NASA Technical Reports Server (NTRS) 20010039415: Approach for Uncertainty Propagation and Robust Design in CFD Using Sensitivity Derivatives

16 Pages·0.9 MB·English

by  NASA Technical Reports Server (NTRS)

AIAA 2001-2528
Approach for Uncertainty Propagation and
Robust Design in CFD Using Sensitivity Derivatives
Michele M. Putko*
Perry A. Newman**
Arthur C. Taylor III ***
Lawrence L. Green****
Department of Mechanical Engineering
Old Dominion University
Norfolk, VA 23529
Multidisciplinary Optimization Branch
NASA Langley Research Center
Hampton, VA 23681
Presented at the AIAA 15thComputational Fluid Dynamics Conference
June 11-14, 2001, Anaheim, CA
-7
*LTC, US Army. PhD Candidate, Old Dominion University, Norfolk, VA 2352
**Senior Research Scientist, NASA Langley Research Center, Hampton VA 23681
***Associate Professor, Old Dominion University, Norfolk, VA 23529
****Research Scientist, NASA Langley Research Center, Hampton VA 23681,Senior Member AIAA, [email protected]

AIAA 2001-2528
APPROACH FOR UNCERTAINTY PROPAGATION AND
ROBUST DESIGN IN CFD USING SENSITIVITY DERIVATIVES
Michele M. Putko*
Old Dominion University, Norfolk, VA 23529
Perry. A. Newman t
NASA Langley Research Center, Hampton, VA 23681
Arthur C. Taylor 11I3
Old Dominion University, Norfolk VA 23529
Lawrence. L. Green '_
NASA l_xmgley Research Center, Hampton, VA 23681
Abstract
This paper presents an implementation of the approximate statistical moment method for uncertainty propagation
and robust optimization for aquasi I-D Euler CFD code. Given uncertainties in statistically independent, random,
normally distributed input variables, a first- and second-order statistical moment matching procedure isperformed to
approximate the uncertainly in the CFD output. Efficient calculation of both first- and second-order sensitivity
derivatives is required. In order to assess the validity of the approximations, the moments are compared with
statistical moments generated through Monte Carlo simulations. The uncertainties in the CFD input variables are
also incorporated into a robust optimization procedure. For this optimization, statistical moments involving first-
order sensitivity derivatives appear in the objective function and system constraints. Second-order sensitivity
derivatives are used in a gradient-based search to successfully execute a robust optimization. The approximate
methods used throughout the analyses are found to be valid when considering robustness about input parameter
mean values.
Nomenclature
A nozzle area Minf free-stream Mach number
a geometric shape parameter Mt target inlet Mach number
b geometric shape parameter N sample size
b vector of independent input variables Pb normalized nozzle static back (outlet) pressure
F vector of CFD output functions Q vector of flow-field variables (state variables)
g vector of conventional optimization constraints q mass flux through nozzle
k number of standard deviations qt target mass flux through nozzle
M Mach number at nozzle inlet R vector of state equation residuals
M vector of Mach number at each grid point V nozzle volume
Vt target nozzle volume used for optimization
x normalized axial position within nozzle
*LTC, US Army, Ph.D. Candidate, Department of Mechanical (Y standard deviation
Engineering, mputko @tabdemo.larc.nasa.gov variance
tSenior Research Scientist, Muhidisciplinary Optimization Branch,
M/S 159, [email protected]
-:Associate Professor, Deparlment of Mechanical Engineering, superscript:
ataylor @lions.odu.edu
§Research Scientist, Multidisciplinary Optimization Branch, M/S
159, AIAA senior member, [email protected] mean value
This paper isdeclared awork of the U.S. Government and isnot
subject to copyright protection in the United Stales.
