Paper ID: ETC2019-263 Proceedings of 13th European Conference on Turbomachinery Fluid dynamics & ThermodynamicsETC13, April 8-12, 2019; Lausanne, Switzerland

GRADIENT-FREE AND GRADIENT-BASED OPTIMIZATIONOF A RADIAL TURBINE

N. Lachenmaier1 - D. Baumgartner2 - H.-P. Schiffer3 - J. Kech11MTU Friedrichshafen GmbH, Maybachplatz 1, 88045 Friedrichshafen, Germany

2Technical University of Munich, Chair of Structural Analysis, Arcisstr. 21, 80333 Munich,Germany

3Technical University of Darmstadt, Gas Turbines and Aerospace Propulsion,Otto-Berndt-Straße 2, 64287 Darmstadt, Germany

ABSTRACTA turbocharger’s radial turbine has a strong impact on the fuel consumption and tran-sient response of internal combustion engines. This paper summarizes the efforts to designa new radial turbine aiming at high efficiency and low inertia by applying two differentoptimization techniques to a parametrized CAD model. The first workflow wraps 3Dfluid and solid simulations within a meta-model assisted genetic algorithm to find an ef-ficient turbine subjected to several constraints. In the next step, the chosen turbine isre-parametrized and fed into the second workflow which makes use of a steepest descentalgorithm to further fine-tune the design. This requires the computation of gradients withrespect to the CAD parametrization, which is done by calculating and combining surfacesensitivities and design velocities. Both methods are applied successfully, i.e the first de-livers a well performing turbine, which, by the second method, is further improved by0.34% in efficiency.

KEYWORDSTurbocharger, Radial Turbine, Optimization, Adjoint sensitivity, Efficiency, Inertia

NOMENCLATURELatineext Extrapolated relative error [−]fe Eigenfrequency

[1/s]

fj Cost function j [−]h Specific enthalpy

[J/kg

]J Inertia

[kg ·m2

]K Stiffness matrix [−]m Mass flow

[kg/s

]m Rotor mass [kg]M Mass matrix [−]n Rotation rate [rpm]p Pressure [Pa]s Parameter set [−]T Temperature [K]v Velocity

[m/s

]zg Center of gravity [m]

Greekκ Isentropic exponent [−]η Efficiency [−]σ Stress tensor [MPa]Subscriptsgb gradient-basedgf gradient-frees Static quantityt Stagnation quantityAbbreviationsCFD Computational Fluid DynamicsCHT Conjugate Heat TransferCSM Computational Structural MechanicsGA Genetic algorithmRANS Reynolds-averaged Navier StokesSTL Standard Tessellation Language

OPEN ACCESSDownloaded from www.euroturbo.eu

1 Copyright © by the Authors

INTRODUCTIONThe stationary and transient performance of large combustion engines is shaped by the de-

sign of its turbocharger system, i.e. its compressor and turbine. In the present paper a radialturbine is designed with focus on both engine efficiency and transient response. These goals aredirectly linked to the turbine efficiency and its inertia. Furthermore, a high life expectancy isrequested which constrains the stress levels.

This problem is tackled with two different, subsequent design steps: The first is a gradient-free multidisciplinary optimization based on a meta-model assisted evolutionary algorithm. Itcan be considered standard in the sense that there are numerous publications that use comparablemethods to improve rotor designs, see Verstraete [2008], Muller et al. [2013], Peter [2016] orKhairuddin and Costall [2018]. This method is capable of exploring the design space, however,its success is limited as the calculation effort scales with the number of design parameters, i.e.increasing the design space or, respectively, the potential for improvement comes with greatcomputational cost.

To overcome this major drawback, the global optimization in the first step is combinedwith a gradient-based optimization in a second step. Additionally, the gradients in the secondstep are computed following an adjoint technique, which has the advantage that the effort periteration does not scale with the number of parameters and promises fast design improvements.Differentiating every available cost function with high accuracy and subsuming all of them in asingle optimization is, however, challenging, which might be why few authors have presentedcomparable multidisciplinary design chains. Yet, there are some and the potential for designimprovements has proven to be considerable, see Verstraete et al. [2017a] or Schwalbach et al.[2018].

