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Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rh

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VARIATIONAL MULTISCALE STABILIZED FEM FORMULATIONS FOR STOCHASTIC ADVECTION-DIFFUSION EQUATIONS

V. A. BADRI NARAYANAN and

NICHOLAS ZABARAS

Materials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering169 Frank H. T. Rhodes Hall

Cornell University

Ithaca, NY 14853-3801

Email: zabaras@cornell.edu, bnv2@cornell.edu

URL: http://www.mae.cornell.edu/zabaras

- Multiscale systems of engineering importance – Introduction and examples
- Variational Multiscale approach – Basic ideas, importance of uncertainty
- Mathematical modeling of uncertainty - Probability preliminaries - Spectral stochastic expansions - Support-space (or) Stochastic Galerkin method
- Governing equations for natural convection under Boussinesq assumptions
- Function space specifications and weak formulation
- Variational multiscale decomposition and derivation of stabilized formulations for energy, momentum and continuity equations
- Numerical examples

Flow past an aerofoil

Atmospheric flow in Jupiter

Solidification process

Modeling of dendrites at small scale, fluid flow and transport at large scale

Astro-physical flows, effects of gravitational and magnetic fields

Large scale turbulent structures, small scale dissipative eddies, surface irregularities

- Presence of a variety of spatial and time scales - commonality
- Varied applications – Engineering, Geophysical, Materials
- Boundary conditions, material properties, small scale behavior inherently are uncertain

IDEA BEHIND VARIATIONAL MULTISCALE - VMS

Subgrid model

- Green’s function
- Residual free bubbles
- MsFEM “Hou et al.”
- TLFEM “Hughes et al.”

Small scale behavior – statistical resolution

Micro-constitutive laws from experiments, theoretical predictions

Large scale Residual

Subgrid scale solution

Physical model

Solidification process

- FEM
- FDM
- Spectral

Large scale behavior – explicit resolution

Where does uncertainty fit in ?

Resolved model

WHY STOCHASTIC MODELING IN VMS ?

Surroundings uncertainty

Model uncertainty

- Uncertain boundary conditions
- Inherent initial perturbations
- Small scale interactions

- Imprecise knowledge of governing physics
- Models used from experiments

Solidification microscale features

- Uncertainty in codes
- Machine precision errors
- Not accounted for in analysis here

- Material properties fluctuate – only a statistical description possible

Material uncertainty

Computational uncertainty

Probability space – A triplet - - Collection of all basic outcomes of the experiment- - Permutation of the basic outcomes- - Probability associated with the permutations

Random variable – a function

Sample space

Real interval

Stochastic process – a random function at each space and time point

Notations:

SPECTRAL STOCHASTIC EXPANSIONS

- Series representation of stochastic processes with finite second moments

Karhunen-Loeve expansion

- Mean of the stochastic process
- Coefficient dependant on the eigen-pairs of the covariance kernel of the stochastic process
- Orthogonal random variables

- Covariance kernel required – known only for inputs
- Best possible representation in mean-square sense

Generalized polynomial chaos expansion

- Coefficients dependant on chaos-polynomials chosen
- Chaos polynomials chosen from Askey-series (Legendre – uniform, Jacobi – beta)

SUPPORT-SPACE/STOCHASTIC GALERKIN

- Joint probability density function of the inputs

- The input support-space denotes the regions where input joint PDF is strictly positive

Triangulation of the support-space

Any function can be represented as a piecewise polynomial on the triangulated support-space

- Function to be approximated

- Piecewise polynomial approximation over support-space

L2 convergence – (mean-square)

Error in approximation is penalized severely in high input joint PDF regions. We use importance based refinement of grid to avoid this

h = mesh diameter for the support-space discretization

q = Order of interpolation

Momentum equation boundary conditions

Energy equation boundary conditions

- Temperature gradients are small
- Constant fluid properties except in the force term
- viscous dissipation negligible

Function spaces for deterministic quantities

- Spatial domain

- Time interval of simulation [0,tmax]

