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Introduction
I do research in numerical analysis and scientific computing, with emphasis on finite element methods for some boundary and interface conditions. Main applications are solid mechanics and fluid mechanics.
Additional affiliations
September 2018 - present
Education
September 2002 - December 2005
Publications
Publications (136)
The finite‐element method (FEM) is a well‐established procedure for computing approximate solutions to deterministic engineering problems described by partial differential equations. FEM produces discrete approximations of the solution with a discretisation error that can be quantified with a posteriori error estimates. The practical relevance of e...
We are interested in heterogeneous domain decomposition methods to couple partial differential equations in space-time. The coupling can be used to describe the exchange of heat or forces or both, and has important applications like fluid-structure or ocean-atmosphere coupling. Heterogeneous domain decomposition methods permit furthermore the reuse...
This textbook provides an accessible introduction to the mathematical foundations of the finite element method for a broad audience. The author accomplishes this, in part, by including numerous exercises and illustrations. Each chapter begins with a clear outline to help make complex concepts more approachable without sacrificing depth. Structurall...
Prior to discretizing the Poisson problem, we need to introduce fundamental concepts of the finite element method, with one of the simplest finite elements: the first-order (piecewise linear) Lagrange finite element on a simplex.
This chapter presents Nitsche method as a prototype of an alternative method to handle essential boundary conditions, and, moreover, as a prototype of non-standard finite element method, for which numerical analysis is helpful to fix some issues.
Now we combine the two previous chapters to present the standard method to discretize Poisson’s problem with Lagrange finite elements. Particular emphasis is made on how to handle the non-homogeneous Dirichlet boundary condition.
This chapter is about practical situations where the user is interested to estimate the error due to finite element approximation.
This chapter presents a simple introduction to Signorini conditions for a membrane equation, as a prototype of non-linear non-smooth boundary conditions that model contact.
This chapter details a simple diffusion model as a prototype of a boundary value problem based on a linear, scalar, stationary partial differential equation.
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 hig...
This study is concerned with the finite element approximation of the elastoplastic torsion problem. We focus on the case of a nonconstant source term, which cannot be easily recast into an obstacle problem as can be done in the case of a constant source term. We present a simple formulation that penalizes the constraint directly on the gradient nor...
This work focuses on the numerical performance of the Nitsche-based Finite Element Method for dynamic unilateral contact problems combined with two implicit one-step time-marching schemes. The non-linear contact boundary conditions cause irregularities, which may lead to unstable performance and potential divergence during simulations. By focusing...
Error control by means of a posteriori error estimators or indica-tors and adaptive discretizations, such as adaptive mesh refinement, have emerged in the late seventies. Since then, numerous theoretical developments and improvements have been made, as well as the first attempts to introduce them into real-life industrial applications. The present...
Presentation on frictional contact problems with emphasis on the control of the discretization error.
Error control by means of a posteriori error estimators or indicators and adaptive discretizations, such as adaptive mesh refinement, have emerged in the late seventies. Since then, numerous theoretical developments and improvements have been made, as well as the first attempts to introduce them into real-life industrial applications. The present i...
This study investigates the computational efficacy of HHT-and TR-BDF2 schemes in addressing dynamic frictionless unilateral contact challenges between an elastic structure and a rigid obstacle. The application of combinations of Nitsche's method with these schemes is explored for managing unilateral contact conditions. An examination of the converg...
Presentation at the MOX seminar of some past and recent research about frictional contact
This work focuses on the numerical performance of HHT-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and TR-BDF2 schemes for dynamic frictionl...
Keloids are fibroproliferative disorders described by excessive growth of fibrotic tissue, which also invades adjacent areas (beyond the original wound borders). Since these disorders are specific to humans (no other animal species naturally develop keloid-like tissue), experimental in vivo/in vitro research has not led to significant advances in t...
Errors in biomechanical simulations arise from modelling and discretization. Discretization errors can be quantified with a posteriori error estimates. The relevance of such error estimates for practical biomechanics problems is rarely adressed. In this contribution, we propose an implementation of a posteriori error estimates targeting a user-defi...
