sparse grid collocation schemes for stochastic convection materials process design and control...

Sparse grid collocation schemes for Sparse grid collocation schemes for stochastic convectionstochastic convection

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Nicholas Zabaras and Baskar Ganapathysubramanian

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://mpdc.mae.cornell.edu/




All physical systems have an inherent associated randomness

SOURCES OF UNCERTAINTIES

•Multiscale material information – inherently statistical in nature.

•Uncertainties in process conditions

•Input data

•Model formulation – approximations, assumptions.

Why uncertainty modeling ?

Assess product and process reliability.

Estimate confidence level in model predictions.

Identify relative sources of randomness.

Provide robust design solutions.

Engineering component

Heterogeneous random

Microstructural features

MOTIVATION

Process

Control?




Problem of interest

Investigate the effects of input uncertainties in natural convection problems.

- Results in more realistic modeling

- Leads to topology design for enhanced heat and mass transfer problems

Consider a 2D natural convection system.

Interested in the effects of three kinds of input uncertainties

a) Uncertainties in boundary conditions

b) Uncertainties in boundary topology

c) Uncertainties in material properties




S

Sample space of elementary events

Real line

Random

variable

MAP

Collection of all possible outcomes

Each outcome is mapped to a

corresponding real value

Interpreting random variables as functions

A general stochastic process is a random field with variations along space and time – A

function with domain (Ω, Τ, S)

REPRESENTING RANDOMNESS:1

1. Interpreting random variables

2. Distribution of the random variable

Ex. Inlet velocity, Inlet temperature

1 0.1o

3. Correlated data

Ex. Presence of impurities, porosity

Usually represented with a correlation function

We specifically concentrate on this.




REPRESENTING RANDOMNESS:2

1. Representation of random process

- Karhunen-Loeve, Polynomial Chaos expansions

2. Infinite dimensions to finite dimensions

- depends on the covarience

Karhunen-Loèvè expansion

Based on the spectral decomposition of the covariance kernel of the stochastic process

Random process Mean

Set of random variables to

be found

Eigenpairs of covariance

kernel

• Need to know covariance

• Converges uniformly to any second order process

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Index

Eig

enva

lue

5 10 15 200

5

10

15

Set the number of stochastic dimensions, N

Dependence of variables

Pose the (N+d) dimensional problem




KARHUNEN-LOEVE EXPANSION

1

( , , ) ( , ) ( , ) ( )i ii

X x t X x t X x t

Stochastic process

Mean function

ON random variablesDeterministic 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

1( , , ) fn( , , , , )NX x t x t




Problem definition

Represent input uncertainties in terms of N random variables. This is possible usually due to the ‘finite dimensional noise assumption’1

1) I. Babuska, R. Tempone, G. E. Zouraris, Galerkin finite elements approximation of stochastic finite elements, SIAM J. Numer. Anal. 42 (2004) 800–825

g(y) =

The dependant variables (T,u,p) depend on these N random variables. Reformulate the problem in terms of these N variables

where g is the appropriate input stochastic process




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

Stochastic collocation: Results in decoupled equations




SPECTRAL STOCHASTIC REPRESENTATION

: ( , , )X x t

A stochastic process = spatially, temporally varying random function

CHOOSE APPROPRIATE BASIS FOR THE

PROBABILITY SPACE

HYPERGEOMETRIC ASKEY POLYNOMIALS

PIECEWISE POLYNOMIALS (FE TYPE)

SPECTRAL DECOMPOSITION

COLLOCATION, MC (DELTA FUNCTIONS)

GENERALIZED POLYNOMIAL CHAOS EXPANSION

SUPPORT-SPACE REPRESENTATION

KARHUNEN-LOÈVE EXPANSION

SMOLYAK QUADRATURE, CUBATURE, LH




GENERALIZED POLYNOMIAL CHAOS Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input

0

( , , ) ( , ) (ξ( ))i ii

Z x t Z x t

Stochastic output

Askey polynomials in inputDeterministic functions

Stochastic input

1( , , ) fn( , , , , )NX x t x t

Askey polynomials ~ type of input stochastic process

Usually, Hermite, Legendre, Jacobi etc.




CURSE OF DIMENSIONALITY 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 )




COLLOCATION TECHNIQUES

Spectral Galerkin method: Spatial domain is approximated using a finite element discretization

Stochastic domain is approximated using a spectral element discretization

Coupled equationsDecoupled equations

Collocation method: Spatial domain is approximated using a finite element discretization

Stochastic domain is approximated using multidimensional interpolating functions




WHY ARE THE EQUATIONS DECOUPLED?

