Aerospace Science and Technology 15 (2011) 334–342

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Aerospace Science and Technology

www.elsevier.com/locate/aescte

Curvature driven two-dimensional multi-objective optimization of compressorblade sections

Lars Sommer ∗, Dieter Bestle

Engineering Mechanics and Vehicle Dynamics, Brandenburg University of Technology, 03013 Cottbus, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 August 2009Received in revised form 27 April 2010Accepted 12 August 2010Available online 19 August 2010

Keywords:AerodynamicsAero engineCompressor airfoilCurvatureMulti-objective optimization

This paper introduces a new curvature based design parameterization of two-dimensional high pressurecompressor blade sections to be used in a multi-criteria aerodynamic design optimization process. Thesuction side of the airfoil section is represented by its curvature distribution which is described by aB-spline curve. The coordinates are then derived by numerical integration. The camber line as well aspressure side are obtained by adding a thickness distribution perpendicularly to the camber line. Thethickness distribution itself is modeled as a B-spline curve as well and varied during optimization. Inorder to achieve smoothness between leading edge, trailing edge and pressure side, respectively, specificconditions for the control points of the thickness distribution are derived. The resulting compressorblade section is optimized w.r.t. pressure loss and working range using process integration and a geneticalgorithm. The results show major improvements over a manually “optimized” datum design regardingboth criteria.

© 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Using optimization techniques for designing blades of compres-sors and turbines is becoming state of the art. This allows tospeed-up development processes but also demands knowledge andexperience not only on optimization but also process automationand design parameterization. The latter is the key of every op-timization problem and has a major influence on the outcome.Therefore, this paper introduces a new alternative design parame-terization approach applicable to compressor blade sections whichis then implemented into an optimization process in order to showits capabilities compared to a classical camberline-thickness ap-proach.

For parameterizing high pressure compressor blade sections,the camber line-thickness approach has proven to be advantageousdue to its simplicity and low number of design parameters [6,4].However, it lacks of direct control on suction and pressure side. Forthis reason other authors prefer direct parameterization of pres-sure and suction side. Keskin [9] splits the blade section into acircular leading edge and B-spline representations of pressure andsuction side. A similar approach is used by Rai [13] who separatesthe leading edge into two elliptical arcs. In all these approachesno direct influence on the curvature especially of the more criticalsuction side can be taken. The high influence of curvature and itschange on turbine blade aerodynamics, however, has been shown

* Corresponding author.E-mail addresses: lars.sommer@tu-cottbus.de (L. Sommer), bestle@tu-cottbus.de

(D. Bestle).

1270-9638/$ – see front matter © 2010 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.ast.2010.08.008

in [16]. Curvature constraints on suction sides of high pressurecompressor blade sections derived by a camber line-thickness pa-rameterization proved the ability to shape curvature-smooth suc-tion sides, but degrade aerodynamic performance [14]. A directparameterization for turbine blade sections based on curvature dis-tributions of suction and pressure side has been presented in [7].

A similar approach will be used in this paper, where curvaturebased methods and classical camber line-thickness parameteriza-tions are combined in order to present a design method with su-perior aerodynamic potential. The profile build-up starts with a B-spline representation of the curvature along the suction side wherethe control points can be moved almost arbitrarily by an optimiza-tion algorithm in order to shape the suction side. In the same waythe thickness distribution along the blade section is described by aB-spline with free control points (Section 2.1). The suction side isderived from its curvature by numerical integration (Section 2.2),whereas the pressure side can be found analytically (Section 2.3).Special treatment is necessary to obtain smoothness of trailing andleading edges (Sections 2.4 and 2.5) resulting in restrictions for thethickness distribution. However, they can be handled easily by con-trol point manipulation (Section 2.6). The optimization problem isdefined in Section 3 and solved by an integrated design processdescribed in Section 4. The optimal trade-off solutions determinedin Section 5 are superior to the given reference design.

2. Two-dimensional blade parameterization

Designing high-pressure compressor blades is a challengingtask, especially since 3D-CFD (Computational Fluid Dynamics) isvery time and resource consuming. Therefore, splitting the flow

L. Sommer, D. Bestle / Aerospace Science and Technology 15 (2011) 334–342 335

Fig. 1. Blade section parameterization.

field into two coupled steady state 2D-fields according to Wu [17]and designing blades on a 2D-CFD basis is still state of the art inindustrial preliminary compressor design [8,10]. It could be shownin [5] that after stacking these optimized profiles the resulting air-foil will also show comparable improvements in 3D-CFD analyses.Typically, the 2D blade section is described in the angle preservingm′, θ -system, however, for simplicity of notation in the following(x, y) is used.

