Exploring the fitness landscape of a realistic turbofan rotor blade optimization
EExploring the fitness landscape of a realisticturbofan rotor blade optimization
Jakub Kmec and Sebastian Schmitt Palack´y University,Olomouc, Czech Republic [email protected] Honda Research Institute Europe GmbH,Offenbach, Germany [email protected]
Abstract.
Aerodynamic shape optimization has established itself as avaluable tool in the engineering design process to achieve highly effi-cient results. A central aspect for such approaches is the mapping fromthe design parameters which encode the geometry of the shape to be im-proved to the quality criteria which describe its performance. The choicesto be made in the setup of the optimization process strongly influencethis mapping and thus are expected to have a profound influence on theachievable result. In this work we explore the influence of such choices onthe effects on the shape optimization of a turbofan rotor blade as it canbe realized within an aircraft engine design process. The blade qualityis assessed by realistic three dimensional computational fluid dynamics(CFD) simulations. We investigate the outcomes of several optimizationruns which differ in various configuration options. We compare the resultsfrom the covariance matrix adaptation evolutionary strategy (CMA-ES)with the outcome of a particle swarm optimization (PSO). We also inves-tigate the changes induced by a different initialization of the CMA-ESand by a variation of its population size. A particular focus is put on thevariation of the results if we use different number of degrees of freedomfor parametrization of the rotor blade geometry. For all such variations,we generally find that the achievable improvement of the blade qualityis comparable for most settings and thus rather insensitive to the detailsof the setup. On the other hand, even supposedly minor changes in thesettings, such as using a different random seed for the initialization ofthe optimizer algorithm, lead to very different shapes. Optimized shapeswhich show comparable performance usually differ quite strongly in theirgeometries over the complete blade. Our analyses indicate that the fit-ness landscape for such a realistic turbofan rotor blade optimization ishighly multi-modal with many local optima, where very different shapesshow similar performance.Published as: Kmec J., Schmitt S. (2019) Exploring the Fitness Landscape of a Real-istic Turbofan Rotor Blade Optimization. In: Rodrigues H. et al. (eds) EngOpt 2018Proceedings of the 6th International Conference on Engineering Optimization. EngOpt2018.https://doi.org/10.1007/978-3-319-97773-7 46 a r X i v : . [ c s . C E ] O c t J. Kmec and S. Schmitt
Aerodynamic shape optimization using computational fluid dynamics (CFD) is widelyused for creating efficient designs in engineering applications such as the automotive orthe aerospace domain [1]. Global optimization techniques enable unbiased search forimproved shapes and are capable of yielding conceptional new designs. However, theyusually require a significant number of CFD simulations for assessing the quality of theproposed designs. This constitutes a substantial drawback for extensive application ofglobal optimization techniques in actual design cycles since realistic CFD simulationsare usually rather complex and have a large time and resource demand. Difficulties arisein that context due to the many options available to the designer for setting up sucha numerical design optimization. These include the choice of the optimizer algorithm,specific configuration of the parameters for the optimizer algorithm, representation ofthe design and design changes to be used, and many more.In practice, one usually employs a specific setup with some more or less well mo-tivated choices. Then optimization is performed and depending on the results somerefinements are done and a new optimization run is performed. This is iterated until asatisfactory result is obtained. It is not possible to assess whether the results obtainedin this manner are representative. In particular, it is not clear if the results could havebeen much better, if a slightly different configuration was chosen. Or, if a rather dif-ferent design could have been obtained which still has a similarly good fitness value.The answers to such questions are of course strongly linked to the qualitative behaviorof the fitness function.In this work, we explore the fitness landscape of a realistic turbofan simulationcase. We investigate the outcome of different optimization runs in which we vary theconfiguration settings. We compare the achievable fitness values obtained with a co-variance matrix adaptation evolutionary strategy (CMA-ES) [4] to a particle swarmoptimization (PSO) [3].Another key ingredient to shape optimization approaches is the choice how to rep-resent a specific design and in particular how to generate sets of different designs to beevaluated during the optimization. In a typical engineering application, a valid base-line design is readily available and the design process focuses on improving that design.Deformation methods are especially suited and well-established for such situations. Acrucial decision to be taken by the engineer is the choice of the deformation methodand the details of its realization. In the context of evolutionary aerodynamic shape op-timization, shape morphing methods are popular such as free-form deformation (FFD)and radial basis functions (RBF) based deformation methods, or shape-function meth-ods like Hicks-Henne (HH) functions [2].
