STOCHASTIC MODELING OF MULTISCALE SYSTEMS

STOCHASTIC MODELING OF MULTISCALE SYSTEMS
NICHOLAS ZABARAS Materials Process Design and Control Laboratory Sibley School of Mechanical & Aerospace Engineering 188 Frank H T Rhodes Hall Cornell University Ithaca, NY 14853 URL:

OUTLINE Motivation: coupling multiscaling and uncertainty analysis
Mathematical representation of uncertainty Variational multiscale method (VMS) Stochastic support method Stochastic convection-diffusion equations Computing PDFs of microstructures (Maximum Entropy) Sparse grid collocation methods (Smolyak quadrature) Natural convection on rough surfaces Diffusion in stochastic heterogeneous media Future research directions

NEED FOR UNCERTAINTY ANALYSIS
Uncertainty is everywhere From NIST From Intel website From GE-AE website From DOE Porous media Silicon wafer Aircraft engines Material process Variation in properties, constitutive relations Imprecise knowledge of governing physics, surroundings Simulation based uncertainties (irreducible)

WHY UNCERTAINTY AND MULTISCALING ?
Uncertainties introduced across various length scales have a non-trivial interaction Current sophistications – resolve macro uncertainties Micro Meso Macro Use micro averaged models for resolving physical scales Imprecise boundary conditions Initial perturbations Physical properties, structure follow a statistical description

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 Sigma-algebra Probability measure 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)

KARHUNEN-LOEVE EXPANSION
ON random variables Stochastic process Mean function 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
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.

SUPPORT-SPACE REPRESENTATION
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

VARIATIONAL MULTISCALE METHOD WITH ALGEBRAIC SUBGRID MODELLING
Application : deriving stabilized finite element formulations for advection dominant problems

VARIATIONAL MULTISCALE HYPOTHESIS
EXACT SOLUTION COARSE SOLUTION INTRINSICALLY COUPLED SUBGRID SOLUTION H COARSE GRID RESOLUTION CANNOT CAPTURE FINE SCALE VARIATIONS THE FUNCTION SPACES FOR THE EXACT SOLUTION ALSO SHOW A SIMILAIR DECOMPOSITION In the presence of uncertainty, the statistics of the solution are also coupled for the coarse and fine scales

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 COMPUTATIONAL SUBGRID MODELS ALGEBRAIC 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

FINAL COARSE FORMULATION
VMS – ILLUSTRATION [NATURAL CONVECTION] Mass conservation Momentum conservation Energy conservation Constitutive laws DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS DERIVE SUBGRID OBTAIN ASGS FINAL COARSE FORMULATION

WEAK FORM OF EQUATIONS Energy function space Test Trial Energy equation – Find such that, for all , the following holds VMS hypothesis: Exact solution = coarse scale solution + fine scale (subgrid) solution DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS DERIVE SUBGRID OBTAIN ASGS FINAL COARSE FORMULATION

ENERGY EQUATION – SCALE DECOMPOSITION Energy equation – Find and such that, for all and , the following holds Coarse scale variational formulation Subgrid scale variational formulation These equations can be re-written in the strong form with assumption on regularity as follows DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS DERIVE SUBGRID OBTAIN ASGS FINAL COARSE FORMULATION

ELEMENT FOURIER TRANSFORM
RANDOM FIELD DEFINED IN WAVENUMBER SPACE RANDOM FIELD DEFINED OVER THE DOMAIN SPATIAL MESH Addressing spatial derivatives NEGLIGIBLE FOR LARGE WAVENUMBERS  SUBGRID APPROXIMATION OF DERIVATIVE DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS DERIVE SUBGRID OBTAIN ASGS FINAL COARSE FORMULATION

ASGS [ALGEBRAIC SUBGRID SCALE] MODEL
STRONG FORM OF EQUATIONS FOR SUBGRID CHOOSE AND APPROPRIATE TIME INTEGRATION ALGORITHM TIME DISCRETIZED SUBGRID EQUATION TAKE ELEMENT FOURIER TRANSFORM

MODIFIED COARSE FORMULATION Assume the solution obeys the following regularity conditions By substituting ASGS model in the coarse scale weak form A similar derivation ensues for stochastic Navier-Stokes DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS DERIVE SUBGRID OBTAIN ASGS FINAL COARSE FORMULATION