A blade profile can be separated into three parts: suction side,pressure side, and circular trailing edge, Fig. 1. The camber linecan be regarded as the center line running from the stagnationpoint to the center of the trailing edge circle where the thicknessis superposed perpendicularly.

2.1. B-spline representations of suction side curvature and bladethickness

For numerical investigations of the flow field around the com-pressor blade the CFD-code MISES [18,3] will be used. MISES sepa-rates the flow field into a non-viscous outer flow field described byEuler equations and a viscous boundary layer being modeled as asystem of nonlinear boundary layer equations and coupled to theouter field via a Newton–Raphson method. When looking at theEuler equations in polar form, e.g. in [2], major terms are related tocurvature. This expresses a decisive correlation between curvatureand aerodynamics [16]. In general it can be said that smooth cur-vature distributions will be advantageous for flows around bladeprofiles.

This is the motivation for developing a blade section parameter-ization based on the curvature of the suction side. For a parametriccurve x(s), y(s) curvature κ can be calculated as

κ(s) = 1

r= x′ y′′ − x′′ y′

(x′2 + y′2)3/2(1)

where ( )′ = d( )/ds means differentiation w.r.t. curve parameter s.In order to have enough design freedom, the curvature functionis modeled as a parametric curve [s(u), κ(u)], u ∈ [0,1]. Generally,parametric curves may be described as B-splines [12]:[

x(u)

y(u)

]=

n−1∑i=0

Ni,p(u)P i, u ∈ [0,1], (2)

which can be modified by moving the n control points P i =[xi, yi]T. In our application x represents the arc length s and yrepresents the curvature κ of the suction side, Fig. 2. The basisfunctions of B-splines are polynomials of degree p and defined re-cursively by

Ni, j(u) = u − Ui

Ui+ j − UiNi, j−1(u)

+ Ui+ j+1 − u

U − UNi+1, j−1(u), j = 0,1, . . . , p, (3)

i+ j+1 i+1

Fig. 2. B-spline representations of suction side curvature and half-thickness distri-bution.

Fig. 3. Suction side parameterization.

and

Ni,0(u) ={

1 for Ui � u < Ui+1,

0 else.(4)

The knots Ui , i = 0,1, . . . , p + n, are summarized in the so-calledknot vector U = [0, . . . ,0, U p+1, . . . , Un−1,1, . . . ,1] where the firstand last knots are repeated (p + 1)-times, respectively, and the re-maining knots are filled with non-descending values. Due to themultiple knots the B-splines have end point interpolation proper-ties, meaning that e.g. the first and last control points are equal tothe first and last curve points, respectively.

In our case, we describe the suction side curvature distribution[s(u), κ(u)]T as a cubic B-spline (p = 3) with n = 7 control pointsPκ

i and a uniform knot vector

U := U κ = [0,0,0,0,1/4,1/2,3/4,1,1,1,1]T. (5)

Partly, the free form leading edge is already contained in the cur-vature distribution. For structural and manufacturing reasons, itshould have a least curvature radius rI which can be automati-cally guaranteed by choosing Pκ

0 = [0,−1/rI]T for the first controlpoint, Fig. 2.

As second, very important quantity for the flow field the thick-ness distribution may be described by a B-spline (2) where x againrepresents the arc length s and y denotes the half-thickness h,Fig. 2. Here, the half-thickness distribution [s(u),h(u)]T is mod-eled as a B-spline of degree p = 4 with n = 9 control points P h

iand knot vector

U := U h = [0,0,0,0,0,1/5,2/5,3/5,4/5,1,1,1,1,1]T. (6)

2.2. Calculation of suction side coordinates

In order to derive coordinates (xSS, ySS) of the suction side fromthe above curvature distribution, an infinitesimally small section ofthe suction side is considered in Fig. 3. This can be regarded as acircular arc with length ds and radius R(s) = 1/κ(s). According tods = R dϕ the change of slope angle ϕ is then given as

ϕ′(s) = dϕ = 1 ≡ κ(s). (7)

ds R(s)

336 L. Sommer, D. Bestle / Aerospace Science and Technology 15 (2011) 334–342

Fig. 4. Camber line CL and half-thickness h(s).

