0000000000279269
AUTHOR
Franz Chouly
Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche
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 experiments.
Parameter identification problem in bimaterial human skin and sensitivity analysis : Uncertainties in biomechanics of skin
International audience; The proposed paper concerns the prediction of the numerical response of a biomechanical structure submitted to an unknown external loading state. The methodology is based on homogeneous and then heterogeneous structures such as healthy or pathological cutaneous tissues that can be mechanically tested in vivo under a patchy knowledge of boundary conditions. Experimental data corresponding to the extension of a piece of skin located between two pads with displacement enslavement, represent input data to the numerical model. Data are reaction force on one pad and displacement field between the two pads and all around. The numerical model consists of a representation of …
A review on some discrete variational techniques for the approximation of essential boundary conditions
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…
Computing bi-tangents for transmission belts
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 geometrical and is based on the determination of the normal vector to the tangent line. This approach turns out to be more robust than the first one and is valid for any configuration of ellipses.
Méthodes de lagrangien et de Nitsche pour l'approximation numérique des conditions de contact avec frottement
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 existants entre la méthode de Nitsche, le lagrangien stabilisé et le lagrangien augmenté sont alors analysés.
A few remarks about the computer implementation and the verification of some hyperelastic constitutive laws and an illustration with the mechanical response of an artery
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 constitutive equations have been defined in the literature by means of strain components or strain invariants isotropic hyperelastic models. This short technical note recalls how to derive the weak form for hyperelasticity equations, and how this weak form can be linearized. It also presents a semi-analytical solution for a simplified geometry re…
Residual a posteriori error estimation for frictional contact with Nitsche method
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 …
An introductive course to some numerical approximation methods for ordinary and partial differential equations
These lectures notes were written for students of the first year of the international Master ‘Maths for Physics’ at the University of Burgundy in 2021, as a part of the course called ‘Numerical Methods’. It was designed for a total of 22 hours of classes. The main objective of this course is to provide a first insight into numerical methods to solve mathematical problems inspired from physics and engineering. It focuses both on the mathematical fundations and justifications of numerical methods and on practical aspects related to their implementation.
Mixed and Nitsche's discretizations of frictional contact-mechanics in fractured porous media
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…
Analyse numérique MIGS 1re Année
Il s'agit d'un cours d'analyse numérique qui a été dispensé à l'Université de Bourgogne en Master MIGS 1re Année (en 2019-2020, et 2020-2021). Il traite principalement de la résolution itérative de systèmes linéaires symétriques par gradient conjugué, et aussi de la résolution de systèmes linéaires mal posés par moindres carrés. Il vient en complément aux cours d'analyse numérique et d'optimisation que les étudiants ont suivi auparavant.
HHT-alpha and predictor-corrector scheme for dynamic contact problem by Nitsche's method
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.
Mechanical parameters identification of keloid and surrounding healthy skin using Digital Image Correlation measurements in vivo
International audience; The human skin behaves as an elastic membrane initially prestressed but not uniformly. The presence of anatomical sites favorable to the appearance of some tumors, a keloid in our case, while other sites never develop them attests to the importance of the mechanical environment of the tissue. Thus, a mechanical characterization of the tumored skin is necessary to understand the keloid expansion from a mechanical point of view. Our case study consists in modeling a bi-material structure composed of a keloid skin surrounded by healthy skin located on upper left arm of a young female. From the experimental measurements in vivo, by combining force sensor, displacement se…
Sur la prise en compte de quelques conditions aux limites avec la méthode des éléments finis
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.
Removing the saturation assumption in Bank-Weiser error estimator analysis in dimension three
International audience; 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 test case.
Modelling Keloids Dynamics: A Brief Review and New Mathematical Perspectives
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 advances in this field. One possible approach could be to combine in vitro human models with calibrated in silico mathematical approaches (i.e., models and simulations) to generate new testable biological hypotheses related to biological mechanisms and improved treatments. Since these combined approaches do not really exist for keloid disorders, in…
HHT-α and TR-BDF2 schemes for dynamic contact problems
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 clustering package for nucleotide sequences using Laplacian Eigenmaps and Gaussian Mixture Model.
International audience; In this article, a new Python package for nucleotide sequences clustering is proposed. This package, freely available on-line, implements a Laplacian eigenmap embedding and a Gaussian Mixture Model for DNA clustering. It takes nucleotide sequences as input, and produces the optimal number of clusters along with a relevant visualization. Despite the fact that we did not optimise the computational speed, our method still performs reasonably well in practice. Our focus was mainly on data analytics and accuracy and as a result, our approach outperforms the state of the art, even in the case of divergent sequences. Furthermore, an a priori knowledge on the number of clust…