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A STOCHASTIC VARIATIONAL MULTISCALE METHOD FOR DIFFUSION IN HETEROGENEOUS RANDOM MEDIA

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A STOCHASTIC VARIATIONAL MULTISCALE METHOD FOR DIFFUSION IN HETEROGENEOUS RANDOM MEDIA

N. ZABARAS AND B. VELAMUR ASOKAN

Materials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall

Cornell University

Ithaca, NY 14853-3801

Email: zabaras@cornell.edu

URL: http://mpdc.mae.cornell.edu/

WHY UNCERTAINTY AND MULTISCALING ?

- Uncertainties introduced across various length scales have a non-trivial interaction
- Current sophistications – resolve macro uncertainties

Micro

Meso

Macro

- Imprecise boundary conditions
- Initial perturbations

- Use micro averaged models for resolving physical scales

- Physical properties, structure follow a statistical description

EXAMPLE 2: DIFFUSION IN A RANDOM MICROSTRUCTURE

- DIFFUSION COEFFICIENTS OF INDIVIDUAL CONSTITUENTS NOT KNOWN EXACTLY
- A MIXTURE MODEL IS USED

THE INTENSITY OF THE GRAY-SCALE IMAGE IS MAPPED TO THE CONCENTRATIONS

DARKEST DENOTES b PHASE

LIGHTEST DENOTES a PHASE

- Motivation: coupling multiscaling and uncertainty analysis
- Mathematical representation of uncertainty
- Variational multiscale method (VMS)
- Application of VMS with explicit subgrid model
- Stochastic multiscale diffusion equation
- Increasing efficiency in uncertainty modeling techniques
- Sparse grid quadrature, support-space method
- Future directions

UNCERTAINTY ANALYSIS TECHNIQUES

- Monte-Carlo : Simple to implement, computationally expensive
- Perturbation, Neumann expansions : Limited to small fluctuations, tedious for higher order statistics
- Sensitivity analysis, method of moments : Probabilistic information is indirect, small fluctuations

- Spectral stochastic uncertainty representation
- Basis in probability and functional analysis
- Can address second order stochastic processes
- Can handle large fluctuations, derivations are general

RANDOM VARIABLES = FUNCTIONS ?

- Math: Probability space (W, F, P)

Sample space

Probability measure

Sigma-algebra

- Random variable

- : Random variable

- A stochastic process is a random field with variations across space and time

SPECTRAL STOCHASTIC REPRESENTATION

- A stochastic process = spatially, temporally varying random function

CHOOSE APPROPRIATE BASIS FOR THE PROBABILITY SPACE

GENERALIZED POLYNOMIAL CHAOS EXPANSION

HYPERGEOMETRIC ASKEY POLYNOMIALS

SUPPORT-SPACE REPRESENTATION

PIECEWISE POLYNOMIALS (FE TYPE)

KARHUNEN-LOÈVE EXPANSION

SPECTRAL DECOMPOSITION

SMOLYAK QUADRATURE, CUBATURE, LH

COLLOCATION, MC (DELTA FUNCTIONS)

ON random variables

Mean function

Stochastic process

Deterministic functions

- Deterministic functions ~ eigen-values , eigenvectors of the covariance function
- Orthonormal random variables ~ type of stochastic process
- In practice, we truncate (KL) to first N terms

- Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input

Stochastic input

Askey polynomials in input

Stochastic output

Deterministic functions

- Askey polynomials ~ type of input stochastic process
- Usually, Hermite, Legendre, Jacobi etc.

