Document
Proceedings of DETC98 ASME 1998 Computers In Engineering Conference September 13-16, 1998, Atlanta, GA
DETC98/CIE-6006
HIGH-LIFT OPTIMIZATION DESIGN USING NEURAL NETWORKS
ON A MULTI-ELEMENT AIRFOIL
Karlin R. Roth Roxana M. Greenman Aerospace Engineer Aerospace Engineer NASA Ames Research Center NASA Ames Research Center Moffett Field, California 94035-1000, U. S. A.
Moffett Field, California 94035-1000, U. S. A.
Tel: 650-604-6678, Fax: 650-604-2238 Tel: 650-604-3997, Fax: 650-604-2238 E-mail: kroth@mail.arc.nasa.gov E-mail: rgreenman@mail.arc.nasa.gov ABSTRACT pressure coefficient, Cp =- ( P - P=)/ qoo The high-lift performance of a multi-element airfoil was pressure difference C e .,is optimized by using neural-net predictions that were trained chord C using a computational data set. The numerical data was gener- D drag force ated using a two-dimensional, incompressible, Navier-Stokes L lift force algorithm with the Spaiart-Allmaras turbulence model. Because M pitching moment it is difficult to predict maximum lift for high-lift systems, an ol overlap empirically-based maximum lift criteria was used in this study 1 2 to detemaine both the maximum lift and the angle at which it q_ freestream dynamic pressure, q_ - _o_V_ occurs. Multiple input, single output networks were trained t'ms root-mean-square p_ V .oc using the NASA Ames variation of the Levenherg-Marquardt Reynolds number, R e _ = -- Rec algorithm for each of the aerodynamic coefficients (lift, drag, V freestream velocity la_o and moment). The artificial neural networks were integrated (I angle of attack with a gradient-based optimizer. Using independent numerical deflection angle simulations and experimental data for this high-lift configura- coefficient of viscosity g tion, it was shown that this design process successfully opti- mized flap deflection, gap, overlap, and angle of attack to density P maximize lift. Once the neural networks were trained and inte- grated with the optimizer, minimal additional computer Subscripts resources were required to perform optimization runs with dif- f flap ferent initial conditions and parameters. Applying the neural max maximum networks within the high-lift rigging optimization process s slat reduced Me amount of computational time and resources by oo freestream value 83% compared with traditional gradient-based optimization pro- cedures for multiple optimization runs.
INTRODUCTION The design of an aircraft's high-lift system is a crucial part NOMENCLATURE of the design phase of commercial and military airplanes since Ca drag coefficient, C a - D/(q_c) this system controls the takeoff and landing performance. The Ct lift coefficient, C l - L/(q=c) importance of a well designed high-lift system is seen by C m moment coefficient, C,. -- M/(q=c 2) increased payloads which also increase the operational 1 Copyright © 1998 by ASME
flexibility byextending ranges andby decreasing take-off and
landing distances. Traditionally, high-lift desi_s have been accomplished by extensive wind tmmel and flight test programs which are expensive and difficult due to the large design space.
Recently, computational fluid dynamics (CFD) has been incor- porated in high-lift design (Ying, 1996). For high-lift applica- tions, CFD can also be expensive because the entire design space is large, grids must be generated around geometrically- .----_"Rk- _. _'¢ gap complex l-figh-lift devices, and complex flow phenomena must wing chord line " - deflecti-r
be resolved. In order to achieve optimum, rapid designs, new .IL"-q o
tools for speedy and efficient analysis of high-lift configurations flap undeflected overlap -"_._ are required. For these tools to be effective, they need to be functional in all areas of design including wind tunnel, CFD, b) Definition of flap rigging parameters" and flight.
Figure 1 Flap-Edge Geometry.
Artificial neural networks am a collection (or network) of tunnel test was used to train neural nets, the results had a predic- simple computational devices which are modeled after the tive accuracy equal to or better than the experimental data. The architecture of biological nervous systems. The ability of neural success of the NASA Ames neural network application for networks to accurately learn and predict nonlinear multiple wind-tunnel data prompted this current study (Greenman, 1998) input and output relationships makes them a promising tech- to use optimization with neural networks to optimize high-lift nique in modeling nonlinear aerodynamic data. Computational aerodynamics of a mnlti-element airfoil.
