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MAY/JUNE 2001 15
S
cientists strive to develop and understand
materials (and the processes undertaken
to manufacture them) by associating
properties with a material’s microstruc-
ture. These properties can include chemical
composition, crystal structure, crystalline orien-
tation, magnetic susceptibility, and thermal con-
ductivity. Even single-phase materials have mi-
crostructures that affect properties significantly.
For instance, steel is composed of grains that
have the same chemical composition and crystal
structure, but metallurgists can vary their shape,
size, and degree of alignment by a judicious
choice of processing parameters. For example,
the materials in a transformer core are manufac-
tured so that their grains are primarily crystal-
lographically aligned to optimize mechanical and
magnetic properties. Materials often contain de-
fects and second phases in small fractions that
can ultimately affect integrity and reliability.
Because microstructures are such complex en-
sembles of materials and crystallography, few re-
searchers have attempted to correlate known mi-
crostructures with their properties.
1–3
Instead, they
have used mean-field and other spatial averaging
techniques to produce approximate models for ma-
terials behavior. In some cases, mean-field models
work well, but we can’t expect them to be predic-
tive for cases when material properties depend on
extremes of statistical or stereological distributions
or the spatial correlations of microstructural fea-
tures. In such cases—and even for cases where
mean-field calculations do pertain—direct com-
putations that use all pertinent microstructural in-
formation should provide a useful tool for mi-
crostructure property analysis. We have created
such a tool—OOF (named for its object-oriented
finite elements). Several groups are using it to in-
vestigate material behavior,
4–10
and it received a
Technology of the Year award from Industry Week
magazine in 1999. This article presents an
overview of OOF and some of its algorithms that
are in the public domain. OOF was developed at
the NIST Center for Theoretical and Computa-
tional Materials Science and is freely available on
the Web at www.ctcms.nist.gov/oof.
Program overview
OOF consists of two programs: oof and
ppm2oof. Conceptually, ppm2oof combines
OOF: AN IMAGE-BASED
FINITE-ELEMENT ANALYSIS OF
MATERIAL M
ICROSTRUCTURES
Determining a material’s macroscopic properties given its microscopic structure is of
fundamental importance to materials science. The authors describe two public-domain
programs that jointly predict macroscopic behavior. The programs start from an image of
the microstructure and end with results from finite-element calculations.
M ATERIALS
S CIENCE
S
TEPHEN A. LANGER AND EDWIN R. FULLER, JR.
National Institute of Standards and Technology
W. C RAIG CARTER
Massachusetts Institute of Technology
1521-9615/01/$10.00 © 2001 IEEE
16 COMPUTING IN SCIENCE & ENGINEERING
microstructural data, in the form of experimental
or simulated micrographs, with fundamental ma-
terials data (such as elastic moduli and so on) and
materials physics models. The result is a repre-
sentation of a material in terms of C++ objects.
Practically, ppm2oof reads an image in the
portable pixel map format, assigns material prop-
erties to the microscopic features in the image,
and creates a finite-element mesh for subsequent
computation, all with an easy-to-use graphical
interface. (The ppm format is simple to read, and
many programs can convert from popular image
formats to it.) oof takes ppm2oof’s object rep-
resentation and performs virtual experiments that
employ the same graphical user interface to com-
pute effective macroscopic physical properties or
illuminate the microstructural physics.
OOF’s user interface is designed to be flexible,
expandable, and user friendly. The interface has
command windows, which present menus of
variables, functions, and submenus; graphics win-
dows, which present images and graphical tools
for manipulating the images; and function win-
dows, which set function arguments and serve as
a quick way to perform common operations.
Every action the user takes has a text representa-
tion, which the program can save in a log file.
The user can edit this log file and read it back
into the program, allowing for scripted sessions.
Currently, there are two versions of OOF. The
original OOF solves linear thermoelasticity prob-
lems with a constant temperature field and in-
cludes simple models of fracture, damage, and
ferroelastic domain switching. The “thermal”
OOF extends the original version by allowing the
temperature to vary in space, and it solves for
thermal diffusion as well as elasticity. This arti-
cle concentrates on the original elastic program.