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 friction models. We derive residual a posteriori error estimates for each friction model, following (Chouly et al., in IMA J...
This study is concerned with the elastoplastic torsion problem, in dimension $n\geq1$, and in a polytopal, convex or not, domain. In the physically relevant case where the source term is a constant, this problem can be reformulated using the distance function to the boundary. We combine the aforementioned reformulation with a Nitsche-type discretiz...
We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates are established, and illustrated with various numerical experiment...
In this chapter, we detail the formulation for the contact between two elastic bodies, in the small deformation framework. This is mostly a preliminary step before the study of contact in large strains that involve much more technical difficulties in terms of formulation and implementation. Indeed, we focused up to now on the Signorini problem, stu...
This chapter is devoted to the finite element approximation of the Signorini problem (3.12) described in previous Chap. 3. We start with the simplest possible setting just to point out the main difficulty that appears in the convergence analysis of many methods. We then present discretizations based on a direct approximation of the variational ineq...
This chapter is about the Coulomb frictional contact model. As we have already seen for Tresca friction, when taking into account friction in addition to the contact model, there are supplementary nonlinearities that have to be taken into account. Remark there exist simplified and/or different models for friction: Tresca friction (see the previous...
As a preliminary step before going into Coulomb friction, we focus on the numerical approximation of Tresca friction in this chapter. The Signorini problem with Tresca friction leads to a well-posed variational inequality of the second kind. As a result, this setting makes easier the mathematical analysis of some numerical approximations, as it was...
Contact and friction problems, as well as many other problems involving elliptic partial differential equations and some specific boundary conditions, are formulated and studied appropriately using fractional order Sobolev spaces. This functional setting allows to provide some meaningful reformulations of the original mathematical models (in strong...
As we have seen in the previous Chap. 5, it may not be the best idea to approximate directly the variational inequality for practical solving. As a result, many reformulations of Problem (3.12) have been studied since the 1970s, which aim at making easier the implementation and numerical resolution for contact, especially for problems more difficul...
Mixed methods were among the first ones to be formulated for contact, along with penalty methods: see, e.g., the pioneering works of J. Haslinger in 1977, or of F. Brezzi, W. W. Hager and P.A. Raviart in 1978, see also the monographs and references therein. Indeed, such formulations, that involve a Lagrangian, are very natural for constrained probl...
In many engineering applications, the elastic bodies in contact may undergo large transformations. In this case, a setting such as presented in Chap. 3 and in Chap. 10 is no longer valid, since it was restricted to small displacements and small strain. Taking into account large displacements and large strain, for instance in the hyperelastic framew...
In this chapter, we present the Signorini problem in small strain elasticity, as a first step before going into its numerical approximation and into more complex contact and friction problems. Indeed, Signorini conditions are the simplest conditions that allow to model appropriately frictionless contact between an elastic body and a rigid support,...
The aim of this chapter is to give a brief introduction to finite element spaces, and introduce some useful interpolation estimates in fractional order Sobolev spaces which will serve later on to establish corresponding a priori error estimates. For simplicity, we limit the presentation to Lagrange finite elements on triangular or tetrahedral meshe...
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, I...
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 val...
Modélisation numérique en mécanique fortement non linéaire traite des avancées récentes sur le traitement numérique des phénomènes de contact/frottement et d’endommagement.Bien que distincts sur le plan physique, ces phénomènes entraînent tous deux une forte non-linéarité du problème mécanique qui limite la régularité du problème, dorénavant non di...
This work deals with the discretization of single-phase Darcy flows in fractured and de-formable 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 keloids are fibroproliferative disorders described by an excessive growth of fibrotic tissue, which also invades adjacent areas (beyond the original wound borders). Since these disorders are specific to humans (no other animal species naturally develop keloid-like tissue), the experimental in vivo/in vitro research has not lead to significant a...
This study is concerned with
the elastoplastic torsion problem, in dimension $n\geq1$, and in a polytopal, convex or not, domain. In the physically relevant case where the source term is a constant, this problem can be reformulated using the distance function to the boundary. We combine the aforementioned reformulation with a Nitsche-type discreti...
