The effects of regularity on the geometrical properties of Voronoi

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Mar 18, 2014 - Application of controlled Voronoi tessellations in micromechanics modelling. ... and the mean total surface area and edge length per cell, while Gilbert .... 1. Two contrasting 3D Voronoi tessellations: (a) (125 cells) is fully random, ...... values for ˜L (the cell face perimeter normalized by the mean) are 0.7778.
Physica A 406 (2014) 42–58

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Physica A journal homepage: www.elsevier.com/locate/physa

The effects of regularity on the geometrical properties of Voronoi tessellations H.X. Zhu a,∗ , P. Zhang b , D. Balint b , S.M. Thorpe c , J.A. Elliott c , A.H. Windle c , J. Lin b a

School of Engineering, Cardiff University, CF24 3AA, UK

b

Department of Mechanical Engineering, Imperial College London, SW7 2AZ, UK

c

Department of Materials Science and Metallurgy, Cambridge University, CB2 3QZ, UK

highlights • • • • •

Effect of regularity on the properties of a random 3D Voronoi tessellation. Statistical analysis of Voronoi tessellations using 106 cell generations. Probability distributions derived for geometric properties of the tessellations. A simple scheme for generating Voronoi tessellations with regularity control. Application of controlled Voronoi tessellations in micromechanics modelling.

article

info

Article history: Received 4 December 2013 Received in revised form 24 February 2014 Available online 18 March 2014 Keywords: 3D Voronoi tessellations Geometrical properties Regularity Virtual grain structure Micromechanics

abstract This study comprehensively quantifies the effects of regularity on the geometrical properties of a random three-dimensional Voronoi tessellation (VT), where regularity was defined as the ratio of the minimum seed distance to the seed distance of the correlated body-centred cubic lattice. A scheme to generate Voronoi tessellations with controlled regularity is proposed, which was used to simulate 106 cells for a series of regularities. The results were used to derive probability distributions for the properties of the tessellation, including faces and edges per cell, vertex and dihedral cell angles, cell areas and volumes, etc. An understanding of the relation between a simple, measurable parameter characterizing the degree of regularity of a Voronoi tessellation and its geometrical properties is essential in generating virtual microstructures that are statistically representative of reality; the statistical results are also relevant to all other applications involving random Voronoi tessellations. Finally, an application is presented of the proposed Voronoi tessellation generation scheme applied to micromechanical modelling of grain structures with defined regularities for crystal plasticity finite element analysis. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The three-dimensional (3D) Voronoi tessellation (VT) geometric model [1] partitions space into convex polygons, or cells, which fill the available volume completely. The VT model and its variants have been applied in a wide range of science and engineering subjects [2,3], including crystallography [4], materials science [5,6], biology [7], geography [8],



Corresponding author. Tel.: +44 0 2920 874824. E-mail address: [email protected] (H.X. Zhu).

http://dx.doi.org/10.1016/j.physa.2014.03.012 0378-4371/© 2014 Elsevier B.V. All rights reserved.

H.X. Zhu et al. / Physica A 406 (2014) 42–58

43

astronomy [9], management [10] and control [11]. These models have been found extremely useful in micromechanical modelling, e.g. in generating high fidelity virtual cellular structures for mechanics simulations [5,6], and grain structures for large-scale realistic micro-forming analyses [12,13]. Some micromechanical applications require exact microstructure representation using reconstructions based on e.g. Electron Backscatter Diffraction (EBSD) measurements [14]. The VT is employed where a statistically equivalent representation of a real grain structure is sufficient, which obviates the need for laborious and expensive experimental characterization; statistically equivalent representations also facilitate parametric studies to correlate features of the grain morphology to the mechanical behaviour. Other methods for generating statistically equivalent virtual grain structures exist, including the Monte Carlo (Potts) model [15], ellipsoid packing [16], cellular automata [17], phase field [18] and level set [19] methods. In some instances, these methods are coupled to kinetics equations to describe an evolving microstructure, e.g. static recrystallization [20]. The VT model can be interpreted as the product of the isotropic growth process from a spatial distribution of static seeds. The resulting structure is completely and unambiguously determined by the initial distribution of seeds. If seeds are entirely randomly generated, the resulting structure is a Poisson Voronoi tessellation. An important adaptation of the Voronoi tessellation is the ‘hard-sphere’ model, which introduces a minimum exclusion distance between adjacent seeds upon which the tessellation is based. This type of ‘hard-sphere’ model, along with the limiting case of a Poisson Voronoi tessellation for which the exclusion distance is zero, will be considered in this paper. The geometric characteristics of three-dimensional Poisson Voronoi tessellations have been studied extensively. Meijering [4] derived theoretical results for several properties including the mean numbers of faces and edges per cell and the mean total surface area and edge length per cell, while Gilbert [21] enumerated the theoretical variances of the cell volume. Mason et al. [22] derived local relations to evaluate the number of faces of a grain in individual grain clusters. Meanwhile, Mahin et al. [23] and Andrade and Fortes [24] studied characteristics of such cells using computer simulation, the former considering planar sections through the tessellation and the latter considering the cell volume; in each case the simulations were based upon fewer than 10,000 cells. Using larger-scale simulations, Kumar et al. [25] simulated 358,000 Poisson Voronoi cells and examined the distributions of various properties including the numbers of faces and edges per cell and both total surface area and the volume per cell. In a subsequent study, Kumar and Kurtz [26] simulated 377,000 cells and derived distributions of the dihedral and bond angles (the angles between adjacent faces and edges, respectively), as well as the total edge length of a cell and of a cell face. Ferenc and Néda [27] studied the cell size distribution properties for two- and three-dimensional Poisson Voronoi tessellations, and proposed a simpler general form of distribution function, calibrated based on statistical results. Naturally occurring Voronoi tessellations vary significantly in their ‘degree’ of regularity. The higher the minimum exclusion distance for a given number of points in a region, the greater will be the regularity of the corresponding Voronoi tessellation. Studies of Voronoi tessellations with non-zero exclusion distance, which may be regarded as packings of hard spheres, include those by Hanson [28] who considered the cell volume distribution. In addition, for such tessellations based upon sphere packing, Oger et al. [29] derived distributions for the number of faces per cell, the total surface area per cell and the volume per cell, as well as several metric properties for an f -faceted cell. Lucarini [30] examined the statistical properties of random 3D tessellations, which were produced by perturbing cubic lattices with a specified Gaussian noise to individual lattice points. The distribution features of the number of faces, the area and the volume have been reported for tessellations obtained with different strengths of white noise. In Ref. [31], Kumar and Kumaran studied the cell volume distributions of random Voronoi tessellations, and a parameter to evaluate a tessellation’s regularity was proposed based on the statistical variation of the volume distribution. This was based on the fact that the more regular a tessellation is, the narrower its volume distribution. The value can only be determined by a statistical evaluation, not by a geometrical measurement. This paper will build upon previous works in two dimensions [32], and 3D Voronoi tessellations with different degrees of regularity [5,6]. The main objective is an investigation of the relation between the topological and metric characteristics of a Voronoi tessellation and its ‘degree of regularity’, defined here by a parameter a. Previous statistical studies were limited to examinations of particular regularities; Kumar and Kurtz [26] and Oger et al. [29] focused mainly on Poisson (α = 0) and delta (α ∼ 0.7) type Voronoi tessellations, respectively. This study also goes beyond previous studies by examining 106 cells. The statistical results can be of importance for every application involving random Voronoi tessellations. In order to demonstrate the potential application, three dimensional crystal plasticity finite element (CPFE) models were built, where virtual grain structures are generated with controlled regularities using the proposed scheme. 2. Method of analysis 2.1. The seeds Firstly, N points are generated in a central cube that has a volume V0 and periodic boundary conditions. A Cartesian coordinate system is chosen, and points are placed in the cube by deriving x, y and z coordinates independently from pseudorandom numbers1 generated evenly between zero and one. Once the first point has been placed, subsequent points are 1 The ‘rand()’ library function on the Silicon Graphics IRIX6.5 platform is used here.

