NASA Technical Reports Server (NTRS) 20000097390: First-Order Model Management With Variable-Fidelity Physics Applied to Multi-Element Airfoil Optimization

12 Pages·0.61 MB·English

by  NASA Technical Reports Server (NTRS)

AIAA 2000-4886
First-Order Model Management with
Variable-Fidelity Physics Applied to
Multi-Element Airfoil Optimization
N. M. Alexandrov, E. J. Nielsen,
R. M. Lewis, and W. K. Anderson
NASA Langley Research Center
Hampton, VA 23681
8th AIAA/USAF/NASA/ISSMO Symposium
on Multidisciplinary Analysis & Optimization
6-8 September 2000 / Long Beach, CA
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500, Reston VA 20191-4344

AIAA-2000-4886
FIRST-ORDER MODEL MANAGEMENT WITH VARIABLE-FIDELITY PHYSICS
APPLIED TO MULTI-ELEMENT AIRFOIL OPTIMIZATION
N. M. ALEXANDROV*, E. J. NIELSEN t. R. M. LEWIS*' AND W. K. ANi)ERSON _
Abstract structural optimization, tbr instance, can be found in [4].
Accounts of recent efforts in developing methodologies
First-order approximation and model management is a tor variable-complexity modeling are relayed in [5, 61.
methodology for a systematic use of variable-fidelity models The present work concerns an approach, the Approx-
or approximations in optimization. The intent of model man- imation and Model Management Framework (AMMF)
agement is to attain convergence to high-fidelity solutions with [7-10], designed to enable rapid and early integration of
minimal expense in high-fidelity computations. The savings in high fidelity nonlinear analyses and experimental results
terms of computationally intensive evaluations depends on the into the multidisciplinary optimization process. This is
ability of the available lower-fidelity model or a suite of models accomplished by reducing the frequency of performing
to predict the improvement trends for the high-fidelity problem. high-fidelity computations within a single optimization
Variable-fidelity models can be represented by data-fitting ap- procedure.
proximations, variable-resolution models, variable-convergence Until recently, procedures for the use of variable-
models, or variable physical fidelity models. The present
fidelity models and approximations in design had relied
work considers the use of variable-fidelity physics models. We
on heuristics or engineering intuition. In addition, with
demonstrate the performance of model management on an aero- a few exceptions (e.g., [I I], [12]), the analysis of algo-
dynamic optimization of a multi-element airloil designed to rithms had focused on convergence to a solution of the ap-
operate in the transonic regime. Reynolds-averaged Navier-
proximate or surrogate problem ([ 13], [14]). The AMMF
Stokes equations represent the high-fidelity model, while the Eu- methodology discussed here and in related papers is dis-
ler equations represent the tow-fidelity model. An unstructured
tinguished by asystematic approach to alternating the use
mesh-based analysis code FUN2D evaluates functions and sen- of variable-fidelity models that results in procedures that
sitivity derivatives for both models. Model management tbr the
are provably globally convergent to critical points or so-
present demonstration problem yields fivefold savings in terms lutions of the high-fidelity problem.
of high-fidelity evaluations compared to optimization done with
Model management can be, in principle, imposed on
high-fidelity computations alone.
any optimization algorithm and used with any models. In
[15], we considered AMMF schemes based on three non-
Key Words: Aerodynamic optimization, airfoil de- linear programming methods and demonstrated them on
sign. approximation concepts, approximation manage- a 3D aerodynamic wing optimization problem and a 2D
ment, model management, nonlinear programming, sur- airfoil optimization problem. In both cases, Euler analy-
rogate optimization, variable-fidelity sis perlormed on meshes of varying degree of refinement
formed a suite of variable-resolution models. Results in-
Background dicated approximately threefold savings (similar across
the three schemes) in terms of high-fidelity function eval-
Approximations and low-fidelity models have long uations. The AMMF based on the sequential quadratic
been used in engineering design to reduce the cost of opti- programming (SQP) approach was judged to be the most
mization (e.g., [I-3 ]). An overview of approximations in promising for single-discipline problems with a modest
*Member AIAA. Multidisciplinary Optimization Branch, MS 159. NASA Langley Research Center. Hampton. VA, 23681-2199.