- Any function of the inputs, thus can be represented as a function defined over the support-space

FINITE ELEMENT GRID REFINED IN HIGH-DENSITY REGIONS

- SMOLYAK QUADRATURE
- IMPORTANCE MONTE CARLO

JOINT PDF OF A TWO RANDOM VARIABLE INPUT

OUTPUT REPRESENTED ALONG SPECIAL COLLOCATION POINTS

NEED FOR SUPPORT-SPACE APPROACH

- GPCE and Karhunen-Loeve are Fourier like expansions
- Gibb’s effect in describing highly nonlinear, discontinuous uncertainty propagation

Onset of natural convection

[Zabaras JCP 208(1)] – Using support-space method

[Ghanem JCP 197(1)] – Using Wiener-Haar wavelets

- Finite element representation of stochastic processes [stochastic Galerkin method: Babuska et al]
- Incorporation of importance based meshing concept for improving accuracy [support space method]

- Both GPCE and support-space method are fraught with the curse of dimensionality
- As the number of random input orthonormal variables increase, computation time increases exponentially
- Support-space grid is usually in a higher-dimensional manifold (if the number of inputs is > 3), we need special tensor product techniques for generation of the support-space
- Parallel implementations are currently performed using PETSc (Parallel scientific extensible toolkit )

Subgrid scale solution

Coarse scale solution

h

VARIATIONAL MULTISCALE METHOD

Hypothesis

- Exact solution = Coarse resolved part + Subgrid part [Hughes, 95, CMAME]

Induced function space

- Solution function space = Coarse function space + Subgrid function space

Idea

- Model the projection of weak form onto the subgrid function space, calculate an approximate subgrid solution
- Use the subgrid solution to solve for coarse solution