From the approximation of the right angled triangle P Q S differ-ential equations can be formulated for the coordinates (xSS, ySS) ofthe suction side:

x′SS(s) = cosϕ(s), y′

SS(s) = sinϕ(s). (8)

With a prescribed upper integration limit sE Eqs. (7) and (8) canbe solved for 0 � s � sE efficiently and with high accuracy e.g. byusing the numerical integration algorithm ODE45 of MATLAB [11].The initial conditions for the differential equations result from thestagnation point s = 0 in Fig. 1:

ϕ(0) = βI + π/2, xSS(0) = 0, ySS(0) = 0. (9)

The real length of the suction side sE depends on the curvaturedistribution κ(s). In order to fill the prescribed aero block, it isnecessary to find the corresponding value for sE. This may beachieved within the blade optimization process by including sEas an additional design parameter. However, it proved to be morecomplicated than finding sE in an inner optimization loop, wheresE is adapted iteratively.

At the end of the described procedure the suction side is solelyobtained from its curvature where a free-form leading edge is partof the result.

2.3. Calculation of camber line and pressure side coordinates

Camber line and pressure side can be calculated in a purelyanalytical manner from the suction side coordinates and the half-thickness distribution h(s) in Section 2.2. Additionally, its firstderivative h′(s) is required which can be directly computed fromthe B-spline for h(s). Generally, the k-th derivative of a B-spline(2) is given as

dk

duk

[x(u)

y(u)

]=

n−k−1∑i=0

Ni,p−k(u)P (k)i , u ∈ [0,1], (10)

which again is a B-spline but with reduced order (p − k) andadapted control points

P (k)i = p − k + 1

Ui+p+1 − Ui+k

(P (k−1)

i+1 − P (k−1)i

). (11)

In our case h′(s) will be a B-spline of degree p = 3 with eightcontrol points and the knot vector

U := U h′ = [0,0,0,0,1/5,2/5,3/5,4/5,1,1,1,1]T (12)

obtained from (6) by omitting the first and last knots.The thickness of the section is defined perpendicularly to the

camber line as the distance between suction and pressure side(Fig. 4). Since the camber line is unknown a priori, an infinitesi-mal section of it is approximated by the circular arc CL with angle

Fig. 5. Trailing edge.

dβ and unknown slope angle β . An auxiliary circle with constantdistance h from the camber line arc together with the suctionside approximately defines a right angled triangle P Q T with an-gle ϕ(s) − β(s), where the slope angle ϕ(s) of the suction side SSis known from numerical integration of Eq. (7). From the trianglewe find the relation

sin(ϕ(s) − β(s)

) = dh(s)

ds= h′(s) (13)

and thus the camber line angle

β(s) = ϕ(s) − arcsin h′(s). (14)

The camber line point V (xCL, yCL) can then be found from coor-dinates (xSS, ySS) of point P on the suction side by the analyticalexpression[

xCLyCL

]=

[xSSySS

]+ h

[sinβ

− cosβ

]. (15)

The same holds for point W (xPS, yPS) on the pressure side of theblade section:[

xPSyPS

]=

[xSSySS

]+ 2h

[sinβ

− cosβ

]. (16)

In order to get proper leading and trailing edges for the meanwhilegiven suction and pressure sides, they have to be considered inmore detail in the following.

2.4. Trailing edge

Although a trailing edge is not needed for the used CFD-codeMISES, a circular edge is modeled for geometric consistency wheretransitions to both suction and pressure side are supposed tobe tangential. Tangential transition from the suction side to thetrailing edge can be easily obtained (Fig. 5), however, tangen-tial smoothness between trailing edge and pressure side will re-sult in conditions reducing the design freedom of the thicknessdistribution. The center M(xM , yM) of the trailing edge circle di-rectly results from point Q on the suction side with coordinates(xSS(sE), ySS(sE)) and slope angle ϕE ≡ ϕ(sE) computed in Sec-tion 2.2. By attaching the radius rE perpendicularly to the suctionside, we get according to Fig. 5[

xM

y

]=

[xSS(sE)

y (s )

]+ rE

[sinϕE

− cosϕ

]. (17)

M SS E E

L. Sommer, D. Bestle / Aerospace Science and Technology 15 (2011) 334–342 337

The half thickness hE at the end of the thickness distribution inFig. 2 then results from the prescribed trailing edge radius rE andthe prescribed blade outlet angle βE. Fig. 5 shows the relation

hE ≡ h(sE) = rE cos(βE − ϕE). (18)