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H.X. Zhu et al. / Physica A 406 (2014) 42–58

accepted only if they are located no nearer than a minimum allowed distance, δ , from all other points; that is, if the spheres of equal diameter δ centred on each point do not overlap. In accordance with the maintenance of periodic boundary conditions, each point is also copied into the surrounding 26 cubes. This process is continued until N points have been specified. If the value of δ , and, hence, the diameter of the spheres, is set to zero, then the points are distributed randomly in space. 2.2. The definition of the regularity parameter The regular tetrakaidecahedral cell with planar faces is an approximation to the ‘Kelvin’ cell, which partitions space with minimum surface area for identical cells [2]. An array of such tetrakaidecahedral cells represents a fully-ordered 3D Voronoi tessellation, which is based upon points arranged in a body-centred cubic (BCC) lattice. Each tetrakaidecahedral cell possesses 14 planar sides, of which 8 are regular hexagons and 6 are square.2 In order to fit N tetrakaidecahedral cells into a given volume V0 , the distance, d0 , between each point and its 26 nearest neighbours (and, hence, the diameter of the ‘hard spheres’) must be equal to:

√  6

d0 =

2

3



V0



N

2

.

(1)

To construct a random Voronoi tessellation with N cells in the volume V0 (hence, to randomly place N identical spheres in the Volume V0 ), the minimum distance between points, δ , (hence, the diameter of the spheres) should be less than d0 ; otherwise, it is impossible to incorporate N spheres. In order to quantify the regularity of the resulting 3D tessellation, a parameter, α , is introduced as follows [5,6]:

α=

δ

(2)

d0

where α = 1 (i.e. δ = d0 ) for a lattice of regular tetrakaidecahedral cells. For a fully-random (Poisson Voronoi) tessellation, α = 0 (δ = 0).   It is noted that while the packing density [28,29] of the identical spheres, which is given by

N π δ3 6V0

in the above notation,

may vary, the packing density of the Voronoi cells is always unity. The diameter of the spheres is not, therefore, an indication of the ‘average size’ of the Voronoi cells, but rather a limit on the smallest Voronoi cell, which cannot be smaller than the sphere within. For α values of 0, 0.7, and 1, respectively, the equivalent packing densities are 0, 0.2333, and 0.68017. While packing densities as high as 0.74 may be achieved using hexagonal close-packed (HCP) and face-centred cubic (FCC) lattices, the maximum packing densities attained by random packings of identical hard spheres has been placed empirically at around 0.64 [33]. For this reason, the body-centred cubic lattice (with a packing density of 0.68017) has been chosen here, in preference to the higher density lattices quoted above, as the more natural ordered limit for 3D Voronoi tessellations incorporating random irregularity. This choice of limiting structure also approximates, as mentioned previously, a lattice of true ‘Kelvin’ cells which minimizes the surface area for a partitioning of space using identical cells; the plane-sided tetrakaidecahedral cell is frequently used as a model for natural systems such as foams [34–39]. 2.3. Constructing the tessellations Once the points have been distributed according to the procedure described, the 3D Voronoi tessellation for this arrangement may be derived. For a given point, its Voronoi cell is defined by firstly constructing the lines connecting this point to every other, and, secondly, bisecting each of these lines by its perpendicular plane. The required cell is then given by the smallest polyhedron which is both bounded by these planes and which contains the chosen point; there being, typically, a relatively small number of nearby points yielding planes bounding the cell. The Voronoi cell for every point in the distribution is generated, according to the above definition, using software developed for this study. Throughout the construction process, periodic boundary conditions are maintained at the faces of the cube in which the points have been specified. Fig. 1 shows two tessellations: one of which has been generated in this way for α = 0, and the other represents the theoretical ordered limit of α = 1. Further details on the Voronoi tessellation algorithms can be found in [40,41]. 3. Results Except where stated the statistical results presented are based upon 3D tessellations, each containing 125 polyhedra or cells, which have known values of the regularity parameter, α . For each value of α adopted, the statistical properties were obtained on the basis of approximately 106 cells in total. The statistical results obtained are presented as follows: Figs. 2 and 3 show the distributions in the numbers of faces per cell, and sides per cell face, respectively. Fig. 4 illustrates the dependency of f µ(Ff ) upon f ; where µ(Ff ) is the mean

2 The quadrilateral faces of the true ‘Kelvin’ cell are planar, and the hexagonal faces are curved.

H.X. Zhu et al. / Physica A 406 (2014) 42–58

45

Fig. 1. Two contrasting 3D Voronoi tessellations: (a) (125 cells) is fully random, and corresponds to α = 0; while (b) (128 cells) represents our ordered limit of α = 1.

a

b

0.25

0.20 Probability, P

Probability, P

0.20

α = 0.0

0.15 0.10 0.05

0.10

0.00 0

6

12

18 f

24

30

36

0

d

0.25

6

12 ↑

0.25

18 f

24

30

36

30

36

P = 1.0

0.20 Probability, P

0.20 Probability, P

α = 0.3

0.15

0.05

0.00

c

0.25

α = 0.5

0.15 0.10 0.05

α = 0.7 α = 1.0

0.15 0.10 0.05

0.00

0.00 0

6

12

18 f

24

30

36

0

6

12

18 f

24

Fig. 2. The distributions in the number of faces per cell, f , belonging to 3D Voronoi tessellations having varying values for α ; the two-parameter gamma function fits (see Table S1 in the Supplementary material (see Appendix A)) are shown as solid curves. Superimposed on (d) is the result for α = 1.0.

number of faces per cell in the neighbouring cells of an f -faceted cell. Figs. 5 and 6 show the distributions in the normalized individual edge lengths and total cell edge lengths, respectively. The variation in the mean normalized total edge length of an f -faceted cell with f is shown in Fig. 7. The cell surface areas are considered in Figs. 8 and 9, in which the distributions in normalized total surface area per cell and the variation in the mean normalized total surface area of an f -faceted cell with f are respectively shown. The distributions in the normalized volume per cell are shown in Fig. 10 and Fig. 11, which demonstrate the variation in the mean normalized volume of an f -faceted cell with f . An application of the proposed Voronoi tessellation generation scheme in micromechanical modelling for grain structures with defined regularities is shown in Fig. 12.