VARIATIONAL MULTISCALE BASICS

DERIVE THE WEAK FORMULATION FOR THE GOVERNING EQUATIONS

PROJECT THE WEAK FORMULATION ON COARSE AND FINE SCALES

SOLUTION FUNCTION SPACES ARE NOW STOCHASTIC FUNCTION SPACES

COARSE WEAK FORM

FINE (SUBGRID) WEAK FORM

ALGEBRAIC SUBGRID MODELS

COMPUTATIONAL SUBGRID MODELS

REMOVE SUBGRID EFFECTS IN THE COARSE WEAK FORM USING STATIC CONDENSATION

APPROXIMATE SUBGRID SOLUTION

NEED TECHNIQUES TO SOLVE STOCHASTIC PDEs

MODIFIED MULTISCALE COARSE WEAK FORM INCLUDING SUBGRID EFFECTS

VMS HYPOTHESIS

AFFINE CORRECTION

DERIVE WEAK FORM

COARSE-TO-SUBGRID MAP

DEFINE PROBLEM

MODEL MULTISCALE HEAT EQUATION

in

on

in

THE DIFFUSION COEFFICIENT K IS HETEROGENEOUS AND POSSESSES RAPID RANDOM VARIATIONS IN SPACE

- OTHER APPLICATIONS
- DIFFUSION IN COMPOSITES
- FUNCTIONALLY GRADED MATERIALS

FLOW IN HETEROGENEOUS POROUS MEDIA INHERENTLY STATISTICAL

DIFFUSION IN MICROSTRUCTURES

VMS HYPOTHESIS

AFFINE CORRECTION

DERIVE WEAK FORM

COARSE-TO-SUBGRID MAP

DEFINE PROBLEM

STOCHASTIC WEAK FORM

such that, for all

: Find

- Weak formulation

- VMS hypothesis

Exact solution

Subgrid solution

Coarse solution

EXPLICIT SUBGRID MODELLING: IDEA

DERIVE THE WEAK FORMULATION FOR THE GOVERNING EQUATIONS

PROJECT THE WEAK FORMULATION ON COARSE AND FINE SCALES

COARSE WEAK FORM

FINE (SUBGRID) WEAK FORM

COARSE-TO-SUBGRID MAP EFFECT OF COARSE SOLUTION ON SUBGRID SOLUTION

AFFINE CORRECTION SUBGRID DYNAMICS THAT ARE INDEPENDENT OF THE COARSE SCALE

LOCALIZATION, SOLUTION OF SUBGRID EQUATIONS NUMERICALLY

FINAL COARSE WEAK FORMULATION THAT ACCOUNTS FOR THE SUBGRID SCALE EFFECTS

VMS HYPOTHESIS

AFFINE CORRECTION

DERIVE WEAK FORM

COARSE-TO-SUBGRID MAP

DEFINE PROBLEM

SCALE PROJECTION OF WEAK FORM

such that, for all

Find

and

and

- Projection of weak form on coarse scale

- Projection of weak form on subgrid scale

EXACT SUBGRID SOLUTION

COARSE-TO-SUBGRID MAP

SUBGRID AFFINE CORRECTION

SPLITTING THE SUBGRID SCALE WEAK FORM

- Subgrid scale weak form

- Coarse-to-subgrid map

- Subgrid affine correction

ASSUMPTIONS:

NUMERICAL ALGORITHM FOR SOLUTION OF THE MULTISCALE PDE

COARSE TIME STEP

SUBGRID TIME STEP

1

1

ũC

ūC

Coarse solution field at end of time step

Coarse solution field at start of time step

ûF

ELEMENT

COARSE MESH

RANDOM FIELD DEFINED OVER THE ELEMENT

FINITE ELEMENT PIECEWISE POLYNOMIAL REPRESENTATION

USE GPCE TO REPRESENT THE RANDOM COEFFICIENTS

- Given the coefficients , the coarse scale solution is completely defined

VMS HYPOTHESIS

AFFINE CORRECTION

DERIVE WEAK FORM

COARSE-TO-SUBGRID MAP

DEFINE PROBLEM

COARSE-TO-SUBGRID MAP

ELEMENT

COARSE MESH

ANY INFORMATION FROM COARSE TO SUBGRID SOLUTION CAN BE PASSED ONLY THROUGH

BASIS FUNCTIONS THAT ACCOUNT FOR FINE SCALE EFFECTS

INFORMATION FROM COARSE SCALE

COARSE-TO-SUBGRID MAP

SOLVING FOR THE COARSE-TO-SUBGRID MAP

START WITH THE WEAK FORM

APPLY THE MODELS FOR COARSE SOLUTION AND THE C2S MAP

AFTER SOME ASSUMPTIONS ON TIME STEPPING

THIS IS DEFINED OVER EACH ELEMENT, IN EACH COARSE TIME STEP

VMS HYPOTHESIS

AFFINE CORRECTION

DERIVE WEAK FORM

COARSE-TO-SUBGRID MAP

DEFINE PROBLEM

SOLVING FOR SUBGRID AFFINE CORRECTION

START WITH THE WEAK FORM

- WHAT DOES AFFINE CORRECTION MODEL?
- EFFECTS OF SOURCES ON SUBGRID SCALE
- EFFECTS OF INITIAL CONDITIONS

CONSIDER AN ELEMENT

IN A DIFFUSIVE EQUATION, THE EFFECT OF INITIAL CONDITIONS DECAY WITH TIME. WE CHOOSE A CUT-OFF

- To reduce cut-off effects and to increase efficiency, we can use the quasistatic subgrid assumption

- Based on the indices in the C2S map and the affine correction, we need to solve (P+1)(nbf) problems in each coarse element
- At a closer look we can find that
- This implies, we just need to solve (nbf) problems in each coarse element (one for each index s)

VMS HYPOTHESIS

AFFINE CORRECTION

DERIVE WEAK FORM

COARSE-TO-SUBGRID MAP

DEFINE PROBLEM

MODIFIED COARSE SCALE FORMULATION

- We can substitute the subgrid results in the coarse scale variational formulation to obtain the following
- We notice that the affine correction term appears as an anti-diffusive correction
- Often, the last term involves computations at fine scale time steps and hence is ignored

- Stochastic investigations
- Example 1: Decay of a sine hill in a medium with random diffusion coefficient
- The diffusion coefficient has scale separation and periodicity
- Example 2: Planar diffusion in microstructures
- The diffusion coefficient is computed from a microstructure image
- The properties of microstructure phases are not known precisely [source of uncertainty]
- Future issues

