Image Anal Stereol 2017;36:5-13 doi:10.5566/ias.1656
Original Research Paper
DESCRIPTION OF THE 3D MORPHOLOGY OF GRAIN BOUNDARIES
IN ALUMINUM ALLOYS USING TESSELLATION MODELS
GENERATED BY ELLIPSOIDS
ONDŘEJ ŠEDIVÝB,1,2, JULES M. DAKE3, CARL E. KRILL III3, VOLKER SCHMIDT1 AND
ALEŠ JÄGER2
1Institute of Stochastics, Ulm University, Helmholtzstrasse 18, 89069 Ulm, Germany; 2Institute of Physics,
Czech Academy of Sciences, Na Slovance 1999/2, 18221 Prague, Czech Republic; 3Institute of Micro and
Nanomaterials, Ulm University, Albert-Einstein-Allee 47, 89081 Ulm, Germany
e-mail: ondrej.sedivy@uni-ulm.de, jules.dake@uni-ulm.de, carl.krill@uni-ulm.de, volker.schmidt@uni-ulm.de,
jager@fzu.cz
(Received November 1, 2016; revised February 3, 2017; accepted February 8, 2017)
ABSTRACT
Parametric tessellation models are often used to approximate complex grain morphologies of polycrystalline
microstructures. A big advantage of such models is the substantial reduction in disk space required to store
large, three-dimensional data sets, especially when compared with voxel-based alternatives. By selection of
an appropriate tessellation model, a reasonably small loss of information on the real grain shapes can usually
be achieved. Special attention has recently been devoted to models based on ellipsoidal approximations fitted
to each grain. Faces of these tessellations are portions of quadric surfaces whose parameters can be derived
easily. In this paper, we deal with geometric features of the structure, notably curvatures and dihedral angles,
which are closely related to the kinetics of grain growth. These characteristics are computed for ellipsoid-
based tessellations fitted to two different aluminum alloys with nominal composition Al-3 wt% Mg-0.2 wt% Sc
and Al-1 wt% Mg. The results are then compared with estimations based on meshed empirical data. We
observe that the model offers more consistent estimations of grain shape characteristics than do the meshed
empirical data. Precise description of grain boundaries by the model is also promising with respect to possible
applications of these tessellations in stochastic space-time modeling of grain growth.
Keywords: curvature, dihedral angle, grain boundary, polycrystalline material, tessellation.
INTRODUCTION
The explanation of microstructural changes
induced in polycrystalline metals by various kinds
of thermomechanical treatments is a prerequisite
for understanding their macroscopic behavior. We
are particularly interested in modeling the spatial
arrangements of grain boundaries in these materials,
since the grain boundaries are of paramount
importance in explaining many physical phenomena.
Obviously, the morphology of grain boundaries is
directly related to mechanical response, thermal and
electrical conductivity, magnetism and many other
properties of polycrystalline materials (Grimvall,
1999).
From a mathematical point of view, the grain
microstructure of polycrystalline materials can be
described by tessellations, as the latter are divisions
of space into non-overlapping regions (Chiu et al.,
2013). Many parametric tessellation models have been
invented to describe cellular structures in various
fields of research (Okabe et al., 2010). Concerning
materials science, great interest has been devoted to
the application of convex tessellation models, in which
the grains are convex polyhedra; see, e.g., Lyckegaard
et al. (2011). The simplicity of these models allows
a simple evaluation of size and shape characteristics
of the grains (Lautensack and Zuyev, 2008) and
relatively fast and accurate fitting to empirical data
(Spettl et al., 2016). However, in connection with
recent progress in microscopic research, the interest
of scientists has significantly increased regarding more
general tessellations with curved boundaries, which
can better describe real grain shapes.
In Altendorf et al. (2014), several such models
were applied to martensitic and bainitic steels. These
models are based on isotropic or anisotropic grain
growth, where the tessellations are constructed on
the basis of an initial approximation of the grains
by balls or ellipsoids. Optimized methods for fitting
these models to real data were presented in Alpers
et al. (2015), Teferra and Graham-Brady (2015) and
Šedivý et al. (2016). These results have shown that, in
particular, the ellipsoid-based tessellation models are
able to describe a great variety of grain microstructures
with very high precision.
