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Using dual response surfaces to reduce variability in launch vehicle design: A case study
Reliability Engineering and System Safety 91 (2006) 407–412 www.elsevier.com/locate/ress
Using dual response surfaces to reduce variability in launch vehicle design: A case study Ozgur Yeniaya, Resit Unalb,*, Roger A. Lepschc b
a Department of Statistics, Hacettepe University, Ankara, Turkey Engineering Management and Systems Engineering Department, Old Dominion University, Norfolk, VA 23693, USA c Vehicle Analysis Branch, NASA Langley Research Center, Hampton, VA 23681, USA
Received 10 June 2004; accepted 12 February 2005 Available online 6 May 2005
Abstract Space transportation system conceptual design is a multidisciplinary process containing considerable element of risk. Uncertainties from one engineering discipline may propagate to another through linking parameters and the final system output may have an accumulation of risk. This may lead to significant deviations from expected performance. An estimate of variability or design risk therefore becomes essential for a robust design. This study utilizes the dual response surface approach to quantify variability in critical performance characteristics during conceptual design phase of a launch vehicle. Using design of experiments methods and disciplinary design analysis codes, dual response surfaces are constructed for the mean and standard deviation to quantify variability in vehicle weight and sizing analysis. Next, an optimum solution is sought to minimize variability subject to a constraint on mean weight. In this application, the dual response surface approach lead to quantifying and minimizing variability without much increase in design effort. q 2005 Elsevier Ltd. All rights reserved. Keywords: Uncertainty; Response surface methods; D-Optimal designs, Optimization
1. Introduction Conceptual engineering design of future launch vehicles is a multidisciplinary process containing considerable element of risk. Uncertainties from one discipline may propagate to another through linking parameters leading to significant deviations from the expected performance. Therefore, an estimate of this variability, together with the expected performance characteristic value is necessary for multidisciplinary optimization for a robust design. Robust design in this study is defined as a solution that minimizes variability subject to constraints on mean performance characteristics. This study applies the dual response surface approach to quantify variability in vehicle empty weight (without propellants) for a launch vehicle weights and sizing case study. Weight is a critical performance characteristic in * Corresponding author. Fax: C1 757 683 5640. E-mail address: [email protected] (R. Unal).
0951-8320/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2005.02.007
aerospace design since it affects performance and cost. Reducing variability in vehicle weight is also an important objective due to safety issues related to design margins such as in propulsion and landing gear.
2. Multidisciplinary conceptual design In launch vehicle conceptual design studies, system performance can generally be determined by the use of computerized analysis tools available in many engineering disciplines. However, these complex sizing and performance evaluation analysis codes utilize iterative algorithms. In many cases, they are expensive and difficult to integrate and use directly for multidisciplinary design optimization. An alternative is to construct mathematical models that approximate the relationships between performance characteristics and design variables. These approximation models, called response surface models, are then used to integrate the disciplines using mathematical programming methods and for multidisciplinary system level optimization.
408
O. Yeniay et al. / Reliability Engineering and System Safety 91 (2006) 407–412
3. Response surface methods Y s Z c0 C A second-order approximation model of the form given in Eq. (1) is commonly used in approximation model building since in many cases it may adequately model the response surface especially if the region of interest is sufficiently limited. Y^ Z b0 C
k X
bi x i C
iZ1
k X
bii x2i C
k XX
bij xi xj
(1)
i!j
iZ1
In Eq. (1), the xi terms are the input design variables that influence the response (performance characteristics) Y, and b0, bi and bij are estimated regression coefficients. The cross terms represent two-parameter interactions, and the square terms represent second-order non-linearity. There are various response surface techniques that may be utilized to sample the design space efficiently to construct a secondorder approximation model. Some of these are central composite designs [1–3] and D-Optimal designs [4,5]. Response surface methods using these designs have been applied to various multidisciplinary design optimization problems [6–8]. The main advantage is that response surface methods can aid multidisciplinary design integration, and provide rapid design analysis and optimization capability in many applications. However, in these (and most) response surface model building applications, only the mean value for the performance characteristic (Y) was studied and optimized ignoring design risk or variability due to uncertain design parameter values (noise).
