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International audience; In this paper, we adapt a previously developed poromechanical formulation to model the perfusion of myocardium during a cardiac cycle. First, a complete model is derived in 3D. Then, we perform a dimensional reduction under the assumption of spherical symmetry and propose a numerical algorithm that enables us to perform simulations of the myocardial perfusion throughout the cardiac cycle. These simulations illustrate the use of the proposed model to represent various physiological and pathological scenarios, specifically the vasodilation in the coronary network (to reproduce the standard clinical assessment of myocardial perfusion and perfusion reserve), the stenosis of a large coronary artery, an increased vascular resistance in the microcirculation (microvascular disease) and the consequences of inotropic activation (increased myocardial contractility) particularly at the level of the systolic flow impediment. Our results show that the model gives promising qualitative reproductions of complex physiological phenomena. This paves the way for future quantitative studies using clinical or experimental data.

International audience; In the context of cardiac muscle modeling, the availability of the myosin heads in the sarcomeres varies over the heart cycle contributing to the Frank-Starling mechanism at the organ level. In this paper, we propose a new approach that allows to extend the Huxley'57 muscle contraction model equations to incorporate this variation. This extension is built in a thermodynamically consistent manner, and we also propose adapted numerical methods that satisfy thermodynamical balances at the discrete level. Moreover, this whole approach-both for the model and the numerics-is devised within a hierarchical strategy enabling the coupling of the microscopic sarcomere-level equations with the macroscopic tissue-level description. As an important illustration, coupling our model with a previously proposed simplified heart model, we demonstrate the ability of the modeling and numerical framework to capture the essential features of the Frank-Starling mechanism.

We propose a method to discover differential equations describing the long‐term dynamics of phenomena featuring a multiscale behavior in time, starting from measurements taken at the fast‐scale. Our methodology is based on a synergetic combination of data assimilation (DA), used to estimate the parameters associated with the known fast‐scale dynamics, and machine learning (ML), used to infer the laws underlying the slow‐scale dynamics. Specifically, by exploiting the scale separation between the fast and the slow dynamics, we propose a decoupling of time scales that allows to drastically lower the computational burden. Then, we propose a ML algorithm that learns a parametric mathematical model from a collection of time series coming from the phenomenon to be modeled. Moreover, we study the interpretability of the data‐driven models obtained within the black‐box learning framework proposed in this paper. In particular, we show that every model can be rewritten in infinitely many different equivalent ways, thus making intrinsically ill‐posed the problem of learning a parametric differential equation starting from time series. Hence, we propose a strategy that allows to select a unique representative model in each equivalence class, thus enhancing the interpretability of the results. We demonstrate the effectiveness and noise‐robustness of the proposed methods through several test cases, in which we reconstruct several differential models starting from time series generated through the models themselves. Finally, we show the results obtained for a test case in the cardiovascular modeling context, which sheds light on a promising field of application of the proposed methods., We propose a method to discover differential equations describing the long‐term dynamics of phenomena featuring a multiscale behavior in time, starting from measurements taken at the fast‐scale. Our methodology is based on a synergetic combination of data assimilation, used to estimate the parameters associated with the known fast‐scale dynamics, and machine learning, used to infer the laws underlying the slow‐scale dynamics. By exploiting the scale separation between the fast and the slow dynamics, we propose a decoupling of time scales that allows to drastically lower the computational burden.

International audience; We consider a previously proposed general nonlinear poromechanical formulation, and we derive a linearized version of this model. For this linearized model, we obtain an existence result and we propose a complete discretization strategy - in time and space - with a special concern for issues associated with incompressible or nearly-incompressible behavior. We provide a detailed mathematical analysis of this strategy, the main result being an error estimate uniform with respect to the compressibility parameter. We then illustrate our approach with detailed simulation results and we numerically investigate the importance of the assumptions made in the analysis, including the fulfillment of specific inf-sup conditions.

International audience; Considering a large class of muscle contraction models accounting for actin-myosin interaction, we present a mathematical setting in which solution properties can be established, including fundamental thermodynamic balances. Moreover, we propose a complete discretization strategy for which we are also able to obtain discrete versions of the thermodynamic balances and other properties. Our major objective is to show how the thermodynamics of such models can be tracked after discretization, including when they are coupled to a macroscopic muscle formulation in the realm of continuum mechanics. Our approach allows to carefully identify the sources of energy and entropy in the system, and to follow them up to the numerical applications.