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ŠEDIVÝ O ET AL: Description of the grain boundary morphology
With the advent of new tessellation models used
in materials science, there is a need for thorough
investigation of geometric properties of these models.
Their simple parametric form often allows to express
geometric characteristics by explicit formulas. Thus,
as another benefit of these models, they can replace
further processing of the data needed to obtain the
smooth surfaces from which these characteristics are
usually estimated.
The estimation of geometric characteristics
from ellipsoid-based tessellation models is the
main contribution of this paper. While previous
research has mostly concentrated on representing
real microstructures by these models, significantly
fewer results are available concerning geometric
characterization of the models. We show that size and
shape characteristics, which are commonly used for
the statistical description of this kind of data, can be
analytically computed from the model parameters. At
first, we focus on volumes, surface areas and lengths of
the grains, grain boundaries and triple grain junctions,
respectively. Further, we deal with the estimation
of surface curvatures defined on grain boundaries
and dihedral angles defined on triple junctions.
We also provide a comparison with estimations by
conventional methods from empirical data.
The aforementioned characteristics have a
straightforward application in modeling real grain
growth. The knowledge of mean curvature allows
to predict the capillary-driven migration of grain
boundaries, while dihedral angles are needed for
the description of constraints at the triple junctions
(Gottstein and Shvindlerman, 2010). The size
characteristics are useful for quantification of the
microstructural changes caused by the migration of
grain boundaries.
EXPERIMENTAL BACKGROUND
We consider two experimental data sets obtained
by different experimental techniques. Sample I is
an aluminum alloy with nominal composition Al-
3 wt% Mg-0.2 wt% Sc. A sample with square cross
section was processed by eight passes of equal
channel angular pressing (ECAP) at room temperature
and subsequently annealed for 1 h at 400◦C. These
processes led to a microstructure with well-defined fine
grains and with mean grain size of 1.9 µm3; see Fig. 1a.
A 3D image of the microstructure of Sample I was
obtained by electron backscatter diffraction (EBSD).
In this method, the sample is repeatedly exposed to
two different beams, an electron beam for scanning
the surface of the sample and a focused ion beam for
sectioning the material (Schwartz et al., 2009). In such
a manner, a 3D image is obtained from a stack of 2D
images. The size of the final 3D image, resulting from
preprocessing and alignment of the section planes,
was 21.6× 28.9× 12.0 µm3 and the voxel size was
0.13 µm3. For more details on sample preparation and
data acquisition, see Šedivý et al. (2016), Šedivý and
Jäger (2017).
Sample II was taken from an aluminum alloy
with nominal composition Al-1 wt% Mg. A cylindrical
specimen of the material was annealed at 400◦C for
1 h. This process led to a microstructure with mean
grain size of 0.36mm3; see Fig. 1b.
A 3D image of the microstructure of Sample
II was obtained by 3D X-ray diffraction (3DXRD)
microscopy. In this nondestructive technique, the
material is illuminated with a monochromatic X-ray
beam and rotated up to 360◦, such that crystallographic
information from the whole volume is obtained. The
experiment was performed at the synchrotron radiation
facility SPring-8. The final cylindrical data had an
approximate radius of 0.8mm and height of 2.7mm,
and the voxel size was 53 µm3. Details on data
acquisition and reconstruction can be found in Dake
et al. (2016).
TESSELLATION MODELS
Consider a countable collection of closed sets,
T = {Ci ⊂ R3}, such that
1. C̊i∩C̊ j = /0 for all i 6= j, where C̊i is the interior of
the set Ci,
2.
⋃
iCi = R3,
3. T is locally finite, i.e., #{Ci ∈T : Ci∩B 6= /0}<∞
for all bounded B⊂ R3.
Then T is called a tessellation of the space R3. A
tessellation can be viewed as a division of space into
non-overlapping sets, which are usually called cells or
grains. For more details, see, e.g., Chiu et al. (2013).
