Data-driven reduced-order models for the representation of polycrystalline microstructures

1
Data-driven reduced-order models for the
representation of polycrystalline
microstructures Nicholas Zabaras, Cornell
University Baskar Ganapathysubramanian, Iowa
State University Zheng Li, Cornell
University/Dalian University of
Technology zabaras_at_cornell.edu baskarg_at_iastate.ed
u http//mpdc.mae.cornell.edu/
2
MOTIVATION
PROVIDE LOW-DIMENSIONAL REPRESENTATION OF
MICROSTRUCTURE SPACES
Process
Applications (i) Identify microstructures that
have extremal properties. (ii) Identify
processing sequences that lead to desired
microstructures and properties.
Microstructure representations
Property-process space
Process-structure space
A1000
Property-structure space
f
e
Taylor factor along TD
C
A80
b
a
R
a
d
3. Rolling followed by drawing
2. Rolling
1. Drawing
Process paths
Taylor factor along RD
A100
3
MOTIVATION
PROVIDE LOW-DIMENSIONAL REPRESENTATION OF
MICROSTRUCTURE SPACES
Applications (i) Identify effect of
microstructurural variability on macro-scale
properties. (ii) Design and control
microstructural statistics that result in
tailored behavior. (iii) Provide more rigorous
bounds and pdf
Limited information
Provide reliable bounds
Property variability induced by topological
uncertainty in microstructures
4
THE COMPLETE MODEL
Preprocessing, define metric
Input microstructures
Low dimensional points from the pair-wise
distances
Grain size distribution
Convex hull of the points The low dimensional
model
Transform hull to unit hypercube
Reconstruction for arbitrary points on the
hypercube
5
LINEAR MODEL REDUCTION STRATEGY

an
a1
a2
..

Convert variability of property/microstructure to
variability of coefficients.
PCA based approaches find the smallest coordinate
representation of the data . but assumes that
the data lies in a linear vector space
3D data
PCA
6
LINEAR MODEL REDUCTION STRATEGY
PCA based methods assume that the data lies in a
linear vector space
Only guaranteed to discover the true structure of
data lying on a linear subspace of the high
dimensional input space
of eigen vectors
As the number of input samples increases, PCA
based approaches tend to overestimate the
dimensionality of the reduced representation. Beco
mes computationally challenging
of samples
NONLINEAR APPROACHES TO MODEL REDUCTION IDEAS
FROM IMAGE PROCESSING, PSYCOLOGY
7
Manifold learning approach
Given some experimental correlation that the
microstructure/property variation
satisfies. Construct several plausible images
of the microstructure/property. Each of these
images consists of , say, n pixels. Each image
is a point in n dimensional space. But each and
every image is related. That is, all these
images lie on a unique curve (manifold) in
Rn. Can a low dimensional parameterization of
this curve be computed? Strategy based on a
variant of the manifold learning problem.
Different microstructure realizations satisfying
some experimental correlations
8
AN INTUITIVE PICTURE OF THE STRATEGY
Input data lies on a curved surface in a
high-dimensional space. Key is to unravel and
smooth out this curve to construct a
low-dimensional representation
This unraveling and smoothing corresponds to a
topological transformation that preserves some
notion of the geometry of the manifold.
Defining the appropriate manifold and identifying
properties of the manifold Defining the
appropriate transformation that results in the
low-dimensional equivalent space.

• J. B. Tenenbaum, V. De Silva, J. C. Langford, A
  global geometric framework for nonlinear
  dimension reduction, Science 290 (2000),
  2319-2323
• S Roweis, L. Saul, Nonlinear Dimensionality
  Reduction by Locally Linear Embedding, Science
  290 (2000) 2323--2326

9
KEY CONCEPT

• Geometry can be preserved if the distances
  between the points are preserved Isometric
  mapping.
• The geometry of the manifold is reflected in the
  geodesic distance between points
• First step towards reduced representation is to
  construct the geodesic distances between all the
  sample points

