2014 14th International Conference on Control, Automation and Systems (ICCAS 2014) Oct. 22-25, 2014 in KINTEX, Gyeonggi-do, Korea Response Surface Smoothing for Wind Tunnel Testing Based on Design of Experiment with Subspace Partitioning 2* l 3 Dongoo Lee , Jaemyung Ahn and Se-Yoon Oh 1
2
Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 305-70 I, Korea (
[email protected]) Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea (
[email protected]) * Corresponding author 3 Agency for Defense Development, Daejeon, 305-152, Korea (seyoonoh
[email protected])
This paper presents a new approach in resolving discontinuities between subspaces when the subspace partitioning method is applied to measure aerodynamic coefficients. The discontinuities between response surfaces in each subspace are hard to resolve in an analytical method, but it can be smoothened by overlapping between subspaces and integrating with proper interpolation. The traditional One Factor at a Time (OFAT) method is used for gathering aerodynamic force and moment coefficient data before applying the Design of Experiments (DOE) method for the purpose of later validation of the suggested method. In each subspace, the Central Composite Design (CCD) is used to reduce experimental points. The Separated Response Surface Model compared to the Integrated Response Surface Model. Quality of fit of the Developed Integrated Model calculated with OFAT data for validation. The test model base is the lambda wing configuration of the UCAV 1303. The study was carried out at the low speed wind tunnel of the Agency for Defense Development (ADD-LSWT).
Abstract:
Keywords:
Wind Tunnel, DOE, Subspace, CCD polynomial equation, but sub-region can be sufficiently expressed as low-order response surface model. In other words, even if aerodynamic coefficients have highly non-linear properties in the whole experimental space, it can be adequately modeled with number of subspaces that have linear property [1]. Since aerodynamic force and moment coefficients of airplane have highly non-linearity property, a response surface model with limited order over whole experiment range result in a significant difference with measured data at the points. Thus, dividing whole experiment range with some subspaces make each response surface model in subspaces more accurate. There is still many practical issues for how to divide subspaces for accurate response surface model [3]. The practical issues for applying subspace partitioning method shown in Table
1. INTRODUCTION
In contrast to traditional productivity paradigm of wind tunnel testing until late 20th-century that was focused on high volume and improving quality of individual data points, Design of Experiment (DOE) based wind tunnel testing concentrates on improving precision of response model developed from measured data [2]. DOE is actively applied to wind tunnel testing at NASA Langley Research Center for the purpose of reducing total operating time and cost [2-4]. In Korea, increased demand of wind tunnel testing for developing complicated flight vehicles and insufficient wind tunnel testing facilities also leads to utilize DOE actively. In this study, an efficient method to model aerodynamic force and moment coefficients through wind tunnel testing based on DOE is represented. Subspace partitioning and Central Composite Design (CCD) are utilized to determine experimental sections and points. Response Surface Modeling (RSM) is applied to quantify the relationship between experimental variables that are angle of attack and side-slip angle in each subspace. Response surface models in each subspace integrated into the final surface resolving discontinuities between subspaces.
l.
Table 1 Practical issues for applying subspace partitioning
------I 2
2. METHODOLOGY
3
2.1 Subspace Partitioning
Subspace partitioning is a very effective method to model unexpected responses for coefficients modeling. The characteristic of subspace is that generated low-order response surface in the subspace is more accurate than non-divided model that should cover bigger range of interest. This is because it is hard to model whole experimental space with low-order 978-89-93215-06-9 95560/14/$15 ©rCROS
4
Issues for subspace partitioning How many subspaces should we divide whole experimental space to adequately fit coefficients? How do we select the subspace boundaries? By what criteria should the adequacy of a fit be judged? How should the forces and moments be predicted on subspace boundaries if there are RSM discontinuities across those boundaries?
In this paper, a new approach to address the issues labeled as 4 in Table 1 is proposed. 207
In the overlapped area, two surfaces are integrated base on linear interpolation. The interpolation method could vary with weighting decisions for the surfaces.
2.2 Central Composite Design
Central Composite Design (CCD) is utilized to decide experimental points in each subspace. CCD is one of the efficient methods to reduce experimental time and cost to develop response surface model compared to traditional factorial design. Experiment executed in factorial, axial and center points with some replications in a center point to improve precision of the experiment. Then second order response surface models are developed.
1:;�r1ap(x)
= I(
x- Pi P,+1 - P,.
J+ ( 1+1
Pi+1 -X P,+1 - P,
J
(2.3.3) In this paper, values in the overlapped area are calculated as equation (2.3.3) and the area is described in Figure l.
