# Find GBPD (Metric-Based Approach)¶

## Group (Subgroup)¶

Statistics (Crystallographic)

## Description¶

This Filter computes the grain boundary plane distribution (GBPD) like that shown in Fig. 1. It should be noted that most GBPDs presented so far in literature were obtained using a method based on partition of the grain boundary space into bins, similar to that implemented in the Find GBCD Filter. This Filter calculates the GBPD using an alternative approach adapted from the one included in the Find GBCD (Metric-based Approach) Filter and 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). Briefly, the GBPD is probed at evenly distributed sampling directions (similarly to Find GBCD (Metric-based Approach) Filter) and areas of mesh segments with their normal vectors deviated by less than a limiting angle ρp from a given direction are summed. If nS is the number of crystal symmetry transformations, each boundary plane segment is represented by up to 4 × nS equivalent vectors, and all of them are processed. It is enough to sample the distribution at directions corresponding to the standard stereographic triangle (or, in general, to a fundamental region corresponding to a considered crystallographic point group); values at remaining points are obtained based on crystal symmetries. After summing the boundary areas, the distribution is normalized. First, the values at sampling vectors are divided by the total area of all segments. Then, in order to express the distribution in the conventional units, i.e., multiples of random distribution (MRDs), the obtained fractional values are divided by the volume v = (A nS) / (4π), where A is the area of a spherical cap determined by ρp. 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, and n stands for the number of grain boundaries (not the number of mesh triangles) in the considered network. The errors can be calculated either as their absolute values, i.e., ε × f or as relative errors, i.e., 100% × ε. The latter are computed in a way that if the relative error exceeds 100%, it is rounded down to 100%.

## Parameters¶

Name Type Description
Phase of Interest int32_t Index of the Ensemble for which to compute GBPD; boundaries having grains of this phase on both its sides will only be taken into account
Limiting Distance float ρp as defined above
Number of Sampling Points int32_t The approximate number of sampling directions
Exclude Triangles Directly Neighboring Triple Lines bool Only interiors of Faces are included in GBPD
Output Distribution File File Path The output file path (extension .dat, GMT format)
Output Distribution Errors File File Path The output file path (extension .dat, GMT format)
Save Relative Errors Instead of Their Absolute Values bool What type of errors to save (see above description for more detail)

Image + Triangle

## Required Objects¶

Kind Default Name Type Component Dimensions Description
Ensemble Attribute Array CrystalStructures uint32_t (1) Enumeration representing the crystal structure for each Ensemble
Feature Attribute Array AvgEulerAngles float (3) Three angles defining the orientation of the Feature in Bunge convention (Z-X-Z)
Feature Attribute Array Phases int32_t (1) Specifies to which phase each Feature belongs
Face Attribute Array FaceLabels int32_t (2) Specifies which Features are on either side of each Face
Face Attribute Array FaceNormals double (3) Specifies the normal of each Face
Face Attribute Array FaceAreas double (1) Specifies the area of each Face
Feature Face Attribute Array FaceLabels int32_t (2) Specifies to which phase each Face Feature belongs
Vertex Attribute Array NodeTypes int8_t (1) Specifies the type of node in the Geometry

## Format of Output Files¶

Output files are formatted to be readable by GMT plotting program. The first line is always "0.0 0.0 0.0 0.0". 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

 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¶

Please see the description file distributed with this Plugin.