4. Dual response surfaces Response surface methodology is designed to construct an approximation model for the response Y. This approximation model is then used to determine the best values of design parameters that optimize the response or the critical quality characteristic. In such problems typically the focus is on the mean value of Y where variance is assumed to be small and constant. However, if these assumptions do not hold, only constructing a response surface model for the mean may not be adequate and optimization results can be misleading. Kim and Lin [9] note that the assumption of constant variance is often not practically valid and classical response surface methods can be misleading in many cases. An alternative to classical response surface methods is the ‘dual response surface’ approach. Initially proposed by Vining and Myers [10,24], this approach seeks to build two models; one for the mean and one for the standard deviation [11–15]. Ym Z b0 C
k X iZ1
bi x i C
k X iZ1
bii x2i C
k XX i!j
bij xi xj C 3m
(2)
k X iZ1
ci xi C
k X iZ1
cii x2i C
k XX
cij xi xj C 3s
(3)
i!j
The dual response surface approach simultaneously considers the estimated mean (m) and the standard deviation (s) as a measure of design risk. The steps in the dual response surface approach can be summarized as: 1. Conduct the experiments to sample the design space using a response surface technique such as central composite design or a D-Optimal design, 2. Repeat the experiments over the range of uncertain design (noise) parameters to capture variability, 3. Compute the mean and the variance for each experiment and construct response surfaces for both the mean and variance, 4. Determine the best values of xi to optimize the mean response subject to a constraint on the variance or determine the best values of xi to minimize the variance subject to a constraint on mean response.
5. Applying dual response surfaces for a robust design Many studies in the literature successfully utilized the dual response methodology for robust design and optimization studies that considered both mean and variability. In their leading article, Vining and Myers [10] demonstrated the use of dual response surfaces in optimizing variability while satisfying conditions on a desired target. They used data taken from Box and Draper [24] for a printing machine in applying coloring ink package labels. Luner [16] utilized dual response surfaces in determining a robust combination of factor levels that reduced variability around a target using a Roman-style catapult. Menon et al. [25] applied dual response surfaces for robust design of a spindle motor. A constrained minimization of the cogging response of the spindle motor was sought, the constraint being that the variance be within a specified limit. Their results showed a 25% reduction in variance of the clogging torque. Kim and Rhee [26] utilized the dual response surfaces in a study to optimize the penetration in gas metal arc welding seeking to determine welding process parameter values to produce target value with minimized variance. Their results showed that the optimal conditions determined through using the dual response surface approach had a lower possibility of producing defective welds. Dhavlikar et al. [17] presented an application of dual response surfaces to determine robust condition for minimization of out of roundness error of work pieces for centerless grinding operation. They observed that the combined Taguchi and dual response surface approach utilized led to successful identification of optimum process parameter values.
O. Yeniay et al. / Reliability Engineering and System Safety 91 (2006) 407–412
In this study, the dual response surface approach is applied to the weights and sizing analysis of a wing-body launch vehicle case study with the objective of optimizing variability in weight.
6. Wing-body launch vehicle configuration case study to optimize empty weight An application of dual response surfaces is described for a weights and sizing study of a rocket-powered, singlestage-to-orbit launch vehicle. The vehicle has been sized to perform a 25,000 lb payload delivery to the International Space Station from Kennedy Space Center. Near term structures and subsystem technologies are assumed in its design. The vehicle has a wing-body configuration with a slender, round cross-section fuselage and a clipped delta wing. The delta wing has elevon control surfaces for aerodynamic roll and pitch control. Small vertical fins, called tip fins, are located at the wing tips for directional control and a body flap extends rearward from the lower base of the fuselage to provide additional pitch control. The purpose in this study is to determine the best values of the engineering design parameters (that satisfy aerodynamic constraints) to minimize variability in vehicle empty weight (weight without propellants). This case study involved the following steps.
