, 2017

We study a fictitious domain approach with Lagrange multipliers to discretize Stokes equations on... more We study a fictitious domain approach with Lagrange multipliers to discretize Stokes equations on a mesh that does not fit the boundaries. A mixed finite element method is used for fluid flow. Several stabilization terms are added to improve the approximation of the normal trace of the stress tensor and to avoid the inf-sup conditions between the spaces of the velocity and the Lagrange multipliers. We generalize first an approach based on eXtended Finite Element Method due to Haslinger-Renard (SIAM J Numer Anal 47(2):1474–1499, 2009) involving a Barbosa-Hughes stabilization and a robust reconstruction on the badly cut elements. Secondly, we adapt the approach due to Burman-Hansbo (Comput Methods Appl Mech Eng 199(41–44):2680–2686, 2010) involving a stabilization only on the Lagrange multiplier. Multiple choices for the finite elements for velocity, pressure and multiplier are considered. Additional stabilization on pressure (Brezzi-Pitkaranta, Interior Penalty) is added, if needed. ...

Lecture Notes in Computational Science and Engineering, 2013

We present new high-order Alternating Direction Implicit (ADI) schemes for the numerical solution... more We present new high-order Alternating Direction Implicit (ADI) schemes for the numerical solution of initial-boundary value problems for convection-diffusion equations with cross derivative terms. Our approach is based on the unconditionally stable ADI scheme proposed by Hundsdorfer [12]. Different numerical discretizations which lead to schemes which are fourth-order accurate in space and secondorder accurate in time are discussed. on a rectangular domain Ω ⊂ R 2 , supplemented with initial and boundary conditions. In (1.1),

Proceedings of the 5th International Conference on Jets, Wakes and Separated Flows (ICJWSF2015), 2016

Over the last few years, superhydrophobic (SH) surfaces have been receiving an increasing attenti... more Over the last few years, superhydrophobic (SH) surfaces have been receiving an increasing attention in many scientific areas by virtue of their ability to enhance flow slip past solid walls and reduce the skin-friction drag. In the present study, a global linear-stability analysis is employed to investigate the influence of the SH-induced slip velocity on the primary instability of the 2D flow past a circular cylinder. The flow regions playing the role of 'wavemaker' are identified by considering the structural sensitivity of the unstable mode, thus highlighting the effect of slip on the global instability of the considered flow. In addition, a sensitivity analysis to slip-induced base-flow modifications is performed, revealing which areas of the cylinder surface provide a stabilising/destabilising effect when treated with a SH coating.

SIAM Journal on Scientific Computing, 2017

We study the numerical approximation of the boundary stabilization of the Navier-Stokes equations... more We study the numerical approximation of the boundary stabilization of the Navier-Stokes equations with mixed Dirichlet/Neumann boundary conditions, around an unstable stationary solution in a two dimensional domain. We first derive a semidiscrete controlled system, coming from a finite element approximation of the Navier-Stokes equations, which is new in the literature. We propose a new strategy for finding a boundary feedback control law able to stabilize the nonlinear semidiscrete controlled system in the presence of boundary disturbances. We determine the best control location. Next, we study the degree of stabilizability of the different real generalized eigenspaces of the controlled system. Based on that analysis, we determine an invariant subspace Zu and the projection of the controlled system onto Zu. The projected system is used to determine feedback control laws. Our numerical results show that this control strategy is quite efficient when applied to the Navier-Stokes system for a Reynolds number Re = 150 with boundary perturbations.

SIAM Journal on Control and Optimization, 2019

We study the stabilization of a fluid-structure interaction system around an unstable stationary ... more We study the stabilization of a fluid-structure interaction system around an unstable stationary solution. The system consists of coupling the incompressible Navier-Stokes equations, in a two dimensional polygonal domain with mixed boundary conditions, and a damped Euler-Bernoulli beam equations located at the boundary of the fluid domain. The control acts only in the beam equations. The feedback is determined by stabilizing the projection of the linearized model onto a finite dimensional invariant subspace. Here we have resolved two important challenges for applications in this field. One is the fact that we prove a stabilization result around a non zero stationary solution, which is new for such fluid-structure interaction systems. The other one is that the feedback laws that we determine do not depend on the Leray projector used to get rid of the algebraic constraints of partial differential equations. This is essential for numerical aspects.

Mathematics in Industry, 2012

In this short paper we are concerned with the von Neumann stability analysis of a compact high-or... more In this short paper we are concerned with the von Neumann stability analysis of a compact high-order finite difference scheme for option pricing in the Heston stochastic volatility model. We first review results on the unconditional stability in the case of vanishing correlation and then present some new results on the behavior of the amplification factor for non-zero correlation.

