Variable Fidelity Optimization of Required Power of Rotor Blades:Investigation of Aerodynamic Models and their Application

Gunther WilkeGerman Aerospace Center (DLR) Braunschweig, Institute of Aerodynamics and Flow Technology

Lilienthalplatz 7, 38108 Braunschweig, Germany, gunther.wilke@dlr.de

Abstract

An investigation of aerodynamic models with varying fidelity is performed with respect tothe required power for different rotor blade geometries. Concluding from this investigationlow, mid and high fidelity methods are selected. For hover these comprise a finite state in-flow model, Euler computations on coarse meshes and highly resolved RANS computations,both with periodic boundaries. For forward flight, a prescribed wake model, single bladedEuler computations and a Chimera setup for RANS computations are selected. A compara-tive study between between single fidelity and variable fidelity surrogate based optimization isperformed, where resource savings of up to 64.8% are seen for forward flight optimizations.

1 INTRODUCTION

The aerodynamic optimization of helicopter rotorsis highly complex due the unsteady flow phenomena.In hover, the flow field is dominated by the blade vor-tices and the downwash of the previous blade, while inforward flight transonic effects and dynamic stall areadded. The fluid-structure coupling is mandatory toaccount for the aero-elastic effects of the problem, re-quiring a great amount of computational power for theunsteady analysis and optimization of rotor blades. Inhis survey, Ganguli [14] depicts two routes for the op-timization of rotor blades, either gradient based localoptimization, or the application of genetic algorithmsof global scope within surrogate models of the goalfunction.

The first route is taken by Le Pape and Beaumier[28], who use a gradient optimizer and a CFD solverto find new blade plan form shapes for a 7A and ER-RATO blade. They compute the gradient using finitedifferences, while Choi et al. [32], as well as Dumontet al. [1] use the adjoint methodology to compute thegradient of the goal function more quickly.

The other route, surrogate based rotor optimization,is split into two major surrogate types, artificial neu-ral networks and Kriging. Artificial neural networks areemployed by Sajial et al. [19] to reduce the hub loadsthrough a higher harmonic control device. Massaro etal. [4] apply a genetic algorithm to an artificial neuralnetwork to minimize the required power of a rotor us-ing a lifting line model as well as a panel method cou-pled with a wake model. Performing high fidelity CFDcomputations, Johnson and Barakos [10] develop anoptimization framework consisting of an artificial neu-ral network and a genetic algorithm to optimize thepitching moments of a UH-60 blade in forward flight.Kriging, the other majorly used surrogate, is utilizedby Glaz et al. [8] to reduce vibratory hub loads. Theycompute their aerodynamics using a rational function

approach, which models the unsteady aerodynamicsof rotors with flaps. Vu et al. [25] applies Krigingfor the optimization of rotor airfoils of a BO-105 rotorcomputed with a 2D panel code. A combined aero-dynamic, aero-acoustic, CFD oriented optimization isundertaken by Chae et al. [30] for a total of 19 designvariables. Another high fidelity CFD optimization for ahover and a forward flight test case is carried out byImiela [17].

A different surrogate based procedure is gone byCollins [9], who couples two surrogates models ofvariable fidelity (VF) together using response surfaceto quickly access the goal functions of vibratory hubloads and torque. He uses a lifting line model as wellas hybrid CFD/wake coupled method for the simula-tion of the rotor aerodynamics.

This work continues Collins’s idea, where the sur-rogate model is mostly created using low fidelity sim-ulations, such as the blade element theory, and thencalibrated with CFD data to yield high fidelity results.Recent approaches for variable fidelity surrogates areeither Co-Kriging or Hierarchical Kriging. The firstapproach fuses high and low fidelity data togetherthrough a combined correlation matrix, while Hierar-chical Kriging is based upon supplying the trend func-tion from a low fidelity model instead of a polynomialfunction. Forrester et al. [3] as well as Yamazaki andMavripilis [34] perform Co-Kriging based drag mini-mization of a wing and of a NACA0012 airfoil, respec-tively. The later also takes advantage of gradient infor-mation. Hierarchical Kriging is used by Han and Görtz[15] for aero-loads prediction, and Xiong et al. [33]show the application of it to surrogate based optimiza-tion for two synthetic and one engine piston problem.

The optimization performed in this paper focuses onKriging based optimization of rotor blades using vari-able fidelity methods. The advantage of Kriging mod-els over artificial neural networks is that they featurethe metric of expected improvement. This metric takes

into account the goal function value, as well as the un-certainty of the model. Jones et al. [11] are the firstto build a surrogate based optimization framework ex-ploiting the expected improvement function, referredto as Efficient Global Optimization (EGO) algorithm.