fluid dynamics in conjunction with neural networks mad optimi- zation may help reduce the time and resources needed to accu- This paper describes a process which allows CFD to impact rately define the optimal aerodynamics of an aircraft including high-lift design. This process has three phases: 1) generation of high-lift. Essentially, the neural networks will reduce the the training database using CFD; 2) training of the neural net- amount of data required to define the aerodynamic characteris- works; and 3) integration of the trained neural networks with an tics of an aircraft while the optimizer will allow the design space optimizer to capture and search the high-lift design space. In to be easily searched for extrema.
this study, an incompressible two-dimensional Navier-Stokes solver is used to compute the flowfield about the three-element Recently, neural networks have been applied to a wide airfoil shown in Figure 1. The selected airfoil is a cross-section range of problems in the aerospace industry. For example, neu- of the Flap-Edge model (Storms, 1997) that was tested in the 7- ral networks have been used in aerodynamic performance opti- by 10-Foot Wind Tunnel No. 1 at the NASA Ames Research mization of rotor blade design (LaMarsh et al., 1992). The study Center. Within the CFD database for this flap optimization prob- demonstrated that for several rotor blade designs, neural net- lem, there are two different slat deflection settings and for each works were advantageous in reducing the time required for the of these, 27 different flap riggings (refer to Figure lb) are com- optimization. Failer mid Schreck (1995) successfully used neu- puted for ten different angles of attack. The neural networks are ral networks to predict real-time three-dimensional unsteady trained by using the flap riggings and angles of attack as the separated flowfields mad aerodynamic coefficients of a pitching inputs and the aerodynamic forces as the outputs. The neural wing. It has also been demonstrated that neural networks are networks are defined to be successfully trained to predict the capable of predicting measured data with stffficient accuracy to aerody_mmic coefficients when given a set of inputs that are not enable identification of instranmntation system degradation in the training set, the outputs are predicted within the experi- (McMillen et al., 1995). Steck and Rokhsaz (1997) demon- mental error. The experimental error of the total lift coefficient strated that neural networks can be successfully trained to pre- (C t) is +0.02 for C t <0.95Ct_ = and +0.06 for C2 > 0.95C_, .
dict aerodynamic forces with sufficicnt accuracy for design and Finally, the trained neural networks are integrated with the opti- modeling. Rai m_d Madavan (1998) demonstrated the feasibility mizer to allow the design space to be easily searched for points of applying neural networks to aerodynamic design of turboma- of interest. It will be shown that this enhanced design process chineD' airfoils.
minimizes the cost and time required to accurately optimize the high-lift flap rigging.
Neural networks have been used at NASA Ames Research Center to mininlize the an_ount of data required to define the A brief description of the training set generation is pre- aerodynamic performance characteristics of a wind tunnel sented in the next section, including grid generation, the govern- model (Jorgensen and Ross, 1997 and Ross et at., 1997). It was ing equations, maximum lift criteria, and the flow solver. Next, shown that when only 50% of the data acquired from the wind the neural network training is discussed followed by the optimi- 2 Copyright (q) 1998 by ASME
zation process. Theresults arethenpresented, fromwhich the
effectiveness of optimization withneural networks asa toolto ,._;-',, ,, ;............ :----:-. i------.----_. ..... ..... i I i i _. _t;¢,,',i.',,/,;,.',, ,, ,, , _.'-', '_ __.--r--!- -T'"r--'"; '-+ ........... ] i 7:7rli.I.','7/:7',','/,,7 _, reduce resources required in aerodynamic design isdiscussed.
' _-_'j *-"'-'" _......... :--- ' ..... _-"4. 1_,'; '1I""'t'_9'1""'*; i \ ", , "',.- --: . ---Z ..... "'"-'_---:- -___.,?t:',,,o,,,,,,,/,".t.', "$ '_<'-t" 4 ....... .: ....... ___-.:--.f_..._ ..... --_..:I_ :,:.:,_;_I:_<,,_,., ,_,;>', TRAINING SET GENERATION Z"_ >,,.c-_ >-_.,-. :-:±::.,. "-i ..... _......... q'- ,,- _'-_.,;_',,,,,_z,_' _-f,-_ .z_ <':_'_ _;4: i::T_-::_:-- _--::-:-;-:::-=:-: ..... :::::-t" -:---_:l_._;;_r,;'_}__._/_';'; ' Geometry Definition Extensive wind-tmmel iIwestigations (Storms, 1997) have -: " ;_>;._z -Z 'z:, been carried out for the Flap-Edge geometry shown in Figure 1.