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Introduction data uncertainties through CFD code to estimate output
uncertainties. The FOSM approximation is then used to
Gradient-based optimization of complex perform sample robust optimizations. For
aerodynamic configurations and their components, demonstration purposes, we assume that the input
utilizing high-fidclity Computational Fluid Dynamics uncertainty quantification isgiven by independent
(CFD) tools, continues as a very active area of research normally-distributed random variables, and we
(see, for example, Ref. I, 2, and 3). In most of the demonstrate the strategy of Ref. 10as applied to aCFD
CFD-based aerodynamic optimization and design code module. This strategy is also applicable to
studies to date, the input data and parameters have been correlated and/or non-normally distributed variables;
assumed precisely known; we refer to this as however, the analysis and resulting equations become
deterministic or conventional optimization. When more complex.
statistical uncertainties exist in the input data or The gradient-based robust optimization
parameters, however, these uncertainties affect the demonstrated herein requires second-order SD from the
design and therefore must be accounted for in the CFD code. A companion paper, Ref. 12, presents,
optimization. Such optimizations under uncertainty discusses, and demonstrates the efficient calculation of
have been studied and used in structural design second-order SD from CFD code using a method
disciplines (see, for example, Ref. 4, 5, 6, 7and 8); we proposed, but not demonstrated, in Ref 13. This
refer to these as non-deterministic or robust design method, uscd herein, incorporates FO SD obtained by
optimization procedures. both forward-mode and reverse-mode differentiation in
Sensitivity derivatives (SD) of CFD code output, a non-iterative scheme to obtain SO SD.
with respect to code input and parameters, contain To date, the only other demonstration or application
information which can be used to direct the of gradient-based, robust optimization involving
optimization search; that is, the objective and constraint advanced or high-fidelity (nonlinear) CFD code that we
gradients are functions of the CFD SD. Such SD can have found was just recently presented in Ref. 14and
also be used to accurately approximate the CFD output 15. Thc analytical statistical approximation of their
in a small region, such as that near the mean value of a objective function for robust optimization also required
random variable. In Ref. 9, it is shown that a statistical second-order SD. However, these studies employed a
First Order Second Moment _OSM) method and direct numerical random sampling technique to
Automatic Differentiation (AD) can be used to compute expected values at each optimization step in
efficiently propagate input uncertainties through finite order to avoid the second-order SD. An example of
element analyses to approximate output uncertainty. linear aerodynamics involved in multidisciplinary
This uncertainty propagation method is demonstrated performance optimization subject to uncertainty is
hcrein for CFD code. found in Ref. 16.
An integrated strategy for mitigating the effect of Two other aspects need to be pointed out in regard
uncertainty in simulation-based design is presented in to the robust optimization demonstrations for CFD code
Ref. 10; this strategy consists of uncertainty modules prescntcd herein and also in Ref. 14and 15.
quantification, uncertainty propagation, and robust First, the sources of uncertainty considered were only
design tasks or modules. Two approaches are discussed those due to code input parameters involving geometry
there for propagating uncertainty through sequential and/or flow conditions; i.e., due to sources external to
analysis codes: an extreme condition approach and a the CFD code simulation. Other computational
statistical approach. Both approaches can be efficiently simulation uncertainties, such as those due to physical,
implemented using SD. For CFD code, thc former mathematical and numerical modeling approximations
approach is demonstrated inRef. 11, whereas the latter (see Ref. 17and 18) - esscntially internal model error
approach is demonstrated herein using second moment and uncertainty sources, were not considered. That is,
approximations and SD. These uncertainty propagation the discrete CFD code analysis results were taken to be
methods have been developed and are being deterministically "certain" herein. Ultimately, all of
investigated as an alternative to propagation by direct these modeling sources of error and uncertainty must bc
Monte Carlo simulation for potentially expensive CFD assessed and considered. Sensitivity derivatives can
analyses. also aid inthis assessment (Ref. 19) since the adequacy
The present paper shows how the approximate of an internal model's (i.e., algorithm,_urbulcncc, etc.)
statistical second moment methods, FOSM and the prediction capability generally depends, to some extent,
Second Order Second Moment (SOSM) counterpart, on the modeling parameter values specified as input.
