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The average grain radius is about 1µm. 3D Voronoi tessellations usually include large numbers of relatively small entities such as small Voronoi grid edges and faces which can result in a refined mesh.
The mesh refinements can increase the total number of elements and significantly increase the time needed for explicit computation.
Because the crystallographic grain orientations are random, grain will show elastic anisotropy [4].
Some cohesive elements are embedded into grains; others are embedded along grain boundary.
Kgrain and Kgb denotes the K value of grain and grain boundary respectively.

A UFG structure produced by ECAP has a large volume fraction of grain boundaries, as compared with a coarse-grained (CG) structure, and is also characterized by a high concentration of defects (point and linear ones) in grain boundaries and in their vicinity, with a decreasing number of dislocations inside grains [1].
(5) A decrease in the density of a substance participating in the mass transfer can take place due to an increase in the number of dislocations and vacancies.
Their average grain size was 15, 0.3 and 0.1 µm, respectively (fig. 2).
Their average grain size was 50 and 0.3 µm, in the initial condition and in the ECAP-processed condition, respectively.
The smaller is the grain size and the higher is the contact temperacture, the stronger is this effect.

Grain refinement is directly related to the number of active nuclei in the melt.
If the melt temperature is low enough to avoid dissolution of the embryos, their life time is long enough to promote a high density of nuclei in the melt, thus leading to the development of large number of globular grains with small grain size [10].
The sample presents mainly homogeneous a-Al globular grains with average grain size around 50 mm and a few rosette like grains.
Isothermal processing at 620 and 640ºC led to the formation of more heterogeneous structures, with larger a-Al grains and higher number of rosettes.
In this case, due the high cooling rate of the melt, the number of nuclei developed by cavitation was smaller, therefore grain refinement was not so efficient.

The result showed that it had excellent grain refining performance for commercially pure aluminum in 800˚C.
So far, AlTiB master alloy is still the most commonly used grain refiner in Al and Al alloys [3].
Research has shown that, AlTiC master alloy is considered as the most promising grain refiner of Al and Al alloy[6].
(a) 740˚C (b)800˚C (c)900˚C Figure 5 Refinement picture Figure 5 shows, AlTiC master alloy prepared at 800 ℃ has the best refinement, all of equiaxed grain, size can reach an aluminum (average grain size 0.026mm2) requirements; at 740 ℃, the temperature is low, refinement of AlTiC is bad, most of columnar crystals; at 900 ℃, the temperature is high, morphological of TiAl3 in the AlTiC master alloy changed, lowered refining effect, the center sites were equiaxed, columnar grain edges.
Reif, Development of Al-Ti-C grain refiners containing Tic.

This research looks at the recovery and recrystallization processes during annealing after cryo-rolling, and compares the formation of sub-grains and recrystallized grains to those where rolling was performed at room temperature.
During subsequent annealing, the driving force for recrystallization is increased with a decrease in the rolling temperature, with the cryo-rolled material having a greater number of nucleation seed and consequently, a fine grained recrystallized microstructure.
The 75% cryo-rolled sample displays flat, elongated grains and shows a hardness of 50HV, while the 75% cold rolled sample shows a less-distorted grain morphology and has a hardness of 47HV.
Recrystallized grain is circled.
This UFG structure does still contain cell boundaries within some of the grains.

The wüstite grains in the grain growth stage have a columnar cell structure.
Our measurements indicate that the number fraction of high angle boundaries with rotation angle θ > 25 o decreased and the fraction of low angle boundaries with rotation angle θ ≤ 25 o increased during the continuous oxidation process.
In the LC steel sample , the ratio of the number of magnetite grains to the number of wüstite grains at the interface is less than in the pure Fe samples and the size of the magnetite grains is larger than in the pure Fe samples.
These grains have a wedge shape and are embedded within the magnetite grains.
Moreover, the hematite grains may span over two magnetite grains.

The affair number of transaction item set in data set is called supporting number, which is recorded as.
(2) Reducing the number of support degree computing, so the number is close to the frequent itemsets number
(3) Using the number of sub itemsets to get the maximum decomposition
Granular computing table Grain Information grain Binary system Grain size [A] {002,004} 10100000 2 [B] {001,004,005,007,008,009} 1E+08 6 [C] {003,005,006,007,008,009} 1011111 6 [D] {001,002,003,004,006,008,009} 1.11E+08 7 [E] {001,008} 1E+08 2 From the table we can see, all the grain size reaches the minimum support degree, so they usually are one group.
Two groups of frequent item set table after granular computing Grain Information grain Binary system Grain size [A] {002,004} 10100000 2 [B] {001,004,005,007,008,009} 1E+08 6 [C] {003,005,006,007,008,009} 1011111 6 [D] {001,002,003,004,006,008,009} 1.11E+08 7 [E] {001,008} 1E+08 2 The number of itemsets is obtained by the new frequent itemsets combination: [B,C,D], [B,C,E], [B,D,E].

To satisfy this condition, the operation of at least 5 independent slip systems in each grain is necessary or having a sufficiently large number of grain boundaries, allowing various shifts and rotations that take place, e.g., during the superplastic flow, is necessary [7].
Doing so we take into account that the size of sub-grains is much smaller than that of grains.
We replace the sum of cosines of angles by the product of the number of boundaries (the upper limits of the sums) and the cosine of the average angle.
Then the number of boundaries is defined as М g = L/ dg and M f = L/ df, where dg and df are the average size of grains and low-angle fragments, respectively.
Size of low-angle fragments and grains.

The grain is distributed with high-density β″ phases, and the grain boundary phases are spherical and intermittently distributed.
And from a quantitative point of view, β" The number of phases has the highest density.
The kind, morphology, and the number of strengthening phase can effectively affect the final comprehensive properties of the alloy.
To obtain good IGC resistance, the grain boundary is usually required.
At this time, a large number of continuous precipitated phases are produced at the grain boundary, forming a continuous corrosion channel.

The grains are elongated along the direction parallel to RD.
The selected area iffraction patteren indicates those grains are high-tilge grain boundaris.
(b) grain-refined sample by ARB
Hirsch pointed out that the BDT temperature depends on the number of dislocation sources and spacing along a crack front.
It indicates the BDT temperature decreases with decreasing the dislocation source spacing, that is, when the number of dislocation sources increases, the BDT temperature deceases.