An example problem
Figure 1 illustrates a typical microstructure.
Suppose that the overall (macroscopic) thermo-
elastic response of the microstructure in Figure 1
is a property of interest. If the two phases in Fig-
ure 1 have different elastic stiffnesses, then the av-
erage elastic response to an externally applied dis-
tortion will depend on the distortion’s direction.
The stresses in the interior will not be distributed
homogeneously and will depend in detail on the
boundary conditions. Figure 2 exhibits the stress
patterns generated by distorting this microstruc-
ture in the vertical and horizontal directions. For
this example’s purposes, we used fictitious mater-
ial constants for the two phases and took them
both to be isotropic with Poisson’s ratios of 0.33.
The Young’s modulus of the light material was 10
times that of the dark, and its thermal expansion
coefficient was three times as large.
If the phases in Figure 1 have differing coeffi-
cients of thermal expansion, thermal strains will
be anisotropic and inhomogeneous, even if the
component materials are isotropic and homoge-
neous. The microstructure’s overall response will
have the symmetry of the composite architecture.
Combining images with constitutive
models
The purpose of ppm2oof is to assign proper-
ties to features in a material’s image and to gen-
erate a finite-element mesh representing the ma-
terial. This involves approximations from the
first step; the digital image is a discretized ap-
proximation to the real material. Boundaries be-
tween grains in the real material might be
smooth (have no sharp corners) and distinct (de-
lineate a sudden transition from one material to
another), but the boundaries between features
in the image might be jagged (due to pixeliza-
tion) and fuzzy (due to limited experimental res-
olution). Fuzzy images almost always require
some user input and judgment. With image pix-
elization, it’s almost never a good idea to make
the finite-element mesh resolve all the details in
an image, because that would also resolve pix-
elization artifacts.
Figure 1. A micrograph of a lamellar directionally solidified eutectic
of NiO (lighter phase) and yttria-stabilized ZrO
2
(darker phase).
The image’s width is approximately 12
µ
m. (Figure data courtesy of
T.C. Isabell and V.P. Dravid.)
MAY/JUNE 2001 17
Identifying features in the image
The first order of business when using ppm2oof
is to identify the image’s features and assign mate-
rial properties to them. Identifying features means
selecting groups of pixels, to which the user may
give names and store for later retrieval.
The ppm2oof program includes a number of
tools for choosing pixels. Demography tools se-
lect pixels within a given range of gray or RGB
(red, green, and blue) values. Users can set the
minimum and maximum values explicitly or
choose a target pixel interactively with the
mouse, leading to the selection of all the pixels
within a specified range of the target’s color.
The demography tools ignore the selected
pixels’ locations. In contrast, the burn algorithms
select sets of contiguous pixels. A burn starts at a
given pixel and spreads outward like a (relatively
benign) forest fire, selecting pixels as it goes.
Specified “flammabilities” determine whether
the selection spreads from one pixel to the next.
(Our burn and demography tools are examples
of simple image segmentation algorithms. More
sophisticated methods appear elsewhere.
11
)
The micrograph in Figure 1, although typical, is
actually not a good candidate for the demography
or burn tools because the boundaries are not dis-
tinct. Figure 3 shows the results of a burn and a
roughly comparable demography application. If it
is possible to modify an image to make the bound-
aries more distinct, then the selection tools be-
come more precise. In Figure 4, we applied the
same selection tools as in Figure 3, but we applied
them after 10 iterations of a nonlinear smoothing
operation designed to reduce noise while preserv-
ing boundaries. (Interpreting the image’s gray val-
ues as a field
φ
on a lattice, we take a set number of
time steps of Euler’s method on an anisotropic dif-
fusion equation suggested by Allen Tannenbaum:
(1)
where
α
is a parameter between 0 and 1.)