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 hi...
We study the Nitsche-based finite element method for contact with Coulomb friction considering both static and dynamic situations. We provide existence and/or uniqueness results for the discretized problems under appropriate assumptions on physical and numerical parameters. In the dynamic case, existence and uniqueness of the space semi-discrete pr...
This paper deals with the coupling between one-dimensional heat and wave equations in unbounded subdomains, as a simplified prototype of fluid-structure interaction problems. First we apply appropriate artificial boundary conditions that yield an equivalent problem, but with bounded subdomains, and we carry out the stability analysis for this coupl...
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 trait...
This study is concerned with the elastoplastic torsion problem and its standard finite element approximation using piecewise affine Lagrange finite elements. In the case of a polytopal convex domain in dimension n=1,2,3 we obtain an H1-error bound of order h for the solution. For a nonconvex domain, we obtain also an error estimate.
In this short technical note, we are interested in the constitutive equations used to model macroscopically the mechanical function of soft tissues under mechanical loading. Soft tissues have the ability to undergo large elastic reversible deformations under quasi-static loading and are usually modelled using hyperelastic constitutive laws. Several...
This study is concerned with the elastoplastic torsion problem and its standard finite element approximation using piecewise affine Lagrange finite elements. In the case of a polytopal convex domain in dimension n ≥ 2 we obtain a H1-error bound of order h for the solution. For a non convex domain we obtain an order h^3/4. This improves the existing...
We develop a novel a posteriori error estimator for the L2 error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi-discretization scheme using a rational approximation which allows to reformulate the fractional problem into a family of non-fractio...
L’objectif de ce chapitre est de faire une présentation détaillée de développements récents sur l’approximation des conditions de contact des solides déformables en petites et grandes déformations. Les principes de base des méthodes de lagrangien, lagrangien augmenté, lagrangien stabilisé, de pénalité et de Nitsche sont exposés. Les liens étroits e...
In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, t...
In this poster, we present our new hierarchical a posteriori error estimation method for the spectral fractional Laplacian equation with homogeneous Dirichlet boundary condition. The numerical results show the adaptively refined mesh we obtain using our method as well as a convergence comparison where we can notice the sharpness of the estimator an...
We present two primal methods to weakly discretize (linear) Dirichlet and (nonlinear) Signorini boundary conditions in elliptic model problems. Both methods support polyhedral meshes with nonmatching interfaces and are based on a combination of the hybrid high-order (HHO) method and Nitsche’s method. Since HHO methods involve both cell unknowns and...
We study the Nitsche-based finite element method for contact with Coulomb friction considering both static and dynamic situations. We provide existence and/or uniqueness results for the discretized problems under appropriate assumptions on physical and numerical parameters. In the dynamic case, existence and uniqueness of the space semi-discrete pr...
The aim of this work is to characterize the mechanical parameters governing the in-plane behavior of human skin and, in particular, of a keloid-scar. We consider 2D hyperelastic bi-material model of a keloid and the surrounding healthy skin. The problem of finding the optimal model parameters that minimize the misfit between the model observations...
This paper deals with the coupling between one-dimensional heat and wave equations in unbounded subdomains, as a simplified prototype of fluid-structure interaction problems. First we build artificial boundary conditions for each subproblem so as to solve it numerically in a bounded subdomain. Then we devise an optimized Schwarz-in-time (or Schwarz...
We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes the main novelty of our study. In addition, we present a numerical comparison of the Bank-Weiser and residual estimators for a three-dimensional te...
We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes the main novelty of our study. In addition, we present a numerical comparison of the Bank-Weiser and residual estimators for a three-dimensional te...
In this note, we determine the bi-tangents of two rotated ellipses, and we compute the coordinates of their points of tangency. For these purposes, we develop two approaches. The first one is an analytical approach in which we compute analytically the equations of the bi-tangents. This approach is valid only for some cases. The second one is geomet...
