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This work addresses the mathematical analysis – by means of asymptotic analysis – of a penalisation strategy for the full discretisation of elastic wave propagation problems in quasi-incompressible media that has been recently developed by the authors. We provide a convergence analysis of the solution to the continuous version of the penalised problem towards its formal limit when the penalisation parameter tends to infinity. Moreover, as a fundamental intermediate step we provide an asymptotic analysis of the convergence of solutions to quasi-incompressible problems towards solutions to purely incompressible problems when the incompressibility parameter tends to infinity. Finally, we further detail the regularity assumptions required to guarantee that the mentioned convergence holds.

When solving 2D linear elastodynamic equations in homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. The case of rigid boundary condition was treated by Burel [Ph.D. thesis, Universite Paris Sud-Paris XI (2014)] and Burel et al. [Numer. Anal. Appl. 5 (2012), pp. 136— 143]. However the case of traction free boundary condition was proven by Martinez et al. [J. Sci. Comput. 77 (2018), pp. 1832-1873] to be unstable if a straightforward approach is used. Then an adequate functional framework as well as a time domain mixed formulation to circumvent these issues was presented. In this work we first review the formulation presented by Martinez et al. [J. Sci. Comput. 77 (2018), pp. 1832-1873] and propose a subsequent discretised formulation. We provide the complete stability analysis of the corresponding numerical scheme. Numerical results that illustrate the theory are also shown.

The goal of this work is to derive a reliable stable and accurate inverse problem strategy for reconstructing cardiac output blood flow entering the ascending aorta from pressure measurements at a distal site of the arterial tree, assumed here to be the descending aorta. We assume that a reduced one-dimensional model of the aorta can be linearized around its steady state, resulting in a wave system with absorbing boundary condition at the outlet. Using this model, we attempt to reconstruct the inlet flow from a pressure measurement at the distal outlet. First, we investigate the observability of the problem and prove that the inversion of the input-output operator for the flow and pressure in the space of time-periodic solutions is ill-posed of degree one. We then develop a variational approach where we minimize the discrepancy between measurements and a simulated state and penalize the error with respect to a periodic state. It is shown that the penalty strategy is convergent and provides an efficient solution for the minimization. Numerical results illustrate the robustness of our approach to noise and the potential of our method to reconstruct inlet flow from real pressure recordings during anesthesia.

The aim of this work is to provide a mathematical model and analysis of the excitation and the resulting shear wave propagation in acoustic radiation force (ARF)-based shear wave cardiac elastography. Our approach is based on asymptotic analysis; more precisely, it consists in considering a family of problems, parametrised by a small parameter inversely proportional to the excitation frequency of the probes, the viscosity and the velocity of pressure wave propagation. We derive a simplified model for the expression of the ARF by investigating the limit behaviour of the solution when the small parameter goes to zero. By formal asymptotic analysis – an asymptotic expansion of the solution is used – and energy analysis of the nonlinear elastodynamic problem, we show that the leading-order term of the expansion is solution of the underlying, incompressible, nonlinear cardiac mechanics. Subsequently, two corrector terms are derived. The first is a fast-oscillating pressure wave generated by the probes, solution of a Helmholtz equation at every time. The second corrector term consists in an elastic field with prescribed divergence, having a function of the first corrector as a source term. This field corresponds to the shear acoustic wave induced by the ARF. We also confirm that, in cardiac mechanics, the presence of viscosity in the model is essential to derive an expression of the shear wave propagation from the ARF, and that this phenomenon is related to the nonlinearity of the partial differential equation.

In this work, we construct an efficient numerical method to solve 3D Maxwell’s equations in coaxial cables. Our strategy is based upon a hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objectives is to validate numerically generalized telegrapher’s models that are used to simplify the 3D Maxwell equations into a 1D problem. This is the object of the second part of the article.

In this work we provide a novel energy-consistent formulation for the classical 1D formulation of blood flow in an arterial segment. The resulting reformulation is shown to be suitable for the coupling with a lumped (0D) model of the heart that incorporates a reduced formulation of the actin-myosin interaction. The coupling being consistent with energy balances, we provide a complete heart-circulation model compatible with thermodynamics hence stable numerically and informative physiologically. These latter two properties are verified by numerical experiments.

In this work, we extend the Arlequin method, a multiscale and multimodel framework based on overlapping domains and energy partitions for reliable modeling and flexible simulation of transient prob...

The aim of this work is to propose a novel, fast solver for the Poisson problem discretized with high-order spectral element methods (HO-SEM) in canonical geometries (rectangle in two dimensions, r...