Proposed time discretization scheme for general two-phase poromechanical model.Discrete energy estimate with the total free energy of the mixture, in a general nonlinear framework.Numerical examples with representative test problems. We consider a general nonlinear poromechanical model, formulated based on fundamental thermodynamics principle, suitable for representing the coupling of rapid internal fluid flows with large deformations of the solid, and compatible with a wide class of constitutive behavior. The objective of the present work is to propose for this model a time discretization scheme of the partitioned type, to allow the use of existing time schemes and possibly separate solvers for each component of the model, i.e. for the fluid and the solid. To that purpose, we adapt and extend an earlier proposed approach devised for fluid-structure interaction in an Arbitrary Lagrangian-Eulerian framework. We then establish an energy estimate for the resulting time scheme, in a form that is consistent with the underlying energy principle in the poromechanical formulation, up to some numerical dissipation effects and some perturbations that we have carefully identified and assessed. In addition, we provide some numerical illustrations of our numerical strategy with test problems that present typical features of large strains and rapid fluid flows, and also a case of singular transition related to total drainage. An example of challenging application envisioned for this model and associated numerical coupling scheme concerns the perfusion of the heart.

International audience; In this paper we propose a muscle tissue model -- valid for striated muscles in general, and for the myocardium in particular -- based on a multi-scale physiological description. This model extends and refines an earlier-proposed formulation by allowing to account for all major energy exchanges and balances, from the chemical activity coupled with oxygen supply to the production of actual mechanical work, namely, the biological function of the tissue. We thus perform a thorough analysis of the energy mechanisms prevailing at the various scales, and we proceed to propose a complete discretization strategy -- in time and space -- respecting the same balance laws. This will be crucial in future works to adequately model the many important physiological -- normal and pathological -- phenomena associated with these energy considerations.

International audience; Parameter estimation from non-invasive measurements is a crucial step in patient-specific cardiac modeling. It also has the potential to provide sig- nificant assistance in the clinical diagnosis of cardiac diseases through the quantification of myocardial material heterogeneity. In this paper, we for- mulate a novel Reduced-order Unscented Kalman Filter (rUKF) applied to the left ventricular (LV) nonlinear mechanical model based on cubic-Hermite finite elements. Material parameters in the widely-employed transversely isotropic Guccione's constitutive law are successfully identified for both ho- mogeneous and heterogeneous cases. We conclude that the four parameters in Guccione's law can be uniquely and correctly determined in silico from noisy displacement measurements of material points located on the myocar- dial surfaces. The future application of this novel and effective approach to real clinical measurements is thus promising.

International audience; Effective shell finite elements are of utmost interest in engineering practice, and more particularly triangular shell elements are often needed to handle complex geometries. The quadrilateral elements of the MITC family have been shown to be effective for general asymptotic behaviors, but a triangular MITC shell finite element of comparable performance remains to be found. The main difficulty is to reduce the locking phenomenon arising in bending dominated situations without destroying the ability of the procedure to accurately capture membrane dominated and mixed behaviors. In this context, we have designed a simple numerical rank test aimed at assessing the reliability of MITC triangular elements. This test led to selecting some specific MITC6 tying schemes, which have been further assessed using some meaningful test problems. The assessment results showed that the MITC6a shell element is a most promising candidate, although it may feature some non-physical membrane spurious modes in some specific shell problems.

International audience; We present a patient-specific biomechanical modeling framework and an initial case study for investigating cardiac amyloidosis (CA). Our patient-specific heartbeat simulations are in good agreement with the data, and our model calibration indicates that the major effect of CA in the biophysical behavior lies in a dramatic increase of the passive stiffness. We also conducted a preliminary trial for predicting the effects of pharmacological treatments – which is an important clinical challenge – based on the model combined with a simple venous return representation. This requires further investigation and validation, albeit provides some valuable preliminary insight.

International audience; We propose a new sequential estimation method for making an electrophysiology model patient-specific, with data in the form of level sets of the electrical potential. Our method incorporates a novel correction term based on topological gradients, in order to track solutions of complex patterns. Our assessments demonstrate the effectiveness of this approach, including in a realistic case of atrial fibrillation.

International audience; We propose a variational formulation suitable for coupling a shell and a surrounding softer 3D medium. This is intended to model the behavior of a reinforcing sheet as frequently occurs in applications. The crucial issue of the kinematical coupling conditions is addressed by taking into account the rotation degrees of freedom of the shell. We then perform an asymptotic analysis of the coupled model in which the distinction between bending-dominated and membrane-dominated shells arises. For the limit solutions we identify coupling kinematical conditions that do not involve rotations. Finally we present some numerical results that illustrate our discussions.

International audience; We present an inf-sup test for general mixed shell finite element discretizations. The test is useful in the thorough evaluation of a shell finite element discretization scheme. We apply the test to the MITC shell elements and find that these elements pass the test.

