Construction of weighted crystallographic orientations capturing a given orientation density function

To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initiall...

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Bibliographic Details
Published in:Journal of materials science Vol. 52; no. 4; pp. 2077 - 2090
Main Authors: Schaeben, Helmut, Bachmann, Florian, Fundenberger, Jean-Jacques
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2017
Springer
Springer Nature B.V
Subjects:
Characterization and Evaluation of Materials
Chemistry and Materials Science
Classical Mechanics
Computer simulation
Crystallites
crystallization
Crystallography
Crystallography and Scattering Methods
data collection
Datasets
Deformation
Density
Dirichlet problem
Economic models
Fourier series
Fourier transforms
Kernels
least squares
Linearity
Materials Science
mathematical models
Numerical analysis
Orientation
Original Paper
Polymer Sciences
Recrystallization
Series expansion
Solid Mechanics
Specific gravity
Truncation errors
zirconium
Orientation Density Function (ODF)
Poussin Kernel
Kernel Density Estimate
Dirichlet Kernel
Electron Back Scatter Diffraction (EBSD)
ISSN:0022-2461, 1573-4803
Online Access:Get full text
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Summary:To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function and in particular its non-negativity, or to preserve the unbiased estimates of the first Fourier coefficients up to a given finite order. Methods are presented how to construct a much smaller set of weighted orientations such that their kernel density estimate approximates the initial estimate. To preserve its overall shape the de la Vallée Poussin kernel is applied as it is the only known non-negative kernel with a finite Fourier series expansion avoiding truncation errors. If the first Fourier coefficients are to be preserved the Dirichlet kernel applies as it is the only kernel providing unbiased estimates of the Fourier coefficients up to any given finite order. The weights are determined numerically by resolving a least squares or a maximum likelihood problem. Due to the linearity of kernel density estimation and the Fourier transform the approaches in spatial and spectral domain are related to each other in a unique complementary way. For an exemplary practical application we use a large EBSD dataset of about 80.000 orientations from a recrystallized low alloyed Zirconium sheet. Our methods reduce the size of the dataset by about 99.75 % to the order of 200 weighted orientations supporting a secondary approximate distribution with a volume portion of crystallites oriented differently than initially of less than 10 % .
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ISSN:0022-2461
1573-4803
DOI:10.1007/s10853-016-0496-1