The principal aim of this work is to provide an adapted numerical scheme for the approximation of elastic wave propagation in incompressible solids. We rely on high-order conforming finite element with mass lumping for space discretisation and implicit/explicit, second-order, energy-preserving time discretisation. The time step restriction only depends on the shear wave velocity and at each time step a Poisson problem must be solved to account for the incompressibility constraint that is imposed by penalisation techniques.

In this paper we study a dispersive shallow water type model derived from the free surface compressible Navier-Stokes system. The compressible effects allow to capture the acoustic-like waves propagation and can be seen as a relaxation of an underlying incompressible model. The first interest of such a model is thus to capture both acoustic and water waves. The second interest lies in its numerical approximation. Indeed, at the discrete level, the pseudo-compressibility terms circumvent the resolution of an elliptic equation to capture the non-hydrostatic part of the pressure. This drastically reduces the cost of the numerical resolution of dispersive models especially in 2d and 3d.

International audience; In this work we present and analyse a time discretisation strategy for linear wave equations t hat aims at using locally in space the most adapted time discretisation among a family of implicit or explicit centered second order schemes. The proposed family of schemes is adapted to domain decomposition methods such as the mortar element method. They correspond in that case to local implicit schemes and to local time stepping. We show that, if some regularity properties of the solution are satisfied and if the time step verifies a stability condition, then the family of proposed time discretisations provides, in a strong norm, second order space-time convergence. Finally, we provide 1D and 2D numerical illustrations that confirm the obtained theoretical results and we compare our approach on 1D test cases to other existing local time stepping strategies for wave equations.

International audience; In this work we tackle the modeling of non-perfectly conducting thin coaxial cables. From the non-dimensionnalised 3D Maxwells equations, we derive, by asymptotic analysis with respect to the (small) transverse dimension of the cable, a simplified effective 1D model and an effective reconstruction procedure of the electric and magnetic fields. The derived effective model involves a fractional time derivatives that accounts for the so-called skin effects in highly conducting regions.

International audience; This paper deals with the construction of a family of fourth order, energy consistent, explicit time discretizations for dissipative linear wave equations. The schemes are obtained by replacing the inversion of a matrix, that comes naturally after using the technique of the Modified Equation on the second order Leap Frog scheme applied to dissipative linear wave equations, by explicit approximations of its inverse. The stability of the schemes are studied using an energy analysis and a convergence analysis is carried out. Numerical results in 1D illustrate the space/time convergence properties of the schemes and their efficiency is compared to more classical time discretizations.

International audience; We are interested in reconstructing the initial condition of a wave equation in an unbounded domain configuration from measurements available in time on a subdomain. To solve this problem, we adopt an iterative strategy of reconstruction based on observers and time reversal adjoint formulations. We prove the convergence of our reconstruction algorithm with perfect measurements and its robustness to noise. Moreover, we develop a complete strategy to practically solve this problem on a bounded domain using artificial transparent boundary conditions to account for the exterior domain. Our work then demonstrates that the consistency error introduced by the use of approximate transparent boundary conditions is compensated by the stabilization properties obtained from the use of the available measurements, hence allowing to still be able to reconstruct the unknown initial condition.; Nous nous intéressons au problème de recontruction de la donnée initiale pour uné equation d'ondes posée en domaine non bornéà l'aide de mesures en temps sur un sous domaine. Pour résoudre ce problème, nous utilisons une approche itérative de reconstruction basée sur une formulation type observateurs et retournement temporel. On prouve la convergence de l'algorithme de reconstruction dans le cas de mesures non bruitées et sa stabilité par rapport à des mesures bruitées. De plus, nous développons une méthodologie complète pour résoudre numériquement le problème en domaine borné en utilisant des conditions transparentes (absorbantes) sur les frontières artificielles bornant le domaine. Nous montrons en particulier que l'erreur de consistance introduite par les conditions absorbantes (non exactes) est compensée par les propriétés de stabilisation obtenues grâce aux mesures, ce qui permet d'être toujours en mesure de reconstituer la donnée initiale.

International audience; We follow a formal homogenization approach to investigate the effects of mechanical deformations in electrophysiology models relying on a bidomain description of ionic motion at the microscopic level. To that purpose, we extend these microscopic equations to take into account the mechanical deformations, and proceed by recasting the problem in the framework of classical two-scale homogenization in periodic media, and identifying the equations satisfied by the first coefficients in the formal expansions. The homogenized equations reveal some interesting effects related to the microstructure - and associated with a specific cell problem to be solved to obtain the macroscopic conductivity tensors - in which mechanical deformations play a non-trivial role, i.e. do not simply lead to a standard bidomain problem posed in the deformed configuration. We then present detailed numerical illustrations of the homogenized model with coupled cardiac electrical-mechanical simulations - all the way to ECG simulations - albeit without taking into account the abundantly-investigated effect of mechanical deformations in ionic models, in order to focus here on other effects. And in fact our numerical results indicate that these other effects are numerically of a comparable order, and therefore cannot be disregarded.

