RALLABHANDI AND MAVRIS 为 Actual by Predicted Plot Summary of Fit 04- RSquare 0983488 0982834 Root Meon Squore Error 0.006542 Mean of Response 023B486 Observations (or Sum Wgts) 256 03 Analysis of Variance Source DF Sum of Squares Mean Sausre F Ratio Model 9 10941814 0.1215761627999 Error 246 0.0183708 0100075p,hsF C.Total 255 1.1125522 .0001* Parameter Estimates Term Estimsle std Error t Ratio Prob=相 Intercept 0.3526925 0021633 16.300001 FPR 011324800009G .1141e0101* 0FFR-253)FPR-253 0.0861182 0.00228 37.760001* 2 OPR 0.00157650000145 10880001 WteyThrust Predicted FPR2.53THR.1.05) 0.18823430.026635 7.07<0001* P=0001RSg-0.98RMSE-0.00B6 EXTR 0.07178490.014492 4.95e0001 (FPR-253)(EXTR-1) -00m8167002665 -25600111 THR 0.02758760014492 19000581 (THR.105)'FPR-2.5310PR-30) 0.01330110.007147 18600639 (THR-1.05THR-105) -0.887741 0.486117 18300590 a)Model fit summary after removing inconsequential variables Quantiles Moments D 100 0%maximum 7409 Mean 00817948 95 7333 38450412 975% E3u slg日m6sn 02403151 -020 00% g042 aB5%Men05550495 760 a5 wer 95%Mean 039146 0.15 500 0.185 255 250 quortie 2642 0.10 100 .5232 5% .7559 005 8 0 b)Model fit error Quantiles Moments ◆ 100.0%mexmum 6,.866 05522769 995 686 Std Dev 975% B.49 Sld后t Mean 0.378478 025 900 4.771 upper 95%Mesn 13156185 020 750% qu 3.412 -02010 65 500% 1,381 250% 2.550 100% 3426 010 25% 5734 05% -5.826 005 00% mininun 5828 964202 c)Model representation error Fig.6 Engine-weight-regression model. weight used in this study is 1.2.For the one-stage-fan CDFS VCE a desired total range of 4000 n mile.The first 1500-n-mile segment of (i.e.,with FPR <1.8),the scaling factor is assumed to be 1.042,and the mission is the subsonic leg (including climb),with a cruise speed for the two-stage-fan CDFS VCE(i.e.,with FPR 1.8).the scaling of Mach 0.92.The remainder of the mission includes the supersonic factor is chosen as 1.3.These scaling factors had to be incorporated climb,a supersonic cruise at a Mach number in the range of 1.6-1.8. because WATE cannot model the VCE components.For example. and descent.This approximates the realistic missions for which a the bypass door must be modeled as a simple duct so that the weight supersonic business jet would be desirable for accelerated travel. of the moving parts is not captured.The same is the case with the Because of the Federal Aviation Administration(FAA)regulations variable-area bypass injector(VABD).The CDFS-stage weight is not on noise and supersonic travel over land,an aggressive supersonic computed properly,because associated frames,disks,bearings,etc. mission was not chosen;this will be studied in subsequent are not accounted for correctly.These factors have been treated efforts.With the desired total range for the mission being manually for a baseline engine,and the differences were combined 4000 n mile.approximately 2500 n mile is needed for the supersonic into a scale factor to be applied to an MFTF model that otherwise has cruise (including the descent).In this study,the takeoff gross the same cycle parameters of the VCE models.The authors realize weight (TOGW)is set by the GA for every run;therefore,the that these scale factors are somewhat subjective.But due to the lack aircraft flies a 1500-n-mile subsonic leg and then flies as far as it can of a proper tool to model complex VCE propulsion system weights, in the supersonic leg,sometimes exceeding the desired 4000-n-mile calibrated scale factors for baseline design were used in this study. threshold and sometimes falling short.Scaling of the engine,on the other hand,is performed through the use of the engine deck created by NPSS and the scaling laws imbedded in FLOPS. 5.Mission Analysis which scale the engine data to the thrust level that is necessary to FLOPS is used to perform mission analysis.The mission execute the mission performance module and takeoff and landing investigated in this study is a split subsonic/supersonic mission with module
weight used in this study is 1.2. For the one-stage-fan CDFS VCE (i.e., with FPR <1:8), the scaling factor is assumed to be 1.042, and for the two-stage-fan CDFS VCE (i.e., with FPR 1:8), the scaling factor is chosen as 1.3. These scaling factors had to be incorporated because WATE cannot model the VCE components. For example, the bypass door must be modeled as a simple duct so that the weight of the moving parts is not captured. The same is the case with the variable-area bypass injector (VABI). The CDFS-stage weight is not computed properly, because associated frames, disks, bearings, etc., are not accounted for correctly. These factors have been treated manually for a baseline engine, and the differences were combined into a scale factor to be applied to an MFTF model that otherwise has the same cycle parameters of the VCE models. The authors realize that these scale factors are somewhat subjective. But due to the lack of a proper tool to model complex VCE propulsion system weights, calibrated scale factors for baseline design were used in this study. 