The model is a three-element airfoil consisting of a 12%c LB- 546 slat, NACA 632-215 Mod B main element and a 30%c Fowler flap where c is chord and is equal to 0.761 m (30.0 inches) for the undeflected (clean, all high-lift components stowed) airfoil. As mentioned, two-different slat deflection angles that are computed, six and twenty-six degrees. Each slat Figure 2 Grid around three-element airfoil (every other has a gap of gaPs = 2.0%c and an overlap of ol s = -0.05%c. In point shown for clarity).
this present study, only the results of the six-degree slat deflec- tion data set are presented (detailed results for 6, = 26.0 ° are presented by Greenman (1998)). For the computational data Governing Equations and Numerical Methods base, 27 different flap riggings are created for each slat configu- In order to obtain solutions for the computational training ration. The flap riggings are combinations of the following flap data, 54 configurations are solved at 10 different angles of ira deflection, _ p, and overlap defined in Figure lb. The flap attack. This study is performed for two-dimensional flows since deflection angles are 8f = 25.0 ° , 29.0 ° , and 39.5 ° . The three it is less computationally intensive than three-dimensional prob- gap settings are gapf = 1.50, 2.10, and 2.70%c whereas the over- lems and allows the itwestigation of many parameters. The lap settings are o{f = 0.40, 1.00, and 1.50%c. All gap mad over- incompressible Navier-Stokes equations in two-dimensional lap values in this paper are expressed in terms of percent clean generalized coordinates are solved using INS2D-UP (Rogers chord, %c. The range of angle of attack varies from and Kwak, 1990, 1991) flow solver. This code has been used 0.0 ° _<a _<22.0 ° andRe c = 3.7 million in this study.
extensively to predict high-lift multi-element airfoil flows.
INS2D-UP uses an artificial compressibility approach to couple Grid Generation the mass and momentum equations. The convective terms are The grids around the tlaree-element airfoil are generated differenced using a third-order accurate upwind biased flux- using OVERMAGG (Rogers, 1997) which is an automated splitting. The equations are solved using a generalized minimum script system used to perform overset multi-element airfoil grid residual implicit scheme. Since the flow is turbulent, the Spalan- generation. OVERMAGG takes as input the surface definition Allmaras (Spalart and Allmaras, 1992) turbulence model is used of the individual elements of the airfoil. Then it creates a surface in this study for closure. The Spalart-Allmaras turbulence model grid for each individual element by generating and redistributing has been successfully used to compute flowfields associated points from the given surface definition. It calls the HYPGEN with high-lift multi-element airfoils (Rogers, 1993 mid code (Chan et at., 1993) to generate volume grids about each Dominik, 1994).
element. The finite difference volume grid is generated in the normal direction of the surface by solving a set of hyperbolic Maximum Lift Criteria partial differential equations. OVERMAGG also automatically The determination of maximum lift is one of the most calls the PEGSUS code (Subs and Tramel, 1991) to tmite the important results of any high-lift wing design study. Figure 3 individual meshes into an overset grid system which is the final shows the computed lift coefficient versus angle of attack for output of OVERMAGG.
one high-lift setting. The solid symbols show that the computed solutions do not display the characteristic increase m c t with Figare 2 shows the grid system that is used. A grid resolu- increasing angle of attack up to Ct, " . For angles of attack tion study (Greenman, 1998) is conducted to determine the grid beyond that point, the lift coefficient s_ould decrease. Valarezo density required to solve the physical flow features. As a result, and Chin (1994) reported a hybrid method that couples cost- a total of 121,154 grid points are used consisting of a 242 x 81 effective computational fluid dy_amics teclmology with empiri- C-grid around the slat; a 451 x 131 C-grid around the main ele- cally-observed phenomenon in order to predict maximum ment; and a 351 x 121 embedded grid around the flap wllich is lift (C_) for complex multi-element wing geometries. Their used to help resolve the merging wake in this region. The nor- semi-empirical Ct_ criteria for multi-element airfoils or mal wall spacing for all grids is 5 x 10 -6 chords.
wings, designated t_e pressure difference role, is applied to the computational training data set. The pressure difference role 3 Copyright (d) 1998 by ASME Table 1 Design Space for 6 s = 6.0 degrees 5f = 25.0 deg. -- 29.0 deg.
8f = 39.5 deg.
0.4 1.0 1.5 N 0.4[ 1.0115/ 0.41 1.011.51 Lower Upper Design Bound Bound Variables
,.;li iili i: INi]
,-< NNN
25.0 38.5
qi: iNiiNiil
gap]- 1.50 2.70
-NN:N]
overlap]- 0.40 1.50 a) Method 1 o_ 0.0 10.0 N! o.41 1.5
1.o ).5
:::5:;::::::::::'J ,.,.,.,.,.,.,:.,.
f 'a_`.5. 1,%0"4 _ll'0 ;';t 1= !iiii|!iii!i lOI illli!!