Before this variable fidelity optimization procedurecan be applied, research of the applicability of thevarious available aerodynamic models has to be per-formed. This is the first part of this paper, which fo-cuses on the analysis of these models with respect tothe goal function of required power dependent on theblade geometry for two flight conditions. After suitablemodels are identified, they are applied in in a variablefidelity optimization, which is benchmarked with a sin-gle fidelity optimization using the same optimizationstrategy.

2 AERODYNAMIC MODELS

One key discipline for the proper analysis of heli-copter rotor blades is the modeling of the aerodynam-ics. However, as a regular helicopter blade is a slen-der wing with a high aspect ratio, the elastic effects arenot negligible. On top of that, for a proper comparisonof the performances between two rotors, the implicitconstraint of a trimmed aircraft becomes necessary.To account for the elastic effects, as well as ensuringa proper trimmed condition, the comprehensive codeHOST [7], majorly developed by Eurocopter, is used.HOST features a quasi 1D-beam elastic model ex-tended from Bernoulli’s theory. The trim procedure forstatic flight conditions, such as hover and cruise flight,is given by a Newton-Raphson method. As most com-prehensive codes, HOST features some simple aero-dynamics, but at the same time allows the coupling tomore sophisticated methods, which are covered in thefollowing.

2.1 Blade Element Theory

The idea of the blade element theory (BET) is touse look-up tables of the aerodynamic coefficients ofthe individual airfoil sections on the blade and thenintegrating them spanwise, as well as over azimuth.One major difficulty of the BET is the proper finding ofthe induced inflow angle, which is required to computethe sectional angle of attack. For hover flight, the BETcan be extended with the momentum theory to pro-duce more realistic results, while in forward flight gen-eralized inflow models improve the aerodynamics. Adetailed description is found in Leishman [22]. HOSTfeatures some semi-empirical corrections, mentionedin [13]. Nevertheless, even these improvements donot account for all physical phenomena, and particu-lar 3D effects such as blade vortex interaction (BVI)are neglected or only semi-empirically modeled.

2.2 Finite State Inflow Models

HOST features a finite state inflow model (FISUW)developed by Basset et al. [26] which uses a modi-fied accelerated potential theory to compute the inflowmore realistically. The advantage of this approachcompared to the classical inflow models is that thenumber of applied polynomials and harmonics can beset for the azimuthal and radial directions. The in-flow velocity is determined by the forces on the blade,which are given from the look-up tables. Thus, an it-erative procedure is applied to find an equilibrium be-tween aerodynamic forces and the inflow velocities.Yet, FISUW is also not able to capture any BVI effects,but the downwash is accounted for more realistically.

2.3 Wake Vortex Modeling

There are two general variants of wake vortex mod-eling; prescribed and free. The principle is derivedfrom the lifting line theory, where the bound circula-tion of the rotor is computed from the forces, whichthen yields the induced velocities. The difference be-tween prescribed and free wake models is given bythe description of the vortex field. While the pre-scribed model moves the vortices according to a semi-empirical definition, as done in the HOST moduleMETAR [6], [13], a free wake model uses the inducedvelocities to move the wake filaments downstream,which results in a better representation of the wake.Latter approach has been implemented in the MESIRcode by Michea [24]. The disadvantage of free wakemodels is that in practical applications they are lessstable than prescribed wake models. In general, wakemodels allow the computation of BVI effects, but as-sume incompressible, inviscid and irrotational flow inthe wake.

2.4 Panel Methods

At the DLR, a code called UPM is developed by Yinand Ahmed [36], which does not use airfoil polars tocompute the loads, but a panel code, which allows therepresentation of the blade as a surface, rather thana line. The panel method is based upon the potentialtheory, which assumes incompressible, inviscid and ir-rotational flow, just as the vortex models. UPM itselfincludes an unsteady free wake model, which is cou-pled with the panel method. It is successfully appliedto aero-acoustic problems in [35]. One strength of thisapproach is that BVI effects can be captured, whiletheoretically requiring less computational effort thanCFD methods. Compressible effects are accountedfor by Glauert’s correction (1/

√|M2 − 1|) in UPM. A

disadvantage of UPM is that it can not be coupled withHOST (yet), and thus elastic effects are neglected.However, a trim procedure is contained within UPM.

