multipoint and multiobjective optimization of a...

Research ArticleMultipoint and Multiobjective Optimization of a CentrifugalCompressor Impeller Based on Genetic Algorithm

Xiaojian Li1 Zhengxian Liu1 and Yujing Lin2

1Department of Mechanics Tianjin University Tianjin 300072 China2School of Mechanical and Aerospace Engineering Kingston University London London SW15 3DW UK

Correspondence should be addressed to Zhengxian Liu zxliutjueducn

Received 25 July 2017 Accepted 6 September 2017 Published 15 October 2017

Academic Editor Filippo Ubertini

Copyright copy 2017 Xiaojian Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The design of high efficiency high pressure ratio andwide flow range centrifugal impellers is a challenging taskThe paper describesthe application of a multiobjective multipoint optimization methodology to the redesign of a transonic compressor impeller forthis purpose The aerodynamic optimization method integrates an improved nondominated sorting genetic algorithm II (NSGA-II) blade geometry parameterization based on NURBS a 3D RANS solver a self-organization map (SOM) based data miningtechnique and a time series based surge detection method The optimization results indicate a considerable improvement to thetotal pressure ratio and isentropic efficiency of the compressor over the whole design speed line and by 53 and 19 at designpoint respectively Meanwhile surge margin and choke mass flow increase by 68 and 14 respectively The mechanism behindthe performance improvement is further extracted by combining the geometry changes with detailed flow analysis

1 Introduction

The reliance on numerical methods in the aerodynamicdesign process of turbomachinery components has consider-ably increased in the last decades Nowadays ComputationalFluid Dynamics (CFD) codes have matured to a level wherethey are capable of not only providing a substantial insightinto the three-dimensional flow field in turbomachines butalso calculating aerodynamic performances of the machines[1ndash3] Meanwhile higher pressure ratio demand impellerefficiency and compressor map width are in severe trade-off relations for transonic impellers [4] Essentially theaerodynamic design of transonic impellers is amultiobjectiveproblem It is a challenging task to design a high pressure ratiocentrifugal impeller for high efficiency and wide operatingrange at the same time [5]

In order to obtain better designs and reduce designcost automated design optimization of centrifugal impellerhas received a widespread attention in recent years Guoet al [6] conducted an automated design optimization ofa high pressure ratio centrifugal impeller by integrating

an evolution algorithm 3D blade parameterization methodCFD solver technique and data mining technique Verstraeteet al [7] combined a genetic algorithm with an artificialneural network (ANN) to optimize a centrifugal compres-sor Hyun-Su et al [8] carried out the optimal design ofimpeller for a centrifugal compressor under the influence offlow-induced vibration using fluid-structure interaction andresponse surface method (RSM) Although much progresshas beenmade in this area the research on themultiobjectivedesign optimization of a blade is still insufficient especiallyfor very high pressure centrifugal impellers which has totake into account the trade-off among objective functionsIn addition most published optimization design techniqueswere performed at one single operating point usually designpoint with the danger of serious deterioration of the perfor-mance at off-design conditions such as poorer surge marginand smaller swallowing capacity

The optimization for the whole speed line is evenmore challenging especially near surge condition (certainimpellers require good efficiency near the surge and mostcompressors need good surge margin) as CFD convergence

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 6263274 18 pageshttpsdoiorg10115520176263274

2 Mathematical Problems in Engineering

is often not guaranteed under or near such a conditionDemeulenaere et al [9] performed a multipoint optimiza-tion of a turbocharger compressor wheel They simulatedentire compressor stage including diffuser and housing withsignificantly increased computational time and effort Theircriterion of surge improvement is higher pressure ratiosand the method is open to questions Pini et al [10] did ashape optimization of a supersonic turbine cascade at off-design conditions These two studies propose a pseudoob-jective function by summing up all the penalty terms andthe original performance objectives with weighting factorswhich is not strictly multiobjectives at multipoints Thedesign variables are so strong and complex to affect theperformances that the more variables must be considered inthe optimization design However there are less analyticalexpressions available to directly correlate the design variableswith the performance at present Data mining techniques areconsidered to be able to provide a possible way to extractsome useful information from the design space and makethe optimization problems in an accessible way By detectingthe features of the data set such as parameter correlationsdata mining method can help to gain the mechanism overthe performance improvement by optimized designs Guo etal [6 11] applied SOM-based data mining to optimizationresults of turbomachinery Jeong et al [12] conducted adata mining for aerodynamic design space However limitedresearch on data mining in turbomachinery design meansthat the process of data mining and data mining results arestill not very clear

The purpose of this paper is to develop a multipoint andmultiobjective design optimizationmethod for high pressureratio impellers to achieve better aerodynamic performancesat both design and off-design conditions and with a wideroperating range The remaining of this paper is organizedas follows first a time series autoregressive (AR) model isdeveloped to predict surge point from the CFD simulation ofa single impeller flow passage and validated by experimentalresults A self-organization map (SOM) is then carried outon samples of CFD results to explore the relation betweenobjective functions and design parameters A total of 27 keydesign variables selected by the SOM are then employed inthe follow-upmultiobjective optimization Finally the resultsfrom the optimization are shown and discussed and someconclusions and remarks are drawn

2 Time Series Based Surge Detection

Surge is instability of the centrifugal compressors and isassociated with strong unsteadiness of the inlet and outletpressures and temperatures of the compressor Thereforemonitoring the time based signal by adopting a Fast FourierTransform analysis makes it possible to state the instantwhen the compressor starts to surge and to highlight thetypical frequency peak related to surge occurrence [13ndash15]However there is no universal standard for the upper limitof pressure pulsation amplitude in surge detection makingit inconvenient to use In CFD simulation though a veryaccurate CFD model will be the perfect candidate near surgefor the rise of numerical instabilities due to the large temporal

and spatial gradients related to the actual flow physics In realdesign environment often only a single impeller flow passageis employed in numerical simulation in order to reducecomputational time and effort As a result the numericalcalculation may become unstable and not convergent beforeor after real surge because of the simplification made to thereal compression system in CFD model This makes surgedetection of a new impeller design or judging the surgemargin relative to the baseline impeller difficult

21 AutoregressiveModel Here we propose a newmethod forsurge detection in CFD it is based on autoregressive (AR)statistical pattern recognition algorithms [16] by monitoringmodel residual variances As this paper was first written wefound a similar approach which was employed for early surgewarning in axial compressors [17]

The slope of pressure rise is a reliable indicator ofcompressor stability If the slope is positive the compressorwill be unstable The maximum or peak compressor pressureratio thus defines the stability limit Numerical studies of axialcompressors [18 19] showed that this criterion gives goodresults in predicting compressor stall Centrifugal compres-sors may be able to operate into the left of the peak pressureratio but compressors will be inminor surge in this condition[20] and compressor outlet and inlet flow conditions will bepulsating and unsteady Just like in experimental detection ofsurge the unsteadiness of numerical simulation results couldbe utilized to find the peak compressor pressure ratio and thushelp to find stability limit of a compressor

In the surge detection one may use the frequenciesor amplitude of the unsteady signal These two parametersare however dimensional so their values depend directlyon compressor size and speed making them unsuitable asuniversal thresholds in surge detection By contrast the timeseries model methods such as AR and ARMA are based onmonitoring model residual variances which is independentof specified compressor impellers and speed

Autoregression is a data processing technique that iscommonly used in constructing a model from time sequencedata for extracting underline trends For a stationary timeseries 119909119905 (119905 = 1 2 119873) autoregressive process producesthe following results

119909119905 =119902sum119894=1

120601119894119909119905minus119894 + 119886119905 119886119905 sim NID (0 1205902119886) (1)

where 120601119894 is model coefficients 119886119905 the random component ofthe model and 1205902119886 the variance If 119909119905 is not zero-mean thenthe process can be similarly carried out by introducing a newtime series