I I .tf_ li .............
2. (4) l_iiil ,,,,""',
:'! N!i!!i 14)1
coefficient. The design variables in tiffs study are chosen to be 2:;2:1:2:2:1:) the flap deflection, gap, overlap, and angle of attack. The bounds
,iiSiNIz61
2.: i_ii_!i!i!! 16 ! N!!
on the design space (shown in Table 1) are chosen to be the same as the design space that are used to train the neural net- b) Method 5 o.41 1.ol 1.5I works with the exception that for optimization cases without constraints, the angle of attack is bounded to et --- 10.0 ° since i_ 0'4' 1"01 1 1
•
i_i_i_.::i_i_i: 12 this is near the range where maximum lift is predicted to occur by the pressure difference role for most of the configurations. To
1 t22t
123 l
start the optimization, the initial values of the design variables are arbitrarily chosen.
c) Method 9 Method 9. Method 9 is used to train the neural networks cases that are used to train the mural Figure 7 Computational which are integrated with the optimizer. Method 9 contains only networks.
74% of the entire configurations in the training set (Figure 7).
Five different optimization runs are shown in Table 2. Each of these runs has different initial or starting values (orig) of the data is presented to the neural networks to lean0 required to design variables (DV). Gradient based optimizers do not guaran- train the neural networks. The results of this learning curve are tee that the maximum which is found is the global maximum of presented by Greenman (1998). It was determined that 250 iter- the design space; it only guarantees an improvement. Thus, dif- ations was optimal for this stud)'.
ferent starting values of the design variables are used to search the entire design space. Tile first optimization run, 9-A, has the Even though the computational database that is used is initial design variables set to the lower bounds. Whereas, the sparse, a study (Greenman, 1998) was conducted to see how second run, 9-B, has the initial values set to the upper bounds of much further the training set can be reduced and still allow the tile design space. In the ttfird run, 9-C, the initial conditions are neural networks to predict within the acceptable error. Several set to the average value of the lower and upper bounds. The last subsets of the computational data were used to train the neural two runs have arbitrary initial values to test different regions of nets. It was shown that carefully selecting configurations to omit the design space. With this optimization procedure, the design from the training set, neural networks can be trained with only space can he easily searched with several optimization runs 50-74% of the entire data set to accurately predict the aerody- because each run only requires several seconds of CPU time. A namic characteristics of a multi-element airfoil (Greemnan, total of 28.6 CPU seconds are used for these five optimizatiun 1998). Method 1 designates the training set which contains all runs.
the computed data (Figure 7). Figure 7 shows additional training methods that are successful in training the neural networks and In this study, the optimizer found 2 different maximums.
that are presented in this paper. Here, the shaded boxes represent The smaller of the two maximums is fotmd using the initial the cases that are in the training set whereas the numbers in tile design variables of Runs 9-B through 9-E. Tile modified high- wtfite boxes and in the parentheses are the cases that are omitted lift rigging is 3f = 38.5 ° , gap/= 2.04%c, o(f = 1.50%c, and from the training set.
(x = 10.0 ° mid has Ct = 4.11. The other maximum for this particular study is just slightly higher at C_ = 4.13. The modi- High-Lift Flap Setting Optimization fied values of the design variables for this case are 6f = 38.5 ° , The high-lift system is optimized by maximizing the lift gapf= 2.01%c, o!f= 0.56%c, and o_ = 10.0 ° . The flap deflec- 6 Copyright © 1998 by ASME Table 2 Optimization Results with Method 9 as the Training Set
A%
CPU CI CI A% Cl Cl A% diff
Run DV
orig rood orig orig orig rood rood rood rood (sec) NN INS2D NN INS2D 1NS2D 9-A _f 25.0 38.5 2.04 2.04 0.0 4.13 4.03 2.48 -14.8 6.9 gapf 1.50 2.01 olf 0.40 0.56 o_ 0.0 10.0 9-B _f 38.5 38.5 3.54 3.56 -0.56 4.11 4.00 2.75 -14.5 3.3 gap/ 2.70 2.04 olf 1.50 1.50 10.0 I0.0 9-C _f 32.0 38.5 3.19 3.20 -0.31 4.11 4.00 2.75 -14.5 6.9 gap/ 2.10 2.04 olf 0.95 1.50 ot 5.0 10.0 9-D _f 30.0 38.5 3.02 2.96 2.02 4.11 4.00 2.75 -14.5 5.5 gap./, 1.90 2.04 olf 0.75 1.50 4.0 10.0 9-E _f 27.0 38.5 2.51 2.47 1.62 4.11 4.00 2.75 -14.5 6.0 gap/ 2.10 2.04 olf 0.50 1.50 o_ 2.0 10.0 tion for both instances is optimal at the upper bound. The mod- predicted and computed C l are compared mad the percent dif- ified gaps are free variables (the variable lies between the upper ference (k%) is shown in Table 2. The initial configurations and lower bounds) m_d close to each other, whereas the over- have lower errors than the modified configurations. In Run 9-A laps are quite different. The smaller maximtun has the overlap there is zero error and only one case has an error greater than at the tipper bound whereas the larger maximum has it as a free 2%. Modified configurations have prediction errors greater variable. Both configurations have the magle of attack to be than 2%. The pressure difference rule is applied to these cases optimal at the upper bound.
