Three-dimensional reconstruction of statistically optimal unit cells of polydisperse particulate
composites from microtomography
H. Lee,
1
M. Brandyberry,
1
A. Tudor,
1
and K. Matouš
2,
*
1
Computational Science and Engineering, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801, USA
2
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA
Received 19 August 2009; published 9 December 2009
In this paper, we present a systematic approach for characterization and reconstruction of statistically
optimal representative unit cells of polydisperse particulate composites. Microtomography is used to gather
rich three-dimensional data of a packed glass bead system. First-, second-, and third-order probability functions
are used to characterize the morphology of the material, and the parallel augmented simulated annealing
algorithm is employed for reconstruction of the statistically equivalent medium. Both the fully resolved prob-
ability spectrum and the geometrically exact particle shapes are considered in this study, rendering the opti-
mization problem multidimensional with a highly complex objective function. A ten-phase particulate com-
posite composed of packed glass beads in a cylindrical specimen is investigated, and a unit cell is reconstructed
on massively parallel computers. Further, rigorous error analysis of the statistical descriptors probability
functions is presented and a detailed comparison between statistics of the voxel-derived pack and the repre-
sentative cell is made.
DOI: 10.1103/PhysRevE.80.061301 PACS numbers: 45.70.-n, 05.20.-y, 46.65.+g, 87.59.-e
I. INTRODUCTION
Computational methods based on particulate packs are
commonly used in variety of scientific disciplines. For ex-
ample, particulate packs have been used in modeling of het-
erogeneous materials, such as solid propellants 1,2, granu-
lar media 3, protein folding 4, and low-temperature
phases of matter such as liquids, crystals, and glasses 5.
Moreover, packing problems are common in information
theory 6 and many different branches of pure mathematics
7. A study of these systems in a computational framework
usually starts with a model of the morphology, such as a
packing algorithm. Therefore, a packing algorithm to guide
these models has been in the forefront of mathematical and
scientific investigations for many decades, and this fascina-
tion led to development of several packing codes that are
capable of producing high-quality polydisperse heteroge-
neous packs 8–10.
Recently, new developments in three-dimensional 3D
imaging using microtomography micro-CT have also ush-
ered in the rapid expansion of statistical modeling techniques
that investigate the morphology and the microstructure char-
acterization of widely used material systems, such as propel-
lants 11,12, glass beads 13, paper 14, and engineered
cementitious composites 15, just to name a few. An ex-
ample of the complicated microstructures obtained from the
micro-CT can be seen in Fig. 1.
The subsequent statistical characterization is usually per-
formed, for both computationally and/or tomographically ob-
tained packs, in order to understand the internal structure of
these systems. The need for such understanding and impor-
tance of the higher-order statistics start with early work of
Bernal 16 who investigated the geometrical structure of
liquids using the radial distribution function. Significance of
statistical description galvanized several research communi-
ties in condensed matter physics and far beyond, with appli-
cations in non-Gaussian noise as a tool to study disordered
materials 17, and application of Minkowski functionals in
analysis of background cosmic radiation 18, just to name a
few. Moreover, the analysis of higher-order statistics, in the
guise of the analysis of x-ray speckle, is increasingly gaining
attention among condensed matter physicists 19.
Unfortunately, both computationally and/or tomographi-
cally derived packs are often too large to be uniformly re-
solved in practical numerical simulations of combustion phe-
nomena 1, nonlinear viscoelastic response of a binder 20,
or damage evolution along the particle-matrix interface 21.
Therefore, many researchers have devoted their attention to
finding a statistically optimal unit cell. Povirk 22 proposed
a method for determining periodic microstructures in two
dimensions that are statistically similar to more complex,
random, two-phase microstructures by using a certain statis-
tical descriptor function. Yeong and Torquato 23,24 pro-
posed a method for the reconstruction of random media
based on two-point probability functions using simulated an-
nealing SA and two-dimensional material slices. Bochenek
and Pyrz 25 also used the simulated annealing procedure in
conjunction with a pair correlation function and a stress in-
teraction parameter to reconstruct a unit cell in three dimen-
sions. However, the simple pair correlation function used in
their work did not represent the disparate particle modes, and
thus, the probability spectrum optimized in 25 was re-
stricted. Zeman and Šejnoha 26 examined three consider-
ably different material systems, a fiber composite, a woven
composite, and a masonry, to demonstrate the reconstruction
using two-point probability functions and the lineal path
function, yet again, only two-dimensional images of micro-
structures have been employed.
Recently, Jiao et al. 27,28 reconstructed a three-
dimensional realization of Fontainebleau sandstone and a
boron-carbide or aluminum composite from two-dimensional
*
Corresponding author; kmatous@nd.edu
PHYSICAL REVIEW E 80, 061301 2009
1539-3755/2009/806/06130112 ©2009 The American Physical Society 061301-1