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 convection system

One can construct the stochastic solution by solving at the M deterministic points




CONSTRUCTING OPTIMAL INTERPOLATING FUNCTIONS

Two issues with constructing accurate interpolating functions:

1) Choice of optimal points to sample at

2) Constructing multidimensional polynomial functions

Analysis of optimal points for one dimensional functions

Consider the one D function

Need to approximate this function through a polynomial interpolant

Sample the function at a finite set of points

Construct the interpolant such that

The interpolant can be written as

As the number of sampling points increases the approximating quality of the polynomial improves




CONSTRUCTING OPTIMAL INTERPOLATING FUNCTIONS

As the number of sampling points increases the approximating quality of the polynomial improves. This is irrespective of how one chooses the sampling points.

But uniform convergence is not guaranteed

To choose optimal distribution of points to ensure uniform convergence, must need a notion of the approximating quality of the polynomial.

The Best approximating polynomial is defined such that

Every interpolation function can be related to the best approximation polynomial through its Lebesgue constant

where the Lebesgue constant is

Note that the Lebesgue constant depends only on the node distribution and not on the function.

One can find distribution of points that minimize the Lebesgue constants




FROM ONE DIMENSION TO HIGHER DIMENSIONS

Gauss points and Chebyshev points have small Lebesgue constants and can be used as nodes to construct the interpolation function1.

Can come up with error bounds for these distribution of points. The interpolation error while using n Chebyshev sampling points is given by

1. A. Klimke, Uncertainty Modeling using Fuzzy Arithmetic and Sparse Grids, PhD Thesis, Universitt Stuttgart, Shaker Verlag, Aachen, 2006.

From this optimal one dimensional interpolation function, straightforward extension to multiple dimensions using the concept of tensor products.

This quickly becomes impossible to use. For instance, if N=10 dimensions ans we were to use n=2 points in each dimension, we would require 210 points to interpolate this function.

Look at better ways to sample these points




SMOLYAK ALGORITHM

LET OUR BASIC 1D INTERPOLATION SCHEME BE SUMMARIZED AS

IN MULTIPLE DIMENSIONS, THIS CAN BE WRITTEN AS

( ) ( )i

i i

i i

xx X

U f a f x

1 11

1 1

( )( ) ( ) ( , , )d di id

i i i id d

i ii i

x xx X x X

U U f a a f x x

TO REDUCE THE NUMBER OF SUPPORT NODES WHILE MAINTAINING ACCURACY WITHIN A LOGARITHMIC FACTOR, WE USE SMOLYAK METHOD

1

0 11

, 1,

0, ,

( ) ( ) ( )( )d

i i id

iiq d q d

i q

U U U i i i

A f A f f

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

( ) ( )i

i i

i i

xx X

U f a f x

1

0 11

, 1,

0, ,

( ) ( ) ( )( )d

i i id

iiq d q d

i q

U U U i i i

A f A f f

Uni-variate interpolation

Multi-variate interpolation

Smolyak interpolation

ORDER SC FE

3 1581 1000

4 8801 10000

5 41625 100000

D = 10

Accuracy the same as tensor product

Within logarithmic constant

Increasing the order of interpolation increases the number of points sampled




SMOLYAK ALGORITHM: REDUCTION IN POINTS

ORDER SC FE

3 1581 1000

4 8801 10000

5 41625 100000

D = 10

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

Results in multiple orders of magnitude reduction in the number of points to sample




SMOLYAK ALGORITHM: Numerical illustration

Interpolating smooth anisotropic functions.

Investigate interpolation accuracy as ρ increases. As ρ increases the function becomes steeper in one direction

Error defined as deviation of interpolant from actual function

For ρ =1000, the function is very anisotropic. The sparse collocation method uses 3329 points to construct an approximate solution with an error of 3x10-5.

The full tensor product method uses 263169 points to get the same level of accuracy




SMOLYAK ALGORITHM: Numerical illustration

Interpolating discontinuous functions.

Increasing number of sampling points

Function has a discontinuity in the y direction.

32769 points required to construct interpolant with error 3x10-3.

Issues:

Notice that the smolyak method uniformly samples both dimensions.

Can the number of sampling points be further reduced by choosing points adaptively in different directions based on the behavior of the function?