FLOW PAST A CIRCULAR CYLINDER
NO-SLIP TRACTION FREE RANDOM UINLET NO-SLIP INLET VELOCITY ASSUMED TO BE A UNIFORM RANDOM VARIABLE KARHUNEN-LOEVE EXPANSION YIELD A SINGLE RANDOM VARAIBLE THUS, GENERALIZED POLYNOMIAL CHAOS  LEGENDRE POLYNOMIALS (ORDER 3 USED) Investigations: Vortex shedding, wake characteristics

FULLY DEVELOPED VORTEX SHEDDING
Mean pressure First LCE coefficient Second LCE coefficient Wake region in the mean pressure is diffusive in nature Also, the vortices do not occur at regular intervals [Karniadakis J. Fluids. Engrg]

VELOCITIES AND FFT FFT YIELDS A MEAN SHEDDING FREQUENCY OF 0.162
FFT SHOWS A DIFFUSE BEHAVIOR IMPLYING CHANGING SHEDDING FREQUENCIES MEAN VELOCITY AT NEAR WAKE REGION EXHIBITS SUPERIMPOSED FREQUENCIES

VARIATIONAL MULTISCALE METHOD WITH EXPLICIT SUBGRID MODELLING FOR MULTISCALE DIFFUSION IN HETEROGENEOUS RANDOM MEDIA

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 DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS COARSE-TO-SUBGRID MAP FINAL COARSE FORMULATION AFFINE CORRECTION

STOCHASTIC WEAK FORM Weak formulation : Find such that, for all
VMS hypothesis Exact solution Coarse solution Subgrid solution DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS COARSE-TO-SUBGRID MAP FINAL COARSE FORMULATION AFFINE CORRECTION

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

SCALE PROJECTION OF WEAK FORM
Find and such that, for all 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 DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS COARSE-TO-SUBGRID MAP FINAL COARSE FORMULATION AFFINE CORRECTION

SPLITTING THE SUBGRID SCALE WEAK FORM
Coarse-to-subgrid map Subgrid affine correction

NATURE OF MULTISCALE DYNAMICS
ASSUMPTIONS: NUMERICAL ALGORITHM FOR SOLUTION OF THE MULTISCALE PDE COARSE TIME STEP SUBGRID TIME STEP 1 1 ũC ūC Coarse solution field at start of time step Coarse solution field at end of time step ûF

REPRESENTING COARSE SOLUTION
COARSE MESH ELEMENT 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

COARSE-TO-SUBGRID MAP
COARSE MESH ELEMENT ANY INFORMATION FROM COARSE TO SUBGRID SOLUTION CAN BE PASSED ONLY THROUGH INFORMATION FROM COARSE SCALE BASIS FUNCTIONS THAT ACCOUNT FOR FINE SCALE EFFECTS COARSE-TO-SUBGRID MAP DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS COARSE-TO-SUBGRID MAP FINAL COARSE FORMULATION AFFINE CORRECTION

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

INTRODUCE A SUBSTITUTION
BCs FOR THE COARSE-TO-SUBGRID MAP INTRODUCE A SUBSTITUTION CONSIDER AN ELEMENT

SOLVING FOR SUBGRID AFFINE CORRECTION
START WITH THE WEAK FORM CONSIDER AN ELEMENT WHAT DOES AFFINE CORRECTION MODEL? EFFECTS OF SOURCES ON SUBGRID SCALE EFFECTS OF INITIAL CONDITIONS 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 DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS COARSE-TO-SUBGRID MAP FINAL COARSE FORMULATION AFFINE CORRECTION

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 DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS COARSE-TO-SUBGRID MAP FINAL COARSE FORMULATION AFFINE CORRECTION

DIFFUSION IN A RANDOM MICROSTRUCTURE
A MIXTURE MODEL IS USED AS AN EXAMPLE OF GENERATING A HETEROGENEOUS DISTRIBUTION OF CONDUCTIVITY WE ASSUME THAT THE DIFFUSION COEFFICIENTS OF INDIVIDUAL CONSTITUENTS ARE NOT KNOWN EXACTLY THE INTENSITY OF THE GRAY-SCALE IMAGE IS MAPPED TO THE CONCENTRATIONS DARKEST DENOTES b PHASE LIGHTEST DENOTES a PHASE

RESULTS AT TIME = 0.05 MEAN FIRST ORDER GPCE COEFF
SECOND ORDER GPCE COEFF RECONSTRUCTED FINE SCALE SOLUTION (VMS) FULLY RESOLVED GPCE SIMULATION

