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Asymptotic Expansion of Transmission Eigenvalues for Anisotropic Thin Layers
No full textInternational audienceWe study the asymptotic expansion of transmission eigenvalues for anisotropic thin layers. We establish a rigorous second order expansion for simple transmission eigenvalues with respect to the thickness of the layer. The convergence analysis is based on a generalization of Osborn's Theorem to non-linear eigenvalue problems by Moskow [19]. We also provide formal derivation in more general cases validating the obtained theoretical result
Fourier representation of the diffusion MRI signal using layer potentials
No full textInternational audienceThe diffusion magnetic resonance imaging signal arising from biological tissues can be numerically simulated by solving the Bloch-Torrey partial differential equation. Numerical simulations can facilitate the investigation of the relationship between the diffusion MRI signals and cellular structures. With the rapid advance of available computing power, the diffusion MRI community has begun to employ numerical simulations for model formulation and validation, as well as for imaging sequence optimization.Existing simulation frameworks use the finite difference method, the finite element method, or the Matrix Formalism method to solve the Bloch-Torrey partial differential equation. We propose a new method based on the efficient evaluation of layer potentials. In this paper, the mathematical framework and the numerical implementation of the new method are described. We demonstrate the convergence of our method via numerical experiments and analyze the errors linked to various model and simulation parameters. Since our method provides a Fourier-type representation of the diffusion MRI signal, it can potentially facilitate new physical and biological signal interpretations in the future
Wave propagation in one-dimensional quasiperiodic media
No full textInternational audienceThis work is devoted to the resolution of the Helmholtz equation −(µ u) − ρ ω 2 u = f in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed in [10, 19] for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation
Flow-Lenia: Towards open-ended evolution in cellular automata through mass conservation and parameter localization
No full textInternational audienceLenia is a family of cellular automata (CA) generalizing Conway's Game of Life to continuous space, time and states. Lenia has attracted a lot of attention because of the wide diversity of self-organizing patterns it can generate. Among those, some spatially localized patterns (SLPs) resemble life-like artificial creatures. However, those creatures are found in only a small subspace of the Lenia parameter space and are not trivial to discover, necessitating advanced search algorithms. We hypothesize that adding a mass conservation constraint could facilitate the emergence of SLPs. We propose here an extension of the Lenia model, called Flow Lenia, which enables mass conservation. We show a few observations demonstrating its effectiveness in generating SLPs with complex behaviors. Furthermore, we show how Flow Lenia enables the integration of the parameters of the CA update rules within the CA dynamics, making them dynamic and localized. This allows for multi-species simulations, with locally coherent update rules that define properties of the emerging creatures, and that can be mixed with neighbouring rules. We argue that this paves the way for the intrinsic evolution of self-organized artificial life forms within continuous CAs
Low cycle fatigue lifetime prediction of superplastic shape memory alloy structures: Application to endodontic instruments
No full textInternational audienceIn this paper we propose a methodology for a fast numerical determination of low cycle fatigue lifetime of superelastic shape memory alloy structures. This method is based on the observation that generally, in low cycle fatigue, shape memory alloy (SMA) structures are subject to loadings that lead to a confined non-linear behaviour at stress concentration points, such as notches. Numerical fatigue lifetime prediction requires the computation of the mechanical state at critical points. However, classical computational methods, like the non-linear finite element method, lead to a prohibitive computation time in a non-linear cyclic framework. To overcome this issue, we propose to use fast prediction methods, based on localization laws. Following the determination of the stabilized behaviour, an energetic fatigue criterion is applied. The numerical fatigue life prediction model is validated experimentally on SMA endodontic instruments
Effect of scanning speed on fatigue behavior of 316L stainless steel fabricated by laser powder bed fusion
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A fast procedure to predict the notch strength with the coupled criterion