2.3 Integrating Response Surfaces
Since the models in each subspace developed independently, there always exists discontinuities between subspaces. Hocking proposed simple method to resolve discontinuities between subspaces by fixing one variable making the response surface as a function of one variable [5], and the method applied in wind tunnel testing [3] but it is not applicable to making three-dimensional integrated response surface model. However, modeling aerodynamic coefficients in three-dimensional space is necessary to analyze aircraft effectively. The existing subspace partitioning method has a limitation for integrating response surfaces. Smoothed response surfaces should satisfy the following conditions. When functions of each response surface labeled as
Fig. 1 Overlapping between subspaces 3. EXPERIMENT 3.1 Experiment Device
The Blended Wing Body Unmanned Combat Air Vehicle (BWB-UCAV) which is slightly modified from UCAV 1303 is used for the wind tunnel testing. The specification and shape of the plane are presented in Figure 2. The experiment was carried out at the low speed wind tunnel in Agency for Defense Development (ADD-LSWT), Republic of Korea.
J;, and there exist m subspaces in
n-dimensional space,
For all boundary (xl'···'xn), f(xl,···,x,)=f+I(XI'···'x,), i= I, ···, m-1 (2.3.1)
i = I, ···, m- I,
k = I, ···, n
•
Airfoil: NACA 64A21 0
•
Wino Span: 1000
•
Center line: 590.5
•
MAC: 352.2
•
Reference area: 0.259 m2
•
mm mm
mm
Leading edge sweep angle : 47·
•
If the number of subspaces is small so number of boundaries is small, above conditions can be satisfied by applying constraints to the response surface models, but we are unaware of the general solution of the conditions, yet. Therefore, modification to traditional subspace partitioning method is needed. Smoothing of response surface boundaries for subspaces requires the overlap of subspaces. It means the boundaries between subspaces should not a line but a surface. Overlapped surfaces make possible to integrate two surfaces into one smoothed model with some calculation such as linear interpolation. Ranges of subspaces for the integrated model set as equally partitioned in assumption of there is no prior information for partitioning. The range is different with separated model that is presented as Table 2 because it is impossible to make a subspace contains only linear part when overlap of subspaces exist. The integrated response surface guarantee a continuous-surface for whole experimental range.
•
Moment reference point
: 301.1
mm
(from apex)
Twist angle:
O·
Fig. 2 Specification of BWB-UCAY. 3.2 OFAT Experiment and Subspaces
We measured six aerodynamic force and moment coefficients for a full range of interest as getting experimental data. Traditional One Factor at a Time (OFAT) method used for the measurement. Steps for Angle of Attack and Angle of Sideslip were 1 degree and 2 degrees each. The measurement result shown at Figure 3. The acquired data used for validating an accuracy of developed response surface model and determining where to divide subspaces according to the angle of attack to maximize linearity in each subspace based on overall data tendency. Experiment range is not split with angle of sideslip to minimize the number of subspaces. Subspaces are divided at 4° and 9° of angle of attack result in total three subspaces as Table 2. As mentions in chapter 2.3, the subspace division described
208
here is for separated response surface model. For integrated response surface model, division is not based on prior data so subspaces are equally divided with angle of attack. Table 2 Range of Subspaces (Separated model) Alpha
Subspace
Beta
Low
High
High
Low
1
_3°
4°
0°
20°
2
4°
9°
0°
20°
3
9°
25°
0°
20°
CD
surfaces is in Figure 4. Red dots in figure 4 stand for data from previous screening test and blue dots represent experimental points decided by CCD.
CY
Cl
CY
Cn
Cm
0
0
0
CL
AoA
Cl
0.8 0.6 0.4
0.02 . 0.01
Fig. 4 Response surface model for coefficients with 3 subspaces.
0
0.2 0
-0.01 20
20
AoS
CL
I
20
0
CD
0
10 AoA
0
AoS
To validate response surfaces in each subspace, R-square values are calculated. Table 3 show the values. 10 AoA
0
0
Table 3 R2 value of response surface(3 subspaces)
Cn
Cm
Subspace Coefficient
o
0
AoA
o
0
AoA
Fig. 3 Measured aerodynamic force and moment data.
1
2
3
CD
0.996
0.997
0.996
Cy
0.973
0.968
0.917
CL
0.999
0.998
0.989
C1
0.993
0.990
0.904
Cm
0.906
0.926
0.869
Cn
0.934
0.853
0.768
Table 4 Significant first and second order response surface terms(3 subspaces) 2 � �2 � Response Subspace
3.3 Response Surface Model with Subspaces
The response surface models in each subspace generated by employing 'fit' function of MATLAB with second order. ANOVA('anova' function of MATLAB) is applied to find significant terms for the response surface. Table 4 identifies the significant first and second order terms identified from the ANOYA analysis. Graphical representation of second order
a
CD
1 2 3
209
a
0
a
0
0
0
0
0
0 0
0
0
Cy
0
1
0
2 3 CL
1 2 3
C1
1
0
0
0
0
0
0
0
0
0
0
3 1 2 3 Cn
0
0
0
0
0
0
0
0
0
0
, , , , , , , , , '-__-=___�=__� --;!.. .____. __________ ........-c.c ..............................;
AoA
_"_
0
2
Cm
0
0
3
0
0
1
_
"-.
