Challenges of Uncertainty Quantification for Computational Aerodynamic Applications

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Challenges of Uncertainty Quantification for
Computational Aerodynamic Applications

• Robert W. Walters
• Professor and Department Head
• Aerospace and Ocean Engineering
• Virginia Polytechnic Institute and State
  University
• NCSU ACE Workshop, May 31-June 1, 2006

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Acknowledgements

• NASA Langley Research Center
• Dr. Luc Huyse, SwRI
• Professor Roger Ghanem
• Dr. Serhat Hosder

A man with one watch knows what time it is. A
man with two watches is never quite
sure. Segals Law
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Some Key Challenges

• Characterization of model uncertainty
• Turbulence, transition, multi-phase,
  thermo-chemical non-equilibrium flows
• Discretization error for complex geometries
• Computational expense of
• non-deterministic CFD simulations
• Parameter Uncertainty (better data)
• e.g. Chemical reaction rates

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Motivation for Aerodynamic UQ

• Robust aerodynamics optimization
• Aerodynamic designs insensitive to uncertainty
• Multi-disciplinary risk-based design
• Less expensive designs (e.g lower weight)
• Solutions computed within acceptable bounds
• Output sensitivity to parameters (ranking, DOX)

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Factor-of-Safety (FOS) and Probabilistic Design
Approaches
Traditional FOS Approach
Aero Tools Data
Structures Tools Data
Probabilistic Approach
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Aerodynamic UQ Perspective

• Two research groups have a long history in
• non-deterministic methods applications
• Structures
• Dynamics and Control
• The fluid dynamics (Aero/CFD) community has been
  essentially deterministic (until recently)
• Vast amount of research in the form of algorithm
  development and applications in the aerospace
  industry exists
• Sandia (DAKOTA), NASA, SwRI (NESSUS), numerous
  universities

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Sources of Uncertainty/Error
Turbulence modeling is the single most important
limitation to obtaining accurate simulations to
many flows of engineering interest. W. Oberkampf
and F. Blottner, Sandia National Laboratory
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Aerodynamic Drag

• An Important Performance Parameter for
  Aircraft
• Drag Reduction Fuel Efficiency
  Reduction in Direct Operating Costs
• Range Equation

CD Drag coefficient CL Lift Coefficient
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Uncertainty goals in Aerodynamic Performance
parameters (Hemsch, 2001)

• 1 drag count 0.0001
• On Concorde one drag count increase required 2
  passengers out of total capacity (90-100
  passengers)

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DLR-F4 Wing-body geometry used in the AIAA 1st
Drag Prediction Workshop
7.2 Million nodes
Grid source 1st Drag Prediction Workshop,
Courtesy of Cessna Aircraft Company
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Experimental Results for CL0.5 and M0.75
(Hemsch, 2001)

• Experimental Results obtained in three different
  wind tunnels (NLR, ONERA, and DRA)

• The above observed scatter values are
  approximately twice of those reported for each
  wind tunnel

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Uncertainty Estimates in Measured Quantities for
Each Wind Tunnel (Wahls, 2001)
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Total drag coefficient results obtained at CL0.5
and M0.75 from with different codes (Hemsch,
2001)
(Mean)
Solution Index
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Drag polar results from 1st AIAA Drag Prediction
Workshop
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Summary of Grid Dimensions

• 1st Drag Prediction Workshop
• Geometry DLR-F4 wing-body
• Grids
• Structured
• Baseline 3.26 million nodes (supplied grid)
• Refined7.17 million nodes (Cessna Aircraft
  Company)
• 2. Unstructured
• Baseline1.6 million nodes (9.7 tetrahedral
  cells)
• Refined13 million nodes (77.6 tetrahedral
  cells)
• 3rd Drag Prediction Workshop
• Geometry DLR-F6 wing-body and DLR-F6 with
  fairing (DLR-F6 FX2B)
• Grids - same number of grid points for both
  geometries
• - multiple grids supplied by different users
• 1. Structured (NASA Langley multi-block point 1-1
  point matching)
• (i) 2.6 million nodes (coarse), (ii) 9.2
  million nodes (medium)
• (iii) 18.0 million nodes (medium-fine), (iv)
  30.8 million nodes (fine)
• 2. Unstructured (NASA Langley)
• (i) 5. 3 million nodes (coarse), (ii)
  14.3 million nodes (medium)

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2nd Drag Prediction Workshop Geometry
DLR-F6 Wing-Body and DLR-F6 Wing-Body-Nacelle-Pylo
n
Picture Source Rakowitz, 2nd Drag Prediction
Workshop proceedings
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3rd Drag Prediction Workshop Wing-Body Geometries