According to Eq. (13) the prescribed blade outlet angle βE alsopoints a condition on the derivative

h′E ≡ h′(sE) = sin(ϕE − βE) (19)

due to βE ≡ β(sE). The position of the blend point Q ∗ on the trail-ing edge circle is determined by Eq. (16), however, for tangentialcontinuity an additional condition has to be set up. Fig. 5 showsthe circular trailing edge for a general situation. Obviously, thepressure side will be tangential in Q ∗ only if the angle �T ∗ Q ∗ P∗is identical to �T Q P = (βE − ϕE). Since then one side and twoangles of the triangles P Q T and P∗ Q ∗T ∗ are identical, they haveto be congruent. Therefore, segment Q T = (RCL,E + hE)dβ has thesame length as Q ∗T ∗ = (RCL,E −hE)dβ which in case of hE �= 0 canbe fulfilled only for dβ = 0, i.e. T T ∗‖Q Q ∗ . Division by ds resultsin a vanishing derivative

β ′(sE) = 0. (20)

The derivative is given by differentiation of Eq. (14) w.r.t. s andsubstitution with Eqs. (7) and (13), i.e.

β ′ = ϕ′ − h′′√

1 − h′2≡ κ − h′′

cos(ϕ − β). (21)

Applied to condition (20) this results in a restriction on the secondderivative of the thickness distribution, i.e.

h′′E = κE cos(ϕE − βE) (22)

where κE = κ(sE) is the suction side curvature, ϕE is known fromSection 2.2 and βE is the prescribed blade outlet angle. Summa-rizing Eqs. (18), (19) and (22), the following conditions have to befulfilled by the thickness distribution in Fig. 2 at the end sE:

hE = rE cos(βE − ϕE),

h′E = sin(ϕE − βE),

h′′E = κE cos(ϕE − βE). (23)

2.5. Leading edge

On the suction side the leading edge above the stagnation pointSP in Fig. 1 is already part of the curvature parameterization andthe numerical integration described in Section 2.2. In principle, thecorresponding pressure side also including this part of the leadingedge could be obtained by evaluating Eq. (16). However, it turnedout that without using very restrictive conditions for the thicknessdistribution this leads to major problems in the leading edge re-gion like sharp corners and bumpy shapes. Therefore, the alwaysacceptable suction side part between SP and B is reflected at anaxis defined by the stagnation point SP and the blade inlet angleβI (Fig. 6). The arc length sB of point B is used as a free designvariable within the optimization process.

The remaining part of the pressure side is gained by applyingthe thickness function according to Eq. (16). For good aerodynamiccharacteristics a smooth transition from leading edge to pressureside at the blend point B∗ is necessary. For tangential continu-ity this leads to analogous considerations as in Section 2.4 for thetrailing edge and results in similar conditions as Eqs. (23) whichhave to be fulfilled by the thickness distribution:

hB = ySS(sB) cosβI − xSS(sB) sin βI,

h′B = sin(ϕB − βI),

h′′B = κB cos(ϕB − βI). (24)

Fig. 6. Leading edge.

The value of the curvature κB = κ(sB) may be taken from the cur-vature distribution in Fig. 2. The first equation results from theposition of point B(xSS(sB), ySS(sB)) in Fig. 6:

xSS(sB) = d cosβI − hB sinβI,

ySS(sB) = d sinβI + hB cosβI. (25)

Since the leading edge is a region of high flow velocities, addi-tionally curvature continuity is desirable. This means, Eq. (1) mustgive equal results at the blend point B∗ from both left and righthand sides, i.e. the mirrored leading edge and the attached pres-sure side. Tangential continuity already yields a continuous slopeangle at B∗ resulting in equal left and right sided derivatives x′, y′analogously to (8). Thus, curvature continuity can be achieved ifalso the second derivatives[

x′′PS

y′′PS

]=

[x′′

SSy′′

SS

]+ 2h′′

[sinβ

− cosβ

]+ 4h′β ′

[cosβ

sinβ

]

+ 2hβ ′′[

cosβ

sinβ

]+ 2hβ ′2

[− sin β

cosβ

](26)

of the pressure side (16) match the second derivatives of the mir-rored leading edge LE at the blend point. In analogy to the suctionside, Eqs. (8) yield

x′LE = cos(2βI − ϕ), y′

LE = sin(2βI − ϕ) (27)

for the first derivatives where the slope angle 2βI − ϕ is used forthe mirrored leading edge, see Fig. 6. By differentiation we obtainthe second derivatives[

x′′LE

y′′LE

]B∗

= κB

[sin(2βI − ϕB)