International audience; We propose a direct method for computing modal coupling coefficients - due to geometrically nonlinear effects - for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. These coupling coefficients arise when considering a discrete expansion of the unknown displacement onto the eigenmodes of the linear operator. The evolution problem is thus projected onto the eigenmodes basis and expressed as an assembly of oscillators with quadratic and cubic nonlinearities. The nonlinear coupling coefficients are directly derived from the finite element formulation, with specificities pertaining to the shell elements considered, namely, here elements of the ''Mixed Interpolation of Tensorial Components'' family (MITC). Therefore, the computation of coupling coefficients, combined with an adequate selection of the significant eigenmodes, allows the derivation of effective reduced-order models for computing - with a continuation procedure - the stable and unstable vibratory states of any vibrating shell, up to large amplitudes. The procedure is illustrated on a hyperbolic paraboloid panel. Bifurcation diagrams in free and forced vibrations are obtained. Comparisons with direct time simulations of the full FE model are given. Finally, the computed coefficients are used for a maximal reduction based on asymptotic nonlinear normal modes (NNMs), and we find that the most important part of the dynamics can be predicted with a single oscillator equation.

International audience; We analyze the performances of two types of Luenberger observers -- namely, the so-called Direct Velocity Feedback and Schur Displacement Feedback procedures, originally devised for elasto-dynamics -- to estimate the state of a fluid-structure interaction model for hemodynamics, when the measurements are assumed to be restricted to displacements or velocities in the solid. We first assess the observers using hemodynamics-inspired test problems with the complete model, including the Navier-Stokes equations in Arbitrary Lagrangian-Eulerian formulation, in particular. Then, in order to obtain more detailed insight we consider several well-chosen simplified models, each of which allowing a thorough analysis -- emphasizing spectral considerations -- while illustrating a major phenomenon of interest for the observer performance, namely, the added mass effect for the structure, the coupling with a lumped-parameter boundary condition model for the fluid flow, and the fluid dynamics effect per se. Whereas improvements can be sought when additional measurements are available in the fluid domain in order to more effectively deal with strong uncertainties in the fluid state, in the present framework this establishes Luenberger observer methods as very attractive strategies -- compared, e.g., to classical variational techniques -- to perform state estimation, and more generally for uncertainty estimation since other observer procedures can be conveniently combined to estimate uncertain parameters.

We address the issue of parameter variations in POD approximations of time-dependent problems, without any specific restriction on the form of parameter dependence. Considering a parabolic model problem, we propose a POD construction strategy allowing us to obtain some a priori error estimates controlled by the POD remainder – in the construction procedure – and some parameter-wise interpolation errors for the model solutions. We provide a thorough numerical assessment of this strategy with the FitzHugh − Nagumo 1D model. Finally, we give detailed illustrations of the approach in two parameter estimation applications, the first in a variational estimation framework with the FitzHugh − Nagumo model, and the second with a beating heart mechanical model for which we employ a sequential estimation method to characterize model parameters using real image data in a clinical case.

International audience; Computational electrophysiology is a very active field with tremendous potential in medical applications, albeit leads to highly intensive simulations. We here propose a surface-based electrophysiology formulation, motivated by the modeling of thin structures such as cardiac atria, which greatly reduces the size of the computational models. Moreover, our model is specifically devised to retain the key features associated with the anisotropy in the diffusion effects induced by the fiber architecture, with rapid variations across the thickness which cannot be adequately represented by naive averaging strategies. Our proposed model relies on a detailed asymptotic analysis in which we identify a limit model and establish strong convergence results. We also provide detailed numerical assessments which confirm an excellent accuracy of the surface-based model -- compared with the reference 3D model -- including in the representation of a complex phenomenon, namely, spiral waves.

International audience We present the fundamental principles of data assimilation underlying the Verdandi library, and how they are articulated with the modular architecture of the library. This translates -- in particular -- into the definition of standardized interfaces through which the data assimilation library interoperates with the model simulation software and the so-called observation manager. We also survey various examples of data assimilation applied to the personalization of biophysical models, in particular for cardiac modeling applications within the euHeart European project. This illustrates the power of data assimilation concepts in such novel applications, with tremendous potential in clinical diagnosis assistance.

International audience; The objective of this paper is to propose and assess an estimation procedure - based on data assimilation principles - well-suited to obtain some regional values of key biophysical parameters in a beating heart model, using actual Cine-MR images. The motivation is twofold: (1) to provide an automatic tool for personalizing the characteristics of a cardiac model in order to achieve predictivity in patient-specific modeling, and (2) to obtain some useful information for diagnosis purposes in the estimated quantities themselves. In order to assess the global methodology we specifically devised an animal experiment in which a controlled infarct was produced and data acquired before and after infarction, with an estimation of regional tissue contractility - a key parameter directly affected by the pathology - performed for every measured stage. After performing a preliminary assessment of our proposed methodology using synthetic data, we then demonstrate a full-scale application by first estimating contractility values associated with 6 regions based on the AHA subdivision, before running a more detailed estimation using the actual AHA segments. The estimation results are assessed by comparison with the medical knowledge of the specific infarct, and with late enhancement MR images. We discuss their accuracy at the various subdivision levels, in the light of the inherent modeling limitations and of the intrinsic information contents featured in the data.

