Sparse grid collocation schemes for
stochastic convection
Nicholas Zabaras and Baskar Ganapathysubramanian
Materials Process Design and Control Laboratory
Sibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes Hall
Cornell University
Ithaca, NY 14853-3801
Email: zabaras@cornell.edu
URL: http://mpdc.mae.cornell.edu/
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MOTIVATION
All physical systems have an inherent associated randomness
SOURCES OF UNCERTAINTIES
Process
•Multiscale material information –
inherently statistical in nature.
•Uncertainties in process conditions
Engineering
component
•Input data
•Model formulation – approximations,
assumptions.
Why uncertainty modeling ?
Heterogeneous
random
Microstructural
features
Assess product and process reliability.
Estimate confidence level in model predictions.
Identify relative sources of randomness.
Control?
Provide robust design solutions.
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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
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REPRESENTING RANDOMNESS:1
Interpreting random variables as functions
1. Interpreting random variables
2. Distribution of the random variable
Random
variable x
Ex. Inlet velocity, Inlet temperature
MAP
S
Real line
Sample space of
elementary events
Collection of all
possible outcomes
  o 1  0.1x 
Each outcome is
mapped to a
corresponding real
value
A general stochastic process is a random field
with variations along space and time – A
function with domain (Ω, Τ, S)
3. Correlated data
Ex. Presence of impurities, porosity
Usually represented with a correlation function
We specifically concentrate on this.
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REPRESENTING RANDOMNESS:2
Karhunen-Loèvè expansion
1. Representation of random process
Based on the spectral decomposition of the
covariance kernel of the stochastic process
- Karhunen-Loeve, Polynomial Chaos
expansions
2. Infinite dimensions to finite dimensions
Random
process
- depends on the covarience
Mean
15
1
• Need to know covariance
0.9
0.8
0.7
10
0.6
Eigenvalue
Set of random
variables to
be found
0.5
0.4
• Converges uniformly to
any second order process
Eigenpairs of
covariance
kernel
5
0.3
0.2
0.1
0
0
10
20
30
40
50
60
0
5
10
15
20
Index
Set the number of stochastic dimensions, N
Dependence of variables
Pose the (N+d) dimensional problem
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KARHUNEN-LOEVE EXPANSION

X ( x, t ,  )  X ( x, t )   X i ( x, t )xi ( )
i 1
ON random variables
Deterministic functions
Stochastic Mean
process function
 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
X ( x, t ,  )  fn( x, t , x1 ,
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,xN )
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Problem definition
Represent input uncertainties in terms of N random variables. This is
possible usually due to the ‘finite dimensional noise assumption’1
g(y) =
where g is the appropriate input stochastic process
The dependant variables (T,u,p) depend on these N random variables.
Reformulate the problem in terms of these N variables
1) I. Babuska, R. Tempone, G. E. Zouraris, Galerkin finite elements approximation of stochastic finite elements,
SIAM J. Numer. Anal. 42 (2004) 800–825
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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
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SPECTRAL STOCHASTIC REPRESENTATION
 A stochastic process = spatially, temporally varying random
function
X : ( x, t , )
CHOOSE APPROPRIATE
BASIS FOR THE
PROBABILITY SPACE
HYPERGEOMETRIC ASKEY POLYNOMIALS
GENERALIZED POLYNOMIAL
CHAOS EXPANSION
SUPPORT-SPACE
REPRESENTATION
PIECEWISE POLYNOMIALS (FE TYPE)
SPECTRAL DECOMPOSITION
KARHUNEN-LOÈVE
EXPANSION
COLLOCATION, MC (DELTA FUNCTIONS)
SMOLYAK QUADRATURE,
CUBATURE, LH
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GENERALIZED POLYNOMIAL CHAOS
 Generalized polynomial chaos expansion is used to
represent the stochastic output in terms of the input
X ( x, t ,  )  fn( x, t , x1 ,

