Construction of weighted crystallographic orientations capturing a given orientation density function
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- 作者：Helmut Schaeben ; Florian Bachmann…
- 刊名：Journal of Materials Science
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：52
- 期：4
- 页码：2077-2090
- 全文大小：
- 刊物类别：Chemistry and Materials Science
- 刊物主题：Materials Science, general; Characterization and Evaluation of Materials; Polymer Sciences; Continuum Mechanics and Mechanics of Materials; Crystallography and Scattering Methods; Classical Mechanics;
- 出版者：Springer US
- ISSN：1573-4803
- 卷排序：52
文摘
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 \(10\,\%\).