1. Materials Process Design and Control Laboratory SIBLEY SCHOOL OF MECHANICAL ENGINEERING CORNELL UNIVERSITY HOME PAGE – http://people.cornell.edu/pages/bnv2 WORK PAGE -- http://mpdc.mae.cornell.edu BADRINARAYANAN VELAMUR ASOKAN VARIATIONAL MULTISCALE METHOD FOR STOCHASTIC THERMAL AND FLUID FLOW PROBLEMS

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

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

6. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory RANDOM VARIABLES = FUNCTIONS ?  Math: Probability space ( , 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

8. Materials Process Design and Control Laboratory SPECTRAL STOCHASTIC REPRESENTATION  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

9. Materials Process Design and Control Laboratory KARHUNEN-LOEVE EXPANSION Stochastic process Mean function ON random variables 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. Materials Process Design and Control Laboratory GENERALIZED POLYNOMIAL CHAOS  Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input Stochastic output Askey polynomials in input Deterministic functions Stochastic input  Askey polynomials ~ type of input stochastic process  Usually, Hermite, Legendre, Jacobi etc.

11. Materials Process Design and Control Laboratory SUPPORT-SPACE REPRESENTATION  Any function of the inputs, thus can be represented as a function defined over the support-space JOINT PDF OF A TWO RANDOM VARIABLE INPUT FINITE ELEMENT GRID REFINED IN HIGH-DENSITY REGIONS OUTPUT REPRESENTED ALONG SPECIAL COLLOCATION POINTS – SMOLYAK QUADRATURE – IMPORTANCE MONTE CARLO

12. Materials Process Design and Control Laboratory VARIATIONAL MULTISCALE METHOD WITH ALGEBRAIC SUBGRID MODELLING  Application : deriving stabilized finite element formulations for advection dominant problems

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

14. Materials Process Design and Control Laboratory VARIATIONAL MULTISCALE BASICS DERIVE THE WEAK FORMULATION FOR THE GOVERNING EQUATIONS PROJECT THE WEAK FORMULATION ON COARSE AND FINE SCALES COARSE WEAK FORM FINE (SUBGRID) WEAK FORM ALGEBRAIC SUBGRID MODELS COMPUTATIONAL SUBGRID MODELS APPROXIMATE SUBGRID SOLUTION REMOVE SUBGRID EFFECTS IN THE COARSE WEAK FORM USING STATIC CONDENSATION MODIFIED MULTISCALE COARSE WEAK FORM INCLUDING SUBGRID EFFECTS SOLUTION FUNCTION SPACES ARE NOW STOCHASTIC FUNCTION SPACES NEED TECHNIQUES TO SOLVE STOCHASTIC PDEs

15. Materials Process Design and Control Laboratory 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

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

17. Materials Process Design and Control Laboratory  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 ENERGY EQUATION – SCALE DECOMPOSITION DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS DERIVE SUBGRID OBTAIN ASGS FINAL COARSE FORMULATION

18. Materials Process Design and Control Laboratory ELEMENT FOURIER TRANSFORM  Element Fourier transform SPATIAL MESH RANDOM FIELD DEFINED OVER THE DOMAIN RANDOM FIELD DEFINED IN WAVENUMBER SPACE  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

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

21. Materials Process Design and Control Laboratory FLOW PAST A CIRCULAR CYLINDER RANDOM U INLET TRACTION FREE NO-SLIP  Investigations: Vortex shedding, wake characteristics 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)

22. Materials Process Design and Control Laboratory FULLY DEVELOPED VORTEX SHEDDING  Mean pressure  Second LCE coefficient  First 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. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory VARIATIONAL MULTISCALE METHOD WITH EXPLICIT SUBGRID MODELLING FOR MULTISCALE DIFFUSION IN HETEROGENEOUS RANDOM MEDIA

25. Materials Process Design and Control Laboratory MODEL MULTISCALE HEAT EQUATION in on in THE DIFFUSION COEFFICIENT K IS HETEROGENEOUS AND POSSESSES RAPID RANDOM VARIATIONS IN SPACE FLOW IN HETEROGENEOUS POROUS MEDIA  INHERENTLY STATISTICAL DIFFUSION IN MICROSTRUCTURES OTHER APPLICATIONS – DIFFUSION IN COMPOSITES – FUNCTIONALLY GRADED MATERIALS DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS COARSE-TO- SUBGRID MAP FINAL COARSE FORMULATION AFFINE CORRECTION

26. Materials Process Design and Control Laboratory STOCHASTIC WEAK FORM : Find such that, for all  Weak formulation  VMS hypothesis Exact solution Coarse solution Subgrid solution DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS COARSE-TO- SUBGRID MAP FINAL COARSE FORMULATION AFFINE CORRECTION

27. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory SCALE PROJECTION OF WEAK FORM Find such that, for all and  Projection of weak form on coarse scale  Projection of weak form on subgrid scale DEFINE PROBLEM DERIVE WEAK FORM VMS HYPOTHESIS COARSE-TO- SUBGRID MAP FINAL COARSE FORMULATION AFFINE CORRECTION EXACT SUBGRID SOLUTION COARSE-TO-SUBGRID MAP SUBGRID AFFINE CORRECTION

29. Materials Process Design and Control Laboratory SPLITTING THE SUBGRID SCALE WEAK FORM  Coarse-to-subgrid map  Subgrid affine correction  Subgrid scale weak form

30. Materials Process Design and Control Laboratory NATURE OF MULTISCALE DYNAMICS ũCũC ūCūC ûFûF 1 1 Coarse solution field at start of time step Coarse solution field at end of time step ASSUMPTIONS: NUMERICAL ALGORITHM FOR SOLUTION OF THE MULTISCALE PDE COARSE TIME STEP SUBGRID TIME STEP

31. Materials Process Design and Control Laboratory 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

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

33. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory BCs FOR THE COARSE-TO-SUBGRID MAP INTRODUCE A SUBSTITUTION CONSIDER AN ELEMENT

35. Materials Process Design and Control Laboratory 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

36. Materials Process Design and Control Laboratory 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

37. Materials Process Design and Control Laboratory 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  PHASE LIGHTEST DENOTES  PHASE

38. Materials Process Design and Control Laboratory RESULTS AT TIME = 0.05 MEAN FIRST ORDER GPCE COEFF SECOND ORDER GPCE COEFF RECONSTRUCTED FINE SCALE SOLUTION (VMS) FULLY RESOLVED GPCE SIMULATION

39. Materials Process Design and Control Laboratory RESULTS AT TIME = 0.2 MEAN FIRST ORDER GPCE COEFF SECOND ORDER GPCE COEFF RECONSTRUCTED FINE SCALE SOLUTION (VMS) FULLY RESOLVED GPCE SIMULATION

40. Materials Process Design and Control Laboratory HIGHER ORDER TERMS AT TIME = 0.2 FOURTH ORDER GPCE COEFF FIFTH ORDER GPCE COEFF RECONSTRUCTED FINE SCALE SOLUTION (VMS) FULLY RESOLVED GPCE SIMULATION THIRD ORDER GPCE COEFF

41. Materials Process Design and Control Laboratory ISSUES IN EXTENSION TO CONVECTION ROBUST DESIGN PROBLEMS FRACTIONAL TIME-STEP METHODS FOR STOCHASTIC CONVECTION-DIFFUSION PROBLEMS (SPECIAL CASE)

42. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory ALGORITHM  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  Velocity at time step k is denoted as and is assumed to be zero for all negative k

45. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory STOCHASTIC LID DRIVEN CAVITY L = 1 U = unif[0.9, 1.1] COMPARISON WITH GHIA MEAN X-VELOCITY MEAN X-VELOCITY (STAB) FIRST COEFF U-x FIRST COEFF U-x (STAB)

48. Materials Process Design and Control Laboratory STOCHASTIC LID DRIVEN CAVITY SECOND COEFF U-x THIRD COEFF U-x MEAN Y-VELOCITY FIRST COEFF U-y SECOND COEFF U-y THIRD COEFF U-y

49. Materials Process Design and Control Laboratory SUGGESTIONS FOR FUTURE RESEARCH

50. Materials Process Design and Control Laboratory 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

51. Materials Process Design and Control Laboratory SPARSE GRID QUADRATURE  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)

52. Materials Process Design and Control Laboratory 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

53. Materials Process Design and Control Laboratory COLLOCATION POINTS FOR 3D RANDOM INPUT LEVEL 0 LEVEL 1 LEVEL 2 LEVEL 3 LEVEL 4 WHAT ABOUT HIGHER DIMENSIONAL INPUTS –D = 10 ORDERCCFE 315811000 4880110000 541625100000 (CC, FE) = (1, 8) (CC, FE) = (7, 8) (CC, FE) = (25, 25) (CC, FE) = (69, 64) (CC, FE) = (177, 125)

54. Materials Process Design and Control Laboratory MULTISCALE RELATED 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 IN COUPLING MULTIPLE EQUATION MODELS, STATISTICS MUST BE CONSISTENT TRANSFERRING DATA, STATISTICS ACROSS LENGTH SCALES USING INFORMATION THEORY

55. Materials Process Design and Control Laboratory PUBLICATIONS  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