46

H.X. Zhu et al. / Physica A 406 (2014) 42–58

b

0.4

α = 0.0

Probability, P

0.3 0.2 0.1

c

0

2

4

6

d α = 0.5

0.3 Probability, P

0.2

0.0

8 10 12 14 16 n

0.4

0.2

0

2

4

6

P = 0.43 ↑ 0.4



8 10 12 14 16 n P = 0.57

α = 0.7 α = 1.0

0.3

0.1 0.0

α = 0.3

0.1

Probability, P

0.0

0.4 0.3

Probability, P

a

0.2 0.1

0

2

4

6

8 10 12 14 16 n

0.0

0

2

4

6

8 10 12 14 16 n

Fig. 3. The distributions in the number of sides per cell face, n, belonging to 3D Voronoi tessellations of increasing regularity. Also shown are the twoparameter gamma function fits (solid curves) based upon the parameter values given in Table S2. The probabilities for square and hexagonal faces when α = 1.0 are shown in (d).

The data in Figs. 2, 3, 6, 8 and 10 have been fitted using both one-3 and two-parameter4 gamma distribution functions. In either case, the parameter values are derived from the mean, µ, and the variance, µ2 , of the data, where the variance is, in general, given by

µ2 =

 (nj − µ)2 f (nj )

(5)

j

in which f (nj ) refers to the distribution function of the specific data (see Figs. 2, 3, 6, 8 and 10). The values calculated for a and b, where a is common to both the one- and two-parameter fits, are listed in Tables S1, S2, S5, S7, and S9 in the Supplementary material (see Appendix A) for the different values of α employed. It is noted that, while the values for the mean and variance quoted in these tables refer to the distribution as a whole, the parameters a and b have been calculated on the basis of the truncated distributions as shown in the associated figures. In order to compare the quality of the fit obtained using the one-parameter gamma function with that achieved using two parameters, we define:

  δ1 = max ffit1 − fobs    δ2 = max ffit2 − fobs 

(6) (7)

3 The gamma distribution function using one parameter, a, is given by:

Px,x+dx =

aa xa−1 −ax e dx γ (a)

x>0

(3)

where a = µ2 /µ2 ; µ and µ2 represent the mean and variance respectively. 4 The gamma distribution function using two parameters, a and b, is given by:

Px,x+dx =

x a −1 b a γ ( a)

e−ax dx

x>0

where a = µ2 /µ2 ; b = µ2 /µ; µ and µ2 represent the mean and variance, respectively.

(4)

H.X. Zhu et al. / Physica A 406 (2014) 42–58

a

b

500

400

300

300 200

α = 0.0 α = 1.0

100 0

c

500

400

200

α = 0.3 α = 1.0

100 0

6

12

18 f

24

30

0

36

d

500

400

300

300

200

0

6

12

18 f

200

α = 0.5 α = 1.0

100

24

30

36

500

400

0

47

α = 0.7 α = 1.0

100 0

6

12

18 f

24

30

36

0

0

6

12

18 f

24

30

36

Fig. 4. The mean number of faces per cell in the neighbouring cells of an f -faceted cell, µ(Ff ), multiplied by f and plotted against f . The solid lines represent fits to the Aboav–Weaire law, as given in Eq. (14), using the parameters in Table S3. In each case, the 14-sided tetrakaidecahedral cell result is also shown for comparison.

E1 =

 (ffit1 − fobs )2

(8)

n

E2 =

 (ffit2 − fobs )2

(9)

n

where n is the number of each data group (see Figs. 2, 3, 6, 8 and 10), ffit1 and ffit2 are the one- and two-parameter gamma function fits, and fobs is the actual probability for the data group obtained in the simulation. The values obtained for δ1 , E1 , δ2 and E2 are given in Tables S1, S2, S5, S7, and S9. In the remaining figures, the data have been fitted where suitable functions have been identified. In each of these cases the function is represented in the relevant figure and table in the Supplementary material (see Appendix A), and described in the associated section of the discussion. When α takes a value of 0.7 or lower, it has always been found possible to generate N random polygons in the volume V0 . Upon increasing α to 0.75, however, (that is, upon setting the diameter of the spheres to be 0.75d0 ) this is no longer the case. For this reason, the statistical data in Figs. 2–11 only extend up to an α value of 0.7. For reference purposes, however, results for α values of both 0.75 and 0.8 are included in the appropriate tables in the Supplementary material (see Appendix A). 4. Discussion 4.1. The number of faces per cell The distributions in the number of faces per cell, f , are shown in Fig. 2. As the value of α (hence, the regularity of the tessellation) increases, the distributions become progressively more narrow; the limiting case being the single value of f =14 when α = 1, as overlaid in (d). The distributions have been fitted using both the one- and two-parameter gamma functions for varying α , and the latter has consistently been found to provide a significantly better fit. The corresponding parameter values are listed in Table S1, along with the fitting errors and values for the distribution mean, µ(f ) and µ2 (f ). The mean number of faces per cell, µ(f ), is observed to decrease as the tessellation regularity increases. Comparing with previous studies, Oger et al. [29] also obtained distributions for f in tessellations of varying regularity (measured according to the sphere packing fraction). They similarly observed a narrowing of the distributions with increasing regularity, and the corresponding fall in µ2 (f ) with regularity appears consistent in form with our own. In addition, they observed an approximately linear decrease in the mean, µ(f ), from 15.3 to 14.5 over a packing fraction range

