[Optimisation d'un motif de ramification au cours de la transformation de phase martensitique]
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.
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.
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@article{CRMATH_2016__354_6_639_0,
author = {Dondl, Patrick and Heeren, Behrend and Rumpf, Martin},
title = {Optimization of the branching pattern in coherent phase transitions},
journal = {Comptes Rendus. Math\'ematique},
pages = {639--644},
publisher = {Elsevier},
volume = {354},
number = {6},
year = {2016},
doi = {10.1016/j.crma.2016.03.013},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.03.013/}
}
TY - JOUR AU - Dondl, Patrick AU - Heeren, Behrend AU - Rumpf, Martin TI - Optimization of the branching pattern in coherent phase transitions JO - Comptes Rendus. Mathématique PY - 2016 SP - 639 EP - 644 VL - 354 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.03.013/ DO - 10.1016/j.crma.2016.03.013 LA - en ID - CRMATH_2016__354_6_639_0 ER -
%0 Journal Article %A Dondl, Patrick %A Heeren, Behrend %A Rumpf, Martin %T Optimization of the branching pattern in coherent phase transitions %J Comptes Rendus. Mathématique %D 2016 %P 639-644 %V 354 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2016.03.013/ %R 10.1016/j.crma.2016.03.013 %G en %F CRMATH_2016__354_6_639_0
Dondl, Patrick; Heeren, Behrend; Rumpf, Martin. Optimization of the branching pattern in coherent phase transitions. Comptes Rendus. Mathématique, Tome 354 (2016) no. 6, pp. 639-644. doi : 10.1016/j.crma.2016.03.013. http://www.numdam.org/articles/10.1016/j.crma.2016.03.013/
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