can be used in conjunction with SD to propagate input
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Seconads,discussiendRef.15,uncertainty (FOSMaridSOSMw)heretherequireSdDareobtained
classificatiownithrespectotanevent'ismpac(tfrom byhandorbyAD(seeRef.9and20).Thefirststepin
performanlcoesstocatastrophaicn)dfrequenc(fyrom bothFOandSOanalyseisstoapproximattheeCFD
everydaflyuctuatiotnoextremerlyares)etstheproblem systemoutpustolutionosfinteresintTaylorseries
formulatioanndsolutiopnrocedureS.tructural form.Theseapproximatioanrseformedtoestimattehe
reliabilitytechniquteyspicallydeawl ithrisk outpuvtalueforsmaldleviationosftheinput.
assessmoefnint frequebnuttcatastrophfaiciluremodes, Giveninputrandomvariablebs={bl.....b,}with
identifyintghemosptrobablpeoin(tMPPo)ffailure meanb={hi.....b,} andstandadrdeviations,
anditssafetiyndex.Herew, eareaddressitnhge ab={_b,....a.b,,},theCFDoutpuftunctionF,,first-
assessmoefnetverydaoyperationfalulctuatioonns andsecond-ordTearylosrerieaspproximatioanres
performanlcoessn,otcatastrophCeo.nsequenwtlye, givenby
aremosctoncernewdithaeroperformanbceehavior
duetoprobabfleluctuationi.se,.,neatrhemeanof FO:
probabilidtyensitfyunction(spdf).Structural
t/
reliabilityassessmiesnmtosctoncernewdith F(b)=F(b)+£_F(b i -bi) (I)
improbabcleatastropheivcentsi.,e.,probabiliitnythe i=1rib,
tailsofthepdf.Simultaneocuosnsideratioofnboth SO:
typesofuncertainitsydiscusseindRef.16.
-- " 3F
InRef.10anintegratemdethodolofgoyrdcaling
F(b) = F(b) + i:l£_i (bi -- bi) -k
withuncertainitnyasimulation-badseesdigins (2)
proposeadnddemonstraftoerdalinkagemechanism
designT.heintegratesdtrategoyfRef.10formitigating
theeffecotfuncertainitnyclude(sa)uncertainty
quantificatio(nb,)uncertainptyropagatioann,d(c) where both first and second derivatives are evaluated at
robusdtesignT.hepresensttudyutilizetshestrategy the mean values, b.
proposeindRef.10b,utdiffersinregartdouncertainty One then obtains expected values for the mean (firsl
propagatiaonndapplicatioHn.erew, eareconsidering moment) and variance (second moment) of the output
theinfluencoefuncertainitnyCFDcodeinputt;hatis, function, F, which depend on the SD and input
theeffecotfuncertainitnyinputgeometoryn variances, _h2. (Recall the variance is equivalent to the
aerodynamshicape-desoigpntimizatioanndtheeffect square of the standard deviation.) The mean of the
ofuncertainitnyflowconditionosndesigfnorflow output function, F, and standard deviation %-, are
control. approximated as
FO:
Integrated Statistical Approach
=F(b)
Our implementation of the three aspects of the
integrated strategy of Ref. 10are as follows: (3)
=+(aF f
Uncertainty Quantification
In this study, we consider the influence of
uncertainty in CFD input parameterization variables.
We have assumed that these input variables are SO:
statistically independent, random, and normally
distributed about a mean value. This assumption not _:F{g)+±'_ _ _2F (4)
only simplifies the resulting algebra and equations, but 2! j--1 =
also serves to quantify input uncertainties.