International audience; We propose a numerical analysis of Proper Orthogonal Decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and -- whereas a general uniform continuity bound seems out of reach -- we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound -- and the effectiveness of the POD approach altogether -- for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

International audience; We propose an observer-based approach to circumvent the issue of unbounded approximation errors -- with respect to the length of the time window considered -- in the discretization of wave-like equations in bounded domains, which covers the cases of the wave equation per se and of linear elasticity as well as beam, plate and shell formulations, and so on. Namely, taking advantage of some measurements available on the system over time, we adopt a strategy inspired from sequential data assimilation and by which the discrete system is dynamically corrected using the discrepancy between the solution and the measurements. In addition to the classical cornerstones of numerical analysis made up by stability and consistency, we are thus led to incorporating a third crucial requirement pertaining to observability -- to be preserved through discretization. The latter property warrants exponential stability for the corrected dynamics, hence provides bounded approximation errors over time. Special care is needed to establish the required observability at the discrete level, in particular due to the fact that we focus on an original observer method adapted to measurements of the main variable, whereas measurements of the time-derivative -- admissible, of course, albeit less frequent in practical systems -- lead to a stability analysis in which existing results can be more directly applied. We also provide some detailed application examples with several such wave-like equations, and the corresponding numerical assessments illustrate the performance of our approach.

International audience; The objective of this work is to address the formulation of an adequate model of the external tissue environment when studying a portion of the arterial tree with fluid-structure interaction. Whereas much work has already been accomplished concerning flow and pressure boundary conditions associated with truncations in the fluid domain, very few studies take into account the tissues surrounding the region of interest to derive adequate boundary conditions for the solid domain. In this paper, we propose to model the effect of external tissues by introducing viscoelastic support conditions along the artery wall, with two--possibly distributed--parameters that can be adjusted to mimic the response of various physiological tissues. In order to illustrate the versatility and effectiveness of our approach, we apply this strategy to perform patient-specific modeling of thoracic aortae based on clinical data, in two different cases and using a distinct fluid-structure interaction methodology for each, namely an Arbitrary Lagrangian-Eulerian (ALE) approach with prescribed inlet motion in the first case and the coupled momentum method in the second case. In both cases, the resulting simulations are quantitatively assessed by detailed comparisons with dynamic image sequences, and the model results are shown to be in very good adequacy with the data.

See also erratum DOI:10.1051/cocv/2011001; International audience; We propose a general reduced-order filtering strategy adapted to Unscented Kalman Filtering for any choice of sampling points distribution. This provides tractable filtering algorithms which can be used with large-dimensional systems when the uncertainty space is of reduced size, and these algorithms only invoke the original dynamical and observation operators, namely, they do not require tangent operator computations, which of course is of considerable benefit when nonlinear operators are considered. The algorithms are derived in discrete time as in the classical UKF formalism - well-adapted to time discretized dynamical equations - and then extended into consistent continuous-time versions. This reduced-order filtering approach can be used in particular for the estimation of parameters in large dynamical systems arising from the discretization of partial differential equations, when state estimation can be handled by an adequate Luenberger observer inspired from feedback control. In this case, we give an analysis of the joint state-parameter estimation procedure based on linearized error, and we illustrate the effectiveness of the approach using a test problem inspired from cardiac biomechanics.

International audience; In this paper we apply specific data assimilation methods in order to estimate regional contractility parameters in a biomechanical heart model, using as measurements real Cine MR images obtained in an animal experiment. We assess the effectiveness of this estimation based on independent knowledge of the controlled infarcted condition, and on late enhancement images. Moreover, we show that the estimated contractility values can improve the model behavior in itself, and that they can serve as an indicator of the local heart function, namely, to assist medical diagnosis for the post-infarct detection of hypokinetic or akinetic regions in the myocardial tissue

International audience; When establishing and analyzing model-parameter dependent mixed formulations, it is common to consider required ellipticity and inf-sup conditions for the continuous and discrete problems. However, in the modeling of some important categories of problems, like in the analysis of plates and shells, the ellipticity condition usually considered does not naturally hold, and the inf-sup condition can only be stated in an abstract form and can hardly be evaluated analytically. In this paper we present a new and practical ellipticity condition which together with the inf-sup condition guarantees that (i) when the model parameter goes to zero, the limit problem solution is uniformly approached, and (ii) an optimal finite element discretization has been established (for the interpolations used). In practice, a numerical test might be performed to see whether the proposed ellipticity condition is satisfied.

