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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 choice for the discretization of the contact dual cone in order to account for complex fracture networks with corners and intersections. It leads to local expressions of the contact conditions and to efficient semi-smooth nonlinear solvers. On the other hand, such a mixed formulation requires to satisfy a compatibility condition between the discrete spaces restricting the choice of the displacement space and potentially leading to sub-optimal accuracy. This motivates the investigation of two alternative formulations based either on a stabilized mixed formulation or on the Nitsche's method. These three types of formulations are first investigated theoretically in order to enhance their connections. Then, they are compared numerically in terms of accuracy and nonlinear convergence on a coupled poromechanical model.

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 in different situations.

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.

Fractional powers of the Laplacian operator are important tools in the modeling and study of non-local phenomena. Several numerical challenges arise from the discretization of these operators due to their non-local nature. For example, a direct discretization via finite element methods can lead to dense and possibly large linear systems. One way to circumvent this density is by using a rational scheme combined with a finite element method. In this talk we describe a novel local a posteriori estimator for the finite element discretization error measured in the L2 norm that can be used to perform adaptive mesh refinement. This estimator is adapted from the strategy introduced by Bank and Weiser and can be used with any rational approx- imation scheme such as best uniform rational approximations or schemes based on the Dunford–Taylor formula. Especially, our estimator preserves the locality and robustness of the Bank–Weiser estimator and preserves the parallel nature of rational approximations. In addition, oour method can be combined with an estimator for the rational approximation error to obtain a more complete description of the discretization errors. Finally, we use an implementation in the FEniCSx finite element software to demonstrate the performances of our method on several numerical experiments including three–dimensional prob- lems.

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.

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 this brief review we start by summarising the biology of these disorders, then present various types of mathematical and computational approaches used for related disorders (i.e., wound healing and solid tumours), followed by a discussion of the very few mathematical and computational models published so far to study various inflammatory and mechanical aspects of the keloids. We conclude by discussing the open problems and opportunities offered in the context of keloid disorders by such combined in vitro/in silico approaches, and the need for multidisciplinary research to enable clinical progress.

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 formulations, and recall what is known about their stability and convergence properties.

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 representing the mechanical response of an artery modelled under pressure loading.

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.

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.

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.

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.

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, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open source implementation based on the FEniCS Project finite element software is provided as supplementary material.

Section of the proceedings of FEniCS 2021

Section of the proceedings of FEniCS 2021

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.

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 the bi-material domain geometry with neo-hookean behaviors. The boundary conditions and loadings of the experimental extension test are imposed. The materials parameters have been identified by inverse method starting from a constrained cost function minimizing the difference between the calculated displacements field and experimental displacements field obtained by digital image correlation and taking into account the reaction force as a constraint. An analysis of the model sensitivity to material parameters is presented.

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 sensor and Digital Image Correlation techniques, we perform a mechanical analysis to characterize the mechanical stress fields over the entire area and on the interface ‘healthy skin/keloid skin’. Since the mechanical behavior of the tumorous skin is unknown, many physical models can be implemented and assessed very easily inside the specific digital software to fit with the real data. Once a set of mechanical parameters for both the healthy skin and the keloid skin are identified, the stress fields around the keloid are calculated. Next steps consist in determining matching preferential directions in order to define as precisely as possible the specifications of a device for preventing the growth of keloids.

We present probabilistic approaches aiming at the selection of the best constitutive model and to identify their parameters from experimental data. These parameters are always associated with some degree of uncertainty. It is therefore important to study how this statistical uncertainty in parameters propagates to a safety-critical quantity of interest in the output of a model. Efficient Monte Carlo methods based on variance reduction techniques (Sensitivity Derivatives Monte Carlo methods [Hauseux et al. 2017] and MultiLevel Monte Carlo [Giles 2015] methods) are employed to propagate this uncertainty for both random variables and random fields. Inverse and forward problems are strongly connected. In a bayesian setting [Matthies et al. 2017], developing methods that reduce the number of evaluations of the forward model to an absolute minimum to achieve convergence is crucial for tractable computations. Numerical results in the context of soft tissue biomechanics are presented and discussed.