[email protected]+
tMember AIAA, Computational Modeling and Simulation Branch. MS 128. NASA Langley Research Center. Hampton. VA, 23681-2199.
e.j [email protected],
Member AIAA. ICASE, MS 132C. NASA Langley Research Center, Hampton. VA. 2368 I-2199, buckaroo@i case. edu. This aulhor's
research was supported by the National Aeronautics and Space Administration under NASA Contract No, NAS 1-97046 while the author was in
residence at the Institute for Computer Applications in Science and Engineering (ICASE). NASA Langley Research Center. Hampton. VA 2368 I.
Member AIAA, Computational Modeling and Simulation Branch. MS 128, NASA Langley Research Center. Hampton, VA, 23681-2199.
w. [email protected] .gov.
Copyright Q2000 by the American Institute of Aeronautics and Astronautics. Inc, No copyright is asserted in the United States under Title 17. U.S.
Code. The U.S. Government has a royahy-free liccnse to exercise all rights under the copyright claimed herein for Government Purposes. All other
rights are reserved by the copyright owner.
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number of design variables, as well as for certain fl_rmu- f(J,_,) > f(.r,. + ._,.). Otherwise the step is rejected. The
lations of the MDO problem. (, trust region (chosen because it interacts naturally with
The study in [15] has served as a proof of con- the bound constraints) isdecreased if pc is small. Experi-
cepl for AMMF in the case where low-fidelity models ence suggests that "small" be taken as less that 10-r'. if
are represented by data-fitting approximations (kriging, p,: is close to one or greater than one (this indicates excel-
splines and polynomial response surfaces) or variable- lent predictive properties of the model), the trust region is
resolution models. The present work considers, arguably, doubled. Otherwise, it is left unchanged.
the most challenging combination of the high and low- The conditions
fidelity models within a single optimization procedure -
that of variable-fidelity physics models. The performance a,(a',) = f(,r,) (2)
of the first-order model management is demonstrated on
U'a<.(.rc) = Vf(a'_) (3)
an aerodynamic optimization of a multi-element airloil.
Variable-fidelity models are represented by an unstruc-
are known as the first-order consistency conditions, which
tured mesh-based analysis run in viscous and inviscid
we will discuss presently.
modes.
In the next section, we describe the AMMF under in- In conventional optimization, a,. is usually a linear or
vestigation and discuss the points of interest for the cur- quadratic model of the objective f. AMMF replaces this
rent study. We then present the demonstration problem, local. Taylor series approximation by an arbitrary model
lollowed by a discussion of the numerical experiments required to satisfy the consistency conditions (2)-(3). Re-
and results. gardless of the properties of the low-fidelity model, the
consistency conditions tbrce it to behave as a first-order
Taylor series approximation at points where they are sat-
AMMF under investigation
isfied. Solving the subproblem of minimizing a,, is itself
For the present demonstration, the optimal design an iterative procedure that now requires the function and
problem is represented by the bound constrained nonlin- derivative inlormation from the low-fidelity model.
ear programming problem: First-order AMMF methods can be shown to converge
to critical points or solutions of the high-fidelity problem
min f(.r)
(1) under appropriate standard assumptions of continuity and
s.t. / <.r < ,,
boundedness of the constituent functions and derivatives
where .r is the vector of design variables, the objective (see [9], tbr instance), given that the consistency condi-
f is acontinuously different|able nonlinear function, and tions (2)-(3) arc imposed at each maior iterate .r,..