International audience; This paper is motivated by the modeling of blood flows through the beating myocardium, namely cardiac perfusion. As in other works, perfusion is modeled here as a flow through a poroelastic medium. The main contribution of this study is the derivation of a general poroelastic model valid for a nearly incompressible medium which experiences finite deformations. A numerical procedure is proposed to iteratively solve the porous flow and the nonlinear poroviscoelastic problems. Three-dimensional numerical experiments are presented to illustrate the model. The first test cases consist of typical poroelastic configurations: swelling and complete drainage. Finally, a simulation of cardiac perfusion is presented in an idealized left ventricle embedded with active fibers. Results show the complex temporal and spatial interactions of the muscle and blood, reproducing several key phenomena observed in cardiac perfusion.

International audience; Starting from the presentation of a unified perspective of segmentation with deformable models and data assimilation with images, we propose a data assimilation procedure designed to dynamically estimate 3D positions and velocities of the myocardium along the heart cycle, using data consisting of contours extracted from image sequences. We assess this procedure with a test problem based on a realistic computational heart model, and with synthetic data produced from reference simulations by creating binary images of the myocardium. The automatic meshing library CGAL is then employed to create contour meshes for each snapshot, and these meshes are directly used in the model measurement comparisons. This approach gives very satisfactory qualitativeand quantitative results.

International audience; Effective triangular shell elements are of utmost interest in engineering practice, and the MITC6a element - a 6 node quadratic general shell element of the MITC family - has been shown to significantly reduce the locking phenomena arising in bending dominated behaviours. However, for some specific combinations of midsurface geometry and boundary conditions, the MITC6a element features some non-physical displacement modes with vanishing membrane strain energy. This phenomenon is thoroughly analyzed, and a remedy based on a stabilized bilinear form is proposed. Detailed numerical tests are included and the results demonstrate the good performance of the proposed method both for membrane and bending dominated problems.

International audience; We describe an approach that we propose to model the electromechanical behavior of the heart, and to use the model in a data assimilation procedure in order to perform an identification of the parameters and state. The modeling of the heart tissue is based on an electrically-activated contraction law formulated via multiscale considerations and is consistent with various physiological and thermomechanical key requirements. The global heart system also incorporates a simplified lumped modeling of the blood compartments. We report on numerical simulations and on validations of our model in reference and pathological conditions. Furthermore, the data assimilation procedure is intended to give access to quantities of interest for diagnosis purposes, and we present some promising results in this direction.

International audience; The incompressible singularity found in 3D elasticity when Poisson's ratio approaches 1/2 is not present in classical shell models, nor in the limit models obtained from 3D elasticity when performing an asymptotic analysis with respect to the thickness parameter. However, some specific shell models - such as the 3D-shell model - do retain the incompressible singularity. These observations raise the issue of how adequately shell models can represent incompressible conditions, which this paper aims at investigating. We first perform a combined asymptotic analysis of 3D elasticity with respect to both the thickness parameter and Poisson's ratio and we obtain a commuting property, which is very valuable as a justification of the concept of an "incompressible shell", and substantiates the use of classical shell models with incompressible materials.We then show that the 3D-shell model does not enjoy a similar commuting property; nevertheless we propose a simple modification of this model for which commuting is obtained, hence consistency with incompressibility is recovered. We also illustrate our discussions with some numerical results.

International audience; We focus on a family of shell elements which are a direct generalization of the shell elements most commonly used in engineering practice. The elements in the family include the effects of the through-the-thickness normal stress and can be employed to couple directly with surrounding media on either surfaces of the shell. We establish the "underlying" mathematical model of the shell discretization scheme, and we show that this mathematical model features the same asymptotic behaviors "when the shell thickness becomes increasingly smaller" as classical shell models. The question of "locking" of the finite element discretization is also briefly addressed and we point out that, for an effective finite element scheme, the MITC approach of interpolation is available.

International audience; We consider a family of shell finite elements with quadratic displacements across the thickness. These elements are very attractive, but compared to standard general shell elements they face another source of numerical locking in addition to shear and membrane locking. This additional locking phenomenon - that we call ''pinching locking'' - is the subject of this paper and we analyse a numerical strategy designed to overcome this difficulty. Using a model problem in which only this specific source of locking is present, we are able to obtain error estimates independent of the thickness parameter, which shows that pinching locking is effectively treated. This is also confirmed by some numerical experiments of which we give an account.

