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Number of faces versus relative grain size for: a – parabolic growth regime of grain boundary controlled grain growth; b – linear growth regime of triple junction controlled growth; c – exponential growth regime of quadruple junction controlled growth.
However, the rage of observed grain sizes, the maximum in the number of faces and the mean number of faces increases from normal grain growth via triple line controlled to quadruple junction controlled grain growth.
On the other hand, also from a simple argument the relation between number of quadruple points resp. triple lines and number of grain faces can be obtained.
Considering the grains of a network as polyhedra, they obey the well-known Euler theorem relating the number of faces s to the number of edges e (triple lines) and vertices v (quadruple points) by .
Number of quadruple points v vs. number of grain boundary faces s for: a – normal grain growth; b – triple junction controlled growth; c – quadruple junction controlled growth.

However, grain growth will reduce the area of grain boundary in order to reduce the grain boundary energy.
The main contents of this paper are as follows: the influence of changed interface energy in the alloy system on the growth process of recrystallized grain, the analysis of grain number and grain size.
It can be clearly seen that from Fig. 2 (a) to (d), with the increase of interfacial energy, the number of recrystallized grains is less and less, which indicates that the average grain size increases with the increase of interfacial energy under the same conditions, which is mean, the increase of interfacial energy can promote the growth of recrystallized grains.
The speed of grain boundary movement determines the grain growth rate, and the grain boundary energy is the decisive factor of grain boundary movement speed.
Conclusion The simulated results show that the number of recrystallized grains in AZ31 Mg alloy system decreases and the average grain size increases with the increase of interface energy.

(Liu [4])An uncertain variable is a measurable function from an uncertainty space (,,) to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event.
(Liu [4]) The uncertainty distribution Φ of an uncertain variable is defined by for any real number .
Model parameters : volume of products sold by farmers ;: cost of opening grain elevators ; : cost of opening final processor ;: maximum capacity of final processor ; : unit penalty cost at grain elevator ;: unit handling cost at grain elevator ; :distance from grain elevators to final processor ; : unit transportation cost ;: arbitrarily set large number; : maximum allowable distance ;m: minimum number of established grain elevator; n: minimum number of established final processor;: confidence level not over cost; C.
Constraints (7) and (8) maintain a minimum number of grain elevators and final processors.
Repeat the second to sixth steps a given number of cycles.

As documented by a number of researchers, the cell structure first recovers in the sense of eliminating dislocations of opposite sign, and certain cells grow sufficiently more quickly than all others that they become new grains [3, 4].
In this study, we varied the misorientation distribution via the orientation spread, which was quantified as the grain reference orientation deviation (GROD), defined as: (1) where N is the total number of (sub-)grains in a digital material, is the orientation of a (sub-)grain with index i, and is the central orientation over all (sub-)grains.
It is clear that the number of abnormally large grains increases with increasing GROD.
The number of observed abnormal subgrains after 107 MCS divided by the initial number of subgrains in the system, PAsGG, as a function of mean misorientation.
The use of facilities supported by the MRSEC at CMU under contract under award number DMR-0520425 is gratefully acknowledged.

The grain refinement after processing by ECAP is complex resulting in the formation of a bimodal grain structure with high fraction of fine grains already after the first or second pass.
The mean dislocation density rD calculated from Eq. 1 is plotted in Fig. 3 as a function of the number of ECAP passes.
This results in inhomogeneous microstructure in specimens after different number of rotations.
Consequently, the grain fragmentation in the centre and near Fig. 3 Dislocation density in AZ31 specimens subjected to different number of ECAP passes the edge of the specimen differs with increasing number of HPT rotations.
The difference in grain sizes is smeared out with increasing number of HPT rotations and nearly the same microstructure consisting of grains of the average size of 250 nm is observed in central (Fig. 4c) and peripheral region (Fig. 4d) of the sample after 15 HPT revolutions.

Based on the experimental results, grain growth exponent n at each peak temperature can be drawn by applying regression analysis on sequence number of (D1n-D2 n) and (t1－t2): n=3.7±0.39.
The random orientation numbers Sr, 0One is the number of MC simulation of the given metallurgical process.
In the present studies, the step number of MC simulation is from 1000 to 50,000.
Otherwise, the number of MCS might be too large to run in the computer.

The influence of the grain size distribution on the grain growth process has been studied.
The growth of a single grain in a system of grains was postulated by Hillert in 1965, [1].
Simultaneous GBM and GRC, a linear combination of both case 1 and case 2, may be given as )(),,(),( 2211 tdNtrGadttrfadr θ+ = (1) where the aiis are the "compensation coefficients" that indicate the proportion in which the two growth mechanisms combine, f(r,t)=f(r,θ,t) is a curvature driven mechanism, ),,( trG θ grain misorientation angle driven mechanism and dN(t) is the increment of the number coalescence mechanisms during an infinitesimal time interval.
Considering the fact many other mechanisms bring about the variations in the size of the grain, the grain size in the system at any time, t, is independent of the number of coalescence mechanisms (stochastic counting processes) up to that time instant.
The number of coalescence between the a grain and its neighbour is given 1 1 12)( i i i ii i tN θ θΔ θ θθ = − = (6) where 1iθ is the misorientation angle at the initial time, t=0, and 2iθ is the misorientation angle after time, t.

Whereas, the misorientation angle analysis shows that the number of 20°~45° misoriented boundaries is higher around the Goss grains than around other texture components.
Especially, many researchers have investigated the texture and grain size of primary recrystallized microstructure related with grain growth inhibitors.
Therefore, this small difference is unlikely enough to explain the influence of CSL boundaries for the abnormal grain growth of Goss grain.
Whereas, the misorientation angle analysises show that the number of 20°~45° misoriented boundaries is higher around the Goss grains than around other texture components.
Therefore, it could be concluded that the high frequency of 20°~45° misoriented grain boundaries of Goss grain have a priority of selective grain growth causing the secondary recrystalliztion of Goss grain.

Simulation algorithm and procedure At first the grain structure is mapped onto a two-dimensional random number lattice.
Here the random numbers should assume numbers between 1 and 64.
A grain was defined as a collection of points that have the same orientation number.
In other words, two adjacent grid points having the same orientation number are considered to be a part of the same grain.
End t = t+1MCS No No No Yes Yes Yes Fig.1 Flow chart of grain growth simulation in HAZ Growth Dynamic Input the value of Q: Input the step number: Display Simulation Total number Avg.

Grain boundaries which separate such grains are the CSL-boundaries.
Grain tolerance angle was 5°, minimum grain size was 5 points.
A home made software was developed to count the number of good and undesired junctions.
IQ means the "goodness" of the Kikuchi-images formed during EBSDmeasurement, and is given by a number.
The better the Kikuchi image, the higher this number will be.