International audience; In this work we present a method for the computation of numerical solutions of 2D homogeneous isotropic elastodynamics equations by solving scalar wave equations. These equations act on the potentials of a Helmholtz decomposition of the displacement field and are decoupled inside the propagation domain. We detail how these equations are coupled at the boundary depending on the nature of the boundary condition satisfied by the displacement field. After presenting the case of rigid boundary conditions, that presents no specific difficulty, we tackle the challenging case of free surface boundary conditions that presents severe stability issues if a straightforward approach is used. We introduce an adequate functional framework as well as a time domain mixed formulation to circumvent these issues. Numerical results confirm the stability of the proposed approach.

In Grote and Sim (Efficient PML for the wave equation. Preprint, arXiv:1001.0319 [math:NA], 2010; in: Proceedings of the ninth international conference on numerical aspects of wave propagation (WAVES 2009, held in Pau, France, 2009), pp 370–371), a PML formulation was proposed for the wave equation in its standard second-order form. Here, energy decay and $$L^2$$ stability bounds in two and three space dimensions are rigorously proved both for continuous and discrete formulations with constant damping coefficients. Numerical results validate the theory.

International audience; The aim of this paper is to provide a complete mathematical analysis of the periodic homogenization procedure that leads to the macroscopic bidomain model in cardiac elec-trophysiology. We consider space-dependent and tensorial electric conductivities as well as space-dependent physiological and phenomenological non-linear ionic models. We provide the nondimensionalization of the bidomain equations and derive uniform estimates of the solutions. The homogenization procedure is done using 2-scale convergence theory which enables us to study the behavior of the non-linear ionic models in the homogenization process .

A family of fourth order coupled implicit-explicit time schemes is presented as a special case of fourth order coupled implicit schemes for linear wave equations. The domain of interest is decomposed into several regions where different fourth order time discretization are used, chosen among a family of implicit or explicit fourth order schemes. The coupling is based on a Lagrangian formulation on the boundaries between the several non conforming meshes of the regions. A global discrete energy is shown to be preserved and leads to global fourth order consistency in time. Numerical results in 1d and 2d for the acoustic and elastodynamics equations illustrate the good behavior of the schemes and their potential for the simulation of realistic highly heterogeneous media or strongly refined geometries, for which using everywhere an explicit scheme can be extremely penalizing. Accuracy up to fourth order reduces the numerical dispersion inherent to implicit methods used with a large time step, and makes this family of schemes attractive compared to second order accurate methods.

International audience; This paper concerns the space/time convergence analysis of conservative two-steps time discretizations for linear wave equations. Explicit and implicit, second and fourth order schemes are considered, while the space discretization is given and satisfies minimal hypotheses. The convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value.; Ce travail concerne l'analyse de convergence espace/temps de schémas en temps à deux pas pour les équations d'onde linéaires. Sont considérés des schémas explicites et implicites, d'ordre deux et quatre, tandis que la discrétisation spatiale est donnée et satisfait des hypothèses minimales. L'analyse de convergence est faite par techniques d'énergie et est valide si le pas de temps est borné par une quantité dépendant des paramètres de discrétisation spatiale. En plus de montrer la convergence pour des schémas d'ordre quatre récemments introduits, la nouveauté de ce travail réside dans le fait que les estimations ne dépendent pas de la différence entre le pas de temps et sa plus grande valeur admissible.

This paper presents an original approach to design perfectly matched layers (PML) for anisotropic scalar wave equations. This approach is based, first, on the introduction of a modified wave equation and, second, on the formulation of general "perfectly matched" transmission conditions for this equation. The stability of the transmission problem is discussed by way of the adaptation of a high frequency stability (necessary) condition, and we apply our approach to construct PML suited for any convex domain with straight boundaries. A new variational formulation of the problem, including a Lagrange multiplier at the interface between the physical and the absorbing domains, is then set and numerical results are presented in 2D and 3D. These results show the efficiency of our approach when using constant damping coefficients combined with high order elements.