This nonlinear smoothing operation is one of
a handful of image modification routines built
into ppm2oof—it isn’t meant to be a general-
∂
∂
=
−+
++
()
φ
φφ φφφ φφ
αφ φ
t
xx y x y xy yy x
xy
22
22
13
2
1
/
Figure 2. The results of OOF calculations on Figure 1. The mesh has
3,798 nodes and 7,440 elements. Stresses increase as color
progresses from black to red to yellow to white. This example uses
fictional material parameters, so the absolute values and units for
the stresses are not meaningful. (a) Stress
σ
xx
in response to strain
ε
xx
= 1 percent. (b) Stress
σ
yy
in response to strain ε
yy
= 1 percent—
the microstructure’s influence is clear. (c) Hydrostatic stress in
response to thermal strain with free boundaries.
(a)
(b)
(c)
18 COMPUTING IN SCIENCE & ENGINEERING
purpose image manipulation package, although
it does have a number of tools that we have
found to be useful in working on micrographs.
In addition, ppm2oof can import many differ-
ent versions of an image into an image gallery
and work on them all simultaneously. A user can
therefore enhance different features of an image
with external programs and then select any of
the visible features in one of a set of overlaid im-
ages in ppm2oof. All the images in an image
gallery share the set of selected pixels, the mate-
rial properties assigned to them, and the finite-
element mesh (which is created from the mate-
rials specified in the selection algorithms).
Once the user makes a selection, he or she can
modify it in ppm2oof with one of several com-
mands. For example, expand selects all the pixels
within a specified distance of the current set of
selected pixels. Despeckle recursively selects all
pixels with a specified minimum number of se-
lected neighbors. The inverses of these opera-
tions—shrink and elkcepsed (“despeckle” spelled
backwards)—reduce the selected set’s size. Fig-
ure 5 is Figure 4c after using both despeckle and
elkcepsed.
Assigning material properties to image
features
After selecting pixels in ppm2oof, the user
can assign material properties and a corre-
sponding grayscale value to them to form a ma-
terial image (see Figure 6). OOF understands lin-
ear elasticity, thermal expansion, and thermal
conductivity. Although it always performs calcu-
lations in two dimensions (either in plane stress
or plane strain), OOF materials can have any 3D
crystal symmetry and any 3D crystallographic
orientation. The material parameters always
have their full 3D form.
OOF’s object-oriented nature makes it rela-
tively easy to add new materials. For example, al-
though OOF can handle general crystal symme-
try with 21 elastic constants, the user interface
for such a material would be a mess. Therefore,
OOF started with only two types of elastic ma-
terials: isotropic and anisotropic. The anisotropic
material class provided no way to set the material
parameters, but had all the rest of the machinery
necessary for constructing finite-element stiff-
ness matrices, computing stress and strain, and
so forth. When the need arose for specific
anisotropic materials (hexagonal, trigonal, and so
on), we easily added them by deriving new ma-
terial classes from the anisotropic base class.
Generating and refining meshes
In most finite-element programs, you specify a
problem’s geometry, and the computer fits a mesh
to mathematical boundaries. However, when
working with micrographs, the geometry is not
so well defined. One approach would be to force
the boundaries to be well defined, but even in Fig-
ure 5, where the image is smoothed and the se-
lection despeckled, you would not want to claim
that the red region is a precise representation of
the ZrO
2
phase in Figure 1. Remember that the
image is an approximate representation of the
physical system, the material image is an approx-
Figure 3. The upper left corner of Figure 1, showing regions selected for (a) burning and (b) demography. The selected
pixels appear in red—the burn started from a pixel near the center of the red region in (a).
(a) (b)
MAY/JUNE 2001 19
imation of the image, and the generated mesh is
an approximation to the material image. Forcing
the boundary to be well defined and then approx-
imating the boundary by the boundaries of the fi-
nite elements just introduces another level of ap-
proximation into the existing hierarchy.
A second approach would be to take every
pixel in the material image and create one
quadrilateral or two triangular elements from it.
ppm2oof can do this (with triangular elements
Figure 5. The image from Figure 4c, after despeck-
ling the selection and applying elkcepsed.
Despeckle has filled in small holes, and elkcepsed
has eliminated small islands and peninsulas.