In the following, both methodologies will be presented: In the next chapter, the gradient-free framework is detailed and a turbine is designed. In the subsequent chapter, this turbine isfurther improved by applying the gradient-based workflow.

GRADIENT-FREE OPTIMIZATION

CSM, CFD and CHT mesh

Run CFD (adiabatic)

Run CHT

Run CSM

Geometry

Automized design evaluationwithin optimization process

• design parameters 𝒔

• ሶ𝑚• 𝑇𝑡,𝑖𝑛(𝑝𝑡,𝑖𝑛)• Rotation rate 𝑛• Turbine power 𝑃𝑇

• Maximum rotationrate 𝑛𝑚𝑎𝑥

• Tshaft

• 𝜂𝑡𝑠 , 𝜂𝑡𝑡

• 𝑓𝑒, 𝜎, 𝐽𝑧, 𝑚, 𝑧𝑔

Figure 1: Gradient-free setup

Fig. 1 visualizes the gradient-freeworkflow that will be detailed in this firstpart of the paper. Initially, a fluid and so-lid geometry are created and meshed ba-sed on a parametrized CAD-model and aparameter set s. The simulation is per-formed in three steps: First, an adiabaticCFD (Computational Fluid Dynamics)simulation is conducted to calculate theefficiency η. Second, the solid mesh isactivated to facilitate a CHT (ConjugateHeat Transfer) calculation that is initia-lized with the aforementioned CFD flowfield. Finally, the CSM (ComputationalStructural Mechanics) is run, i.e. thetemperature field in the solid, the surfacepressure distribution and the rotation ratenmax are used to calculate both the eigenfrequencies and the stresses. Furthermore, rotor mass

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m, center of gravity zg and inertia Jz are calculated. The workflow is wrapped within an op-timization software, that implements a meta-model assisted genetic algorithm described anddetailed at the end of this chapter.

Geometry parametrization

gradient-free

gradient-based

z

Figure 2: Meridional view

Thickness, 𝜃𝑔𝑓 , 𝜃𝑔𝑏

Outlet Inlet

Hub

Shroud𝛽𝑜𝑢𝑡

𝛽𝑖𝑛

-z

Figure 3: Thickness and θ-distribution

The CAD geometry is built using the 3D modeling software CAESES®. The blade ismodeled in three steps:

1. A meridional view of the blade is created using B-Splines including leading edge, trailingedge, hub and shroud contour, fig. 2. In the same step, the turbine back and outlet are designed.

2. The blade camber surface is created based on the meridional surface by specifying a θ-distribution at hub, fig. 3, which is controlled via inlet angle βin and outlet angle βout plus twoweighting points (red). The θ-distribution solely depends on the axial position z, and describesthe peripheral angle. Consequently, the position x of every point on the camber surface isfully defined by its radius r and its axial position z

(x := x

(r, θ(z)

)). The major benefit of

this camber description is, that it leads to radially fibered blades, fig. 4, that prevent bendingstresses within the blades due to centrifugal forces.

3. Two splines are used to define a thickness distribution at both hub and shroud, fig. 3. Athird spline is used to continuously control the blending of the spanwise thickness.

Figure 4: Radially fibered blades Figure 5: Solid (l.) and fluid (r.) body

The model includes several further important features, fig. 5: Internally, the largest possiblefillet radius is calculated based on the smallest distance of two adjacent blades. The latter mostlydepends on the chosen number of blades and its hub thickness. This result is used to design thevariable fillet connection of blade and hub geometry and is automatically chosen as large as

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Model part Parameters Gradient-free setup Gradient-based setup

Leading edge 3 3Trailing edge 2 4

Meridional contour Shroud contour 4 4Hub contour 4 4Axial length 1 1

Camber θ-curve 5 9At hub 8 8

At shroud 4 4Thickness Hub-to-shroud distribution 4 4

Leading edge ellipticity 0 0Trailing edge ellipticity 3 3

Scallops 3 3Number of blades 1 0

Sum 42 47

Table 1: Variable parameters for gradient-free and gradient-based setup

possible to minimize the stresses in the fillets. Additionally, the model includes parameters tocontrol the scallops, which are used to steer the inertia of the rotor and the stresses at the turbinerear side. The fluid geometry includes the gap between heat shield and rotor, the shroud tipgap can be defined and both length and diameter of the diffuser can be set. However, theseparameters are fixed during optimization. In total, ngf = 42 parameters, including the numberof blades, are varied during optimization, see tab. 1.