The four common parameters included in aerodynamics and weights and sizing analysis were the fineness ratio (defined as the fuselage length divided by diameter), the wing area, the tip fin area, and the body flap area. The other two parameters included were ballast-weight and mass-ratio for weights and sizing analysis and angle of attack and elevon deflection for aerodynamics analysis. In this study, ballast-weight and mass-ratio were taken as the noise variables. Noise variables are those that one has little control on the values during operation. Therefore, one design goal is to reduce the variability in performance (empty weight in this case) due to the uncertain values of these parameters. The ranges for the six weights and sizing parameters are given in Table 1. Table 1 Weights and sizing parameters and ranges Range 4 10 0.5 0 0 7.75
6.2. Construct the experimental design matrix The next step is to construct an experimental design matrix to efficiently sample the design space and construct a second-order response model for four parameters. Carpenter [18] has conducted a study comparing the performance of various experimental design methods for approximation model building in terms of the quality of fit in the region studied and in terms of the number of design points required. His findings suggest that the D-optimality criterion is a good approach for constructing experimental designs for design studies using computerized engineering design and analysis codes. Other studies in the literature also found that the D-optimality criterion provides a rational means for creating experimental designs for an irregularly shaped response surfaces [6]. Therefore, D-Optimal designs are used in this study to construct an experimental design to sample the design space [2,4,8,19,20]. The JMPw software was utilized to construct the D-Optimal design matrix given in Table 2. With this design matrix, the four engineering design parameters are studied at three levels (values) as represented in coded form by K1, 0 and C1. As an example, a K1 for fineness ratio corresponds to 4 (lower bound), a 0 corresponds to 5.5 (mid value) and C1 corresponds to 7 (upper bound). These coded values are then transformed into actual parameter values to be used in conducting the analysis. 6.3. Conduct the matrix experiments
6.1. Identify design variables and feasible ranges
Design parameters Fineness ratio (FR) Wing area ratio (WA) Tip fin area ratio (TP) Body flap area ratio (BF) Noise parameters Ballast weight (BL) Mass ratio (MR)
409
7 20 3 1 0.04 8.25
In this study, all geometry and subsystem packaging of the SSV were performed using a NASA-developed geometry-modeling tool. The Aerodynamic Preliminary Analysis System (APAS) was used to determine vehicle aerodynamics. The weights and sizing analyses were performed using the NASA-developed Configuration Sizing (CONSIZ) weights/sizing package. This process was continued for the 45 rows of the D-Optimal design matrix, each of which corresponds to a vehicle design generating the data. An orthogonal array based simulation approach is used to simulate the variability resulting from noise factors [21]. For each of the 45 rows, experiments were repeated at five points corresponding to the five-point design matrix shown below to simulate the variability due to the uncertain values of the two noise variables (Ballast weight and MassRatio). From the weight analysis, data points for empty weight were obtained (Table 2). Using this data, mean empty weights (m) and corresponding standard deviations (s), as a measure of design risk due to the uncertain values of BL and MR, were computed. 6.4. Construct the second-order response surface model Least-squares regression analysis is then used to determine the coefficients of the second order, dual response surface models for both the mean empty weight
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O. Yeniay et al. / Reliability Engineering and System Safety 91 (2006) 407–412
Table 2 Analysis results
1 2 3 4 « 42 43 44 45
BL MR
K1 K1
C1 K1
K1 C1
C1 C1
0 0
FR
WA
TP
BF
EWT1
EWT2
EWT3
EWT4
EWT5
K1 K1 K1 K1 « 1 1 1 1
K1 K1 K1 K1 « 1 1 1 1
K1 K1 0 1 « K1 0 1 1
K1 1 1 K1 « 1 0 K1 1
234,798 244,264 269,903 287,841 « 216,406 225,895 236,433 241,231
266,104 277,534 308,403 329,905 « 246,297 257,754 270,443 276,283
269,950 281,649 313,455 335,853 « 250,762 262,683 275,933 281,946
309,903 324,255 363,252 390,673 « 289,312 303,939 320,256 327,754
268,058 279,592 310,966 332,920 « 248,572 260,244 273,221 279,152
and the standard deviation. Eq. (4) below shows the secondorder response surface model for the standard deviation. Y^ s Z 28643 K 9513 !FR C 6828 !WA C 3643 !TP 2
C 542 !BF C 4753 !FR K 3273 !WA !FR C 267 !WA2 K 1614 !TP !FR K 302 !BF !FR K 207 !BF !WA K 450 !BF2
ð4Þ
Table 3 displays regression analysis results. The model fit is very good in this case with an indicated Adjusted R-square value of 0.9965. Eq. (5) shows the second-order response surface model for the mean weight and model coefficients. Table 4 displays regression analysis results for the mean weight model. The model fit is also very good in this case with an indicated Adjusted R-Square value of 0.9944. Y^ m Z 271646 K 66300 !FR C 43696 !WA C 24725 !TP C 5096 !BF C 35505 !FR2 K 17905 !WA !FR K 8993 !TP !FR K 724 !BF !FR C 506 !BF !TP K 1716 !BF2
ð5Þ
These dual response surface equations can be used to rapidly determine the effect of varying design parameter values on the (weights) performance characteristics and for optimization. 6.5. Determine design parameter values that optimize the response
(risk) subject to a constraint on weight. Therefore, the optimization problem was set up as follows: Min Y^ s subject to Y^ m % 331; 704 K1%FR%1 K1%WA%1 K1%TP%1 K1%BF%1 This optimization problem was solved by utilizing two approaches; Excelw Solver and Genetic Algorithms. Solver, available in Microsoft Excelw, is a gradient-based optimization algorithm. Genetic Algorithms mimic natural search and selection processes leading to the survival of fittest individuals. In the range of optimization techniques, Genetic Algorithms occupy a gap between calculus-based methods and random search [22]. They are in general less efficient than gradient-based methods, however they can be applied to multimodal problems with discontinuous topologies [23]. The GA results in this study were obtained using a floating point Genetic Algorithm for minimization problems. Table 5 summarizes the results. Solver and Table 4 Mean weight model regression statistics R-square Adjusted R-square RMS error
The purpose in this design study was to determine the optimum values of the design parameters to considering both the mean empty weight and variance. To obtain a robust design, one would seek to minimize the variability
Table 5 Optimization results
Table 3 Standard deviation model regression statistics
FR WA TP BF Standard deviation Mean EWT
R-square Adjusted R-square RMS error
0.9976 0.9965 3961.58
Design variable
0.9962 0.9944 30,857.2
Optimized value Excel solver
Genetic algorithm
0.454844 K1 K1 K1 16,164 186,466
0.455912 K1 K1 K1 16,164 186,457
O. Yeniay et al. / Reliability Engineering and System Safety 91 (2006) 407–412 Table 6 Mean versus standard deviation plot
411
and the variability are minimized at the same solution for this case study. Even though this is a desirable result, one would expect that for many optimization problems, there would be a trade off between the mean and variance.