− cos(2βI − ϕB)

](28)

for the mirrored leading edge at the blend point B∗ . They must beequal to Eq. (26) at the blend point s = sB which can be simplifiedusing β ′

B = 0 in analogy to Eq. (20):[x′′

LEy′′

LE

]B∗

!=[

x′′SS(sB)

y′′SS(sB)

]+ 2h′′

B

[sinβI

− cosβI

]

+ 2hBβ ′′B

[cosβIsinβI

]. (29)

With the second derivatives of the suction side

x′′SS(sB) = −κB sinϕB , y′′

SS(sB) = κB cosϕB (30)

resulting from (8) and h′′B from (24), as well as addition theorems

for trigonometric functions, Eq. (29) yields

κB

[sin(2βI − ϕB)

− cos(2βI − ϕB)

]

!= κB

[sin(2βI − ϕB)

− cos(2β − ϕ )

]+ 2hBβ ′′

B

[cosβIsinβ

](31)

I B I

338 L. Sommer, D. Bestle / Aerospace Science and Technology 15 (2011) 334–342

which can only be fulfilled for arbitrary hB �= 0 if the secondderivative β ′′

B of the camber line angle (14) equals zero at theblend point:

β ′′B = κ ′

B − h′′′B

cos(ϕB − βI)− κBh′′

Bsin(ϕB − βI)

cos2(ϕB − βI)

!= 0. (32)

With h′′B from Eq. (24) the third derivative of half-thickness h at

the blend point must then read as

h′′′B = κ ′

B cos(ϕB − βI) − κ2B sin(ϕB − βI) (33)

as a further constraint for h(s) at the starting point s = sB of thethickness distribution in the blend point B∗ .

2.6. Boundary conditions for the thickness B-spline

The boundary conditions (24) and (33) at the blend point s = sB

and (23) at the blend point s = sE for the thickness distributionin Fig. 2 can be automatically satisfied by a proper choice of thecontrol points. Setting the first control point to

P h0 =

[sBhB

](34)

will fulfill the first equation of boundary conditions (24). Thesecond equation can be fulfilled by a correct choice of the sec-ond control point. Since P h,(1)

0 is identical to the first derivative[sB , hB ] = [ds/du,dh/du]T at s = sB of the B-spline [12], Eq. (11)with degree p = 4, derivative k = 1 and knot vector (6) yields

P h,(1)0 =

[sBhB

]= 20

(P h

1 − P h0

). (35)

With Eq. (34) and the first derivative of parametric curves

h′ = dh

ds= dh

du

du

ds= h

s(36)

we obtain the h-coordinate of the second control point P h1 =

[P h1,s, P h

1,h]T from (35) as

P h1,h = hB + h′

B

(P h

1,s − sB), (37)

where the s-coordinate P h1,s may be chosen arbitrarily and h′

B isgiven by condition (24).

The same approach applies for the second derivative h′′B in con-

ditions (24) determining the third control point. From Eq. (11) weget

P h,(2)0 =

[sB

hB

]= 15

(10

(P h

2 − P h1

) −[

sB

hB

])(38)

where P h,(1)1 is obtained from (11) as well and P h,(1)

0 is alreadyknown from (35). The second derivative

h′′ = d

ds

(h′) = d

ds

(h

s

)= d

du

(h

s

)du

ds= sh − hs

s3(39)

yields h = s2h′′ + hs/s. With prescribed s-coordinate P h2,s and h′′

Bfrom (24), the h-coordinate of the third control point is found bymanipulation of (38):

P h2,h = P h

1,h + h′′B

s2B

150+ hB

sB

(P h

2,s − P h1,s

). (40)

The fourth control point is obtained from condition (33) with

h′′′ = d

ds

(h′′) = s2

...h − 3ssh + 3hs2 − sh

...s

s5(41)

in the same manner resulting in

P h3,h = 5

2P h

2,h − 3

2P h

1,h + hB

50

+ 1

500sB

[h′′′

B s4B + 3sB hB − 3s2

BhB

sB

+ 10hB(50P h

3,s − 125P h2,s + 75P h

1,s − sB)]

(42)

with prescribed s-coordinate P h3,s and h′′′

B from (33).Analogously, the last three control points are obtained from the

boundary conditions (23) for the trailing edge. The last controlpoint itself then reads as

P h8 =

[sEhE

]. (43)

According to Eq. (10) the number of control points reduces withevery derivative. The first derivative of the end point s = sE thenfollows from (11) as

