SIBLEY SCHOOL OF MECHANICAL ENGINEERING CORNELL UNIVERSITY

1. 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

2. 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

3. OUTLINE • 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

4. NEED FOR UNCERTAINTY ANALYSIS • 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)

5. 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

6. 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

7. 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

8. 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)

9. KARHUNEN-LOEVE EXPANSION 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

10. 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.

11. 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

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

13. 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

14. 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

15. 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

16. FINAL COARSE FORMULATION 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

17. FINAL COARSE FORMULATION 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

18. FINAL COARSE FORMULATION 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

19. 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

20. FINAL COARSE FORMULATION 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

21. 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

22. 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]

23. 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

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

25. FINAL COARSE FORMULATION 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

26. FINAL COARSE FORMULATION 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

27. 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

28. FINAL COARSE FORMULATION 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

29. SPLITTING THE SUBGRID SCALE WEAK FORM • Subgrid scale weak form • Coarse-to-subgrid map • Subgrid affine correction

30. 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 end of time step Coarse solution field at start of time step ûF

31. REPRESENTING COARSE SOLUTION 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

32. FINAL COARSE FORMULATION 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

33. 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

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

35. FINAL COARSE FORMULATION 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

36. FINAL COARSE FORMULATION 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

37. 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

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

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

40. 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

41. ISSUES IN EXTENSION TO CONVECTION ROBUST DESIGN PROBLEMS FRACTIONAL TIME-STEP METHODS FOR STOCHASTIC CONVECTION-DIFFUSION PROBLEMS (SPECIAL CASE)

42. EXTENSION TO DESIGN PROBLEMS • 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

43. FORMULATION • 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

44. ALGORITHM • 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

45. 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

46. ALGORITHM • 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

47. STOCHASTIC LID DRIVEN CAVITY 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

48. STOCHASTIC LID DRIVEN CAVITY SECOND COEFF U-x MEAN Y-VELOCITY THIRD COEFF U-x FIRST COEFF U-y THIRD COEFF U-y SECOND COEFF U-y

49. SUGGESTIONS FOR FUTURE RESEARCH

50. UNCERTAINTY RELATED • 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