Eventually, the heat transfer calculations are substantially simplified as fluid and solid bodyshare the same periodic surfaces, i.e. the fluid-solid-interface has a perfect overlap, see fig. 6.

Figure 6: Mesh for CFD and CHT Figure 7: Tetrahedrals (CSM, l.), polyhe-drals (CHT, r.)

MeshesThe solid geometry for CSM is meshed with ALTAIR SIMLABTM producing roughly 80.000

tetrahedral second order elements. Tab. 2 presents a mesh study with three different meshes. ARichardson extrapolation (Celik et al. [2008]) is used to calculate an estimate for the asymptoticvalues φext of the eigenfrequency fe and von Mises stresses σ. The relative extrapolated erroreext =

∣∣(φfine − φext)/φext∣∣ shows, that a high uncertainty of σfillet is to be expected. This is notsurprising as small radii lead to high stress concentration and therefore require many elements.

Both fluid geometry for CFD and solid geometry for CHT are meshed with unstructuredpolyhedrals using STAR-CCM+ meshing capabilities. Prism layers with a stretching of 1.2are used to resolve the boundary layer maintaining y+ ≈ 1 leading to ca. 1 million cells intotal. The flow outlet is elongated with a mesh extrusion. Tab. 3 summarizes a mesh study thatonce more uses a Richardson extrapolation to get an estimate for the asymptotic solution. Therelative extrapolated error is considered reasonably small.

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# elements fe [Hz] σfillet [MPa] σback [MPa]19051 6179.0 466.1 458.136419 6155.4 478.0 459.2

φfine, 83585 6141.0 489.2 462.0φext 6130.0 516.7 466.0

eext in % 0.18 5.31 0.85

Table 2: Mesh study on CSM mesh

# cells pt,in [Pa] Tt,out [K] ηts230325 320590.0 620.1 0.8349572710 320475.0 617.9 0.8451

φfine, 1089061 320282.6 617.2 0.8460φext 320194.7 616.3 0.8463

eext in % 0.03 0.14 0.03

Table 3: Mesh study on CFD mesh

Even though block-structured meshes offer a higher ratio of accuracy to computational cost,unstructured meshes are preferred here, as they offer the flexibility to mesh the fillets, scallopsand heatshield gap. Furthermore, a node-conformal mesh at the interface between solid andfluid can be created, fig. 6. This facilitates a robust simulation of the turbine temperature field.

Simulation setup

Simulation step Relative wall clock time #CPUsGeometry generation 0.04 1

Meshing 0.34 2CFD 0.29 24CHT 0.23 24CSM 0.10 12

Total run time 1.00(47 min)

Table 4: Wall clock time within g.-f. se-tup

After the meshes have been generated, an adi-abatic CFD calculation in STAR-CCM+ is con-ducted. The boundary conditions are set as fol-lows: The inlet massflow m and the back pressureps,out are fixed. In absence of a volute, an inletflow angle αin has to be set and is adjusted duringsimulation to assure that the turbine is deliveringthe target power PT = m∆ht. The angle is incre-ased when the current power is too high and viceversa. Obviously, this way pt,in is not known a pri-ori, which is connected to the turbine inlet temperature Tt,in in piston engines. A simple, linearway (Pucher and Zinner [2012], p.30) is used to model this dependency in order to set the inlettemperature dynamically

Tt,in := Tt,in(pt,in) = TZ

[1− κ− 1

κ·(

1− pt,inpZ

)], (1)

where TZ and pZ are in-cylinder temperature and pressure at valve opening. The flow is assumedfully turbulent. The Reynolds number based on the inlet width b is Re ≈ 2.2 · 105, which iswhy a turbulent velocity profile is set at the inlet, i.e. v(r) = vmax · (1 − 2 r

b)1/8, with r as

the distance from the mid of the inlet channel. The k-ω-SST -model is used for closure of theRANS equations.