P h,(1)7 =

[sEhE

]= 20

(P h

8 − P h7

)(44)

resulting in

P h7,h = hE − h′

E

(sE − P h

7,s

)(45)

after applying Eq. (36) where P h7,s is a free variable again. In order

to get proper characteristics of the decreasing rear part of the half-thickness distribution, instead of the s-coordinate of control pointP h

6 the h-coordinate P h6,h is prescribed and analogously to (40) we

find

P h6,s = P h

7,s − s3Eh′′

E

150hE+ sE

hE

(P h

6,h − P h7,h

). (46)

3. Optimization task

Stating a proper problem formulation including parameteriza-tion, optimization criteria and constraints is crucial for optimiza-tion processes. The parameterization is already performed by usingthe B-spline representations of both distributions κ(s) and h(s) inFig. 2. The control points are marked as dots and the arrows in-dicate their corresponding design freedom within the optimizationprocess, where the restrictions of Section 2.6 have been alreadytaken into account. Additionally, metal angles βI and βE as well asblend point position sB are varied by the optimizer. This yields atotal of 20 design variables resulting in the design vector

p = [Pκ

1,s, Pκ2,s, Pκ

3,s, Pκ3,κ , Pκ

4,s, Pκ4,κ , Pκ

5,s, Pκ5,κ , Pκ

6,κ ,

sB , P h1,s, P h

2,s, P h3,s, P h

4,s, P h4,h, P h

5,s, P h6,h, P h

7,s, βI, βE]T

. (47)

Engineering problems typically depend on more than one op-timization goal [1]. Instead of combining the goals in a singleobjective function, often done by summation with some weightingfactors, it is favorable to use genetic multi-objective optimizationalgorithms. For contradicting goals, the result is then not just asingle optimum, but a set of optimal trade-off solutions.

In our case a compressor blade section is to be optimized withrespect to the aerodynamic pressure loss at design point and work-ing range at the same time. These objectives are best derived fromthe so-called loss polar where pressure loss ω is drawn againstincidence angle αI , Fig. 7. The non-dimensional pressure loss coef-ficient

ω = pisen0,E − p0,E

p0,I − pI(48)

determines the aerodynamic properties of the blade section [18].The difference of isentropic total pressure pisen and mass averaged

0,E

L. Sommer, D. Bestle / Aerospace Science and Technology 15 (2011) 334–342 339

Fig. 7. Schematic loss polar.

total pressure p0,E at the exit is referred to the difference of totalpressure p0,I and static pressure pI , the dynamic head, at the inletof the cascade.

It is not advisable to evaluate the working range by the widthof the loss polar as seen in Fig. 7, since a complete calculation ofthe loss polar curve would require too much CPU resources dur-ing optimization. Instead, the loss at two chosen incidence angles(αDP

I −αCHI ) and (αDP

I +αSTI ) for choke CH and stall ST cases is

calculated with CFD. The maximal value of both, i.e.

γ = max{ωCH,ωST}, (49)

is then minimized as representative for the working range duringoptimization. Experience shows that the choice of equal αCH

I =αST

I = αI makes the polar more symmetric around the designpoint, but by choosing asymmetric variations of the incidence an-gle it would be also possible to shift the polar to the left or rightin order to meet specific operating requirements.

With parameter vector (47) and objectives (48), (49) the vectoroptimization problem reads as

minp

[ωDP

γ

](50)

where geometrical and aerodynamical constraints

c1 := |ϕE,ref − ϕE|!� ε1, c2 := Aref − A

!� 0,

c3 := |αE,ref − αE|!� ε2, c4 := HSS

E − HSSE,ref

!� 0,

c5 := φnc!� 0 (51)

must be fulfilled. The first constraint c1 ensures that exit angleϕE of the suction side lies within a span ε1 of a prescribed valueϕE,ref where reasonable flow deflection may be expected. The sec-ond constraint limits the minimum cross section area for structuralreasons. Constraint c3 restricts the exit flow angle to a range ε2around a desired value αE,ref and c4 ensures a proper bound-ary layer form factor for compressible flows at the trailing edgeof the suction side. Since it is necessary to evaluate the designat three different incidence angles it is required that all threeCFD-calculations converge marked by φnc = 0. In order to forcethe evolutionary algorithm into the right direction also in caseof infeasible designs, the artificial constraint c5 counts the non-converged CFD-solutions by φnc.

