12.16. Compute GBCD (Metric-Based Approach)

Group (Subgroup)

Statistics (Crystallography)

Description

This Filter computes a section through the five-dimensional grain boundary character distribution (GBCD) for a fixed misorientation, using a metric-based approach rather than binning. An example of such a section is shown in Fig. 1.

Unlike the Compute GBCD filter, which partitions the boundary space into discrete bins, this filter uses a continuous metric-based approach described by K. Glowinski and A. Morawiec in Analysis of experimental grain boundary distributions based on boundary-space metrics, Metall. Mater. Trans. A 45, 3189-3194 (2014). The metric-based approach avoids discretization artifacts and can produce smoother distributions, especially with limited data.

The metric-based approach defines distance measures in the boundary space so that boundaries with similar geometry are considered “close” and boundaries with different geometry are considered “distant,” taking crystal symmetry into account. The procedure has two stages:

1. Misorientation selection: Boundary segments whose misorientation is within a limiting angular distance ρ_m of the user-specified fixed misorientation are selected.

2. Normal probing: The distribution is sampled at evenly distributed normal directions (see Fig. 2). For each sampling direction, the areas of boundaries whose normals fall within ρ_p of that direction are summed.

The limiting distances ρ_m and ρ_p should be chosen based on the resolution, amount, and quality of the data. The result is normalized and expressed in multiples of a random distribution (MRD).

Image
Fig. 2: End-points (drawn in stereographic projection) of sampling directions used for probing distribution values; the number of points here is about 1500. Additionally, distributions are probed at points lying at the equator (marked with red); this is helpful for some plotting software.

This Filter also calculates statistical errors of the distributions using the formula

ε = ( f n v )^1/2

where ε is the relative error of the distribution function at a given point, f is the value of the function at that point, n stands for the number of grain boundaries (not the number of mesh triangles) in the considered network, and v denotes the volume restricted by ρ_m and ρ_p. The errors can be calculated either as their absolute values, i.e., ε × f (Fig. 3a) or as relative errors, i.e., 100% × ε (Fig. 3b). The latter are computed in a way that if the relative error exceeds 100%, it is rounded down to 100%.

Image

Format of Output Files

Output files are formatted to be readable by GMT plotting program. The first line contains the fixed misorientation axis and angle. Each of the remaining lines contains three numbers. The first two columns are angles (in degrees) describing a given sampling direction; let us denote them col₁ and col₂, respectively. The third column is either the value of the GBCD (in MRD) for that direction or its error (in MRD or %, depending on user’s selection). If you use other software, you can retrive spherical angles θ and φ of the sampling directions in the following way:

θ = 90° - col₁

φ = col₂

Then, the directions are given as [ sin θ × cos φ , sin θ × sin φ , cos θ ].

Feedback

In the case of any questions, suggestions, bugs, etc., please feel free to email the author of this Filter at kglowinski at ymail.com

Limiting Distances

The Limiting Distances parameter selects the maximum angular deviations used when selecting boundary segments. The choices are pairs of (misorientation radius, plane-inclination radius):

• 3 deg. Misorientations; 7 deg. Plane Inclinations [0]: Use a 3-degree misorientation radius and a 7-degree plane-inclination radius.

• 5 deg. Misorientations; 5 deg. Plane Inclinations [1]: Use a 5-degree radius for both misorientations and plane inclinations.

• 5 deg. Misorientations; 7 deg. Plane Inclinations [2]: Use a 5-degree misorientation radius and a 7-degree plane-inclination radius.

• 5 deg. Misorientations; 8 deg. Plane Inclinations [3]: Use a 5-degree misorientation radius and an 8-degree plane-inclination radius.

• 6 deg. Misorientations; 7 deg. Plane Inclinations [4]: Use a 6-degree misorientation radius and a 7-degree plane-inclination radius.

• 7 deg. Misorientations; 7 deg. Plane Inclinations [5]: Use a 7-degree radius for both misorientations and plane inclinations.