International audience; 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 binary images, are taken into account by a multilevel subelement embedding algorithm applied to elements of the background mesh that are cut by the surface/interfaces. To stabilize the system matrix when elements are cut by the surface/interfaces with very small intersections , we propose to move the background node(s) of the concerned cut elements by a distance proportional to the element size. This approach is simple but it can avoid the stability issues in such situations. Moreover, this approach does not include additional parameters as, e.g. ghost penalty method [1]. Dirichlet boundary conditions can be implicitly imposed on the surface using Lagrange multipliers, whereas traction or Neumann boundary conditions, which is/are applied on parts of the surface, can be distributed to the background nodes using shape functions. The implementation is verified by convergence studies with optimal rates. To verify the reliability of the method, it is applied to various needle insertion simulations (e.g. for biopsy or brachytherapy) into brain and liver models while considering frictional interactions between the needle and the tissue. Numerical results show that the present method can make the discretization independent from geometric description, and it can avoid the complexity of mesh generation of complex geometries while retaining the accuracy of the standard Finite Element Method. The proposed methodology is very suitable for real-time and patient specific simulations as it improves the simulation accuracy by automatically, and properly, taking the geometry of the simulated object into account.References[1] E. Burman et al, Int J Numer Meth Engg, vol. 104, no. 7, pp. 472-501, 2015.

International audience; The aim of this paper is to characterize the mechanical behavior of cutaneous tissues with the long-term goal to prevent keloid development in patients. Keloids are non cancerous tumors [1] that grow continuously on the skin for several reasons. These specific tumors affect 11 million patients of all ages and are particularly frequent in the Asian and African populations. The evolution of keloids is related to genetic, biological, biophysical and biomechanical factors. Here we focus on the biomechanical aspects, such as the state of stress inside and surrounding the keloid, which is believed to significantly affect the keloid's growth [2]. We present a patient-specific methodology for the identification of hyper-elastic model parameters of a keloid-skin composite model of a patient. A keloid is observed on a patient using an ultrasound and optical microscopy. The 3D imageof the surface is use to develop the geometrical model of the keloid. To effectively characterize the mechanical behavior of the skin-keloid composite we investigate several hyper-elastic material models. The model parameters for the healthy skin and the keloid are determined using an inverse analysis approach; specifically, the model parameters are determined by minimizing the mismatch between the numerical and the experimentally obtained displacement fields. The experimental results are obtained in-vivo using a custom-made extensometer [3] and a digital image correlation (DIC) technique.

International audience; As keloids are a consequence of abnormal wound healing process, the fibroblast proliferate excessively in dermis (Ogawa & Orgill, Mechanobiology of Cutaneous Wound Healing and Scarring, in Bioengineering Research of Chronic Wounds, 2009) and develop an unsightly and uncomfortable tumor that replaces the healthy skin. The structure and the properties of keloids highly differe from healthy skin, in particular the mechanical properties. Several anatomic sites are known as pro-keloid sites. Others don’t develop any keloid (Ogawa & Orgill, Mechanobiology of Cutaneous Wound Healing and Scarring, in Bioengineering Research of Chronic Wounds, 2009). In addition to genetic and biological causes, the mechanical solicitation of the in vivo skin induces the growth of keloid. Furthermore, this is highly related to the site and the range of body motion. This assumption is well documented by (Ogawa, et al., 2012). The design of a medical device should improve the prevention of the pathology but the mechanical components of the stress fields have to be determine in order to explain the mechanical process of keloid growth.

International audience; We investigate numerically the vibrational behavior of a simple finite element model that stands for an elastic container filled with an inviscid fluid. 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. A parametric study allows to characterize the system's response to dimensionless parameters in terms of eigenfrequencies. Then an insight into the mode shapes is provided, with a discussion on the presence and the behavior of singular modes caused by specific boundary conditions in the fluid.

International audience; We adopt the generalized local B-bar method to deal with the locking phenomenon in Reissner-Mindlin shell. The local element-wise projection saves computational effort, and projected basis functions of different orders are used to achieve good accuracy. The skew-symmetric Nitsche method is introduced for boundary conditions imposing and patch coupling. It has an advantage of unconditional stability w.r.t. the stabilization parameter, i.e. it is parameter-free.

In this work we present and analyse a stabilised finite element method for a fictitious domain formulation of a transient heat equation posed on an irregular domain. First, the problem is reformulated as a variational problem with constraints in a larger, simpler domain. In order to have freedom to choose the spaces for primal variable and Lagrange multiplier, we introduce a stabilising term whose consistency error can be shown to be of optimal order. We show that the fully discrete problem is stable and enjoys optimal convergence under the extra assumption that the initial condition has been chosen appropriately. This requirement is not linked to the stabilised character of the method, as the numerical experiments confirm.