Figure 6. A material image. The two colors corre-
spond to the two material properties assigned to
the micrograph in Figure 1. The few remaining
small light regions inside dark areas are artifacts
of the selection tools used, and user intervention
can easily clean them up.
(a)
(b)
(c)
Figure 4. A portion of Figure 1 after applying (a)
a nonlinear smoothing operation and repeating
the (b) burn and (c) demography selections from
Figure 3.
20 COMPUTING IN SCIENCE & ENGINEERING
through its “simple_mesh” menu), but this ap-
proach has two drawbacks. First, it almost always
creates far too many elements—the interiors of
large homogeneous regions should be dis-
cretized more coarsely than inhomogeneous re-
gions. Second, it resolves the jagged edges of the
pixels on the boundaries between materials, as
in Figure 7, even when the boundary in the real
material is smooth. This can result in pixeliza-
tion errors.
12
The approach ppm2oof’s adaptive mesh rou-
tines use is to subdivide triangles and move nodes
to minimize a functional E of the mesh. E is min-
imized by equilateral triangles that overlie a ho-
mogeneous set of pixels in the material image.
Thus, at any given level of refinement, the edges
of the triangles approximate the interfaces in the
material image as well as they can on the length
scale of the triangles. The user determines the
length scale by deciding when to stop the refine-
ment process—the caveat is that he or she should
generally not refine below the pixel level. There
is no need to describe the material boundaries as
mathematical curves—the homogeneity part of
E automatically finds the boundaries. Further-
more, as long as the mesh is not too fine, it will
smooth out the sharp corners of the pixels.
The functional E is a sum of two terms for
each triangular element:
E =
α
E
hom
+ (1 –
α
)E
shape
(2)
where
α
is a tunable parameter between 0 and 1,
and E
hom
and E
shape
are functionals that depend on
the element’s homogeneity and shape, respectively.
If
α
= 1, the mesh triangles tend to be highly acute,
because nodes move to put the triangle edges on
the material boundaries. If
α
= 0, the triangles tend
to be equilateral (providing good finite-element
convergence properties), but their positions are
uncorrelated with the material microstructure. A
value of
α
= 0.3 seems to work well in practice.
We compute the homogeneity term E
hom
by sep-
arating the pixels in the material image into N cat-
egories, where all the pixels in each category have
the same material type and parameters and belong
to the same pixel groups. Then for each mesh tri-
angle T, we compute the fraction a
i
(T) of its area
that overlies pixels in category i. E
hom
is defined by
.
(3)
If a triangle lies over only one category of pixel,
then it is homogeneous, a
i
(T) = 1 for some i, and
that triangle makes no contribution to E
hom
. If a
triangle contains equal areas of each pixel cate-
gory, then a
i
= 1/N for all i, and that triangle’s
contribution is the maximum value, 1.
The shape term E
shape
in Equation 2 is defined
by
(4)
where A
T
is the area of triangle T, and L
T
is its
perimeter. The parenthesized expression is 0 for
equilateral triangles and 1 for degenerate trian-
gles with collinear vertices.
Figure 8 shows how creating a mesh for the
sample problem might proceed. ppm2oof con-
tains a number of tools for mesh manipulations;
their most effective order of application might
vary from image to image. The process shown
here is typical but does not illustrate the full range
of tools. For simplicity, Figure 8 shows only the
upper left corner of the image. First, in Figure 8a,
the program creates a uniform mesh with a size
the user chooses to be roughly the size of the
largest features to be resolved. Figure 8b shows
the results of 10 iterations of an annealing proce-
dure. Note that many nodes have moved to the
internal material boundaries and that the edges
of the elements are beginning to follow the edges
of the materials. The annealing is a Monte Carlo
algorithm, where nodes are moved at random,
and moves are accepted if they reduce the total E.
In each iteration, the program attempts to move
each node once, choosing the node’s displacement
E
A
L
T
T
T
shape
=−
∑
1
36
3
2
E
aT
N
i
i
N
T
hom
()
/
=
−
−
=
∏
∑
1
11
1
Figure 7. Jagged pixelized representation of smooth boundaries,
taken from Figure 6. The finite-element mesh should not resolve
the sharp corners on such boundaries.