2.5 CFD

In this paper, CFD methods refer to the solutionof the Euler and RANS equations using the FLOWercode [20] of the German Aerospace Center (DLR).The Euler equations neglect viscous effects, but areable to capture 3D effects, as well as compressibleflows. Eventually the shock locations are not matchedas well with Euler as with RANS. The RANS equationsadditionally account for viscous terms through a turbu-lence model. In this work the Wilcox k − ω turbulencemodel is used. For more details on the CFD setup,see Imiela [17]. The advantage of FLOWer coupledwith a comprehensive code is that it yields high fidelityresults [27], but is computationally expensive, and dueto numerical viscosity added for stability, vortices tendto dissolve too fast and certain BVI aspects may notbe modeled.

2.6 Comparison

Guided from the previous discussion, Table 5 inthe appendix gives a concluding overview of the abili-ties and points out the weaknesses of each individualmethod. In this paper, these methods are categorizedinto low-, mid- and high fidelity methods based upontheir theoretical abilities to model all physical effectsof rotor aerodynamics properly. The BET and BETcoupled with momentum theory (BEMT), generalizedinflow models and finite state inflow models are cat-egorized as low fidelity methods. These are usuallysuited for preliminary design as well flight mechani-cal considerations. Wake modeling with the BET orpanel methods are mid-range fidelity, while solving theRANS equations is considered high fidelity. It has tobe noted though, that even RANS has drawbacks. Thenumerical dissipation is still eminent, and turbulencemodeling is not the best option for dynamic stall sim-ulations. Other methods, such as vorticity transportmodels (VTM) [23], or further developed CFD meth-ods like time-spectral and large or detached eddy sim-ulations (LES/DES) [31] are not taken into account inthis study.

3 INVESTIGATION OF AERODYNAMIC MODELS

The goal of the first part of this work is to identifysuitable aerodynamic models to be applied later onfor the variable fidelity optimization. Therefore, an in-vestigation of the described models from Section 2 isperformed with four different design parameters variedin two flight conditions. These parameters are takenfrom Imiela [18], namely tip twist for a linear twist dis-tribution, tip chord tapering, horizontal and vertical off-set of the tip, here synonymously used for anhedraland sweep. Fig. 1 depicts the design variables forclarity. The base line rotor is a 7A rotor [5] with 2.1m

z

x

zy

tip

tip

ch

ord

swee

p

anh

edra

l

root airfoil

tip airfoil

twist

quarter chord

y

x

Radius

root chord

OA213 OA209

airfoil transition

Figure 1: The parameter anhedral, chord, sweep andtwist for the modification of a 7A rotor.

Parameter baseline lower uppervalue bound bound

anhedral [∗cref ] 0.0 -1.0 1.0tip chord [∗cref ] 1.0 0.5 1.5sweep [∗cref ] 0.0 -1.0 1.0twist [◦] -4.32 -20.0 0.0

Table 1: parameter range. reference chord lengthcref = 0.14m.

radius, and the beginning of the tip is set at 80.6 % ro-tor radius = 1.6926m. In Table 1, the range and initialvalue of each parameter are summarized. The rootchord length is adjusted to match a thrust equivalentweighted chord length of cref = 0.14m when the tipchord length is modified. For a robust meshing pro-cedure, the trailing edge is tapered, instead of a tap,which is featured by the original 7A rotor. As for thestructural beam model, only the quarter chord positionis adjusted, stiffness and inertia properties remain thesame for all configurations.

The two investigated flight conditions are hover andforward flight, both trimmed for a vertical force of4400N with a tip Mach number of 0.646. In forwardflight the advance ratio is µ = 0.38 corresponding to aflight Mach number of 0.2468, while a virtual fuselagedrag of 530N has to be overcome.

The investigation is organized as follows; at first thebaseline rotor (7A) is computed, secondly a parametervariation study is performed, where each parameter ismodified individually, and finally a genetic algorithm isapplied to directly optimize on the different models tofind the respective optima. The different models aredesignated as follows:

• BET: Computations based on the blade elementtheory. In forward flight the Pitt and Peters [29]generalized inflow model is used

• BEMT: BET coupled with the momentum theory,only valid for hover

• FISUW: BET coupled with a finite state inflowmodel

• PWAKE: BET coupled with a prescribed wakemodel

• FWAKE: BET coupled with a free wake model

• EU: Euler computations on a coarse mesh. Inhover a periodic boundary condition is used, inforward flight only a single blade is modeled, thusneglected the effects of other blades.