,xN )
Stochastic input
Z ( x, t ,  )   Z i ( x, t ) i (ξ( ))
i 0
Stochastic
output
Askey polynomials in input
Deterministic functions
 Askey polynomials ~ type of input stochastic process
 Usually, Hermite, Legendre, Jacobi etc.
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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 )
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COLLOCATION TECHNIQUES
Spectral Galerkin method: Spatial domain is approximated using a finite
element discretization
Stochastic domain is approximated using a
spectral element discretization
Decoupled equations
Coupled equations
Collocation method: Spatial domain is approximated using a finite element
discretization
Stochastic domain is approximated using
multidimensional interpolating functions
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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
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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
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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
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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
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
1. A. Klimke, Uncertainty Modeling using Fuzzy Arithmetic and Sparse Grids, PhD Thesis, Universitt Stuttgart, Shaker Verlag,
Aachen, 2006.
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SMOLYAK ALGORITHM
LET OUR BASIC 1D INTERPOLATION SCHEME BE SUMMARIZED AS
Ui( f ) 
a
xi  X i
x
i
f ( xi )
IN MULTIPLE DIMENSIONS, THIS CAN BE WRITTEN AS
(U i1 
 U id )( f ) 

xi1 X i1

xid X id
(axi1 
 axid ) f (x i1 ,
, x id )
TO REDUCE THE NUMBER OF SUPPORT NODES WHILE MAINTAINING
ACCURACY WITHIN A LOGARITHMIC FACTOR, WE USE SMOLYAK METHOD
U 0  0, i  U i  U i 1 ,
i  i1 
Aq ,d ( f )  Aq 1,d ( f )   (i1 
 id
id )( f )
i q
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
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SMOLYAK ALGORITHM
Extensively used in statistical mechanics
Uni-variate interpolation
Provides a way to construct interpolation
functions based on minimal number of
points
Ui( f ) 
a
xi  X i
x
i
f ( xi )
Multi-variate interpolation
(U i1 
Univariate interpolations to multivariate
(U i1   U id )( f )  
interpolations
xi1 X i1
 U id )( f ) 

xid X id
( axi1 


(axi1 
 axid ) f (x ,
, x id )
xi1 X i1
xid iX id
1
Smolyak interpolation
U 0  0, i  U i  U i 1 ,
Accuracy the same as tensor
product
i  i1 
Aq ,d ( f )  Aq 1,d ( f )   (i1 
 id
id )( f )
i q
D = 10
Within logarithmic constant
Increasing the order of interpolation increases
the number of points sampled
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ORDER
SC
FE
3
1581
1000
4
8801
10000
5
41625
100000
Materials Process Design and Control Laboratory