119910119905 = 119909119905 minus 120583119909 (2)

where

120583119909 = 1119873119873sum119894=1

119909119894 (3)

Mathematical Problems in Engineering 3

is themean of 119909119905Themodel coefficients 120601119894 and the variance1205902119886 can be calculated by the following method

Autocovariance 119877119896 function119877119896 = 1119873 minus 119896

119873sum119905=119896+1

119910119905119910119905minus119896 (119896 = 0 1 2 119873 minus 1) (4)

Coefficients f = [1206011 1206012 sdot sdot sdot 120601119902]119879 are calculated by

f = (y119879y)minus1 y119879z (5)

where

z = [119910119902+1 119910119902+2 sdot sdot sdot 119910119873]119879

y =[[[[[[[[

119910119902 119910119902minus1 sdot sdot sdot 1199101119910119902+1 119910119902 sdot sdot sdot 1199102 119910119873minus1 119910119873minus2 sdot sdot sdot 119910119873minus119902

]]]]]]]] (6)

The variance is given by

1205902119886 = 1119873 minus 119902119873sum119905=119902+1

(119910119905 minus119902sum119905=1

120601119894119910119905minus119894)2

(7)

By monitoring the variance of compressor unsteady signalone may judge whether the compressor surges or becomesunstable

22 Surge Detection Method For the outlet boundary condi-tion mass flow rate condition is not appropriate as the con-vergence of CFD cannot be guaranteed near surge conditionresulting in premature sinusoidal waves of flow parametersThus a static pressure condition is adopted and the signal ofmass flow rate at LE is monitored It tends to be a periodicwave but not sinusoidal

The rise of static pressure at outlet boundary is flattenedout near surge condition a dichotomy or sectional methodis applied to the determination of the maximum outletstatic pressure near surge condition This method will bedemonstrated in the next section with an example Theflow chart of surge detection is presented in Figure 1 Aninitial static pressure is specified at impeller outlet beforea steady calculation If this calculation is not convergentthe CFD simulation is switched to unsteady calculationand the temporal mass flow rate signal at LE is monitoredThe AR model is then applied to this time series signal todetermine whether the compressor surges at the set pressurelevel and this is followed by a dichotomy method to finda new static pressure between previous converged steadyand present unsteady pressures at the outlet The detectionprocess continues until a preset minimum pressure rise isreached and the last steady point is regarded as the last stablepoint before surge

Table 1 SRV2AB rotor design parameters

Shaft speed 50000 rpmDesign mass flow rate 255 kgsImpeller tip radius 112mmDiffuser outlet radius 2128mmRotor tip speed 586msRotor pressure ratio 62 1Blade number fullsplitter 1313LE hub radius 30mmLE tip radius 78mmBlade angle LE tip 265 deg (from tangential)Blade angle TE 52 deg (from tangential)Exit blade height 87mmDiffuser inclination againstradial 13 deg

Tip clearance 05mm at inlet to 03mm at exit

23 Surge Detection of SRV2AB Impeller This surge detectionapproach is applied to a higher pressure centrifugal impellerSRV2AB impeller [21 22]

The surge detection validation is carried out by thecomparison of surge line between CFD prediction and exper-imental results of the SRV2AB impeller with vaneless diffuserwhich serves as the baseline for the subsequent optimizationThe basic design parameters of this compressor are given inTable 1

Amesh independence investigation was firstly conductedby coarse medium and fine meshes with total mesh numberof 03 09 and 17 million respectively (for one bladechannel) The mesh has a structured H-O topology andthe minimum value of 119910+ is less than 10 The size of thefirst cell to wall is 5 times 10minus6m As shown in Figure 2 themediummesh seems to provide a good compromise betweenaccuracy and computational efficiency and hence this meshis used for the rest of the study (see Figure 3) whereinthe gird numbers in 119868 119869 119870 direction is 73 53 and 201respectively The grid number in impeller clearance is 13The 3D Reynolds-averaged NavierndashStokes (RANS) equationsare applied using the commercial software of NUMECAFineThe Spalart-Allmaras turbulencemodel which is highlyefficient and suitable for the 3D flow with strong pressuregradient moderate curvature and separating flows such astransonic compressor internal flow [11 23] is adopted to takethe turbulence effects into account The spatial discretizationused for the computation is central scheme and the timeintegration scheme is an explicit four-step Runge-Kuttaalgorithm

The sketch of dichotomy to reach the highest outletstatic pressure at surge point is shown in Figure 4 For theimpellers generated in the optimization process the outletstatic pressure near surge were first set to be 511300 pa521300 pa 531300 pa and 541300 pa according to Figure 1 Anew value of static pressure was then obtained by dichotomyin between 531300 pa and 541300 pa and so onThe smalleststep of pressure rise Δ119901 was set to be 200 pa This value


CFD steady calculation forN iterations

Convergence

CFD unsteady calculationmonitoring time based signal of massflow rate at LE

AR model

Surge

The last steady point issurge point

Steady SurgeSteady

Yes

No

No

Yes

A initial value of static pressure p0 at impeller outlet

and the value ofsignal is prettysmall

６ＬＣＨ＝ asymp 0and the value ofsignal is at a normallevel

６ＬＣＨ＝ asymp 0

６ＬＣＨ＝ gt 0

A new value of

through dichotomystatic pressure pi

p0 minus pi lt Δp

Figure 1 The flow chart of surge detection method

is large but was used to speed up the detection In orderto compensate the effect of a relatively large value of theminimum pressure increment the surge point was actuallylocated by decreasing the last detected stable mass flow

rate value by 25 of the difference between the last twomass flow rates The corresponding pressure ratio is thenevaluated through a third-order polynomial interpolationIn the detection CFD steady iteration was 2000 the upper


CoarseMediumFine

2

25

3

35

4

45

5

55

6

65

Tota

l pre

ssur

e rat

io

24 25 26 27 28 29 323Mass flow rate (kgs)

Figure 2 Mesh independence investigation

Inlet

Splitter blade Hub

Full blade Outlet

Figure 3 Aerodynamic computational domain and grid of SRV2AB(single passage)

511300 521300 531300 541300Unit pa

Figure 4 The sketch of dichotomy to reach the highest staticpressure at surge point

border of variance was set to be 120575 = 5119886 The spatialdiscretization and time integration schemes applied to theunsteady computation are the same as above in this sectionCourant-Friedrich-Levy (CFL) number globally scales thetime-step sizes used for the time-marching scheme of theflow solver A higher value of the CFL number results in afaster convergence but will lead to divergence if the stabilitylimit is exceeded [24] Typical values of CFL are 1sim10 andthe maximum CFL number reached in unsteady calculationsis set to be 3 The physical time-step size is recommendedto be Δ119905 le 110 sdot Nb(11989960) for turbomachinery where 119899denotes shaft speed (rpm) Nb is blade number [25] and hereit is 50119899 s (equals 0001 second) The CFL and physical time

Last stable pointFirst unstable point

1

15

2

25

3

Inle

t mas

s fol

w ra

te (k

gs)

5 10 15 20 25 300Number of time steps

Figure 5 Time based mass flow rate signal at LE

Autocovariance last stable pointPartial autocovariance last stable pointAutocovariance first unstable pointPartial autocovariance first unstable point

2010 155Order of AR model

minus03minus02minus01

0010203040506070809

1

Valu

e

22 asymp 0

55 asymp 0

Figure 6 Autocovariance and partial autocovariance versus orderof AR model

step seem to provide a good trade-off between computationalstability and efficiency after trial