• NS: Navier-Stokes computations. Just as EU, ex-cept with 15 points in the boundary layer included.

• FNS: Navier-Stokes computations on a fine mesh.In hover a periodic boundary condition is used, inforward flight a four-blade chimera setup is used.

In general, hover is considered quasi-steady, while for-ward flight is handled as an unsteady computation.FNS is considered the highest fidelity available, eventhough better models exist. For the specific details onthe discretization of each method, check Table 6 in theappendix.

3.1 Baseline Rotor

The baseline rotor is computed for each individualmethod in hover and forward flight. Fig. 2 visual-izes these results along with the results of the puremomentum theory denoted MT. The momentum the-ory defines the absolute possible minimum for therequired power for any rotor of this radius, as it as-sumes inviscid, irrotational and incompressible flow ofa stream tube.

In hover, the BEMT is in close agreement with theFNS computations, while the wake couple methodsgenerally under predict the required power. The in-duced power from the FNS computation is 66.45kWin hover, which is mostly captured by the EU method.However, PM predicts a required power close to therequired power given by the momentum theory. Thisis a questionable result, as the momentum theory as-sumes a constant inflow over the disc, which the 7Arotor does not have. The over prediction of requiredpower of the NS method is attributed to the higher nu-merical dissipative nature of coarse meshes, and aroughly resolved boundary layer.

In forward flight, the FWAKE method is in closeagreement with FNS. The prediction from the PM is

MT

BE

T

BE

MT

FIS

UW

PW

AK

E

FW

AK

E

PM

EU

NS

FN

S

0

20

40

60

80

100

120

140

Hover Forward Flight

methods

req

uir

ed

po

we

r [k

W]

Figure 2: Bar chart of the required power computedby each method for hover and forward flight.

rather high. Reason for this is the lack of properly ac-counting for compressibility effects, as well as missingaero-elastic torsion, which is not modeled. This re-sults in very high pitch control angles, leading to an in-crease in drag. The EU method computes about 20%less required power than FNS, which is in the range ofthe inviscid power computed by FNS, making it a validresult within the physical assumptions. The NS meth-ods over predicts the power, just as it does in hover, forthe same reasons; coarse dissipative grid and roughlyresolved boundary layer. BET, FISUW and PWAKEover predict the required power. For the first two meth-ods, this comes from the general methodology, whilefor PWAKE the prescribed wake does not match aswell as a free wake, leading to an offset in power.

For the optimization process, in particular in the VFcontext, the right position of the optimum in the designspace is important, not so much the absolute value.The assumption is that it can be computed afterwardswith a high fidelity method in order to obtain the rightabsolute value. Thus, all future results concerning thegoal function of required power, will be divided by thebaseline value of the respective method and flight con-dition.

3.2 Sensitivity Study

The goal of the sensitivity study is to practicallygrasp the behavior of the goal function with respectto geometry changes. Fig. 4 to Fig. 11 depict the pa-rameter variations in hover and forward flight with thelegend given in Fig. 3. At a first glance at these fig-ures one can see that similar methods, BET, BEMT,FISUW, or PWAKE, FWAKE, or PM, or EU, NS, FNS,yield similar tendencies of parameter preference stem-ming from the similarity in their physical assumptions.Most trends of FNS are captured by EU fairly well,which in turn means that for these goal functions vis-cous effects are of minor importance. This statementmay not be valid when an airfoil optimization is in-

cluded.

Figure 3: Legend for the sensitivity figures.

In hover, the results are widespread. The anhedralparameter (Fig. 4) only seems to have an effect onthe structure when using low fidelity methods (BET,BEMT, FISUW), while wake coupled methods capturea different trend, as the CFD methods. The optimumof the anhedral parameter for the wake models lies at−1 with a power reduction to 82% and 75% for PWAKEand FWAKE respectively, but is intentionally not plot-ted in Fig. 4. The benefits of an anhedral or dihedral inhover are greater than in forward flight (Fig. 5), wherethe slight aerodynamic gain is lost due to the aero-elastic effects. Points beyond an absolute value of 0.3anhedral are not convergeable for some methods, asseen in the Fig. 5.

The trend of the tip chord parameter (Fig. 6, Fig. 7),which tapers the blade, is captured similarly for allmethods, and flight conditions. As it offloads the bladeon the outboard tip, the induced velocity is reduced, aswell as the viscous drag, if modeled, at the tip, whichhas the greatest moment arm.