International audience; Dans ce résumé, nous présentons une nouvelle formulation non biaisée du problème de contact frottant entre deux corps élastiques qui se base sur la méthode de Nitsche. Nous présentons, d’abord, l’étude faite en petites déformations pour la méthode non biaisée. Ensuite, nous introduisons une extension de la méthode de Nitsche pour les grandes déformations en hyper-élasticité pour les deux versions : baisée ou non. La méthode est testée à travers plusieurs cas tests classiques.

International audience; The aim of this paper is to evaluate the stress fields in cutaneous tissues with the long-term goal to prevent keloid development in patients. Keloids are non cancerous tumors that expand continously on the skin for a number of reasons. These specific tumors affect 11 million more patients of all age, every year and are particularly frequent in the asian and african populations. The evolution of keloids is related to genetic, biological, biophysical and biomechanical factors and are considered \complex biological systems". Keloids develop due to the excessive proliferation of fibroblasts on several specific anatomical sites on the skin, that can be identified from the value of the retractability of the tissue. Keloids grow like expansive scars on the skin surface. The relation between their development and the mechanical stress fields in and around the Keloid were studied in some papers but to our knowledge one of the few attempts at formulating a growth criterion is due to who studied the influence of the local state of stress on the keloid propagation direction. The paper presents a patient-specific methodology to identify the stress field. A keloid was observed on a patient using several imaging modalities, namely ultrasound and optical microscopy. A 3D image of the surface of the keloid was used to develop a 3D geometrical model of the keloid under consideration. The hyperelastic properties of the keloid and of the healthy skin were obtained through independent testing, on the same patient using an custom-made extensometer. A three-dimensional numerical model of the keloid-skin composite was built in order to investigate the nature of the stress field in the vicinity of the keloid-skin interface. The validity of the simulation is determined by complementary data that have been recorded during the mechanical tests.The mechanical results obtained from the numerical model are compared to displacement fields identified by Digital Image Correlation.

We propose a new procedure for clustering nucleotide sequences based on the "Laplacian Eigenmaps" and Gaussian Mixture modelling. This proposal is then applied to a set of 100 DNA sequences from the mitochondrially encoded NADH dehydrogenase 3 (ND3) gene of a collection of Platyhelminthes and Nematoda species. The resulting clusters are then shown to be consistent with the gene phylogenetic tree computed using a maximum likelihood approach. This comparison shows in particular that the clustering produced by the methodology combining Laplacian Eigenmaps with Gaussian Mixture models is coherent with the phylogeny as well as with the NCBI taxonomy. We also developed a Python package for this procedure which is available online.

The present study aims at comparing different computational models used for simulating the fluid-structure interaction within an in-vitro setup resembling simplified major obstruction of pharyngeal airway. Due to the nature of the problem, i.e. air flow passing over a deformable latex surface, a fully coupled fluid-structure interaction algorithm is used. A comparison is made between two finite element models for the solid domain, one using shell and the other using volume elements. The material properties of these models follow a hyperelastic behavior. For the fluid part, laminar and various turbulence models such as standard k-epsilon, Shear Stress Transport, SSG Reynolds Stress and BSL Reynolds Stress are compared. We evaluate the efficiency of the models and how close to the experimental data are their results. The predictions of the structural model containing volume elements showed better consistency with the experimental data. In addition, the results obtained with the standard k-epsilon turbulence model were the least deviated among all turbulence models.

International audience; This 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.

International audience; We 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.

National audience; Les répliques expérimentales de plis vocaux sont couramment utilisées dans l'étude de la production de parole. La caractérisation mécanique expérimentale in-situ des plis vocaux permet la validation de modèles numériques. Ces modèles peuvent alors servir à décrire des comportements vibratoires plus complexes et plus difficilement observables lors de la mesure. Les simulations présentées ici reposent sur une méthode éléments finis avec une formulation variationnelle du couplage hydro-élastique. Un premier calcul hyper-élastique simule le gonflement du latex soumis à la pression d'eau, ce résultat est ensuite reporté comme terme de précontrainte dans l'analyse modale des vibrations de petite amplitude du système hydro-élastique, permettant ainsi de caractériser l'influence de la pression d'eau sur les résonances mécaniques de la maquette simulée. Nous nous intéresserons ici aux différentes étapes nécessaires à la formulation variationnelle du problème complet, leur validation et la discussion sur les hypothèses réalisées. Les résultats d'études de convergence et études paramétriques du modèle numérique seront présentés et comparés aux résultats expérimentaux obtenus par une méthode de caractérisation mécanique des maquettes de plis vocaux, présentés dans une autre communication.

