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VARIATIONAL MULTISCALE METHOD FOR STOCHASTIC THERMAL AND FLUID FLOW PROBLEMS

BADRINARAYANAN VELAMUR ASOKAN

SIBLEY SCHOOL OF MECHANICAL ENGINEERING

CORNELL UNIVERSITY

HOME PAGE – http://people.cornell.edu/pages/bnv2

WORK PAGE -- http://mpdc.mae.cornell.edu

ACKNOWLEDGEMENTS

SPECIAL COMMITTEE

- Prof. NICHOLAS ZABARAS
- Prof. SUBRATA MUKHERJEE
- Prof. SHANE HENDERSON

FUNDING SOURCES

- AFOSR (AIRFORCE OFFICE OF SCIENTIFIC RESEARCH), NATIONAL SCIENCE FOUNDATION
- CORNELL THEORY CENTER
- SIBLEY SCHOOL OF MECHANICAL ENGINEERING

- Motivation: coupling multiscaling and uncertainty analysis
- Mathematical representation of uncertainty
- Variational multiscale method (VMS)
- Application of VMS with algebraic subgrid model
- Stochastic convection-diffusion equations: (example for illustration) Navier-Stokes
- Application of VMS with explicit subgrid model
- Stochastic multiscale diffusion equation
- Issues for extension to convection process design problems
- Suggestions for future work

- Uncertainty is everywhere

From DOE

From GE-AE website

From Intel website

From NIST

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

- Imprecise boundary conditions
- Initial perturbations

- Use micro averaged models for resolving physical scales

- 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

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

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

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 – ILLUSTRATION [NATURAL CONVECTION]

Mass conservation

Momentum conservation

Energy conservation

Constitutive laws

FINAL COARSE FORMULATION

VMS HYPOTHESIS

OBTAIN ASGS

DERIVE WEAK FORM

DERIVE SUBGRID

DEFINE PROBLEM

VMS HYPOTHESIS

OBTAIN ASGS

DERIVE WEAK FORM

DERIVE SUBGRID

DEFINE PROBLEM

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

VMS HYPOTHESIS

OBTAIN ASGS

DERIVE WEAK FORM

DERIVE SUBGRID

DEFINE PROBLEM

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

VMS HYPOTHESIS

OBTAIN ASGS

DERIVE WEAK FORM

DERIVE SUBGRID

DEFINE PROBLEM

ELEMENT FOURIER TRANSFORM

- Element Fourier transform

RANDOM FIELD DEFINED OVER THE DOMAIN

RANDOM FIELD DEFINED IN WAVENUMBER SPACE

SPATIAL MESH

- Addressing spatial derivatives

NEGLIGIBLE FOR LARGE WAVENUMBERS SUBGRID

APPROXIMATION OF DERIVATIVE

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

VMS HYPOTHESIS

OBTAIN ASGS

DERIVE WEAK FORM

DERIVE SUBGRID

DEFINE PROBLEM

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

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]

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

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

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

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

ISSUES IN EXTENSION TO CONVECTION ROBUST DESIGN PROBLEMS

FRACTIONAL TIME-STEP METHODS FOR STOCHASTIC CONVECTION-DIFFUSION PROBLEMS (SPECIAL CASE)

- Till now, we have seen techniques for direct analysis of stochastic thermal and fluid flow problems
- Extensions to practical design problems
- D.O.F typically of the order of millions (say 1M)
- A fluid-flow design problem requires at least 20 direct solves (10 direct + 10 sensitivity)
- With a stabilized (U, P) formulation, we will end up with a coupled algebraic system with 4M D.O.F (serious issue)
- It is prudent to derive alternatives to stabilized stochastic finite element methods stochastic fractional time-step methods

- Most fractional time step schemes follow a projection approach

- Pressure does not appear in the continuity equation, it is a constraint
- Essential algorithm
- Solve the momentum equation without the pressure term (yields some velocity field that defies continuity)
- Project the velocity field such that continuity is satisfied

- Velocity at time step k is denoted as and is assumed to be zero for all negative k

- Find the intermediate velocity

- Solve for a fictitious pressure field such that the resultant velocity satisfies continuity

- The above process involves the solution of a fictitious pressure Poisson equation

FRACTIONAL TIME-STEP GPCE IMPLEMENTATION

- Consider the stochastic Navier-Stokes equations with uncertainty in boundary (or) initial conditions

- Expand the stochastic velocity and pressure in their respective GPCEs

- Using the orthogonality of the Askey polynomials, we can write the momentum equation as (P+1) coupled PDEs

- The r-th GPCE coefficient of velocity at k-th time step is denoted as

- Solve for intermediate velocities

- We can further write these equations in terms of individual velocity components (Thus, D(P+1) scalar equations)
- Project the intermediate velocity to satisfy continuity

COMPARISON WITH GHIA

MEAN X-VELOCITY

U = unif[0.9, 1.1]

L = 1

MEAN X-VELOCITY (STAB)

FIRST COEFF U-x (STAB)

FIRST COEFF U-x

SECOND COEFF U-x

MEAN Y-VELOCITY

THIRD COEFF U-x

FIRST COEFF U-y

THIRD COEFF U-y

SECOND COEFF U-y

- 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

COLLOCATION POINTS FOR 3D RANDOM INPUT

LEVEL 0

LEVEL 2

LEVEL 1

(CC, FE) = (1, 8)

(CC, FE) = (7, 8)

(CC, FE) = (25, 25)

LEVEL 3

LEVEL 4

- WHAT ABOUT HIGHER DIMENSIONAL INPUTS
- D = 10

(CC, FE) = (69, 64)

(CC, FE) = (177, 125)

- 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

- B. Velamur Asokan and N. Zabaras, "A stochastic variational multiscale method for diffusion in heterogeneous random media ", Journal of Computational Physics, submitted in revised form.
- B. Velamur Asokan and N. Zabaras, "Using stochastic analysis to capture unstable equilibrium in natural convection", Journal of Computational Physics, Vol. 208/1, pp. 134-153, 2005
- B. Velamur Asokan and N. Zabaras, "Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations", Journal of Computational Physics, Vol. 202/1, pp. 94-133, 2005
- B. Velamur Asokan and N. Zabaras, "Stochastic inverse heat conduction using a spectral approach", International Journal for Numerical Methods in Engineering, Vol. 60/9, pp. 1569-1593, 2004

THANK YOU

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