Mean vs Std. Devn. 16220
Std.Devn.
16210 16200
Acknowledgements
16190 16180
This work was supported by National Aeronautics and Space Administration, Langley Research Center, under Grant No. NAG-1-01086.
16170 16160 186000
186200
186400
186600
186800
187000
187200
Mean
Mean EWT
Standard deviation
187,000 186,900 186,800 186,700 186,600 186,500 186,400 186,300 186,200 186,100 186,080 186,466
16,180 16,175 16,171 16,168 16,165 16,164 16,165 16,167 16,175 16,196 16,217 16,164
the Genetic Algorithm have lead to almost identical results in this case. Since the goal in this study was to search for optimization results considering both the mean and the variability, a mean and variance graph is plotted below. Table 6 indicates that both the mean and the variability are minimized at x 0 Z(0.455,912; K1; K1; K1) for this case study.
7. Concluding remarks This case study utilized dual responses to estimate variability resulting from uncertain values of several design parameters for a weights and sizing analysis of a wing-body launch vehicle. Using D-Optimal designs and disciplinary design analysis codes, dual response surfaces were constructed for the mean and standard deviation to quantify variability in vehicle empty weight. An orthogonal arraybased simulation approach was used to simulate the variability resulting from noise factors. Next, an optimum solution was sought to minimize variability subject to a constraint on mean weight. In this application, the dual response surface approach lead to quantifying and reducing variability without much increase in design effort. The generation of additional data points needed to construct a second response surface for the standard deviation was not costly since the analysis was already set up to run the design points for constructing the response surface for mean empty weight. The results showed that both the mean
References [1] Montgomery DC. Design and analysis of experiments. New Jersey: Wiley; 1991. [2] Myers RH. Response surface methodology. Boston, MA: Virginia Commonwealth University, Allyn and Bacon Inc.; 1971. [3] Khuri AI, Cornell JA. Response surfaces: designs and analyses. New York: Marcel Dekker Inc.; 1987. [4] Craig JA. D-optimal design method: final report and user’s manual, USAF Contract F33615-78-C-3011, FZM-6777, General Dynamics, Forth Worth Div.; 1978. [5] Roux WJ, Stander N, Haftka R. Response surface approximations for structural optimization. In: AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimization; September, 1996. Paper No. AIAA-96-4042. [6] Giunta AA, Balabanov V, Haim D, Grossman B, Mason WH, Watson LT. Wing design for high speed civil transport using doe methodology. In: AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimization; September, 1996. AIAA Paper 96-4001. [7] Lepsch RA, Stanley DO, Unal R. Dual-fuel propulsion in single stage advanced manned launch system vehicle. J Spacecraft Rockets 1995; 32(3):417–25. [8] Unal R, Lepsch RA, McMillin ML. Response surface model building and multidisciplinary optimization using d-optimal designs. In: Annual AIAA/NASA/ISSMO symposium on multidisciplinary analysis and optimization; September 1998. AIAA-98-4759. [9] Kim K, Lin DK. Dual response surface optimization: a fuzzy modeling approach. J Qual Technol 1998;30(1):1–10. [10] Vining GG, Myers RH. Combining Taguchi and response surface philosophies: dual response approach. J Qual Technol 1990;22(1): 38–45. [11] Lin DK, Tu W. Dual response surface optimization. J Qual Technol 1995;27(1):34–9. [12] Del Castillo E, Montgomery DC. Nonlinear programming solution to the dual response surface problem. J Qual Technol 1993;25(3):199–204. [13] Del Castillo E, Fan SK, Semple J. Dual response surface optimization. J Qual Technol 1997;29(3):347–53. [14] Tong LI. Optimizing dynamic multi-response problems using the dual response surface method. Qual Eng 2001;14(1):115–25. [15] Tang LC, Xu K. A unified approach for dual response surface optimization. J Qual Technol 2002;34(4):437–47. [16] Luner JJ. Achieving continuous improvement with dual response surface approach. Qual Eng 1994;6(1):691–705. [17] Dhavlikar MN, Kulkarni MS, Mariappan V. Combined Taguchi and dual response method for optimization of a centerless grinding operation. J Mater Process Technol 2002;5875:1–5. [18] Carpenter WC. Effect of design selection on response surface performance, NASA contractor report 4520; June 1993. [19] Mitchell TJ. An algorithm for the construction of d-optimal experimental designs. Technometrics 1974;16(2):203–11.