Euclidian dist
Geodesic dist
Pt A
Pt B
10
MATHEMATICAL ISSUES/DETAILS
1) The set of input data lies on a manifold
embedded in a high dimensional space. Define the
appropriate manifold and identifying properties
of the manifold
2) Identifying the intrinsic dimensionality of
the manifold
3) Constructing a reliable transformation of
points on the manifold to a low dimensional
surrogate space.
11
THE MANIFOLD ? ? Rn
Definition By a microstructure x, we mean a
pixelized representation of a topology or
property variation (with n pixels)
Definition Let ? denote the set of
microstructures satisfying a set of statistical
relations (limited experimental information).
Denote this set of limited information as
? is a subset of Rn. ? is a curve in this
space We show that with appropriate choice of a
distance metric, ? is a compact manifold
12
DEFINING THE DISTANCE METRIC
The distance measure, ?, based on how much the
microstructures vary. Two choices A) When the
experimental statistics are explicitly known, the
distance metric can be defined as the difference
in some statistical correlation between two
microstructures. B) When only snap shots of the
data is provided, utilize euclidean metric as the
distance metric
(A) is useful when the statistical difference in
the microstructures/property variations are
important (B) is useful when we want to
incorporate the effect of rotation and scaling
13
PROPERTIES OF THE MANIFOLD ? ? Rn
The key to a reasonable dimension reduction is a
good choice of the distance measure
Any choice of functions are allowable as long as
they satisfy the metric properties
Show a sequence of properties that the manifold
satisfies
a) (?, ? ) is a metric space.
b) (?, ? ) is a bounded.
c) (?, ? ) is dense.
d) (?, ? ) is complete.
e) (?, ? ) is a compact metric space1,2.

• B. Ganapathysubramanian and N. Zabaras, "A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models",
  Journal of Computational Physics, Vol. 227, pp.
  6612-6637, 2008
• J. R. Munkres, Topology, Second edition,
  Prentice-Hall, 2000.

14
PROPERTIES OF THE MANIFOLD ? ? Rn
Why do we have to show compactness?
Compactness is a very strong condition for a
manifold
In these problems when the data satisfies some
correlations or has some structure it is
straightforward
The basic conditions is that the manifold must be
unraveled The manifold must not have holes or any
singularities Compactness ensures these
well-behaved properties But the strict
compactness condition can be relaxed scope for
future work
15
MAPPING A COMPACT MANIFOLD TO A LOW_D SET
Map close points on the manifold to close point
on the low dimensional space Map points far away
on the manifold to points further away to each
other in the low dimensional space This results
in a isometric transformation of the manifold
embedded in a high dimensional space to its low
dimensional counterpart
Euclidian
Geodesic
Pt A
Pt B
16
MAPPING A COMPACT MANIFOLD TO A LOW_D SET
Have no notion of the geometry of the manifold to
start with. Hence cannot construct true geodesic
distances!
Approximate the geodesic distance using the
concept of graph distance ?G(i,j) the distance
of points far away is computed as a sequence of
small hops. This approximation, ?G,
asymptotically matches the actual geodesic
distance ??. In the limit of large number of
samples1,2. (Theorem 4.5 in 1)
Based on results on graph approximations to
geodesics2.

• B. Ganapathysubramanian and N. Zabaras, "A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models",
  Journal of Computational Physics, Vol. 227, pp.
  6612-6637, 2008
• M.Bernstein, V. deSilva, J.C.Langford,
  J.B.Tenenbaum, Graph approximations to geodesics
  on embedded manifolds, Dec 2000

17
PAIRWISE DISTANCES TO LOW-D POINTS

• Given the N unordered sample points
  (microstructures, property maps, )
• Compute the geodesic distance between each pair
  of samples ??(i,j) .
• Given the pairwise distance matrix between N
  objects, compute the location of N points, ?i
  in ?d such that the distance between these points
  is arbitrarily close to the given distance matrix
  ?? . Basic premise of group of statistical
  methods called Multi Dimensional Scaling1 (MDS)