MAY/JUNE 2001 21
Figure 8. Steps in the meshing process. (a) The initial uniform mesh overlaid on the material image. (b) The mesh after
10 annealing steps. (c) The triangles with the largest E are divided in two. (d) The mesh after 10 more annealing steps
and one interface refinement. (e) After further annealing and refinement, the mesh closely follows the material bound-
aries. (f) A final refinement resolves some of the smaller features. In (e) and (f) the triangles are colored according to the
material type that they inherit from the pixels.
(a)
(c)
(e)
(b)
(d)
(f)
22 COMPUTING IN SCIENCE & ENGINEERING
from a Gaussian with a given width
δ
(here, 1
pixel width). The method is called annealing be-
cause it can, in fact, operate at a nonzero effective
temperature in which moves that increase E are
accepted with a thermal probability. In practice,
this has not proved to be terribly useful.
In Figure 8c, triangles with E greater than 0.3
have been divided in two, with the direction of the
separatrix chosen to minimize the total E of the re-
sult. We chose the 0.3 threshold so that a reason-
able number (approximately 20 percent) of the tri-
angles were divided. Figure 8d shows the mesh
after a further 10 annealing steps, and the applica-
tion of the refine interface command. This subdi-
vides all triangles that have neighbors of a differ-
ent material type. ppm2oof determines a triangle’s
type from the types of the pixels underneath it, ei-
ther by choosing the pixel at the center of the tri-
angle or by voting, with each pixel getting a vote
proportional to the area of its intersection with the
triangle. In this image, almost all triangles are in-
terface triangles at this stage of the process.
Figure 8e shows the result of another 10 an-
nealing steps, another interface refinement, 10
more annealing steps, an edge swap (in which the
diagonals of quadrilaterals formed by two trian-
gles are swapped, if it lowers E), and one more an-
nealing with a smaller
δ
. Here the triangles are
colored according to the material properties that
they inherit from the pixels. The mesh follows the
overall boundaries quite well on the triangle size’s
scale. Further refinement is needed to resolve
smaller features, at the risk of resolving individ-
ual pixels. The small light dot within a dark re-
gion near the upper left corner of Figure 8d is not
reproduced in Figure 8e, but it has affected the
boundary’s shape in that region. Similarly, the
narrow isthmus between the upper and lower
portions of the large light island in the upper por-
tion of Figure 8d is not resolved in Figure 8e.
Whether details such as these should be resolved
is a choice the user must make. If necessary, he or
she can define a restricted “active area” and refine
and anneal only those elements in those regions.
In fact, this example improves significantly with
one more iteration of interface refinement and
annealing, as Figure 8f shows.
Virtual experiments
oof is the part of OOF that performs virtual ex-
periments on the meshes ppm2oof produces. It is
designed with the expectation that users can per-
form useful computations by intuitively setting
boundary conditions and distortions. In its most
simple usage, oof is a basic finite-element solver.
It can calculate stresses, strains, and thermal dis-
tortions and display the results. This does not dif-
fer from other finite-element solvers, but because
elements in oof are instances of a programmable
object, each element can develop a behavior that
the current solution or the user can alter.
For example, one element in OOF is called the
Griffith element. The Griffith element is designed
to compare the stored elastic energy within the
element to the amount of energy released by
fracturing the element and creating surface area.
The Griffith element can “mutate” when the lo-
cal conditions suggest that it should fracture. The
mutated element has a new compliance that re-
flects the softening that would occur due to a
crack proceeding through the element.