International audience; We 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.

Obstructive Sleep Apnea Syndrome is caused by frequent episodesof upper airway obstruction during sleep.Physical modelling allows a better insight into this phenomenon which is needed to improve the efficiency of treatments.So, the aim was to build and to validate an algorithm fornumerical simulations of the interaction betweenthe soft tissue and the airflow at the origin of an apneic episode.So as to reduce the computational cost, simplificatory assumptionshave been stated. First, for the soft tissue, the finite element methodallows a realistic prediction of their behaviour. Small perturbations framework combined with linear elasticity theory enable fast computationsof their mechanical response. Then, the numerical simulation of theairflow is based on an asymptotic formulation of the Navier-Stokesequations (Reduced Navier-Stokes / Prandtl), which makes easierthe numerical solving.For the validation of both the assumptions and the numerical solving method,an in-vitro setup has been used. It allows the reproduction, in a controlledenvironment, of an interaction between the airflow and a deformablestructure, similar to the one that occurs near the tongue during the beginningof an obstruction. An accurate measure of the deformation of the airflow ducthas been obtained thanks to a digital camera. A set of quantitative comparisons hasrevealed that the error between predictions and measurements is small, inspite of the simplifications formulated.Finally, to get closer to clinical reality, upper airway models of fourapneic patients have been built from sagittal radiographies. Comparisonsbetween simulations from pre-operative and post-operative radiographieshave showed that the predictions were in agreement with the effectsof the surgical gesture. Moreover, they put forward some limitations ofthe method, due to the complexity of the phenomenon., Le Syndrome d'Apnées Obstructives du Sommeil est caractérisépar la survenue fréquente d' épisodes d' obstructiondes voies aériennes supérieures. L'intérêt d'une modélisation physiqueest qu'elle autorise une compréhensionplus fine du phénomène, et laisse espérer une améliorationdes traitements. Le but a donc été de concevoir, puisde valider, un algorithme de simulation numérique del'interaction entre les tissus vivants et le fluxd'air à l'origine d'un épisode apnéique. Afin d'allégerles calculs et de réduire le temps de simulation, des hypothèsessimplificatrices ont été envisagées. D'une part, en ce qui concerneles tissus vivants, la méthode des éléments finis permet uneprédiction réaliste de leur déformation. Le cadredes petites perturbations et de l'élasticité linéaire impliquede plus un calcul rapide de la réponse mécanique. D'autre part, la simulationde l'écoulement d'air se fait via une formulationasymptotique des équations de Navier-Stokes (équationsde Navier-Stokes Réduites / Prandtl), qui facilite la résolutionnumérique.Afin de valider hypothèses physiques et méthode de résolutionnumérique, une maquette in-vitro a été utilisée. Celle-cipermet de reproduire, dans des conditions contrôlées, uneinteraction entre flux d'air et paroi déformable analogueà celle qui se produit à la base de la langue en début d'obstruction.Une mesure précise de la déformation du conduit d'écoulementest obtenue à l'aide d'une caméra digitale.Une série de comparaisons quantitatives a montré qu'en dépit des simplifications effectuées, l'erreur entre prédictionet mesures est faible.Finalement, pour se rapprocher de la réalité clinique,des modèles de voies aériennes supérieures de quatre patients apnéiques ontété construits à partir de radiographies sagittales.Des comparaisons entre simulations à partir de radiographiespré-opératoires et post-opératoires ont montréque les prédictions étaient globalement cohérentes avec les conséquencesdu geste chirurgical. Elles ont pû égalementmettre en évidence certaines limites de notre approche,dûes à la complexité du phénomène.

