Participants: F. Alauzet [correspondant]

The interpolation stage is a crucial step for time-dependent mesh adaptive simulations. Indeed, if the accuracy of the solution is spoiled during this stage then it is lost for ever. In the past, we have demonstrated the superiority of conservative interpolation for 2D compressible flow simulations. Here, we develop the 3D extension of the method on tetrahedral meshes. The same conclusion arose. The main difficulty was to design a fast, accurate and robust mesh intersection algorithm.

Participants: F. Alauzet [correspondant] and G. Olivier

The main difficulty arising in numerical simulations with moving geometries is to handle the displacement of the domain boundaries, i.e.,the moving bodies. Only vertices displacement is not sufficient to achieve complex movement such as shear. We proved that the use of edge swapping allows us to achieve such complex displacement. We therefore developed an ALE formulation of this topological mesh modification to preserve the solver accuracy and convergence order  .

Participants: F. Alauzet [correspondant], J. Castelneau, L. Marechal, D. Marcum and A. Loseille

We design a new method to generate structured boundary layer meshes which are mandatory to accurately compute compressible flows a high Reynolds number (several millions). It couple the specification of the optimal boundary layer from the geometry boundary and moving mesh techniques to extrude the boundary layer in an already existing mesh. The main advantage of this approach is its robustness, i.e.,at each step of the algorithm we have always a valid mesh.

Participants: T. Grosges [correspondant], D. Barchiesi, Michael François

Développement de nouveaux procédés de génération de clés de chiffrement, dits “exotiques”, basé sur des processus physiques, multi-échelles, multi-domaines assurant un niveau élevé de sécurité. Application du maillage adaptatif au calcul du couplage lumière-matériaux avec contrôle d'erreur. Application du maillage avec estimateur d'erreur sur l'entropie du système. Etude (locale) de la réduction d'entropie des clés tout au long de la chaîne de création et étude (globale) de l'entropie de l'espace des clés générées, à partir de tests statistiques.

Participants: D. Barchiesi [correspondant], T. Grosges, Houman Borouchaki, Laurence Giraud-Moreau, Sameh Kessentini, Anis Chaari

Développement de modèles et méthodes numériques en spectroscopie et électromagnétisme. Optimisation et adaptation aux études de l'interaction lumière-matière aux échelles nano-micro-macro. Applications des maillages adaptifs et optimisés avec estimateur d'erreur sur le champ physique (intensite électrique et/ou magnétique). Etudes des couplages multi-phyisque (électro-thermique dans les nano-mesostructures, et dans les tissus biologiques).

Participants: S.H. Lo, H. Borouchaki [correspondant], P. Laug

Free-form surfaces and industrial surface forms could nowadays be conveniently generated efficiently by laser-based digitizing techniques or from a CAD graphics system. Automatic algorithms are imperative to decompose complex triangulated objects consisting of thousands to millions of nodal points into simpler surface parts for parameterization and parallel processing. Non-manifold complicated discretized objects will be handled and simple closed or open surface parts (manifolds) are retrieved by means of pure topological considerations. Each of the simple surface part, which is topologically equivalent to an open surface, a sphere or a torus will be decomposed by entirely topological operations into n equal pieces based on a specified geometrical criterion such as surface area, Gaussian curvatures or magnitudes of dihedral angles, etc. Cut planes could be conveniently defined normal to the axes of inertia of the object to be decomposed. The cut plane which produces surface parts which best respect the given criterion will be chosen, and the two bisected surfaces will be made as equal as possible by means of some general balancing mechanisms. This procedure could be repeated as often as necessary until sub-surface parts satisfying the given criterion are obtained. Each piece of the decomposed surfaces having an intact topological boundary can then be processed independently in parallel, and all the pieces could be put back together to recover the original object or partially to represent a certain portion of the object  .

Participants: P.L. George [correspondant], H. Borouchaki, P. Laug

There is a need for finite elements of degree 2 or more to solve various P.D.E. problems. This study discusses a method to construct such meshes in the case of triangular element (in the plane or for a surface) or tetrahedral element (in the volume case), restricting at degree 2. This first part considers the planar case and, to begin with, returns to Bézier curves and Bézier triangles of degree 2. In the case of triangles, the relation with Lagrange P2 finite element is shown. Validity conditions are discussed and some invalid elements are shown while proposing a method to correct them. A construction method is then proposed  .