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H.X. Zhu et al. / Physica A 406 (2014) 42–58

a

b

0.08

0.06 Probability, P

Probability, P

0.06

α = 0.0 0.04 0.02 0.00 0.0

c

1.0

2.0

3.0

4.0

0.00 0.0

5.0

d

0.08

1.0



2.0

α = 0.5 0.04 0.02

3.0

4.0

5.0

4.0

5.0

P = 1.0

0.06 Probability, P

0.06 Probability, P

α = 0.3 0.04 0.02

0.08

0.00 0.0

0.08

α = 0.7 α = 1.0

0.04 0.02

1.0

2.0

3.0

4.0

5.0

0.00 0.0

1.0

2.0

3.0

Fig. 5. The normalized individual edge length, ˜l, distributions for 3D Voronoi tessellations of increasing regularity; where ˜l = l/µ(l), and µ(l) is the individual edge length distribution mean. The data were grouped in equal intervals of width 0.1. In (d), the α = 1.0 result of ˜l = 1.0 (for identical tetrakaidecahedral cells) is also shown.

of 0 to 0.6. We also observe an approximately linear decline in µ(f ) up to our limit α = 0.8 (which equates to a packing fraction of 0.35), albeit with a somewhat steeper gradient since our value for the fully-random case is higher at 15.58 than their figure of 15.3. Other studies have focused upon the Poisson Voronoi (fully-random) case, which is obtained  when α (and the sphere packing fraction) is zero. For this case Meijering [4] derived the theoretical value for µ(f ) of

48π 2 35

+2 ,

i.e. 15.535, which supports our result of 15.58 over the lower figure obtained by Oger et al. as mentioned above. Another study of the Poisson Voronoi case has been carried out by Kumar et al. [25] who obtained a figure for µ(f ) of 15.543 which is very close to the theoretical result; their value for the standard deviation, σ (f ), being 3.335, which is close to our own result of 3.350. They also fitted their distribution in f with the two-parameter gamma function, using parameter values of a = 21.629 and b = 0.720 which are in excellent agreement with our own in Table S1; there being no previous data available for fitting f distributions in tessellations which depart from the fully-random case. Kumar et al., in addition, report that the minimum and maximum number of faces observed in their Voronoi cells are respectively 4 and 36, which is identical to our findings. For α = 0.8 we find that this range has narrowed to the respective limits of 8 and 26, as the tessellations become increasingly regular. 4.2. The number of edges and vertices per cell Since each edge of a non-degenerate 3D Voronoi cell is bounded by two vertices, and each vertex represents a junction between three edges, the following well-known relationship exists between the number of edges, e, and the number of vertices, v , in a cell: 2e = 3v.

(10)

In addition, the Euler relation for a single cell in three dimensions dictates that:

v+f −e=2

(11)

where f is the number of faces in a cell. Combining Eqs. (10) and (11), the number of edges and vertices in a given cell are given by (3f − 6) and (2f − 4) respectively. The distributions in these properties are therefore simply related and identical in form to those shown for the number of faces per cell in Fig. 2; the limiting values for the case of ordered tetrakaidecahedral cells being 36 and 24 for e and v , respectively. The mean values of µ(e) and µ(v) for different α can be calculated from

H.X. Zhu et al. / Physica A 406 (2014) 42–58

a

b

0.5

Probability, P

Probability, P

α = 0.0

0.3 0.2

α = 0.3

0.3 0.2 0.1

0.1 0.0 0.0

0.5

1.0

1.5

2.0

0.0 0.0

2.5

d

0.5

0.5

1.0 ↑

0.5

1.5

2.0

2.5

P = 1.0

0.4 Probability, P

0.4 Probability, P

0.5 0.4

0.4

c

49

α = 0.5

0.3 0.2

α = 0.7 α = 1.0

0.3 0.2 0.1

0.1 0.0 0.0

0.5

1.0

1.5

2.0

0.0 0.0

2.5

0.5

1.0

1.5

2.0

2.5

Fig. 6. The normalized total edge length per cell, T˜ , distributions (T˜ = T /µ(T ), where µ(T ) is the distribution mean). The data were grouped in equal intervals of width 0.1, and the two-parameter gamma function fits (see Table S5) are also shown. The tetrakaidecahedral cell result, whereupon T˜ = 1.0, is plotted for comparison in (d).

a

b

2.5 2.0

2.0

α = 0.0 α = 1.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

c

2.5

0

6

12

18 f

24

30

0.0

36

d

2.5 2.0

2.0

α = 0.5 α = 1.0

1.5

1.0

1.0

0.5

0.5

0

6

12

0

18 f

24

30

36

6

12

18 f

24

30

36

18 f

24

30

36

2.5

1.5

0.0

α = 0.3 α = 1.0

0.0

α = 0.7 α = 1.0

0

6

12

Fig. 7. The variation in the mean normalized total edge length of an f -faceted cell, µ(T˜f ), with f , under increasing α ; the solid lines representing the fitted Desch–Lewis law (also given by Eq. (17)) using the parameters listed in Table S6. In each case, the 14-sided tetrakaidecahedral cell result is also plotted for comparison.

50

H.X. Zhu et al. / Physica A 406 (2014) 42–58

a

b

0.5

0.4

α = 0.0

Probability, P

Probability, P

0.4 0.3 0.2 0.1 0.0 0.0

c

0.5

1.0

1.5

2.0

0.2

0.0 0.0

2.5

d

0.4

0.5

1.0 ↑

0.5 0.4

α = 0.5

Probability, P

Probability, P

α = 0.3 0.3

0.1

0.5

0.3 0.2 0.1 0.0 0.0

0.5

1.5

2.0

2.5

P = 1.0

α = 0.7 α = 1.0

0.3 0.2 0.1

0.5

1.0

1.5

2.0

2.5

0.0 0.0

0.5

1.0

1.5

2.0

2.5

˜ distributions (S˜ = S /µ(S ), where µ(S ) is the distribution mean). The data were grouped in equal Fig. 8. The normalized total surface area per cell, S, intervals of width 0.1, and the two-parameter gamma function fits (see Table S7) are shown. In (d), a comparison is drawn with the tetrakaidecahedral cell result (S˜ = 1.0).

the value of µ(f ) in Table S1. For the Poisson Voronoi case (α = 0), the values for µ(e) and µ(v) are 40.731 and 27.154.