5. Mission Analysis FLOPS is used to perform mission analysis. The mission investigated in this study is a split subsonic/supersonic mission with a desired total range of 4000 n mile. The first 1500-n-mile segment of the mission is the subsonic leg (including climb), with a cruise speed of Mach 0.92. The remainder of the mission includes the supersonic climb, a supersonic cruise at a Mach number in the range of 1.6–1.8, and descent. This approximates the realistic missions for which a supersonic business jet would be desirable for accelerated travel. Because of the Federal Aviation Administration (FAA) regulations on noise and supersonic travel over land, an aggressive supersonic mission was not chosen; this will be studied in subsequent efforts. With the desired total range for the mission being 4000 n mile, approximately 2500 n mile is needed for the supersonic cruise (including the descent). In this study, the takeoff gross weight (TOGW) is set by the GA for every run; therefore, the aircraft flies a 1500-n-mile subsonic leg and then flies as far as it can in the supersonic leg, sometimes exceeding the desired 4000-n-mile threshold and sometimes falling short. Scaling of the engine, on the other hand, is performed through the use of the engine deck created by NPSS and the scaling laws imbedded in FLOPS, which scale the engine data to the thrust level that is necessary to execute the mission performance module and takeoff and landing module. Fig. 6 Engine-weight-regression model. RALLABHANDI AND MAVRIS 43
44 RALLABHANDI AND MAVRIS D.Advanced Genetic Algorithms Table 2 Objective goals and weights In engineering design,there are almost always multiple criteria Objectives Goal value Normalization value that must be considered during the concept selection process. Objectives such as weight,cost,and speed must be balanced against Range,n mile 4200 1.0 each other to find the appropriate combination that will result in a Gross weight.lb×10O0 100 50.0 successful design.Traditionally,these problems are handled by Jet velocity for 7000-ft takeoff,ft/s 900 2.0 Cruise Mach number 1.7 0.0025 creating an aggregate objective function,the so-called overall Shock pressure rise,psf 0.35 0.002 evaluation criterion.However.this method cannot numerically Sonic boom,PL (dB) 88 0.01 quantify how important the objectives are in relation to each other Approach velocity.kt 140 0.1 resulting in designs that do not really best meet the designers goals. Length,ft 140 0.1 Multi-objective genetic algorithms attempt to solve this problem by Static stability penalty 100.0 10.0 using the concept of Pareto-optimality to find nondominated Cabin diameter.ft 69 0.001 solutions.A solution is said to be nondominated when there is no other solution in the space that is better with regard to all decision variables.The set of nondominated solutions,or Pareto front,makes up a hypersurface along which improvement in one objective is a vector of normalization constants.In the present study,all the requires a sacrifice in another.Once this front has been found,the objectives were equally weighted.The conceptual design process for engineer can use it to explore the relationship between the objectives the desired supersonic transport does not have a standard set of so that intelligent decisions can be made. requirements or goals,unlike many traditional subsonic design Several modifications to the conventional GA,such as the strength efforts.Certain well-accepted guidelines,as shown in Table 2,are Pareto evolutionary algorithm(SPEA)[27].have gained widespread used in this study as design goals.The normalization constants were acceptance for use in the multi-objective optimization problem.This determined by calculating the variance of each objective for a algorithm incorporates elitist and population-diversifying character- random population of individuals and then rounding that value to a istics to multi-obiective Pareto optimization.The SPEA2 algorithm convenient number.The normalization values allow for the width of balances the diversification of the population as well as exploits the distribution to be taken into account,thus including a convenient solutions from an archive ofelite designs.However,as the number of way to scale the objectives in an unbiased manner. objectives increases,the proportion of nondominated designs also increases.Recent research [28]has shown that although GA operators work well for two or three objective problems.their 2.Operator Enhancements effectiveness drops significantly when there are a large number of As has been mentioned earlier in this paper,one of the objectives conflicting objectives.The SPEA2 algorithm becomes ineffective of this work is to study the tradeoffs associated with component under these conditions.This is because the proportion of the placement.The GA not only attempts to optimize the component population that is nondominated grows exponentially as the number shapes,but also their placement,implying that the optimization of simultaneous objectives increase.In fact,Deb [29]found that as problem is a mixed discrete/continuous hierarchical problem.It is the number of objectives grows,nearly all solutions become found through experience that the conventional GA crossover nondominated and would therefore have equal fitness values. operators yield very poor performance for such a problem,due to Because of reliance on dominance-based fitness,most multi- excessive chromosome disruption.To overcome this.several objective algorithms have difficulties solving problems with more operator enhancements have been included in the GA.Conventional than two or three responses. GAs use binary encoding to discretize the computational domain According to the definition of nondominance,a solution is before searching for optimum locations.However,application of nondominated if no other solution in the set is better in any objective binary encoding to variables with discrete settings requires without being inferior in at least one other objective.This definition specification of several special cases,which can cause the