4. Process integration and automation

The profile section geometry generation, the CFD-analysis andthe optimization algorithm are integrated in order to perform anautomated optimization process, where the software tool iSIGHT-FD is used. The suction side and profile build-up are realized usingMATLAB where a fast data exchange to iSIGHT-FD exists due to a

Table 1Optimization statistics.

Function evaluations 15 000Failed suction sides 4265A < Aref 1331CFD-calls 9404Not converged designs 2192|αE,ref − αE| > ε 3826

HSSE > HSS

E,ref 1392Feasible designs 4132Duration 2 d 10 h

special interface. The CFD-solver MISES is called via the MATLABDistributed Computing Toolbox offering the possibility to run severalcalculations in parallel on a high performance cluster and thereforeto speed-up the process. Fig. 8 shows the flow chart.

For optimizing the profile section, the multi-objective geneticalgorithm AMGA (Archive-based Micro Genetic Algorithm) [15] is ap-plied which is implemented in iSIGHT-FD. First, the optimizer cre-ates an actual design p and passes it to the design evaluation pro-cess. The suction side is build-up from the B-spline control pointsfor curvature κ . Curvatures κi , derivatives κ ′

i , angles ϕi and coor-dinates (xSS,i, ySS,i) at discrete points along the curve are returnedto iSIGHT-FD for pressure side build-up. As already mentioned inSection 2.2, the suction side build-up implies a fitting routine tofind the optimal suction side length sE. Since this process takesabout two seconds the constraint c1 is checked beforehand, andin case of failure the fitting process is skipped yielding importanttime savings at the beginning of the process. In this case, penaltiesare assigned to all objectives and constraints and the evaluationprocess is interrupted. Otherwise, MATLAB is called to build-up thepressure side and write CFD-input files for all three flow inlet an-gles αDP

I , αCHI and αST

I . The area constraint c2 is calculated andreturned to iSIGHT-FD. In case of a proper section area, the CFD-solver MISES is called and the values of the objectives ωDP and γas well as the constraints c3, c4, c5 are returned to iSIGHT-FD. IfMISES does not converge for at least one evaluation case, artificialvalues are returned for all criteria which could not be calculatedsuch that they do not satisfy the corresponding constraint. Theoptimization loop iterates until the total number of design eval-uations is reached.

5. Optimization results

For this paper an aerodynamic optimization of the midsectionof a stator of a multi-stage high pressure compressor has beencarried out. With an inlet Mach number Ma ≈ 0.8, a Reynoldsnumber Re ≈ 106 and a flow deflection αI − αE ≈ 28◦ , this is ahighly loaded transonic section. The standard settings of the ge-netic optimization algorithm AMGA in iSIGHT-FD are adjusted to15 000 function evaluations, initial population size 1500, popula-tion size 200 during iteration, archive size limit 1500 and Paretosize limit 100.

Table 1 shows the statistics of the optimization run. More thana third of all designs do not reach the CFD-analysis mainly due toa failed suction side. This does not surprise since the numerical in-tegration of the suction side is very sensitive to parameter changeson curvature control points even with tight parameter bounds. Atleast one of the three CFD-analyses did not converge in 23% of thecalls, 15% fail due to high HSS

E -values and 41% do not match the re-quired exit flow angle. The numbers given in Table 1 have not beenstatistically confirmed and may differ from one optimization run tothe other. Nevertheless, other test runs showed similar numbers.

Fig. 9 shows the criteria space of the optimization task wherethe loss at design point ωDP is drawn against the second opti-mization criterion γ , each normalized w.r.t. values of a reference

340 L. Sommer, D. Bestle / Aerospace Science and Technology 15 (2011) 334–342

Fig. 8. Optimization flow chart.

Fig. 9. Criteria space.

Table 2Comparison of results.

Ω K Γ

(A − Aref)/Aref 1.4% 0.8% 4.8%(HSS

E − HSSE,ref)/HSS

E,ref −28.8% −28.4% −27.2%

(ωDP − ωDPref )/ω

DPref −18.1% −17.1% −15.7%

(γ − γref)/γref −27.2% −40.0% −41.8%(αE,ref − αE)/αE,ref −1.2% 0.3% 0.02%

design, respectively. The feasible solutions found during optimiza-tion are plotted as light gray dots, whereas black dots indicatenon-dominated solutions. For comparison the “human optimized”reference design as well as former optimization results of prior in-vestigations regarding a classical camber line-thickness approachwith and without curvature constraints on suction sides [14] areplotted as dark gray dots.