Function spaces for stochastic quantities

Velocity function spaces

Pressure function spaces

Temperature function spaces

- Uncertainty is incorporated in the function space definition
- Solution velocity, temperature and pressure are in general multiscale quantities (as Rayleigh number increases) the computational grid capture less and less information

WEAK FORMULATION – BOUSSINESQ EQNS

Energy equation – weak form

Find such that for all , the following holds

Momentum and continuity equations – weak form

Find such that for all , the following holds

VARIATIONAL MULTISCALE DECOMPOSITION

- Bar denotes large scale/resolved quantity
- Prime denotes subgrid scale/ unresolved quantity

Induced multiscale decomposition for function spaces

- Interpretation
- Large scale function spaces correspond to finite element spaces – piecewise polynomial and hence are finite dimensional
- Small scale function spaces are infinite dimensional

SCALE DESOMPOSED WEAK FORM - ENERGY

Find and such that for all and , the following holds

Small scale strong form of equations

Time discretization rule

- Other techniques to solve for an approximate subgrid solution include:
- Residual-free bubbles, Green’s function approach
- Two-level finite element method – explicit evaluation
- Multiscale FEM – Incorporates subgrid features in large scale weighting function

Subgrid scale solution denotes unresolved part of the solution, hence dominated by large wave number modes!!

Spatial domain

Spatial derivative approximation

Time discretization

Element Fourier transform

Parseval’s theorem

Mean value theorem

STABILIZED FINITE ELEMENT EQUATIONS

Strong regularity conditions

Stabilized weak formulation

where

Time integration has a role to play in the stabilization (Codina et al.)

Stochastic intrinsic time scale (subgrid scale solution has a stochastic model)

CONSIDERATIONS FOR MOMENTUM EQUATION

Picard’s linearization

Fairly accurate for laminar up to transition (moderate Reynolds number flows)

For high Reynolds number flows, the term assumes importance since small scales act as momentum dissipaters

Small scale strong form of equations

Element Fourier transform

Simultaneous solve

Parseval’s theorem

Mean-value theorem

STABILIZED FINITE ELEMENT EQUATIONS

Strong regularity conditions

Stabilized weak formulation

Momentum equation

Continuity equation

where

- Assume the inputs have been represented in Karhunen-Loeve expansion such that the input uncertainty is summarized by few random variables

Galerkin shape function

Generic function

Random coefficient

Spatial domain

GPCE expansion for random coefficients

- Each node has P+1 degrees of freedom for each scalar stochastic process
- Interpolation is accomplished by tensor-product basis functions
- (P+1) times larger than deterministic problems

Random coefficient

Askey polynomial

IMPLEMENTATION ISSUES – SUPPORT SPACE

A stochastic process can be interpreted as a random variable at each spatial point

Two-level grid approach

- Support-space grid
- Mesh dense in regions of high input joint PDF

Spatial domain

Spatial grid

- There is finite element interpolation at both spatial and random levels
- Each spatial location handles an underlying support-space grid
- Highly OOP structure

Element

- Flow past a circular cylinder with uncertain inlet velocity – Transient behavior
- RB convection in square cavity with adiabatic body at the center – uncertainty in the hot wall temperature (simulation away from critical points) - Transient behavior - Simulation using GPCE, validation using deterministic simulation
- RB convection in square cavity – uncertainty in Rayleigh number (simulation about a critical point) - Failure of the GPCE approach - Analysis support-space method - Comparison of prediction by support-space method with deterministic simulations

In the last example, temperature contours do not convey useful information and hence are ignored

- Computational details – 2000 bilinear elements for spatial grid, third order Legendre chaos expansion for velocity and pressure, preconditioned parallel GMRES solver

- Time of simulation – 180 non-dimensional units
- Inlet velocity – Uniform random variable between 0.9 and 1.1
- Kinematic viscosity 0.01
- Time stepping – 0.03 non-dimensional units

No-slip

Inlet

Traction free outlet

No-slip

- Investigations
- Onset of vortex shedding
- Shedding near wake regions, flow statistics

- Mean pressure at t = 79.2
- Vortex shedding is just initiated
- Not in the periodic shedding mode