International audience; The biomechanical parameters of the multiple tissue layer structure of the vocal folds are of primary importance in order to understand their vibrating behaviour during phonation. A lot of models have been used to study the complex inhomogeneous structure of the vocal fold and its influence on the dynamics involved during the flow-induced vibration. However one of the major problem is to link all the different parameters of the physical models with the anatomical and physiological reality. The aim of this paper is to develop and describe a Finite Element Model for the vocal folds, using the open-source software FreeFem++. The study will use different continuum based models with increasing theoretical complexity and geometrical details. Starting with a simple 2D single layered structure using a linear elastodynamics theory, the model will tend - through different steps - to a 3D structure using a hydroelastic coupling. All these models are systematically compared with in-vitro measurements. The focus of this comparison is firstly upon the modal analysis of the vocal fold. A setup of experimental modal analysis, is used to characterize the frequency response of the vocal fold replica. The frequency responses of the measurement and its corresponding finite element model can then be compared.

International audience; Numerical computations from patient-specific medical data have become a key issue now in biomedical engineering. The complexity of the geometry and of the physical problem are generally such that simulation time can be problematically elevated. As a result, numerical achievements which aim at accelerating the computations are of very high interest in this context. We will present in this talk two numerical strategies to speed-up the numerical resolution of fluid-structure interaction problems involving a viscous incompressible fluid and that arise typically in blood flows. These strategies are both based on a partitioned algorithm to solve the coupled problem. In a first part, we will describe a semi-implicit coupling scheme [1], stable irrespectively of the so-called added-mass effect [4] and which allows for conservative time-stepping within the structure. Its efficiency is based on the explicit splitting of the viscous effects and geometrical/convective non-linearities, through the use of the Chorin-Temam projection scheme within the fluid [6]. Stability relies on the implicit treatment of the pressure stresses and on a Robin-based treatment of the viscous coupling, inspired from Nitsche's method [2, 3]. The numerical stability of the scheme is proved theoretically through a priori energy estimation. Numerical results in two and three dimensions illustrate the stability and efficiency of the scheme as well as its potentiality in the context of blood flow simulations. In a second part, we will present a parallel time-marching scheme for coupled parabolic- hyperbolic problems [5], as a prototype of fluid-structure interaction problems involving a linear structure and a viscous fluid. This scheme should allow further speed-up in the numerical simulations. It is obtained through application of the classical Parareal scheme [9] to the parabolic part, while the modified algorithm proposed by Farhat et al. [7, 8] is applied to the hyperbolic part. This hybrid Parareal treatment takes advantage of the partitioned for- mulation of the coupled propagator. No linearity assumption is made on the parabolic side. Numerical evidence shows that the resulting scheme is stable for a wider range of physical and discretization parameters than the classical Parareal.

Obstructive Sleep Apnea Syndrome is caused by frequent episodesof upper airway obstruction during sleep.Physical modelling allows a better insight into this phenomenon which is needed to improve the efficiency of treatments.So, the aim was to build and to validate an algorithm fornumerical simulations of the interaction betweenthe soft tissue and the airflow at the origin of an apneic episode.So as to reduce the computational cost, simplificatory assumptionshave been stated. First, for the soft tissue, the finite element methodallows a realistic prediction of their behaviour. Small perturbations framework combined with linear elasticity theory enable fast computationsof their mechanical response. Then, the numerical simulation of theairflow is based on an asymptotic formulation of the Navier-Stokesequations (Reduced Navier-Stokes / Prandtl), which makes easierthe numerical solving.For the validation of both the assumptions and the numerical solving method,an in-vitro setup has been used. It allows the reproduction, in a controlledenvironment, of an interaction between the airflow and a deformablestructure, similar to the one that occurs near the tongue during the beginningof an obstruction. An accurate measure of the deformation of the airflow ducthas been obtained thanks to a digital camera. A set of quantitative comparisons hasrevealed that the error between predictions and measurements is small, inspite of the simplifications formulated.Finally, to get closer to clinical reality, upper airway models of fourapneic patients have been built from sagittal radiographies. Comparisonsbetween simulations from pre-operative and post-operative radiographieshave showed that the predictions were in agreement with the effectsof the surgical gesture. Moreover, they put forward some limitations ofthe method, due to the complexity of the phenomenon.; Le Syndrome d'Apnées Obstructives du Sommeil est caractérisépar la survenue fréquente d' épisodes d' obstructiondes voies aériennes supérieures. L'intérêt d'une modélisation physiqueest qu'elle autorise une compréhensionplus fine du phénomène, et laisse espérer une améliorationdes traitements. Le but a donc été de concevoir, puisde valider, un algorithme de simulation numérique del'interaction entre les tissus vivants et le fluxd'air à l'origine d'un épisode apnéique. Afin d'allégerles calculs et de réduire le temps de simulation, des hypothèsessimplificatrices ont été envisagées. D'une part, en ce qui concerneles tissus vivants, la méthode des éléments finis permet uneprédiction réaliste de leur déformation. Le cadredes petites perturbations et de l'élasticité linéaire impliquede plus un calcul rapide de la réponse mécanique. D'autre part, la simulationde l'écoulement d'air se fait via une formulationasymptotique des équations de Navier-Stokes (équationsde Navier-Stokes Réduites / Prandtl), qui facilite la résolutionnumérique.Afin de valider hypothèses physiques et méthode de résolutionnumérique, une maquette in-vitro a été utilisée. Celle-cipermet de reproduire, dans des conditions contrôlées, uneinteraction entre flux d'air et paroi déformable analogueà celle qui se produit à la base de la langue en début d'obstruction.Une mesure précise de la déformation du conduit d'écoulementest obtenue à l'aide d'une caméra digitale.Une série de comparaisons quantitatives a montré qu'en dépit des simplifications effectuées, l'erreur entre prédictionet mesures est faible.Finalement, pour se rapprocher de la réalité clinique,des modèles de voies aériennes supérieures de quatre patients apnéiques ontété construits à partir de radiographies sagittales.Des comparaisons entre simulations à partir de radiographiespré-opératoires et post-opératoires ont montréque les prédictions étaient globalement cohérentes avec les conséquencesdu geste chirurgical. Elles ont pû égalementmettre en évidence certaines limites de notre approche,dûes à la complexité du phénomène.