Furthermore, it isnot an unreasonable assumption for
input geometric variables subject torandom (_F2 i=l _ b, = (Ib
manufacturing errors nor for input flow conditions
subject to random fluctuations. where both first and second derivatives are evaluated at
the mean values, b. Note in Eq. (4) that the second-
Uncertainty Prop..agation
order mean output, F, is not at the mean values of
Uncertainty propagation is accomplished by
input b, i.e., F cF(b).
approximate statistical second moment methods
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Equations (3) represent a FO method and Eq. (4) a Therefore, a gradient-based optimization will then
SO method for examining uncertainty propagation. The require second-order SD tocompute the objective and
methods arc straightforward with the difficulty largely constraint gradients. Note that for the SOSM
lying in computation of the SD. The very efficient and approximation, third-order SD would bc required for
effective method used here to obtain such derivatives is these gradicnts.
presented in a companion paper, Ref. 12. The calculation of second-order SD for CFD code,
such as those required for SOSM and robust
Robust Design optimization with FOSM, was demonstrated in Ref. 13;
Conventional optimization for an objective function, the efficient calculation used herein isdemonstrated
Obj, that is a function of the CFD output, F, state and discussed in a companion paper, Ref. 12. Both
variables, Q, and input variables, b, is expressed inEq. hand differentiation and AD via the ADIFOR tool (Ref.
(5). Herein, the CFD state equation residuals, R, are 21, 22 and 23) were used. Both conventional and
represented as an equality constraint, and other system robust optimizations were performed using the
constraints, g, are represented as inequality constraints. Sequential Quadratic Programming (SQP) method
The input variables, b, are precisely known, and all option in the Design Optimization Tools, DOT (Ref.
functions of b are thercforc deterministic. 24).
min Obj, Obj = Obj(F,Q,b) Application to Quasi 1-D Euler CFD
subject to
R(Q,b) = 0 (5) A very simple example has been chosen to
g(F,Q,b) _<0 demonstrate the propagation of input uncertainty
through CFD codc and its effect on optimization. Two
For robust design, the conventional optimization, separate applications are presented; the first involving
Eq. (5), must be treated in aprobabilistic manner. propagation of geometric uncertainties, the second
Given uncertainty in the input variables, b, all functions involving propagation of flow parameter uncertainties.
in Eq. (5) are no longer deterministic. The dcsi...gn Both uncertainty analyses are performed with quasi
variables are now the mean values, b = {bI..... b, }, one-dimensional Euler equations and boundary
where all elements of b" arc assumed statistically conditions describing subsonic flow through a variable
independent and normally distributed with standard area nozzle. The nozzle inlet is located at x= 0with
deviations ¢_b.The state equation residual equality area A(x = 0) = 1;the nozzle outlet isat x= 1.The area
constraint, R, isdeemed to be satisfied at the expected distribution is given by
values of Q and b, that is the mean values Q and b for
the FO approximation. The ob.iective function is cast in A(x) = 1- ax + bx 2.
terms of expected values and becomes afunction of F
and c_-. The other constraints are cast into a The w)lume, V, tx:eupied by the nozzle, isthe
probabilistic statement: the probability that the integration of A(x) over the length x= 0 to x= I
constraints are satisfied is greater than or equal to a a b
V=I---+--,
desired or specified probability, Pk. This probability 2 3
statement is transformcd (sec Ref. 10) into a constraint where aand b are the input geometric parameters.
involving mean values and standard deviations under Three flow parameters arespecified as input
the assumption that variables involved are normally boundary conditions: the stagnation enthalpy, inlet
distributed. The robust optimization can be expressed entropy, and outlet static (back) pressure. The quasi
as i-D Euler equation set is symbolically written as the
state equation in Eq. (5); itsresidual, R is driven to
minObj, Obi= Obj(F ,¢_,Q, b)
(machine) zero for asolution.
subject to For supersonic flow through avariable area nozzlc,
R(Q,b)=0 (6) shock waves generally appear and the flow solution
g(_,_, _) +k_ _<o,
(objective, constraint, etc.) becomes noisy or non-
smooth (see Rcf. 25 and the references cited therein).
where k is the number of standard deviations, og, that Care must be exercised with respect to obtaining and
the constraint g must be displaced in order to achieve using the SD needed for gradient-based optimization
the desired or specified probability, Pk. For the FOSM (Ref. 25 and 26). Therefore, we chose to bypass issues
approximation, standard deviations (_Fand _g are of thc related to this supersonic flow non-smoothness in these
form given inEq. (3) involving first-order SD.
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initialdemonstratioonftshepresenatpproacfohr SO:
uncertainptryopagatiaonndrobusdtesignforCFD
codemodules.