Blade sweep (Fig. 8, Fig. 9) brings an improve-ment of the goal function for all methods and flightconditions. In hover flight, two optima exist for BET,FISUW and the CFD methods; a forward swept bladeis slightly more favorable than a backward swept bladefor EU and FNS. For all other methods, backwardsweep is perceived more valuable. In forward flight,due to strong blade torsion at the tip, only backwardsweep is advantageous. Some values can not becomputed, as trimming the blade proofs to be impos-sible. The PM method, as a result of missing elasticmodeling, leads to an infeasible preference of forwardsweep in forward flight.

A parabola like curve is given for the twist param-eter (Fig. 10, Fig. 11) for all methods in both flightconditions. Yet, the position of the minimum variesacross the methods. The low fidelity methods usuallyhave this optimum at low angles, while with increas-ing fidelity this value rises. However, EU and NS pre-dict a greater twist angle than FNS. This arises fromthe coarse mesh, where more numerical dissipation is

given.Overall, it is noticed that the wake coupled methods

show stronger trends of all parameters than the othermethods. For example the effect on the anhedral pa-rameter in hover grants an improvement of over 19%for PWAKE and over 21% for FWAKE. For FWAKE thisleads to an absolute value for the required power of45.3kW in hover, which is below the theoretical valueof 54.1kW . Results from these methods should thusbe looked upon with skepticism, and are reasons toexclude these models for hover optimizations.

Concluding from this sensitivity study, the down-wash is the main driver for the hover case seen fromthe effect of blade twist, while the lift redistribution to-wards the inboard rotor caused by the blade sweepingand tapering brings the most improvement in forwardflight.

3.3 Direct Optimization

In order to identify suitable models for a variable fi-delity rotor optimization, a direct optimization using thegenetic algorithm SOGA [12] in the Dakota Toolboxis performed for each aerodynamic model and bothflight cases. The advantage of using a genetic algo-rithm is that the likelihood of finding a globally validoptimum is high, yet the major drawback is that it isvery resource consuming and the local accuracy islimited. The criteria for selecting the various meth-ods is based upon their robustness, physical feasiblerepresentation of the required power and the compu-tational time required to execute the simulation. Anexception is made for the four-bladed Chimera com-putation in forward flight using RANS, for which theoptimum is found through the single fidelity optimiza-tion described in Section 4, as direct optimization istoo costly.

3.3.1 Hover

In Table 2, the metrics for the final rotor of eachhover optimization is listed. The column featuring FNSshows the value of this individual rotor configuration,when it is re-computed with FNS. The idea is to seehow much the optima correspond with each other. Inorder to visualize these various parameters, the dif-ferent blade configurations are shown in Fig. 15. Thegain of the anhedral parameter is not captured by thelow fidelity methods. The wake coupled methods all doprefer an anhedral, while the CFD methods tend to adihedral. NS does not go for such a high value of dihe-dral, given the fact that the tip vortex dissipates fasteron the coarse mesh. The EU method has a similartrend as FNS, as it is possible to tune out the numer-ical dissipation more than for NS. This leads to insta-bilities with rotors for which transonic effects occur inhover, yet these are non-optimal configurations at the

-- -- ----

---

+ ++ ++++

++++

EE

EE

E EE

E

E

E

E

* ** ** **

**

**

7A

anhedral

Pow

er %

-1 -0.5 0 0.5 10.9

0.95

1

1.05

1.1

Figure 4: anhedral variation in hover.

- -

-

-

E

EE

E E

**

*

*

*

7A

anhedral

Pow

er %

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

Figure 5: anhedral variation in forward flight.

- - - - - - - - - - -+ + + + + + + + + + +

EE

EE

EE

EE

EE

E

**

**

**

**

**

*

7A

chord

Pow

er %

0.6 0.8 1 1.2 1.40.9

0.95

1

1.05

1.1

Figure 6: chord variation in hover.

- - - - - - - - - - -

E

E

E

E

*

**

*

*

7A

chord

Pow

er %

0.6 0.8 1 1.2 1.40.9

0.95

1

1.05

1.1

Figure 7: chord variation in forward flight.

same time and thus are left out of consideration for theoptimizer. All methods except FWAKE yield a taperedtip, which corresponds to the findings of the sensitivitystudy. From the different behavior of the sweep pa-rameter, one can see that backward or forward sweep-ing of the blade depends on the method. From inves-tigating single individuals of the FNS optimization, it isfound that the required power is a multi-modal functiongiven from forward and backward sweeping. For thetwist parameter, a similar finding to the one given bythe sensitivity study is made; with increasing fidelitythe tip twist angle grows larger, except for FWAKE.The final angle is somewhat larger for EU and NS thanFNS, yet not too far off.