- First order term in Legendre chaos expansion of pressure at t = 79.2
- Vortex shedding is predominant
- Periodic shedding behavior noticed

FULLY DEVELOPED VORTEX SHEDDING

Mean pressure contours

First order term in LCE of pressure contours

Second order term in LCE of pressure contours

- The FFT of the mean velocity shows a broad spectrum with peak at frequency 0.162
- The spectrum is broad in comparison to deterministic results wherein a sharp shedding frequency is obtained
- Mean velocity has superimposed frequencies
- Mean velocity has comparatively lower magnitude than the deterministic velocity (Y-velocities compared at near wake region)

RB CONVECTION - CENTRAL ADIABATIC BODY

- Computational details – 2048 bilinear elements for spatial grid, third order Legendre chaos expansion for velocity, pressure and temperature, preconditioned parallel GMRES solver

Cold wall

- Time of simulation – 1.5 non-dimensional units
- Rayleigh number - 104
- Prandtl number – 0.7
- Time stepping – 0.002 non-dimensional units

Adiabatic body

Insulated

Insulated

Hot wall

- Transient behavior of temperature statistics ( Flow results in paper )

TRANSIENT BEHAVIOR – TEMPERATURE

- Second order term in the Legendre chaos expansion of temperature

- Mean temperature contours
- Steady conduction like state not reached

- First order term in the Legendre chaos expansion of temperature

CAPTURING UNSTABLE EQUILIBRIUM

- Computational details – 1600 bilinear elements for spatial grid

- Time of simulation – 1.5 non-dimensional units
- Rayleigh number – uniformly distributed random variable between 1530 and 1870 (10% fluctuation about 1700)
- Prandtl number – 6.95
- Time stepping – 0.002 non-dimensional units
- Support-space grid – One-dimensional with ten linear elements

Cold wall

Insulated

Insulated

Hot wall

- Simulation about the critical Rayleigh number – conduction below, convection above
- Both GPCE and support-space methods are used separately for addressing the problem
- Failure of Generalized polynomial chaos approach
- Support-space method – evaluation and results against a deterministic simulation

Y-vel

X-vel

Mean X- and Y- velocities determined by GPCE yields extremely low values !! (Gibbs effect)

X-vel

Y-vel

X- and Y- velocities obtained from a deterministic simulation with Ra = 1870 (the upper limit)

PREDICTION BY SUPPORT-SPACE METHOD

Y-vel

X-vel

Mean X- and Y- velocities determined by support-space method at a realization Ra=1870

X-vel

Y-vel

X- and Y- velocities obtained from a deterministic simulation with Ra = 1870 (the upper limit)

- Stabilization for Boussinesq system of natural convection equations in presence of uncertainty in boundary conditions, initial conditions and material properties was derived
- Implementation of the above using Generalized polynomial chaos approach GPCE and the support-space/stochastic Galerkin.
- For systems away from critical points, GPCE approach is highly accurate (Examples 1 and 2). For systems at critical point, the Gibbs effect and other inconsistencies emerge and only support-space approach works
- Ability to capture unstable equilibrium using stochastic analysis was shown

- Principal reference –
- “Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations”, Journal of Computational Physics, in press

OTHER REFERENCES – FURTHER READING

- Le Maitre et al. J. Comp. Phys. 197(1):28—57, 2004
- Ghanem et al. J. Comp. Phys. 181(1):9—44, 2002
- Xiu et al. J. Comp. Phys. 187:137—167, 2003
- Karniadakis et al. J. Sci. Comput. 17:319—338, 2002
- Babuska et al. Comput. Meth. App. Mech. Engrg. 190:6259—6372, 2001
- Xiu et al. J. Fluids Engrg. 125:51—59, 2001
- T.J.R.Hughes. Comput. Meth. App. Mech. Engrg. 127:387—401, 1995
- Codina. Comput. Meth. App. Mech. Engrg. 191(39):4295—4321, 2002
- Codina. Comput. Visual. Sci. 4(3):167-174, 2002
- Wiener. Am. J. Math. 60:897—936, 1938

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