This study concerns the mathematical modeling of anisotropic and transversely inhomogeneous slender piezoelectric bars. Such rod-like structures are employed as passive sensors aimed at measuring the displacement field on the boundary of an underlying elastic medium excited by an external source. Based on the coupled three-dimensional dynamical equations of piezoelectricity in the quasi-electrostatic approximation, a set of limit problems is derived using formal asymptotic expansions of the electric potential and elastic displacement fields. The nature of these problems depends strongly on the choice of boundary conditions, therefore, an appropriate set of constrains is introduced in order to derive one-dimensional models that are relevant to the measurement of a displacement field imposed at one end of the bar. The structure of the first-order electric and displacement fields as well as the associated coupled limit equations are determined. Moreover, the properties of the homogenized material parameters entering these equations are investigated in various configurations. The obtained one-dimensional models of piezoelectric sensors are analyzed, and it is finally shown how they enable the identification of the boundary displacement associated with the probed elastic medium.

The paper addresses the construction of a non spurious mixed spectral finite element (FE) method to problems in the field of computational aeroacoustics. Based on a computational scheme for the conservation equations of linear acoustics, the extension towards convected wave propagation is investigated. In aeroacoustic applications, the mean flow effects can have a significant impact on the generated sound field even for smaller Mach numbers. For those convective terms, the initial spectral FE discretization leads to non-physical, spurious solutions. Therefore, a regularization procedure is proposed and qualitatively investigated by means of discrete eigenvalues analysis of the discrete operator in space. A study of convergence and an application of the proposed scheme to simulate the flow induced sound generation in the process of human phonation underlines stability and validity.

Seismocardiography (SCG) is measurement of sternal acceleration caused by heart beats. Although fiducial cardiac events have been associated with seismocardiogram extrema, the forces that cause the vibrations are unknown. The goal of this study was to create a 3D model of the thorax capable of modelling its vibrations under heart-like forces.

International audience; We consider the reconstruction of internal elastic displacements from ultrasound measurements , which finds applications in the medical imaging modality called elastography. By appropriate interferometry and windowed Fourier transforms of the ultrasound measurements, we propose a reconstruction procedure of the vectorial structure of spatially varying elastic displacements in biological tissues. This provides a modeling and generalization of scalar reconstruction procedures routinely used in elastography. The proposed algorithm is justified using a single scattering approximation and local asymptotic analysis. Its validity is assessed by numerical simulations.

International audience; This paper proposes a formal justification of simplified 1D models for the propagation of electromagnetic waves in thin non-homogeneous lossy conductor cables. Our approach consists in deriving these models from an asymptotic analysis of 3D Maxwell’s equations. In essence, we extend and complete previous results to the multi-wires case.

Within the framework of linear elasticity we assume the availability of internal full-field measurements of the continuum deformations of a non-homogeneous isotropic solid. The aim is the quantitative reconstruction of the associated moduli. A simple gradient system for the sought constitutive parameters is derived algebraically from the momentum equation, whose coefficients are expressed in terms of the measured displacement fields and their spatial derivatives. Direct integration of this system is discussed to finally demonstrate the inexpediency of such an approach when dealing with noisy data. Upon using polluted measurements, an alternative variational formulation is deployed to invert for the physical parameters. Analysis of this latter inversion procedure provides existence and uniqueness results while the reconstruction stability with respect to the measurements is investigated. As the inversion procedure requires differentiating the measurements twice, a numerical differentiation scheme based on an ad hoc regularization then allows an optimally stable reconstruction of the sought moduli. Numerical results are included to illustrate and assess the performance of the overall approach., Comment: 23 pages, 7 figures

International audience; In this work, we focus on the time-domain simulation of the propagation of electromagnetic waves in non-homogeneous lossy coaxial cables. The full 3D Maxwell equations, that described the propagation of current and elec- tric potential in such cables, are classically not tackled directly, but instead a 1D scalar model known as the telegraphist's model is used. We aim at justifying, by means of asymptotic analysis, a time-domain "homogenized" telegraphist's model. This model, which includes a non-local in time op- erator, is obtained via asymptotic analysis, for a lossy coaxial cable whose cross-section is not homogeneous.

International audience; The focus of this article is the transient wave-based detection and identification of defects embedded in isotropic elastic solids using piezoelectric transducers. This work addresses this problem within a comprehensive framework encompassing description of elastic wave propagation within the probed media as well as consideration of the coupling phenomena induced by the transducers. A fundamental reciprocity identity associated with a quasi-static piezoelec- tric model is derived to lay the foundations of ensuing developments and approach of this inverse scattering problem. Modeling of piezoelectric transducers is discussed and application of the proven reciprocity theorem enables the propo- sition of an iterative construction procedure of electric inputs generating waves expected to focus on the sought defects. The characteristic features of the inverse problem considered, which uses piezoelectric sensor-based measurements, are also discussed. Next, the identification problem is investigated by way of an adjoint field-based topological sensitiv- ity approach that permits the construction of a defect indicator function based on the derived reciprocity identity. For simplicity of exposition, the studied configurations involve defects in the form of traction-free cavities. Finally, a set of 2D numerical examples based on the spectral finite-elements method is presented to assess the performances of the proposed approach in identifying embedded defects from electric measurements.