We devise and analyze a Hybrid high-order (HHO) method to discretize unilateral and bilateral contact problems with Tresca friction in small strain elasticity. The nonlinear frictional contact conditions are enforced weakly by means of a consistent Nitsche's technique with symmetric, incomplete, and skew-symmetric variants. The present HHO-Nitsche...
Errors in biomechanics simulations arise from modelling and discretization. Modelling errors are due to the choice of the mathematical model whilst discretization errors measure the impact of the choice of the numerical method on the accuracy of the approximated solution to this specific mathematical model. A major source of discretization errors i...
Le travail proposé concerne la prédiction de la réponse numérique d'une structure biomécanique sou-mise à un état de chargement externe inconnu. La méthodologie s'appuie sur la modélisation de structures homogènes puis hétérogènes, telles que des tissus cutanés sains ou pathologiques accessibles expé-rimentalement et soumis à des conditions aux lim...
La peau humaine se comporte comme une membrane élastique précontrainte. La présence de sites anatomiques favorables à l'apparition de certaines tumeurs, une chéloïde dans notre cas, alors que d'autres en sont systématiquement dépourvus atteste de l'importance de l'environnement mécanique du tissu. Par conséquent, une caractérisation de la peau avec...
We present two primal methods to weakly discretize (linear) Dirichlet and (nonlinear) Signorini boundary conditions in elliptic model problems. Both methods support polyhedral meshes with non-matching interfaces and are based on a combination of the Hybrid High-Order (HHO) method and Nitsche's method. Since HHO methods involve both cell unknowns an...
The aim of this paper is to provide some mathematical results for the discrete problem associated to contact with Coulomb friction, in linear elasticity, when finite elements and Nitsche method are considered. We consider both static and dynamic situations. We establish existence and uniqueness results under appropriate assumptions on physical (fri...
The aim of the present paper is to study theoretically and numerically the Verlet scheme for the explicit time-integration of elastodynamic problems with a contact condition approximated by Nitsche’s method. This is a continuation of papers (Chouly et al. ESAIM Math Model Numer Anal 49(2), 481–502, 2015; Chouly et al. ESAIM Math Model Numer Anal 49...
In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements...
We present the Corotational Cut Finite Element Method for real-time surgical simulation. Users only need to provide a background mesh which is not necessarily conforming to the boundaries/interfaces of the simulated object. The details of the latter, represented by its surface or/and its internal interfaces, which can be directly obtained from bina...
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...
The aim of the present paper is to study theoretically and numerically the Verlet scheme for the explicit time-integration of elastodynamic problems with a contact condition approximated by Nitsche's method. This is a continuation of papers [11, 12] where some implicit schemes (theta-scheme, Newmark and a new hybrid scheme) were proposed and proved...
We investigate numerically the vibrational behavior of an elastic structure containing an inviscid fluid with a filling hole where the pressure is prescribed. The underlying mathematical model is detailed and its spectra is characterized. The finite element method relies upon the added-mass formulation of Morand and Ohayon but avoids the explicit a...
In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements...
The aim of this paper is to provide some mathematical results for the discrete problem associated to contact with Coulomb friction, in linear elasticity, when finite elements and Nitsche method are considered. We consider both static and dynamic situations. We establish existence and uniqueness results under appropriate assumptions on physical (fri...
A simple skew-symmetric Nitsche's formulation is introduced into the framework of isogeometric analysis (IGA) to deal with various problems in small strain elasticity: essential boundary conditions, symmetry conditions for Kirchhoff plates, patch coupling in statics and in modal analysis as well as Signorini contact conditions. For linear boundary...
A simple skew-symmetric Nitsche's formulation is introduced into the framework of isogeometric analysis (IGA) to deal with various problems in small strain elasticity: essential boundary conditions, symmetry conditions for Kirchhoff plates, patch coupling in statics and in modal analysis as well as Signorini contact conditions. For linear boundary...
Errors in biomechanics simulations arise from modeling and discretization. Modeling errors are due to the choice of the mathematical model whilst discretization errors measure the impact of the choice of the numerical method on the accuracy of the approximated solution to this specific mathematical model. A major source of discretization errors is...