Can this be done on-the-fly?




ADAPTIVE SPARSE GRID COLLOCATION

The conventional sparse grid method treats every dimension equally.

Functions may have widely varying characteristics in different directions (discontinuities, steep gradients) or the function may have some special structure (additive, nearly-additive, multiplicative).

The basis proposition of the adaptive sparse grid collocation is to detect these structures/behaviors and treat different dimensions differently to accelerate convergence.

Must use some heuristics to select the sampling points.

Such heuristics have been developed by Gerstner and Griebel

Have to come up with a way to make the Smolyak algorithm treat different dimensions differently.

Generalized Sparse Grids:

Convention sparse grids imposes a strict admissibility condition on the indices. By relaxing this to allow other indices, adaptivity can be enforced.

Admissibility criterion for a set of indices S.

where ej is the unit vector in the j-th direction

1. T. Gerstner, M. Griebel, Numerical integration using sparse grids, Numerical Algorithms, 18 (1998) 209–232.




ADAPTIVE SPARSE GRID COLLOCATION

Finding the most sensitive dimensions:

The generalized sparse grid indices allow one to sample different dimensions differently. To accurately build interpolants using a minimal number of points, most of the points should be concentrated in the directions that have the steepest gradient or have discontinuities

Define directional errors to quantify the notion of sensitivity of each direction. Direction error are the interpolation errors achieved by adding sampling points in that specific direction

Interpolation procedure:

Start from the smallest index.

Add indices in each coordinate direction. Evaluate the function at these new indices. Compute the error between evaluated value and estimated value for each direction.

The direction with the maximal error need more indices. The function is evaluated in this direction




ADAPTIVE SPARSE GRID COLLOCATION: Numerical illustration

Interpolating smooth anisotropic functions.

Investigate interpolation accuracy as ρ increases. As ρ increases the function becomes steeper in one direction

For ρ = 1000, the adaptive sparse grid collocation uses 577 points to generate an interpolation function with error 5x10-

2. The conventional sparse grid collocation uses 1577 points to get the same accuracy.

More points sampled in the y direction




ADAPTIVE SPARSE GRID COLLOCATION: Numerical illustration

Interpolating discontinuous functions.

Function has a discontinuity in the y direction.

The adaptive method uses 559 points to build the interpolation function, while the conventional method uses 3300 points

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Conventional sparse grid

Adaptive sparse grid




SPARSE GRID COLLOCATION METHOD: implementation

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 (w i)

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

Solution Methodology

Use any validated deterministic solution procedure.

Completely non intrusive

0.3010.2600.2200.1800.1400.1000.0600.020

0.3010.2600.2200.1800.1400.1000.0600.020

Std deviation of temperature: Natural convection




REFERENCESSPARSE GRID COLLOCATION

1) E. Novak, K. Ritter, R. Schmitt, A. Steinbauer, On an interpolatory method for high-dimensional integration, J. Comp. Appl. Mathematics, 112 (1999) 215–228.

2) V. Barthelmann, E. Novak, K. Ritter, High-dimensional polynomial interpolation on sparse grids, Adv. Compu. Math. 12 (2000) 273–288.

3) A. Klimke, Uncertainty Modeling using Fuzzy Arithmetic and Sparse Grids, PhD Thesis, Universitt Stuttgart, Shaker Verlag, Aachen, 2006.

4) T. Gerstner, M. Griebel, Numerical integration using sparse grids, Numerical Algorithms, 18 (1998) 209–232.

STOCHASTIC COLLOCATION

1) B. Ganapathysubramanian, N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comp. Physics, submitted for publication.

2) B. Ganapathysubramanian, N. Zabaras, Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the VMS method, JCP, submitted.

3) I. Babuska, F. Nobile, R. Tempone, A stochastic collocation method for elliptic PDEs with random input data, ICES Report 05-47, 2005.

4) D. Xiu, J. S. Hesthaven, High order collocation methods for the differential equation with random inputs, SIAM J. Sci. Comput. 27 (2005) 1118–1139

5) F. Nobile, R. Tempone, C. G. Webster, A sparse grid stochastic collocation method for elliptic PDEs with random input data, preprint.




NUMERICAL EXAMPLES




Natural convection with random boundary conditions: A comparisonT

(y)

= 0

.5

T (

y) =

f (

y,ω

)

Fluid (Pr=1.0) in a square domain

Computational domain [-0.5,0.5]^2

Left wall maintained at T = 0.5

Right wall maintained at a meant temperature, T = -0.5

Temperature varies spatially along the right wall. These temperatures are correlated.