An intermediate conclusion from these results is thatthe simple low fidelity methods are not appropriate foroptimizing anhedral, and the twist parameter is toolow. The wake coupled methods (PWAKE,FWAKEand PM) are sometimes unstable in their prediction, inparticular FWAKE. The final rotor configurations yield

Method Power anh. chord sweep twist FNSBET 0.986 -0.05 0.89 0.97 -1.74 1.003BEMT 0.988 0.39 0.60 0.83 -5.97 0.976FISUW 0.979 -0.04 0.61 0.97 -5.48 0.963PWAKE 0.749 -0.99 0.62 -0.74 -9.16 0.959FWAKE 0.337 -0.98 1.41 -0.99 -2.56 1.045PM 0.767 -1.00 0.89 0.69 -9.43 0.980EU 0.879 0.98 0.64 -0.95 -16.52 0.933NS 0.898 0.25 0.63 -0.97 -16.52 0.938FNS 0.917 0.76 0.54 -0.97 -12.34 0.917

Table 2: Final results of the different optimization inhover.

power requirements close to or even below the the-oretical minimum, which is suspicious. The group ofCFD results agree mostly with each other. However, itshould be kept in mind that numerical dissipation hasa great impact on the tip vortex and thus on the an-hedral parameter, as seen with NS in contrast to EUand FNS. Given these findings, the proposed methodsfor hover optimization in a VFM framework are listed

-- -- --- -- -- ++ ++ +++++++

EEE EE EE EE EE

* * * ** ** *** *

7A

sweep

Pow

er %

-1 -0.5 0 0.5 10.9

0.95

1

1.05

1.1

Figure 8: sweep variation in hover.

--

-

-

-- -

-

E

E

EE

EE

E

E

*

****

7A

sweep

Pow

er %

-1 -0.5 0 0.5 10.9

0.95

1

1.05

1.1

Figure 9: sweep variation in forward flight.

--- ----

---

-

++ +++ +

+

+

E EE EEE

E

E

E

E

E

** ****

* *

*

*

*

7A

twist

Pow

er %

-20 -15 -10 -5 00.9

0.95

1

1.05

1.1

Figure 10: twist variation in hover.

- --

- -

-

E EE

E

EE

E

E

E

** * **

*

7A

twist

Pow

er %

-20 -15 -10 -5 00.9

0.95

1

1.05

1.1

Figure 11: twist variation in forward flight.

on the left side of Table 4.

3.3.2 Forward Flight

In a similar fashion to the hover case, Table 3 sum-marizes the findings from the forward flight optimiza-tion with the blades depicted in Fig. 16. In forwardflight, the results are more in harmony than in hover.BET, FISUW and PM are misleading in terms of thechord parameter, for which the optimization processfeatures a widening of the blade towards the tip. Fur-thermore, PM features an optimum analog to the hoveroptimum seen from the preference for anhedral giventhe lack of structural modeling. The other solutions in-dicate no strong favoring of dihedral, a full backwardsweep, and a tip twist slightly greater than the base-line.

Deducting from these observations, the available(low fidelity) tools for computing rotor blades are moresuitable for (fast) forward flight optimizations than for

Method Power anh. chord sweep twist FNSBET 0.926 0.25 1.18 0.95 -5.48 0.989FISUW 0.866 0.25 1.46 0.86 -5.48 1.009PWAKE 0.928 -0.01 0.57 0.94 -4.52 0.952FWAKE 0.868 0.25 0.69 0.86 -7.70 0.957PM 0.577 -0.99 1.48 -0.58 -2.61 1.380EU 0.904 -0.01 0.84 0.97 -5.48 0.968NS 0.985 -0.04 0.70 0.86 -4.13 0.964FNS 0.941 0.10 0.50 1.00 -5.34 0.941

Table 3: Final results of the different optimizations inforward flight.

hover optimizations. For forward flight, a suitable com-bination of methods for a VF framework given on theright side of Table 4. The prescribed wake is favoredover the free wake model, as it proofed to be morerobust, and required a fraction of the computationaleffort.