During the optimization, the blade geometry has to be changed automatically based onthe parameters proposed by the optimization algorithm. Thus, a blade representationhas to be defined. In this work, we use a deformation approach, where we start from agiven baseline blade design and encode changes to it via the optimization parameters.Specifically, a blade is represented as a collection of sections stacked on top of eachother as shown in Figure 1 (left). Each section (which can be thought of as a 2D airfoil ttps://doi.org/10.1007/978-3-319-97773-7 46
Fig. 1.
Left: 3D baseline blade geometry where the points for the sections are shown asblue dots and the three independently deformable sections near the hub, the mid-spanand the shroud are shown in white. Right: Hicks-Henne (HH) shape functions.In the most general situation, all sections would be allowed to change independentlyto generate a modified blade design. In order to keep the optimization parameters man-ageable, we select only three sections, one close to the hub, one in the mid-span region,and one close to the shroud, which we deform based on the optimization parameters.The in-between sections are linearly interpolated.For each of those independently deformed sections, we allow the following changes:(i) rotation of the section around the leading edge point, (ii) movement of the sectionin the axial-meridional plane, and (iii) deformation of the section profile by addingHH shape functions (see Figure 1 right), which is a well-known representation from 2Dairfoil design, see e.g. [5].The explicit functional form of the HH shape functions weuse is b ( x, x ) = (cid:104) sin (cid:16) πx log(0 . x (cid:17)(cid:105) , (1)where x ∈ [0 ,
1] parametrizes the chord length of each section and x controls thelocation of the maximum of each shape function. The number of shape functions persection N HH / section will be varied in this work and for a given number of N HH / section we use equally spaced locations for the maxima, i.e. x ( i ) = iN HH / section +1 where i =1 , . . . , N HH / section .In order to fully characterize the deformed blade geometry, we need to specifyfor each of the three independently deformable sections the amplitude for each shapefunction, the rotation angle, and the shift in axial and rotational direction. Therefore,the total number of free parameters, which is also the number of search variables forthe optimization problem, is given by N search = 3( N HH / section + 3) . . The turbo-fan simulation setup was inspired by the GE Honda HF120 [6] small busi-ness jet engine operated at cruise condition. We used steadyCompressibleMRFFoam , the J. Kmec and S. Schmittcompressible flow solver from the OpenFOAM CFD suite (specifically, the foam-extend-3.2 version) which was adapted to be more robust [7]. The blade rotation speed was setto a fixed value of 15600rpm and constant mass-flow and radial equilibrium boundaryconditions were employed. We used the kω SST turbulence model. The meshes for allsimulations consisted of about 6 million cells and were created by an in-house softwarebased on the OpenFOAM utility blockMesh . The flow equations were solved on 32 CPUcores in parallel, iterated for 12000 solver iterations in order to arrive at a convergedsolution, and the run-time was typically in the range of 2-4 hours.The flow-domain of the simulation setup and an example output flow field (pressure)at a circumferential section at a fixed radius is shown in Figure 2. In order to stabilizethe CFD solutions we did not include a clearance between rotor blade and shroudgeometry. This is somewhat unrealistic as it qualitatively changes the tip vortex causedby such a gap and thus overestimates the blade efficiency. But for the current studyit seems a valid simplification which should still provide realistic results except for theblade geometry in the tip region.
Fig. 2.
Left: Schematic side view of the flow passage. Right: Exemplary pressure dis-tribution at a cylindrical cut at 90% span height between hub and shroud. In order tomore clearly see the shock forming within the passage, we plot two adjacent passages.Red colors imply high pressure, blue colors imply low pressure.