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[20] Unal R, Stanley DO, Lepsch RA. Parametric modeling using saturated experimental designs. J Parametrics 1996;16(1):3–18. [21] Phadke SM. Quality engineering using robust design. Englewood Cliffs, NJ: Prentice Hall; 1989. [22] Goldberg DE. Genetic algorithms in search, optimization, and machine learning. Reading, MA: Addison Wesley; 1989. [23] Gage PJ. New approaches to optimization in aerospace conceptual design. PhD Dissertation, Stanford University; 1994.
[24] Box GEP, Draper NR. Empirical model building and response surfaces. New York: Wiley; 1987. [25] Menon R, Tong LT, Zhije L, Ibrahim J. Robust design of a spindle motor: a case study. Reliab Eng Syst Safety 2002;75: 313–9. [26] Kim D, Rhee S. Optimization of a GMA welding process using the dual response surface approach. Int J Prod Res 2003;41(18): 4505–15.
Using dual response surfaces to reduce variability in launch vehicle design: A case study Ozgur Yeniaya, Resit Unalb,*, Roger A. Lepschc b
a Department of Statistics, Hacettepe University, Ankara, Turkey Engineering Management and Systems Engineering Department, Old Dominion University, Norfolk, VA 23693, USA c Vehicle Analysis Branch, NASA Langley Research Center, Hampton, VA 23681, USA
Received 10 June 2004; accepted 12 February 2005 Available online 6 May 2005
Abstract Space transportation system conceptual design is a multidisciplinary process containing considerable element of risk. Uncertainties from one engineering discipline may propagate to another through linking parameters and the final system output may have an accumulation of risk. This may lead to significant deviations from expected performance. An estimate of variability or design risk therefore becomes essential for a robust design. This study utilizes the dual response surface approach to quantify variability in critical performance characteristics during conceptual design phase of a launch vehicle. Using design of experiments methods and disciplinary design analysis codes, dual response surfaces are constructed for the mean and standard deviation to quantify variability in vehicle weight and sizing analysis. Next, an optimum solution is sought to minimize variability subject to a constraint on mean weight. In this application, the dual response surface approach lead to quantifying and minimizing variability without much increase in design effort. q 2005 Elsevier Ltd. All rights reserved. Keywords: Uncertainty; Response surface methods; D-Optimal designs, Optimization
1. Introduction Conceptual engineering design of future launch vehicles is a multidisciplinary process containing considerable element of risk. Uncertainties from one discipline may propagate to another through linking parameters leading to significant deviations from the expected performance. Therefore, an estimate of this variability, together with the expected performance characteristic value is necessary for multidisciplinary optimization for a robust design. Robust design in this study is defined as a solution that minimizes variability subject to constraints on mean performance characteristics. This study applies the dual response surface approach to quantify variability in vehicle empty weight (without propellants) for a launch vehicle weights and sizing case study. Weight is a critical performance characteristic in * Corresponding author. Fax: C1 757 683 5640. E-mail address: [email protected] (R. Unal).
0951-8320/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2005.02.007
aerospace design since it affects performance and cost. Reducing variability in vehicle weight is also an important objective due to safety issues related to design margins such as in propulsion and landing gear.
2. Multidisciplinary conceptual design In launch vehicle conceptual design studies, system performance can generally be determined by the use of computerized analysis tools available in many engineering disciplines. However, these complex sizing and performance evaluation analysis codes utilize iterative algorithms. In many cases, they are expensive and difficult to integrate and use directly for multidisciplinary design optimization. An alternative is to construct mathematical models that approximate the relationships between performance characteristics and design variables. These approximation models, called response surface models, are then used to integrate the disciplines using mathematical programming methods and for multidisciplinary system level optimization.
