Title: Modeling diffusion in heterogeneous media: Data driven microstructure reconstruction models, stochastic collocation and the variational multiscale method*
1Modeling diffusion in heterogeneous media Data
driven microstructure reconstruction models,
stochastic collocation and the variational
multiscale method Nicholas Zabaras and Baskar
Ganapathysubramanian Materials Process Design and
Control Laboratory Sibley School of Mechanical
and Aerospace Engineering Cornell
University Ithaca, NY 14853-3801 zabaras_at_cornell.
edu http//mpdc.mae.cornell.edu Work
supported by AFOSR/Computational Mathematics
2TRANSPORT IN HETEROGENEOUS MEDIA
- Thermal and fluid transport in heterogeneous
media are ubiquitous - Range from large scale
systems (geothermal systems) to the small scale -
Complex phenomena - How to represent complex
structures? - How to make them tractable? - Are
simulations believable? - How does error
propagate through them?
- To apply physical processes on these
heterogeneous systems - worst case scenarios
- variations on physical properties
3ISSUES WITH INVESTIGATION TRANSPORT IN
HETEROGENEOUS MEDIA
- Some critical issues have to be resolved to
achieve realistic results. - Multiple length scale variations in the material
properties of the heterogeneous medium - The essentially statistical nature of information
available about the media - Presence of uncertainty in the system and
properties
Only some statistical features can be extracted
4PROBLEM OF INTEREST
Interested in modeling diffusion through
heterogeneous random media
Aim To develop procedure to predict statistics
of properties of heterogeneous materials
undergoing diffusion based transport
- Should account for the multi length scale
variations in thermal properties - Account for the uncertainties in the topology of
the heterogeneous media
- What is given
- Realistically speaking, one usually has access to
a few experimental 2D images of the
microstructure. Statistics of the heterogeneous
microstructure can then be extracted from the
same. - This is our starting point
5OVERVIEW OF METHODOLOGY
2. Microstructure reconstruction
1. Property extraction
Extract properties P1, P2, .. Pn, that the
structure satisfies. These properties are usually
statistical Volume fraction, 2 Poit correlation,
auto correlation
Reconstruct realizations of the structure
satisfying the properties. Monte Carlo, Gaussian
Random Fields, Stochastic optimization ect
Construct a reduced stochastic model from the
data. This model must be able to approximate the
class of structures. KL expansions, FFT and other
transforms, Autoregressive models, ARMA models
Solve the heterogeneous property problem in the
reduced stochastic space for computing property
variations. Collocation schemes VMS
4. Stochastic collocation Variational
multiscale method
3. Reduced model
61. PROPERTY EXTRACTION
7IMAGE PROCESSING
Reconstruction of well characterized
material Tungsten-Silver composite1 Produced by
infiltrating porous tungsten solid with molten
silver
640x640 pixels 198 µm x 198 µm
1. S. Umekawa, R. Kotfila, O. D. Sherby, Elastic
properties of a tungsten-silver composite above
and below the melting point of silver, J. Mech.
Phys. Solids 13 (1965) 229-230
8PROPERTY EXTRACTION
First order statistics Volume fraction 0.2
Second order statistics 2 pt correlation
Digitized two phase microstructure image White
phase- W Black phase- Ag Simple matrix operations
to extract image statistics
92. DATA DRIVEN MODELS FOR MICROSTRUCTURE
RECONSTRUCTION
10MICROSTRUCTURE RECONSTRUCTION
Statistical information available- First and
second order statistics Reconstruct Three
dimensional microstructures that satisfy these
experimental statistical relations
GAUSSIAN RANDOM FIELDS GRF- model interfaces as
level cuts of a function Build a function y(r).
Model microstructure is given by level cuts of
this function. y(r) has a field-field correlation
given by g(r) If this function is known, y(r) can
be constructed as
Uniformly distributed over the unit sphere
Uniformly distributed over 0, 2p)
Distributed according to where
11MICROSTRUCTURE RECONSTRUCTION
- Relate experimental properties to
- Two phase microstructure, impose level cuts on
y(r). Phase 1 if - Relate to statistics
- first order statistics
where
second order statistics
Set , and For the Gaussian Random Field
to match experimental statistics
12MICROSTRUCTURE RECONSTRUCTION FITTING THE GRF
PARAMETERS
Assume a simplified form for the far field
correlation function
Three parameters, ß is the correlation length, d
is the domain length and rc is the cutoff length
Use least square minimization to find optimal fit
133D MICROSTRUCTURE RECONSTRUCTION
20 µm x 20 µm x 20 µm
64x64x64 pixel
40 µm x 40 µm x 40 µm
128x128x128 pixel
200 µm x 200 µm
143. REDUCED MODEL OF THE TOPOLOGICAL DESCRIPTOR
15WHY A REDUCED MODEL?