The optimization maximizes the isentropic efficiency of the rotor which is calculatedfrom the OpenFOAM CFD solutions as η = (cid:16) P T , outlet P T , inlet (cid:17) γ − γ − T T , outlet T T , inlet − P T and T T are the mass-flow averaged total pressure and the total temperatureat the specified location and γ = 1 . f = 1 − η + P (3) ttps://doi.org/10.1007/978-3-319-97773-7 46 η is averaged over the last 1000 flow solver iterations and P repre-sents penalty terms which increase (i.e. worsens) the fitness in case the CFD solutiondoes not converge or the generated blade geometry is not feasible.We employ two different optimization algorithms, the particle-swarm optimization(PSO) [3] and the covariance-matrix adaptation strategies (CMA-ES) [4]. Unless oth-erwise stated, for the CMA-ES we always use a populations size of λ = 12 with µ = 4parents and an initial step size of σ = 0 .
05 in relative units where the maximal allowedvariation is normalized to one (i.e., σ = 0 .
05 amounts to 5% initial variation). For thePSO we use 12 particles (which amounts to 12 evaluations per generation) and theparameters ω = 0 . φ = 1 .
7, and φ = 1 .
4. (see Eq.(6) in [3]).
In order to evaluate the impact of the optimization algorithm, we compared the resultsof the two optimization methods, specifically a ( µ = 4 , λ = 12) CMA-ES and a PSOwith 12 particles. For the blade deformation, we chose N HH / section =9 which lead to atotal of 36 parameters for the optimizer algorithms.Figure 3 shows the progress of the optimization where the efficiency of each ofthe proposed individual geometries is shown as function of the optimization gener-ations. The efficiency is normalized to the efficiency of the baseline geometry, i.e.¯ η = η/η baseline . N o r m a li ze d e ffi c i e n c y ¯ η = η / η b a s e li n e GenerationsCMA-ES 0.950.960.970.980.9911.011.021.031.041.05 0 10 20 30 40 50 60 70 N o r m a li ze d e ffi c i e n c y ¯ η = η / η b a s e li n e GenerationsPSO
Fig. 3.
Normalized efficiency of each evaluated blade geometry as function of optimiza-tion generations for CMA-ES (left) and PSO (right). The red marked solutions justindicate the best solution found so far.As can be seen from the Figure 3, the variation within the population in eachgeneration is much larger for the PSO than for the CMA-ES. This is because we used arather small initial step size for the CMA-ES, but initialized the particles and velocitiesfor the PSO randomly within the complete search domain. Thus the PSO performs amore global search, whereas the CMA-ES is more local. However, when looking at theprogress of the best solution found up to a certain generation, both approaches producevery similar results. Also, the fitness values during later stages of the optimization are J. Kmec and S. Schmittapproaching very similar results. For example, the best normalized efficiencies werefound to be ¯ η PSO = 1 . η CMA − ES = 1 . Fig. 4.
Comparison of three independently deformed sections of the final optimizeddesigns with the baseline design. The plots are for the hub (left), the mid-span (middle),and the shroud (right) sections.
Fig. 5.
The difference between the optimized geometries from PSO (left) and CMA-ES(right) to the baseline design where the baseline geometry is shown and the differenceis color-coded. Green color indicates similar geometries in that region, red indicates thePSO/CMA-ES geometry is displaced more into the picture away from the viewer andblue indicates the PSO/CMA-ES geometry is displaced toward the viewer as comparedto the baseline geometry.
The stochastic optimization algorithms used in this work need an initialization of thepseudo-random number generator which is done by specifying a random seed. The ttps://doi.org/10.1007/978-3-319-97773-7 46 N HH / section =7 which lead to a totalof N search = 30 search parameters. The progress of the best solutions produced by theoptimization runs is shown in Figure 6. It can be seen that the initial progress is quitesimilar but after some initial generations, the three runs diverge. During later stagesthe runs converge again to similar efficiency values, ¯ η seed1 = 1 . η seed2 = 1 . η seed3 = 1 . N o r m a li ze d e ffi c i e n c y ¯ η = η / η b a s e li n e Generationsrandom seed 1random seed 2random seed 3
Fig. 6.
Comparison of best solutions found during the optimization as function of thegeneration of three different CMA-ES optimization runs which just differ in the valueof the random seed used to initialize the optimization algorithm.The comparison of the actual best geometries reveals the same insights as in theprevious sections as the final geometries differ strongly between the runs (see Figure 7).
Fig. 7.
Color coded differences between the three optimized geometries obtained forthree different random seeds 1, 2, and 3 (from left tor right) and the baseline geometry.The color coding of the difference is the same as for Figure 5.