Physically represents the behavior of, say, a resistance heater.

Boundary temperature correlation C(y1,y2) = exp(-c|y1-y2|)

Solve problem using MonteCarlo methods, GPCE and Sparse collocation




Comparison

Mean temperature distribution

Standard deviation of temperature

Sparse grid collocation GPCE MC




Comparison

Computational effort

MonteCarlo: 65000 samples

GPCE: second order expansion

Smolyak: level 6 interpolation

Domain: [-0.5, 0.5]x[-0.5, 0.5]

Grid: 50x50 quad elements

Time steps: 600 dt = 1e-3

MonteCarlo just a means to validate, computationally not feasible

Compare GPCE and Sparse grid collocation methods

All problems solved on 16 nodes of V3 cluster at Cornell Theory Center




ComparisonSparse grid collocation is computationally very efficient for moderate dimensions.

Post processing to obtain higher order statistics is very simple.




Natural convection with random domains

Effect of roughness on natural convection

T (y) = 0.5

T (y) = -0.5

y = f(x,ω)

Thermal evolution of fluid on a rough surface heated from below.

Surface characterization:

Waviness and roughness

Waviness: Large scale variations

Roughness: Small scale perturbations to the surface

Representing roughness:

Roughness represented by two components: PDF of a point above a datum z and the correlation between two points (ACF)

ACF depends on the processing methodology, ex shot peening, sand blasting and milling

PDF is usually assumed to be a Gaussian





Effect of roughness on natural convection

T (y) = 0.5

T (y) = -0.5

y = f(x,ω)

Thermal evolution of fluid on a rough surface heated from below.

C(y1,y2) = exp(-c|y1-y2|)

ACF taken to be a simple exponential correlation

Mean roughness measure is 1/100 of the characteristic length of the domain

The correlation length is set at 0.1

First 8 eigen values represent 96% of the spectrum

Computational domain: [-1, 1]x[-0.5, 0.5]

Grid 200x100

Pr = 6.4 (corresponding to water)

Ra = 5000

Top wall set at T = -0.5

Bottom wall set at T = 0.5





Level 4 sparse grid collocation scheme is used

Number of points = 3937

Computational effort: 8 nodes of V3 on CTC ~ 500 minutes

Temperature and velocity realizations

0.4380.3130.1880.063

-0.063-0.188-0.313-0.438

0.4380.3130.1880.063

-0.063-0.188-0.313-0.438

0.4380.3130.1880.063

-0.063-0.188-0.313-0.438

0.4380.3130.1880.063

-0.063-0.188-0.313-0.438

8.3696.4494.5292.6100.690

-1.230-3.150-5.070

17.53713.326

9.1144.9020.691

-3.521-7.733

-11.944

12.0137.7733.533

-0.707-4.947-9.186

-13.426-17.666

15.70711.772

7.8373.901

-0.034-3.969-7.904

-11.839

0.4380.3130.1880.063

-0.063-0.188-0.313-0.438




Natural convection with random domains: Mean statistics

0.4370.3120.1870.062

-0.062-0.187-0.312-0.437

6.5304.6702.8100.950

-0.911-2.771-4.631-6.492

4.1922.5280.864

-0.800-2.465-4.129-5.793-7.457

0.3550.2530.1520.050

-0.051-0.153-0.254-0.356

Much more diffuse behavior

Temperature Pressure

v velocity u velocity




15.87513.75811.642

9.5257.4085.2923.1751.058

0.3010.2600.2200.1800.1400.1000.0600.020

18.59916.11913.63911.159

8.6796.2003.7201.240

0.8010.6940.5870.4800.3740.2670.1600.053

Natural convection with random domains: Higher order statistics

Temperature Pressure

v velocity u velocity




Natural convection with random domains: Mode shifts and PDF’s

0.3010.2600.2200.1800.1400.1000.0600.020

Location (0,0.25) shows large deviation in temperature. Plot of distribution of temperature and v velocity show a bi-modal nature. Possibility of two distinct modes. Can find most sensitive dimension. Dimension which shows an abrupt change in the variables

Temperature

v velocity




Natural convection with random domains: Large dimensions. experimental data

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

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,

T (y) = 0.5

T (y) = -0.5

y = f(x,ω)




