1. * Work supported by AFOSR/Computational Mathematics
Modeling 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
* Work supported by AFOSR/Computational Mathematics

2. TRANSPORT 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

3. ISSUES 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

4. PROBLEM 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

5. OVERVIEW OF METHODOLOGY
1. Property extraction
2. Microstructure reconstruction
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

6. 1. PROPERTY EXTRACTION

7. IMAGE 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)

8. PROPERTY 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

9. 2. DATA DRIVEN MODELS FOR MICROSTRUCTURE RECONSTRUCTION

10. MICROSTRUCTURE 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, 2π)
Distributed according to where

11. MICROSTRUCTURE 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

12. MICROSTRUCTURE RECONSTRUCTION: FITTING THE GRF PARAMETERS
Assume a simplified form for the far field correlation function
Three parameters, β is the correlation length, d is the domain length and rc is the cutoff length
Use least square minimization to find optimal fit

13. 3D 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

14. 3. REDUCED MODEL OF THE TOPOLOGICAL DESCRIPTOR

15. WHY 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

16. REDUCED MODEL FOR THE STRUCTURE
M microstructure images of nxnxn pixels each
The microstructures are represented as vectors Ii i=1,..,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 =
+ ..+ a1 + a2

17. REDUCED 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

18. CONSTRUCTING 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

19. CONSTRUCTING 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

20. SEQUENTIAL 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

21. THE 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

22. 4. SOLUTION TO THE STOCHASTIC PARTIAL DIFFERENTIAL EQUATION

23. SPDE Definition Governing equation for thermal diffusion
Uncertainty comes in as the random material properties, which depend on the topology of the microstructure
The (N+d) dimensional problem (N stochastic dimensions+ d spatial dimensions) is represented as
The number of stochastic dimensions is usually large ~ 10-20

24. UNCERTAINTY 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

25. COLLOCATION 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

26. DECOUPLED 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

27. SMOLYAK 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

28. SMOLYAK 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
ORDER SC FE 3
1581
8000 4
8801
40000 5
41625
100000
Increasing the order of interpolation increases the number of points sampled

29. SMOLYAK 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
ORDER SC FE 3
1581
1000 4
8801
10000 5
41625
100000

30. SPARSE 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

31. 5. 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.

33. ADDITIVE 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:

34. SUB 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.

35. SUB 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

36. SUB 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

37. SUB 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

38. VERIFICATION OF THE VMS FORMULATION
Reconstructed VMS solutions
Coarse scale VMS solutions
(a)
Fully resolved FEM solution
(b)
(c)
(d)
(e)
Increasing coarse element size

39. OVERVIEW OF METHODOLOGY
1. Property extraction
2. Microstructure reconstruction
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

40. 6. NUMERICAL EXAMPLE

41. MICROSTRUCTURE RECONSTRUCTION
Experimental statistics
Experimental image
3D microstructure
GRF statistics

42. MODEL 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

43. PHYSICAL PROBLEM UNDER CONSIDERATION
Structure size 40x40x40 μm
Tungsten Silver Matrix
Heterogeneous property is the thermal diffusivity.
Tungsten: ρ kg/m3
k 174 W/mK
c 130 J/kgK
Silver: ρ kg/m3
k 430 W/mK
c 235 J/kgK
Diffusivity ratio αAg/αW = 2.5
T= -0.5
T= 0.5
Left wall maintained at -0.5
Right wall maintained at +0.5
All other surfaces insulated

44. COMPUTATIONAL 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

45. FIRST ORDER STATISTICS: MEAN TEMPERATUREb d f a g

46. FIRST ORDER STATISTICS: HIGHER ORDER MOMENTSb e a f

47. SECOND ORDER STATISTICS: MEAN TEMPERATUREb d f g a

48. SECOND ORDER STATISTICS: HIGHER ORDER MOMENTSb c e a f

49. CONCLUSIONS Scope of further research
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 PDF’s at a point due to the application of first and second order constraints

50. REFERENCES
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 , 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 , 2005