π These compare with the theoretical values, as derived by Meijering [4], of 144 (i.e. 40.606) and 9635π (i.e. 27.070) for these 35 respective quantities in the fully-random case; the level of agreement being as for the mean number of faces per cell. 2

2

4.3. The number of sides per cell face The distributions in the number of edges (or sides) per cell face, n, are shown in Fig. 3 for increasing values of α . In the limiting case of tetrakaidecahedral cells, which have only square and hexagonal faces, the values of n must be either 4 or 6 according to the proportions in (d). The distributions are again observed to narrow as the value of the regularity parameter increases, as quantified by the decrease in the variance, µ2 (n), which is reported in Table S2 along with the values for the mean, µ(n), and the fitting parameters (with errors) for one- and two-parameter gamma function fits to the data. As for the distributions in f , the two-parameter gamma functions provided a better fit and the distribution mean is observed to decrease as α increases. Prior studies of the distributions in the number of sides per cell face in 3D Voronoi tessellations have focused upon the Poisson Voronoi (α = 0) case. Kumar et al. [25] derived the distribution in n for this case, although did not attempt a fit to their data. Their reported results of µ(n) = 5.228 and σ (n) = 1.5763 compare with our corresponding values of µ(n) = 5.2309 and  σ (n) = 1.5744. Using theoretical averages derived by Meijering [4] for the mean numbers of faces and

edges per cell, of

48π 2 35

+ 2 and

144π 2 , 35

respectively, the expected value for µ(n) in the Poisson Voronoi case is calculated

to be 5.228 (since edges must be double-counted when associated with individual faces). From the topological relations quoted in Section 4.2, an expression for the mean number of sides per cell face in a given cell (regardless of the regularity of the tessellation) may be derived. By combining Eqs. (10) and (11) and double-counting the edges (sides), one has:

µ(nf ) = 6 −

12 f

(12)

where µ(nf ) represents the mean number of sides per cell face in a cell which has f faces (since the result is dependent upon f ). A straightforward check on the integrity of our results has been made by verifying the validity of this relationship for varying f and α . Oger et al. [29] also reported having performed a similar check in 3D Voronoi tessellations based upon

H.X. Zhu et al. / Physica A 406 (2014) 42–58

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b

2.5

α = 0.0 α = 1.0

2.0

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0.5

0

6

12

18 f

24

30

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36

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α = 0.5 α = 1.0

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0.5

6

12

18 f

6

24

30

36

12

18 f

24

30

36

24

30

36

α = 0.7 α = 1.0

2.0 1.5

0

0

2.5

1.5

0.0

α = 0.3 α = 1.0

2.0

1.5

0.0

51

0.0

0

6

12

18 f

Fig. 9. The variation in the mean normalized total surface area of an f -faceted cell, µ(S˜f ), with f ; the normalization being with respect to µ(S ), the distribution mean. Fits to the linear Desch–Lewis law (Eq. (18) and Table S8) are also shown. For each, the 14-sided tetrakaidecahedral cell result is plotted for comparison purposes.

varying fractions of identical spheres, and likewise Kumar et al. [25] in the specific case of a Poisson Voronoi (α = 0) tessellation. Applying Eq. (12) to the 14-sided tetrakaidecahedral cell reproduces the well-known result of an average of 5.14 sides per cell face for the polyhedron. 4.4. The number of faces in neighbouring cells Historically, correlations between properties of adjacent cells in cellular networks have been examined in terms of a property which we shall denote by µ(Ff ); that is, the mean number of faces in the neighbouring cells of a cell with f faces (where two cells are neighbours if they share a common face). A relationship originating from observations of the grains in sections through polycrystalline materials, and later soap froths, may be extended to three dimensional Voronoi networks using the generalized Aboav–Weaire law [42,43]: f µ(Ff ) = (µ(f ) − aab )f + µ(f )aab + µ2 (f )

(13)

where µ(f ) and µ2 (f ) are respectively the mean and variance of the number of faces per cell, and aab is the ‘Aboav parameter’ which also depends upon the degree of regularity of the tessellation. Eq. (13) has been shown to hold exactly for certain three dimensional ‘columnar’ structures [44,45]. For a given value of the regularity parameter, α , Eq. (13) may be simply written: f µ(Ff ) = g1 f + g2

(14)

where g1 and g2 are constants for the particular value of α . In Fig. 4, the product f µ(Ff ) is plotted against f for varying values of α in order to test the validity of this relationship for our data. A good fit to the linear relationship is observed for each value of α , and the values of g1 and g2 derived using a least squares fit are listed in Table S3. As α increases there is a movement towards the value for the 14-sided tetrakaidecahedral cell result, which is compatible with the choice of a BCC lattice packing limit. Previously, Oger et al. [29], using approximately 4000–4500 cells, also found the Aboav–Weaire law to be applicable to 3D tessellations of varying regularity; their results for packing densities of 0%, 4.44%, 8.90% and 30.0% (corresponding to α values of 0, 0.403, 0.508 and 0.761 respectively) being listed in Table S3 alongside their closest counterpart results. There is reasonable agreement between their findings and ours. In the specific case of a Poisson Voronoi tessellation, Kumar et al. [25], based upon the data for 3729 cells, applied a different function fit of the form:

µ(Ff ) = −0.02f + 16.57

(15)

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H.X. Zhu et al. / Physica A 406 (2014) 42–58

a

b

0.30

0.25

α = 0.0

0.20

Probability, P

Probability, P

0.25

0.15 0.10 0.05 0.00 0.0

c

0.5

1.0

1.5

2.0

2.5

0.15 0.10

0.00 0.0

3.0

d

0.25

0.5

0.30

1.0

1.5



P = 1.0

2.0

2.5

3.0

2.5

3.0

0.25

α = 0.5

0.20

Probability, P

Probability, P

α = 0.3

0.20

0.05

0.30

0.15 0.10 0.05 0.00 0.0

0.30

α = 0.7 α = 1.0

0.20 0.15 0.10 0.05

0.5

1.0

1.5

2.0

2.5

3.0

0.00 0.0

0.5

1.0

1.5

2.0

Fig. 10. The normalized volume per cell, V˜ , distributions (V˜ = V /µ(V ), where µ(V ) is the distribution mean). The data were grouped in equal intervals of width 0.1, and the two-parameter gamma function fits (see Table S9) are shown. In (d), the tetrakaidecahedral cell result, whereby V˜ = 1.0, is included for comparison.

which implies a direct, rather than an inverse, relationship between µ(Ff ) and f subsequently. However, Fortes [46] took their data and found that it could also be fitted to the form shown in Eq. (14), with values for g1 and g2 of 15.95 and 4.45, respectively; these compare reasonably with our own corresponding values of 16.03 and 3.52 for the case of zero α . 4.5. The angle distributions Two sets of angle distributions pertaining to 3D Voronoi tessellations are discussed here: the first of these refers to the dihedral angle (that is, the angle measured between two faces of a cell, in a plane which is perpendicular to both), and the second accounts for the vertex (or ‘bond’) angle (which is the angle between two edges at the vertex). In the case of the dihedral angle, ϕ , Kumar and Kurtz [26] pointed out that the theoretical distribution for a 3D Poisson Voronoi tessellation (α = 0) may be calculated exactly as follows: f (ϕ) =



4 3π 2



(2ϕ(2 + cos 2ϕ) − 3 sin 2ϕ) sin2 ϕ (0 ≤ ϕ ≤ π ) 

2 for which the mean is 120◦ and the variance is π18 −

3 8



(16)

radians squared, which equates to a standard deviation of 23.853◦ .

