TY - JOUR
T1 - Towards the efficient computation of effective properties of microstructured materials
AU - Kreiner, Carl Friedrich
AU - Zimmer, Johannes
AU - Chenchiah, Isaac V.
N1 - Funding Information:
This paper contains a part of CFK’s Diploma thesis [14], written under the supervision of JZ. The work was carried out while JZ was a postdoctoral scholar and IVC a graduate student at the California Institute of Technology. CFK was a Diploma student in mathematics at the Technische Universität München, Germany and visited Caltech as a special student. His stay was financed by the Freistaat Bayern through a stipend of the Bayerische Begabtenförderung. He thanks Kaushik Bhattacharya for support, and the Division of Engineering and Applied Sciences at Caltech for hospitality. The financial support of an NSF-ITR grant (ACI-0204932) is gratefully acknowledged. We thank Kaushik Bhattacharya for encouragement and generous support. The results were presented on the NATO Advanced Research Workshop ‘Nonlinear Homogenization and its Application to Composites, Polycrystals and Smart Materials’ in Kazimierz Dolny, Poland.
PY - 2004/3
Y1 - 2004/3
N2 - 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 (T 4-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. To cite this article: C.-F. Kreiner et al., C. R. Mecanique 332 (2004).
AB - 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 (T 4-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. To cite this article: C.-F. Kreiner et al., C. R. Mecanique 332 (2004).
KW - Configuration de type T
KW - Continuum mechanics
KW - Enveloppe convexe de rang 1
KW - Milieux continus
KW - Rank-one convex hull
KW - T -configuration
UR - http://www.scopus.com/inward/record.url?scp=1542408613&partnerID=8YFLogxK
U2 - 10.1016/j.crme.2004.01.011
DO - 10.1016/j.crme.2004.01.011
M3 - Article
AN - SCOPUS:1542408613
SN - 1631-0721
VL - 332
SP - 169
EP - 174
JO - Comptes Rendus - Mecanique
JF - Comptes Rendus - Mecanique
IS - 3
ER -