International audience; We study the implicit time discretization of piano strings governing equations within the Timoshenko prestressed beam model. Such model features two different waves, namely the flexural and shear waves, that propagate with very different velocities. We present a novel implicit time discretization that reduces the numerical dispersion while allowing the use of a large time step in the numerical computations. After analyzing the continuous system and the two branches of eigenfrequencies associated with the propagating mode{s}, the classical $\theta$-scheme is studied. We present complete {new} proofs of stability using energy-based approaches that provide uniform results with respect to the featured time step. A dispersion analysis confirms that theta=1/12 reduces the numerical dispersion, but yields a severely constrained stability condition for the application considered. Therefore we propose a new theta-like scheme, which allows to reduce the numerical dispersion while relaxing this stability condition. Stability proofs are also provided for this new scheme. Theoretical results are illustrated with numerical experiments corresponding to the simulation of a realistic piano string.; Nous étudions la discrétisation implicite en temps des équations permettant de modéliser les cordes de piano, grâce au modéle de poutre précontrainte de Timoshenko. Ce modéle considére la propagation de deux ondes (de flexion et de cisaillement) à des vitesses très différentes. Nous présentons une nouvelle discrétisation en temps implicite qui permet de réduire la dispersion numérique tout en autorisant un grand pas de temps lors des simulations numériques. Après avoir analysé le système continu et ses deux branches de fréquences propres, associées à des modes propres, le theta-schéma classique est étudié. Nous présentons des preuves nouvelles de stabilité, basées sur une approche énergétique et qui fournissent des estimations uniformes par rapport au pas de temps. Une analyse de dispersion confirme que la valeur theta = 1/12 réduit la dispersion numériques, mais conduit à une condition de stabilité très sévère pour l'application considérée. Nous proposons donc un nouveau schéma de type theta-schéma, qui permet de réduire la dispersion numérique tout en relaxant la restriction sur le pas de temps. Des preuves de stabilité sont également fournies pour ce nouveau schéma. Les résultats théoriques sont illustrés par des expériences numériques correspondant à la simulation d'une corde de piano réaliste.

International audience; The present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element / finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented.

After setting a mixed formulation for the propagation of linearized water waves problem, we define its spectral element approximation. Then, in order to take into account unbounded domains, we construct absorbing perfectly matched layer for the problem. We approximate these perfectly matched layer by mixed spectral elements and show their stability using the “frozen coefficient” technique. Finally, numerical results will prove the efficiency of the perfectly matched layer compared to classical absorbing boundary conditions.

This paper is the first contribution towards the rigorous justification of asymptotic 1D models for the time-domain simulation of the propagation of electromagnetic waves in coaxial cables. Our general objective is to derive error estimates between the “exact” solution of the full 3D model and the “approximate” solution of the 1D model known as the Telegraphist’s equation.

International audience; In this article, elastic wave propagation in a homogeneous isotropic elastic medium with rigid boundary is considered. A method based on the decoupling of pressure and shear waves via the use of scalar potentials is proposed. This method is adapted to a finite elements discretization, which is discussed. A stable, energy preserving numerical scheme is presented, as well as 2D numerical results.; Dans cet article, on s'intéresse à la propagation d'ondes élastiques dans un matériau élastique homogène et isotrope avec une condition d'encastrement. On propose une méthode basée sur le découplage des ondes de pression et de cisaillement via l'utilisation de potentiels scalaires. Cette méthode est adaptée à une discrétisation éléments finis, et on présente un schéma stable préservant une énergie discrète et des résultats numériques en 2D.