• DLR-F6 wing-body (original geometry)
• DLR-F6 wing-body with fairing (DLR-F6 FX2B)
• Original geometry has a separation region in the
  wing trailing-edge-body junction region which
  inhibits asymptotic grid convergence
• A fairing was designed by Vassberg et al. to have
  attached flow in this region

DLR-F6 FX2B (Wing-Body with fairing)
DLR-F6 (Wing-Body)
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Aerodynamic Improvement in DLR-F6 wing trailing
edge-body junction region
Surface Streamline plots from Vassberg et al
(AIAA Paper 2005-4730)
DLR-F6 (Wing-Body)
DLR-F6 FX2B (Wing-Body with fairing)
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HSCT Sensitivity to Drag

• Mission 251 passengers, 5500 n. mi. range,
  Mach 2.4
• 2 counts of drag results in a 56,000 lb increase
  in TOGW
• 2 count drag underprediction results in 120 n.
  mi. overprediction in range

74 Design variables 70 Constraints
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Non-deterministic Analysis (NDA) Methods
Possibilistic Methods
Probabilistic Methods

• Monte Carlo
• Basic, Latin Hypercube, HSS
• Moment Methods
• FOSM, SOSM
• Polynomial Chaos
• Intrusive
• Non-Intrusive

• Interval Analysis
• Sensitivity Derivatives
• CSE, Discrete Adjoint
• Fuzzy Set Theory
• Evidence Theory

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Basics of Polynomial Chaos

• A generic stochastic variable in PC form

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Polynomial Chaos Basics

• Spectral representation of uncertainty over an
  orthogonal set of basis functions
• One solves for the modal values of the basis
  functions
• Provides a complete description of the PDF
• Known convergence properties
• Converges in the L2 sense

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Hermite Polynomials
For a single random variable
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Basics of Polynomial Chaos
Basis functions are orthogonal with respect to a
Weight Function
Inner Product of two functions
Weight function
for Hermite PC
Multi-dimensional Gaussian distribution with unit
variance
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Intrusive Polynomial Chaos

• Procedure
• Replace all random variables or parameters with
  PC expansions in governing equations
• Take the inner product of equations,
  for k0,..,P
• Solve N equations
• N(P1) x ( of deterministic governing
  equations)
• Can be very difficult, expensive, and
  time-consuming to implement for complex problems
• Full N-S simulations of 3-D turbulent flows
  around realistic aerospace vehicles
• Chemically reacting flows
• Multi-system level simulations, etc.

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A Simple Polynomial Chaos
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Turbulent Flow Application
L.Mathelin, M. Houssani, T. Zang, F. Bataille
PC equation for the turbulent dissipation rate
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Non-Intrusive Polynomial Chaos

• Objective Obtain the approximations to PC
  expansion coefficients with no modification to
  the existing deterministic code
• Two commonly used NIPC approaches
• (1) Sampling Based (SB) (2) Quadrature Methods
  (QM)

Project PC expansion equation on to kth basis
Estimate
1. SB By averaging samples of

2. QM By numerical quadrature (Gauss-Hermite
Quadrature, etc)
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Oblique Shock Wave Problem
Mean grid (65x65)
Mref3.0

• q modeled as a normally distributed uncertain
  parameter with a mean value of qmean5 deg. and a
  CoV of 10.
• 
  and
• 10,000 MC simulations
• 2.3 days on an Apple G-5 with dual processors
• 4th order NIPC
• 2 minutes on an Apple G-5 with dual processors

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Typical data comparisons
MC
NIPC
Mean P/Pref
StD P/Pref
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Boundary Layer Statistics
Mom. Thickness
Disp. Thickness
BL Thickness
PC
PC
PC
MC
MC
MC
q (m)
d (m)
d (m)
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Future Directions Issues

• Develop a Non-Intrusive PC method in conjunction
  with Random Field Theory
• Fundamental PC research
• Chaos convergence -gt basis functions
• Adaptive sampling -gt PC coefficients
• Importance sampling -gt sample pre-selection
• Model uncertainty will continue to be a focus
  area for aerodynamic UQ

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Predictive Capabilities (???)
Radio has no future. Heavier-than-air flying
machines are impossible. X-rays will prove to be
a hoax Lord Kelvin, 1899.
By 2000, more than 1,000 people will live and
work on the moon, according to NASA
predictions Omni Future Almanac, 1982.
There will never be a bigger plane built Boeing
engineer after the first flight of the 247, a
twin-engine plane that carried ten people.