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Gamma-convergence results for nematic elastomer bilayers: relaxation and actuation
No full textInternational audienceWe compute effective energies of thin bilayer structures composed of soft nematic elastic liquid crystals in various geometrical regimes and functional configurations. Our focus is on elastic foundations composed of an isotropic layer attached to a nematic substrate where order-strain interaction results in complex opto-mechanical instabilities activated via coupling through the common interface. Allowing out-of-plane displacements, we compute Gamma-limits for vanishing thickness which exhibit spontaneous stress relaxation and shape-morphing behaviour. This extends the plane strain modelling of Cesana and Leon Baldelli [ Math. Models Methods Appl. Sci. (2018) 2863-2904], and shows the asymptotic emergence of fully coupled active macroscopic nematic foundations. Subsequently, we focus on actuation and compute asymptotic configurations of an active plate on nematic foundation interacting with an applied electric field. From the analytical standpoint, the presence of an electric field and its associated electrostatic work turns the total energy non-convex and non-coercive. We show that equilibrium solutions are min-max points of the system, that min-maximising sequences pass to the limit and, that the limit system can exert mechanical work under applied electric fields.Nous calculons les énergies effectives de structures bicouches minces composées de cristaux liquides élastiques nématiques mous dans divers régimes géométriques et configurations fonctionnelles. Nous nous concentrons sur les fondations élastiques composées d'une couche isotrope fixée à un substrat nématique où l'interaction ordre-déformation entraîne des instabilités opto-mécaniques complexes activées par couplage à travers l'interface commune. En autorisant les déplacements hors du plan, nous calculons des limites Gamma pour une épaisseur croissante, qui présentent une relaxation spontanée des contraintes et un comportement de morphing. Cela étend la modélisation de déformation plane de Cesana et Leon Baldelli [ Math. Models Methods Appl. Sci. (2018) 2863-2904], et montre l'émergence asymptotique de fondations nématiques macroscopiques actives entièrement couplées. Par la suite, nous nous concentrons sur l'actionnement et calculons les configurations asymptotiques d'une plaque active sur fondation nématique interagissant avec un champ électrique appliqué. Du point de vue analytique, la présence d'un champ électrique et du travail électrostatique associé rend l'énergie totale non convexe et non coercitive. Nous montrons que les solutions d'équilibre sont des points de min-max du système, que les séquences de min-max passent à la limite et, que le système limite peut exercer un travail mécanique sous champ électrique appliqué
Edge states in rationally terminated honeycomb structures
No full textInternational audienceConsider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type,” generalizing the classical zigzag and armchair edges. We prove that zero-energy/flat-band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulae for flat-band edge states when they exist. We produce strong evidence for the existence of dispersive (nonflat) edge state curves of nonzero energy for most l
Set-Based B-Series
No full textInternational audienceB-series were defined to unify the formalism of solutions for ordinary differential equations defined by series. Runge–Kutta schemes can be seen as truncated B-series, similar to Taylor series. In the prolific domain of reachability analysis, i.e., the process of computing the set of reachable states for a system, many techniques have been proposed without obvious links. In the particular case of uncertain initial conditions and/or parameters in the definition of differential equations, set-based approaches are a natural and elegant method to compute reachable sets. In this paper, an extension to B-series is proposed to merge these techniques in a common formalism—named set-based B-series. We show that the main properties of B-series are preserved. A validated technique, based on Runge–Kutta methods, able to compute such series, is presented. Experiments are provided in order to illustrate the proposed approach
Inside-Outside Duality for Modified Transmission Eigenvalues
No full textInternational audienceWe introduce a new modified spectrum associated with the scattering from penetrable objects using the modified background technique. We prove that the inside-outside duality method allows to reconstruct this spectrum from full aperture far field measurements at a fixed frequency. The method is numerically tested and validated on some synthetic examples
An automatic PML for acoustic finite element simulations in convex domains of general shape
No full textInternational audienceThis article addresses the efficient finite element solution of exterior acoustic problems with truncated computational domains surrounded by perfectly matched layers (PMLs). The PML is a popular nonreflecting technique that combines accuracy, computational efficiency, and geometric flexibility. Unfortunately, the effective implementation of the PML for convex domains of general shape is tricky because of the geometric parameters that are required to define the PML medium. In this work, a comprehensive implementation strategy is proposed. This approach, which we call the automatically matched layer (AML) implementation, is versatile and fully automatic for the end‐user. With the AML approach, the mesh of the layer is extruded, the required geometric parameters are automatically obtained during the extrusion step, and the practical implementation relies on a simple modification of the Jacobian matrix in the elementwise integrals. The AML implementation is validated and compared with other implementation strategies using numerical benchmarks in two and three dimensions, considering computational domains with regular and nonregular boundaries. A three‐dimensional application with a generally shaped domain generated using a convex hull is proposed to illustrate the interest of the AML approach for realistic industrial cases
Benchmarking Quantized Neural Networks on FPGAs with FINN
No full textInternational audienceThe ever-growing cost of both training and inference for state-of-the-art neural networks has brought literature to look upon ways to cut off resources used with a minimal impact on accuracy. Using lower precision comes at the cost of negligible loss in accuracy. While training neural networks may require a powerful setup, deploying a network must be possible on lowpower and low-resource hardware architectures. Reconfigurable architectures have proven to be more powerful and flexible than GPUs when looking at a specific application. This article aims to assess the impact of mixed-precision when applied to neural networks deployed on FPGAs. While several frameworks exist that create tools to deploy neural networks using reduced-precision, few of them assess the importance of quantization and the framework quality. It is used on top of FINN and Brevitas, two frameworks from Xilinx labs, to assess the impact of quantization on neural networks using 2 to 8 bit precisions and weights with several parallelization configurations. The benchmark set up in this work is available in a public repository (https://github.com/QDucasse/nn_benchmark)
Some Aminocyclopropane Chemistry and some Organotitanium Games with Alkynes
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Palladium-Catalyzed Cycloaddition Strategies Towards Medium-Sized Azacycles
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