aiaa 2002-5531 observations on cfd simulation uncertainties

AIAA 2002-5531

9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 1

AIAA 2002-5531OBSERVATIONS ON CFD SIMULATION

UNCERTAINTIES

Serhat Hosder, Bernard Grossman, William H. Mason, and Layne T. Watson

Virginia Polytechnic Institute and State University Blacksburg, VA

Raphael T. HaftkaUniversity of Florida

Gainesville, FL

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization

4-6 September 2002Atlanta, GA

AIAA 2002-5531


Introduction

• Computational fluid dynamics (CFD) as an aero/hydrodynamic analysis and design tool

• CFD being used increasingly in multidisciplinary design and optimization (MDO) problems

• CFD results have an associated uncertainty, originating from different sources

• Sources and magnitudes of the uncertainty important to assess the accuracy of the results

AIAA 2002-5531


• Design uncertainties due to computational simulation error

• Optimization is an iterative procedure subject to convergence error.

• Estimating convergence error may require expensive accurate optimization runs

• Many simulation runs are performed in engineering design. (e.g., design of experiments)

• Statistical analysis of error

Motivation

AIAA 2002-5531


• Find error characteristics of a structural optimization of a high speed civil transport

• Estimate error level of the optimization procedure

• Identify probabilistic distribution model of the optimization error

• Estimate mean and standard deviation of errors without expensive accurate runs

• Improve response surface approximation against erroneous simulation runs via robust regression

Objectives

AIAA 2002-5531


+ x

z Leading edge radius (fixed)

Outboard LE sweep (fixed)

x y

v1

v3

Location of maximum thickness (fixed)

v2

v4

Wing semi span (fixed)

v5: Fuel Weight • 250 passenger aircraft,

5500 nm range, cruise at Mach 2.4

• Take off gross weight (WTOGW) is minimized

• Up to 29 configuration design variables including wing, nacelle and fuselage geometry, fuel weight, and flight altitude

• For this study, a simplified 5DV version is used

High Speed Civil Transport (HSCT)

AIAA 2002-5531


• For each HSCT configuration, wing structural weight (Ws) is minimized by structural optimization (GENESIS) with 40 design variables

• Structural optimization is performed a priori to build a response surface approximation of Ws

Structural optimization of HSCT

AIAA 2002-5531


design

Op

timu

mw

ing

stru

ctu

ralw

eig

ht

(Ws)

5 10 15 2040000

60000

80000

100000

120000

140000

Case 1Case 2

Case 1 Case 2

Average error in Ws

5.51% 5.34%

For Case 2, the initial design point was perturbed from that of Case 1, by factors between 0.1 ~ 1.9

In average, Case 2 has the same level of error as Case 1

The errors were calculated with respect to higher fidelity runs with tightened convergence criteria

Effects of initial design point on optimization error

AIAA 2002-5531


• To get optimization error we need to know true optimum, which is rarely known for practical engineering optimization

• Optimization error = OBJ* - OBJ*true

• To estimate OBJ*true

• Find convergence setting to achieve very accurate optimization

• This approach can be expensive

Estimating error of incomplete optimization

AIAA 2002-5531


x

f(x)

0 1 2 3 4 50.0

1.0

2.0

3.0

= 0.5, = 1 = 1.0, = 1 = 2.0, = 1 = 4.0, = 1

• Widely used in reliability models (e.g., lifetime of devices)• Characterized by a shape parameter and a scale parameter

otherwise

xifx

x

0

0exp1

2

)1

(1

)2

(2

)1

(

22 Variance

Mean

PDF function according to

The Weibull distribution model

AIAA 2002-5531


• Maximum likelihood estimation (MLE) to estimate distribution parameters of the assumed distribution.

• Find to maximize the likelihood function.

2 goodness of fit test

• Comparison of histograms between data and fit

• p-value indicates quality of the fit

nx

xfl

i

n

ii

size of sample for

1

);()(

Estimation of distribution parameters

AIAA 2002-5531


Errors of two optimization runs of different initial design point

s = W1 – Wt

t = W2 – Wt , (Wt is unknown true optimum.)