International audience; In general, shell structural problems can be identified to fall into one of the categories of membrane-dominated, bending-dominated and mixed shell problems. The asymptotic behaviour with a well-defined load-scaling factor shows distinctly into which category a given shell problem falls. The objective of this paper is to present a shell problem and its solution for which there is no convergence to a well-defined load-scaling factor as the thickness of the shell decreases. Such shells are unduly sensitive in their behaviour because the ratio of membrane to bending energy stored changes significantly and indeed can fluctuate with changes in shell thickness. We briefly review the different asymptotic behaviours that shell problems can display, and then present the specific problem considered and its numerical solution using finite element analysis.

International audience; The formulation of nine-node mixed-interpolated shell elements based on a classical mathematical shell theory is presented, taking into account some fundamental considerations for the finite element analysis of shells. The elements are based on the Mixed Interpolation of Tensorial Components (MITC) approach, but the assumed covariant strain fields are applied only for the membrane and shear components. Two different types of elements are considered, depending on whether or not geometric approximations are included in the formulation. The performance of the proposed elements is illustrated with a well-established test problem - the Scordelis-Lo roof.

This article, a companion to the article by Philippe G. Ciarlet on the mathematical modelling of shells also in this issue of Acta Numerica, focuses on numerical issues raised by the analysis of shells.Finite element procedures are widely used in engineering practice to analyse the behaviour of shell structures. However, the concept of ‘shell finite element’ is still somewhat fuzzy, as it may correspond to very different ideas and techniques in various actual implementations. In particular, a significant distinction can be made between shell elements that are obtained via the discretization of shell models, and shell elements – such as the general shell elements – derived from 3D formulations using some kinematic assumptions, without the use of any shell theory. Our first objective in this paper is to give a unified perspective of these two families of shell elements. This is expected to be very useful as it paves the way for further thorough mathematical analyses of shell elements. A particularly important motivation for this is the understanding and treatment of the deficiencies associated with the analysis of thin shells (among which is the locking phenomenon). We then survey these deficiencies, in the framework of the asymptotic behaviour of shell models. We conclude the article by giving some detailed guidelines to numerically assess the performance of shell finite elements when faced with these pathological phenomena, which is essential for the design of improved procedures.

International audience; Based on fundamental considerations for the finite element analysis of shells, we evaluate in the present paper the performance of the MITC general shell elements. We give the results obtained in the analysis of judiciously selected test problems and conclude that the element are effective for general engineering applications.

SUMMARY We show that although no actual mathematical shell model is explicitly used in ‘general shell element’ formulations, we can identify an implicit shell model underlying these nite element procedures. This ‘underlying model’ compares well with classical shell models since it displays the same asymptotic behaviours|when the thickness of the shell becomes very small|as, for example, the Naghdi model. Moreover, we substantiate the connection between general shell element procedures and this underlying model by mathematically proving a convergence result from the nite element solution to the solution of the model. Copyright ? 2000 John Wiley & Sons, Ltd.

This thesis aims at incorporating complex data derived from images into a data assimilation strategy available for mechanical systems. Our work relies on some recent developments that propose a sequential data assimilation method made of a Luenberger filter for the state space and an optimal filter reduced to the remaining parameter space. We aim at performing parameter identification for a biomechanical model of the heart and, within the scope of this application, we formalize the construction of shape discrepancy measurements for two types of data sets: first, the data expected of a processing step of tagged Magnetic Resonance Imaging (tagged-MRI) and, second, more standard data composed by the contours of the object. Initially based on simple distance measurements we enrich these discrepancy measures by incorporating the formalism of currents which enables to embed the contours of the object within the dual of an appropriate space of test functions. For each discrepancy operators we analyze its impact on the observability of the system and, in the case of tagged-MRI, we prove that they are equivalent to a direct measurement of the displacement. From a numerical standpoint, taking into account these complex data sets is a great challenge that motivates the creation of new numerical schemes that provide a more flexible management of the various observation operators. We assess these new means of extracting the rich information contained in the image by identifying in realistic cases the position and the intensity of an infarct in the heart tissue.; Cette thèse est dédiée à l'intégration de données complexes issues de l'imagerie dans une stratégie d'assimilation de données pour des modèles mécaniques. Notre stratégie s'appuie sur des travaux récents proposant une méthode séquentielle d'assimilation de données qui se décompose en un filtre de Luenberger pour l'espace d'état et un filtre optimal réduit à l'espace des paramètres. Nous l'appliquons à l'identification de paramètres pour un modèle biomécanique du cœur et, dans ce cadre, nous formalisons la construction de comparateurs de formes évolués pour deux types de données : d'une part des données extraites d'un traitement de l'Imagerie par Résonance Magnétique (IRM) de marquage tissulaire et, d'autre part, des données plus classiques de type contours de l'objet. D'abord fondés sur des simples distances nous enrichissons ces comparateurs grâce au formalisme des courants permettant d'inscrire le contour de l'objet dans le dual d'un espace de fonctions-test appropriées. Pour chacun des comparateurs nous analysons son impact sur l'observabilité du système et, pour le cas de l'imagerie de marquage tissulaire, nous prouvons qu'ils sont équivalents à une mesure directe du déplacement. D'un point de vue numérique, la prise en compte de ces mesures complexes présente d'importantes difficultées nous poussant à mettre en place des schémas numériques originaux permettant une manipulation plus flexible des différents opérateurs d'observation. Nous profitons de ces nouveaux moyens d'extraction de l'information contenue dans les données d'imagerie pour permettre, dans des cas réalistes, l'identification de la position et de l'intensité d'un infarctus du myocarde.