408
O. Yeniay et al. / Reliability Engineering and System Safety 91 (2006) 407–412
3. Response surface methods Y s Z c0 C A second-order approximation model of the form given in Eq. (1) is commonly used in approximation model building since in many cases it may adequately model the response surface especially if the region of interest is sufficiently limited. Y^ Z b0 C
k X
bi x i C
iZ1
k X
bii x2i C
k XX
bij xi xj
(1)
i!j
iZ1
In Eq. (1), the xi terms are the input design variables that influence the response (performance characteristics) Y, and b0, bi and bij are estimated regression coefficients. The cross terms represent two-parameter interactions, and the square terms represent second-order non-linearity. There are various response surface techniques that may be utilized to sample the design space efficiently to construct a secondorder approximation model. Some of these are central composite designs [1–3] and D-Optimal designs [4,5]. Response surface methods using these designs have been applied to various multidisciplinary design optimization problems [6–8]. The main advantage is that response surface methods can aid multidisciplinary design integration, and provide rapid design analysis and optimization capability in many applications. However, in these (and most) response surface model building applications, only the mean value for the performance characteristic (Y) was studied and optimized ignoring design risk or variability due to uncertain design parameter values (noise).
4. Dual response surfaces Response surface methodology is designed to construct an approximation model for the response Y. This approximation model is then used to determine the best values of design parameters that optimize the response or the critical quality characteristic. In such problems typically the focus is on the mean value of Y where variance is assumed to be small and constant. However, if these assumptions do not hold, only constructing a response surface model for the mean may not be adequate and optimization results can be misleading. Kim and Lin [9] note that the assumption of constant variance is often not practically valid and classical response surface methods can be misleading in many cases. An alternative to classical response surface methods is the ‘dual response surface’ approach. Initially proposed by Vining and Myers [10,24], this approach seeks to build two models; one for the mean and one for the standard deviation [11–15]. Ym Z b0 C
k X iZ1
bi x i C
k X iZ1
bii x2i C
k XX i!j
bij xi xj C 3m
(2)
k X iZ1
ci xi C
k X iZ1
cii x2i C
k XX
cij xi xj C 3s
(3)
i!j
The dual response surface approach simultaneously considers the estimated mean (m) and the standard deviation (s) as a measure of design risk. The steps in the dual response surface approach can be summarized as: 1. Conduct the experiments to sample the design space using a response surface technique such as central composite design or a D-Optimal design, 2. Repeat the experiments over the range of uncertain design (noise) parameters to capture variability, 3. Compute the mean and the variance for each experiment and construct response surfaces for both the mean and variance, 4. Determine the best values of xi to optimize the mean response subject to a constraint on the variance or determine the best values of xi to minimize the variance subject to a constraint on mean response.
5. Applying dual response surfaces for a robust design Many studies in the literature successfully utilized the dual response methodology for robust design and optimization studies that considered both mean and variability. In their leading article, Vining and Myers [10] demonstrated the use of dual response surfaces in optimizing variability while satisfying conditions on a desired target. They used data taken from Box and Draper [24] for a printing machine in applying coloring ink package labels. Luner [16] utilized dual response surfaces in determining a robust combination of factor levels that reduced variability around a target using a Roman-style catapult. Menon et al. [25] applied dual response surfaces for robust design of a spindle motor. A constrained minimization of the cogging response of the spindle motor was sought, the constraint being that the variance be within a specified limit. Their results showed a 25% reduction in variance of the clogging torque. Kim and Rhee [26] utilized the dual response surfaces in a study to optimize the penetration in gas metal arc welding seeking to determine welding process parameter values to produce target value with minimized variance. Their results showed that the optimal conditions determined through using the dual response surface approach had a lower possibility of producing defective welds. Dhavlikar et al. [17] presented an application of dual response surfaces to determine robust condition for minimization of out of roundness error of work pieces for centerless grinding operation. They observed that the combined Taguchi and dual response surface approach utilized led to successful identification of optimum process parameter values.
O. Yeniay et al. / Reliability Engineering and System Safety 91 (2006) 407–412
In this study, the dual response surface approach is applied to the weights and sizing analysis of a wing-body launch vehicle case study with the objective of optimizing variability in weight.