to detem-tine if the modified configurations have a Cp,_s r less than the acceptable value. Examining the outcome, sho_s that The accuracy of the neural network prediction is tested for the pressure difference exceeds the allowable value of both the initial mad modified configurations by generating fl_e Cpa,_ _ = -13.0. All the pressure differences are equal to or appropriate grid and computing the INS2D solution. Then the greater than Ced,/_ = -I 4.5. Some CFD training data may be 7 Copyright g) 1998 by ASME Table 3 Constrained Optimization Results for Method 9 as the Training Set A% AC,, CI CI A% Ct Cl A% diff diff CPU
Run DV
o__,,ri_' mod orig orig ori g mad rood mod rood mad (see) NN IN$2D NN INS2D NN 1NS2D 3.19 3.20 -0.31 3.94 3.86 2.07 -13.0 -13.0 27.3 9-C- _f 32.0 37.5 ACp gapf 2.10 2.08 olf 0.95 0.40 ot 5.0 9.0 3.18 3.20 -0.63 3.94 3.92 0.51 -13.0 -13.4 26.1 9-C- _if 32.0 38.5 opt gapf 2.10 1.5 o!f 0.95 0.4 ft 5.0 8.30 non-physical at the tipper bound of the angle of attack since the racy Again, the neural network that predicts the lift coefficient bound on angle of attack is chosen to be an average value of is trained with the data set that includes the data points that are where maximum lift occurs. Consequently, the neural networks at or below the maximum lift. The optimization runs are again are not properly trained to predict the aerodynamics in this constrained mad the best improvement is shown in Table 3 range. denoted by Run 9-C-opt. The values of the modified design vari- ables are different for the flap deflection, gap, and angle of Constrained O_Dtimization. In order to test whether the attack and are the same for the overlap that in the previous case.
accuracy would get better if the modified configurations were The modified lift coefficient predicted by the neural network restricted within the empirically predicted pre-stall range, the happens to be the same as in the previous optimization run, however, the INS2D value of the modified coefficient is differ- upper bound on the angle of attack design variable is removed.
Instead a constraint is placed on the value of the pressure differ- ent and the error is reduced to only 0.51%. Thus, by constrain- ing the design space that the optimizer is allowed to search and ence, Cud >-13.0. An additional neural network is trained with flap _eflection, gap, overlap, and angle of attack to predict by adding one data point near maximum lift to the training data, the pressure difference. In this case, the entire training data is the prediction error is reduced and all constraints are met. The used to compute C_ , whereas the neural networks that com- predicted and actual pressure difference are close and differ by t'di# pure the aerodynamic coefficients are trained with data only only 0.4. It should be noted that the CPU time required to run a including pre-stall data that is predicted by the pressure differ- constrained optimization run is increased, however, it is still less than 30 seconds as shown in Table 3.
ence rule. The design variables of the optimization runs remain the same as does the objective ftmction. The results of the case that fotmd the best improvement by the optimizer is shown in To get a better understanding of the flow physics, the pres- sure distribution of the modified and oriNnal configurations for Table 3 for Run 9-C-ACp. The modified design variables are optimization Run 9-C-opt are examined. Figure 8a shows the _Sf = 37.5 ° , gapf= 2.08%c, o/f= 0.40%c, and ot = 9.0 ° . The modified angle of attack is lower than in the previous case that modified and original flap positions in relation to the main ele- specified the upper bound to be a = 10.0 ° . The neural network ment trailing edge. Figure 8b shows the pressure distribution of predicted the pressure difference value to be exactly what is cal- the slat, main, and flap elements in a solid line for the modified culated with the INS2D solutio_t and predicted the modified lift configuration. The original configuration was initially at coefficient to be higher than 2% the actual INS2D value. ct = 5.0 ° (plotted in a dotted line) but in order to compare the pressure distributions, the original colffignration is also plotted To fi_rther reduce the prediction error in the modified lift at c_ = 8.3 ° (in a dashed line). The basic shape of the Cp curves coefficient, the INS2D data from this optimal case is added to are similar for all elements for both configurations. The flow is the training data. The neural networks are then re-trained with attached for all elements. The suction pressure on the modified this additional infommfion in hope that it will improve the accu- elements are clearly larger than the origi_ml configuration result- 8 Copyright (CA 1998 by ASME 14U 02 0 _ --T T NN optimization ...... Main ----- Traditional ---- Mod Flap 0.10 - -- - Orig Flap III2ZI::-:: ::::...................