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
Results in multiple orders of magnitude
reduction in the number of points to sample
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D = 10
ORDER
SC
FE
3
1581
1000
4
8801
10000
5
41625
100000
Materials Process Design and Control Laboratory
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
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SMOLYAK ALGORITHM: Numerical illustration
Interpolating discontinuous
functions.
Function has a discontinuity in the y
direction.
32769 points required to construct
interpolant with error 3x10-3.
Increasing number of sampling points
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?
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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.
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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
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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 5x102. The conventional sparse grid
collocation uses 1577 points to get the
same accuracy.
More points sampled in the y direction
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ADAPTIVE SPARSE GRID COLLOCATION: Numerical illustration
Interpolating discontinuous
functions.
1
0.9
0.8
0.7
Function has a discontinuity in the y direction.
0.6
0.5
The adaptive method uses 559 points to build the
interpolation function, while the conventional method
uses 3300 points
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Conventional sparse grid
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Adaptive sparse grid
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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)
Solve the deterministic problem defined by each set of
collocated points
Use any validated deterministic
solution procedure.
Completely non intrusive
POSTPROCESSING
0.301
0.260
0.301
0.260
0.220
0.220
0.180
0.180
0.140
0.140
0.100
0.100
0.060
0.060
0.020
0.020
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
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REFERENCES
SPARSE 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.
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NUMERICAL EXAMPLES
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Natural convection with random boundary conditions: A comparison
Fluid (Pr=1.0) in a square domain
T (y) = 0.5
T (y) = f (y,ω)
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
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Comparison
Sparse grid collocation
GPCE
MC
Mean temperature distribution
Standard deviation of temperature
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Comparison
Computational effort
MonteCarlo: 65000 samples
GPCE: second order expansion
Domain: [-0.5, 0.5]x[-0.5, 0.5]
Grid: 50x50 quad elements
Time steps: 600 dt = 1e-3
Smolyak: level 6 interpolation
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
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Comparison
Sparse grid collocation is computationally very efficient for moderate
dimensions.
Post processing to obtain higher order statistics is very simple.
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Natural convection with random domains
Effect of roughness on natural convection
Thermal evolution of fluid on a rough
surface heated from below.
T (y) = -0.5
Surface characterization:
Waviness and roughness
Waviness: Large scale variations
T (y) = 0.5
y = f(x,ω)
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
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Natural convection with random domains
Effect of roughness on natural convection
Thermal evolution of fluid on a rough
surface heated from below.
T (y) = -0.5
ACF taken to be a simple exponential
correlation
C(y1,y2) = exp(-c|y1-y2|)
Mean roughness measure is 1/100 of the
characteristic length of the domain
T (y) = 0.5
y = f(x,ω)
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
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Natural convection with random domains
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
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0.438
0.313
0.188
0.063
-0.063
-0.188
-0.313
-0.438
8.369
6.449
4.529
2.610
0.690
-1.230
-3.150
-5.070
0.438
0.313
0.188
0.063
0.438
-0.063
0.313
-0.188
0.188
-0.313
0.063
-0.438
-0.063
-0.188
0.438
-0.313
0.313
-0.438
0.188
0.063
-0.063
-0.188
-0.313
-0.438
17.537
13.326
9.114
4.902
0.691
-3.521
-7.733
-11.944
0.438
0.313
0.188
0.063
-0.063
-0.188
-0.313
-0.438
15.707
11.772
7.837
3.901
-0.034
-3.969
-7.904
-11.839
12.013
7.773
3.533
-0.707
-4.947
-9.186
-13.426
-17.666
Materials Process Design and Control Laboratory
Natural convection with random domains: Mean statistics
Pressure
Temperature
0.437
0.312
0.187
0.062
-0.062
-0.187
-0.312
-0.437
v velocity
6.530
4.670
2.810
0.950
-0.911
-2.771
-4.631
-6.492
u velocity
4.192
2.528
0.864
-0.800
-2.465
-4.129
-5.793
-7.457
0.355
0.253
0.152
0.050
-0.051
-0.153
-0.254
-0.356
Much more diffuse behavior
