78
S. Ghosh et al.
Grain equivalent sphere
with diameter, d
Twins
Parent grain
N um be r of T w in s
0
2
4
6
0.025
(a)
(b)
0.015
0.02
0.01
0.005
0 20 40 60 80 100
P a re n t gr a in d ia m e te r (µ m )
Probability density function
Fig. 12 (a) Schematic of the correlation between twins and parent grain; the parent grain has
equivalent diameter d and contains twins with thickness of t, located at a distance x from the parent
grain centroid, and (b) joint probability distribution of the parent grain size (μm) and number of
twins. (Reprinted from: Bagri [29], with permission from Springer)
in [29]. A schematic representing the relation of a twin with the parent grain is
shown in Fig. 12a. There is a wide range of variation in the size and crystallographic
orientations of the twinned grains, as well as the number of twins per grain and their
thickness. Statistical analysis identifies four variables are needed to characterize the
distributions of twins within grains, viz., the parent grain effective diameter d, the
number of twins in parent grain n, the minimum distance x of the twin from the
parent grain centroid, and the twin thickness t. A four dimensional joint probability
distribution P 0 (x, t, n, d) is employed to describe the probability of observing a
twin, given these four parameters. Due to the dependency between these twin
descriptor variables, this joint probability distribution cannot be easily approximated
by sampling from four independent marginal distributions, and therefore, the full
joint distribution must be used to capture the appropriate intercorrelations.
A novel algorithm is developed in [29] for inserting twins in parent grain
microstructure following the joint and conditional probability distribution functions
derived from the 3D EBSD assembly maps. The joint probability distribution of
parent grain size and number of twins and the conditional probability distributions
of both twin distance from parent centroid and twin thickness are expressed as:
P 1 (d, n) =
P 0 (d, n, t, x) dt dx joint PD
(17)
P 2 (x|d = D, n = N) and P 3 (t|d = D, n = N) conditional PD
(18)
where PD refers to the probability distribution.
The process first samples the joint probability distribution space for the number
of twins and the conditional probability distribution space for the twin thickness
and twin distance using statistics from the EBSD data. The four dimensional
probability distribution P 0 (d, n, t, x) necessitates a large dataset to be constructed.
Alternatively, a marginal probability function may be used when a smaller set of
S. Ghosh et al.
Grain equivalent sphere
with diameter, d
Twins
Parent grain
N um be r of T w in s
0
2
4
6
0.025
(a)
(b)
0.015
0.02
0.01
0.005
0 20 40 60 80 100
P a re n t gr a in d ia m e te r (µ m )
Probability density function
Fig. 12 (a) Schematic of the correlation between twins and parent grain; the parent grain has
equivalent diameter d and contains twins with thickness of t, located at a distance x from the parent
grain centroid, and (b) joint probability distribution of the parent grain size (μm) and number of
twins. (Reprinted from: Bagri [29], with permission from Springer)
in [29]. A schematic representing the relation of a twin with the parent grain is
shown in Fig. 12a. There is a wide range of variation in the size and crystallographic
orientations of the twinned grains, as well as the number of twins per grain and their
thickness. Statistical analysis identifies four variables are needed to characterize the
distributions of twins within grains, viz., the parent grain effective diameter d, the
number of twins in parent grain n, the minimum distance x of the twin from the
parent grain centroid, and the twin thickness t. A four dimensional joint probability
distribution P 0 (x, t, n, d) is employed to describe the probability of observing a
twin, given these four parameters. Due to the dependency between these twin
descriptor variables, this joint probability distribution cannot be easily approximated
by sampling from four independent marginal distributions, and therefore, the full
joint distribution must be used to capture the appropriate intercorrelations.
A novel algorithm is developed in [29] for inserting twins in parent grain
microstructure following the joint and conditional probability distribution functions
derived from the 3D EBSD assembly maps. The joint probability distribution of
parent grain size and number of twins and the conditional probability distributions
of both twin distance from parent centroid and twin thickness are expressed as:
P 1 (d, n) =
P 0 (d, n, t, x) dt dx joint PD
(17)
P 2 (x|d = D, n = N) and P 3 (t|d = D, n = N) conditional PD
(18)
where PD refers to the probability distribution.
The process first samples the joint probability distribution space for the number
of twins and the conditional probability distribution space for the twin thickness
and twin distance using statistics from the EBSD data. The four dimensional
probability distribution P 0 (d, n, t, x) necessitates a large dataset to be constructed.
Alternatively, a marginal probability function may be used when a smaller set of