First of all it can be stated that the shown optimization processfinds solutions clearly beating the datum design in both criteria.The loss at design point is up to 18% and the maximum loss atoff-design up to approximately 42% lower. At the same time thefront of non-dominated solutions has a good diversity indicatingthe expected contradiction in both criteria. In the following, threedifferent solutions namely the designs with minimal loss at designpoint Ω , best loss at off-design Γ and a trade-off solution K atthe knee of the Pareto-front are discussed (Table 2). All three de-signs try to minimize the cross section area and get close to thelower bound Aref, especially in cases Ω and K . Design Γ is slightlylarger but has the biggest loss at design point, too. The boundary

Fig. 10. Optimized curvature (a) and half-thickness (b) distributions.

layer shape factor HSSE is well below the upper bound in all three

designs. Design Ω makes the least flow deflection hardly achiev-ing the demanded lower bound of the exit flow angle, whereasthe other two designs are overachievers. So, obviously minimizingarea and flow deflection helps to get a low pressure loss at designpoint.

Fig. 10 displays suction side curvature and half-thickness dis-tributions of the three chosen results. Due to integration suctionside coordinates are very sensitive to curvature changes why evenminor differences in curvature distribution (Fig. 10a) yield ratherdifferent suction sides, Fig. 11.

Bigger differences occur in half-thickness distributions, Fig. 10b.Not only the maximum half-thickness differs but also axial posi-tions of maximum half-thickness. Especially, design Ω with min-imal loss at design point shows an untypically backwards moveddistribution. The other two designs show a more typical distribu-tion with almost same axial position of maximum half-thicknessclose to the middle of the blade section.

L. Sommer, D. Bestle / Aerospace Science and Technology 15 (2011) 334–342 341

Fig. 11. Optimized blade section profiles compared to reference design (dottedlines).

Fig. 12. Mach number distribution around LE.

Analyzing the optimized section geometries (Fig. 11) it can bestated that due to the fitting process the aero block space is per-fectly used allowing best aerodynamic properties. Tangential con-tinuity at trailing edge is fulfilled as well as smooth intersectionat blend point of leading edge and pressure side. On account ofsuction side reflection at leading edge, the minimal radius is main-tained without using additional constraints. Designs K and Γ lookalmost identical whereas design Ω has a unique shape. The suc-tion side of Ω is almost akin to the other two, indicating enoughflow deflection, however, Table 2 shows less flow turning than thereference design and only because of the tolerance ε2 in Eq. (51)the design is acceptable. In order to fulfill area constraint c2, de-sign Ω becomes rather thick in the rear while staying very thinin the front. This will probably yield different moments of inertiacompared to classical designs which indicates that cross sectionarea alone may be not well suited as a structural constraint.

Curvature continuity at the leading edge of suction sides evokesreduction of spikes in Mach number distributions compared to thedatum design, Fig. 12. The small spikes on the left do not re-sult from curvature imperfections but from high flow deflectionat the leading edge. The loss polars are plotted in Fig. 13. As al-ready seen in the criteria space plot, all three optimized designsbeat the datum by far in the evaluated range between choke andstall boundaries. Especially, designs K and Γ suggest significantimprovements in loss at almost every inlet flow angle. Therefore,these two designs have a much wider working range compared todatum profile and design Ω .

6. Summary

This paper introduces a new curvature based two-dimensionalblade section parameterization used within a multi-criteria opti-mization process. Suction side coordinates are derived from a cur-vature distribution by integration, the pressure side is built-up bysuperimposing a half-thickness distribution perpendicularly to the

Fig. 13. Loss polars.

unknown camber line which has to be determined as part of thebuilding-procedure. Due to the parameterization, the suction sidecurvature can be controlled without the need of constraints. As aresult a curvature continuous suction side and a curvature contin-uous pressure side are gained.

Starting from scratch it is possible to find a number of feasi-ble designs with reduced computational cost. The non-dominatedsolutions show low pressure loss at the design point and a wideworking range at the same time. Compared to prior studies bladesection behavior can be improved. Due to the high design freedomfor the half-thickness distribution, various optimal blade shapesare obtained. While minimal losses at design point occur for aftloaded blades, front loaded blade sections show better workingranges.

Acknowledgements

The work presented in this paper is performed within the class“Compressor Technologies and Materials” of the international grad-uate school at the Brandenburg University of Technology, Cottbusand is financed by the state of Brandenburg, Germany. For profes-sional support we specially thank Rolls-Royce Deutschland namelyDr. Marius Swoboda and Dr. André Huppertz.

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