After convergence of the CFD simulation, the polyhedral mesh of the solid region is acti-vated, fig. 7. A constant temperature Tshaft is set at the shaft. Every other wall is assumedadiabatic, i.e. the shaft is the only heat sink and radiation is neglected. As this CHT simulationstarts from a converged flow solution, it converges reasonably fast: Runtime CHT

Runtime adiabatic CFD ≈ 0.8. Thetemperature field is mapped onto the tetrahedral mesh as depicted in fig. 7. Additionally, thesurface pressure distribution is mapped onto the surface of the tetrahedral mesh.

Subsequently, the von Mises stresses within the rotor are calculated with the open-sourcefinite element code CALCULIX (Dhondt and Wittig [1998]) based on the maximum rotationrate nmax, including both the previously calculated temperature and pressure field. While thetemperature has a major impact on the stresses especially in the turbine back, the latter is almostnegligible as it just slightly increases the stresses at the rotor-shaft-connection. The first bladeeigenfrequency fe is calculated based on this pre-stressed state.

5

Lastly, inertia Jz, mass m and center of gravity zg are calculated. Gauss’s theorem is used,e.g.

Jz = ρ

∫V

(x2 + y2) dv = ρ

∫S

(x3/3, y3/3, 0

)> · n da, (2)

to calculate these quantities based on a tessellated STL file of the solid body surface.The respective simulation run times are reported in tab. 4.

Objectives and ConstraintsTwo objective values are maximized during the optimization: The total-to-static efficiency

and the total-to-total efficiency, eq. 3:

ηts =Tt,in − Tt,out

Tt,in ·

(1−

(ps,outpt,in

)κ−1κ

) , ηtt =Tt,in − Tt,out

Tt,in ·

(1−

(pt,outpt,in

)κ−1κ

).

(3)

These cost functions have a high correlation, yet, it is not one, see the final data base in fig.8. Maximizing ηts is equivalent to assuming the exit kinetic energy after diffuser being a loss.However, within modern piston engines the radial turbine is not the last geometry in the exhaustpiping system, i.e. there usually will be an exhaust after treatment (EAT). Consequently, from asystem perspective, the stagnation pressure between turbine diffuser exit and EAT is important.Summarized, assuming having the choice between two designs with the same ηts, then decidingfor the one with the higher ηtt is reasonable.

Apart from these objectives, several constraints are considered:The first eigenfrequency of the blade fe ≥ smargin ·

nEO·60 rpms

nmaxis constrained to lie above a

specific engine order nEO ∈ N (+ safety margin), which is an empirically chosen value. Thisconstraint is set to prevent high cycle fatigue damages during operation.

The inertia Jz of the turbine is constrained as it affects the transient performance of theturbocharger as a lower inertia leads to faster rotor acceleration. In that regard, the compressorinertia is not equally significant as its density is substantially lower due to different materials:3 · ρAluminum ≈ ρInconel.

The rotor mass m is constrained as it slightly affects the manufacturing costs, but moreimportantly influences the rotor dynamics as lower masses lead to smaller rotation orbits assu-ming comparable eccentricity and center of gravity. As such, it prevents the rotor from rubbingagainst the housing during operation even for small tip gaps. For the same reason, the axialposition of the center of gravity zg is constrained as well.

The von Mises stresses σ are constrained at two different locations where peaks are ex-pected: On the fillet surface between two adjacent blades and at the turbine rear side, see fig. 5.The allowable stress is set well below the materials yield strength σy with a large safety marginas the turbine is designed to withstand a high number of load cycles. Furthermore, the stressesare evaluated in an integral sense as in Verstraete et al. [2017a], i.e.

σS =p

√1

|S|

∫S

σp da, p = 10, S ∈ {fillet, back}. (4)

This has a smoothing effect on the non-differentiable max{·}-evaluation, in consequence im-proves the quality of the applied meta models and is less prone to mesh induced noise in theresults.