In the following we focus on tessellations
generated by a locally finite point pattern, P , the
points of which we call the seeds or generators of the
tessellation. The cell Ci corresponding to a given seed,
xi ∈P , is defined as the set of points in R3 that are
closer to xi than to any other seed in P with respect to
an appropriate distance measure d. Formally,
Ci = {y∈R3 : d(y,xi)≤ d(y,x′) for all x′ ∈P} . (1)
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Image Anal Stereol 2017;36:5-13
a) b)
Fig. 1. Grain microstructure of the analyzed samples.
a) Sample of Al-3 wt% Mg-0.2 wt% Sc acquired by 3D
EBSD, b) sample of Al-1 wt% Mg acquired by 3DXRD.
A classical tessellation model is the Voronoi
tessellation, where the distance measure is given by
the Euclidean distance
dV (y,xi) = ‖y−xi‖ . (2)
The cells of a Voronoi tessellation are convex
polyhedra; see, e.g., Chiu et al. (2013). In more general
cases, we furnish each generating point with additional
parameters. Cells of the tessellation are then generated
by a marked point pattern, and the distance measure d
in Eq. 1 is a function of both the points’ locations and
their additional parameters (marks). A first extension
of the Voronoi tessellation is the Laguerre tessellation
or power diagram; see Lautensack and Zuyev (2008).
Its generating marked point pattern is P = {xi,wi} ⊆
R3×R, and the distance measure is given as
dL(y,(xi,wi)) = ‖y−xi‖2−wi. (3)
Here, a real parameter, called power, is assigned to
each cell, affecting its size and shape. As for the
Voronoi tessellation, this definition leads to planar cell
facets. However, the additional parameter allows for
the generation of tessellations with greater variation
in terms of both cell size and shape than Voronoi
tessellations (Jeulin, 2013; Teferra and Graham-Brady,
2015). Actually, any normal tessellation in 3D can
be described by a Laguerre tessellation (Lautensack
and Zuyev, 2008). Note that the function dL can
take on negative values, which simplifies subsequent
considerations. However, positive distances can easily
be obtained simply by shifting all weights by an
appropriate additive constant that is identical for each
seed. This operation does not change the inequality in
Eq. 1.
Voronoi and Laguerre tessellations have
extensively been used in literature to model grain
microstructures; see, e.g., Kumar and Kurtz (1994)
and Lyckegaard et al. (2011). However, modern
methods of microscopic research have reliably proven
that the grain boundaries in real microstructures are
not perfectly planar. This motivates us to apply the
definition given in Eq. 1 with different kinds of
distance measures such that the boundaries between
the cells of tessellations are curved.
Examples of such models were presented in Jeulin
(2013) and Altendorf et al. (2014). Here, the so-
called local Voronoi tessellation is defined using a local
metric
dM(y,(xi,Mi)) = (y−xi)>Mi(y−xi), (4)
where Mi is a real-valued positive definite 3×3 matrix.
Each cell of this tessellation has 6 parameters, which
can be identified with the 6 independent entries of Mi.
In the special case in which Mi is the identity matrix,
this model corresponds to the Voronoi tessellation.
This local metric model is further extended in
Alpers et al. (2015) to obtain the generalized balanced
power diagram (GBPD). This is a tessellation
generated by the local distance measure
dGB(y,(xi,Mi,wi)) = (y−xi)>Mi(y−xi)−wi, (5)
where Mi is a real-valued positive definite matrix and
wi ∈ R. In this model, each cell is defined in terms of
3 coordinates of the seed and 7 additional parameters.
In the special case where Mi is the identity matrix, this
model corresponds to the Laguerre tessellation.
In what follows we will use tessellations obtained
from the local metric dGB, being the most general
representative of a class of models that we entitle
ellipsoid-based tessellations. Obviously, the same
principles apply for its submodels, including the one
obtained from the local metric dM, potentially having
the matrix Mi in a special form such that the number of
free parameters is smaller. Such special cases include
tessellations generated by oblate/prolate ellipsoids or
balls.
Tessellations fitted to empirical data will be
restricted to a bounded spatial domain W ⊂ R3.
Obviously, cells of such tessellations are bounded.
However, our definition allows tessellations to include
lower-dimensional or empty cells (cells with which
no point in real space is associated, because the
corresponding set Ci defined in Eq. 1 is empty). These
cases correspond to a situation in which an observed
grain has no counterpart in the fitted tessellation
model. Often, when a small and irregularly shaped
grain is surrounded by larger grains, then, in the model,
the region containing the small grain will be covered
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ŠEDIVÝ O ET AL: Description of the grain boundary morphology
by tessellation cells fitted to the larger neighbours.