Given N unordered samples
Compute pairwise geodesic distance
Perform MDS on this distance matrix
N points in a low-dimensional space

• T.F.Cox, M.A.A.Cox, Multidimensional scaling,
  1994, Chapman and Hall

18
DIMENSIONALITY OF THE REDUCED SPACE
The intrinsic dimension of an embedded manifold
is estimated using a novel geometrical
probability approach. This work is based on a
powerful result in geometric probability - the
Breadwood-Halton-Hammersley theorem where d is
linked to the rate of convergence of the
length-functional of the minimal spanning tree of
the geodesic distance matrix of the unordered
data points in the high-dimensional space.
Consistent estimates of the intrinsic dimension
d of the sample set are obtained using a very
simple procedure.
19
BHH THEOREM AND LINK TO DIMENSION
The rate of change of the length functional as
more number of points are chosen is related to
the dimensionality of the manifold
with
20
THE REDUCED ORDER STOCHASTIC MODEL
? ? ?d
?? ?n.
Given N unordered samples
N points in a low dimensional space
The procedure results in N points in a
low-dimensional space. The geodesic distance
MDS step (Isomap algorithm1) results in a
low-dimensional convex, connected space2, ? ? ?d.
Using the N samples, the reduced space is given
as
? serves as the surrogate space for ?. Access
variability in ? by sampling over ?. BUT have
only come up with ? ?? map . Need ??? map too

• J. B. Tenenbaum, V. De Silva, J. C. Langford, A
  global geometric framework for nonlinear
  dimension reduction Science 290 (2000),
  2319-2323.
• B. Ganapathysubramanian and N. Zabaras, "A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models",
  Journal of Computational Physics, Vol. 227, pp.
  6612-6637, 2008

21
THE REDUCED ORDER STOCHASTIC MODEL
Only have N pairs to construct ??? map. Various
possibilities based on specific problem at hand.
But have to be conscious about computational
effort and efficiency. Illustrate 3 such
possibilities below that work for two-phase
microstructures
? ? ?d
?? ?n
? ? ?d
?? ?n
2. Local linear interpolation
1. Nearest neighbor map
? ? ?d
?? ?n
3. Local linear interpolation with projection
1. B. Ganapathysubramanian and N. Zabaras, "A
non-linear dimension reduction methodology for
generating data-driven stochastic input models",
Journal of Computational Physics, Vol. 227, pp.
6612-6637, 2008
22
THE REDUCED ORDER STOCHASTIC MODEL

• Algorithm consists of two parts.
• Compute the low dimensional representation of a
  set of N unordered sample points belonging to a
  high dimensional space