6

Optimization results

0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05

Normalized Efficiencyt-s

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

Nor

mal

ized

Eff

icie

ncy t-

t

Figure 8: Comparison of ηts and ηtt

0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00

Normalized Inertia Jz

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

Nor

mal

ized

Eff

icie

ncy

t-t

Infeasible DesignsFeasible Designs

Figure 9: Optimization resultsBased on the previously described workflow, the automated design evaluation is implemen-

ted within the optimization software MODEFRONTIER. In order to sample the design space,first, a design of experiments is set up as uniform latin hypercube delivering 395 successful de-sign evaluations. Upon this database, the optimization is started exploiting one of the software’simplemented meta-model assisted genetic algorithm (GA) called FAST MOGA II (Montroneet al. [2014]). During the optimization, the software sets up four meta-models (polynomialSVDs, neural networks, radial basis functions, kriging) per output function and automaticallychooses the most successful one for every function by comparing their prediction to the evalu-ated designs of the newest generation of designs. ’Success’ is determined by means of a meanerror. Afterwards, the new generation is added to the meta-model training base and the nextgeneration is started. Another 1302 designs are assessed this way. 1697 designs are evaluatedin total with 1105 being infeasible and violating at least one of the constraints, see fig. 9. Thedesign with the highest efficiency ηtt fulfilling all constraints is picked and prepared for thesubsequent design step, described in the next chapter. Figures 4, 5, 6 and 7 show the chosendesign. This design, coincidentally, is the one with the highest total-to-static efficiency ηts, i.e.there is just one rank 1 pareto design.

GRADIENT-BASED OPTIMIZATION

Create geometry

Determine design

velocities𝜕𝑋

𝜕𝑠𝑖

Determine surface

sensitivities𝜕𝑓𝑗

𝜕𝑋

Determine 𝜕𝑓𝑗

𝜕𝑠𝑖=

𝜕𝑓𝑗

𝜕𝑋⋅𝜕𝑋

𝜕𝑠𝑖

Calculate projected descentdirection

𝑘 ≔ 𝑘 + 1

Figure 10: Gradient-based optimizationworkflow

Fig. 10 summarizes the process used todrive a gradient-based optimization of the ra-dial turbine. The central idea is to locally ap-proximate each response function fj linearlyas

fj(s + ∆s) ≈ fj(s) + ∆s · ∂fj∂s

. (5)

A projected steepest descent algorithm is ap-plied to drive an optimization aimed at rai-sing the efficiency ηtt while conforming to se-veral constraints. Calculating ∂fj

∂sfor every

response function in a reasonable timeframeis, however, not straightforward:

7

Using finite differences is too costly as it would require ngb design evaluations to approxi-mate ∂fj

∂s. Hence, the derivative is split via the chain rule

∂fj∂s

=∂fj∂X· ∂X∂s

, (6)

where X refers to the surface mesh node positions and ∂fj∂X

is computed in a single simulationper response function. Still, ngb evaluations are necessary to approximate ∂X

∂s, however, these

operations are cheap. Both steps are detailed below.Unfortunately, not all response functions handled in the gradient-free framework are avai-

lable in this gradient-based setting, i.e. the calculation of the stress sensitivity ∂σ∂X

has not beenincluded yet. One way to implement this has been presented by Verstraete et al. [2017b].

Altered geometry parametrization and simulation setupIn order to investigate whether the method is capable of further improving the design, the

geometry is re-parametrized, i.e. two splines are exchanged to add further degrees of freedom.The two chosen splines describe the turbine trailing edge, and the θ-distribution, fig. 2 and fig.3, sketched in red (old) and green (new). In total, it contains ngb = 47 parameters (reminder:ngf = 42). As the number of blades is a discontinuous parameter, it needs to be set constant.

The CFD simulation setup is adapted for this optimization: Tt,in and αin remain fixed and astagnation pressure at inlet is used, i.e. pt,in is set at the inlet. As m might change as the designchanges with this boundary conditions, it is now handled as constraint.

The run time of this setup changes compared to the gradient-free setup: The primal CFDrun now takes approximately twice as long as going down to machine precision is advisablein the context of the subsequent adjoint calculation. The adjoint run takes approximately threetimes as long as the primal run accounting for two cost functions and the chosen GMRES solver.However, neither a temperature field nor stresses are calculated in this gradient-based setup.