The HexMC® is a carbon epoxy SMC composite made of prepeg patches. Its heterogeneity is pointed o... more The HexMC® is a carbon epoxy SMC composite made of prepeg patches. Its heterogeneity is pointed out using NDT measurements, physico-chemical characterizations and mechanical tests. Tensile tests performed on multi instrumented standard coupons are presented. The coherence between the information retrieved is commented. Displacement fields are first used in order to reveal strain field heterogeneity. These measurements are then used as an input for a procedure developed for identifying elastic properties field and based on the “Equilibrium Gap Method”. The maps of Young moduli identified at the principle and the end of the loading are compared with ultrasonic measurements performed before and after the tensile test.

Communications in Computational Physics, 2011

Within the projection schemes for the incompressible Navier-Stokes equations (namely “pressure-co... more Within the projection schemes for the incompressible Navier-Stokes equations (namely “pressure-correction” method), we consider the simplest method (of order one in time) which takes into account the pressure in both steps of the splitting scheme. For this scheme, we construct, analyze and implement a new high order compact spatial approximation on nonstaggered grids. This approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions (without error on the velocity) which could be extended to more general splitting. We prove the unconditional stability of the associated Cauchy problem via von Neumann analysis. Then we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical solutions. Finally we present detailed numerical tests for the Stokes and the Navier-Stokes equations (including the driven cavity benchmark) to illustrate the theoretical results.

Journal of Fluids and Structures, 2015

The paper extends a fictitious domain finite element method analyzed for Stokes problem in [30] t... more The paper extends a fictitious domain finite element method analyzed for Stokes problem in [30] to the Navier-Stokes equations coupled with a moving solid. The dynamics of the solid is governed by the Newton's laws. The mixed finite element method used cuts element at the interface localized by level-set while preserving an optimal accuracy of the approximation of the normal stress tensor. An algorithm is proposed in order to treat the time evolution of the geometry and numerical tests are given.

Computer Algebra in Scientific Computing CASC’99, 1999

A symbolic procedure for deriving finite difference approximations for partial differential equat... more A symbolic procedure for deriving finite difference approximations for partial differential equations is described. We restrict our study to high-order compact schemes in conservative and non-conservative form.

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

AIP Conference Proceedings, 2010

We present a compact high-order finite difference scheme for option pricing in the well-known Hes... more We present a compact high-order finite difference scheme for option pricing in the well-known Heston stochastic volatility model. The scheme is fourth order accurate in space and second order accurate in time. This is also confirmed by the numerical experiments that we present.

Euro-Par 2010 - Parallel Processing, 2010

We consider the finite element environment Getfem++ 1 , which is a C++ library of generic finite ... more We consider the finite element environment Getfem++ 1 , which is a C++ library of generic finite element functionalities and allows for parallel distributed data manipulation and assembly. For the solution of the large sparse linear systems arising from the finite element assembly, we consider the multifrontal massively parallel solver package Mumps 2 , which implements a parallel distributed LU factorization of large sparse matrices. In this work, we present the integration of the Mumps package into Getfem++ that provides a complete and generic parallel distributed chain from the finite element discretization to the solution of the PDE problems. We consider the parallel simulation of the transition to turbulence of a flow around a circular cylinder using Navier Stokes equations, where the nonlinear term is semi-implicit and requires that some of the discretized differential operators be updated and with an assembly process at each time step. The preliminary parallel experiments using this new combination of Getfem++ and Mumps are presented.

Journal of Computational and Applied Mathematics, 2012

We derive a new high-order compact finite difference scheme for option pricing in stochastic vola... more We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth order accurate in space and second order accurate in time. Under some restrictions, theoretical results like unconditional stability in the sense of von Neumann are presented. Where the analysis becomes too involved we validate our findings by a numerical study. Numerical experiments for the European option pricing problem are presented. We observe fourth order convergence for non-smooth payoff.

Journal of Applied Mathematics and Computing, 2006

International Journal for Numerical Methods in Fluids, 2013

SUMMARYIn the present work, we propose to extend to the Stokes problem a fictitious domain approa... more SUMMARYIn the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by extended finite element method and studied for the Poisson problem in a paper of Renard and Haslinger of 2009. The method allows computations in domains whose boundaries do not match. A mixed FEM is used for the fluid flow. The interface between the fluid and the structure is localized by a level‐set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf‐sup condition between the spaces for the velocity and the Lagrange multiplier. Convergence analysis is given, and several numerical tests are performed to illustrate the capabilities of the method. Copyright © 2013 John Wiley & Sons, Ltd.