The distribution in ϕ has been derived for various values of α from our own data, although the plots are not reproduced here for the sake of brevity. In the Poisson Voronoi case, the data are in excellent agreement with the theoretical distribution (16); while the mean, µ(ϕ), and the standard deviation, σ (ϕ), are respectively measured to be 119.98◦ and 23.87◦ . For the form of the distribution in the Poisson Voronoi case, the reader is referred to the data presented in Ref. [26]. As the value of α increases the distributions become increasingly narrow, the standard deviation, σ (ϕ), falling to 16.22◦ by α = 0.8. In every case, the mean, µ(ϕ), is measured to be very close to the theoretical value of 120◦ . For tetrakaidecahedral cells (α = 1) one third of all angles between face-pairs are 109.471◦ (for hexagonal–hexagonal pairs), while the rest have the value 125.264◦ (square–hexagonal pairs). For the distribution in vertex (or ‘bond’) angles, θ , the mean value, µ(θ ), for a Poisson Voronoi tessellation   may be calculated exactly by combining the well-known expression for the mean angle of an n-sided polygon, of

π (n−2) n

radians,

with the value for the mean number of sides per cell face (µ(n) = 5.228) which has been derived in Section 4.3; the result being µ(θ ) = 111.134◦ . The distributions in θ have been derived for varying α (again not reproduced here in order to limit

H.X. Zhu et al. / Physica A 406 (2014) 42–58

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2.0

α = 0.3 α = 1.0

2.0

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0.0

c

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53

0.0

0

6

12

18 f

Fig. 11. The variation in the mean normalized volume of an f -faceted cell, µ(V˜ f ), with f ; the normalization being with respect to µ(V ), the distribution mean. The solid lines represent fits to the linear Desch–Lewis law (see Eq. (19) and Table S10). In each plot, the 14-sided tetrakaidecahedral cell result is additionally shown.

the number of figures) although no function has been found to provide a suitable fit to the data. In the Poisson Voronoi case (α = 0), the distribution appears in excellent agreement with that derived by Kumar and Kurtz [26] on the basis of a simulation of 377,000 cells. The mean, µ(θ ), and standard deviation, σ (θ ), are calculated to be 111.127◦ and 35.621◦ respectively, which compare closely with the figures obtained by Kumar and Kurtz of µ(θ ) = 111.139◦ and σ (θ ) = 35.658◦ ; in either case the mean is very close to the theoretical value cited above. As the value of α increases, both the mean and standard deviation fall and the distributions narrow. By α = 0.8 the mean has fallen to 110.710◦ and the standard deviation to 28.220◦ . In the limiting case of tetrakaidecahedral cells and α = 1, one third of the angles are 90◦ (due to vertices in the square faces), while the remainder are 120◦ (from hexagon vertices). It is noted that the mean dihedral angle and mean vertex angle compare favourably with the values which are required by the local minimization of surface energy (and, hence, the achievement of equilibrium of local surface-tension forces), these being dihedral angle of 120◦ and vertex angles of 109◦ 028′ 16′′ [47]. In the case of the dihedral angles, the mean is exactly 120◦ , as required, regardless of the tessellation regularity; while the mean vertex angle approaches the required figure of 109◦ 028′ 16′′ more closely as the regularity increases. As quantified by standard deviations (or variances) of the distributions, however, there will be local deviations from these ideal values which generally become greater as the tessellation regularity decreases. Such considerations become important if using Voronoi tessellation as a model for the structure of a real system in which these local minimization conditions may be met, such as a liquid–gas foam system. 4.6. Edge length distributions Fig. 5 shows the distributions obtained in the individual edge lengths of 3D Voronoi cells, normalized by the mean of the distribution in question, µ(l), for varying values of α (where ˜l = l/µ(l)). While no successful fit of the distributions to a function has been found, the probability falls approximately linearly with increasing edge length over most of the valid range when the regularity (and, hence, α ) is small; this being in agreement with the observations of Kumar et al. [25] in the Poisson Voronoi case based upon a simulation of 102,000 cells. As the tessellation regularity increases the distributions become progressively less linear and more narrow, as indicated by the fall in distribution variance, µ2 (˜l), listed in Table S4. For the Poisson Voronoi case, Kumar and Kurtz [26] observe a standard deviation of σ (˜l) = 0.752 (after normalization of their reported result by their value for the mean) which is identical to our result to the same accuracy. The distribution in the total edge length per cell face, i.e. the cell face perimeters, have also been derived, although are not reproduced here. For the Poisson Voronoi case, Kumar and Kurtz [26] presented a distribution based upon a simulation of

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H.X. Zhu et al. / Physica A 406 (2014) 42–58

Grain size distribution 0.21 0.19 0.17 0.14 0.12 0.10 0.08 0.06 0.04 0.02

Grain size distribution Expected Generated Frequency

Frequency

a

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8

D/Dmean

0.36 0.32 0.29 0.25 0.22 0.18 0.14 0.11 0.07 0.04

Expected Generated

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8

D/Dmean

b

c

Fig. 12. Illustration of two Voronoi tessellations with different regularities and the corresponding CPFE models. (a) Comparison between the desired grain size distributions and those generated by VGRAIN; (b) the Voronoi tessellations; (c) CPFE models constructed in ABAQUS from the tessellations, where different colours indicate different crystallographic orientations.

377,000 cells, which appears in agreement with our own. As the value of α increases, the distributions narrow as expected; in the limiting case of tetrakaidecahedral cells, the values for L˜ (the cell face perimeter normalized by the mean) are 0.7778 or 1.1667 for square and hexagonal faces respectively. Referring to the results having previously been obtained for two dimensional Voronoi tessellations [32], the cell face perimeter distributions bear a strong resemblance to the distributions