Another design choice for the optimizer setup is how many evaluations per generationare made. Here we compare two CMA-ES runs with ( µ = 4 , λ = 12) and ( µ = 4 , λ = J. Kmec and S. Schmitt24) and otherwise identical setup. For the blade deformation, we chose two setupswith N HH / section =9 and N HH / section =12 which lead to a total of N search = 36 and N search = 45 search parameters, respectively.The expectation was that the optimization progress would be faster for higher num-ber of evaluations per generation, if measured in terms of generations since a largerregion of the search space could be sampled in each generation and the informationgathered could be used more efficiently. In practice, with a usually large resource de-mand for the CFD simulations, there is trade-off between the parallel resources (orlicenses) utilized and the overall run-time. This leads to a limit on the number ofevaluations running in parallel. Therefore, the more relevant measure for optimizationperformance is in terms of actual evaluations, i.e. number of CFD simulations. N o r m a li ze d e ffi c i e n c y ¯ η = η / η b a s e li n e Number of evaluations N HH / section = 9 λ = 12 λ = 24 N o r m a li ze d e ffi c i e n c y ¯ η = η / η b a s e li n e Number of evaluations N HH / section = 12 λ = 12 λ = 24 Fig. 8.
The efficiency as a function of the number of evaluations for N HH / section =9(left) and N HH / section =12 (right) for λ = 12 and λ = 24.As can be seen in Figure 8, the optimization progress in terms of actual evaluationswas very similar between the λ = 12 and λ = 24 runs for both setups. The λ = 24 runsseem to have slightly larger spread in each generation. It also seems that for largersearch dimension, i.e. for N HH / section =12, a larger population size is beneficial in theinitial phase of the optimization. Both tendencies are somewhat instructive, but inorder to arrive at a really conclusive evidence, single optimization runs as performed inthis work are not sufficient. Instead, one would need to run each optimization multipletimes and compute statistics over the results. However, this was not done due to thealready high resource demand and long run-times of the CFD simulations.Comparing the achieved efficiency and the actual optimized blade geometries, thesame observations as in the previous sections can be made. Even though the fitness val-ues are comparable, ( N HH / section =9: ¯ η λ =12 = 1 . η λ =24 = 1 . N HH / section =12:¯ η λ =12 = 1 . η λ =24 = 1 . N HH / section =9 whereas the geometries are rathersimilar for N HH / section =12 (see Figure 9). A very important choice in the course of blade optimization is the specification of thenumber of HH shape functions per section. It is expected that this parameter has a ttps://doi.org/10.1007/978-3-319-97773-7 46 Fig. 9.
Color coded differences between the baseline geometry and the optimized ge-ometries for N HH / section =9 and λ = 12, N HH / section =9 and λ = 24, N HH / section =12 and λ = 12, and N HH / section =12 and λ = 24 (from left to right). The color coding of thedifference is the same as for Figure 5.strong influence on the outcome of the optimization. Choosing a rather small numberleads to a very rigid blade representation where only a limited number of designs canbe realized which might not include very efficient blades. On the other hand, a verylarge number might allow to represent a plethora of shapes in principle, including veryefficient designs, but the resulting search space has a very high dimension and thus theoptimization problem becomes very hard. This usually leads to a much slower progressof the optimization and threatens that a sufficiently improved design will not be foundwithin a reasonable time and the limited resources for the search.We explored this trade-off by running several optimization runs and varying thenumber of shape functions ranging from N HH / section =3 up to N HH / section =12. Thisamounted to search dimensions ranging from N search = 18 to N search = 45, and weused the CMA-ES with a populations size of either λ = 12 or λ = 24. N o r m a li ze d e ffi c i e n c y ¯ η = η / η b a s e li n e Generations λ = 12 N HH / section = 3 N o r m a li ze d e ffi c i e n c y ¯ η = η / η b a s e li n e Generations λ = 24 N HH / section = 5 Fig. 10.