The reconstruction procedure gives a large set of
3D microstructures The topology of the
reconstructured microstructures are all
different All these structures satisfy the
experimental statistical relations These
microstructures belong to a very large (possibly)
infinite dimensional space. These topological
variations are the inputs to the stochastic
problem The necessity of model reduction arises
Model reduction techniques Most commonly used
technique in this context is Principle Component
Analysis Compute the eigen values of the dataset
of microstructures
16REDUCED MODEL FOR THE STRUCTURE
M microstructure images of nxnxn pixels each The
microstructures are represented as vectors Ii
i1,..,M The eigenvectors of the n3xn3 covariance
matrix are computed The first N eigenimages are
chosen to represent the microstructures
Represent any microstructure as a linear
combination of the eigenimages
I Iavg I1a1 I2a2 I3a3 Inan
an
..
a2
a1
17REDUCED MODEL FOR THE STRUCTURE CONSTRAINTS
Let I be an arbitrary microstructure satisfying
the experimental statistical correlations The PCA
method provides a unique representation of the
image That is, the PCA provides a function
The function is injective but nor
surjective Every image has a unique
mapping But every point
need not define an image in
Construct the subspace of allowable n-tuples
18CONSTRUCTING THE REDUCED SUBSPACE H
Image I belongs to the class of structures? It
must satisfy certain conditions a) Its volume
fraction must equal the specified volume
fraction b) Volume fraction at every pixel must
be between 0 and 1 c) It should satisfy the
given two point correlation Thus the n tuple
(a1,a2,..,an) must further satisfy some
constraints. Enforce these constraints
sequentially
1. Pixel based constraints
Microstructures represented as discrete images.
Pixels have bounds This results in 2n3 inequality
constraints
19CONSTRUCTING THE REDUCED SUBSPACE H
2. First order constraints
The Microstructure must satisfy the experimental
volume fraction
This results in one linear equality constraint on
the n-tuple
3. Second order constraints
The Microstructure must satisfy the experimental
two point correlation. This results in a set of
quadratic equality constraints
This can be written as
20SEQUENTIAL CONSTRUCTION OF THE SUBSPACE
Computational complexity Pixel based constraints
first order constraints result in a simple
convex hull problem Enforcing second order
constraints becomes a problem in quadratic
programming Sequential construction of the
subspace First enforce first order statistics, On
this reduced subspace, enforce second order
statistics Example for a three dimensional space
3 eigen images
21THE REDUCED MODEL
The sequential contraction procedure a subspace
H, such that all n-tuples from this space result
acceptable microstructures
H represents the space of coefficients that map
to allowable microstructures. Since H is a plane
in N dimensional space, we call this the
material plane
Since each of the microstructures in the
material plane satisfies all required
statistical properties, they are equally
probable. This observation provides a way to
construct the stochastic model for the allowable
microstructures
Define such that
This is our reduced stochastic model of the
random topology of the microstructure class
224. SOLUTION TO THE STOCHASTIC PARTIAL
DIFFERENTIAL EQUATION
23SPDE Definition
Governing equation for thermal diffusion
Uncertainty comes in as the random material
properties, which depend on the topology of the
microstructure
The (Nd) dimensional problem (N stochastic
dimensions d spatial dimensions) is represented
as
The number of stochastic dimensions is usually
large 10-20
24UNCERTAINTY ANALYSIS TECHNIQUES
- Monte-Carlo Simple to implement,
computationally expensive - Perturbation, Neumann expansions Limited to
small fluctuations, tedious for higher order
statistics
- Spectral stochastic uncertainty representation
Basis in probability and functional analysis, Can
address second order stochastic processes, Can
handle large fluctuations, derivations are
general - Stochastic collocation Results in decoupled
equations
25COLLOCATION TECHNIQUES
Spectral Galerkin method Spatial domain is
approximated using a finite element
discretization Stochastic domain is
approximated using a spectral element
discretization
Coupled equations
Decoupled equations
Collocation method Spatial domain is
approximated using a finite element
discretization Stochastic domain is
approximated using