The difference of s and t is equal to W1 – W2

x = s – t = (W1 – Wt) – (W2 – Wt ) = W1 – W2

We can fit distribution to x instead of s or t.

• s and t are independent • Joint distribution of g(s)

and h(t)

g(s; 1)

PDF

s, t

h(t; 2)

x=s-t

dsxshsgxf )()(),;( 21

• MLE fit for optimization difference x using f(x)

• No need to estimate Wt

Difference fit to estimate statistics of optimization errors

AIAA 2002-5531


Error in Ws (%)

Fre

qu

en

cie

s

0 10 20 30 40 50 60 700

20

40

60

80

DataExpected by error fitExpected by difference fit

• Difference fit is applied to the differences between Cases 1 and 2, assuming the Weibull model for both cases

• Even when only inaccurate optimizations are available, difference fit can give reasonable estimates of uncertainty of the optimization

Error in Ws (%)

Fre

qu

en

cie

s

0 10 20 30 40 50 60 70 800

20

40

60

80

100

DataExpected by error fitExpected by difference fit

Comparison between error fit and difference fit

AIAA 2002-5531


Cases Case 1 Case 2 Average of Abs(W1-W2) 5941

From data 4458 4321 Error fit

(discrepancy) 4207

(-5.63%) 3952

(-8.54%) Estimate of mean, lb. Difference fit

(discrepancy) 3804

(-14.7%) 3481

(-19.4%)

From data 8383 9799 Error fit

(discrepancy) 7157

(-14.6%) 7505

(-23.4%) Estimate of STD, lb. Difference fit

(discrepancy) 9393

(12.0%) 9868

(0.704%)

p-value of 2 test 0.5494

• Another set of optimization with a difference initial design point was enough, which is straightforward and no more expensive

• The difference fit gave error statistics as accurate as the error fits involving expensive higher fidelity runs

Estimated distribution parameters

AIAA 2002-5531


• Robust regression techniques• Alternative to the least squares fit that may be

greatly affected by a few very bad data (outliers)• Iteratively reweighted least squares (IRLS)

• Inaccurate response surface approximation is another error source in design process Improve response surface approximation by

repairing outliers

Outliers and robust regression

AIAA 2002-5531


• Weighted least squares: Initially all points have same weight of 1.0.

• At each iteration, the weight is reduced for points that are far from the response surface.

• This gradually moves the response surface away from outliers.

• IRLS approximation leaves out detected outliers

x

y

IRL

Sw

eig

htin

g

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5DataLSIRLSweighting

Outliers

1 dimensional example

Iteratively Reweighted Least Squares

AIAA 2002-5531


• Regular weighting functions• Symmetrical• Huber’s function gives non-

zero weighting to big outliers

• Biweight rejects big outliers completely

• Nonsymmetric weighting function• make use of one-sidedness

of error by penalizing points of positive residual more severelyResidual, r

We

igh

ting

,w(r

)

-4 -2 0 2 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6HuberBiweightNIRLS

Weighting functions of IRLS

Nonsymmetric weighting function

Aggressive search by reducing tuning constant B

AIAA 2002-5531


RMSE(lb.) R2 Before repair

6755 (8.7%)

0.9297

IRLS repair

3257 (4.7%)

0.9769

NIRLS repair

3042 (4.1%)

0.9824

All 117 pts repaired

2578 (3.5%)

0.9879

Design

Ws

(lb

)

5 10 15 200

20000

40000

60000

80000

100000

120000

140000

GENESIS: RepairedRS: Without repairRS: IRLS repairRS: NIRLS repair

Comparison of RS before and after repair

Improvements of RS approximation by outlier repair

AIAA 2002-5531


Drag polar results from 1st AIAA Drag Prediction Workshop (Hemsch, 2001)

AIAA 2002-5531


Objective of the Paper

• Finding the magnitude of CFD simulation uncertainties that a well informed user may encounter and analyzing their sources