/ _<.r _< , denotes bound constraints on the design vari- Qualitatively, the reason a first-order AMMF con-
ables. verges to an answer of the high-fidelity problem may be
The first-order AMMF used here for solving (1) is summarized as follows. Although a lower-fidelity model
based on the trust-region strategy, which is a methodol- may not capture a particular feature of the physical phe-
ogy lor the improvement of global behavior of the local nomenon to the same degree of accuracy (or at all) as
model-based optimization algorithms I161.The following its higher-fidelity counterpart, a lower-fidelity model may
pseudo-code describes the AMMF. still have satisfactory predictive properties lbr the pur-
poses of finding a good direction of improvement |'or the
Initialize .r',.._k,,
higher-fidelity model. By imposing the consistency con-
Do until convergence:
ditions. AMMF ensures that at least at the major iterates,
Select amodel a,. such that
the lower-fidelity model provides the same direction of
,, (.,',) : f(._', ) and U<,,.(.r_) : U'f(._,,
descent as the high-fidelity counterpart. Two questions
Solve approximately for ,_, : 2'- .r_.:
arise. How easy isit toimpose the first-order consistency?
min ., (.r, + ._)
How does the method perfl_rm in practice'/
s
s.t. / < .r < ,
The answer to the first question is that imposing the
[I-,II,__<x,
conditions (2)-(3) is straightfl_rward using a correction
Compute p,. -- .ff_', )-.fi3, +._,) technique due to Chang et al. 117] This technique cor-
.f(.,._.)-. _(.r•+._.:_
rects a low-fidelity version Jio of an arbitrary function so
Update _k, and ._',based on p,.
that it agrees to lirst-order with a given high-fidelity ver-
End do
sion fhi. This is done by defining the correction factor J;_
Details of the updating strategy can be found, for instance, as
in [7]. Briefly. the point .r, is accepted if the step s, re- j_/(._.)
suits in a simple decrease in the objective function, i.e., if
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FigureI: Meshfortheviscoumsodel
1/7
Figure 2: Mesh for the inviscid modcl
Given the current design variable vector ._.,.,one builds a high-fidelity model only at the point ._'_..Thus. the high-
first-order model 3,. of 3 about ._:,: fidelity intormation may potentially have to be computed
at every step to re-calibrate the low-fidelity information.
_b(x) = ,3(._',) + T3(._,, )7"(.r - ._',,).
This would lead AMMF to become, at worst, a conven-
The local model of 3 is then used to correct J)o to obtain tional optimization algorithm. At its best, the AMMF
a better approximation ,(.r) of fhi: would be able to take many steps with the corrected low-
fidelity model before resorting to re-calibration with ex-
fhi(.l') = ;3(_')fto(,r) ,_ a(J') -=,3c(;rlflo(x). pensive evaluations. Which scenario actually takes place
depends on the problem at hand.
The corrected approximation a(.r) has the properties that
a(x,.) = fhi(x,.)and XSa(.trc) = X-'fhi(._',.). Zero-order or AMMF has shown promise with low-fidelity models
higher-order corrections are easily constructed as well. represented by data-fitting approximations and variable-
Because the 3-correction can make any two unrelated resolution models. In an attempt to evaluate the poten-
functions match to first order, the framework admits a tial worst-case scenario, we are now considering manag-
wide range of models. In the worst case of performance, ing variable physical fidelity models. We view this model
the subproblem will yield a good predictive step .sofor the combination as the potential worst-case scenario, because
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Pra_O01 I30AeJ92000 iFUN2D __ [J:¢amG01 30At_ 2000 [FUN2D __
Mach number contours, viscous mo;del Mach number contours, inviscid model
'I
1
o75
0.75
0,5
0,25
>.
0
-0.25
-0.5
o 05 0 O5 1
X X
Figure 3: Math number contours for viscous vs. inviscid model
I
:ra_ o01 !31A_2000 MULTI-ELEMENT AIRFOIL: INVISCID FUNCTION DATA
Drag coefficient contours, inviscid
2 Drag coefficient contours, viscous
From 00*i31_00 .[MULTi-ELEMENT AIRFOIL: VISCOUS FUNCTION DATA
15
o75
025
o_ _ _ o_75
021_05
-0005 Y DISP 0 0.005 • " Y-DISP
Figure 4: Drag coefficient contours of the viscous and inviscid models
low-fidelity physics models arc expected not to capture be the Euler equations. The flow solver, FUN2D, used
the behavior of the high-fidelity counterparts accurately, for this study 12)llows the unstructured mesh methodol-
or at all, over some or all regions of interest in the design ogy [I9]. Sensitivity derivatives are provided via a hand-
space. coded adjoint approach [20].