multidimensional interpolating functions
26DECOUPLED EQUATIONS IN STOCHASTIC SPACE
Simple interpolation Consider the function We
evaluate it at a set of points The approximate
interpolated polynomial representation for the
function is Where Here, Lk are the Lagrange
polynomials Once the interpolation function has
been constructed, the function value at any point
yi is just Considering the given natural
diffusion system One can construct the
stochastic solution by solving at the M
deterministic points
27SMOLYAK ALGORITHM
LET OUR BASIC 1D INTERPOLATION SCHEME BE
SUMMARIZED AS
IN MULTIPLE DIMENSIONS, THIS CAN BE WRITTEN AS
TO REDUCE THE NUMBER OF SUPPORT NODES WHILE
MAINTAINING ACCURACY WITHIN A LOGARITHMIC FACTOR,
WE USE SMOLYAK METHOD
IDEA IS TO CONSTRUCT AN EXPANDING SUBSPACE OF
COLLOCATION POINTS THAT CAN REPRESENT
PROGRESSIVELY HIGHER ORDER POLYNOMIALS IN
MULTIPLE DIMENSIONS A FEW FAMOUS SPARSE
QUADRATURE SCHEMES ARE AS FOLLOWS CLENSHAW
CURTIS SCHEME, MAXIMUM-NORM BASED SPARSE GRID AND
CHEBYSHEV-GAUSS SCHEME
28SMOLYAK ALGORITHM
Extensively used in statistical
mechanics Provides a way to construct
interpolation functions based on minimal number
of points Univariate interpolations to
multivariate interpolations
Uni-variate interpolation
Multi-variate interpolation
Smolyak interpolation
Accuracy the same as tensor product Within
logarithmic constant
D 10
Increasing the order of interpolation increases
the number of points sampled
29SMOLYAK ALGORITHM REDUCTION IN POINTS
For 2D interpolation using Chebyshev nodes Left
Full tensor product interpolation uses 256
points Right Sparse grid collocation used 45
points to generate interpolant with comparable
accuracy
D 10
Results in multiple orders of magnitude
reduction in the number of points to sample
30SPARSE GRID COLLOCATION METHOD implementation
Solution Methodology
PREPROCESSING Compute list of collocation points
based on number of stochastic dimensions, N and
level of interpolation, q Compute the weighted
integrals of all the interpolations functions
across the stochastic space (wi)
Use any validated deterministic solution
procedure. Completely non intrusive
Solve the deterministic problem defined by each
set of collocated points
POSTPROCESSING Compute moments and other
statistics with simple operations of the
deterministic data at the collocated points and
the preprocessed list of weights
Std deviation of temperature Natural convection
315. SOLUTION TO THE DETERMINISTIC PARTIAL
DIFFERENTIAL EQUATION
32 THE NECESSITY FOR VARIATIONAL MULTISCALE METHODS
The collocation method reduces the stochastic
problem to the solution of a set of deterministic
equations
- These deterministic problems correspond to
solving the thermal diffusion problem on a set of
unique microstructures. - These heterogeneous
microstructure realizations exhibit property
variations at a much smaller scale compared to
the size of the computational domain - Performing
a fully-resolved calculation on these
microstructures becomes computationally
expensive. - Consider a computational scheme that
involves solving for a coarse-solution while
capturing the effects of the fine scale on the
coarse solution.
33ADDITIVE SCALE DECOMPOSITION
The variation form of the diffusion equation can
be written as
- Assume that the solution can be decomposed into
two scales - Coarse resolvable scale
- Fine irresolvable (but modeled) scale
The variation form of the diffusion equation
decomposes into
34SUB GRID MODELLING
Further decompose fine scale solution into two
parts
Particular solution
Homogeneous solution
- The solution component incorporates the
entire coarse scale solution information and
has no dependence on the coarse scale solution.
- The dynamics of is driven by the
projection of the source term onto the
subgrid scale function space.
35SUB GRID MODELLING II
piecewise polynomial finite element
representation for the coarse solution inside a
coarse element Similar representation for the
fine scale. Move problem from computing values at
finest resolution to computing the shape function
at the finest resolution
Substitute into fine scale variational equation
36SUB GRID MODELLING III
Without loss of generality, we can assume the
following representation for the coarse scale
nodal solutions
A very general representation that incorporates
several well known time integration schemes
Substituting this form for the coarse and fine
scale solutions into the fine scale variational
forms gives
This is valid for all possible combinations of u.