• We study 2-D, turbulent, transonic flow in a converging-diverging channel

• complex fluid dynamics problem• affordable for making multiple runs• known as “Sajben Transonic Diffuser”

in CFD validation studies

AIAA 2002-5531


x/ht

y/h

t

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.00.0

0.5

1.0

1.5

2.0

2.5

y/h

t

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Transonic Diffuser Problem

y/h

t

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x/ht

y/h

t

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.00.0

0.5

1.0

1.5

2.0

2.5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

P/P0i

P/P0i

Strong shock case (Pe/P0i=0.72)

Weak shock case (Pe/P0i=0.82)

Separation bubble

experiment CFD

Contour variable: velocity magnitude

streamlines

AIAA 2002-5531


Uncertainty Sources (following Oberkampf and Blottner)

• Physical Modeling Uncertainty• PDEs describing the flow

• Euler, Thin-Layer N-S, Full N-S, etc.• boundary conditions and initial conditions • geometry representation • auxiliary physical models

• turbulence models, thermodynamic models, etc.• Discretization Error• Iterative Convergence Error• Programming Errors

We show that uncertainties from different sources interact

AIAA 2002-5531


Computational Modeling

• General Aerodynamic Simulation Program (GASP)• A commercial, Reynolds-averaged, 3-D, finite volume

Navier-Stokes (N-S) code• Has different solution and modeling options. An

informed CFD user still “uncertain” about which one to choose

• For inviscid fluxes (commonly used options in CFD)• Upwind-biased 3rd order accurate Roe-Flux scheme • Flux-limiters: Min-Mod and Van Albada

• Turbulence models (typical for turbulent flows) • Spalart-Allmaras (Sp-Al)• k- (Wilcox, 1998 version) with Sarkar’s

compressibility correction

AIAA 2002-5531


Grids Used in the Computations

Grid level Mesh Size (number of cells)

1 40 x 25

2 80 x 50

3 160 x 100

4 320 x 200

5 640 x 400

A single solution on grid 5 requires approximately 1170 hours of total node CPU time on a SGI Origin2000 with six processors (10000 cycles)

y/ht

Grid 2

Grid 2 is the typical grid level used in CFD applications

AIAA 2002-5531


Nozzle efficiency

Nozzle efficiency (neff ), a global indicator of CFD

results:

H0i : Total enthalpy at the inlet

He : Enthalpy at the exit

Hes : Exit enthalpy at the state that would be reached by

isentropic expansion to the actual pressure at the exit

esi

eieff HH

HHn

0

0

AIAA 2002-5531


Uncertainty in Nozzle Efficiency

+ ++ + + +

+ + ++ + + + +

+ + + + +

+

+

x x x x x x x x x x x xx

xx

xx

xx

x

x

++ + + +

+ + + ++ + + + +

+ + +

+ + ++

++

x x x x x x x x x x x x x x x xx

xx

xx

x

x

o

o

o

o

+

xo

+

xo +

+

x

x

Pe/P0i

ne

ff

0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.840.700

0.725

0.750

0.775

0.800

0.825

0.850

0.875

0.900

grid 1grid 2

grid 4grid 3

grid 5

Sp-Al k-

AIAA 2002-5531


Discretization Error by Richardson’s Extrapolation

)( 1 ppexactk hOhff

Turbulence model

Pe/P0i estimate of p (observed order of

accuracy)

estimate of (neff)exact

Grid level

Discretization error (%)

Sp-Al

0.72

(strong shock)

1.322 0.71950

1 14.298

2 6.790

3 2.716

4 1.086

Sp-Al

0.82

(weak shock)

1.578 0.81086

1 8.005

2 3.539

3 1.185

4 0.397

k- 0.82

(weak shock)