0.00 Y 6o -0.10 .,,a -0.20 2O Method 5 * , L , [ , i , -0.30 0 ____1 1 i i _1 I I __ 0.80 0.90 1.00 1.10 1.20 1.30 0 1 2 3 4 5 6 7 8 x/c Optimization Rtms a) Optimized Flap Setting -18.0 -- Mod -16.0 - Figure 9 Comparison of CPU time required for traditional Orig _=5.0 - - - Orig (_ = 8.3 -14.0 - and neural network optimization procedures.
-12.0 - -10.0 to find the greatest improvement. This cm_ be very computation- ally expensive in traditional optimization.
Cp -8.0 Computational Resources -4.0 i._.
The advantage of using neural networks in the optimization -2.0 ....... ?' '_ process versus the traditional optimization process is the turn
o
0.0 around time and the CPU time that is saved for many optimiza- tion runs. In the traditional optimization process, every time that 2.0 the design variables are perturbed, the gradient needs to be cal-
..... i ..... i' 4"00.3 -0.1 0.1 0 3 0.5 0.7 0 9 1.1
1.3 culated to determine the search direction. In order to calculate X/C the gradient, a grid needs to be generated and the aerodynamic b) Pressure Distribution coefficients must be calculated by solving the flowfield with Figure 8 Optimization results for run 9-C-opt (flap INS2D. Even though, the traditional optimization method will settings denoted in Table 3).
have shorter turn around time and CPU time when doing one or two optimization nms, there is no guarantee that one or two optimization runs will find the global maximum. On the con- ing in greater lift. There are interesting features on the original wary, the neural networks will have less overall turn arotmd time and modified flap elements. The sharp spike at the trailing edge and CPU time for many optimization nms and there is no major occurs from the sharp point at the trailing edge of the flap geom- increase in overall turn around or CPU time for additional runs.
etry. The numerical grid comes to a sharp comer at the trailing Once the neural networks are trained, only 5-10 seconds are edge, the flow must accelerate at this point causing the pressure required for each additional optimization run. The CPU time to drop. The multiple spikes that are located at the leading-edge that is used in this optimization study for the different training of the flap element are associated with the original definition of methods used is shown in Figure 9. Also plotted in this figure the geomeUT. The flap at this region is faceted due to the high are the calculated CPU time that would have been used in tl_e ct_,ature. The pressure spikes are representative of what the traditional optimization process. The CPU time for the tradi- flow is actually doing. The flow is tunfing around at these facets tional method is estimated by using the sanle number of func- and accelerating.
tion calls that is used in the neural network oplimization procedure. Then for each iteration it is estimated that the CPU BENEFITS OF NEW PROCESS time will consist of 4.3 seconds to generate a grid and 600 sec- The aerodynamic design space of a nmlti-element airfoil is onds (on a Cray C90) for each flow solution. If more than three very complex and may have many local maximums and mini- optimization runs are executed, then the neural network optimi- mums. When a gradient-based optimizer is used to search the zation procedure should be used. The neural network optimiza- design space, many starting points need to be examined in order 9 Copyright © 1998 by ASME Table 4 Neural Network Optimization Procedure Cost Table 5 Traditional Optimization Procedure Cost CPU Number of Wall Clock Total Cost Generating Training Optimization Total Cost Method Method Hours Training Set CPU time CPU time 8 hour jobs (hours) (dollars) (dollars) CPU (hours) (seconds) (seconds) 94.31 11.79 276.45 30,261.21 40.08 281.0 263.0 6466.50 93.13 7.83 272.96 29,876.98 20.78 207.0 93.5 3353.50 110.31 13.79 328.30 35,391.83 29.69 232.0 176.6 4790.16 required to train each method and used to optimize all five opti- tion procedure curves are nearly flat. Thus, the major mization rims (see Table 4) must also be added to the total wall contributor to the CPU time in the neural network optimization clock time and the charged CPU time. The total cost of the neu- is traimng the neural networks to learn and to predict the aero- ral network optimization procedure is shown Table 4. The major dynamics of the airfoil. Many more optimization runs can be element in the cost is the time and computer resources required executed with this procedure without requiring large additional to set-up the training matrix data. Consequently, it is vet 3 , amount of CPU time. On the other hand, the traditional optimi- important to determine the level of prediction accuracy that is zation procedure will continue to increase at a fairly linear rate required and to choose the proper method to train the neural net- as shown.
works.