Participants: F. Alauzet [correspondant] and O. Pironneau

Adjoints are used in optimization to speed-up computations, simplify optimality conditions or compute sensitivities. Because time is reversed in adjoint equations with first order time derivatives, boundary conditions and transmission conditions through shocks can be difficult to understand. In this work, we analyzed the adjoint equations that arise in the context of compressible flows governed by the Euler equations of fluid dynamics. We showed that the continuous adjoints and the discrete adjoints computed by automatic differentiation agree numerically; in particular the adjoint is found to be continuous at the shocks and usually discontinuous at contact discontinuities by both  .

Participants: F. Alauzet [correspondant], A. Belme and A. Dervieux

We have extended our previous work on goal-oriented mesh adaptation to time dependent simulations. This requires to set up a global fixed point algorithm in which state and adjoint variables problems are solved. The adjoint problem is solved backward in time  .

Participants: M. Picasso, F. Alauzet [correspondant], H. Borouchaki and P.-L. George

Mesh adaptation required the use of second order derivatives, i.e.,the Hessian. For second order numerical scheme, the provided numerical solution is only piecewise linear. Consequently, numerical methods are considered to recover second order derivatives, the famous recovery techniques. In this work, we have investigated several methods. Numerical results on 2D and 3D isotropic and anisotropic meshes indicate that the quality of the results is strongly linked to the mesh topology and that no convergence can be insured in general  .

Participants: P. Laug [correspondant], H. Borouchaki

We propose a general scheme of an indirect approach for generating isotropic and anisotropic geometric meshes of a surface constituted by a conformal assembly of parametric patches, based on the concept of metric. The different steps of the scheme are considered and, in particular, the definition of the geometric metric at each point of the surface (internal to a patch, belonging to an interface or boundary curve, or extremity of such a curve) as well as its corresponding induced metric in parametric domains.

Isotropic or anisotropic geometric metrics can locally produce significant size variations (internal to a patch or across interface curves) and can even be discontinuous across the interface curves. The larger the rate of the mesh size variation, the worse is the shape quality of the resulting mesh. To control this size variation, various methodologies based on metric reduction have been proposed in the case of a continuous isotropic metric. We introduce a novel iterative mesh gradation approach for discontinuous metrics. The approach uses a particular metric reduction procedure in order to ensure the convergence of the gradation process. In particular, we show that in the worst case the anisotropic discontinuous geometric metric map is reduced to an isotropic continuous geometric metric map for which the gradation is controlled  .

Participants: P. Laug, H. Borouchaki [correspondant], E. Renaut

Nous proposons une méthodologie pour simplifier la paramétrisation des surfaces composées de carreaux paramètrés issues généralement des environnements CAO (conception assistée par ordinateur). Cette nouvelle paramétrisation est définie via la construction d'une triangulation adaptative appelée support pour chaque carreau. L'adaptation est gouvernée par le contrôle de l'écart entre la paramétrisation initiale et celle issue de la triangulation support. Ce support est utilisé pour générer les maillages de ces surfaces selon une approche indirecte, dans laquelle le maillage est généré via les domaines des paramètres. Le support permet de déconnecter le mailleur du système CAO, et en ce sens constitue un outil universel pour le maillage de telles surfaces  .

Participants: F. Alauzet, A. Loseille [correspondant] and E. Mbinky

In the past, we have demonstrate that multi-scale anisotropic mesh adaptation is a powerful tool to accurately simulate compressible flow problem and to obtain faster convergence to continuous solutions. But, this was limited to second order numerical scheme. Nowadays, numerous teams are working on the development of very high-order numerical scheme (e.g. of third or greater order): Discontinous Galerkin, Residual Distribution scheme, Spectral method, ...

This work extend interpolation error estimates to higher order numerical solution representation. We have examined the case of third-order accuracy. The first step is to reduce the tri-linear form given by the third order error term into a quadratic form based on the third order derivative. From this local error model, the optimal mesh is exhibited thanks to the continuous mesh framework.