RESULTS AT TIME = 0.2 MEAN FIRST ORDER GPCE COEFF
SECOND ORDER GPCE COEFF RECONSTRUCTED FINE SCALE SOLUTION (VMS) FULLY RESOLVED GPCE SIMULATION

HIGHER ORDER TERMS AT TIME = 0.2
THIRD ORDER GPCE COEFF FOURTH ORDER GPCE COEFF FIFTH ORDER GPCE COEFF RECONSTRUCTED FINE SCALE SOLUTION (VMS) FULLY RESOLVED GPCE SIMULATION

SUPPORT-SPACE – STOCHASTIC GALERKIN
- Joint probability density function of the inputs - The input support-space denotes the regions where input joint PDF is strictly positive Triangulation of the support-space Any function can be represented as a piecewise polynomial on the triangulated support-space - Function to be approximated - Piecewise polynomial approximation over support-space L2 convergence – (mean-square) Error in approximation is penalized severely in high input joint PDF regions. We use importance based refinement of grid to avoid this h = mesh diameter for the support-space discretization q = Order of interpolation

IMPLEMENTATION OF SUPPORT-SPACE
A stochastic process can be interpreted as a random variable at each spatial point Two-level grid approach Support-space grid Mesh dense in regions of high input joint PDF Spatial domain Spatial grid There is finite element interpolation at both spatial and random levels Each spatial location handles an underlying support-space grid Highly OOP structure Element

CAPTURING UNSTABLE EQUILIBRIUM
Computational details – 1600 bilinear elements for spatial grid Time of simulation – 1.5 non-dimensional units Rayleigh number – uniformly distributed random variable between 1530 and 1870 (10% fluctuation about 1700) Prandtl number – 6.95 Time stepping – non-dimensional units Support-space grid – One-dimensional with ten linear elements Cold wall Insulated Insulated Hot wall Simulation about the critical Rayleigh number – conduction below, convection above Both GPCE and support-space methods are used separately for addressing the problem Failure of Generalized polynomial chaos approach Support-space method – evaluation and results against a deterministic simulation

FAILURE OF THE GPCE X-vel Y-vel
Mean X- and Y- velocities determined by GPCE yields extremely low values !! (Gibbs effect) X-vel Y-vel X- and Y- velocities obtained from a deterministic simulation with Ra = 1870 (the upper limit)

PREDICTION BY SUPPORT-SPACE METHOD
X-vel Y-vel Mean X- and Y- velocities determined by support-space method at a realization Ra=1870 X-vel Y-vel X- and Y- velocities obtained from a deterministic simulation with Ra = 1870 (the upper limit)

SPARSE GRID COLLOCATION
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)

SMOLYAK ALGORITHM: SPARSE GRIDS
Full tensor product grid: 289 points Example of using sparse grids to build interpolating functions: Discontinuous functions Left to right: Improving interpolation depth For an error around 2x10-2: Required number of points using sparse grids 3300 Required number of points using full tensor products: Sparse Grid: 65 points Number of points required to construct interpolating functions reduces combinatorially. Reduction more significant as the number of dimensions increases

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

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 Some degradation in accuracy Maximal reduction when the function is assumed to be smooth D = 10 ORDER CC FE 3 1581 1000 4 8801 10000 5 41625 100000

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
Material: Sandstone Experimental correlation for the porosity of the sandstone. Eigen spectrum is peaked. Requires large dimensions to accurately represent the stochastic space Simulated with N= 8 Number of collocation points is (level 4 interpolation) 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 (level 4 interpolation) Sample realizations of temperature at collocation points Numerically computed Eigen spectrum

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

Statistical characterization of microstructures
Can we compute statistical response of a class of microstructures subjected to applied loads based on limited experimental information? Features of a microstructure Grain size (in 3D, grain volume) When a specimen is manufactured, the microstructures at a sample point will not be the same always. Orientation Distribution Function Rodrigues’ representation FCC fundamental region

Technique employed Numerical implementation Problem formulation
Maximum entropy (MAXENT): The probability distribution that maximizes entropy and satisfies the given (experimental/simulation-based) information is the least-biased estimate that can be made. Numerical implementation Problem formulation Limited microstructures computed using phase field simulations We employ the voronoi cell tessellation technique for representing microstructures. Extract features of the microstructure Geometrical: grain size Texture: ODFs Conjugate gradient Compute a PDF of microstructures Entropy Compute bounds on macroscopic properties microstructure feature constraints features of microstructure, I Meshing a statistical class of microstructures using CUBIT Given information about microstructures. We use grain size and texture features