Consequently, an empty cell is produced. A non-
empty, but lower-dimensional cell can arise if the seed
of the diminished cell lies exactly on a facet of the
tessellation.
MODEL FITTING
Fitting of ellipsoid-based tessellations to real data
is a challenging task. For larger data sets, the problem
becomes high dimensional because the number
of parameters in the presented models increases
linearly with the number of grains. However, simple
heuristic approaches exist that usually provide nice
approximations (Altendorf et al., 2014). For ellipsoid-
based tessellations, the heuristic approximation is
based on best-fitting ellipsoids found to the original
grains. The centres of ellipsoids are identified as
centres of mass of the grains, and the principal axes
are found by principal component analysis.
This heuristic approach can be improved further
by optimization techniques that iteratively increase the
accuracy of the fit by modifying the parameters. Such
methods were presented, e.g., in Alpers et al. (2015);
Teferra and Graham-Brady (2015). However, they are
difficult to apply to large data sets. We rather use a
stochastic optimization algorithm based on simulated
annealing presented in Šedivý et al. (2016). In each
step of this algorithm, only a subset of parameters of
one grain is modified. This allows a fast reconstruction
of the modified tessellation, because the change affects
only the immediate vicinity of the modified grain. The
suggested change in parameter values is accepted with
a certain probability in each iteration, such that an
optimal solution is approached.
To assess the quality of the fitted model with
respect to empirical data, we utilize several criteria
that compare the structures in both a metrical and a
topological sense. Assume that the empirical image
data are observed on a voxel grid W ′ ⊂ W within a
bounded window W ⊂ R3. Let I(x) denote the grain
index at a point x in the empirical image data and
IT (x) the index of the cell of the tessellation fitted to
this data. Further, by BI we denote the set of boundary
voxels in the empirical image data, i.e.,
BI = {x ∈W ′ : ∃y ∈W ′, I(x) 6= I(y),‖x−y‖ ≤
√
3}.
The latter condition (with distance measured in voxel
size) expresses the requirement that the corresponding
voxels represented as cubes share at least a point.
By BT we denote an analogous set of boundary
voxels in the tessellation model. The basic discrepancy
measure
DI,T =
#{x ∈W ′ : I(x) 6= IT (x)}
#{x ∈W ′}
(6)
provides information on the fraction of voxels for
which these two indices are different. For a local
quality of fit near the grain boundaries, we use
the distance of the observed grain boundaries to
their modeled counterparts. This distance is evaluated
for each pair of neghbouring boundary points with
different grain index, x,y ∈ BI : I(x) 6= I(y), ‖x−y‖ ≤√
3, and it is computed as
νI,T (x,y) = min
z∈B∗T (I(x),I(y))
∥∥∥∥x+y2 − z
∥∥∥∥ , (7)
where B∗T (i, j) stands for the exact boundary between
cells Ci and C j in the tessellation model.
Finally, we define some topological characteristics.
Denote by Gi ∼ G j the relation that the grains Gi
and G j are neighbours, i.e., there exists a pair of
points x,y ∈ BI : I(x) = i, I(y) = j, ‖x− y‖ ≤
√
3.
Analogously, Ci ∼ C j denotes the neighbourhood
relationship of cells in the fitted tessellation. Let N be
the total number of grains observed in the empirical
image data. Then for the ith grain the quantity
µI,T (i) = #{ j ∈ 1, . . . ,N : (Gi ∼ G j and Ci 6∼ C j)
or (Gi 6∼ G j and Ci ∼ C j)} (8)
describes the neighbourhood fit by counting the
number of disagreements in the list of neighbours in
the observed data and the model, respectively. The
average
µ̄I,T =
1
N
N
∑
i=1
µI,T (i) (9)
is called the mean neighbourhood fit.
SIZE CHARACTERISTICS
The boundaries between cells in the ellipsoid-
based model are parts of quadric surfaces. Using
the parameters from Eq. 5, these quadrics can be
represented as
Qi j(x) = x>Ai jx+Bi jx+Ci j = 0 , (10)
where
Ai j = Mi−M j ,
Bi j =−2(Mixi−M jx j) ,
Ci j = x>i Mixi−x>j M jx j .