Cost Analysis Second, the cost of the traditional optimization procedure is Another advantage of using the neural network optimiza- calculated with the same assumptions. The wall clock time and tion procedure is reduction of cost. There are many factors con- the CPU hours charged are calculated based on the number of tributing to the total cost of a research job including the cost of iterations (or gradient calls) that are made by the optimizer for the engineer support, computer resources, and wall clock turn each optimization ran. For each method that is used in the neu- around time. One of the largest contributors to turn around time ral network optimization procedure, the traditional optimization is waiting for a computer job to he completed especially if the cost is calculated for the same five optimization starting runs.
job executes within a batch queue. The average tum around time The total turn around (wall clock) time that the engineer waits for the computers used in this study at the Numerical Aerospace for the job to be finished is multiplied by $96.15 and is added to Simulation Facility (NAS) at NASA Ames Research Center is the total CPU hours that are charged. The traditional optimiza- 23.45 hours for an eight-hour queue job.
tion procedure is performed on the Cray computer in the batch queue. This is one of the reasons that the cost is higher than the To calculate the cost that is related to the two types of opti- neural network optimization procedure as shown in Table 5.
mization procedures considered, it is assumed that an experi- enced engineer is executing both optimization processes. This The total costs are compared in Figure 10 for the two opti- engineer is familiar with the different components to each pro- mization procedures. For five optimization runs for each train- cess such as grid generation, flow simulation, neural networks, ing method, the neural network optimization procedure does and optimization. The set-up time is assumed to be equal for cost less. Again, if only one or two optimization runs are per- both processes. The engineer is a full time equivalent of formed, then the traditional optimization procedure would cost $200,000 per year and there are 2080 working hours in a year.
less, however, for multiple runs, the neural network optimiza- Thus, there is a charge of $96.15 per hour for an engineer.
tion procedure uses less resources. The biggest advantage now Another expense which must be considered is computer is that many more optimization runs can be performed with the resources. For this comparison, assume the cost of a computing neural network optimization procedure while oNy adding sec- hour is $39.00.
onds to the CPU time and turn around time.
First, the cost of the neural network optimization procedure The neural network optimization procedure should be used is calculated. A grid is generated for each configuration for design because several designs with different constraints or included in the training method and soltltions are calculated for design space can be considered without driving the cost and tun_ 10 different angles of attack for each configuration. The grid around time up. Also, once a design is chosen, the design space generation requires 4.3 CPU seconds per grid and 269.0 CPU can be altered and the optimization procedure can now be per- seconds per flow solution (the co_wergence time is low since formed again with minimum additional cost and turnaround these solutions are below maximum lift). The CPU time time.
10 Copyright © 1998 by ASME REFERENCES 40000 Chan, W. M., Chui, I. T., aJ1d Burdng, P. G., 1993, "User's NN 35000 Traditional Manual for the HYPGEN Hyperbolic Grid Generator and the HGUI Graphical User h]tefface, "NASA TM 108791.
30000 Dominik, C., 1994, "Application of the Incompressible Navier-Stokes Equations to High-Lift Flows", AIAA Paper 94- 25000 1872.
v 20000 Failer, W. E. and Schreck, S. J., 1995, "Real-Time Predic- o tion of Unsteady Aerodynamics: Application for Aircraft Con- 15000 trol and Maneuverability Ezflmncement," IEEE Transactions on NeuralNem,orks, Vol. 6, No. 6, pp. 1461 - 1468.
10000 Gill, P. E., Murray, W., Saunders, M. A., and Wright, M. H., 1994, "User's Guide for NPSOL 5.0: A Fortran Package for Nonlinear Programming," Dept. of Operations Research, Stan- ford University, TR SOL 94, Stanford, CA.
1 5 9 Greenman, R. M., 1998, "Two-Dimensional High-Lift Method Aerodynamic Optimization Using Neural Networks", Stanford UniversiO, Ph.D. Dissertation.