This thesis is embedded in the context of « Structural Health Monitoring ». This method of non-destructive testing aiming at monitoring in-situ an engineered structure is increasingly used in numerous industrial fields, e.g. the aeronautics industry. It is based upon elastic guided waves propagating over large distances. The interactions between incident wave fields and structural defects are gathered through a network of receiving sensors. The dispersive nature of guided waves, combined with the inherent anisotropy of some industrial materials, such as composites, makes the interpretation of the output signals difficult. The goal of this thesis is to provide meaningful numerical tools enhancing the understanding and analysis of propagation and interaction phenomena, appearing during the control experiment. The thesis lies between the physical and mathematical modelling of elastic waves and the construction of relevant numerical schemes, altogether in an innovating industrial context involving complex geometries and materials., Ce sujet de thèse s’inscrit dans le contexte du contrôle intégré des structures, ou « Structural Health Monitoring » (SHM). Cette technique de contrôle non-destructif vise à utiliser un ou plusieurs capteurs, installés dans ou sur la structure d’intérêt. Le contrôle se fait in-situ et de façon périodique, afin d’obtenir des informations sur l’éventuelle apparition de défauts, tels que les défauts de corrosion pour les matériaux métalliques ou les défauts de délaminage pour les matériaux composites. Les données recueillies par les capteurs alimentent une analyse statistique dont le but est d’évaluer la santé de la structure au moment du contrôle, d’estimer son temps de vie restant et de faciliter la prise de décision quant à sa maintenance. Le SHM est de plus en plus présent dans de nombreux domaines industriels, en particulier dans le secteur aéronautique. Aussi le développement de modèles numériques pertinents comme performants constitue un atout majeur dans la conception de ces systèmes. Grâce à leur capacité à se propager sur de très grande distance, l’utilisation de capteurs ultrasonores générant des ondes guidées élastiques est une solution attirante. En pratique, des capteurs piézo-électriques fins, disposés à la surface de la structure, ou éventuellement enfouis pendant le procédé de fabrication, sont utilisés. Ils permettent l’émission et la réception des perturbations ultrasonores. Cependant, la nature dispersive des ondes guidées, combinée avec l’anisotropie inhérente aux matériaux composites rend difficile l’analyse des signaux obtenus lors du contrôle. De plus, proposer une modélisation fine de la propagation de ce type d’onde dans des configurations industrielles faisant intervenir des géométries complexes est une tâche difficile. Ces deux points constituent des obstacles non négligeables au développement de la méthodologie SHM, et l’objectif de cette thèse est de constituer l’ensemble des outils numériques qui permettront de proposer des solutions concrètes à ces problèmes.

This PhD thesis concerns the mathematical modelling and numerical simulation of impulsive Acoustic Radiation Force (ARF)-driven Shear Wave Elastography (SWE) imaging in a prestressed soft tissue, with a specific reference to the cardiac setting. The first part of the manuscript deals with the mathematical modelling of the ARF, the resulting shear wave propagation, and the characterisation of the shear wave velocity in a general constitutive law for the myocardial tissue. We also show some applications to the extraction of fibre orientation in the myocardium and the detection of “synthetic pathologies”. One of the main contributions of this work is the derivation of an original mathematical model of the ARF. In more detail, starting from an accurate biomechanical model of the heart, and based on asymptotic analysis, we infer the governing equation of the pressure and the shear wave field remotely induced by the ARF, and we compute an analytical expression of the source term responsible for the generation of shear waves from an acoustic pressure pulse. In the second part of the PhD thesis, we propose efficient numerical tools for a realistic numerical simulation of an SWE experiment in a nearly-incompressible, pre-stressed, fibered soft tissue. The spatial discretisation is based on high-order Spectral Finite Elements (HO-SEM). Concerning the time discretisation, we propose a novel method adapted to incompressible elasticity. In particular, only the terms travelling at infinite velocity, associated with the incompressibility constraint, are treated implicitly by solving a scalar Poisson problem at each time step of the algorithm. Furthermore, we provide a novel matrix-free, high-order, fast method to solve the Poisson problem, based on the use of the Discrete Fourier Transform.; Les objectifs de cette thèse sont la modélisation mathématique et la simulation numérique de l’élastographie impulsionnelle basée sur la force de radiation acoustique (FRA) dans un tissu mou précontraint, et en particulier le myocarde. La première partie du manuscript concerne la modélisation mathématique de la FRA, la propagation d’ondes de cisaillement qui en résulte et la caractérisation de la vitesse des ondes de cisaillement pour une loi de comportement générale du tissu myocardique. Nous montrons aussi des applications pour l’estimation de l’orientation des fibres cardiaques dans le myocarde et l’évaluation de “pathologies synthétiques ”. Une des contributions principales de ce travail est le développement d’un modèle mathématique original de la FRA. En particulier, à partir d’un modèle biomécanique tridimensionnel du coeur, nous obtenons, à travers une approche asymptotique, les équations qui régissent les champs de pression et de cisaillement induits par la FRA. De plus, nous calculons une expression analytique du terme source responsable de la génération des ondes de cisaillement à partir d’une impulsion acoustique en pression. Dans la deuxième partie de la thèse, nous proposons des outils numériques efficaces pour une simulation numérique réaliste d’une expérience d’élastographie impulsionnelle dans un tissu quasi-incompressible, précontraint et fibré. La discrétisation en espace se base sur des éléments finis spectraux d’ordre élevé. Pour la discrétisation en temps, nous proposons une nouvelle méthode adaptée à l’élasticité incompressible. En particulier, seuls les termes correspondant à des vitesses infinies, associés à la contrainte d’incompressibilité, sont traités implicitement, à travers la resolution d’un problème de Poisson à chaque pas de temps de l’algorithme. En outre, nous proposons une nouvelle méthode d’ordre élevé et efficace pour la résolution d’un problème de Poisson, qui se base sur la transformée de Fourier discrète.