procedure of dominance does not deal with magnitudes.A solution that trades a to become inefficient.To overcome this problem.real-valued genetic large amount of capability in one objective for an infinitesimal algorithms are used in this study amount of gain in another is not penalized by the fitness-assignment Although evolutionary algorithms have been applied to several schemes used by many of the common evolutionary algorithms in engineering design problems,not many studies have looked at literature. optimizing problems that have both the system architecture(discrete) and design (continuous)variables.One specific instance of 1.Multi-Objective Optimization combining discrete and continuous variables in evolutionary optimizers is described by Parmee [31].termed structured genetic In response to the poor performance of the multi-objective evolutionary algorithms available in literature,an advanced genetic algorithms.Unfortunately,the structured GA quickly focused on a single alternative,even for relatively small design hierarchies.The algorithm method has been developed.Rather than optimizing each of the M objectives,the problem is reformulated to solve for the tendency to quickly focus on a small portion of the design space may result in the algorithm overlooking potentially promising solutions. tradeoff between M biased aggregate functions that favor attainment This deficiency was addressed with a method called the hybrid of one particular objective but do not ignore performance of the other structured genetic algorithm [31],which introduced variable M-1 objectives.Goal programming [30]is used in the current study mutation probabilities to the discrete and continuous variables.This as the aggregation technique.In this method,three parameters per method used a bitwise mutation probability of 20%for discrete objective are specified:a goal or ideal value,a normalization value obtained using sampling.and a weighting value to specify variables and a much lower mutation probability of 2%for importance.The goal value corresponds to the minimum level of continuous variables.The large mutation probability applied to the performance in a given metric that would be considered acceptable. discrete variables effectively maintained genetic diversity,but may also prevent efficient convergence because of the mutation The problem is then recast as the minimization of the weighted operator's tendency to move away from good solutions. difference between the actual and goal values,as shown in Eq.(1) Goal programming may be understood to be analogous to a weighted The crossover operator is applied to a postreproduction population to cross genes between its members.The aim of this operator is to target-matching problem: attempt to create better designs by swapping genes (design variables);the idea is to insert a good gene from one design into fm(x)-8m 1) another design and increase its fitness value.Several choices exist for real-variable crossover,such as uniform linear,bilinear (BLX-a) [32],simulated binary (SBX)[33,34],or an enhanced version of where w is a vector of weights,g is a vector of goal values,and n simulated binary (VSBX)[35].Going into the details of each of the
D. Advanced Genetic Algorithms In engineering design, there are almost always multiple criteria that must be considered during the concept selection process. Objectives such as weight, cost, and speed must be balanced against each other to find the appropriate combination that will result in a successful design. Traditionally, these problems are handled by creating an aggregate objective function, the so-called overall evaluation criterion. However, this method cannot numerically quantify how important the objectives are in relation to each other, resulting in designs that do not really best meet the designers goals. Multi-objective genetic algorithms attempt to solve this problem by using the concept of Pareto-optimality to find nondominated solutions. A solution is said to be nondominated when there is no other solution in the space that is better with regard to all decision variables. The set of nondominated solutions, or Pareto front, makes up a hypersurface along which improvement in one objective requires a sacrifice in another. Once this front has been found, the engineer can use it to explore the relationship between the objectives so that intelligent decisions can be made. Several modifications to the conventional GA, such as the strength Pareto evolutionary algorithm (SPEA) [27], have gained widespread acceptance for use in the multi-objective optimization problem. This algorithm incorporates elitist and population-diversifying characteristics to multi-objective Pareto optimization. The SPEA2 algorithm balances the diversification of the population as well as exploits solutions from an archive of elite designs. However, as the number of objectives increases, the proportion of nondominated designs also increases. Recent research [28] has shown that although GA operators work well for two or three objective problems, their effectiveness drops significantly when there are a large number of conflicting objectives. The SPEA2 algorithm becomes ineffective under these conditions. This is because the proportion of the population that is nondominated grows exponentially as the number of simultaneous objectives increase. In fact, Deb [29] found that as the number of objectives grows, nearly all solutions become nondominated and would therefore have equal fitness values. Because of reliance on dominance-based fitness, most multiobjective algorithms have difficulties solving problems with more than two or three responses. According to the definition of nondominance, a solution is nondominated if no other solution in the set is better in any objective without being inferior in at least one other objective. This definition of dominance does not deal with magnitudes. A solution that trades a large amount of capability in one objective for an infinitesimal amount of gain in another is not penalized by the fitness-assignment schemes used by many of the common evolutionary algorithms in literature. 