Design velocity ∂X∂si

Figure 11: Tesselated surfacemesh

▪ Baseline surface▪ Varied surface▪ Design velocity based

on nearest neighborsearch

Figure 12: Designvelocity calculation

Tessellated surface descriptions,i.e. STL files, fig. 11, are usedto calculate the design velocities ∂X

∂siwith a first order finite differencingscheme, where X are the centersof the STL triangles and si, i ∈{1, ..., ngb} is a CAD parameter. Inorder to calculate the sensitivity,two geometries are necessary: Thebaseline geometry X based on theparameter set s = {s1, ..., sngb} andXi, which is varied in one parameter{s1, ..., si−1, si + hi, si+1, ..., sngb}by a small distance hi. The derivative is calculated pointwise

∂x

∂si≈ xi − x

hi, (7)

8

where x ∈ X , xi ∈ Xi. Care has to be taken as to which xi the point x is compared to for thedifference calculation in eq. 7. The nearest neighbor algorithm from PYTHONs SCIPY libraryis used to find the point xi which is located closest to x. The design velocity is calculated as ineq. 7, depicted in blue in fig. 12.

Surface sensitivity ∂fj∂X

The surface mesh sensitivity ∂fj∂X

has to be determined for every response function fj . Theapplied methods differ for the various responses:

The mesh sensitivities of both ηtt and m are determined with the adjoint method, imple-mented in STAR-CCM+, which allows for an efficient calculation of the gradients. Applyingthis method requires the solution of one additional system of equations per response, which aresolved with a Krylov subspace scheme. The code implements the discrete adjoint approach, i.e.the derivation of the adjoint equations is based on the discretized Navier-Stokes-Equations. Thefrozen turbulence assumption is used, i.e. the turbulent viscosity is assumed constant during theadjoint simulation. After calculation of the volume mesh sensitivities, only the surface meshsensitivities are used further. For a general outline on this and related methods, see Peter andDwight [2009]. The total-to-static efficiency ηts is no longer regarded: Gradient-based schemesin a multidisciplinary setting require compromise functions that weight individual objectivesagainst each other, e.g. a weighted linear combination a1 · ηts + a2 · ηtt. In the following,however, we neglect ηts s.t. (a1, a2) = (0, 1).

The first eigenfrequency fe and its mesh sensitivity are calculated with the open-sourcecode KRATOS (KRATOS-Multiphysics [2018]). After the eigenvalue λe = (2πfe)

2 and itseigenvector Φe have been determined, the calculation of the sensitivity with respect to a meshnode position x ∈ X can be found by differentiating the undamped eigenvalue equation to get

∂λe∂x

= Φ>e

(−λe

∂M

∂x+∂K

∂x

)Φe, (8)

where M , K are the finite element mass and stiffness matrix, i.e. no further linear system hasto be solved. In KRATOS, first order finite differences are used to determine both ∂M

∂xand ∂K

∂x.

The mesh sensitivity of nodes not lying on the surface is neglected and assumed zero. For aderivation of eq. 8, see e.g. Firl [2010].

The surface sensitivity of Jz, m and zg can be determined analytically based on the tessel-lated surface. Take a triangle of the triangulated surface and let ∆a be its area. Assume thistriangle being translated by ∆s in surface normal direction n. Then, the surface sensitivity ofe.g. the inertia J is

1

ρ

∂Jz∂n

= lim∆s→0

∫V=∆s·∆a(x

2 + y2) dv∆s

≈ ∆a(x2 + y2) (9)

Gradient calculation ∂fj∂X· ∂X∂si

Fig. 13 exemplarily summarizes the steps described above. On the left, the surface sensi-tivity of the eigenfrequency fe is shown. In the middle, the design velocity of one parametersi describing the blade thickness at shroud is presented. On the right, the pointwise productof these two is visualized, that is calculated using another nearest neighbor search. Integrationalong the surface finally delivers ∂fj

∂si.