The comparison of the best solutions found during the optimization for dif-ferent number of HH functions for λ = 12 (left) and λ = 24 (right) evaluations pergenerations.The best results as a function of the optimization generations are shown in Fig-ure 10. First of all it can be observed that there is no clear trend for the achievableefficiency as function of N HH / section . For λ = 12 the smallest number of shape functionsper section N HH / section =3 indeed gives the lowest improvement, but for example, for0 J. Kmec and S. Schmitt N HH / section =4 a rather large improvement is realized whereas for N HH / section =5 theimprovement is as small as for N HH / section =3. In case of λ = 24, the best efficiency isactually produced with the lowest N HH / section =5.Figure 10 suggests that the optimization runs are not yet converged after 120 gen-erations but instead keep improving with more generations. This is actually expectedfor such population sizes and search dimensions. Therefore, it still could be that thereis a clear relation between the achievable efficiency and N HH / section , but for all prac-tical purposes, with limited computing resources, this is not relevant. The increasedflexibility from a larger number of design parameters is compensated by the increaseddifficulty of the search problem.The preliminary insight from this is that already with a rather small number ofshape functions per section a considerable improvement in the efficiency can be realized.So the choice of N HH / section is not determined by the expected efficiency increase butrather based on the geometrical aspects implied by the blade representation and theavailable optimization run time.We again observe that all the resulting geometries are rather different over largeparts of the blade which can be seen in Figure 11. In particular, the shapes are usuallymuch more curvy and wavy during the optimization for larger values of N HH / section ,but with increasing generations the shapes seem to be smoothed out again even forlarger values of N HH / section (not shown). Fig. 11.
The comparison of the optimized blade geometries for N HH / section =3 (top left), N HH / section =4 (top middle), N HH / section =5 (top right), N HH / section =7 (bottom left), N HH / section =10 (bottom middle), N HH / section =12 (bottom right) where the differenceto the baseline design is color coded and displayed on the blade. The color-coding isthe same as in Figure 5. ttps://doi.org/10.1007/978-3-319-97773-7 46 We studied multiple shape optimization runs for a realistic turbo-fan blade geometrysetup. In order to explore the fitness landscape, we investigated many setups, varyingthe optimization algorithms, the random seed for the initialization, the population sizeand the number of Hicks-Henne shape functions per independently deformable bladesection.The number of Hicks-Henne shape functions does not seem to have a pronouncedinfluence on the achievable efficiency improvement, at least with the realistic constraintthat only a small number of evaluations (i.e. optimization generations) can be done.This most likely directly reflects the trade-off between the increasing flexibility with alarger number of design parameters and the increasing difficulty of the search problem.For the practical application this implies that a small number of HH shape functionscan already be sufficient and geometrical aspects for the location and number of shapefunctions might be equally or maybe even more important.The central finding of this work is that for all the tested variants, comparableimprovements in the rotor efficiency could be achieved but the actual optimized ge-ometries showed substantial variation over the complete blade geometry. This leadsto the conclusion that the fitness landscape of such a realistic turbo-fan optimizationis highly multi-modal with many local minima. Even minor changes (e.g. the randomseed for the initialization of the optimization algorithm) may lead to very differentgeometries with comparable efficiencies.This has strong implications for practical blade design optimization approaches. Onthe one hand, the designer can easily generate a set of design variations which all havecomparable efficiencies by minor changes to the setup. On the other hand, this makessurrogate-assisted optimization approaches [8] more difficult. First of all, training asurrogate model on a highly multi-modal function requires large amounts of data whichare most likely not available thereby reducing the model accuracy drastically. Second,data from finished optimization runs will be dense only around one or a few localminima which makes models trained such data not very helpful for optimizations runswhich end up converging to another local minimum.In the current work we only investigated a single operating condition (fixed mass-flow and rotational speed at cruise condition) and only the rotor passage. A morerealistic scenario should include the downstream stator blade which is know to have astrong influence on the fan blade design. Also, additional operating conditions with dif-ferent mass-flow and rotational speed (e.g. take-off condition) need to be incorporated.The expectation of the authors is that the fitness landscape of such a more realisticsetup will still be multi-modal, probably to a lesser extent than for the current case,but the general findings of this work will still be valid. Such investigations are left forfuture work.
Acknowledgments
The authors thank Hisato Tanaka, Kunio Nakakita, Jiˇr´ı ˇSimonek, Radek M´aca andTom´aˇs Kr´atk´y for valuable discussion.2 J. Kmec and S. Schmitt
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