1.656 0.82889

1 4.432

2 1.452

3 0.461

4 0.146

order of the method

a measure of grid spacing grid level

error coefficient

AIAA 2002-5531


x/ht

y/h

t

-1.0 -0.5 0.0 0.5 1.0 1.5 2.00.98

0.99

1.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

modified experimentaldata points

upper wall contour obtainedwith the analytical equation

upper wall contour of the modified-wallgeometry (cubic-spline fit to the modified data points)

upper wall contour of the modified-wallgeometry (cubic-spline fit to the data points)

x/ht

y/h

t

-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

1.00

1.10

1.20

1.30

1.40

1.50 upper wall contour of the modified-wallgeometry (cubic-spline fit to the data points)

upper wall contour obtainedwith the analytical equation

Error in Geometry Representation

AIAA 2002-5531


Error in Geometry Representation

x/ht

P/P

0i

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.00.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0Pe/P0i=0.72Top Wall

experiment

Sp-Al, Min-Mod, grid 2mw , wall contourfrom modified experimental data

Sp-Al, Min-Mod, grid 2,wall contour from equattion

Sp-Al, Min-Mod, grid 2mw , wall contourfrom experimental data

• Upstream of the shock, discrepancy between the CFD results of original geometry and the experiment is due to the error in geometry representation.

• Downstream of the shock, wall pressure distributions are the same regardless of the geometry used.

AIAA 2002-5531


Downstream Boundary Condition

x/hty/

ht

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.00.5

1.0

1.5

2.0Extended geometry, Sp-Al, Van Albada, grid 3ext, Pe/P0i=0.7468

x/ht

y/h

t

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.00.5

1.0

1.5

2.0Extended geometry, Sp-Al, Van Albada, grid 3ext, Pe/P0i=0.72

x/ht

y/h

t

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.00.5

1.0

1.5

2.0Original geometry, Sp-Al, Van Albada, grid 3, Pe/P0i=0.72

x/ht

y/h

t

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.51.0

1.1

1.2

1.3

1.4

1.5

Extended geometry, Sp-Al,Van Albada, grid 3ext, Pe/P0i=0.72

Original geometry, Sp-Al, Van Albada, grid 3, Pe/P0i=0.72

Extended geometry, Sp-Al, Van Albada,grid 3ext, Pe/P0i=0.7468

x/ht

y/h

t

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.51.0

1.1

1.2

1.3

1.4

1.5

x/ht

y/h

t

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.51.0

1.1

1.2

1.3

1.4

1.5

AIAA 2002-5531


Downstream Boundary Condition

x/ht

P/P

0i

-4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.00.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0Top Wall

experiment

Sp-Al, Van Albada,grid 3ext, Pe/P0i=0.72

Sp-Al, Van Albada, grid 3

Sp-Al, Van Albada,grid 3ext, Pe/P0i=0.7468

Extending the geometry or changing the exit pressure ratio affect:

• location and strength of the shock

• size of the separation bubble

AIAA 2002-5531


the total variation in nozzle efficiency 10% 4%

the difference between grid level 2 and grid level 4

6%

(Sp-Al)

3.5%

(Sp-Al)

the relative uncertainty due to the selection of turbulence model

9%

(grid 4)

2%

(grid 2)

the uncertainty due to the error in geometry representation

0.5%

(grid 3, k-)

1.4%

(grid 3, k-)

the uncertainty due to the change in exit boundary location

0.8%

(grid 3, Sp-Al)

1.1%

(grid 2, Sp-Al)

Uncertainty Comparison in Nozzle EfficiencyStrong Shock

Weak ShockMaximum value of

AIAA 2002-5531


Conclusions• Based on the results obtained from this study,

• For attached flows without shocks (or with weak shocks), informed users may obtain reasonably accurate results

• They may get large errors for the cases with strong shocks and substantial separation

• Grid convergence is not achieved with grid levels that have moderate mesh sizes (especially for separated flows)

• Difficult to isolate physical modeling uncertainties from numerical errors

AIAA 2002-5531


Conclusions• Uncertainties from different sources interact,

especially in the simulation of flows with separation

• The magnitudes of numerical errors are influenced by the physical models (turbulence models) used

• Discretization error and turbulence models are dominant sources of uncertainty in nozzle efficiency and they are larger for the strong shock case

• We should asses the contribution of CFD uncertainties to the MDO problems that include the simulation of complex flows

aiaa 2002-5531 observations on cfd simulation uncertainties

Documents