9

Figure 13: Eigenfrequency sensitivity∥∥∥∂fe∂x ∥∥∥ (left), design velocity

∥∥∥ ∂x∂si∥∥∥ (mid) and pointwise

product∥∥∥∂fe∂x · ∂x∂si∥∥∥ (right)

Optimization resultsThe gradient-based optimization is driven with a projected steepest descent algorithm, that

facilitates the handling of both inequality and equality constraints, see Haftka et al. [2013], ch.5: First, the gradient of the efficiency ηtt is orthogonally mapped into the subspace of all active,linearized constraints to calculate the projected descent direction. Equality constraints, such asthe massflow constraint, are always considered active, while inequality constraints includingparameter boundaries are inactive as long as they are not violated. The second step is a dam-ped correction step that copes with the difference between target value and actual value of therespective constraint. Finally, the chosen descent step is a linear combination of these two, i.e.projection and correction step.

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46Parameter number

-1

0

1

Nor

mal

ized

Gra

dien

t

Eigenfrequency FDEigenfrequency SS+DVInertia FDInertia SS+DVMassflow FDMassflow SS+DV

Figure 14: ∂fe∂s

, ∂Jz∂s

, ∂m∂s

each calculated both via finite differences (FD) and via surfacesensitivities + design velocities (SS+DV)

The success of the optimization is closely linked to the quality of the gradient approxima-tion. Hence, a study based on forward differences is conducted to judge whether the presentedchain of methods exploiting surface sensitivities and design velocities (SS+DV) delivers reaso-nable derivatives. However, determining a gradient via finite differences (FD) is expensive asthe baseline design plus ngb = 47 variations thereof have to be evaluated to facilitate a finitedifference gradient approximation. A comparison for few cost functions is presented in fig. 14,where the dotted lines show the FD gradients while the solid lines represent the derivatives ba-sed on the SS+DV approach for all ngb = 47 parameters. The gradients ∂Jz

∂s, ∂fe∂s

and ∂m∂s

matchreasonably with only few deviating parameters.

The optimization runs successfully, see fig. 15. Within five iterations, the optimizationstagnates having increased ηtt by 0.34%. In iteration four, the eigenfrequency fe drops belowits constraint value and the projected descent algorithm starts its correction steps. However,m drifts away from its target value. This hints either at imprecise gradients or a too stronglydamped correction. Both the initial design and the design from iteration eight are shown in fig.16 for comparison. Fig. 16 shows that the major design change lies in the θ-distribution.

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Figure 15: Optimization run Figure 16: g.-f. design s1 (red) and g.-b. designs8 (green)

CONCLUSIONSTwo workflows to optimize a radial turbine with different advantages and disadvantages

have been presented. They have shown to complement each other well: The gradient-free setupcan be used to sample the design space and find a reasonable turbine design. Furthermore, itoffers the possibility to handle discrete variables such as the number of blades. Yet, the designeris obliged to limit the number of free variables as computational cost becomes an issue, knownas the curse of dimensionality. The gradient-based process allows for an increase in degrees offreedom without majorly affecting the computational costs. In fact, it is negligible comparedto the costs that come along with the gradient-free setup. Within this study, the run time ofthe gradient-based setup turned out to be two orders of magnitude lower. Furthermore, to theauthors knowledge, this is the first time that a multidisciplinary gradient-based design chain forradial turbines has been presented that incorporates inertia, mass and center of gravity.

Future work will focus on three aspects: First, the gradient-free setup will be supplementedwith life expectancy calculations, that necessitate prior stress and temperature field simulations.Second, the gradient-based setup will be complimented with stress sensitivities to allow fora more holistic and comparable design chain. Third, the gradient quality of the flow relatedsensitivities will be improved by dropping the frozen turbulence assumption and introducing amore sophisticated derivative of volume mesh w.r.t. the surface mesh.

ACKNOWLEDGEMENTSThe authors acknowledge the financial support by the Federal Mini-

stry for Economic Affairs and Energy of Germany (BMWi) in the projectGAMMA (project number 03ET1469). We gratefully thank MTU Friedri-chshafen GmbH for the permission to publish this paper and Friendship Sy-stems for their support in building the CAD geometry.

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