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Image Anal Stereol 2017;36:5-13
This implicit representation can be transformed to a
parametric representation of the surface
qi j(u,v) = (x(u,v),y(u,v),z(u,v)) , u,v ∈S ⊂ R2,
and, similarly, for triple grain junctions we can obtain
a parametric representation of the curve
ti jk(u) = (x(u),y(u),z(u)) , u ∈L ⊂ R.
Here, the sets S and L represent the restricted
domains for the parameters such that the
bounded surfaces or curves are generated. For the
parametrization, a general method can be used that
is based on minimal µ-bases (Chen et al., 2007).
Both parametric functions qi j(u,v) and ti jk(u) are
differentiable in each inner point of the boundary
or the triple junction, respectively, because they
are represented by polynomial equations. Using
the aforementioned representations, the basic size
characteristics can be expressed as follows. The
volume of a grain Gi is given by
Vi =
∫
x∈Gi
dx , (11)
the surface area of a boundary between grains Gi and
G j is given by
Si j =
∫
S
∥∥∥∥∂qi j(u,v)∂u × ∂qi j(u,v)∂v
∥∥∥∥d(u,v) , (12)
and the length of a triple junction shared by the grains
Gi,G j and Gk reads
Li jk =
∫
L
∥∥∥∥∂ ti jk(u)∂u
∥∥∥∥du . (13)
LOCAL SHAPE
CHARACTERISTICS
Consider a point x lying at a triple junction shared
by the grains Gi,G j and Gk. The normal vectors to the
three boundary surfaces are
ni j(x) =±∇Qi j(x) ,
nik(x) =±∇Qik(x) ,
n jk(x) =±∇Q jk(x) .
With appropriate signs for the normal vectors, the
dihedral angles can be evaluated as
αi j(x) = arccos
(
1−
‖ñik(x)− ñ jk(x)‖2
2
)
, (14)
etc., where ñi j(x) =
ni j(x)
‖ni j(x)‖ .
To determine the curvature of the surface at a
surface point x, we consider the intersection curves
of the surface with the planes orthogonal to the
surface at point x. Principal curvatures κ1(x),κ2(x) are
defined as maximum and minimum curvature of the
intersection curves at x, the mean curvature is κM(x) =
(κ1(x) + κ2(x))/2, and the Gaussian curvature is
κG(x) = κ1(x)κ2(x).
For the implicit surface given by the equation
Q(x) = 0, we denote the Hessian by H(x), and the
matrix of its cofactors by H∗(x). Then the Gaussian
curvature can be expressed as
κG(x) =
∇Q(x)H∗(x)∇Q(x)>
‖∇Q(x)‖4
, (15)
the mean curvature is
κM(x) =
∇Q(x)H(x)∇Q(x)>−∇Q(x)2 tr(H(x))
2‖∇Q(x)‖3
,
(16)
where tr( ·) is the trace of a matrix, and the principal
curvatures are
κ1,2 = κM(x)±
√
κM(x)2−κG(x) , (17)
see Goldman (2005).
Finally, we explain how the same characteristics
were computed from the empirical data. The grain
volumes were simply computed by counting the
voxels belonging to particular grains. For the other
characteristics, the voxelized grain boundaries were
first transformed to fine triangular surface meshes,
using the filters ‘Quick Surface Mesh’ and ‘Laplacian
Smoothing’ from the DREAM.3D software library
(Groeber and Jackson, 2014). Grain boundary surfaces
were obtained by summing up the areas of the triangles
of these meshes that correspond to given boundaries.
Similarly, the lengths of triple junctions were obtained
by summing up the lengths of the triangle edges that
correspond to these junctions.
The triangular surface meshes were also used for
computing the curvatures. However, the accuracy is
not sufficient near triple junctions. Indeed, during the
Laplacian smoothing, triangles lying in the immediate
vicinity of the junction are influenced by the vertices of
triangles lying on adjacent boundaries. For this reason,
the curvatures were evaluated only on triangles located
at a sufficient distance from the triple junctions.
