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Modeling diffusion in heterogeneous media: Data driven microstructure reconstruction models, stochastic collocation and

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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 14853-3801

zabaras@cornell.edu

http://mpdc.mae.cornell.edu

* Work supported by AFOSR/Computational Mathematics

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

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

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

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

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

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

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

- 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

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

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

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

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

+ ..+

a2

=

a1

+

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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:

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.

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

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

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

VERIFICATION OF THE VMS FORMULATION

Reconstructed VMS solutions

Coarse scale VMS solutions

(a)

Fully resolved FEM solution

(d)

(e)

(c)

(b)

Increasing coarse element size

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

Experimental statistics

Experimental image

3D microstructure

GRF statistics

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

PHYSICAL PROBLEM UNDER CONSIDERATION

Structure size 40x40x40 μm

Tungsten Silver Matrix

Heterogeneous property is the thermal diffusivity.

Tungsten: ρ 19250 kg/m3

k 174 W/mK

c 130 J/kgK

Silver: ρ 10490 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

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

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

- 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

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