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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; 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.

We present a novel strategy to perform estimation for a dynamical mechanical system in standard operating conditions, namely, without ad hoc experimental testing. We adopt a sequential approach, and the joint state-parameter estimation procedure is based on a state estimator inspired from collocated feedback control. This type of state estimator is chosen due to its particular effectiveness and robustness, but the methodology proposed to adequately extend state estimation to joint state-parameter estimation is general, and - indeed -applicable with any other choice of state feedback observer. The convergence of the resulting joint estimator is mathematically established. In addition, we demonstrate its effectiveness with a biomechanical test problem defined to feature the same essential characteristics as a heart model, in which we identify localized contractility and stiffness parameters using measurements of a type that is available in medical imaging.

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.

We consider an elastic model for a curved rod with arbitrary three-dimensional geometry, incorporating shear and membrane as well as bending and torsion effects. We define an approximation procedure based on a discretization by linear Timoshenko beam elements. Introducing an equivalent mixed problem, we establish optimal error estimates independent of the thickness, thereby proving that shear and membrane locking is avoided. The approximation scheme is tested on specific examples and the numerical results confirm the estimates obtained by theory.

International audience; The objective of this paper is to assess a previously-proposed surface-based electrophysiology model with detailed atrial simulations. This model - derived and substantiated by mathematical arguments - is specifically designed to address thin structures such as atria, and to take into account strong anisotropy effects related to fiber directions with possibly rapid variations across the wall thickness. The simulation results are in excellent adequacy with previous studies, and confirm the importance of anisotropy effects and variations thereof. Furthermore, this surface-based model provides dramatic computational benefits over 3D models with preserved accuracy.

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.

This paper presents a numerical study in which several partitioned solution procedures for incompressible fluid-structure interaction are compared and validated against the results of an experimental FSI benchmark. The numerical methods discussed cover the three main families of coupling schemes: strongly coupled, semi-implicit and loosely coupled. Very good agreement is observed between the numerical and experimental results. The comparisons confirm that strong coupling can be efficiently avoided, via semi-implicit and loosely coupled schemes, without compromising stability and accuracy.

International audience; We analyze an observer strategy based on partial --~i.e.~in a subdomain --~measurements of the solution of a wave equation, in order to compensate for unknown initial conditions. We prove the exponential convergence of this observer under a non-standard observability condition, whereas using measurements of the time-derivative of the solution would lead to a standard observability condition arising in stabilization and exact controlability. Nevertheless, we directly relate our specific condition to the classical geometric control condition. Finally, we provide some numerical illustrations of the effectiveness of the approach.; Nous analysons un observateur formulé pour l'équation des ondes et utilisant des mesures de la solution dans un sous-domaine afin de compenser des conditions initiales inconnues. Nous démontrons la convergence exponentielle de cet observateur moyennant une condition d'observabilité non-standard, alors qu'une mesure de la dérivée en temps de la solution conduirait à la forme standard de la condition d'observabilité apparaissant en stabilisation et contrôla\-bilité exacte. Néanmoins, nous établissons un lien direct entre notre condition spécifique et la condition de contrôle géométrique classique.

International audience; Viscoelastic support has been previously established as a valuable modeling ingredient to represent the effect of surrounding tissues and organs in a fluid-structure vascular model. In this paper, we propose a complete methodological chain for the identification of the corresponding boundary support parameters, using patient image data. We consider distance maps of model to image contours as the discrepancy driving the data assimilation approach, which then relies on a combination of (1) state estimation based on the so-called SDF filtering method, designed within the realm of Luenberger observers and well adapted to handling measurements provided by image sequences, and (2) parameter estimation based on a reduced-order UKF filtering method which has no need for tangent operator computations and features natural parallelism to a high degree. Implementation issues are discussed, and we show that the resulting computational effectiveness of the complete estimation chain is comparable to that of a direct simulation. Furthermore, we demonstrate the use of this framework in a realistic application case involving hemodynamics in the thoracic aorta. The estimation of the boundary support parameters proves successful, in particular in that direct modeling simulations based on the estimated parameters are more accurate than with a previous manual expert calibration. This paves the way for complete patient-specific fluid-structure vascular modeling in which all types of available measurements could be used to estimate additional uncertain parameters of biophysical and clinical relevance.

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.

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; We present numerical results obtained with a three-dimensional electromechanical model of the heart with a complete realistic anatomy. The electrical activity of the heart-torso domain is described by the bidomain equations in the heart and a Laplace equation in the torso. The mechanical model is based on a chemically-controlled contraction law of the myofibres integrated in a 3D continuum mechanics description accounting for large displacements and strains, and the main cardiovascular blood compartments are represented by simplified lumped models. We considered a normal case and a pathological condition and the medical indicators resulting from the simulations show physiological values, both for mechanical and electrical quantities of interest, in particular pressures, volumes and ECGs.

International audience; We propose an effective filtering methodology designed to perform estimation in a distributed mechanical system using position measurements. As in a previously introduced method, the filter is inspired from robust control feedback, but here we take full advantage of the estimation specificity to choose a feedback law that can act on displacements instead of velocities and still retain the same kind of dissipativity property which guarantees robustness. This is very valuable in many applications for which positions are more readily available than velocities, as in medical imaging. We provide an in-depth analysis of the proposed procedure, as well as detailed numerical assessments using a test problem inspired from cardiac biomechanics, as medical diagnosis assistance is an important perspective for this approach. The method is formulated first for measurements based on Lagrangian displacements, but we then derive a nonlinear extension allowing to instead consider segmented images, which of course is even more relevant in medical applications.

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.

International audience; We consider the design of finite element methods for the Naghdi shell model in the case when the deformation is bending dominated. Two formulations based on stabilizing techniques are introduced and it is proved that they are stable, hence free from locking. The theoretical estimates are confirmed by numerical benchmark studies.

International audience; Investigating the causes of failure of Fontan circulation in individual patients remains challenging despite detailed combined inva-sive cardiac catheterisation and magnetic resonance (XMR) exams at rest and during stress. In this work, we use a biomechanical model of the heart and Fontan circulation with the components of systemic and pulmonary beds to augment the diagnostic assessment of the patients undergoing the XMR stress exam. We apply our model in 3 Fontan patients and one biventricular "control" case. In all subjects, we obtained important biophysical factors of cardiovascular physiology-contractil-ity, contractile reserve and changes in systemic and pulmonary vascular resistance-which contribute to explaining the mechanism of failure in individual patients. Finally, we used the patient-specific model of one Fontan patient to investigate the impact of changes in pulmonary vas-cular resistance, aiming at in silico testing of pulmonary vasodilation treatments.