The mesh lk)rthe viscous model depicted in Fig. I
Demonstration problem consists of 10449 nodes and 20900 triangles. The mesh
for the inviscid model, shown in Fig. 2, comprises l.q17
We consider aerodynamic optimization of a two-
nodes and 38.q(Jtriangles. The Math number is ,'_I_ =
element airfoil designed to operate in transonic condi-
0.75, the Reynolds number is ll_ = !)× 10_;,the global
tions [18J.The inclusion of viscous effects is very impor-
angle of attack is (_ :- I°.
tant for obtaining physically correct results. Therefore,
the high-fidelity model will be the Reynolds-averaged Fig. 3depicts the Mach number contours for the vis-
Navier-Stokes equations and the low-lidelity model will cous and inviscid model, respectively. The boundary and
4
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Test hi-fieval lo-fieval totatlime factor
PORTwithhi-fisurrogate2sv,ar 15/15 negligible
AMMFwithsurrogate2sv,ar 3/3 18/9 negligible _5
PORTwithhi-fianalyse2sv,ar 14/I3 _ 12hrs
AMMFwithdirecatnalyse2sv,ar 3/3 19/9 _ :2.4h1rs ,1.98
Tablei: AMMFperformanvcsePORT
shear layers are clearly visible in the viscous case. Be- substitutes in the conventional sense, i.e., they were not
cause of the importance of the viscous effects in this prob- used to provide lower-fidelity models. Instead, they sim-
lems, the use of the inviscid equations for the low-fidelity ply served to provide low-cost substitutes for both mod-
model should present an important test Ibr the present ap- els for the problem components in the testing phase. Of
proach. course, such a test would never be conducted in a non-
The objective of this problem is simply to minimize research setting, nor would it be considered for a problem
the drag coefficient by adjusting the global angle of at- with more than afew variables. In our setting, however, it
tack and the y-displacement of the flap. In this study, saved us much time by providing an excellent approxima-
we restrict ourselves to two design variables to enable vi- tion of the actual functions with respect to descent charac-
sualization. The baseline case for both models was con- teristics at a tiny fraction of computational cost. After wc
structed at _ = 1oand zero g-displacement of the flap. ascertained the correctness of our procedures, tests were
Fig. 4 depicts the level sets of the drag coefficient conducted directly with the flow and adjoint solver, with-
for the viscous and inviscid models. The problem ap- out recourse to substitutes, because the substitutes were
pears to support the worse-case scenario: not only is the expected to smooth out the problem to a certain degree.
low-fidelity model not a good representation of the high- The problems were first solved with single-fidelity
fidelity model but, in addition, the descent trends in the models alone by using well-known commercial optimiza-
two models are reversed. The solution for each problem tion software ¶ PORT [21], in order to obtain a baseline
is marked with acircle. Thus the problem provides a good number of function evaluations or iterations to find an
test of the methodology indeed. optimum. The problems were then solved with AMMF.
The computational expense necessary to calculate Identical experiments were conducted with spline substi-
functions and derivatives in the viscous case is consid- tutes and with the actual flow and adjoint solver.
erably greater than that for the inviscid model. We con- For each experiment, performance of AMMF was
ducted our experinaents on an SGP MOrigin 2000 work- evaluated in terms of the absolute number of calls to the
TM
station with four MIPS RISC RIO000 processors. One high and low-fidelity function and sensitivity calculations.
low-fidelity analysis took approximately 23 seconds and Because the time for low-fidelity computations was neg-
one low-fidelity sensitivity analysis took between 70 and ligible in comparison to the high-fidelity computations,
100 seconds. In contrast, one high-fidelity analysis took we estimated the savings strictly in terms of high-fidelity
approximately 21 minutes and one high-tidelity sensitiv- evaluations. Table I summarizes the number of function
ity analysis took between 21 and 42 minutes to compute. (first number) and derivative (second number) computa-
The measures were taken in CPU time. Thus, the time tions expended in PORT and in AMMF.