Statistical class of 3D Aluminium polycrystals
Three statistical Aluminium polycrystal samples generated using phase field simulations Comparison of grain size distributions between a phase field simulation from the representative class and a MaxEnt sample 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 0.05 0.1 0.15 0.2 0.25 Grain volume (voxels) Probability mass function Grain volume distribution using phase field simulations pmf reconstructed using MaxEnt First four statistical moments of grain sizes (volumes)

Representation in Frank-Rodrigues space
ODF reconstruction using MAXENT 5000 10000 15000 20000 25000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Grain volume (voxels) Probability mass function R corr =0.9644 KL=0.0383 Input ODF Representation in Frank-Rodrigues space Grain size distribution of a microstructural sample. Comparison with the MaxEnt distribution Reconstructed samples using MAXENT A microstructural specimen computed from the MaxEnt distribution

Statistical variation of homogenized stress-strain curves.
Statistical variation of properties How to mesh a microstructure? We employ hexahedral elements using Cubit software Homogenization scheme: First order stress averaging Scheme employing Hill’s criterion 1 2 3 x 10 -4 10 20 30 40 50 60 Equivalent strain Equivalent stress (MPa) Mean stress-strain curve Mean std Aluminium polycrystal with rate-independent strain hardening. Pure tensile test. Statistical variation of homogenized stress-strain curves.

HETEROGENEOUS DIFFUSION
a) Two phase materials b) Micro-emulsions, c) porous media, d) ceramics e) Polycrystals f) Foams, blends Different morphology, anisotropy To apply physical processes on these heterogeneous systems worst case scenarios variations on physical properties Aim: To develop a procedure to predict statistics of properties of heterogeneous materials undergoing certain phenomena Most materials in nature are heterogeneous. Examples. Tailored materials to meet certain demands. This leads to functionally graded heterogeneous materials. Examples are concrete, ceramics for armor and heat resistant shields, shock absorbing foams and blends. Major activity in understanding physical phenomena that these heterogeneous materials undergo. First issue is the fact that the material properties are heterogeneous across multiple length scales. AIM is to develop a methodology to stochastically predict the behavior of these materials. This will immediately result in the quantitative information on worst case scenarios, expected variation in properties and dependent variables ect.

OVERVIEW OF METHODOLOGY
Given certain properties P1, P2, .. Pn, that the structure satisfies. These properties are usually statistical: Volume fraction, 2 Point correlation, auto correlation Reconstruct realizations of the structure satisfying the properties. Monte Carlo, Gaussian Random Fields, Stochastic optimization ect STEP 1 STEP 2 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 STEP 3 Consider a microstructure that is a result of a specific process (chemical, thermal or mechanical). This microstructure will satisfy certain properties that are unique/determined by the process leading to it. That is, it will have some specific volume fraction (this is a first order statistic), it may have a certain two point correlation as well as higher order statistics. Now given these properties, one can reconstruct the microstructure (using different techniques, like MonteCarlo methods that simulate the process leading to the microstructure, Gaussian random fields that match the statistics of the microstructure with a randomly generated surface and stochastic optimization techniques that anneal a given structure until it matches the given properties). One thus constructs a large library of all possible allowable microstructures that satisfy the given properties. We denote this the space of allowable microstructures It should be noted that as the number of properties that have to be matched increases, the space of allowable microstructures rapidly shrinks. To compute any statistic of this space of microstructures, one has to sample all the available microstructures. This is computationally intractable. One approach is to construct a reduced model of the space of microstructure. This effectively reduced the infinite dimensional space representing the microstructures to a more tractable finite dimensional space of coefficients that represent the model. This is an area where substantial improvements/research has to be performed. Pertinent questions are: Which is the best representation for the class of microstructures? Can any transform technique be converted to condense the information into manageable parts (ex like FFT, Discrete Cosine transform)? Can techniques from image processing be useful here? (develop optimal autoregressive models, moving average models ect) Here we proceed to use the maximum energy preserving transform: the Karhunen-Loeve transform. Once the reduced model to represent the microstructure class has been developed, we utilize the collocation method to solve the physical stochastic problem