No full textInternational audienceAn efficient procedure is described for applying the Coupled Criterion in order to estimate the notch strength of a structure made of a brittle material. It is based on the representation of the size effect law, which gives the notch strength as a function of a characteristic structure length. A new analytical formulation of this function, which depends on only three parameters, is provided. Three parameters must be identified, requiring only three finite element calculations. The accuracy of the approach is verified by analyzing various notched structures
Surface high harmonic generation on a relativistic plasma mirror for generating intense isolated attosecond field transients
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Revisiting effective acoustic propagation in labyrinthine metasurfaces
No full textInternational audienceWe revisit the modelling of labyrinthine metasurfaces with space coiling design. To do so, we use homogenization theory which allows us to replace the actual structure by a slab filled with an effective, homogeneous and anisotropic, medium. The effective medium is highly anisotropic as the propagation is allowed in one direction only and its effective refractive index is obtained unambiguously thanks to the resolution of a static cell-problem. The result is compared to a classical, two-step, model which follows the intuitive idea that a coiled labyrinth and a slot being its uncoiled version behave the same. Beyond the approximation of such statement (the evanescent fields triggered at the turning regions of the labyrinth are neglected), we stress the difficulty in defining unambiguously the length of the uncoiled labyrinth
Simulation numérique d’écoulements à surface libre avec la Méthode des Eléments-Finis
No full textThis thesis is dedicated to the development of a solver for the subsequent study of ship behavior in nonlinear waves. This dissertation specifically focuses on the numerical simulation of unsteady, incompressible and inviscid free surface flows. The method is based on a two-phase flow, described by incompressible Euler equations. A Level-Set approach is used to capture the interface between the two fluids. This formalism offers the advantage of representing the flow in a global approach, thereby eliminating any discontinuities at the interface. Unlike classical CFD codes commonly employed in the field of naval hydrodynamics, which are usually founded on finite volume/VOF formulations and have limited options for the interpolation order, the proposed method uses a finite-element discretization. This approach allows the implementation of robust, high-order numerical schemes. Numerical scheme stability is ensured in part by the use of a Discontinuous Galerkin method, which allows the use of non centered schemes, but also by the use of a projection method, which consists in solving separately the pressure and the velocity fields. To ensure a high-order numerical scheme in space and time, a new explicit projection method is proposed, where the Hodge decomposition is applied to the velocity derivative. Finally, the numerical results from several representative naval problem cases validate the solver and confirm the effectiveness of the developed method.Cette thèse est consacrée au développement d’un outil de simulation numérique permettant d’étudier, à terme, le comportement d’un navire sur houle fortement non linéaire. Ce travail se concentre spécifiquement sur la simulation numérique d’écoulements à interface mobile, à caractères instationnaires, incompressibles et non-visqueux. La méthode repose sur la modélisation des écoulements bifluides, décrits par les équations d’Euler incompressibles. Une approche Level Set est utilisée pour capturer l’interface entre les deux fluides. Ce formalisme offre l’avantage de représenter l’écoulement dans une approche globale, permettant ainsi de s’affranchir des discontinuités présentes à l’interface. À la différence des codes CFD classiques employés en hydrodynamique navale, qui sont généralement basés sur des formulations Volumes-Finis/VOF limitées sur le choix des ordres d’intégration, la discrétisation spatiale se fait à l’aide d’une méthode Eléments-Finis. Cette dernière autorise l’utilisation de schémas numériques robustes et d’ordre élevé afin d’améliorer la précision des résultats, tout en réduisant le temps de calcul. La stabilité du schéma numérique est assurée en partie par l’utilisation d’une approche Galerkin Discontinu, qui permet de décentrer les schémas de résolution en présence de termes advectifs, et aussi par l’utilisation d’une méthode de projection, qui consiste à résoudre séparément les champs de pression et de vitesse. Pour garantir un schéma numérique d’ordre élevé en espace et en temps, une nouvelle méthode de projection explicite est proposée, où la décomposition de Hodge est appliquée à la dérivée de la vitesse. Finalement, les résultats numériques obtenus, issus de plusieurs cas représentatifs des problématiques navales, permettent de valider l’outil numérique tout en confirmant la puissance de la méthode développée