2) For an arbitrary point ? ? must find the
corresponding point x ?. Compute the mapping
from ???
? ? ?d
?? ?n.
This first step results in a mathematically
unique mapping from High-D space to a lw-D
space The second step has to construct an
(locally) unique mapping from low-D space to
High-D space
23
MODELS OF POLYCRYSTALLINE MATERIALS
Microstructural variations affect macro-scale
properties It is lot more difficult to analyze
than two-phase or multi-phase materials Multiple
layers of representation (a) grain distribution
(b) orientation distribution Continuum
distribution of orientation. Solvable
problem Limit analysis to grain distribution
This is a tricky problem Have to faithfully
encode grain distribution features and should
quickly reconstruct approximate grains
24
CONSTRUCTING INPUT DATA POLYCRYSTALS
Created 1000 polycrystalline microstructures that
satisfy a particular statistical constraint.
Constraint 1 All microstructures must satisfy
a mean grain size of 10.97 µm
Realizations of the polycrystalline
microstructures that satisfy a particular
statistical constraint.
25
STATISTICS OF THE INPUT MICROSTRUCTURES
Even though all the microstructures satisfied a
mean grain size, they showed very varied grain
size distributions
Distribution of the mean grain size.
Distribution of the deviation in grain size.
Mean grain size distribution and one standard
deviation variation
26
DEFINING THE METRIC
As discussed before, the key to good model
reduction and reconstructions is a viable metric.
Since the important feature we are looking to
embed and recreate is the grain sizes, use this
feature as the metric
A normalized grain size feature is the histogram
of the grain size distribution
27
ESTIMATING THE OPTIMAL DIMENSIONALITY
Dimensionality of embedded space computed from
application of the BHH theorem. Connects the
dimensionality of the surface to the length
functional of a graph D 6
The optimal dimensionality asymptotically reaches
a value of 6 with increasing input sample sets.
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THE LOW DIMENSIONAL MODEL
Using MDS on the geodesic distance matrix and
utilizing the optimal dimensionality results in a
representation of the 1000 microstructures as 6
dimensional points. The convex hull of these
points is the reduced model. For computational
simplicity, transform this convex hull into a
unit hypercube 0, 16.
29
THE COMPLETE MODEL
Preprocessing, define metric
Input microstructures
Low dimensional points from the pair-wise
distances
Grain size distribution
Convex hull of the points The low dimensional
model
Transform hull to unit hypercube
Reconstruction for arbitrary points on the
hypercube
30
The Interpolation in Between Microstructures
What we have two microstructures A and B What we
want given a parameter , use
the information of A and B to reconstruct
another microstructure C, which satisfies
Since we defined the grain size distribution as
the metric, this also means that
It is obvious that
Notice If A and B have the same mean grain size,
then based on this equation, C has the same mean
grain size as A and B.
31
The Interpolation Between Grains
What we have two grains G1 and G2 What we
want use the information of G1 and G2 to
reconstruct another grain G3, which satisfy
Here we have two grains, and we want to find
another grain between them. Obviously, simple
interpolation does not make any sense
32
The Interpolation Between Grains on Fourier Space
In this work, we interpolate different grains
spectrally. Here we use the microstructure
function to denote grains. We interpolate
microstructure functions Fourier amplitude, and
then use the phase recovery method to get a
blurred image of the new grain
On Step3, we use Gercherg-Saxton algorithm to do
the phase recovery work. The constraints used
here are the same as the ones in 2-Pt based
reconstruction
We do not need a result with a very high
precision here. To keep the method efficient, we
only loop 100 times for the phase recovery.
33
The Interpolation Between Grains on Fourier Space
The Fourier amplitude images of G1 and G2.
Notice that the location of the new grain is
recomputed (we will talk about it later), which
means the phase information of G1 and G2 is
useless here.
The third image is the interpolation result of
the left two. Obviously, it is a blurred image.
34
The Location of the New Grains
Although, we have already obtained the blurred
images of the new grains, we still need to know
where should we put these grains. This is a
mathematically difficult problem. An algorithm
is proposed to roughly compute the locations of
new grains. Since the desired size of each grain
is known, Assuming all the grain is close to
circle or sphere, the desire radius of each grain
is defined
The location the grains center is
used to denote the grain location. A grain force
between two grains M(i) and M(j) is defined as
Notice that, the distance used here is using the
coordinates of real space
35
The Location of the New Grains

How to solve the optimization problem

Compute the grain force between two
grains And object function
Solve 1 dimension optimization problem Th
en M(i).x M(i).xaSD(i), i1N
We have several choices to solve the one
dimensional optimization problem, like Newtons
method.
N
WholeForcelt Criterion?
Compute search direction of each grain i
Y
over
36
The Blurred Image of New Microstructure
Now, we have obtained the blurred images of each
new grain and their desire locations (M(i).x).
Then we should use these information to
reconstruct a new microstructure.
Here we use B(i) to denote the image of the new
ith grain. B(i)(x) is used to denote the blur
image's value on the position x
Step.1 Compute the center location of
B(i) Step.2 Move each blurred images to its
desire location and get a new blur
image B'(i). Step.3 For each
position x in the microstructure, sort
the values of all B'(i) from big to small
then we set S and Index as
S(k)(x)B'(Index(k))(x),
S(i)(x)gtS(j)(x), for any iltj Step.4 Keep the
first values of S and Index For
each position x, take Index(1)(x) as its grain
B'(i) denotes the image of B(i) after moving.
Keeping the first values is for the
optimization which we will talk about next
37
The Blurred Image of New Microstructure
The left is the image of S(1)(x), for each
position x. Index(1)(x) is considered as the
grain number of each position x. Then the right
is the grain distribution of the new
microstructure.