NATURAL CONVECTION ON ROUGH SURFACES

V1

V2

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

IndexE

igen

valu

e5 10 15 20

0

4

8

12

16

Experimental ACF

0.440.310.190.06

-0.06-0.19-0.31-0.44

0.440.310.190.06

-0.06-0.19-0.31-0.44

0.440.310.190.06

-0.06-0.19-0.31-0.44

Sample realizations of temperature at collocation points

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

0.440.310.190.06

-0.06-0.19-0.31-0.44




NATURAL CONVECTION ON ROUGH SURFACES

0.440.310.190.06

-0.06-0.19-0.31-0.44

FIRST MOMENT

Temperature

Streamlines

0.170.140.120.100.080.060.030.01

7.636.625.604.583.562.541.530.51

Temperature

Y Velocity

SECOND MOMENT

Roughness causes improved thermal transport due to enhanced nonlinearities

Results in thermal plumes

Can look to tailor material surfaces to achieve specific thermal transport




Natural convection in heterogeneous media: large dimensions

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)

T = 0

u=v=0

T = 1

u=v=0

Porous medium

Free fluid

u=v=0

u=v=0




0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Index

Eig

enva

lue

5 10 15 200

5

10

15

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= 8

Number of collocation points is 3937 (level 4 interpolation)

Material: Sandstone

Numerically computed

Eigen spectrum




0.90.80.70.60.40.30.20.1

0.90.80.70.60.40.30.20.1

0.90.80.70.60.40.30.20.1

9.16.43.71.0

-1.7-4.4-7.1-9.8

14.210.16.12.0

-2.0-6.1

-10.1-14.1

12.18.65.11.7

-1.8-5.3-8.7

-12.2

Snapshots at a few collocation points

Temperature y-Velocity0.940.810.690.560.440.310.190.06

7.04.41.8

-0.8-3.4-6.0-8.6

-11.2

FIRST MOMENT

0.0970.0840.0710.0580.0450.0320.0190.006

5.0564.3823.7083.0342.3591.6851.0110.337

SECOND MOMENT

FLOW THROUGH HETEROGENEOUS RANDOM MEDIA

Temperature

Temperature Y velocity

Y velocity

Streamlines




RANDOM TOPOLOGY AND STOCHASTIC COLLOCATION

Investigate diffusion through random heterogeneous media

- Given experimental image, extract features

- Reconstruct 3D microstructures from 2D image

- Construct reduced model for the random topology

- Get statistics of temperature driven by this random topology

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) 229-2302) B. Ganapathysubramanian, N. Zabaras, Modelling diffusion in random heterogeneous media: Data-driven models,stochastic collocation and the variational multiscale method, J. Comp. Physics, submitted




RANDOM TOPOLOGY: Reconstruction

Given 2D slice, reconstruction techniques to construct 3D microstructures:

Gaussian Random Fields, Stochastic optimization, Simulate dannealing ect

Match experimental statistics with reconstructed statistics




RANDOM TOPOLOGY: Model reduction

PCA on the image set. First 10 eigen values represent the structure well

=a1a2

an+ + ..+

I = Iavg + I1a1 + I2a2+ I3a3 + … + Inan

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.

Represent any microstructure as a linear combination of the eigenimages




RANDOM TOPOLOGY: Model reduction

Impose constraints on the set of n-tuples

1) Impose first order constraints : Volume fraction must be matched

2) Impose pixel constraints: Results in a convex hull

3) Sequentially impose higher order constraints on the convex hull to get allowable space of n-tuples

Reduced model




RANDOM TOPOLOGY: STOCHASTIC COLLOCATION

N = 9 dimensions

Level 5 interpolation: 15713 deterministic problems

Each deterministic problem: 128x128x128 elements

Steady state diffusion problem.

Look at effect of imposing first order statistics

Mean statistics: Contours, iso surfaces and slices

Higher order statistics: Isosurfaces of standard deviation, pdf’s at two points and slices of standard deviation




RANDOM TOPOLOGY: STOCHASTIC COLLOCATION

N = 9 dimensions

Level 5 interpolation: 15713 deterministic problems

Each deterministic problem: 128x128x128 elements

Steady state diffusion problem.

Look at effect of imposing up to second order statistics

Mean statistics: Contours, iso surfaces and slices

Higher order statistics: Isosurfaces of standard deviation, pdf’s at two points and slices of standard deviation

sparse grid collocation schemes for stochastic convection materials process design and control...

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