H.X. Zhu et al. / Physica A 406 (2014) 42–58

55

having been obtained for the individual edge lengths in 2D. While a fit to a Gaussian function has been found to provide a reasonable fit in the 2D case (particularly in tessellations of low regularity), it has been found that a Gaussian function provided only a poor fit to the data in the present case. Summing the edge lengths for each Voronoi cell, the distributions in the normalized total lengths per cell, T˜ , are shown in Fig. 6 for various values of α ; where the normalization has been carried out with respect to the mean of each distribution (T˜ = T /µ(T )). The narrowing of the distributions as the tessellation regularity increases is documented by the decrease in variance, µ2 (T˜ ), in Table S5, in which the parameter values for one- and two-parameter gamma function fits are also listed. Interestingly, as is also the case for the distributions in cell perimeters in two dimensions [32], the one-parameter gamma function provides a better fit for low values of α , while the reverse is true for values of α above 0.4 whereupon the two-parameter function is better suited. On the basis of a simulation of 165,000 cells, Kumar and Kurtz [26] fitted their equivalent distribution for the Poisson Voronoi case using the two-parameter gamma function. Taking their value for the standard deviation and normalizing by their distribution mean, we obtain the value for the α = 0 case of σ (T˜ ) = 0.2103; this compares closely with our analogous figure of σ (T˜ ) = 0.2092. Previously, some work ([25,26,29] for example) has been done to establish a linear relationship between several metric properties of 3D Voronoi cells and the number of cell faces f ; these properties include the cell volume, the total cell surface area, and the total cell edge length. This work is generally seen as having its origin both in the study of metal crystal grains by Desch [48], who suggested such a linear law for grain perimeters, and in the work by Lewis [49], who similarly treated the cross-sectional areas of cells in plant tissue; a review may be found in Ref. [50]. In Fig. 7, the mean normalized total edge length of an f -faceted cell, µ(T˜f ), has been plotted against f for increasing α ; where Tf is the total edge length of an f -faceted cell and T˜f = Tf /µ(T ) (µ(T ) being the mean total edge length of the entire distribution for a given α ). In each case the increase in µ(T˜f ) with f appears highly linear, which agrees with the previous findings of Kumar and Kurtz [26] for the Poisson Voronoi (α = 0) case and those of Oger et al. [29] for a range of tessellation regularities. A generalized linear ‘Desch–Lewis’ law for this case may be written:

µ(T˜f ) = h1 f + h2

(17)

where h1 and h2 are constants which depend upon the regularity of the tessellation. Our values for h1 and h2 are given in Table S6, and show a decrease in slope (h1 ) and an increase in intercept (h2 ) with increasing α . In contrast, the results of Oger et al., based upon simulations of 4000–4500 cells, show no consistent trend in the slope and intercept of their linear plots with increasing tessellation regularity. For the Poisson Voronoi case, Kumar and Kurtz, on the basis of 165,000 simulated cells, obtain values of h1 = 0.05515 and h2 = 0.1434 (following normalization by their value for µ(T )) which are very close to our own findings for α = 0. 4.7. The surface area distributions

˜ for varying α , are shown in Fig. 8, where the normalizaOur distributions of the normalized total surface area per cell (S), tion has been performed with respect to the global mean surface area per cell, µ(S ), in each distribution (hence, S˜ = S /µ(S )). As expected, the distribution variances, µ2 (S˜ ), are listed in Table S7. Just as for the total edge length per cell distributions, we have found that the lower regularity tessellations could be fitted marginally better using the one-parameter gamma function than its two-parameter counterpart, the latter being conversely more suitable at higher α (the parameter values and fitting errors are also given in Table S7). In the Poisson Voronoi case, Kumar et al. [25] fitted their distribution for total surface area per cell, obtained from a simulation of 102,000 cells, using the two-parameter gamma function. Normalizing their results for the standard deviation in this case by their reported mean gives a result of σ (S˜ ) = 0.2512, which compares with our own figure of σ (S˜ ) = 0.2414. In order to determine the applicability, or otherwise, of the generalized Desch–Lewis law, the mean normalized total surface area of an f -faced cell, µ(S˜f ), is plotted against f for varying α in Fig. 9; where Sf represents the total surface area of an f -faced cell and S˜f = Sf /µ(S ). The relationship in each case is seen to be approximately linear, and hence consistent with the Desch–Lewis law, in agreement with the finding of Oger et al. [29] for a series of tessellation regularities and Kumar and Kurtz [26] for the Poisson Voronoi case. The plots have been fitted, using the method of least squares, to a linear law as follows:

µ(S˜f ) = m1 f + m2

(18)

where m1 and m2 depend upon the tessellation regularity and are given in Table S8 for increasing α . Our results generally indicate a decrease in the slope, m1 , and an increase in the intercept, m2 , of the fitting line with increasing α . From the data of Oger et al., there is also some evidence for a decrease in slope and an increase in intercept with increasing tessellation regularity. In the Poisson Voronoi case (α = 0), Kumar and Kurtz reported comparable values of m1 = 0.05329 and m2 = 0.1721 after normalization by their value for the mean. In addition, although not reproduced here, the distributions in the surface area per cell face have been derived for varying α . In the Poisson Voronoi case, the distribution appears in agreement with that derived by Kumar et al. [25] on the basis

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H.X. Zhu et al. / Physica A 406 (2014) 42–58

of 102,000 simulated cells. As α increases there is some narrowing of the distribution, and the value for A˜ (the cell surface area normalized by the distribution mean) in the case of tetrakaidecahedral cells is either 0.5227 or 1.3580 for square and hexagonal faces, respectively. No function has been found, however, which provides a satisfactory fit to the data. For the mean normalized area of an n-sided cell face, µ(A˜ n ), a roughly linear dependence upon n is observed; hence, the Desch–Lewis relationship is again obeyed. This is in agreement with the findings of Kumar et al. [25], who also observed a linear trend in the Poisson Voronoi case (the relationship was not studied by Oger et al.). As in the other applications of the Desch–Lewis law which have been discussed, a general decrease in slope and increase in the intercept of the fitted line is observed with increasing α . 4.8. The volume distributions The normalized volume per cell, V˜ , distributions are shown in Fig. 10 for varying α ; each normalization being carried out with respect to the mean cell volume for the distribution, µ(V ) (hence, V˜ = V /µ(V )). The distributions are seen to narrow as the value of α increases, as confirmed by the decrease in the distribution variance, µ2 (V˜ ), shown in Table S9, the distributions have again been fitted using both the one- and two-parameter gamma functions, the results of which are compared in Table S9. As for the distributions in both the total edge length and the total surface area per cell, the two-parameter gamma function has been found to provide a better fit to the cell volume distribution for higher regularity tessellations, while the single-parameter counterpart is marginally better when α is low. Each of Kiang [51], Andrade and Fortes [24], and Oger et al. [29] fitted the volume per cell distribution for Poisson Voronoi (α = 0) tessellations using the one-parameter gamma function, using parameter values (here denoted by a) of 6.18, 5.56 and (again) 5.56, respectively; our value for a being 6.13. Kumar et al. [25], on the other hand, found that the Poisson Voronoi cell volume distribution could be best fitted using the two-parameter gamma function, while Hanson [28] suggested the use of a ‘Maxwell speed distribution function’ for this purpose (which Kumar et al. concluded to be less suitable than the gamma functions). In the Poisson Voronoi case our standard deviation is 0.4019, by comparison with Kumar et al.’s value of σ (V˜ ) = 0.4193 following normalization by their mean. The mean normalized volume of an f -faceted cell, µ(V˜ f ), is plotted against f for varying α in Fig. 11, where Vf is the volume of an f -faceted cell and V˜ f = Vf /µ(V ). The trends are approximately linear, although a greater degree of nonlinearity appears to be in evidence than was the case for the analogous plots for both the total edge length and total surface area per cell. A fit has again been made to a generalization of the linear Desch–Lewis law:

µ(V˜ f ) = r1 f + r2

(19)

where r1 and r2 , as determined by the least squares method, are listed in Table S10. In agreement with the findings of Oger et al. [29], we observe a decrease in slope and an increase in intercept for a fitted line with increasing tessellation regularity. Kumar et al. [25] fitted their data for the same quantities (in Poisson Voronoi tessellations of 102,000 cells) using instead a power law of the form: µ(Vf ) = 0.0164f 1.498 . 5. Application in micromechanics modelling Table S9 provides the relationship between the tessellation’s regularity and the cell volume distribution characteristics. The distribution function parameter a increased monotonically as the regularity α increased. An empirical relation can be fit to the data pairs (a, α ) in Table S9, given by:

α(a) = A(z (a) − z0 )k+nz (a) ,

a0 ≤ a,

(20)

where z (c ) = a/am and z0 = a0 /am . The constants can be calibrated for a best fit to the data in a least squares sense as a0 = 6.139, am = 370, A = 2.4, k = 0.33 and n = 1.5. This descriptive relation reveals the correlation of the regularity parameter and the distribution parameter of the one-parameter gamma distribution. This is valuable for applications where the volume distribution is a more intuitive property than the regularity. In micromechanics analyses, this can be used to study how a grain size distribution affects the corresponding structure’s mechanical properties. Grains are virtually represented by cells in a three-dimensional Voronoi tessellation. Note that Eq. (2) can be rewritten as

δ = α d0 .

(21)

Accordingly, the minimum seed distance can be obtained. In summary, the following scheme can be used to generate virtual grain structures: Step 1. Specify a desired grain size distribution by defining the parameter a of the one-parameter gamma distribution. Step 2. Derive the regularity value α according to the empirical relation given by Eq. (2). Step 3. Obtain the minimum seed distance. Step 4. Generate seeds and hence the tessellation. A grain structure generation software system, VGRAIN, has been developed using the above scheme to generate random Voronoi tessellations with specified distribution features (see [40,41,52,53]). Fig. 12 shows two virtual grain structures and their corresponding crystal plasticity finite element (CPFE) models, defined using VGRAIN, which is a software

H.X. Zhu et al. / Physica A 406 (2014) 42–58

57

implementation (with graphical user interface) that makes use of the above algorithm. Distribution parameter values a = 6.2 and a = 38.1, respectively, generated the tessellations shown in Fig. 12b that have grain size distributions accurately matching the expected distributions shown in Fig. 12a. The resultant tessellations shown in Fig. 12b were imported into the finite element software Abaqus, as shown in Fig. 12c, where different colours represent the different crystallographic orientations that have been randomly assigned. A user defined material subroutine for deformation according to finite strain crystal plasticity theory was written and used with grain structures generated by VGRAIN, such as that shown in Fig. 12c, in various micromechanical studies [40,41,52,53,13]. Crystal plasticity finite element analyses of tension of a planar twophase steel polycrystal including grain boundary decohesion and sliding, and three-dimensional polycrystalline micropillar compression can be found in Refs. [52] and [53], respectively. 3D random irregular Voronoi tessellations were also used as structural models to study the mechanical properties of open-cell foams, and the geometrical properties were related to the mechanical properties by Zhu et al. [5,6]. They [5,6] found that the degree of cell regularity, α , strongly affects the mechanical properties of cellular materials. Further applications of the Voronoi tessellation algorithms to study the micromechanics of open-cell foams can be found in Refs. [37] and [38]. 6. Conclusions A regularity parameter, α , has been adopted, with the intention of providing a convenient method of quantifying the regularity of a 3D Voronoi tessellation between limits of 0 and 1; the former corresponding to a fully-random (Poisson Voronoi) tessellation, and the latter to a completely ordered array of identical, regular, tetrakaidecahedral cells. Alternatively, considering such tessellations to be a product of the planar interfaces between an underlying packing of hard spheres, the aforementioned limits correspond respectively to spheres of zero diameter, and to spheres of the maximum diameter possible, within the volume available, to a packing of identical spheres on a body-centred cubic lattice. The effects of varying the value of this parameter has then been investigated by simulating 3D Voronoi tessellations incorporating a total of 106 cells for each of a series of α values, and gathering the statistics for an extensive range of both topological and metric properties. In many cases functions have been fitted for purposes of modelling the data, and comparisons have been made with related work in the associated literature. Where appropriate, the probability distributions have been fitted using each of the described one- and two-parameter gamma functions and the results compared quantitatively. For the distributions in the numbers of both faces and edges per cell, and those in the number of sides per cell face, the two-parameter gamma function afforded a better fit to the data for these topological properties. In the cases of the distributions of each of total edge length per cell, total surface area per cell, and volume per cell, however, the one-parameter function proved better for low regularity tessellations (α below about 0.5), with the two-parameter counterpart becoming conversely more suitable as the regularity increased. Each of these distributions, along with those in the angles and cell face perimeters, were observed to become progressively more peaked and narrow as the value of α , and hence the regularity of the tessellations, increased. No suitable functions were found which could be used to represent the distributions in each of the single edge lengths, the cell face areas, and both the dihedral and vertex angles. While a Gaussian function was applied to the distribution in cell face perimeters, only an approximate fit to the data was achieved. For a cell which has a given number of faces, f , we have investigated the variation with f in the mean total edge length, the mean total surface area, and the mean volume. Each of these was fitted using the linear Desch–Lewis law for a range of tessellation regularities (values of α ). In addition, this law was found to apply to the dependency of the mean area of an n-sided cell face upon the value of n. For the variation in the mean perimeter of an n-sided face with n, on the other hand, we used our own generalized function to fit the data. In order to describe the variation in the mean number of faces of the neighbouring cells of an f -faced cell, µ(Ff ), the Aboav–Weaire linear law is often applied to the dependency of the product f µ(Ff ) upon f ; the application of this law having been supported by our data for a range of values of α . Finally, the probability that the kth nearest neighbour of a given point is used when constructing its cell face during the tessellation has been modelled, for varying tessellation regularity, using an extension of the equation which was proposed by Hanson. Whilst the regularity parameter α offers a convenient measure to quantify the degree of uniformity of 3D Voronoi tessellations, the reported statistical data and distribution fit can be applied in any context involving 3D Voronoi tessellations. The demonstrated virtual grain structure generation scheme, implemented in freely available software, VGRAIN, is a feasible approach to using the reported statistical data for real engineering applications. In this scheme, a virtual grain structure can be represented by a random Voronoi tessellation having the desired grain size distribution for e.g. crystal plasticity finite element analyses (CPFE). Appendix A. 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