6. Wing-body launch vehicle configuration case study to optimize empty weight An application of dual response surfaces is described for a weights and sizing study of a rocket-powered, singlestage-to-orbit launch vehicle. The vehicle has been sized to perform a 25,000 lb payload delivery to the International Space Station from Kennedy Space Center. Near term structures and subsystem technologies are assumed in its design. The vehicle has a wing-body configuration with a slender, round cross-section fuselage and a clipped delta wing. The delta wing has elevon control surfaces for aerodynamic roll and pitch control. Small vertical fins, called tip fins, are located at the wing tips for directional control and a body flap extends rearward from the lower base of the fuselage to provide additional pitch control. The purpose in this study is to determine the best values of the engineering design parameters (that satisfy aerodynamic constraints) to minimize variability in vehicle empty weight (weight without propellants). This case study involved the following steps.
The four common parameters included in aerodynamics and weights and sizing analysis were the fineness ratio (defined as the fuselage length divided by diameter), the wing area, the tip fin area, and the body flap area. The other two parameters included were ballast-weight and mass-ratio for weights and sizing analysis and angle of attack and elevon deflection for aerodynamics analysis. In this study, ballast-weight and mass-ratio were taken as the noise variables. Noise variables are those that one has little control on the values during operation. Therefore, one design goal is to reduce the variability in performance (empty weight in this case) due to the uncertain values of these parameters. The ranges for the six weights and sizing parameters are given in Table 1. Table 1 Weights and sizing parameters and ranges Range 4 10 0.5 0 0 7.75
6.2. Construct the experimental design matrix The next step is to construct an experimental design matrix to efficiently sample the design space and construct a second-order response model for four parameters. Carpenter [18] has conducted a study comparing the performance of various experimental design methods for approximation model building in terms of the quality of fit in the region studied and in terms of the number of design points required. His findings suggest that the D-optimality criterion is a good approach for constructing experimental designs for design studies using computerized engineering design and analysis codes. Other studies in the literature also found that the D-optimality criterion provides a rational means for creating experimental designs for an irregularly shaped response surfaces [6]. Therefore, D-Optimal designs are used in this study to construct an experimental design to sample the design space [2,4,8,19,20]. The JMPw software was utilized to construct the D-Optimal design matrix given in Table 2. With this design matrix, the four engineering design parameters are studied at three levels (values) as represented in coded form by K1, 0 and C1. As an example, a K1 for fineness ratio corresponds to 4 (lower bound), a 0 corresponds to 5.5 (mid value) and C1 corresponds to 7 (upper bound). These coded values are then transformed into actual parameter values to be used in conducting the analysis. 6.3. Conduct the matrix experiments
6.1. Identify design variables and feasible ranges
Design parameters Fineness ratio (FR) Wing area ratio (WA) Tip fin area ratio (TP) Body flap area ratio (BF) Noise parameters Ballast weight (BL) Mass ratio (MR)
409
7 20 3 1 0.04 8.25
In this study, all geometry and subsystem packaging of the SSV were performed using a NASA-developed geometry-modeling tool. The Aerodynamic Preliminary Analysis System (APAS) was used to determine vehicle aerodynamics. The weights and sizing analyses were performed using the NASA-developed Configuration Sizing (CONSIZ) weights/sizing package. This process was continued for the 45 rows of the D-Optimal design matrix, each of which corresponds to a vehicle design generating the data. An orthogonal array based simulation approach is used to simulate the variability resulting from noise factors [21]. For each of the 45 rows, experiments were repeated at five points corresponding to the five-point design matrix shown below to simulate the variability due to the uncertain values of the two noise variables (Ballast weight and MassRatio). From the weight analysis, data points for empty weight were obtained (Table 2). Using this data, mean empty weights (m) and corresponding standard deviations (s), as a measure of design risk due to the uncertain values of BL and MR, were computed. 6.4. Construct the second-order response surface model Least-squares regression analysis is then used to determine the coefficients of the second order, dual response surface models for both the mean empty weight
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Table 2 Analysis results
1 2 3 4 « 42 43 44 45
BL MR
K1 K1
C1 K1
K1 C1
C1 C1
0 0
FR
WA
TP
BF
EWT1
EWT2
EWT3
EWT4
EWT5
K1 K1 K1 K1 « 1 1 1 1
K1 K1 K1 K1 « 1 1 1 1
K1 K1 0 1 « K1 0 1 1
K1 1 1 K1 « 1 0 K1 1
234,798 244,264 269,903 287,841 « 216,406 225,895 236,433 241,231