This thesis aims at incorporating complex data derived from images into a data assimilation strategy available for mechanical systems. Our work relies on some recent developments that propose a sequential data assimilation method made of a Luenberger filter for the state space and an optimal filter reduced to the remaining parameter space. We aim at performing parameter identification for a biomechanical model of the heart and, within the scope of this application, we formalize the construction of shape discrepancy measurements for two types of data sets: first, the data expected of a processing step of tagged Magnetic Resonance Imaging (tagged-MRI) and, second, more standard data composed by the contours of the object. Initially based on simple distance measurements we enrich these discrepancy measures by incorporating the formalism of currents which enables to embed the contours of the object within the dual of an appropriate space of test functions. For each discrepancy operators we analyze its impact on the observability of the system and, in the case of tagged-MRI, we prove that they are equivalent to a direct measurement of the displacement. From a numerical standpoint, taking into account these complex data sets is a great challenge that motivates the creation of new numerical schemes that provide a more flexible management of the various observation operators. We assess these new means of extracting the rich information contained in the image by identifying in realistic cases the position and the intensity of an infarct in the heart tissue.; Cette thèse est dédiée à l'intégration de données complexes issues de l'imagerie dans une stratégie d'assimilation de données pour des modèles mécaniques. Notre stratégie s'appuie sur des travaux récents proposant une méthode séquentielle d'assimilation de données qui se décompose en un filtre de Luenberger pour l'espace d'état et un filtre optimal réduit à l'espace des paramètres. Nous l'appliquons à l'identification de paramètres pour un modèle biomécanique du cœur et, dans ce cadre, nous formalisons la construction de comparateurs de formes évolués pour deux types de données : d'une part des données extraites d'un traitement de l'Imagerie par Résonance Magnétique (IRM) de marquage tissulaire et, d'autre part, des données plus classiques de type contours de l'objet. D'abord fondés sur des simples distances nous enrichissons ces comparateurs grâce au formalisme des courants permettant d'inscrire le contour de l'objet dans le dual d'un espace de fonctions-test appropriées. Pour chacun des comparateurs nous analysons son impact sur l'observabilité du système et, pour le cas de l'imagerie de marquage tissulaire, nous prouvons qu'ils sont équivalents à une mesure directe du déplacement. D'un point de vue numérique, la prise en compte de ces mesures complexes présente d'importantes difficultées nous poussant à mettre en place des schémas numériques originaux permettant une manipulation plus flexible des différents opérateurs d'observation. Nous profitons de ces nouveaux moyens d'extraction de l'information contenue dans les données d'imagerie pour permettre, dans des cas réalistes, l'identification de la position et de l'intensité d'un infarctus du myocarde.