Figure 10 Comparison of total cost for the neural Greenman, R. M., Chetmg, S., mad Tu, E. L., 1998, "Cou- network and traditional optimization procedure.
pled Navier-Stokes and Optimizer Analysis of a Transonic Wing," Journal of Aircraft, Vol. 35, No. 3, pp. 362-369.
Jorgensen, C. C. and Ross, J. C., "System and method for CONCLUSIONS Modeling the Flow Performance Features of an Object," U. S.
An enhanced design process was developed wlfich inte- Patent No. 5,649,064, July 1997.
grates neural network and optimizer technologies together with LaMarsh, W. J., Walsh, J. L., and Rogers, J. L., 1992, a computational database. The process is modular, allowing "Aerodynamic Performance Optimization of a Rotor Blade insertion of emerging neural network, optimization, and CFD Using a Neural Network as the Analysis," AIAA Paper 92-4837.
technoloNes within its framework. This design process was McMillen, R. L., Steck, J. E., and Rokhsaz, K., 1995, tested for a typical high-lift design problem to optimize flap rig- "Application of an Artificial Neural Network as a Flight Test ging for maximum lift. Initial studies showed that although opti- Data Estimator," Journal of Aircraft, Vol. 32, No. 5, pp. 1088 - mization could be conducted using a sparse traii_ing dataset, 1094.
unconstrained optimization of the high-lift system produced Norgaard, M., Jorgensen, C. G., and Ross, J. C., 1997, unacceptably high errors. Due to the complexity of the high-lift "Neural-Network Prediction of New Aircraft Design Coeffi- flow physics near the maximum lift condition, an empirically cients," NASA TM 112197.
based constraint, which identifies COlUfigurations at the maxi- Rai, M. M. mad Madavan, N. K., 1998, "Application of Arti- nmm lift condition within the computational database, was ficial Neural Networks to the Design of Turbomachinery Air- required in order to achieve accurate neural net predictions for foils," AIAA Paper 98-1003.
this design problem. Using the empirical constraint together Rogers, S. E., 1993, "Progress in High-Lift Aerodynamic with an iterative optimization procedure which re-inserted the Calculations," AIAA Paper 93-0194.
optimized configuration into the training database and repeated Rogers, S. E., 1997, "Manual for the OVERMAGG Script the optimization produced an optimal configuration with only System," NASA Ames Research Center.
0.5% error. A cost analysis was conducted by comparing the Rogers, S. E. and Kwak, D., 1990, "'An Upwind Differenc- optimization with neural networks procedure to the traditional ing Scheme for the Time Accurate Incompressible Navier- optimization procedure. It was found that the optimization with Stokes Equations," A/AA Journal, Vol. 28, No. 2, pp. 253 - 262.
neural networks procedure resulted in a reduction of turnaround Rogers, S. E. and Kwak, D., 1991, "An Upwind Differenc- time, CPU time, and cost if more than two optimization nms ing Scheme for the Steady State Incompressible Navier-Stokes were conducted. Using the optimization procedure, the average cost reduction is 83%. Equations," Journal of Applied Numerical Mathematics, Vol. 8, pp. 43 - 64.
Ross, J. C., JoNenson, C. C., and Norgaard, M., 1997, ACKNOWLEDGM ENTS "Reducing Wind Tmme] Data Requirements Using Neural Net- This work WOtlld not have been possible without the helpfld works" NASA TM 112193.
discussions mid suggestions from Dr. James C. Ross, Dr. Stuart Spalart, P. R. and Allmaras, S. R., 1992, "A One-Equation E. Rogers, mad Dr. Charles C. Jorgensen at NASA Ames Turbulence Model for Aerodynan_c Flows," AIAA Paper 92- Research Center.
11 Cop),right © 1998 by ASME 0439.
Steck, J.E.andRokhsaz, K., 1997, "Some Applications of
Artificial Neural Networks in Modeling of Nonlinear Aerody- namics and Flight Dynamics," AIAA Paper 97-0338.
Storms, B., 1997, Private communication.
Subs, N. E. and "l_mel R. W., 1991, "PEGSUS 4.0 User's Manual," AEDC-TR-91-8.
Valarezo, W. O. and Chin, V. D., 1994, "Method of Predic- tion of Wing Maximum Lift," Journal of Aircraft, Vol. 31, No.
1, pp. 103-109.
Ying, S. X., 1996, "High Lift Challenges and Directions for CFD," AIAA/NPU AFM Conference Proceedings, CNna.
12 Copyright © 1998 by ASME