A similar difficulty arises with the dihedral angles,
which are to be evaluated from the normal vectors
at triple junctions, see Eq. 14. However, the normal
9
ŠEDIVÝ O ET AL: Description of the grain boundary morphology
vectors can hardly be estimated near triple junctions
with sufficient accuracy. This can be overcome by
computing the normal vectors at a certain distance
from the junction, where the influence of adjacent
boundaries on the surface mesh is negligible.
RESULTS
Both experimental data sets were approximated
with ellipsoid-based tessellations using the simulated
annealing methodology described in Šedivý et al.
(2016). With a stopping condition defined therein, the
algorithm was terminated after roughly 22 million
iterations for Sample I and 5 million for Sample II,
respectively, when no improvement in the fit was
observed within a certain number of consecutive
iterations. Table 1 summarizes the basic characteristics
for evaluating the goodness of fit. From these
results we see that the model provides a very good
approximation to the empirical data. Expressed by
the percentage of correctly assigned voxels, the fit
is approximately 92% for Sample I and 95% for
Sample II.
In addition, we assess the quality of the model
by the functional characteristic given in Eq. 7. The
latter offers better insight into local discrepancies
between the observed grain boundaries and interfaces
between cells in the fitted tessellation. The results
of this comparison, depicted in Fig. 2, reveal that,
for the majority of testing points on observed grain
boundaries, the distance to the nearest interface in the
model is within one voxel in both samples.
Concerning the size characteristics, we present
results for grain volumes and surface areas of grain
boundaries in Fig. 3. For the empirical data, these
characteristics were computed from voxelized data
(volumes) or meshed data (surface areas). For the
model data, the characteristics were computed by
numerical integration of the analytical formulas,
see Eqs. 11, 12 and 13. Despite the fact that
we deal with two different materials observed at
very different length scales, the plots in Fig. 3
reveal a strong similarity in the distributions of the
size characteristics. Moreover, the difference between
estimation based on empirical data and computed from
model data is very small.
Further, we focus on dihedral angles. For both
empirical and model data, these angles were computed
on a set of testing points on the triple grain boundary
junctions. For each point, we have a triplet of angles
summing up to 360◦. The distributions of dihedral
angles are compared in Fig. 4a. Dihedral angles
measured from empirical data have a sharper peak
about an equilibrating value of 120◦. This result is to
be expected, as quadric surfaces are less accurate in
modeling real grain boundaries near triple junctions,
where the boundaries are typically more highly curved
due to specific constraints.
Table 1. Statistics for the quality of fit of
approximations to empirical data by ellipsoid-
based tessellations generated by the metric dGB. 1st
row: percentage of correctly assigned voxels; 2nd-4th
row: percentage of cells with at most 0/1/2 incorrect
neighbours, respectively; 5th row: mean number of
incorrect neighbours per grain.
Sample I Sample II
(1−DI,T ) · 100% 92.03 95.03
% of cells with µI,T (·) = 0 35.67 49.48
% of cells with µI,T (·)≤ 1 70.54 80.59
% of cells with µI,T (·)≤ 2 87.59 94.67
µ̄I,T 1.20 0.79
0 1 2 3 4 5
0
.0
0
.5
1
.0
1
.5
Distance (voxels)
D
e
n
s
it
y
Sample I − empirical vs. model data
Sample II − empirical vs. model data
Fig. 2. Empirical distribution of the nearest distance
from testing points on observed grain boundaries to
the modeled grain boundaries.
Finally, we present results for curvatures. Similar
to dihedral angles, curvatures were computed on a set
of testing points on the grain boundaries. In Fig. 4b we
present the results for the Gaussian curvature. Because
the grain boundaries are nearly planar, most of the
curvatures are concentrated around zero. For better
readability of the plots, we include only those values
that exceed a threshold of 0.01 (in absolute value).
Such a step has no effect other than lowering the peak
near zero arising from the kernel estimation of the
density.