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Natural convection with random domains: Higher order statistics
Pressure
Temperature
15.875
13.758
11.642
9.525
7.408
5.292
3.175
1.058
0.301
0.260
0.220
0.180
0.140
0.100
0.060
0.020
v velocity
u velocity
18.599
16.119
13.639
11.159
8.679
6.200
3.720
1.240
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0.801
0.694
0.587
0.480
0.374
0.267
0.160
0.053
Materials Process Design and Control Laboratory
Natural convection with random domains: Mode shifts and PDF’s
0.301
0.260
0.220
0.180
0.140
0.100
0.060
0.020
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
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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
T (y) = -0.5
Lower surface maintained at a higher
temperature
Rayleigh-Benard instability causes
convection
T (y) = 0.5
y = f(x,ω)
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,
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NATURAL CONVECTION ON ROUGH SURFACES
Experimental ACF
Experimental correlation for the surface roughness
1
Eigen spectrum is peaked. Requires large
dimensions to accurately represent the stochastic
space
0.8
0.6
V2
Simulated with N= 20 (Represents 94% of the
spectrum)
0.4
0.2
Number of collocation points is 11561 (level 4
interpolation)
0.44
0.44
0.31
0.31
0.19
0.19
0.06
0.06
-0.06
-0.06
-0.19
-0.19
-0.31
-0.31
-0.44
-0.44
0.44
0.31
0.19
0.06
-0.06
-0.19
-0.31
-0.44
CORNELL
U N I V E R S I T Y
0
0.5
1
1.5
2
V1
Numerically computed
Eigen spectrum
Sample realizations
of temperature at
collocation points
16
12
Eigenvalue
0.44
0.31
0.19
0.06
-0.06
-0.19
-0.31
-0.44
0
8
4
0
5
10
15
20
Index
Materials Process Design and Control Laboratory
NATURAL CONVECTION ON ROUGH SURFACES
FIRST MOMENT
SECOND MOMENT
Temperature
Temperature
0.17
0.14
0.12
0.10
0.08
0.06
0.03
0.01
0.44
0.31
0.19
0.06
-0.06
-0.19
-0.31
-0.44
Streamlines
Y Velocity
7.63
6.62
5.60
4.58
3.56
2.54
1.53
0.51
Roughness causes improved thermal transport due to enhanced nonlinearities
Results in thermal plumes
Can look to tailor material surfaces to achieve specific thermal transport
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
Natural convection in heterogeneous media: large dimensions
Alloy solidification, thermal insulation, petroleum prospecting
Look at natural convection through a realistic sample of heterogeneous material
u=v=0
Square cavity with free fluid in the middle
part of the domain. The porosity of the
material is taken from experimental data1
T=1
u=v=0
T=0
Free fluid
Left wall kept heated, right wall cooled
Numerical solution procedure for the
u=v=0 deterministic procedure is a fractional time
stepping method
Porous medium
u=v=0
1. Reconstruction of random media using Monte Carlo methods, Manwat and Hilfer, Physical Review E. 59 (1999)
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
FLOW THROUGH HETEROGENEOUS RANDOM MEDIA
Experimental correlation for the porosity of the
sandstone.
Material: 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)
15
1
0.9
0.8
0.7
0.6
0.5
Eigen
spectrum
0.4
0.3
Eigenvalue
10
Numerically
computed
5
0.2
0.1
0
0
10
20
30
40
50
60
0
5
10
15
20
Index
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
FLOW THROUGH HETEROGENEOUS RANDOM MEDIA
Temperature
Snapshots at a few collocation points
Temperature
9.1
6.4
3.7
1.0
-1.7
-4.4
-7.1
-9.8
0.9
0.8
0.7
0.6
0.4
0.3
0.2
0.1
14.2
10.1
6.1
2.0
-2.0
-6.1
-10.1
-14.1
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U N I V E R S I T Y
12.1
8.6
5.1
1.7
-1.8
-5.3
-8.7
-12.2
7.0
4.4
1.8
-0.8
-3.4
-6.0
-8.6
-11.2
Streamlines
Temperature
0.9
0.8
0.7
0.6
0.4
0.3
0.2
0.1
Y velocity
0.94
0.81
0.69
0.56
0.44
0.31
0.19
0.06
y-Velocity
0.9
0.8
0.7
0.6
0.4
0.3
0.2
0.1
FIRST MOMENT
SECOND MOMENT
0.097
0.084
0.071
0.058
0.045
0.032
0.019
0.006
Y velocity
5.056
4.382
3.708
3.034
2.359
1.685
1.011
0.337
Materials Process Design and Control Laboratory
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) 229-230
2) B. Ganapathysubramanian, N. Zabaras, Modelling diffusion in random heterogeneous media: Data-driven models,stochastic
collocation and the variational multiscale method, J. Comp. Physics, submitted
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
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
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
RANDOM TOPOLOGY: Model reduction
Represent any microstructure as a linear combination of the
eigenimages
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
PCA on the image set. First 10 eigen Thus the n tuple (a1,a2,..,an) must further satisfy some constraints.
values represent the structure well
=a1
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U N I V E R S I T Y
+a2
+ ..+ an
Materials Process Design and Control Laboratory
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
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
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
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory
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
CORNELL
U N I V E R S I T Y
Materials Process Design and Control Laboratory