The time based mass flow rate signals at LE monitoredat the last stable point and the first unstable point for designspeed (50000 rpm) are presented in Figure 5 It shows aperiodic but not sinusoidal wave for the first unstable pointThe autocovariance and partial autocovariance versus orderof AR model based on the time signal is given in Figure 6




minus02

0

02

04

06

08

1

12

14

16

Varia

nce

Surge frequency = 625 HZ

Surge Surge

Figure 7 Variance versus number of time steps

where 12059322 asymp 0 for the last stable point and 12059355 asymp 0 for the firstunstable point which indicates that the AR model is suitablefor the time based signals and their order are 119902 = 1 and119902 = 4 respectively As shown in Figure 7 variance of thetime based signal fluctuates periodically for the first unstablepoint The distance between two adjacent main peaks showsthe surge period (frequency) which is corresponding to themain frequency (largest pulsation amplitude) in the FastFourier Transform analysis The variance of a stable point isabout zero while that for an unstable point fluctuates withlarge pulsation amplitude providing a convenient approachto detect instability

Figure 8 compares predicted and measured surge linesof the SRV2AB impeller with vaneless diffuser at 5000040000 and 30000 rpm corresponding to 100 80 and60 of design speed Generally the prediction matches theexperimental results quite well at all operating speeds Thisgives confidence in the surge detection method

3 Data Mining Based onSelf-Organizing Map for Optimizationof Design Space of SRV2AB

In the turbomachinery design using CFD-based optimiza-tion it is important to determine a small number of keydesign variables from design space to simplify design prob-lem The information about the design space such as trade-off between objective functions the relations between designvariables and objective functions and why performance ofthe optimized designs has been improved will be useful forthis purpose by eliminating the design variables which donot have a large influence on the objective functions therebythe efficiency as well as the reliability of optimization process

ExperimentCalculationMeasured surge linePredicted surge line

30K rpm

40K rpm

50K rpm

1

2

3

4

5

6

Tota

l pre

ssur

e rat

io

08 1 12 14 16 18 2 22 24 26 28 306Mass flow rate (kgs)

Figure 8 Comparison of predicted and measured total pressureratio of SRV2AB impeller with vaneless diffuser

may be greatly improved Furthermore it is preferable for adesigner to provide some alternative or suboptimum solu-tions for the decision making of the final design The processto extract information from the design space by detecting thefeatures of the optimization results is called ldquodata miningrdquoThis paper deals with the data mining technique based onSOM

31 Initial Design Variables and Objective Functions In termsof 3D parameterization of impeller geometry refer to authorsrsquoprevious work [26] Figure 9 shows the control points ofendwalls ((119885 119877) two variables for one control point) andblade camber curves (120579 one variable for one control point)For the endwalls 13 control points are applied to the hub and 8for the shroudThe blade profile is parameterized by the rootand the tip sections of both full and splitter blades (4 sectionsin total) and other camber sections are determined by linearinterpolation from root and tip sections 8 control points areselected for each full blade section while 5 for each splitterblade section

Table 2 shows variable names and corresponding num-bers of control points Not all the variables of control pointsare active in the optimization Table 3 shows active variablenumbers range of variations and constraints In total 45design variables are selected for the initial design spaceaccording to the constraints ofmechanical design of SRV2AB

The impeller will be optimized at 50000 rpm for maxi-mizing isentropic efficiency and total pressure ratio at designoperating condition (255 kgs) while striving for smaller


(SZ8 SR8)(HZ13 HR13)

(HZ12 HR12)

TE

LE

(HZ6 HR6)

(SZ1 SR1)

(HZ1 HR1)

(a) Endwalls

Blade camber curvesControl points

ＭＪＦＣＮＮＬＬＩＩＮ1

ＯＦＦＬＩＩＮ1

(b) Root section

ＯＦＦＮＣＪ1

ＭＪＦＣＮＮＬＮＣＪ1


(c) Tip section

Figure 9 Control points of endwalls and blade camber curves

surge and larger choke mass flows The corresponding math-ematical expression of aerodynamic optimization of SRV2ABis as follows

max 120578isdesign 120576totdesign choke

min surge

st 119899 = 50000 rpm(8)

32 Self-Organization Map SOM expresses the informationin a qualitative way and visualizes not only the relationbetween design variables and objective functions but alsothe trade-off between the objective functions It employs anonlinear projection algorithm from high to low-dimensionsand a clustering technique This projection is based on self-organization of a low-dimensional array of neurons In theprojection algorithm the weights between the input vectorand the array of neurons are adjusted to represent featuresof the high-dimensional data on the low-dimensional mapFigure 10 shows the schematic map of SOM where119872 is the

Table 2 Variable names and corresponding numbers of controlpoints

Variable name Variable number119867119885119894 (119894 = 1 2 13) 1ndash13119867119877119894 (119894 = 1 2 13) 14ndash26119878119885119894 (119894 = 1 2 8) 27ndash34119878119877119894 (119894 = 1 2 8) 35ndash42120579fullroot119894 (119894 = 1 2 8) 43ndash50120579fulltip119894 (119894 = 1 2 8) 51ndash58120579splitterroot119894 (119894 = 1 2 5) 59ndash63120579splittertip119894 (119894 = 1 2 5) 64ndash68

number of neurons and 119899 is equal to the dimension of inputvector Each neuron is connected to adjacent neurons by aneighborhood relation and usually forms two-dimensionalhexagonal (see Figure 10(b)) topology

The learning algorithm of SOM starts with finding thebest-matching unit (winning neuron) which is closest to the


Weight vector

Input layer

Output layer

mi = [mi1 mi2 min] (i = 1 M)

SOM neuron

minus04

minus02

0

02

04

06

(a) (b)

Figure 10 Schematic map of SOM

Table 3 Active variable numbers range of variations and con-straints in initial design space

Activevariablenumber

Range of variation

Hub7 8 9 10 11 (minus1sim1) (11986711988513 minus 1198671198851)20 21 2223 24 (minus1sim1) (11986711987713 minus 1198671198771)

Shroud29 30 3132 33 34 (minus1sim1) (11987811988513 minus 1198781198851)

37 38 39 40 (minus1sim1) (11987811987713 minus 1198781198771)Full root

44 45 4647 48 49

50(minus3sim3) sdot (120579fullroot8 minus 120579fullroot1)

Full tip51 52 5354 55 5657 58

(minus3sim3) sdot (120579fulltip8 minus 120579fulltip1)Splitterroot

59 60 6162 63 (minus3sim3) sdot (120579splitterroot5 minus 120579splitterroot1)

Splitter tip 64 65 6667 68 (minus3sim3) sdot (120579splittertip5 minus 120579splittertip1)

Constraints (1) Inactive variables are unchanged in the optimization (2)Diffuser inclination against radial equals 13 deg (3) Shaft speed is 50000 rpm

input vector x and the 119895th winning neuron is selected as theone having minimal distance value

10038171003817100381710038171003817x minusm11989510038171003817100381710038171003817 = min 1003817100381710038171003817x minusm119896

1003817100381710038171003817 (119896 = 1 119872) (9)

Once the best-matching unit is determined the weightvectors are adjusted not only for the best-matching unit butalso for its neighbors As shown in Figure 11 based on thedistance the best-matching unit and its neighboring neurons(situated on the cross of the solid lines the weight vectors ofneurons represent their locations) become closer to the inputvector x The adjusted topology is represented with dashed