The objective of this thesis is the assessment of a biomechanical heart model using experimental data, and the investigation of clinical applications with patient-specific modeling. At the 1D level we aimed at reproducing physiological experiments with myocardial fiber contraction. For the 3D validation we performed in cooperation with a clinical partner an experiment with animals (pigs) in order to obtain data in the healthy stage and after creating a myocardial infarct. We showed that our model can reproduce the pressures and motion of a healthy heart and that the infarct can be represented by changing only the parameters directly related to the pathology. The objective of the first clinical application was to predict the short-term effect of the Cardiac Resynchronization Therapy (CRT) by means of an increase of 'max LV dp/dt'. The model personalization was performed using patients MRI and pressure data in the baseline condition - prior to CRT. Then we fixed the values of all parameters and applied electrical activation patterns according to the pacing modes considered. We obtained a very good prediction of max LV dp/dt using various pacing patterns in 3 clinical cases. This preliminary clinical validation shows that the modeling of CRT is a very promising approach as an assistance to therapy planning. The second application is based on the adaptation of data assimilation methods developed in the MACS team at INRIA. We performed joint state-parameter estimation with real image data. We showed the effectiveness of these algorithms in automatic model personalization and that the estimated contractility values can serve as an indicator of the local heart function.; L'objectif de cette thèse porte sur la validation d'un modèle biomécanique du cœur à base de mesures expérimentales, et sur des investigations concernant des applications cliniques en modélisation personnalisée. Dans le cadre 1D, nous avons reproduit des expériences physiologiques de contraction d'une fibre cardiaque. Concernant la validation 3D, nous avons effectué dans une collaboration clinique une expérience avec des animaux (cochons), afin d'obtenir des données saines et après création d'infarctus. Nous avons démontré que notre modèle est capable de représenter le cœur sain, et que l'infarctus peut être correctement modélisé en modifiant uniquement les paramètres directement concernés par la pathologie. Dans la première application clinique nous avons démontré que le modèle est prédictif sur l'effet immédiat d'une "Cardiac Resynchronization Therapy" (CRT), sur le critère du "max LV dp/dt". La personnalisation du modèle a été réalisée au moyen des données d'IRM et de pressions au stade 'baseline' (avant la CRT). Puis, tous les paramètres du modèle étant figés, nous avons appliqué les schémas d'activation électrique correspondant aux modes de CRT. Nous avons obtenu une prédiction très proche des indicateurs mesurés sur 3 patients avec plusieurs schémas d'activation. Cette étude clinique préliminaire montre que la modélisation de CRT est très prometteuse comme assistance à la planification thérapeutique. Dans une deuxième application nous avons adapté des méthodes d'assimilation de données développées dans l'Equipe-Projet MACS à l'INRIA, à des fins d'aide au diagnostic. Nous avons ainsi réalisé une estimation conjointe état-paramètres en utilisant des données réelles (IRM). Nous avons démontré l'efficacité de ces méthodes, et l'intérêt des paramètres de contractilité estimés comme indicateurs de la fonction du myocarde.

The objective of this thesis is the assessment of a biomechanical heart model using experimental data, and the investigation of clinical applications with patient-specific modeling. At the 1D level we aimed at reproducing physiological experiments with myocardial fiber contraction. For the 3D validation we performed in cooperation with a clinical partner an experiment with animals (pigs) in order to obtain data in the healthy stage and after creating a myocardial infarct. We showed that our model can reproduce the pressures and motion of a healthy heart and that the infarct can be represented by changing only the parameters directly related to the pathology. The objective of the first clinical application was to predict the short-term effect of the Cardiac Resynchronization Therapy (CRT) by means of an increase of 'max LV dp/dt'. The model personalization was performed using patients MRI and pressure data in the baseline condition - prior to CRT. Then we fixed the values of all parameters and applied electrical activation patterns according to the pacing modes considered. We obtained a very good prediction of max LV dp/dt using various pacing patterns in 3 clinical cases. This preliminary clinical validation shows that the modeling of CRT is a very promising approach as an assistance to therapy planning. The second application is based on the adaptation of data assimilation methods developed in the MACS team at INRIA. We performed joint state-parameter estimation with real image data. We showed the effectiveness of these algorithms in automatic model personalization and that the estimated contractility values can serve as an indicator of the local heart function.; L'objectif de cette thèse porte sur la validation d'un modèle biomécanique du cœur à base de mesures expérimentales, et sur des investigations concernant des applications cliniques en modélisation personnalisée. Dans le cadre 1D, nous avons reproduit des expériences physiologiques de contraction d'une fibre cardiaque. Concernant la validation 3D, nous avons effectué dans une collaboration clinique une expérience avec des animaux (cochons), afin d'obtenir des données saines et après création d'infarctus. Nous avons démontré que notre modèle est capable de représenter le cœur sain, et que l'infarctus peut être correctement modélisé en modifiant uniquement les paramètres directement concernés par la pathologie. Dans la première application clinique nous avons démontré que le modèle est prédictif sur l'effet immédiat d'une "Cardiac Resynchronization Therapy" (CRT), sur le critère du "max LV dp/dt". La personnalisation du modèle a été réalisée au moyen des données d'IRM et de pressions au stade 'baseline' (avant la CRT). Puis, tous les paramètres du modèle étant figés, nous avons appliqué les schémas d'activation électrique correspondant aux modes de CRT. Nous avons obtenu une prédiction très proche des indicateurs mesurés sur 3 patients avec plusieurs schémas d'activation. Cette étude clinique préliminaire montre que la modélisation de CRT est très prometteuse comme assistance à la planification thérapeutique. Dans une deuxième application nous avons adapté des méthodes d'assimilation de données développées dans l'Equipe-Projet MACS à l'INRIA, à des fins d'aide au diagnostic. Nous avons ainsi réalisé une estimation conjointe état-paramètres en utilisant des données réelles (IRM). Nous avons démontré l'efficacité de ces méthodes, et l'intérêt des paramètres de contractilité estimés comme indicateurs de la fonction du myocarde.