?. {w, G. B}, L. , C. , C. Des-anodes-´-echantillon et al., 2 est utilisée afin de caractériser séparément chacune des phases de l'ensemble aléatoire (h) est estimée comme la moyenne obtenue sur les n images segmentées Le vecteur h est pris dans 10 cônes (en coordonnées cylindriques) de directions uniformément réparties dans le plan. L'´ etude de la covariance permet dans un premier temps de valider l'hypothèse d'isotropie des matériauxmatériauxétudiés Comme le suggère la Fig. 3.9, la covariance est inchangée par changement d'orientation du vecteur h. Ce comportement est observé pour l'ensemble des trois phases de l'ensemble deséchantillonsdeséchantillons De ce fait, Cette erreur est liéè a la fraction volumique (P = f i selon le formalisme défini en Sec. 2.5) et elle est calculée en considérant la totalité des n images segmentées pour chaqué echantillon. 3.3.2 Covariances Moment d'ordre deux

M. Mis, E. Oeuvre, . Le, and . De-courant-mousse-imprégnée-par-la-céramique, LST-Pores) puis laminée (Fig. 4.2a) Ils seront notés 'SF-LST-X' (pour Single Foam), o` u X représente le numéro associé associéà l'´ echantillon. D'autres ontétéontété réalisésréalisésà partir de deux mousses juxtaposées, imprégnées par la céramique puis laminés (Fig. 4.2b et Fig. 4.2c), ils seront référencées comme 'DF-LST-X' (pour Double Foam). Enfin, en raison d'un manque de conductivité de ces couches (vraisemblablement lié aux propriétés en bulk du LST utilisé dans le projet), il a ´ eté décidé d'´ etudier l'influence de l

. Cet-ajout-permet-d-'augmenter-la-conductivitéconductivitéélectronique-de-la-céramique, Leséchantillons Leséchantillons associés sont notésDF-LST-Ni-X') selon qu'ils ontétéontété réalisésréalisésà partir d'une (resp. deux) mousse(s) La remarque concernant la séparation deséchellesdeséchelles caractéristiques reste valable après l'incorporation de nickel dans la phase céramique. La céramique est toujours vue comme un milieu homogènehomogènè a l'´ echelle de la mousse. Dans un second temps, nous nous plaçonsplaçonsà l'´ echelle de la céramique de longueur caractéristique l ceramique ? 500 nm. Des acquisitions de la microstructure de ces matériauxmatériauxà cetté echelle sont disponibles (Fig. 4.4). CeséchantillonsCeséchantillons sont nommés 'LST-X', o` u X représente le numéro associéassociéà l'´ echantillon, dans le cas d'une céramique (LST-Pores) comme c'est le cas Fig, leséchantillonsleséchantillons associés sont notés 'LST-Ni-X'

. Cependant, les images obtenues (Fig. 4.1a) sont difficilement exploitables du point de vue de la segmentation en raison d'un manqué

L. Sur-les-images-acquisesàacquisesà-l-'´-echelle-de-la-mousse, M. Nicral, and . Cylindre, La phase céramique transpara??tpara??t quandàquandà elle en gris intermédiaire (avec localement des petites zones claires au sein de la céramique lorsqu'il y a présence de nickel c.f. Fig. 4.2d, Fig. 4.2e et Fig. 4.2f) La phase poreuse, traduisant l'absence de ré-´ emissionélectroniqueemissionélectronique, est en noir. On remarque qu'une fine couche sombre d'´ epaisseur < 20 nm entoure l'ensemble de la phase métallique (NiCrAl) Il s'agit d'une couche d'alumine qui se forme naturellement autour de la mousse. Dans la suite, ` a l'´ echelle de la mousse, seront considérées uniquement les images de type Fig. 4.1b, acquises suitè a la campagne d'acquisition collaborative entre le CdM et le CMM Pour les images acquisesàacquises`acquisesà l'´ echelle de la céramique (Fig. 4.4), les pores apparaissent en noir et le LST en gris. Suitè a une analyse EDXàEDXà fort grossissement (Fig. 4.3), les particules de nickel ontétéontété formellement identifiées. Elles apparaissent en gris clair/ blanc au sein de la céramique (pour leséchantillonsleséchantillons LST-Ni, seuls IST-14-i IST-14-s IST-15-i IST-15-s IST-24-i IST-24-s IST-25-i IST-25-s IST-28-i IST-28-s IST-29-i IST-29-s IST-32-i IST-32-s IST-33

A. Effectuer, = G 0 ij (q)A j (q) pour q = 0 et A i (q = 0) := E i

. Dans-c, ´ etape 3. est une boucle effectuée pour tous les modes de Fourier q o` u G 0 ij (q) l'opérateur de Green continu (voir Eq. 9.7) est calculécalculéà la volée. L'espace mémoire total alloué est constitué d'un champ vectoriel A et de la microstructuré etudiée. Le champ vectoriel A devient le lieu de stockage successif du champ de polarisation dans l'espace direct [´ etape 1.] puis dans l'espace de Fourier [´ etape 2 Le critère de convergencè a l'´ etape 5. doitêtredoitêtre revu car il n'est pas pertinent d'utiliser les critères type conservation du courant (voir Eq. 9.11) dans un algorithme in-situ, Il représente ensuite le champélectrique champélectrique E dans le domaine de Fourier [´ etape 3.] et dans l'espace direct [´ etape 4 Ces critères sont remplacés en pratique par le calcul de la différence 1. En tout point x, Effectuer A(x) := div P(x) o` u P i (x) = [?(x) ? ? 0 ][E ? grad A(x)] ; Calculer ? 1 selon l'Eq. 9.11

. Cependant, De plus, l'inversion de l'opérateur Laplacien est la seule opération effectuée dans le domaine de Fourier, elle s'effectue sous la forme d'une division par |k| 2 (sauf pour A(q = 0) = 0) [´ etape 3 Le champ scalaire A est utilisé pour stocker successivement différentes quantités. Dans un premier temps la divergence du champ de polarisation div P est exprimée dans le domaine direct [´ etape 1 La parallélisation dans l'´ etape 1. nécessite une attentionparticulì ere puisqu'il s'agit d'une opération non locale. Cependant, cette nouvelle implémentation réduit le nombre de FFTs par itération de 2 * d (o` u d ? {2; 3} représente la dimension duprobì eme) ` a 2 FFTs par itération. En plus de cetté economie notable du CPU, nous notons la réduction d, Ensuite la partie périodique du potentiel ? est calculée dans le domaine de Fourier [´ etape 3.] puis dans le domaine réel [´ etape 4 Nous avons implémenté cette stratégie dans le code Fortran parallélisé

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