Best matching unit

X

Figure 11 Adjustment of the best-matching unit and its neighbors

linesThe adjustment of weight vector near the best-matchingunit can be formulated as follows

m119896 (119905 + 1) = m119896 (119905) + ℎ119895 (119873119895 (119905) 119905) (x minusm119896)119896 = 1 2 119872 (10)

where m119896(119905) is the weight vector of the 119896th neuron at 119905iteration The amount of adjustment depends on the degreeof similarity between a neuron and the input representedby (x minus m119896) and is scaled by the function ℎ119895(119873119895(119905) 119905) thatplays the role of a ldquolearning raterdquo This ldquolearning raterdquo iscalled the neighborhood kernel [27] It is a function ofboth time (iteration step) and the winning neuron spatialneighborhood 119873119895(119905) This spatial neighborhood is a time-dependent function that defines the set of neurons that aretopographically close to the winning neuron The neurons inthe spatial neighborhood adjust their weights according tothe same learning rule but with amounts depending on theirposition with respect to the winner


minus0514

00531

062 Total_pressure_ratio

(a) 120576119905design

minus0913

minus0337

0239 Isentropic_efficiency

(b) 120578isdesign

minus022

0465

115 Surge_flow

(c) surge

minus0936

minus00912

0753 Choke_flow

(d) choke

Figure 12 SOMs colored by objective functions of initial design space

Repeating this learning algorithm the weight vectorsbecome smooth not only locally but also globally Thus thesequence of the vectors in the original space results in asequence of the corresponding neighboring neurons in thetwo-dimensional map

Once the high-dimensional data projected on the two-dimensional regular grid the map can be used for visual-ization and data mining The same location on each compo-nent map corresponds to the same SOM neuron which iscolored according to its related neutron component valuesBy comparing the behavior of the color pattern in the sameregion one can analyze the correlations among parametersParameters are correlated if there exist similar color patternsin the same region of the corresponding component maps

33 SOM-Based Data Mining for Choosing Design Space TheSOM was carried out by an in-house code on initial designspace to select key design parameters in the design spacefor further optimization 100 impellers generated in randomwere simulated in which 49 parameters (4 objectives and45 active design variables) were analyzed All the employedparameters are normalized (variance is normalized to one)25 SOM neutrons were used in component map Figure 12

presents SOMs colored by 4 objective functions showing anonlinear relation no region exists with good performancefor all the objectives This means that it is difficult to improveall the objectives without any compromise A trade-off needsto make among the 4 objectives For example the clusters inbottom right-hand corner are a better choice for total pressureratio 120576totdesign surge mass flow surge and choke massflow choke but bad in isentropic efficiency 120578isdesign SOMscolored by 120578isdesign and choke show an inverse distributionof colors so they are in a severe trade-off relation Onephysical example is a high trim impeller which has a largechoke flow but tends to have low peak efficiency Anotherpotential conflict of objectives involves surge and 120578isdesignSOM colored by 120576totdesign on the other hand shows diagonalbehavior

Figure 13 shows SOMs for some active design variables inthe initial design space Variable 33 (Figure 13(c)) represents1198781198857 and controls the exit blade height of impeller Largervalue of variable 33 means smaller blade tip width and viceversa (refer to Figure 9) The cluster with small values ofvariable 33 is located in the bottom right-hand corner andbrings good performance of 120576totdesign surge and chokebut deteriorates 120578isdesign The largest value of variable 33


minus0938

minus00552

0827 Variable_21

(a) Variable 21minus0528

minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33

(c) Variable 33minus0309

000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44

(e) Variable 44minus0896

minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58

(g) Variable 58minus0634

00894

0813 Variable_68

(h) Variable 68

Figure 13 SOMs colored by active design variables of initial design space


situates on middle left and benefits 120578isdesign and surge whilepenalizing 120576totdesign and choke Figure 13(f) combined withFigure 12 shows the influence of variable 52 or 120579fulltip2 whichis the wrap angle of full blade at tip section Figure 13(e)(variable 44 120579fullroot2) indicates the influence of the wrapangle of full blade at root section It shows an approximateinverse distribution of color pattern with surge indicating apositive correlation between increasing value of variable 44and decreasing surge Variable 58 (Figure 13(g)) represents120579fulltip8 which governs the back sweep of full blade at tip sec-tion (large value of variable 58 represents large sweepback)SOMs colored by variable 58 and 120578isdesign show similar colorpatterns implying an approximate linear relation between thetwo Both smaller and larger values of variable 58 (located inbottom right-hand and upper left-hand corner resp) bringgood performance of 120576totdesign this is due to smaller sweep-back tending to increase the tangential component of absolutevelocity to raise total pressure while larger sweepback triesto achieve the same by improving efficiency Figure 13(h)(variable 68 120579splittertip5) shows the similar effects Variable21 (Figure 13(a)) variable 22 (Figure 13(b)) and variable 40(Figure 13(d)) show severe nonlinear relations with the fourobjective functions while variable 22 and variable 40 aremore sensitive as the ranges of legends in Figures 13(b) and13(d) are small Thus variable 22 and variable 40 are keptwhile variable 21 is ignored in the final design space

Based on data mining results one may derive the follow-ing conclusions (1) Total pressure ratio isentropic efficiencysurge mass flow and choke mass flow are in severe trade-offrelations and compromises are needed in optimization (2)The performance of high pressure ratio impellers is sensitiveto the tip and root sections parameters located near theleading edge of full blade especially for surge margin Theparameters located near the trailing edge of full and splitterblades also have large effects on the performance The rangesof the control points of these parameters in optimizationneed careful planning (3) Some insensitive variables maybe ignored in the design space to simplify the optimizationand to improve the efficiency as well as the robustness ofoptimization

Table 4 shows the active variables and their ranges in thefinal design space that were used in the optimization Thenumber of active variables reduces from 45 to 27 Figure 14presents active variables of endwalls and blade camber curvesin final design space in which the arrows indicate themovingdirections of the active variable in the optimization

4 Aerodynamic Optimization of SRV2AB

41 Optimization Method A multipoint and multiobjectivedesign optimization method based on an improved NSGA-II genetic algorithm [26] the time series surge detectionand data mining technique discussed earlier is applied tothe optimization of SRV2AB impeller at design shaft speed(50000 rpm) The four objectives are higher efficiency pres-sure ratio larger choke flow and smaller surge flow A Pareto-based ranking was adopted for the 4 objectives

The flow chart of this multipoint and multiobjectivedesign optimization method is shown in Figure 15 The

Table 4 Active variable numbers and range of variations in finaldesign space


Range of variation

Hub 9 10 (minus1sim1) (11986711988513 minus 1198671198851)20 22 23 (minus1sim1) (11986711987713 minus 1198671198771)

Shroud 31 32 33 (minus1sim1) (11987811988513 minus 1198781198851)39 40 (minus1sim1) (11987811987713 minus 1198781198771)

Full root 44 45 4648 50 (minus3sim3) sdot (120579fullroot8 minus 120579fullroot1)

Full tip 52 53 5455 58 (minus3sim3) sdot (120579fulltip8 minus 120579fulltip1)

Splitterroot 59 61 63 (minus3sim3) sdot (120579splitterroot5 minus 120579splitterroot1)Splitter tip 64 65 67

68 (minus3sim3) sdot (120579splittertip5 minus 120579splittertip1)Constraints (1) Same as in Table 3

Table 5 Settings of the genetic algorithm amp CFD

Population size 30Maximum iteration number 20Crossover probability 1198751198881 09Crossover probability 1198751198882 03Mutation probability 1198751198981 02Mutation probability 1198751198982 005Convergence criterion of CFD 10minus5

right-hand part of the chart presents the core optimizationalgorithm based on the genetic algorithm The grid gen-eration and CFD solution were automatically carried outusing Autogrid and FineTurbo fromNUMECA by templatesrespectively The prediction of total pressure ratio and isen-tropic efficiency at design operating conditionwas conductedby imposing designmass flow (255 kgs) boundary conditionat vaneless diffuser outlet Chokemass flow rate was obtainedby specifying a low outlet static pressure of 151300 pa Thedetermination of surge mass flow is carried out by the surgedetection method introduced earlier The initial design spaceis presented in Table 3 and the final design space in Table 4Table 5 shows some essential parameters in the geneticalgorithm Parallel computing technique was applied to theoptimization process It included parallel CFD tool and themultithreaded genetic algorithm in which the parallel CFDcode runsThe optimization was carried out on a workstationwith 48-core Xeon(R) E5-2670 processor and the CPUutilization was about 86 percent The total computationaltime required to run the optimization was about 1400 hoursAfter the optimization the optimum geometry is selectedfrom Pareto-optimal front solutions of the last iteration withthe smallest surge mass flow