- There are two important modeling considerations that were neglected for the dynamic subgrid model
- Effect of the subgrid component of the initial conditions on the evolution of the reconstructed fine scale solution
- Better models for the initial condition specified for the C2S map (currently, at time zero, the C2S map is identically equal to zero implying a completely coarse scale formulation)
- In order to avoid the effects of C2S map, we only store the subgrid basis functions beyond a particular time cut-off (referred to herein as the burn-in time)
- These modeling issues need to be resolved for increasing the accuracy of the dynamic subgrid model

EXAMPLE 2: DIFFUSION IN A RANDOM MICROSTRUCTURE

- DIFFUSION COEFFICIENTS OF INDIVIDUAL CONSTITUENTS NOT KNOWN EXACTLY
- A MIXTURE MODEL IS USED

THE INTENSITY OF THE GRAY-SCALE IMAGE IS MAPPED TO THE CONCENTRATIONS

DARKEST DENOTES b PHASE

LIGHTEST DENOTES a PHASE

FIRST ORDER GPCE COEFF

MEAN

SECOND ORDER GPCE COEFF

RECONSTRUCTED FINE SCALE SOLUTION (VMS)

FULLY RESOLVED GPCE SIMULATION

FIRST ORDER GPCE COEFF

MEAN

SECOND ORDER GPCE COEFF

RECONSTRUCTED FINE SCALE SOLUTION (VMS)

FULLY RESOLVED GPCE SIMULATION

HIGHER ORDER TERMS AT TIME = 0.2

FOURTH ORDER GPCE COEFF

THIRD ORDER GPCE COEFF

FIFTH ORDER GPCE COEFF

RECONSTRUCTED FINE SCALE SOLUTION (VMS)

FULLY RESOLVED GPCE SIMULATION

- THE EXAMPLES USED ASSUME A CORRELATION FUNCTION FOR INPUTS, USE KARHUNEN-LOEVE EXPANSION GPCE (OR) SUPPORT-SPACE
- PHYSICAL ASPECTS OF AN UNCERTAINTY MODEL, DERIVATION OF CORRELATION, DISCTRIBUTIONS USING EXPERIMENTS AND SIMULATION

ROUGHNESS

PERMEABILITY

- AVAILABLE GAPPY DATA
- BAYESIAN INFERENCE
- WHAT ABOUT THE MULTISCALE NATURE ?

- BOTH GPCE AND SUPPORT-SPACE ARE SUCCEPTIBLE TO CURSE OF DIMENSIONALITY
- USE OF SPARSE GRID QUADRATURE SCHEMES FOR HIGHER DIMENSIONS (SMOLYAK, GESSLER, XIU)
- FOR VERY HIGH DIMENSIONAL INPUT, USING MC ADAPTED WITH SUPPORT-SPACE, GPCE TECHNIQUES

- If the number of random inputs is large (dimension D ~ 10 or higher), the number of grid points to represent an output on the support-space mesh increases exponentially
- GPCE for very high dimensions yields highly coupled equations and ill-conditioned systems (relative magnitude of coefficients can be drastically different)
- Instead of relying on piecewise interpolation, series representations, can we choose collocation points that still ensure accurate interpolations of the output (solution)

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

SPARSE GRID COLLOCATION METHOD

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

USING THE COLLOCATION METHOD FOR HIGHER DIMENSIONS

- Flow through heterogeneous random media

Alloy solidification, thermal insulation, petroleum prospecting

Look at natural convection through a realistic sample of heterogeneous material

Square cavity with free fluid in the middle part of the domain. The porosity of the material is taken from experimental data1

Left wall kept heated, right wall cooled

Numerical solution procedure for the deterministic procedure is a fractional time stepping method

1. Reconstruction of random media using Monte Carlo methods, Manwat and Hilfer, Physical Review E. 59 (1999)

FLOW THROUGH HETEROGENEOUS RANDOM MEDIA

Experimental correlation for the porosity of the sandstone.