Because the user can intervene and modify an
element’s properties within oof, users can con-
duct hypothetical or “what-if” experiments on a
particular microstructure and its possible modi-
fications. For example, a user might surmise the
effect of a crack by simply creating a crack in the
microstructure; he or she might determine the ef-
fect of an unknown residual stress by increment-
ing its value and determining what effect such an
increase has on, for instance, the elastic energy
density in abutting microstructural regions.
oof has a fairly advanced graphical display of
results that a user can save and incorporate into
presentations or papers (as in Figure 2). It has
built-in functions designed to perform simple
statistical analysis on all elements or specified
groups of elements. For example, using published
data
13
for the coefficients of thermal expansion
of NiO and yttria-stabilized ZrO
2
and measured
crystallographic orientations for lamellar mi-
crostructures such as that in Figure 1, calculated
residual stress distributions in each phase are in
excellent agreement with those measured by X-
ray diffraction.
14
oof also has methods for data
output in a form that can act as input to other
programs and perform post-calculation analysis
such as a particular microstructure’s reliability.
15
A separate program, oof2abaqus, can convert
OOF data files into a form that Abaqus, a com-
mercial finite-element program, can read.
W
ork is currently under way on
OOF2, which is a major rewrite
of OOF that will be extendible to
a wide variety of problems. In
particular, it will handle any problem in which the
divergence of some generalized flux is a general-
MAY/JUNE 2001 23
ized external force and that flux is a linear combi-
nation of fields and gradients of fields. This en-
compasses thermoelasticity, piezoelectricity, and
heat and mass diffusion, among other topics. Non-
linear solvers will let OOF2 users experiment with
various nonlinear models, such as plasticity and fer-
roelasticity. OOF2 will use higher-order adaptive
triangular and quadrilateral elements, and although
OOF1 is entirely written in C++, OOF2 will be a
mix of C++ and Python, giving it greater flexibility.
OOF1 is meant to be easily extendible to new ma-
terial types, but in our experience it has only been
easy for a subset of OOF’s authors, not to mention
users. In OOF2, users will be able to add new ma-
terial types, new fields, and couplings between
fields by writing some simple Python code.
Acknowledgments
We thank Edwin Garcia, Mark Locatelli, Andrew Reid,
Andrew Roosen, Nita Parekh, and Daniel Vlacich for
valuable contributions. The oof2abaqus program is
provided as a convenience to OOF users and should not be
construed as an endorsement of Abaqus. NIST does not
endorse any commercial products. OOF is supported in part
by the NIST Center for Theoretical and Computational
Materials Science and the US Department of Energy’s
Advanced Turbine Systems Program, under contract DE-
A101-99EE41381.
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Stephen A. Langer is a physicist in the Mathematical
and Computational Sciences Division of the Informa-
tion Technology Laboratory at the US National Insti-
tute of Standards and Technology. His research inter-
ests include theories and simulations of liquid crystals,
glasses, ferrofluids, biomembranes, and foams, as well
as computational methods for analyzing material
microstructure. He received his PhD in physics from
Cornell University. Contact him at the Information
Technology Lab., Nat’l Inst. of Standards and Technol-
ogy, Gaithersburg, MD 20899-8910; stephen.langer@
nist.gov.
W. Craig Carter is the Lord Foundation Associate Pro-
fessor of Materials Science and Engineering at the Mass-
achusetts Institute of Technology. His research interests
are computational and mathematical developments in
materials science, especially microstructural evolution
and the fundamental properties of microstructures. He
received a PhD in materials science from the University
of California, Berkeley. Contact him at the Dept. of Ma-
terials Science and Eng., Massachusetts Inst. of Tech-
nology, Cambridge, MA 02139-4307; ccarter@mit.edu.
Edwin R. Fuller, Jr. is a research physicist in the Ce-
ramics Division at the National Institute of Standards
and Technology. His research interests include meso-
scopic computer simulations of micromechanical be-
havior of heterogeneous, stochastic microstructures,
and theoretical modeling of fracture behavior and
toughening mechanisms in brittle and quasi-brittle ma-
terials. He received a BS and PhD in physics from the
University of North Carolina at Chapel Hill and the Uni-
versity of Illinois at Champaign-Urbana, respectively.
He is a fellow of the American Ceramic Society. Con-
tact him at the Materials Science and Eng. Lab., Nat’l
Inst. of Standards and Technology, Gaithersburg, MD
20899-8521; edwin.fuller@nist.gov.
