Comptes Rendus
Towards the efficient computation of effective properties of microstructured materials
Comptes Rendus. Mécanique, Volume 332 (2004) no. 3, pp. 169-174.

An algorithm for partially relaxing multiwell energy densities, such as for materials undergoing martensitic phase transitions, is presented here. The detection of the rank-one convex hull, which describes effective properties of such materials, is carried out for the most prominent nontrivial case, namely the so-called Tk-configurations. Despite the fact that the computation of relaxed energies (and with it effective properties) is inherently unstable, we show that the detection of these hulls (T4-configurations) can be carried out exactly and with high efficiency. This allows in practice for their computation to arbitrary precision. In particular, our approach to detect these hulls is not based on any approximation or grid-like discretization. This makes the approach very different from previous (unstable and computationally expensive) algorithms for the computation of rank-one convex hulls or sequential-lamination algorithms for the simulation of martensitic microstructure. It can be used to improve these algorithms. In cases where there is a strict separation of length scales, these ideas can be integrated at a sub-grid level to macroscopic finite-element computations. The algorithm presented here enables, for the first time, large numbers of tests for T4-configurations. Stochastic experiments in several space dimensions are reported here.

Nous présentons dans cette Note un algorithme de relaxation partielle de densités d'énergie à plusieurs puits, comme pour la modélisation de matériaux subissant des transitions de phase « martensitiques ». La détection de l'enveloppe rang-un convexe, qui décrit les propriétés effectives de tels matériaux, est menée à bien pour le cas non trivial le plus connu, c'est-à-dire les configurations Tk. Bien que le calcul d'énergies relaxées (et donc de propriétés effectives) soit naturellement instable, nous montrons que la détection de ces enveloppes (configurations T4) peut être effectuée de façon exacte très efficacement. En pratique, cela permet leur calcul à une précision arbitraire. En particulier, notre approche pour la détection de ces enveloppes n'est basée sur aucune approximation ou discrétisation. Ceci la démarque des autres algorithmes (instables et coûteux) de calcul d'enveloppes rang-un convexes ou de lamination séquentielle pour la simulation de microstructures martensitiques. Notre méthode peut être utilisée pour améliorer ces derniers. Dans les cas où il y a une stricte séparation des échelles, ces idées peuvent être utilisées à un niveau inférieur dans des calculs macroscopiques de type éléments finis. La méthode présentée ici permet pour la première fois un grand nombre de tests pour la configuration T4. Nous rendons compte également d'expériences stochastiques en plusieurs dimensions.

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DOI: 10.1016/j.crme.2004.01.011
Keywords: Continuum mechanics, Rank-one convex hull, Tk-configuration
Mot clés : Milieux continus, Enveloppe convexe de rang 1, Configuration de type Tk

Carl-Friedrich Kreiner 1; Johannes Zimmer 1; Isaac V. Chenchiah 1

1 Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany
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Carl-Friedrich Kreiner; Johannes Zimmer; Isaac V. Chenchiah. Towards the efficient computation of effective properties of microstructured materials. Comptes Rendus. Mécanique, Volume 332 (2004) no. 3, pp. 169-174. doi : 10.1016/j.crme.2004.01.011. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2004.01.011/

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