Optimization of the branching pattern in coherent phase transitions

Patrick Dondl; Behrend Heeren; Martin Rumpf

doi:10.1016/j.crma.2016.03.013

Mathematical problems in mechanics/Calculus of variations

Optimization of the branching pattern in coherent phase transitions

Presented by: the Editorial Board

Patrick Dondl ¹; Behrend Heeren ²; Martin Rumpf ²

¹ Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg im Breisgau, Germany
² Institute for Numerical Simulation, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany

Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 639-644.

Abstract
Résumé

Branching can be observed at the austenite–martensite interface of martensitic phase transformations. For a model problem, Kohn and Müller studied a branching pattern with optimal scaling of the energy with respect to its parameters. Here, we present finite element simulations that suggest a topologically different class of branching patterns and derive a novel, low-dimensional family of patterns. After a geometric optimization within this family, the resulting pattern bears a striking resemblance to our simulation. The novel microstructure admits the same scaling exponents, but results in a significantly lower upper energy bound.

Un motif de ramification est observé à l'interface austénite–martensite au cours de la transformation de phase martensitique. Kohn et Müller étudient un motif de ramification qui reflète les exposants optimaux de l'énergie en fonction de ses paramètres. Nous présentons ici des simulations par la méthode des éléments finis qui suggèrent une classe de ramifications ayant une topologie différente, et déduisons une nouvelle famille de ramifications, de faibles dimensions. Après optimisation géométrique au sein de cette famille, le motif résultant présente une ressemblance remarquable avec les résultats de notre simulation. Le nouveau motif possède les mêmes exposants d'échelle optimaux, mais fournit une constante significativement inférieure dans le contrôle de l'énergie.

Received: 2015-12-18
Accepted: 2016-03-17
Published online: 2016-04-16

DOI: 10.1016/j.crma.2016.03.013

Author's affiliations:

Patrick Dondl ¹; Behrend Heeren ²; Martin Rumpf ²

¹ Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg im Breisgau, Germany
² Institute for Numerical Simulation, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany

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     title = {Optimization of the branching pattern in coherent phase transitions},
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Patrick Dondl; Behrend Heeren; Martin Rumpf. Optimization of the branching pattern in coherent phase transitions. Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 639-644. doi : 10.1016/j.crma.2016.03.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.03.013/

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