Symmetrized Bingham distribution for representing texture: parameter estimation with respect to crystal and sample symmetries

Niezgoda, Stephen R.; Magnuson, Eric A.; Glover, Jared

doi:10.1107/s160057671600649x

Cited by 6 publications

(5 citation statements)

References 25 publications

(36 reference statements)

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“…Niezgoda et al (2016) propose modelling the data‐generating mechanism via a density function that is invariant under group operations representing crystal and specimen symmetries. In this setting, an appropriate density function must be invariant for the equivalence class

false[g false] = false{q_{c} * g * q_{s} false| q_{s} \in Q_{s}, q_{c} \in Q_{c} false}

where ∗ is a binary operator representing quaternion multiplication and Q c and Q s are groups representing the crystal and specimen symmetries of a crystal, respectively.…”

Section: Model Formulationmentioning

confidence: 99%

“…However, this model may only be adequate if the distribution of the orientations does not contain symmetries, which are a feature of the ODFs of polycrystalline materials of interest. Niezgoda et al (2016) propose modelling the data-generating mechanism via a density function that is invariant under group operations representing crystal and specimen symmetries. In this setting, an appropriate density function must be invariant for the equivalence class…”

Section: Model Formulationmentioning

confidence: 99%

“…Specimen, sample, or statistical symmetry reflects symmetry of the sample that is maintained during processing operations, such as uniaxial tension of cylindrical sample or rolling reduction of a rectangular plate (Bunge, 2013). This application and previous methods are described in Niezgoda and Glover (2013) and Niezgoda, Magnuson, and Glover (2016) among others, where the symmetric Bingham distribution is proposed as an appropriate model for ODFs since it is able to account for the structure of the data.…”

Section: Introductionmentioning

confidence: 99%

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Bayesian inference for polycrystalline materials

Matuk

Chkrebtii

Niezgoda

2021

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Polycrystalline materials, such as metals, are comprised of heterogeneously oriented crystals. Observed crystal orientations are modelled as a sample from an orientation distribution function (ODF), which determines a variety of material properties and is therefore of great interest to practitioners. Observations consist of quaternions, 4dimensional unit vectors reflecting both orientation and rotation of a single crystal. Thus, an ODF must account for known crystal symmetries as well as satisfy the unit length constraint. A popular method for estimating ODFs non-parametrically is symmetrized kernel density estimation. However, disadvantages of this approach include difficulty in interpreting results quantitatively, as well as in quantifying uncertainty in the ODF. We propose to use a mixture of symmetric Bingham distributions as a flexible parametric ODF model, inferring the number of mixture components, the mixture weights, and scale and location parameters based on crystal orientation data. Furthermore, our Bayesian approach allows for structured uncertainty quantification of the parameters of interest. We discuss details of the sampling methodology and conclude with analyses of various orientation datasets, interpretations of parameters of interest, and comparison with kernel density estimation methods.

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false[g false] = false{q_{c} * g * q_{s} false| q_{s} \in Q_{s}, q_{c} \in Q_{c} false}

where ∗ is a binary operator representing quaternion multiplication and Q c and Q s are groups representing the crystal and specimen symmetries of a crystal, respectively.…”

Section: Model Formulationmentioning

confidence: 99%

Section: Model Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bayesian inference for polycrystalline materials

Matuk

Chkrebtii

Niezgoda

2021

Stat

Self Cite

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show abstract

“…Clustering of crystal orientations must account for crystal symmetry, which implies that a (mis)orientation is only known up to the action of elements of the proper point group (Krakow et al, 2017b). Recently a number of authors have considered the statistics of such ambiguous rotations (Arnold et al, 2018;Chen et al, 2015a;Niezgoda et al, 2016) and hierarchical clustering of (mis)orientations in the presence of crystal symmetry has been demonstrated (Krakow et al, 2017a). Further, a model based clustering algorithm accommodating symmetry, based on a mixture of Von-Mises Fisher or Watson distributions and with parameters estimated using expectation maximization, has also been reported for orientations (Chen et al, 2015b;Chen et al, 2015a).…”

Section: Figurementioning

confidence: 99%

Density-based clustering of crystal orientations and misorientations and the orix python library

Johnstone¹,

Martineau²,

Crout³

et al. 2020

Preprint

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Crystal orientation mapping experiments typically measure orientations that are similar within grains and misorientations that are similar along grain boundaries. Such (mis)orientation data will cluster in (mis)orientation space and clusters are more pronounced if preferred orientations or special orientation relationships are present. Here, cluster analysis of (mis)orientation data is described and demonstrated using distance metrics incorporating crystal symmetry and the density based clustering algorithm DBSCAN. Frequently measured (mis)orientations are identified as corresponding to grains, grain boundaries or orientation relationships, which are visualised both spatially and in three-dimensional (mis)orientation spaces. A new open-source python library, orix, is also reported.

show abstract

“…Clustering of crystal orientations must account for crystal symmetry, which implies that a (mis)orientation is only known up to the action of elements of the proper point group (Krakow et al, 2017b). Recently a number of authors have considered the statistics of such ambiguous rotations (Arnold et al, 2018;Chen et al, 2015a;Niezgoda et al, 2016), and hierarchical clustering of (mis)orientations in the presence of crystal symmetry has been demonstrated (Krakow et al, 2017a). Furthermore, a model-based clustering algorithm accommodating symmetry, based on a mixture of von Mises-Fisher and Watson distributions and with parameters estimated using expectation maximization, has also been reported for orientations (Chen et al, 2015a,b).…”

Section: Introductionmentioning

confidence: 99%

Density-based clustering of crystal (mis)orientations and the orix Python library

et al. 2020

View full text Add to dashboard Cite

Crystal orientation mapping experiments typically measure orientations that are similar within grains and misorientations that are similar along grain boundaries. Such (mis)orientation data cluster in (mis)orientation space, and clusters are more pronounced if preferred orientations or special orientation relationships are present. Here, cluster analysis of (mis)orientation data is described and demonstrated using distance metrics incorporating crystal symmetry and the density-based clustering algorithm DBSCAN. Frequently measured (mis)orientations are identified as corresponding to similarly (mis)oriented grains or grain boundaries, which are visualized both spatially and in three-dimensional (mis)orientation spaces. An example is presented identifying deformation twinning modes in titanium, highlighting a key application of the clustering approach in identifying crystallographic orientation relationships and similarly oriented grains resulting from specific transformation pathways. A new open-source Python library, orix, that enabled this work is also reported.

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Symmetrized Bingham distribution for representing texture: parameter estimation with respect to crystal and sample symmetries

Cited by 6 publications

References 25 publications

Bayesian inference for polycrystalline materials

Bayesian inference for polycrystalline materials

Density-based clustering of crystal orientations and misorientations and the orix python library

Density-based clustering of crystal (mis)orientations and the orix Python library

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