Fidelity hover forward flightlow FISUW PWAKE

8 cpus 1 cpummid EU EU

5 cpuh ≈ 75 cpuh ≈2,250 FISUW 4,500 PWAKE

high FNS FNS160 cpuh ≈ 32 EU 2600 cpuh ≈ 35 EU≈ 72,000 FISUW ≈ 157,500 PWAKE

Table 4: Choices of low-,mid and high fidelity for eachflight condition and their CPU times. (cpus=cpu sec-onds, cpum=cpu minutes, cpuh=cpu hours)

4 OPTIMIZATION FRAMEWORK.

In order to take advantage of the findings of the pre-vious section, a framework for VF optimization is de-veloped. It features a similar approach as a regularsurrogate based optimization (SBO) framework as ex-plained in depth in [2], or [11]. For completeness, theVFM process is sketched in Fig. 12. A design of ex-periments (DoE) for high and low fidelity data is per-formed to gather the initial samples required for cre-ating the surrogate model. Optionally, the low fidelityoptimum is found and this point is sampled in the highfidelity DoE. After the VF surrogate model is created,an iterative loop is started, where the design with thehighest expected improvement is searched and com-puted with the HF simulation. The obtained result is in-serted into the surrogate model and the loop is startedover until the optimization is finished.

Opposing to the Co-Kriging methodology applied in[3] and [34], this VF Kriging method is inspired by Hier-archical Kriging seen in [15] and [33]. The low fidelitymodel is built from a single fidelity Kriging of an arbi-trary polynomial for the trend function:

(1) f̂LFM (~x) = ~fv(~x)T ~βLFM + εLFM (~x)

where f̂ is the predictor value, ~x the design vector, ~fv

the polynomials, ~β the polynomial coefficients and εthe error modeled by the Gaussian process. For theHierarchical Kriging model, the polynomial trend is ex-changed with the low fidelity model:

(2) f̂HFM (~x) = f̂LFM (~x)ρ+ ~xT ~βHFM + εHFM (~x)

This way, the low fidelity model f̂LFM is included by ascaling parameter ρ, and a first order bridge functionis given by ~xT ~β. Here, for the regular Kriging the poly-nomial order is set to two. If the amount of sampledpoints is insufficient for determining all coefficients, thepolynomial order is reduced, while for the HierarchicalKriging the bridge is removed.

The strategy for finding the optimum in the surro-gate model is simplified by running a full factorial DoE

HF Sim

LF Sim

DoE HF

DoE LF

Create LFM

Create VFM

Optimize

Find optimum in VFM

Compute Optimum withTruth function

Update VFM

HF Sim

finished? Best bladeNo Yes

Create LFM LFM Optimum

Figure 12: VFM Framework for the fusion of low(LFM) and high (HFM) fidelity methods.

with eight samples in each spatial direction, and start-ing the local searcher by Hooke and Jeeves [16] fromthe best point found. This is somewhat a brute forcemethod and is only considered feasible for this smallnumber of parameters of four, and should not be usedfor more parameters. The method is also appliedfor tuning the hyper parameters of the Kriging model,which optimize the concentrated likelihood function.

5 SURROGATE BASED OPTIMIZATION

In this section the optimization framework is appliedbased on a low and a mid-fidelity method. The low tomid fidelity optimization is considered more difficult asthe trend function is further away than for the mid tohigh fidelity optimization. For this test, several initialsampling strategies are used.

The point sampling starts the update process withonly one selected point in the DoE. For single fidelityoptimization using the mid-fidelity method only, thispoint is the 7A rotor, while for variable fidelity optimiza-tion, the best low fidelity optimum is used, as predictedby the low fidelity surrogate model.

The hyper cross sampling additionally computesneighboring points left- and right of the optimum foreach direction of the design space similar to finite dif-ferences.

For comparison with Monte-Carlo samplings, tencentral voronoi tessellated (CVT) cubes [21] are gen-erated containing 12 samples each.

The DoE for the low fidelity model is a full factorialcube with four samples in each direction, leading to256 samples in total.

10 5 0 5 10 15iterations

0.86

0.88

0.90

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0.94

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curr

ent

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ion

Design of Experiments Optimization Cycle

EGO PointEGO HypercrossEGO 12 CVT avg

VEGO PointVEGO HypercrossVEGO 12 CVT avg

Figure 13: Comparison single (EGO) and variable fi-delity (VEGO) optimization in hover.

Fig. 13 and Fig. 14 graph the best goal functionvalue sampled so far over the current iteration in theprocess for hover and forward flight optimizations, re-spectively. The single fidelity SBO is marked EGO inthe merit of the optimizer developed by Jones et. al[11], while the variable fidelity SBO is referred to asVEGO (Variable fidelity EGO). The first iteration rep-resents the first update sample, while all previous iter-ations belong to the DoE. The graphs might descentbefore an actually optimization takes place, becausesome points in the DoE might already be better thanothers.