This manuscript is concerned with a posteriori error estimation for the finite element discretization of standard and fractional partial differential equations as well as an application of fractional calculus to the modeling of the human meniscus by poro-elasticity equations. In the introduction, we give an overview of the literature of a posteriori error estimation in finite element methods and of adaptive refine- ment methods. We emphasize the state–of–the–art of the Bank–Weiser a posteriori error estimation method and of the adaptive refinement methods convergence results. Then, we move to fractional partial differential equations. We give some of the most common discretization methods of fractional Laplacian operator based equations. We review some results of a priori error estimation for the finite element discretization of these equations and give the state–of–the–art of a posteriori error estimation. Finally, we review the literature on the use of the Caputo’s fractional derivative in applications, focusing on anomalous diffusion and poro-elasticity applications. The rest of the manuscript is organized as follow. Chapter 1 is concerned with a proof of the reliability of the Bank–Weiser estimator for three–dimensional problems, extending a result from the literature. In Chapter 2 we present a numerical study of the Bank–Weiser estimator, provide a novel implementation of the estimator in the FEniCS finite element software and apply it to a variety of elliptic equations as well as goal-oriented error estimation. In Chapter 3 we derive a novel a posteriori estimator for the L2 error induced by the finite element discretization of fractional Laplacian operator based equations. In Chapter 4 we present new theoretical results on the convergence of a rational approximation method with consequences on the approximation of fractional norms as well as a priori error estimation results for the finite element discretization of fractional equations. Finally, in Chapter 5 we provide an application of fractional calculus to the study of the human meniscus via poro-elasticity equations. Ce manuscrit traite d’estimation d’erreur a posteriori pour la discrétisation d’équations aux dérivées partielles standard et fractionnaires par les méthodes éléments finis ainsi que de l’application de l’analyse fractionnaire à la modélisation du ménisque humain par les équations de poro-élasticité. Dans l’introduction, nous donnons un aperçu de la littérature sur l’estimation d’erreur a posteriori pour les méth- odes éléments finis et des méthodes de raffinement adaptatif. Nous insistons particulièrement sur l’état de l’art de la méthode d’estimation d’erreur a posteriori de Bank-Weiser et sur les résultats de convergence des méthodes adaptatives. Ensuite, nous nous intéressons aux équations aux dérivées partielles fractionnaires. Nous présentons certaines méthodes de discrétisation d’équations basées sur l’opérateur Laplacien fractionnaire et donnons l’état de l’art sur l’estimation d’erreur a posteriori. Finalement, nous donnons un aperçu de la littérature concernant les applications de la dérivée fractionnaire au sens de Caputo en nous concentrant sur le phénomène de diffusion anormale et les applications en poro-élasticité.