M-SERVE and P-SERVE
79
statistical information is available. The marginal probability function corresponds
to a joint probability function of the parent grain size and the number of twins and
the conditional probability distributions of the twin distance and the twin thickness,
respectively. In this step the joint probability distribution in Fig. 12b is used for
parent grain and number of twins per parent, while the conditional probability
distributions are employed for the twin distance from the parent centroid and twin
thickness, respectively. In [29] it is demonstrated that for a given d and n, t is
uncorrelated with x and is assumed independent. The four dimensional distribution
space may then be approximated as:
P 0 (d, n, t, x) ≈ P 1 (d, n) P 2 (x|d = D, n = N) P 3 (t|d = D, n = N)
(19)
where D and N correspond to maximum sample sizes for grain size and number of
twins, respectively.
For generating twin descriptors in step 6, the probability distributions P 1 , P 2 , P 3
are sampled, and a twin is inserted into the parent grain SEVM with selected
characteristics described in [29]. The steps in this algorithm are as follows:
1. With an acceptance-rejection algorithm [54], sample the joint probability distribution for the number of twins and grain diameter and the conditional probability
distributions for the twin thickness and twin distance, using the statistics obtained
from the EBSD data.
2. Upon determination of n, d, t and x, the plane represented by the Miller index
(111) at a distance x from the parent centroid is located.
3. The voxels that belong to selected parent grain and are within a distance
t
2 from
the mid-thickness (111) plane are identified via a search algorithm.
4. The rotation matrix of the twin in the specimen frame R is obtained from the
rotation matrix of the parent grain R parent and the rotation matrix of the twin with
respect to the parent grain R twin as:
R = R parent R twin
(20)
5. The Bunge Euler angles of voxels that belong to the twins are determined by the
components of R.
6. By repeating the above steps, a set of twins is inserted in the parent grain
microstructure. Finally, the statistically equivalent virtual microstructure is
reconstructed using the new set of voxel Euler angles.
This virtual microstructure generation procedure becomes the basis for the
construction of the polycrystalline M-SERVE.
3.2.1 Validation of the SEVM Generation Method
The statistically equivalent virtual microstructure generation algorithm is implemented in a computer code that interfaces with the DREAM.3D software. The
model and algorithms are validated using the EBSD data of the Ni-based superalloy
79
statistical information is available. The marginal probability function corresponds
to a joint probability function of the parent grain size and the number of twins and
the conditional probability distributions of the twin distance and the twin thickness,
respectively. In this step the joint probability distribution in Fig. 12b is used for
parent grain and number of twins per parent, while the conditional probability
distributions are employed for the twin distance from the parent centroid and twin
thickness, respectively. In [29] it is demonstrated that for a given d and n, t is
uncorrelated with x and is assumed independent. The four dimensional distribution
space may then be approximated as:
P 0 (d, n, t, x) ≈ P 1 (d, n) P 2 (x|d = D, n = N) P 3 (t|d = D, n = N)
(19)
where D and N correspond to maximum sample sizes for grain size and number of
twins, respectively.
For generating twin descriptors in step 6, the probability distributions P 1 , P 2 , P 3
are sampled, and a twin is inserted into the parent grain SEVM with selected
characteristics described in [29]. The steps in this algorithm are as follows:
1. With an acceptance-rejection algorithm [54], sample the joint probability distribution for the number of twins and grain diameter and the conditional probability
distributions for the twin thickness and twin distance, using the statistics obtained
from the EBSD data.
2. Upon determination of n, d, t and x, the plane represented by the Miller index
(111) at a distance x from the parent centroid is located.
3. The voxels that belong to selected parent grain and are within a distance
t
2 from
the mid-thickness (111) plane are identified via a search algorithm.
4. The rotation matrix of the twin in the specimen frame R is obtained from the
rotation matrix of the parent grain R parent and the rotation matrix of the twin with
respect to the parent grain R twin as:
R = R parent R twin
(20)
5. The Bunge Euler angles of voxels that belong to the twins are determined by the
components of R.
6. By repeating the above steps, a set of twins is inserted in the parent grain
microstructure. Finally, the statistically equivalent virtual microstructure is
reconstructed using the new set of voxel Euler angles.
This virtual microstructure generation procedure becomes the basis for the
construction of the polycrystalline M-SERVE.
3.2.1 Validation of the SEVM Generation Method
The statistically equivalent virtual microstructure generation algorithm is implemented in a computer code that interfaces with the DREAM.3D software. The
model and algorithms are validated using the EBSD data of the Ni-based superalloy