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Reconstruction of Grains in Polycrystalline Materials From Incomplete Data Using Laguerre Tessellations

Published online by Cambridge University Press:  30 April 2019

Lukas Petrich*
Affiliation:
Institute of Stochastics, Faculty of Mathematics and Economics, Ulm University, 89069 Ulm, Germany
Jakub Staněk
Affiliation:
Department of Mathematics Education, Faculty of Mathematics and Physics, Charles University, 18675 Prague, Czech Republic
Mingyan Wang
Affiliation:
Institute of Functional Nanosystems, Faculty of Engineering, Computer Science and Psychology, Ulm University, 89081 Ulm, Germany
Daniel Westhoff
Affiliation:
Institute of Stochastics, Faculty of Mathematics and Economics, Ulm University, 89069 Ulm, Germany
Luděk Heller
Affiliation:
Institute of Physics, Academy of Sciences of Czech Republic, 18221 Prague, Czech Republic
Petr Šittner
Affiliation:
Institute of Physics, Academy of Sciences of Czech Republic, 18221 Prague, Czech Republic
Carl E. Krill III
Affiliation:
Institute of Functional Nanosystems, Faculty of Engineering, Computer Science and Psychology, Ulm University, 89081 Ulm, Germany
Viktor Beneš
Affiliation:
Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University, 18675 Prague, Czech Republic
Volker Schmidt
Affiliation:
Institute of Stochastics, Faculty of Mathematics and Economics, Ulm University, 89069 Ulm, Germany
*
*Author for correspondence: Lukas Petrich, E-mail: lukas.petrich@uni-ulm.de

Abstract

Far-field three-dimensional X-ray diffraction microscopy allows for quick measurement of the centers of mass and volumes of a large number of grains in a polycrystalline material, along with their crystal lattice orientations and internal stresses. However, the grain boundaries—and, therefore, individual grain shapes—are not observed directly. The present paper aims to overcome this shortcoming by reconstructing grain shapes based only on the incomplete morphological data described above. To this end, cross-entropy (CE) optimization is employed to find a Laguerre tessellation that minimizes the discrepancy between its centers of mass and cell sizes and those of the measured grain data. The proposed algorithm is highly parallel and is thus capable of handling many grains (>8,000). The validity and stability of the CE approach are verified on simulated and experimental datasets.

Type
Software and Instrumentation
Copyright
Copyright © Microscopy Society of America 2019 

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