• Defection of the new microstructure
• The grain boundary is not smooth enough
• Some grains are seriously concave.
• The grain sizes are close to the desired grain
  sizes, but still could be closer.

• Issues with the algorithm
• The blurred red image is close to a new
  microstructure grain, but still not consistent of
  grains (with the value of 0 or 1).
• There is no perfect location for each new grain

38
Phase-Field Algorithm
Here, the phase-field is simply introduced.

• A set of Q continuous order parameters
  is defined at a given time t at
  each position r.
• The thermodynamics of the simulation algorithm
  drives that, within each grain, one and only one
  field variable takes on a value of 1, and all
  other order parameters have the value zero.
• Therefore, the orientation of a given grain can
  be specified by the index q of the order
  parameter in that
  grains interior.

Phase-field mainly process

• 1 Initialize on each node r in grid
  ,t0, Initial TimeStep0
• 2 TimeStepTimeStep1, compute the partial
  derivate with t of
• 3 Renew
• 4 If (TimeStepltMaxTimeStep) goto step 2
• else simulation is over

39
Phase-Field Algorithm

• To avoid grain coalescing, Skrikanth
  Vedantam and B. S. V. Pantnaik used a sparse data
  structure, which is also introduced in this work.
• Sparse data structure During the simulation
  process, only a finite number are stored on a
  lattice point. Each is associated with a index
  which denotes the orientation of field variable.
  When we renew field variable of a grid r (Step
  3), all the orientations of r and rs neighbors
  are computed, and only the largest Q
  orientations values are stored on the lattice
  point r. This method is very efficient to avoid
  coalescing.

The function below is used to compute the partial
derivate with t.
Based on our experience, we can control the grain
size effectively by adjusting parameter In this
report, we take for a grain q, we take where
RealSize is the temporary size of grain q on each
time step, DesiredSize is the desire size of this
grain
40
The Optimization of New Microstructure
To use phase-field to optimize the
microstructure, the blurred microstructure is
taken as the initial condition
Recall that S keeps the first biggest values
on each position, and Index keeps their
correlation grain number, to generate the initial
condition
Because of the sparse data structure, only the
non-zero value and its correlation need to be
stored on each position
Since the phase-field algorithm is used here to
optimize the microstructure and not for modeling
grain growth, it is not important to run it for
too many time steps. The max time step is set as
100 in this work After the optimization with
phase-field, the phase-field result is taken as
the final new microstructure.
41
The Interpolation Result
42
CONCLUSIONS
Data-driven non-linear reduced order models of
microstructures developed. Can access the effect
of microstuctural variability in stochastic
analysis. Can now provide probability
distribution functions instead of just
bounds. Very significant when performing
computationally demanding operations searching,
contouring - in intrinsically high-dimensional
property-process-structure spaces Naturally
coupled with statistical learning and
unsupervised classification strategies to
effectively estimate optimal processing routes
for tailored materials

• Z. Li, B. Ganapathysubramanian, N. Zabaras,
  Low-dimensional models for microstructure
  representation A data-driven approach, Acta
  Materiala, in preparation.
• B. Ganapathysubramanian and N. Zabaras,
  "Modelling diffusion in random heterogeneous
  media Data-driven models, stochastic collocation
  and the variational multi-scale method", Journal
  of Computational Physics, Vol. 226, pp. 326-353,
  2007
• B. Ganapathysubramanian and N. Zabaras, "A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models",
  Journal of Computational Physics, Vol. 227, pp.
  6612-6637, 2008