• 8 deg. Misorientations; 8 deg. Plane Inclinations [6]: Use an 8-degree radius for both misorientations and plane inclinations.

Required Input Sources

This filter operates on a grain-boundary surface mesh and requires the following upstream steps:

• Triangle Geometry with Face Labels, Face Normals, Face Areas, and Node Types – produced by a surface meshing filter such as Quick Surface Mesh, followed by Compute Triangle Normals and Compute Triangle Areas.

• Feature Euler Angles / Feature Phases – feature-level averages produced by Compute Average Orientations and Compute Feature Phases.

• Crystal Structures – ensemble-level array read from EBSD data or created by Create Ensemble Info.

Input Parameter(s)

Parameter Name	Parameter Type	Parameter Notes	Description
Phase of Interest	Scalar Value	Int32	Index of the Ensemble for which to compute GBCD; boundaries having grains of this phase on both its sides will only be taken into account
Fixed Misorientation	Vector of Float32 Values	Order=x,y,z,w	Axis-angle representation of the misorientation of interest. Angle value should be in degrees.
Limiting Distances	Choices		The max angles from within which boundary segments are selected for the misorientations and plane inclinations
Number of Sampling Points (on a Hemisphere)	Scalar Value	Int32	The approximate number of sampling directions
Exclude Triangles Directly Neighboring Triple Lines	Bool		If checked, only interiors of Faces are included in GBCD
Save Relative Errors Instead of Their Absolute Values	Bool		Whether or not to save the distribution errors as relative (if exceeds 100%, then rounded down) or absolute
Triangle Geometry	Geometry Selection	Triangle	The complete path to the triangle geometry

Input Triangle Geometry Vertex Data

Parameter Name	Parameter Type	Parameter Notes	Description
Node Types	Array Selection	Allowed Types: int8 Comp. Shape: 1	Specifies the type of node in the Geometry

Input Triangle Geometry Face Data

Parameter Name	Parameter Type	Parameter Notes	Description
Face Labels	Array Selection	Allowed Types: int32 Comp. Shape: 2	Specifies which Features are on either side of each Face
Face Normals	Array Selection	Allowed Types: float64 Comp. Shape: 3	Specifies the normal of each Face
Face Areas	Array Selection	Allowed Types: float64 Comp. Shape: 1	Specifies the area of each Face

Input Triangle Geometry Face Feature Data

Parameter Name	Parameter Type	Parameter Notes	Description
Feature Face Labels	Array Selection	Allowed Types: int32 Comp. Shape: 2	Specifies which original Features are on either side of each boundary Feature

Input Feature Data

Parameter Name	Parameter Type	Parameter Notes	Description
Average Euler Angles	Array Selection	Allowed Types: float32 Comp. Shape: 3	Three angles defining the orientation of the Feature in Bunge convention (Z-X-Z)
Phases	Array Selection	Allowed Types: int32 Comp. Shape: 1	Specifies to which phase each Feature belongs

Input Ensemble Data

Parameter Name	Parameter Type	Parameter Notes	Description
Crystal Structures	Array Selection	Allowed Types: uint32 Comp. Shape: 1	Enumeration representing the crystal structure for each Ensemble

Output Parameters

Parameter Name	Parameter Type	Parameter Notes	Description
Output Distribution File	FileSystemPath		The output distribution file path (extension .dat, GMT format)
Output Distribution Errors File	FileSystemPath		The output distribution errors file path (extension .dat, GMT format)

References

[1] K. Glowinski and A. Morawiec, Analysis of experimental grain boundary distributions based on boundary-space metrics, Metall. Mater. Trans. A 45, 3189-3194 (2014)

Example Pipelines

(05) SmallIN100 GBCD Metric.d3dpipeline

This pipeline depends on previous pipelines in the Small IN100 reconstruction pipeline series.

License & Copyright

Please see the description file distributed with this Plugin.

DREAM3D-NX Help

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