EXAMPLE: THERMAL DIFFUSION THROUGH TUNGSTEN—SILVER MATRIX
MC-Potts model, generate microstructures database. Apply the KL transform First 9 eigen values are enough Tungsten-silver composite image1 We provide a specific example to illustrate this methodology. A tungsten-Silver matrix is considered. The image shows a experimental pictograph of the microstructural sample. Using a Monte Carlo Potts model for the physical process (sintering) that leads to the metal matrix, a set of 500 microstructures were generated that satisfied the volume fraction and pair correlation properties of the experimental image. The KL transform was applied to this class of images. The first 9 images described about 95% of the energy spectrum. The figure plots the eigen spectrum as well as the first few eigen images. 1. S. Umekawa, R. Kotfila and O.D. Sherby, Elastic properties of a tungsten-silver composite above and below the melting point of silver, J. Mech. Phys. Solids 13 (1965)

REDUCED MODEL FOR THE STRUCTURE
Represent any microstructure as a linear combination of the eigen-images I = Iavg + I1a1 + I2a2+ I3a3 + … + Inan = + ..+ an a1 + a2 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 higher order statistics Thus the n tuple (a1,a2,..,an) must further satisfy some constraints. Using the KL transform the reduced model can be constructed. That is any microstructure in the class of microstructures is a linear combination of the Eigen Images. But the reduced model spans an extension of the microstructure space considered. i.e. There exist linear combination of the eigen images that do not belong to the class of microstructure (they do not satisfy the properties of this class). Thus the coefficients of the reduced model must satisfy some constraints so the the resulting images satisfy the properties.

REDUCED MODEL FOR THE STRUCTURE
Constraints on the coefficients Construct the Convex Hull of the set of linear inequalities. This is the allowable set of coefficients. This represents the space of allowable microstructures In this space all the structures are equiprobable. This represents a stochastic space in (n-1) dimensions. Actually a plane in n dimensions, Call this the ‘material plane’ The coefficients must satisfy the constraints. This results in several linear equalities and inequalities. The convex hull of the allowable plane is solved for. It is shown in the first figure. This represents the space of all allowable coefficients. This can further be reduced by one dimension using the equality condition. This represents the ‘material’ plane. The coefficients in this plane result in microstructures that satisfy the properties that define the microstructure. No other coefficients satisfy these properties. This plane completely represents the space of allowable microstructures. Note that all the microstructures (i.e points in the ‘material’ plane) are equally probable. Thus the space of allowable microstructures is a stochastic space of (n-1) (here shown as2) dimensions, where n is the number of eigen images used to represent the class.

PHYSICAL PROBLEM UNDER CONSIDERATION
Structure size 20x20x20 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

COLLOCATION SCHEME: SAMPLE REALIZATIONS
First column: conductivity Second column: Temperature

MEAN STATISTICS Temperature isosurfaces
The mean statistic of the diffusion problem is shown. The are fluctuations in the temperature iso-surfaces caused by the variations in the thermal conductivity of the matrix. The temperature is uniform near the boundaries due to the applied Dirichlet boundary conditions. Mean temperature: No variations closer to the surfaces, significant variations inside Mean distribution of silver

SECOND ORDER STATISTICS
Temperature slice Property slice This slide shows the second moment of the statistics. The largest deviation of the temperature occurs near the middle plane of the domain. Slices of the temperature and the thermal diffusivity are shown. It is interesting to note that the regions of large temperature fluctuations straddle regions of large property fluctuations. This is illustrated by plotting the isosurfaces of large deviations in temperature and properties. When they are superimposed, notice that the regions of large thermal deviations are in between regions of large property variations. Left, isosurface of temperature deviation Right, isosurface of properties

HIGHER ORDER STATISTICS
This picture plots higher order statistics. Two points are chosen. One is a point where the temperature deviation is the maximum and other is where it is minimum. At both points, the thermal profile for the COMPLETE space of microstructure is plotted. Note the variation in the temperature. In the former case substantial variations in the temperature are seen.

Future research directions
Algorithms to address the curse-of-dimensionality Adaptivity in the support space, adaptive sparse-grid quadrature rules, SPDE model reduction, etc. Stochastic multiscale advection-diffusion-reaction Stochastic multiscale modeling in materials Information-theoretic algorithms for coupling statistics across length scales Robust design techniques Interface stochastic and statistical (Bayesian) computation

STOCHASTIC MODELING OF MULTISCALE SYSTEMS

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