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Image Anal Stereol 2017;36:5-13
0
.0
0
0
.0
5
0
.1
0
0
.1
5
0
.2
0
0
.2
5
0
.3
0
D
e
n
s
it
y
−4 −2 0 2 4
Logarithm of grain volume in Sample I (cubic microns)
−4 −2 0 2 4
Logarithm of grain volume in Sample II (cubic millimeters)
Sample I − empirical data
Sample I − model data
Sample II − empirical data
Sample II − model data
a)
0
.0
0
.5
1
.0
1
.5
2
.0
2
.5
D
e
n
s
it
y
0.0 1.0 2.0 3.0
Surface area of grain boundary in Sample I (squared microns)
0.0 0.5 1.0 1.5
Surface area of grain boundary in Sample II (squared millimeters)
Sample I − empirical data
Sample I − model data
Sample II − empirical data
Sample II − model data
b)
Fig. 3. Comparison of the size characteristics in the
empirical and the model data. a) Grain volume, b)
surface area of grain boundary.
A typical feature of the empirical estimation is that
negative values prevail. This is a consequence of the
nature of the surface meshing procedure, in which an
initial mesh is produced by drawing triangles on the
voxel faces, and this mesh is iteratively smoothed. On
the contrary, Gaussian curvatures evaluated from the
model data are rather slightly positive, which is typical
for the smooth quadric surfaces generated by the
model. This is especially obvious for Sample II, where
the majority of values lie within a short range centered
about a positive value. A graphical representation of
the curvatures on a subset of the empirical data is given
in Fig. 5.
0 50 100 150
0
.0
0
0
0
.0
0
5
0
.0
1
0
0
.0
1
5
0
.0
2
0
0
.0
2
5
Dihedral angle (degrees)
D
e
n
s
it
y
Sample I − empirical data
Sample I − model data
Sample II − empirical data
Sample II − model data
a)
−1.0 −0.5 0.0 0.5 1.0
0
2
4
6
8
1
0
1
2
Gaussian curvature
D
e
n
s
it
y
Sample I − empirical data
Sample I − model data
Sample II − empirical data
Sample II − model data
b)
Fig. 4. Comparison of the local shape characteristics
of the empirical and model data. a) Dihedral angle,
b) Gaussian curvature. Dashed vertical line stands for
an equilibrating value of 120◦ in subfigure (a) and zero
Gaussian curvature in subfigure (b), respectively.
CONCLUSIONS
Our results confirm that ellipsoid-based
tessellations are quite precise in describing
polycrystalline microstructures. In particular, grain
boundaries can be very well fitted by quadric surfaces,
which arise as interfaces between cells in these models.
Besides the commonly used discrepancy measures
(Table 1), we support this claim by a functional
characteristic that directly measures distances between
the empirical and modeled grain boundaries on a dense
set of testing points (Fig. 2).
11
ŠEDIVÝ O ET AL: Description of the grain boundary morphology
a) b)
Fig. 5. Gaussian curvature κM(x) of grain boundaries
in a cutout of Sample I. a) Empirical estimation based
on a triangular surface mesh, b) estimation obtained
from the ellipsoid-based tessellation model. Extreme
curvatures near triple junctions in the empirical
estimation are coloured gray.
A simple description of the modeled grain
boundaries by quadric surfaces allows a tractable
evaluation of size and shape characteristics of the
grains, grain boundaries and triple junctions. Our
results suggest that there is very good agreement
between characteristics computed from both modeled
and empirical image data. A great benefit of the
model is that it can substitute the pre-processing
operations that are required to obtain smooth grain
boundaries. Moreover, the quadric surfaces resulting
from the model are much more suitable for obtaining
nice and consistent approximations of curvatures
and dihedral angles between grain boundaries in
comparison to estimations from meshed data, for
which the smoothing parameters are difficult to
calibrate properly.
ACKNOWLEDGEMENTS
This research was partially financed by the project
GBP108/12/G043, Czech Science Foundation. We
thank Leoš Polı́vka of the Institute of Physics of
the Czech Academy of Sciences for the preparation
of Sample I and 3D EBSD data acquisition. In
addition, we are grateful to the Japan Synchrotron
Radiation Research Institute for the allotment of beam
time on beamline BL20XU of SPring-8 (Proposal
2013A1506), and we thank Dmitri Molodov of the
Institute of Physical Metallurgy and Metal Physics,
RWTH Aachen, for providing the Al-1 wt% Mg
specimen.
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