Participants: L. Moreau [correspondant], A. Cherouat, H. Borouchaki

Développement d'une méthode de remaillage adaptatif surfacique 3D permettant de raffiner de déraffiner le maillage localement autour de l'outil sphérique au cours des simulations numériques de formage incrémental.

Participants: L. Moreau [correspondant], A. Cherouat, H. Borouchaki

Développement de méthodes de remaillage adaptatif surfacique 3D, développement de méthodes d'interpolation et transfert des champs, interfaa̧ge avec le code EF Abaqus et application sur des exemples concrets de mise en forme de structure métalliques et composites.

Participants: F. Alauzet [correspondant] and G. Olivier

We focused on the extension of the multi-scale anisotropic mesh adaptation to unsteady flows. It leads to the development of a global fixed point mesh adaptation algorithm and space-time error estimates. Moreover, the mesh adaptation methodology has been extended to the case of moving meshes simulations  .

Participants: P. Laug, H. Borouchaki [correspondant]

A wide range of surfaces can be defined by means of composite parametric surfaces as is the case for most CAD modelers. There are, essentially, two approaches to meshing parametric surfaces: direct and indirect. Popular direct methods include the octree-based method, the advancing-front-based method and the paving-based method working directly in the tridimensional space. The indirect approach consists in meshing the parametric domain and mapping the resulting mesh onto the surface. Using the latter approach, we have proposed a general meshing scheme which consists in discretizing each interface curve and meshing each parametric domain according to the above boundary discretizations. Complex surfaces such as a car engine or a complete aircraft are composed of thousands of patches, and meshing these surfaces using the above sequential scheme can be inefficient. We propose a parallel version of the general meshing scheme while naturally balancing the load to each processor.

Participants: H. Borouchaki, P.-L. George [correspondant], L. Marechal

Le comportement en complexité des algorithmes de triangulation sur les "gros” maillage nous amène à utiliser les algorithmes de renumérotation de type Hilbert qui minimise les défauts de cache. Cette technique est également utilisée comme aide à l'optimisation des "gros" maillages. L'algorithme de renumérotation est multi-cœurs.

Des triangulations de plusieurs dizaines de millions de sommets sont construites en utilisant un "simple" ordinateur. La vitesse d'insertion trole le million de tétraèdre à la seconde.

Participants: L. Moreau [correspondant], H. Borouchaki, A. Cherouat

Mise au point d'une méthode d'acquisition de la morphologie du buste féminin, développement d?algorithmes de reconstruction de surface 3D d'un buste féminin à partir de photos numériques dans l'objectif de modéliser le comportement du sein en modes statiques et dynamiques.

Participants: H. Borouchaki, P.-L. George [correspondant], P. Laug, L. Marechal

On étudie les conditions assurant la validité géométrique des élémenst finis usuels de degré 1 et 2. La formulation éléments finis mne conduisant pas toujours à une conclusion simple, on formule les éléments finis sous leur forme de Bézier. Ceci conduit à exhiber des conditions suffisantes (parfois nécessaires et suffisantes) de validité des éléments, c'est-à-dire de la positivité de leur jacobien. Pour les éléments de degré 2, on donne l'interprétion géométrique de ces conditions. Les éléments étudiés sont le triangle à 3 nœuds, le triangle à 6 nœuds, le quadrilatère à 4 nœuds et les quadrilatère à 8 et 9 nœuds, le tétraèdre à 4 n|ouuds et le tétraèdre à 10 nœuds puis les hexaèdres à 8, 27 et 20 nœuds.

Le cas du simplexe de dimension quelconque et de degré quelconque est traité.

Quelques remarques sur les maillages de surface de degré 2 indiquent quelques pistes à suivre ou à éviter lors de la construction de tels éléments.

Participants: J. Castelneau, A. Loseille [correspondant], L. Maréchal

Dans le cadre du projet ILab, des nouveaux algorithmes de visualisation et de modifications interactives des maillages courbes et hybrides ont été développés. En effet, une des principales difficultés dans la génération de maillages courbes reste la visualisation. Il est également nécessaire de disposer d'algorithmes de corrections interactifs car les maillages de surfaces initiaux (de degré 2) sont pour la plupart faux.