Eigen spectrum is peaked. Requires large dimensions to accurately represent the stochastic space

Simulated with N= 20

Number of collocation points is 11561 (level 4 interpolation)

Material: Sandstone

Numerically computed

Eigen spectrum

FLOW THROUGH HETEROGENEOUS RANDOM MEDIA

FIRST MOMENT

Snapshots at a few collocation points

Temperature y-Velocity

Temperature

Y velocity

Streamlines

SECOND MOMENT

Temperature

Y velocity

USING THE COLLOCATION METHOD FOR HIGHER DIMENSIONS

2. Flow over rough surfaces

Thermal transport across rough surfaces, heat exchangers

Look at natural convection through a realistic roughness profile

Rectangular cavity filled with fluid.

Lower surface is rough. Roughness auto correlation function from experimental data2

Lower surface maintained at a higher temperature

Rayleigh-Benard instability causes convection

Numerical solution procedure for the deterministic procedure is a fractional time stepping method

T (y) = -0.5

T (y) = 0.5

y = f(x,ω)

2. H. Li, K. E. Torrance, An experimental study of the correlation between surface roughness and light scattering for rough metallic surfaces, Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies II,

NATURAL CONVECTION ON ROUGH SURFACES

Experimental ACF

Experimental correlation for the surface roughness

Eigen spectrum is peaked. Requires large dimensions to accurately represent the stochastic space

Simulated with N= 20 (Represents 94% of the spectrum)

Number of collocation points is 11561 (level 4 interpolation)

Numerically computed Eigen spectrum

Sample realizations of temperature at collocation points

NATURAL CONVECTION ON ROUGH SURFACES

FIRST MOMENT

SECOND MOMENT

Temperature

Temperature

Streamlines

Y Velocity

Roughness causes improved thermal transport due to enhanced nonlinearities

Results in thermal plumes

Can look to tailor material surfaces to achieve specific thermal transport

We have a class of microstructures which share certain features between each other. We want to compute statistical variability of certain diffusion related fields, such as temperature, within this class of microstructures. The variability in the microstructural class is due to variations in grain sizes.

SUB PROBLEMS

- 1. How do you compute the class of microstructures?
- - MaxEnt
- 2. How do you interrogate this class of microstructures for diffusion problems?
- - Stochastic collocation schemes
- 3. How do you compute microstructures at collocation points?
- - POD method

Compute collocation points based on N=20(99.9% energy)

Microstructure samples computed using MaxEnt

Use POD technique to compute most energetic eigen microstructures

+

+

+

Each of the collocation samples is interrogated for computing the required statistical field such as temperature. The statistics account for topological variability within the class of microstructures

+ …

Compute microstructures at collocation points as a linear combination of the eigen microstructures

Temperature

DIFFUSION IN MICROSTRUCTURES INDUCED BY TOPOLOGICAL UNCERTAINTY

Statistical samples of microstructure at computed using maximum entropy technique. Collocation point microstructures are derived from this

Diffusion on the class of microstructures computed at collocation points

Limited set of input microstructures computed using phase field technique

Variability of temperature at a fixed point across the entire class of microstructures

STOCHASTIC MULTISCALING: Open problems

- A TYPICAL MULTISCALE PROCESS IS CHARACTERIZED BY PHYSICS AT VARIOUS LENGTH SCALES
- VMS IS ESSENTIALLY A SINGLE GOVERNING EQUATION MODEL
- HOW TO COMBINE VMS WITH OTHER COARSE-GRAINING TYPE, MULTISCALE METHODS
- HOW TO ADAPTIVELY SELECT MULTISCALE REGIONS : POSTERIORI ERROR MEASURES, CONTROL THEORY

TRANSFERRING DATA, STATISTICS ACROSS LENGTH SCALES USING INFORMATION THEORY

IN COUPLING MULTIPLE EQUATION MODELS, STATISTICS MUST BE CONSISTENT

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