2M
Geometric Uncertainty ProPagation
I 1' 212
For the discussion of geometric uncertainty
propagation, geometric shape parameters a and b will "M2=t-g ' + "-g"b +0.50---g-_ + (10)
represent the statistically independent random input
variables, b. The Mach number distribution through the o32M 2 2
nozzle, M, is viewed here as a component of the state
variable. Q; its value at the inlet, M, is the CFD output,
F. Applying the approach previously outlined (recall
B
Eq. (3) and (4)) yields the following first- and second- Predictions of M(a,b), M, and CMfor FO (Eq. (7)
order approximations of the output function, M. and (9)) and SO (Eq.(8) and (10)) are compared with
CFD solutions and Monte Carlo analyses based on CFD
solutions, as given and discussed in the results section.
Input random variables: b={a,b}
Robust Shape Optimiz_atj_on
Applying the conventional optimization previously
CFD output function: F={M] described yields
FO Taylor series:
min Obj, Obj = Ob.ifM,a,b)
M(a,b) = M(_,b) +OM(a -_) + _M (b -b) (7) subject to
da db R(M,a,b) = 0 (11)
V(a,b) <0,
SO Taylor series: where the system constraint, V, is a constraint on the
nozzle volume anddepends only on a and b; and our
+3M b objective does not explicitly depend on M.
M(a.b)=M(_,b)+_aM(a-g) --_-(-b)+
Applying the robust optimization previously
described yields
+3_2aMa--_(a -_)(b- -b-)+0.5( [33-2_,M.- (a __)2 )+ ('8) rain Obj, Obj = Obj(M,oM,_ ,b)
subject to
+0.5 (b-b) 2 R(M',g,b) = 0 (12)
V( 5, b )+ k_v < O,
where
m
The mean, M, and standard deviation ¢h4of the
output function are expressed as
r_v2 = ['_a ' [-fib-- _ (13)
FO: With a and b subject to statistical uncertainties
(which may be due to measurement, manufacturing,
= M(5,b) etc.), V becomes uncertain. Since V is linearly
dependent on a and b, it is also normally distributed.
(9)
Therefore, its standard deviation, O'v,is given exactly
oM =-(;3xM%j+1 2x(3.Mr]2
by Eq. (13).
To demonstrate the optimizations, a simple target-
matching problem is selected; a unique answer is
obtained when an equality volume constraint is
enforced. The CFD code isrun for given a and b; the
resulting M(a,b) and corresponding V(a,b) are taken as
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thetargevtalueMs tandVt,respectivelFyo. rthis
min Obj Obj = Obj(M,Minf, Pb)
conventionoapltimizatiotnh,eobjectivfeunctioannd
subject to
constrainfutnctionforVofEq.(11)become R(M,Minf, Pb) = 0 (16)
q(Minf, Pb) _<0,
Obj(M,a,b=)[M(a,b-)Mr]2
V(a,b)-Vt=0
where q is aconstraint on the mass flux through the
enforceads (14) nozzle.
V(a,b)-Vt
< 0 and Vt - V(a,b) _<0 The robust optimization is expressed as
for the convenience of the optimizer. rain Obj, Obj = Obj( M ,OM,Minf, Pb )
subject to
For robust optimization using the FOSM
R( M, Minf, Pb ) = 0 (17)
approximation, the corresponding objective and
constraints on V of Eq. (12) become q(Min f,Pb )+ k_q< 0.
For the free stream Mach number, Minf, and the
obj (_,oM,_ ,g) = [_ (5 ,g)- Mt] 2+OM2
nozzle back pressure, Pb, subject to statistical
V( 5, g )-Vt + ko'v =0
uncertainties, the mass flux, q, becomes uncertain.
similarly enforced as (15)
Since q is dependent on Minf and Pb, its standard
V(5, g ) - Vt + ko'v < 0 deviation, (rq,may be approximated by
and Vt - V(_, b ) - kov -<0.