42 Results and Discussions The overall performances ofbaseline impeller and optimal impeller are presented inTable 6 It shows an improvement of both total pressure ratio


LE

TE

002

004

006

008

01

012

R (m

)

0minus01 minus005Z (m)

(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section

Figure 14 Active variables of endwalls and blade camber curves in final design space

(by 53) and isentropic efficiency (by 19) Meanwhilesurge margin shows a considerable improvement by 68while choke mass flow rises by 14 as wellThe whole designspeed line at 50000 rpm shows improvements for all the 4objectives

Figure 16 compares the performances of the optimalimpeller with the baselines at three speeds of 50000 rpm40000 rpm and 30000 rpm At all speeds compressor surgeis improved Combined with a higher choke flow at thehighest speed this makes the new design particularly suitablefor turbocharger application where a wide flow range isrequired and choke flow at low pressure ratios is relatively

Table 6The overall performances of baseline and optimal impellersat 50000 rpm

Parameters Baseline Optimal Improvement120576totdesign 57 60 53120578isdesign () 800 815 19surge (kgs) 237 221 68choke (kgs) 287 291 14

unimportant Compressor peak efficiency and peak pressureratio both increase at the off-design speeds


Start

Preprocessing

3D parameterization

Initial design space

SOMs based data mining technique

Final design spaceOptimization based on genetic algorithm

CandidatesImproved genetic

algorithm (NSGA-II)

Grid generationFitness values

Optimum geometry

End

CFD solutiontotal pressure ratioisentropic efficiencysurge mass flowchoke mass flow

The smallest surge mass flow Pareto-optimal front

solutions of the lastiteration

Figure 15 Flow chart of multipoint multiobjective optimization method

Baseline measuredBaseline predictedOptimal predicted

Measured surge linePredicted surge lineOptimal surge line

30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io

1 15 2 25 305Mass flow rate (kgs)

(a) Total pressure ratio

BaselineOptimal

30K rpm 40K rpm 50K rpm

45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)


(b) Isentropic total-to-total efficiency

Figure 16 Comparison of off-design performance between baseline and optimal impellers


0

002

004

006

008

01

012

014

016

018

02

022

R (m

)

minus01 minus005 0 005minus015Z (m)

(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr

(b) Blade camber curves at root section

BaselineOptimal

0

05

1

15Th

eta

16 1814dmr

(c) Blade camber curves at tip section

Figure 17 Comparison of geometry between baseline and optimal impeller

These results demonstrate the power of this multipointandmultiobjective optimizationmethodA single-point opti-mization is less likely to get similar results Note that thismul-tipoint and multiobjective optimization is only conducted atdesign shaft speed (50000 rpm) and it could further improveoff-design performance if the same technique is applied toboth design and off-design speeds This will significantlyincrease computational time and resources (about 3 times ofthis study) A trade-off is needed between the effort and thegains

Figure 17 compares the geometry of the baseline andthe optimal impeller Figure 17(a) shows that the meridional

passage (variable 22) is reduced after the inducer throat inthe optimal impeller This explains why the choke flow ofthe impeller is reduced at the two lower speeds but not atthe highest speed At the low pressure ratios exducer passagearea also plays a rule in impeller choking while inducer throatalone dictates the impeller choke flow at high speeds Figures17(b) and 17(c) show the profile or camber changes of the fulland splitter blades at root and tip sections At the leading edgeof the full blade (variables 44 and 52) both sections displaya larger blade angle and higher blade turning contributingto a higher efficiency and a smaller inducer area that isunfavourable for surge The reduction of full blade throat is


I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal

Figure 18 Entropy distributions at streamwise sections and streamlines

compensated by a more open splitter (variables 59 and 64)The compressor stability is improved by the reduced impellertip width (variable 33) and a smaller diffuser gap

The backsweep angles (variables 50 and 63) are slightlyreduced in the optimal impeller at its hub for both fullblade and splitter blade and this increases impeller workand pressure ratios Because the same angle is not reduced atthe shroud the reduction at the hub decreases the diffusionimbalance between the shroud and the hub resulting inmore uniform impeller outflowThis compensates the higherdiffuser inlet velocity caused by the reduced hub backsweepangle

It is concluded that the design variablesrsquo optimization fora better performance corresponds to the analysis in SOM Itproves that SOM analysis is effective in extracting the keydesign variables and their effects and the proposed optimiza-tion method was able to find a proper trade-off between allthe objectives

The flow field at design operating condition is analyzedFigure 18 presents the comparison of entropy distributions atdifferent streamwise sections and streamlines of tip leakagebetween the baseline and optimal impeller In section I alarge high entropy region is found at midspan near suctionside (circled in red) of the baseline impeller and the corre-sponding area in the optimal impeller is considerably smallerThis is due to the weaker and detached LE shockwavemovingback toward LE see Figure 19While the higher blade turningof the new impeller at LE area strengthens the tip leakage flowand thus higher loss is found in the tip region at Section I theloss stops at Section III

From Figure 19 it can be seen that impeller exit flow isnow more uniform circumferentially between the two flowchannels

Figure 20 presents the spanwise performance of theimpeller 10mm downstream from the impeller exit It showsthat both the total pressure ratio and the isentropic efficiencyhave increased in the regions ranging from the hub to 80span but reduced near the shroud

5 Conclusions and Remarks

Amultipoint andmultiobjective design optimization strategyof centrifugal impeller is proposed by integrating a geneticalgorithm 3D geometry parameterization CFD tools timeseries based surge detection method and SOMs based datamining technique This approach was successfully applied tohigh pressure ratio centrifugal impeller SRV2AB for highertotal pressure ratio and better efficiency at design operatingcondition and smaller surge mass flow and larger choke massflow at design speed The main conclusions are drawn asfollows

(1) A time series based surge detectionmethod was intro-duced It uses autoregression to detect compressor instabilityand is independent of the impeller geometry shaft speed andboundary conditionThis method was successfully applied toSRV2AB impeller surge prediction

(2) By SOM-based data mining on initial design spacetrade-off relations between objective functions and correla-tions among design variables and objective functions werevisualized and analyzed The key design variables were thenidentified and kept in the final design space

(3) The optimization improves the overall performanceof the impeller at whole design speed line and widens thecompressor flow range At off-design speeds and compressorefficiency surge margin is also enhanced The mechanismbehind the performance improvement is further explained bycombining geometry changes with detailed flow analysis

The surge detection method still needs to be checkedby experiment and by more applications It is currentlycomputational intensive It essentially detects surge by themacroscopic time based signal instead of local flow featuressuch as the development of stall cells which may be moreefficient The method needs further refinement

Another future work concerns with the method ofselecting the data for data mining Results of data miningtechniques may depend on the used data For the consistencyof information obtained from data mining a robust dataselection method is necessary


0 02 04 06 08 10 12 14Relative Mach number

(a) Baseline 60 span


(b) Optimal 60 span


(c) Baseline 95 span


(d) Optimal 95 span

Figure 19 Relative Mach number contour at 60 and 95 spans

Nomenclature

AR Autoregressive processesARMA Autoregressive moving average

processes119886 Amplitude of time based signal119886119905 White noise seriesdeg DegreeGA Genetic algorithmLE Leading edgeMA Moving average processes Mass flow rate

choke Choke mass flowm119894 Weight vectorsurge Surge mass flow (last stable mass flow near

surge condition)NID Normally and independently distributed119899 Shaft speed (rpm)119875 Probability1198751198881 Upper border of crossover probability1198751198882 Lower border of crossover probability1198751198981 Upper border of mutation probability1198751198982 Lower border of mutation probability119901 Pressure


BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise

35 4 45 5 55 6 65 7 753Total pressure ratio


BaselineOptimal

06 08 1Isentropic efficiency

0

02

04

06

08

1

Nor

mal

ized

span

wise

(b) Isentropic efficiency

Figure 20 Spanwise performance 10mm downstream of impeller

119902 Order of AR model119877 Radial coordination (mm)119877119896 Autocovariance functionSOM Self-organization mapTE Trailing edge119909119905 Original time series signal119910119905 Zero-mean time series119885 Axial coordination (mm)

Greek Symbols

120575 Upper border of variance in AR model120576 Total pressure ratio120578 Isentropic efficiency120579 Azimuthal angle (rad)120583119909 Mean value120588119896 Autocorrelation function1205902 Variance120601119894 Coefficients of AR model120593119896119896 Partial autocovariance functionSubscripts

119888 Crossoverchoke Choke conditiondesign Design conditionfull Full bladeis Isentropic119898 Mutationroot Root sectionsplitter Splitter bladesurge Surge condition

tot Total conditionstip Tip section

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by National Natural ScienceFoundation of China (Grant no 11672206) and State KeyLaboratory of Internal Combustion Engine Burning (Grantno K201607)

References

[1] P G Tucker ldquoComputation of unsteady turbomachinery flowspart 1mdashprogress and challengesrdquo Progress in Aerospace Sciencesvol 47 no 7 pp 522ndash545 2011

[2] L He and J Yi ldquoTwo-scale methodology for URANSlargeeddy simulation solutions of unsteady turbomachinery flowsrdquoJournal of Turbomachinery vol 139 no 10 p 101012 2017

[3] D Pasquale G Persico and S Rebay ldquoOptimization of tur-bomachinery flow surfaces applying a CFD-based throughflowmethodrdquo Journal of Turbomachinery vol 136 no 3 p 0310132014

[4] R Hunziker H Dickmann and R Emmrich ldquoNumerical andexperimental investigation of a centrifugal compressor with aninducer casing bleed systemrdquo Journal of Power and Energy vol215 no 6 pp 783ndash791 2005

[5] H Krain ldquoReview of centrifugal compressorrsquos application anddevelopmentrdquo Journal of Turbomachinery vol 127 no 1 pp 25ndash34 2005


[6] Z Guo Z Zhou L Song J Li and Z Feng ldquoAerodynamicAnalysis and Multi-Objective Optimization Design of a HighPressureRatioCentrifugal Impellerrdquo inProceedings of theASMETurbo Expo 2014 Turbine Technical Conference and Expositionp V02DT42A013 Dusseldorf Germany 2014

[7] T Verstraete Z Alsalihi and R A Van den BraembusscheldquoMultidisciplinary optimization of a radial compressor formicrogas turbine applicationsrdquo Journal of Turbomachinery vol132 no 3 pp 1291ndash1299 2007

[8] K Hyun-Su S Yoo-June and K Youn-Jea ldquoOptimal designof impeller for centrifugal compressor under the influenceof fluid-structure interaction ASMEJSMEKSME joint fluidsengineering conferencerdquo in Proceedings of the Optimal Designof Impeller for Centrifugal Compressor Under the Influenceof Fluid-Structure Interaction ASMEJSMEKSME Joint FluidsEngineering Conference p 02248 2015

[9] A Demeulenaere J Bonaccorsi D Gutzwiller L Hu andH Sun ldquoMulti-disciplinary multi-point optimization of a tur-bocharger compressor wheelrdquo in Proceedings of the ASMETurboExpo 2015 Turbine Technical Conference and Exposition pV02CT45A020 Montreal Quebec Canada 2015

[10] M Pini G Persico and V Dossena ldquoRobust adjoint-basedshape optimization of supersonic turbomachinery cascadesrdquo inProceedings of the ASME Turbo Expo 2014 Turbine TechnicalConference and Exposition p V02BT39A043 Dusseldorf Ger-many 2014

[11] Z Guo L Song J Li G Li and Z Feng ldquoResearch onmeta-model based global design optimization and data miningmethodsrdquo in Proceedings of the ASME Turbo Expo 2015 TurbineTechnical Conference and Exposition p V02CT45A007 Mon-treal Quebec Canada 2015

[12] S Jeong K Chiba and S Obayashi ldquoData mining for aerody-namic design spacerdquo Journal of Aerospace Computing Informa-tion amp Communication vol 2 no 11 pp 452ndash469 2005

[13] N Gonzalez Dıez J P Smeulers L Tapinassi A S Del Grecoand L Toni ldquoPredictability of rotating stall and surge in acentrifugal compressor stage with dynamic simulationsrdquo inProceedings of the ASME Turbo Expo 2014 Turbine TechnicalConference and Exposition p V02DT44A032 Dusseldorf Ger-many 2014

[14] J Galindo J Serrano R C Guardiola and C Cervello ldquoSurgelimit definition in a specific test bench for the characterizationof automotive turbochargersrdquo Experimental Thermal amp FluidScience vol 30 no 5 pp 449ndash462 2006

[15] S Marelli C Carraro G Marmorato G Zamboni and MCapobianco ldquoExperimental analysis on the performance of aturbocharger compressor in the unstable operating region andclose to the surge limitrdquo Experimental Thermal amp Fluid Sciencevol 53 no 2 pp 154ndash160 2014

[16] R Yao and N Pakzad S ldquoAutoregressive statistical patternrecognition algorithms for damage detection in civil structuresrdquoMechanical Systems amp Signal Processing vol 31 no 8 pp 355ndash368 2012

[17] L Fanyu L Jun D Xu S Dakun and S Xiaofeng ldquoStall warn-ing approach with application to stall precursor-suppressedcasing treatmentrdquo in Proceedings of the ASME Turbo Expo2016 Turbomachinery Technical Conference and Exposition pV02DT44A038 Seoul South Korea 2016

[18] C Hah D C Rabe and A R Wadia ldquoRole of tip-leakage vor-tices and passage shock in stall inception in a swept transoniccompressor rotorrdquo in Proceedings of the ASMETurbo Expo 2004

Power for Land Sea and Air pp 545ndash555 Vienna Austria2004

[19] C Hah J Bergner and H Schiffer ldquoShort length-scale rotatingstall inception in a transonic axial compressor criteria andmechanismsrdquo in Proceedings of the ASME Turbo Expo 2006Power for Land Sea and Air pp 61ndash70 Barcelona Spain 2006

[20] X Zheng Z Sun T Kawakubo and H Tamaki ldquoExperimentalinvestigation of surge and stall in a turbocharger centrifugalcompressor with a vaned diffuserrdquo Experimental Thermal andFluid Science vol 82 pp 493ndash506 2017

[21] G Eisenlohr H Krain F Richter and V Tiede ldquoInvestigationsof the flow through a high pressure ratio centrifugal impellerrdquoin Proceedings of the ASME Turbo Expo 2002 Power for LandSea and Air pp 649ndash657 Amsterdam The Netherlands 2002

[22] M Zangeneh N Amarel K Daneshkhah and H KrainldquoOptimization of 621 pressure ratio centrifugal compressorimpeller by 3D inverse designrdquo in Proceedings of the ASME 2011Turbo Expo Turbine Technical Conference and Exposition pp2167ndash2177 Vancouver British Columbia Canada 2011

[23] F Ning and L Xu ldquoNumerical investigation of transoniccompressor rotor flow using an implicit 3d flow solver with one-equation spalart-allmaras turbulence modelrdquo in Proceedings ofthe ASME Turbo Expo 2001 Power for Land Sea and Air pV001T03A054 New Orleans LA USA 2001