It follows that each of the quantities in the
brackets above must equal 0
37SUB GRID MODELLING IV
This gives the variational form for the sub-grid
basis functions
The strong form for the fine-scale basis function
is then given by
The solution of the fine scale evolution equation
can then be input into the coarse scale solution
to get the coarse scale evolution equation
38VERIFICATION OF THE VMS FORMULATION
Reconstructed VMS solutions
Coarse scale VMS solutions
(a) Fully resolved FEM solution
(d)
(e)
(c)
(b)
Increasing coarse element size
39OVERVIEW OF METHODOLOGY
2. Microstructure reconstruction
1. Property extraction
Extract properties P1, P2, .. Pn, that the
structure satisfies. These properties are usually
statistical Volume fraction, 2 Poit correlation,
auto correlation
Reconstruct realizations of the structure
satisfying the properties. Monte Carlo, Gaussian
Random Fields, Stochastic optimization ect
Construct a reduced stochastic model from the
data. This model must be able to approximate the
class of structures. KL expansions, FFT and other
transforms, Autoregressive models, ARMA models
Solve the heterogeneous property problem in the
reduced stochastic space for computing property
variations. Collocation schemes VMS
4. Stochastic collocation Variational
multiscale method
3. Reduced model
406. NUMERICAL EXAMPLE
41MICROSTRUCTURE RECONSTRUCTION
Experimental statistics
Experimental image
3D microstructure
GRF statistics
42MODEL REDUCTION
Principal component analysis
Constructing the reduced subspace and the
stochastic model
- Enforcing the pixel based bounds and the linear
equality constraint (of volume fraction) was
developed as a convex hull problem. A primal-dual
polytope method was employed to construct the set
of vertices. - Enforcing the second order constraints was
performed through the quadratic programming tools
in the optimization toolbox in Matlab. - Two separate cases are considered in this
- example. In the first case, only the first-order
constraints (volume fraction) are used to
reconstruct the subspace H. In the second case,
both first-order as well as second-order
constraints (volume fraction and two-point
correlation) are used to construct the subspace H.
First 9 eigen values from the spectrum chosen
43PHYSICAL PROBLEM UNDER CONSIDERATION
Structure size 40x40x40 µm Tungsten Silver
Matrix Heterogeneous property is the thermal
diffusivity. Tungsten ? 19250 kg/m3 k 174
W/mK c 130 J/kgK Silver ? 10490 kg/m3 k
430 W/mK c 235 J/kgK Diffusivity ratio aAg/aW
2.5
T -0.5
T 0.5
Left wall maintained at -0.5 Right wall
maintained at 0.5 All other surfaces insulated
44COMPUTATIONAL DETAILS
The construction of the stochastic solution
through sparse grid collocation level 5
interpolation scheme used Number of deterministic
problems solved 15713
Computational domain of each deterministic
problem 128x128x128 pixels
Each deterministic problem solution solved on a
8 8 8 coarse element grid (uniform hexahedral
elements) with each coarse element having 16 16
16 fine-scale elements.
The solution of each deterministic VMS problem
about 34 minutes, In comparison, a fully-resolved
fine scale FEM solution took nearly 40 hours.
Computational platform 40 nodes on local Linux
cluster Total time 56 hours
45FIRST ORDER STATISTICS MEAN TEMPERATURE
e
c
d
b
f
a
g
46FIRST ORDER STATISTICS HIGHER ORDER MOMENTS
d
c
b
e
a
f
47SECOND ORDER STATISTICS MEAN TEMPERATURE
e
c
d
b
f
g
a
48SECOND ORDER STATISTICS HIGHER ORDER MOMENTS
d
c
b
e
a
f
49CONCLUSIONS
A new model for modeling diffusion in random
two-phase media. A general methodology was
presented for constructing a reduced-order
microstructure model for use as random input in
the solution of stochastic partial differential
equations governing physical processes The twin
problems of uncertainty and multi length scale
variations are decoupled and comprehensively
solved Scope of further research Using more
sophisticated model reduction techniques to build
the reduced-order microstructure model, Extending
the methodology to arbitrary types of
microstructures as well as developing models of
advection-diffusion in random heterogeneous
media.
Comparison of temperature PDFs at a point due to
the application of first and second order
constraints
50REFERENCES
- B. Ganapathysubramanian and N. Zabaras, "Sparse
grid collocation methods for stochastic natural
convection problems", Journal of Computational
Physics, in press - 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, submitted - S. Sankaran and N. Zabaras, "Computing property
variability of polycrystals induced by grain size
and orientation uncertainties", Acta Materialia,
in press - B. Velamur Asokan and N. Zabaras, "A stochastic
variational multiscale method for diffusion in
heterogeneous random media", Journal of
Computational Physics, Vol. 218, pp. 654-676,
2006 - B. Velamur Asokan and N. Zabaras, "Using
stochastic analysis to capture unstable
equilibrium in natural convection", Journal of
Computational Physics, Vol. 208/1, pp. 134-153,
2005