2 . (]8)
Note that for _,_= 6b= 0 in Eq. (15), the
conventional optimization isobtained. Also, inthe
Since q is not a linear function of Minf and Pb, Eq. (18)
probabilistic statement of the constraint on V, it is
is not exact (unlike the previous example where _v was
assumed that the desired volume is less than or equal to
Vt. exactly known).
To demonstrate the optimizations, a simple target-
matching problem is again chosen. The CFD code is
Flow Parameter Uncertainty Propagation
run for given Minf and Pb; the resulting M and
A second example of uncertainty in CFD involves
corresponding q are taken as the target values Mt and
fluctuations in input flow parameters. For the
qt, respectively. For this conventional optimization, the
discussion of flow parameter uncertainty propagation,
objective function and constraint functions of Eq. (16)
the free-stream Mach number, Minf, and the nozzle
static back pressure, Pb, will be taken as statistically are
independent random variables. Specifying the free-
Obj(M,Minf, Pb) = [M(Minf, Pb) - Mr] 2
stream Mach number sets the stagnation enthalpy. The
q(Minf, Pb) - qt = 0
Mach number distribution through the nozzle, M, is
enforced as (19)
again viewed as a component of the state variable, Q;
q(Minf, Pb) -qt <0 and qt - q(Minf, Pb) < 0.
its value at the inlet, M, is the CFD output, F.
Applying the approach previously outlined yields
For robust optimization using the FOSM
equations which are similar toEq. (7) through (10) but
approximation, the corresponding objective and
with
constraint on q of Eq. (17) can be shown as
Input random variables: b={Minf, Pb}
Obj = Obj(M,CrM, Minf, Pb )
CFD output function: F=[M}. = [M (Minf, Pb )- Mt]2+ 6M2
w q(Minf, Pb )-qt +koq =0
Again, predictions of M, M, and OMfor FO and SO
enforced as (2O)
approximations are compared with CFD solutions and
Monte Carlo analyses based on CFD solutions, as given q(Minf, Pb )-qt +kclq- < 0
and discussed in the next section.
and qt-q(Minf, Pb )-ko. -<O.
Robust Design for Flow Control
Again note that for gM_,f= Opb= 0 in Eq. (20), the
The conventional optimization is expressed as
conventional optimization is obtained. Also, in the
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probabilistsictatemeonfttheconstraionntq,itis nonlinear. At larger deviations from the mean, a linear
assumethdatthedesiremdassfluxislessthanorequal approximation for M(Minf, Pb) loses accuracy.
toqt.
1,4
Sample Results & Discussion
1.2
Presentation and discussion of results for the sample
quasi I-D Euler CFD problems are divided into four
topics: function approximations, uncertainty
propagation, pdf approximations, and robust
M 1
•"" • CFD
optimization. For the first three topics, the M
approximations are assessed by comparison with direct
CFD simulations.
- t -----so
0.8
Function Approximations
It is important to assess the Taylor series output
function approximations with direct nonlinear CFD
code simulations prior to presenting uncertainty 0.6
-0.3 -0.15 0 0.15 0.3
propagation. If the CFD output function, M, is quasi-
a-_
linear with respect to the input variables of interest, one
can expect first-order approximations to be reasonably
good; that is, the FO moments given by Eq. (3) should Fig. I. Comparison of Function Approximations vs.
match well with the moments produced by a Monte
CFD Solution, Input Variable b = b.
Carlo simulation. For a more nonlinear system, one
naturally expects better accuracy with second order
approximations; that is, uncertainty analyses which
include SO terms should yield results which better
1.4
predict the statistical moments produced by the Monte
Carlo simulation.