[24] D X Wang and X Huang ldquoSolution stabilization and conver-gence acceleration for the harmonic balance equation systemrdquoJournal of Engineering for Gas Turbines amp Power 2017

[25] Ansys Fluent 160 Userrsquos Guide ldquoAnsys Incrdquo 2015[26] X Li Y Zhao Z Liu and H Chen ldquoThe optimization of

a centrifugal impeller based on a new multi-objective evo-lutionary strategyrdquo in Proceedings of the ASME Turbo Expo2016 Turbomachinery Technical Conference and Exposition pV02CT39A022 Seoul South Korea 2016

[27] K J Cios W Pedrycz and R W Swiniarski Data MiningMethods for Knowledge Discovery Kluwer Academic Publisher1998

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is often not guaranteed under or near such a conditionDemeulenaere et al [9] performed a multipoint optimiza-tion of a turbocharger compressor wheel They simulatedentire compressor stage including diffuser and housing withsignificantly increased computational time and effort Theircriterion of surge improvement is higher pressure ratiosand the method is open to questions Pini et al [10] did ashape optimization of a supersonic turbine cascade at off-design conditions These two studies propose a pseudoob-jective function by summing up all the penalty terms andthe original performance objectives with weighting factorswhich is not strictly multiobjectives at multipoints Thedesign variables are so strong and complex to affect theperformances that the more variables must be considered inthe optimization design However there are less analyticalexpressions available to directly correlate the design variableswith the performance at present Data mining techniques areconsidered to be able to provide a possible way to extractsome useful information from the design space and makethe optimization problems in an accessible way By detectingthe features of the data set such as parameter correlationsdata mining method can help to gain the mechanism overthe performance improvement by optimized designs Guo etal [6 11] applied SOM-based data mining to optimizationresults of turbomachinery Jeong et al [12] conducted adata mining for aerodynamic design space However limitedresearch on data mining in turbomachinery design meansthat the process of data mining and data mining results arestill not very clear

The purpose of this paper is to develop a multipoint andmultiobjective design optimizationmethod for high pressureratio impellers to achieve better aerodynamic performancesat both design and off-design conditions and with a wideroperating range The remaining of this paper is organizedas follows first a time series autoregressive (AR) model isdeveloped to predict surge point from the CFD simulation ofa single impeller flow passage and validated by experimentalresults A self-organization map (SOM) is then carried outon samples of CFD results to explore the relation betweenobjective functions and design parameters A total of 27 keydesign variables selected by the SOM are then employed inthe follow-upmultiobjective optimization Finally the resultsfrom the optimization are shown and discussed and someconclusions and remarks are drawn

2 Time Series Based Surge Detection

Surge is instability of the centrifugal compressors and isassociated with strong unsteadiness of the inlet and outletpressures and temperatures of the compressor Thereforemonitoring the time based signal by adopting a Fast FourierTransform analysis makes it possible to state the instantwhen the compressor starts to surge and to highlight thetypical frequency peak related to surge occurrence [13ndash15]However there is no universal standard for the upper limitof pressure pulsation amplitude in surge detection makingit inconvenient to use In CFD simulation though a veryaccurate CFD model will be the perfect candidate near surgefor the rise of numerical instabilities due to the large temporal

and spatial gradients related to the actual flow physics In realdesign environment often only a single impeller flow passageis employed in numerical simulation in order to reducecomputational time and effort As a result the numericalcalculation may become unstable and not convergent beforeor after real surge because of the simplification made to thereal compression system in CFD model This makes surgedetection of a new impeller design or judging the surgemargin relative to the baseline impeller difficult

21 AutoregressiveModel Here we propose a newmethod forsurge detection in CFD it is based on autoregressive (AR)statistical pattern recognition algorithms [16] by monitoringmodel residual variances As this paper was first written wefound a similar approach which was employed for early surgewarning in axial compressors [17]

The slope of pressure rise is a reliable indicator ofcompressor stability If the slope is positive the compressorwill be unstable The maximum or peak compressor pressureratio thus defines the stability limit Numerical studies of axialcompressors [18 19] showed that this criterion gives goodresults in predicting compressor stall Centrifugal compres-sors may be able to operate into the left of the peak pressureratio but compressors will be inminor surge in this condition[20] and compressor outlet and inlet flow conditions will bepulsating and unsteady Just like in experimental detection ofsurge the unsteadiness of numerical simulation results couldbe utilized to find the peak compressor pressure ratio and thushelp to find stability limit of a compressor

In the surge detection one may use the frequenciesor amplitude of the unsteady signal These two parametersare however dimensional so their values depend directlyon compressor size and speed making them unsuitable asuniversal thresholds in surge detection By contrast the timeseries model methods such as AR and ARMA are based onmonitoring model residual variances which is independentof specified compressor impellers and speed

Autoregression is a data processing technique that iscommonly used in constructing a model from time sequencedata for extracting underline trends For a stationary timeseries 119909119905 (119905 = 1 2 119873) autoregressive process producesthe following results

119909119905 =119902sum119894=1

120601119894119909119905minus119894 + 119886119905 119886119905 sim NID (0 1205902119886) (1)

where 120601119894 is model coefficients 119886119905 the random component ofthe model and 1205902119886 the variance If 119909119905 is not zero-mean thenthe process can be similarly carried out by introducing a newtime series

119910119905 = 119909119905 minus 120583119909 (2)

where

120583119909 = 1119873119873sum119894=1

119909119894 (3)




119873sum119905=119896+1

119910119905119910119905minus119896 (119896 = 0 1 2 119873 minus 1) (4)


f = (y119879y)minus1 y119879z (5)

where

z = [119910119902+1 119910119902+2 sdot sdot sdot 119910119873]119879

y =[[[[[[[[


]]]]]]]] (6)


1205902119886 = 1119873 minus 119902119873sum119905=119902+1

(119910119905 minus119902sum119905=1

120601119894119910119905minus119894)2

(7)













Convergence


AR model

Surge


Steady SurgeSteady

Yes

No

No

Yes






A new value of


p0 minus pi lt Δp





CoarseMediumFine

2

25

3

35

4

45

5

55

6

65

Tota

l pre

ssur

e rat

io



Inlet

Splitter blade Hub

Full blade Outlet


511300 521300 531300 541300Unit pa




1

15

2

25

3

Inle

t mas

s fol

w ra

te (k

gs)






0010203040506070809

1

Valu

e

22 asymp 0

55 asymp 0







minus02

0

02

04

06

08

1

12

14

16

Varia

nce


Surge Surge







30K rpm

40K rpm

50K rpm

1

2

3

4

5

6

Tota

l pre

ssur

e rat

io









(HZ12 HR12)

TE

LE

(HZ6 HR6)

(SZ1 SR1)

(HZ1 HR1)

(a) Endwalls




(b) Root section

ＯＦＦＮＣＪ1



(c) Tip section




min surge

st 119899 = 50000 rpm(8)







Weight vector

Input layer

Output layer

mi = [mi1 mi2 min] (i = 1 M)

SOM neuron

minus04

minus02

0

02

04

06

(a) (b)




Range of variation




44 45 4647 48 49


Full tip51 52 5354 55 5657 58







1003817100381710038171003817 (119896 = 1 119872) (9)


Best matching unit

X



m119896 (119905 + 1) = m119896 (119905) + ℎ119895 (119873119895 (119905) 119905) (x minusm119896)119896 = 1 2 119872 (10)



minus0514

00531


(a) 120576119905design

minus0913

minus0337


(b) 120578isdesign

minus022

0465

115 Surge_flow

(c) surge

minus0936

minus00912

0753 Choke_flow

(d) choke








minus0938

minus00552

0827 Variable_21


minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33


000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44


minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58


00894

0813 Variable_68

(h) Variable 68











Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







119873sum119905=119896+1

119910119905119910119905minus119896 (119896 = 0 1 2 119873 minus 1) (4)


f = (y119879y)minus1 y119879z (5)

where

z = [119910119902+1 119910119902+2 sdot sdot sdot 119910119873]119879

y =[[[[[[[[


]]]]]]]] (6)