per Iow-lidelity evaluation may be considered negligible Given the dissimilarity between the high-lidelity and
compared to that required for a high-fidelity evaluation. low-fidelity model, we were initially surprised to lind
that the AMMF performed well: it consistently yielded
Numerical results approximately fivefold savings in terms of high-fidelity
computations. The result held both for the spline substi-
We conducted the following computational experi- tutes and the actual functions. Optimization applied to
ments. Because our test problem has expensive function both cases produced nearly identical iterates.
evaluations, we first built splint substitutes both for the Following an analysis of the results, we concluded that
viscous and the inviscid model. Error analysis indicated the savings were not surprising after all. For our combi-
that the spline lit was highly satisfactory for both mod- nation of models, the 3-correction worked extremely well.
els. It should be emphasized that we did not use these This is illustrated in Fig. 5. The plot on the left shows the
'liTheuseofnamesofcommercial sc.flwareinthispaper isforaccuratereportinganddoesnotconstituteanofficialendorsement, eitherex-
pressedorimplied,ofsuchproductsbytheNationalAeronauticsandSpaceAdminislrationorInstituteforComputerApplicationstoScienceand
Engineering.
5
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rtarc,eooi!31Aug2OO0IMULTI-ELIEUENTAIRFOIL:CORRECTEDLO-FIDATA
Drag coefficient contours, corrected inviscid
2 Drag coefficient contours, viscous
F_ 001!_1AL__00 IMULI_ELI[MENT All'FOIL:VISCOUSFUNCTIONDkTA _
/i •
151_4_,""'-- _0.0065/
I-/o.oO65_
125 _ 0.006,.6"-'--'1
125 OB6//0 _.._ _
< _1o.o 92/_____0 oo9_ J
0O0 0,_0
,-I 001_1
O. 5 -n nl_, _ _ _^a_'_oq._'-__.'_ _
0.5
025 ;....u.u _ . -- .u258_
0_005 -0005 0 0005
Y-DISP Y-DISP
Figure 5: Drag coefficient level sets of the viscous and corrected inviscid models
level sets of the high-fidelity model with the solution. The Given the present results, we are cautiously optimistic
plot on the right depicts the level sets of the Iow-lidelity about several much larger test cases (e.g.. _! variables)
model .4-corrected at the initial point. The initial point is that are currently under investigation. Large problems
marked by a square. We note that the correction is not must be tested carefully in AMMF in order to ascertain
applied to the entire feasible region during iterations of that its performance is not in some measure an artifact of
optimization algorithm. Here, we applied the correction the problem dimensionality. This does not appear to be
tothe entire region to visualize the affect of the correction the case, because AMMF was prcviously tested on prob-
on the low-fidelity function. The figure clearly shows that lems with over ten variables. However. the tests currently
the correction, using the function and derivative informa- conducted with realistic physical models should prove
tion at the anchor point (at this iteration - the initial point), more conclusive, regardless of the outcome.
reversed the trend of the low-fidelity model, allowing the The performance of AMMF with variable-fidelity
optimizer to find the next iterate in the left upper corner physics models raises a number of intriguing questions
of the plot, marked by a circle. Similar analysis can be about the nature of the corrections and an optimal choice
conducted for all iterations. In fact, AMMF located the of low-fidelity models for a large set of problems. These
solution (n = 1.6305 °, flap !/-displacement = -0.00-18) questions are currently under investigation.
of the high-tidelity problem already at the next iteration.
The high-fidelity drag coefficient at the initial point was Acknowledgments
(,initial = 0.0171, the high-fidelity drag coefficient at
/)
We would like to thank Clyde Gumhert for his invalu-
the solution was ('final __().III48, a decrease of approxi-
able help with the graphics.
mately 13.15(/_.
Concluding remarks References
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