This PhD thesis concerns the mathematical modelling and numerical simulation of impulsive Acoustic Radiation Force (ARF)-driven Shear Wave Elastography (SWE) imaging in a prestressed soft tissue, with a specific reference to the cardiac setting. The first part of the manuscript deals with the mathematical modelling of the ARF, the resulting shear wave propagation, and the characterisation of the shear wave velocity in a general constitutive law for the myocardial tissue. We also show some applications to the extraction of fibre orientation in the myocardium and the detection of “synthetic pathologies”. One of the main contributions of this work is the derivation of an original mathematical model of the ARF. In more detail, starting from an accurate biomechanical model of the heart, and based on asymptotic analysis, we infer the governing equation of the pressure and the shear wave field remotely induced by the ARF, and we compute an analytical expression of the source term responsible for the generation of shear waves from an acoustic pressure pulse. In the second part of the PhD thesis, we propose efficient numerical tools for a realistic numerical simulation of an SWE experiment in a nearly-incompressible, pre-stressed, fibered soft tissue. The spatial discretisation is based on high-order Spectral Finite Elements (HO-SEM). Concerning the time discretisation, we propose a novel method adapted to incompressible elasticity. In particular, only the terms travelling at infinite velocity, associated with the incompressibility constraint, are treated implicitly by solving a scalar Poisson problem at each time step of the algorithm. Furthermore, we provide a novel matrix-free, high-order, fast method to solve the Poisson problem, based on the use of the Discrete Fourier Transform.; Les objectifs de cette thèse sont la modélisation mathématique et la simulation numérique de l’élastographie impulsionnelle basée sur la force de radiation acoustique (FRA) dans un tissu mou précontraint, et en particulier le myocarde. La première partie du manuscript concerne la modélisation mathématique de la FRA, la propagation d’ondes de cisaillement qui en résulte et la caractérisation de la vitesse des ondes de cisaillement pour une loi de comportement générale du tissu myocardique. Nous montrons aussi des applications pour l’estimation de l’orientation des fibres cardiaques dans le myocarde et l’évaluation de “pathologies synthétiques ”. Une des contributions principales de ce travail est le développement d’un modèle mathématique original de la FRA. En particulier, à partir d’un modèle biomécanique tridimensionnel du coeur, nous obtenons, à travers une approche asymptotique, les équations qui régissent les champs de pression et de cisaillement induits par la FRA. De plus, nous calculons une expression analytique du terme source responsable de la génération des ondes de cisaillement à partir d’une impulsion acoustique en pression. Dans la deuxième partie de la thèse, nous proposons des outils numériques efficaces pour une simulation numérique réaliste d’une expérience d’élastographie impulsionnelle dans un tissu quasi-incompressible, précontraint et fibré. La discrétisation en espace se base sur des éléments finis spectraux d’ordre élevé. Pour la discrétisation en temps, nous proposons une nouvelle méthode adaptée à l’élasticité incompressible. En particulier, seuls les termes correspondant à des vitesses infinies, associés à la contrainte d’incompressibilité, sont traités implicitement, à travers la resolution d’un problème de Poisson à chaque pas de temps de l’algorithme. En outre, nous proposons une nouvelle méthode d’ordre élevé et efficace pour la résolution d’un problème de Poisson, qui se base sur la transformée de Fourier discrète.