In hover, the gain from the variable fidelity approachis less than in forward flight likely due to the more com-plex goal function and the greater difference betweenthe low- an mid fidelity model. Still, for both flightcases, it is understood that the variable fidelity op-timization is better when the deterministic samplingsof a point and hyper cross around the low fidelityoptimum are used. In hover, the random 12 pointCVT sampling reduces the number of optimization cy-cles, as the function is more complex than in forwardflight. However, for the random 12 point CVT sam-pling, the superiority of variable to single fidelity van-ishes. Reason for this is the saturation of informa-tion with increasing number of samples. The singlefidelity model, based upon pure Kriging, is more flexi-ble as it can adjust the polynomials of the trend model.For Hierarchical Kriging, the trend function cannot betweaked as much, as it is fixed by the low fidelitymodel.

Nevertheless, for a small number of parametersand a rare amount of samples the variable fidelity ap-proach succeeds for both flight conditions. The pointsampling yields the smallest total required number of

10 5 0 5 10 15iterations

0.90

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curr

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EGO PointEGO HypercrossEGO 12 CVT avg

VEGO PointVEGO HypercrossVEGO 12 CVT avg

Figure 14: Comparison single (EGO) and variable fi-delity (VEGO) optimization in forward flight.

computations for single as well as variable fidelity op-timizations. Comparing the required resources, 69.2%in hover and 35.2% in forward flight of the mid-fidelitysamples are required to reach the optimum in contrastto the single fidelity optimizations. As for the accu-racy of the results, the optimal configurations obtainedfrom the individual runs are in harmony with the onesfrom the direct optimization from Section 3.3. They areslightly better by less than 2% for all cases and in thesame region of the design parameters.

6 CONCLUSIONS

An investigation containing a range of computa-tional methods for rotor aerodynamics ranging fromthe blade element theory to computational fluid dy-namics has been performed. The focus was on theanalysis of the required power with respect to geom-etry changes of the rotor blades. Therefore, the 7Arotor has been computed for all methods as a refer-ence solution. Following that, a sensitivity analysis, aswell as an optimization using a genetic optimizer havebeen performed, to see which method best finds thetrend of the high fidelity model.

In hover, the anhedral parameter was perceivedvery differently, while chord tapering and sweep hadsimilar behavior among the methods. The twist pa-rameter followed a similar trend, yet the optimal loca-tion varied strongly across the methods.

The agreement of these parameters was generallybetter in forward flight across the various methods.Only the panel method, which did not include elasticdeformations, yielded questionable results. The plainblade element theory, as well as the finite state inflowmodel did not taper the chord in the full optimization.Optimizations with the other methods returned similarblades with similar performances.

Concluding from this investigation low, mid and highfidelity models have been selected for the hover andthe forward flight case. For hover these models are afinite state inflow model, Euler CFD computations ona coarse mesh, and Navier-Stokes computations on afine grid. Opposing to this, a prescribed wake model,single blade Euler computations on a coarse mesh,and a four-bladed Chimera setup using the Navier-Stokes equations are selected for forward flight op-timization. While in forward flight a single optimumexists which is in compliance with most methods, inhover at least two optima exist, which also vary acrossthe models. This makes hover a more difficult case interms of optimization.

A variable fidelity framework based on HierarchicalKriging has been designed and tested for the samefour parameters from the initial investigation using lowand mid fidelity methods. For both approaches, vari-able as well as single fidelity optimizations, it wasseen that starting only with one initial sample and pro-ceeding with sampling the expected improvement inthe surrogate yielded the least amount of samples re-quired in total. The variable fidelity approach helps toreduce the required computational resources throughthe decrease of high fidelity computations to 69.2%and 35.2% in hover and forward flight, respectively.The greater reduction in forward flight is explained withthe less complex goal function. An effect of surrogatemodel saturation has been seen, where single andvariable fidelity models become equally good for anincreasing number of initial samples. No advantagefrom variable to single fidelity is seen for the initial ran-dom 12 point sampling, independent of the flight con-dition.

7 OUTLOOK

Currently the mid-high fidelity optimization is under-way, and is expected to work better than low-mid fi-delity optimizations. Future work will include a broaderrange of design parameters, as well as constraintmulti-objective optimizations.

References

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Figure 16: Best blades of the different optimizations in forward flight.