Subspace 2
...�...
Subspace 3
Fig. 4 Overlapped CCD points with two variables (3 subspaces) [7]
0
0
CD(R-'ql1are:O.99862)
0
0
0
0 0
1 2
0
Subspace
0
0
0
0
0
0
0 Cl(R-square:O.90·13-1)
CL(R-square:O.99844)
3.4 Integrated Model
We develop a new approach to smooth boundaries of subspaces divided by AoA that are independently made surfaces resolving discontinuities between them. When dividing subspaces, we assumed there is no prior information about coefficient data so range of subspaces set to equal for all three(four) spaces. Basic idea of integrating is interpolation of two overlapped boundaries between subspaces but number of overlapped boundaries in a region could be three or four (or more) if we divide subspaces with more variables. Red dots in figure 5 stand for data from previous screening test (measured by OFAT) and blue dots represent experimental points decided by CCD. The configuration of experimental points as overhead-view is presented in figure 4. The exact experimental result at the CCD points were not measured in the test, so the data reproduced with interpolation from OFAT results. Since there is exactly duplicated experimental points among subspaces in overlapped CCD configuration, the final number of experimental points are 7n + 1 where n is the number of subspaces. Table 5 shows R-square values of integrated response surface model and Table 6 shows average residual values of integrated response surface model for 3-subspaces and 4-subspaces cases.
10 AoS
Cm(R-square:O.91331 )
0
10
.J
x
AoS
0
1O AoA
20
C'n(R-square:O.82745)
10 AoA
Fig. 5 Integrated response surface model for coefficients with 3 subspaces. Table 5 R2 value of integrated response surface RL value RL value Coefficient (3 subspaces) (4 subspaces) CD
0.99894
0.99616
Cy
0.92985
0.91382
CL
0.99914
0.99856
C1
0.90643
0.94826
Cm
0.92399
0.93949
Cn
0.76828
0.82125
Table 6 Average Residual of integrated response surface
210
Coefficient
Average Residual (3 subspaces)
Average Residual (4 subspaces)
CD
0.0031931
0.0018364
Cy
0.0022536
0.0008596
CL
0.0071040
0.0034729
C1
0.0017375
0.0007940
Cm
0.0029399
0.0015633
Cn
0.0007053
0.0003011
[6]
[7]
4. CONCLUSION
ACKNOWLEGEMENT
Executing wind tunnel testing with subspace partitioning combined with DOE technique resulted in significantly reduced total experiment time and cost for modelling aerodynamic force and moment coefficients. In this paper, integrating subspaces divided by one variable that is Angle of Attack is discussed. Techniques for integrating subspaces with multi-dimensional independent variables should be treated for a more general subspace partitioning case. Also, criteria for subspace partitioning and connecting subspaces should be more formulated in mathematical form to standardize the method of applying DOE based testing. Additional DOE method including CCD that is discussed in this paper could be used with subspace partitioning method. More study about integrating subspaces should be conducted considering where to partition and how to interpolate overlapped area.
This research was supported by the "Optimal Experimental Design and Modeling Techniques for Wind Tunnel Testing" funded by the Agency for Defense Development, Korea (ADD-12-01-08-18).
REFERENCES
[1]
[2]
Montgomery D. C., "Design and Analysis of Experiment", 7th ed., John Wiley & Sons., New York., 2009. DeLoach, R., "The modern design of experiments - A technical and marketing framework," 21st Aerodynamic Measurement
Technology and Ground Testing Conference,
[3]
American Institute of Aeronautics and Astronautics, 2000. DeLoach, R., and Erickson, G., "Low-Order Response Surface Modeling of Wind Tunnel Data Over Truncated Inference Subspaces,"
41st Aerospace Sciences Meeting and Exhibit,
[4]
American Institute of Aeronautics and Astronautics, 2003. DeLoach, Richard. "Analysis of Variance in the Modern Design of Experiments." American
Institute of Aeronautics (AIAA) 1111 (2010): 2010. [5]
and
D. G. Lee, J. M. Ahn, and S. Y. Oh, "Design of Experiment based wind tunnel testing for modeling aerodynamic force and moment coefficients using subspace and central composite design", Korean Society for Aeronautical Space Sciences Conference, 2013. D. G. Lee, J. M. Ahn, and S. Y. Oh, "Design of Experiment based wind tunnel testing for modeling aerodynamic force and moment coefficients using subspace and central composite design", The Korea Institute of Military Science and Technology Conference, 2014
Astronautics
Hocking, R.R., "Methods and Applications of Linear Models, Regression and the Analysis of Variance", John Wiley & Sons., New York., 1996.
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