Figures I-4 show that for F=M(a,b), M behaves as a
quasi-linear function in the neighborhood of (_, b ), 1.2
whereas for F=M(Minf, Pb), M is more nonlinear inthe
neighborhood of (Minf, Pb ). In these figures,
approximations of the CFD output functions, M(a,b) M
and M(Minf, Pb), using the first- (FO) and second-order
t
(SO) Taylor series (as given in Eq. (7) and (8) for
• CFD
M(a,b)), are compared to direct solution of the Euler
CFD. In each example, two traces were considered
...... FO
0.8
through the design space. Trace I varied the first input
variable, while the second remained fixed at its mean _SO
value, and vice versa for trace 2. The required first-
and second-order SD needed for construction of the FO
0.6 _ _ ....
and SO approximations were obtained by hand -0.6 -0.3 0 0.3 0.6
differentiation and AD as discussed and presented in
Rcf. 12. b--_
Nonlinear behavior of the CFD result is reasonably
Fig. 2. Comparison of Function Approximations vs.
well approximated by the SO result in all plots;
CFD Solution, Input Variable a = _.
however, there does appear to be an inflection point in
thc CFD results given in Fig. 3. Note that the linear FO
result is a good approximation in the geometric
example; the flow parameter example is more
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Carlo results. The standard statistical analyses used
were from MicroSoft ® Excel 2000 and the random
1.1 number generator MZRAN used was from Ref. 27.
Tables I and 2 give results for the mean (first moment)
and Tables 3 and 4 give results for the standard
deviation (second moment) value comparisons. The
1.05
input deviations (Oa and (Jb) or (_.umr and Opb), are taken
to be equal and given in the second column of each
M
table. The third column in each table gives the result
from the Monte Carlo simulation, where the sample
size (N) used was 3,000. The Monte Carlo error in its
predicted mean is _M/_", which is given in the fourth
column of Tables I and 2, The FO and SO approximate
0.95
predictions are given in the last two columns of each
table as percent difference from the Monte Carlo
results•
0.9
-0.70 -035 0.00 035 0.70
Table 1. Percent Difference from Monte Carlo (MC)
Minf - Minf for FO and SO Predictions of M (5, b)
Input o M % Error _ diff w/MC % diff w/MC
Fig. 3. Comparison of Function Approximations vs.
CFD Solution, Input Variable Pb = Pb. Case Ga=GI, MC MC FO Predict SO Predict
1.4 I 0.01 0.4041 0.0187 -0.0105 0.0656
2 O.02 0.4040 0.0379 0.0716 0.1531
1.2 l Wt, . * CFD _- 3 0.04 0.4054 0.0756 -0.2867 0.0383
..... // 4 0.06 0.4055 0.1142 -0.3012 0.4301
I
5 0.08 0.4096 0.1557 -1.3078 -0.0209
/
Table 2. Percent Difference from MC for FO and SO
Predictions of M (Minf, Pb)
Input _ M % Error %diffw/MC %diffw/MC
0.8 Caseo_li._On.,, MC MC FO Predict SO Predict
1 0.01 0.3933 0.0056 0.0037 -0.0269
2 0.02 0.3932 0.0114 0.0187 -0.1034
0.6
-0.30 -0.15 0.00 0.15 0.30 3 0.04 0.3898 0.0229 0.8917 0.3991
Pb-Pb
4 0.06 0.3889 0.0364 1.125I 0.0141
Fig. 4. Comparison of Function Approximations vs.
CFD Solution, Input Variable Minf = Minf.
Table 3. Percent Difference from MC for FO and SO
Predictions of ¢J.. Geometric Example
Uncertainty Propagation
Approximation of the statistical first and second
Input o" (JM %diff w/MC %diff w/MC
moments is done using equations Eq. (9) and (10)
(geometric example), and corresponding equations for Case _.=Ob MC FO Predict SO Predict
the flow parameter example. Again, both first- and 1 0.01 0,0102 -0.5773 -0.5708
second-order SD are required and the prediction is
2 0.02 0.0207 -1.7026 -1.6769
straightforward, given these derivatives. An
independent verification of these approximate mean and 3 0.04 0.0414 -1.5794 -1.4766
standard deviation values is obtained here using direct 4 0.06 0.0625 -2.2590 -2.0296
Monte Carlo simulation with the quasi [-D Euler CFD
5 0•08 O.O853 -4.3987 -4.0001
c(_le and standard statistical analyses of these Monte
8
American Institute of Aeronautics and Astronautics