266,104 277,534 308,403 329,905 « 246,297 257,754 270,443 276,283
269,950 281,649 313,455 335,853 « 250,762 262,683 275,933 281,946
309,903 324,255 363,252 390,673 « 289,312 303,939 320,256 327,754
268,058 279,592 310,966 332,920 « 248,572 260,244 273,221 279,152
and the standard deviation. Eq. (4) below shows the secondorder response surface model for the standard deviation. Y^ s Z 28643 K 9513 !FR C 6828 !WA C 3643 !TP 2
C 542 !BF C 4753 !FR K 3273 !WA !FR C 267 !WA2 K 1614 !TP !FR K 302 !BF !FR K 207 !BF !WA K 450 !BF2
ð4Þ
Table 3 displays regression analysis results. The model fit is very good in this case with an indicated Adjusted R-square value of 0.9965. Eq. (5) shows the second-order response surface model for the mean weight and model coefficients. Table 4 displays regression analysis results for the mean weight model. The model fit is also very good in this case with an indicated Adjusted R-Square value of 0.9944. Y^ m Z 271646 K 66300 !FR C 43696 !WA C 24725 !TP C 5096 !BF C 35505 !FR2 K 17905 !WA !FR K 8993 !TP !FR K 724 !BF !FR C 506 !BF !TP K 1716 !BF2
ð5Þ
These dual response surface equations can be used to rapidly determine the effect of varying design parameter values on the (weights) performance characteristics and for optimization. 6.5. Determine design parameter values that optimize the response
(risk) subject to a constraint on weight. Therefore, the optimization problem was set up as follows: Min Y^ s subject to Y^ m % 331; 704 K1%FR%1 K1%WA%1 K1%TP%1 K1%BF%1 This optimization problem was solved by utilizing two approaches; Excelw Solver and Genetic Algorithms. Solver, available in Microsoft Excelw, is a gradient-based optimization algorithm. Genetic Algorithms mimic natural search and selection processes leading to the survival of fittest individuals. In the range of optimization techniques, Genetic Algorithms occupy a gap between calculus-based methods and random search [22]. They are in general less efficient than gradient-based methods, however they can be applied to multimodal problems with discontinuous topologies [23]. The GA results in this study were obtained using a floating point Genetic Algorithm for minimization problems. Table 5 summarizes the results. Solver and Table 4 Mean weight model regression statistics R-square Adjusted R-square RMS error
The purpose in this design study was to determine the optimum values of the design parameters to considering both the mean empty weight and variance. To obtain a robust design, one would seek to minimize the variability
Table 5 Optimization results
Table 3 Standard deviation model regression statistics
FR WA TP BF Standard deviation Mean EWT
R-square Adjusted R-square RMS error
0.9976 0.9965 3961.58
Design variable
0.9962 0.9944 30,857.2
Optimized value Excel solver
Genetic algorithm
0.454844 K1 K1 K1 16,164 186,466
0.455912 K1 K1 K1 16,164 186,457
O. Yeniay et al. / Reliability Engineering and System Safety 91 (2006) 407–412 Table 6 Mean versus standard deviation plot
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and the variability are minimized at the same solution for this case study. Even though this is a desirable result, one would expect that for many optimization problems, there would be a trade off between the mean and variance.
Mean vs Std. Devn. 16220
Std.Devn.
16210 16200
Acknowledgements
16190 16180
This work was supported by National Aeronautics and Space Administration, Langley Research Center, under Grant No. NAG-1-01086.
16170 16160 186000
186200
186400
186600
186800
187000
187200
Mean
Mean EWT
Standard deviation
187,000 186,900 186,800 186,700 186,600 186,500 186,400 186,300 186,200 186,100 186,080 186,466
16,180 16,175 16,171 16,168 16,165 16,164 16,165 16,167 16,175 16,196 16,217 16,164
the Genetic Algorithm have lead to almost identical results in this case. Since the goal in this study was to search for optimization results considering both the mean and the variability, a mean and variance graph is plotted below. Table 6 indicates that both the mean and the variability are minimized at x 0 Z(0.455,912; K1; K1; K1) for this case study.
7. Concluding remarks This case study utilized dual responses to estimate variability resulting from uncertain values of several design parameters for a weights and sizing analysis of a wing-body launch vehicle. Using D-Optimal designs and disciplinary design analysis codes, dual response surfaces were constructed for the mean and standard deviation to quantify variability in vehicle empty weight. An orthogonal arraybased simulation approach was used to simulate the variability resulting from noise factors. Next, an optimum solution was sought to minimize variability subject to a constraint on mean weight. In this application, the dual response surface approach lead to quantifying and reducing variability without much increase in design effort. The generation of additional data points needed to construct a second response surface for the standard deviation was not costly since the analysis was already set up to run the design points for constructing the response surface for mean empty weight. The results showed that both the mean
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