This thesis aim to modelize network made of coaxial and multi-conductors cables.It could be mathematically represent with the Maxwell equations which deals on electromagnetic waves propagating in the network or an electrical circuit whose unknowns - the electrical potentials and currents - satisfy the telegrapher's equation on each branches and the Kirchhoff's laws on each knots.The first method is enough general to integrate many defaults but numerically too expansive for the application we have in mind, namely non destructive testing. The second one is not obtained from the Maxwell theory and it is valid if and only if the cable are perfect (cylindrical, lossless...). We derive some 1D models generalizing the usual telegrapher's equation and Kirchhoff's rules from Maxwell's equation. This new models integrate plenty of defects (geometry, losses, skin-effect, materials' characteristics varying) and are derive via asymptotic analysis (classical ones, multi-scales ones, matched ones) by considering very small parameters (transverse dimensions of the cables relative to length of the cables, conductivity of the dielectric part relative to the metal of the inner-wires, size of the junction part relative to the whole network).One of the mathematical difficult is due to the fact that the geometry we will consider (sections of the cables, junctions) are not simply connected. Thus we will generalize usual tools such as the Helmholtz decompositions.; Cette thèse porte sur la modélisation d'un réseau de câbles coaxiaux et multi-conducteurs. Ce dernier peut être mathématiquement traduit par les équations aux dérivées partielles de Maxwell qui régissent la propagation des ondes électromagnétiques en son sein ou par un modèle type circuit électrique d'inconnues - les potentiels et courants électriques- qui vérifient sur les branches du circuit l'équation des télégraphistes et sur les noeuds les lois de Kirchhoff.Si la première méthode est assez générale pour comprendre toutes sortes de défauts, elle néanmoins trop couteuse pour les applications que nous avons en tête, à savoir le contrôle non destructif. La seconde quant à elle est obtenue par une modélisation non issue de la théorie de Maxwell et est valide que si les câbles sont parfaits (cylindriques, sans pertes...). Nous avons établi diverses modèles 1D venant généraliser l'équation des télégraphistes et les lois de Kirchhoff pour y incorporer diverses défauts (géométrie, pertes, effet de peau, caractéristique des matériaux variables) tant sur les câbles que dans les jonctions. Ceux-ci sont obtenus via des analyses asymptotiques (classiques, multi-échelles, raccordées) des équations 3D de Maxwell en considérant certains paramètres (dimensions transverses des câbles par rapport à leurs longueurs, conductivité du milieu diélectrique par rapport à celle du métal des âmes, petite taille de la zone de jonction par rapport à l'ensemble du réseau) extrêmement petits.Une des difficultés mathématiques tient en ce que les domaines que nous prendrons en compte (sections des câbles, jonctions) ne sont aucunement simplement connexes, nous obligeant ainsi à remanier quelques outils standard tel les décompositions en potentiels.

This thesis aim to modelize network made of coaxial and multi-conductors cables.It could be mathematically represent with the Maxwell equations which deals on electromagnetic waves propagating in the network or an electrical circuit whose unknowns - the electrical potentials and currents - satisfy the telegrapher's equation on each branches and the Kirchhoff's laws on each knots.The first method is enough general to integrate many defaults but numerically too expansive for the application we have in mind, namely non destructive testing. The second one is not obtained from the Maxwell theory and it is valid if and only if the cable are perfect (cylindrical, lossless...). We derive some 1D models generalizing the usual telegrapher's equation and Kirchhoff's rules from Maxwell's equation. This new models integrate plenty of defects (geometry, losses, skin-effect, materials' characteristics varying) and are derive via asymptotic analysis (classical ones, multi-scales ones, matched ones) by considering very small parameters (transverse dimensions of the cables relative to length of the cables, conductivity of the dielectric part relative to the metal of the inner-wires, size of the junction part relative to the whole network).One of the mathematical difficult is due to the fact that the geometry we will consider (sections of the cables, junctions) are not simply connected. Thus we will generalize usual tools such as the Helmholtz decompositions., Cette thèse porte sur la modélisation d'un réseau de câbles coaxiaux et multi-conducteurs. Ce dernier peut être mathématiquement traduit par les équations aux dérivées partielles de Maxwell qui régissent la propagation des ondes électromagnétiques en son sein ou par un modèle type circuit électrique d'inconnues - les potentiels et courants électriques- qui vérifient sur les branches du circuit l'équation des télégraphistes et sur les noeuds les lois de Kirchhoff.Si la première méthode est assez générale pour comprendre toutes sortes de défauts, elle néanmoins trop couteuse pour les applications que nous avons en tête, à savoir le contrôle non destructif. La seconde quant à elle est obtenue par une modélisation non issue de la théorie de Maxwell et est valide que si les câbles sont parfaits (cylindriques, sans pertes...). Nous avons établi diverses modèles 1D venant généraliser l'équation des télégraphistes et les lois de Kirchhoff pour y incorporer diverses défauts (géométrie, pertes, effet de peau, caractéristique des matériaux variables) tant sur les câbles que dans les jonctions. Ceux-ci sont obtenus via des analyses asymptotiques (classiques, multi-échelles, raccordées) des équations 3D de Maxwell en considérant certains paramètres (dimensions transverses des câbles par rapport à leurs longueurs, conductivité du milieu diélectrique par rapport à celle du métal des âmes, petite taille de la zone de jonction par rapport à l'ensemble du réseau) extrêmement petits.Une des difficultés mathématiques tient en ce que les domaines que nous prendrons en compte (sections des câbles, jonctions) ne sont aucunement simplement connexes, nous obligeant ainsi à remanier quelques outils standard tel les décompositions en potentiels.