1. Multi-Objective Optimization In response to the poor performance of the multi-objective evolutionary algorithms available in literature, an advanced genetic algorithm method has been developed. Rather than optimizing each of the M objectives, the problem is reformulated to solve for the tradeoff between M biased aggregate functions that favor attainment of one particular objective but do not ignore performance of the other M-1 objectives. Goal programming [30] is used in the current study as the aggregation technique. In this method, three parameters per objective are specified: a goal or ideal value, a normalization value obtained using sampling, and a weighting value to specify importance. The goal value corresponds to the minimum level of performance in a given metric that would be considered acceptable. The problem is then recast as the minimization of the weighted difference between the actual and goal values, as shown in Eq. (1). Goal programming may be understood to be analogous to a weighted target-matching problem: Vix XM m1 wm fmx gm nm p1=p (1) where wm is a vector of weights, gm is a vector of goal values, and nm is a vector of normalization constants. In the present study, all the objectives were equally weighted. The conceptual design process for the desired supersonic transport does not have a standard set of requirements or goals, unlike many traditional subsonic design efforts. Certain well-accepted guidelines, as shown in Table 2, are used in this study as design goals. The normalization constants were determined by calculating the variance of each objective for a random population of individuals and then rounding that value to a convenient number. The normalization values allow for the width of the distribution to be taken into account, thus including a convenient way to scale the objectives in an unbiased manner. 2. Operator Enhancements As has been mentioned earlier in this paper, one of the objectives of this work is to study the tradeoffs associated with component placement. The GA not only attempts to optimize the component shapes, but also their placement, implying that the optimization problem is a mixed discrete/continuous hierarchical problem. It is found through experience that the conventional GA crossover operators yield very poor performance for such a problem, due to excessive chromosome disruption. To overcome this, several operator enhancements have been included in the GA. Conventional GAs use binary encoding to discretize the computational domain before searching for optimum locations. However, application of binary encoding to variables with discrete settings requires specification of several special cases, which can cause the procedure to become inefficient. To overcome this problem, real-valued genetic algorithms are used in this study. Although evolutionary algorithms have been applied to several engineering design problems, not many studies have looked at optimizing problems that have both the system architecture (discrete) and design (continuous) variables. One specific instance of combining discrete and continuous variables in evolutionary optimizers is described by Parmee [31], termed structured genetic algorithms. Unfortunately, the structured GA quickly focused on a single alternative, even for relatively small design hierarchies. The tendency to quickly focus on a small portion of the design space may result in the algorithm overlooking potentially promising solutions. This deficiency was addressed with a method called the hybrid structured genetic algorithm [31], which introduced variable mutation probabilities to the discrete and continuous variables. This method used a bitwise mutation probability of 20% for discrete variables and a much lower mutation probability of 2% for continuous variables. The large mutation probability applied to the discrete variables effectively maintained genetic diversity, but may also prevent efficient convergence because of the mutation operator’s tendency to move away from good solutions. The crossover operator is applied to a postreproduction population to cross genes between its members. The aim of this operator is to attempt to create better designs by swapping genes (design variables); the idea is to insert a good gene from one design into another design and increase its fitness value. Several choices exist for real-variable crossover, such as uniform linear, bilinear (BLX-) [32], simulated binary (SBX) [33,34], or an enhanced version of simulated binary (vSBX) [35]. Going into the details of each of the Table 2 Objective goals and weights Objectives Goal value Normalization value Range, n mile 4200 1.0 Gross weight, lb 1000 100 50.0 Jet velocity for 7000-ft takeoff, ft/s 900 2.0 Cruise Mach number 1.7 0.0025 Shock pressure rise, psf 0.35 0.002 Sonic boom, PL (dB) 88 0.01 Approach velocity, kt 140 0.1 Length, ft 140 0.1 Static stability penalty 100.0 10.0 Cabin diameter, ft 6.9 0.001 44 RALLABHANDI AND MAVRIS