1205902119886 = 1119873 minus 119902119873sum119905=119902+1

(119910119905 minus119902sum119905=1

120601119894119910119905minus119894)2

(7)













Convergence


AR model

Surge


Steady SurgeSteady

Yes

No

No

Yes






A new value of


p0 minus pi lt Δp





CoarseMediumFine

2

25

3

35

4

45

5

55

6

65

Tota

l pre

ssur

e rat

io



Inlet

Splitter blade Hub

Full blade Outlet


511300 521300 531300 541300Unit pa




1

15

2

25

3

Inle

t mas

s fol

w ra

te (k

gs)






0010203040506070809

1

Valu

e

22 asymp 0

55 asymp 0







minus02

0

02

04

06

08

1

12

14

16

Varia

nce


Surge Surge







30K rpm

40K rpm

50K rpm

1

2

3

4

5

6

Tota

l pre

ssur

e rat

io









(HZ12 HR12)

TE

LE

(HZ6 HR6)

(SZ1 SR1)

(HZ1 HR1)

(a) Endwalls




(b) Root section

ＯＦＦＮＣＪ1



(c) Tip section




min surge

st 119899 = 50000 rpm(8)







Weight vector

Input layer

Output layer

mi = [mi1 mi2 min] (i = 1 M)

SOM neuron

minus04

minus02

0

02

04

06

(a) (b)




Range of variation




44 45 4647 48 49


Full tip51 52 5354 55 5657 58







1003817100381710038171003817 (119896 = 1 119872) (9)


Best matching unit

X



m119896 (119905 + 1) = m119896 (119905) + ℎ119895 (119873119895 (119905) 119905) (x minusm119896)119896 = 1 2 119872 (10)



minus0514

00531


(a) 120576119905design

minus0913

minus0337


(b) 120578isdesign

minus022

0465

115 Surge_flow

(c) surge

minus0936

minus00912

0753 Choke_flow

(d) choke








minus0938

minus00552

0827 Variable_21


minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33


000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44


minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58


00894

0813 Variable_68

(h) Variable 68











Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Convergence


AR model

Surge


Steady SurgeSteady

Yes

No

No

Yes






A new value of


p0 minus pi lt Δp





CoarseMediumFine

2

25

3

35

4

45

5

55

6

65

Tota

l pre

ssur

e rat

io



Inlet

Splitter blade Hub

Full blade Outlet


511300 521300 531300 541300Unit pa




1

15

2

25

3

Inle

t mas

s fol

w ra

te (k

gs)






0010203040506070809

1

Valu

e

22 asymp 0

55 asymp 0







minus02

0

02

04

06

08

1

12

14

16

Varia

nce


Surge Surge







30K rpm

40K rpm

50K rpm

1

2

3

4

5

6

Tota

l pre

ssur

e rat

io









(HZ12 HR12)

TE

LE

(HZ6 HR6)

(SZ1 SR1)

(HZ1 HR1)

(a) Endwalls




(b) Root section

ＯＦＦＮＣＪ1



(c) Tip section




min surge

st 119899 = 50000 rpm(8)







Weight vector

Input layer

Output layer

mi = [mi1 mi2 min] (i = 1 M)

SOM neuron

minus04

minus02

0

02

04

06

(a) (b)




Range of variation




44 45 4647 48 49


Full tip51 52 5354 55 5657 58







1003817100381710038171003817 (119896 = 1 119872) (9)


Best matching unit

X



m119896 (119905 + 1) = m119896 (119905) + ℎ119895 (119873119895 (119905) 119905) (x minusm119896)119896 = 1 2 119872 (10)



minus0514

00531


(a) 120576119905design

minus0913

minus0337


(b) 120578isdesign

minus022

0465

115 Surge_flow

(c) surge

minus0936

minus00912

0753 Choke_flow

(d) choke








minus0938

minus00552

0827 Variable_21


minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33


000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44


minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58


00894

0813 Variable_68

(h) Variable 68











Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





CoarseMediumFine

2

25

3

35

4

45

5

55

6

65

Tota

l pre

ssur

e rat

io



Inlet

Splitter blade Hub

Full blade Outlet


511300 521300 531300 541300Unit pa




1

15

2

25

3

Inle

t mas

s fol

w ra

te (k

gs)






0010203040506070809

1

Valu

e

22 asymp 0

55 asymp 0







minus02

0

02

04

06

08

1

12

14

16

Varia

nce


Surge Surge







30K rpm

40K rpm

50K rpm

1

2

3

4

5

6

Tota

l pre

ssur

e rat

io









(HZ12 HR12)

TE

LE

(HZ6 HR6)

(SZ1 SR1)

(HZ1 HR1)

(a) Endwalls




(b) Root section

ＯＦＦＮＣＪ1



(c) Tip section




min surge

st 119899 = 50000 rpm(8)







Weight vector

Input layer

Output layer

mi = [mi1 mi2 min] (i = 1 M)

SOM neuron

minus04

minus02

0

02

04

06

(a) (b)




Range of variation




44 45 4647 48 49


Full tip51 52 5354 55 5657 58







1003817100381710038171003817 (119896 = 1 119872) (9)


Best matching unit

X



m119896 (119905 + 1) = m119896 (119905) + ℎ119895 (119873119895 (119905) 119905) (x minusm119896)119896 = 1 2 119872 (10)



minus0514

00531


(a) 120576119905design

minus0913

minus0337


(b) 120578isdesign

minus022

0465

115 Surge_flow

(c) surge

minus0936

minus00912

0753 Choke_flow

(d) choke








minus0938

minus00552

0827 Variable_21


minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33


000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44


minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58


00894

0813 Variable_68

(h) Variable 68











Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







minus02

0

02

04

06

08

1

12

14

16

Varia

nce


Surge Surge







30K rpm

40K rpm

50K rpm

1

2

3

4

5

6

Tota

l pre

ssur

e rat

io









(HZ12 HR12)

TE

LE

(HZ6 HR6)

(SZ1 SR1)

(HZ1 HR1)

(a) Endwalls




(b) Root section

ＯＦＦＮＣＪ1



(c) Tip section




min surge

st 119899 = 50000 rpm(8)







Weight vector

Input layer

Output layer

mi = [mi1 mi2 min] (i = 1 M)

SOM neuron

minus04

minus02

0

02

04

06

(a) (b)




Range of variation




44 45 4647 48 49


Full tip51 52 5354 55 5657 58







1003817100381710038171003817 (119896 = 1 119872) (9)


Best matching unit

X



m119896 (119905 + 1) = m119896 (119905) + ℎ119895 (119873119895 (119905) 119905) (x minusm119896)119896 = 1 2 119872 (10)



minus0514

00531


(a) 120576119905design

minus0913

minus0337


(b) 120578isdesign

minus022

0465

115 Surge_flow

(c) surge

minus0936

minus00912

0753 Choke_flow

(d) choke








minus0938

minus00552

0827 Variable_21


minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33


000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44


minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58


00894

0813 Variable_68

(h) Variable 68











Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






(HZ12 HR12)

TE

LE

(HZ6 HR6)

(SZ1 SR1)

(HZ1 HR1)

(a) Endwalls




(b) Root section

ＯＦＦＮＣＪ1



(c) Tip section




min surge

st 119899 = 50000 rpm(8)







Weight vector

Input layer

Output layer

mi = [mi1 mi2 min] (i = 1 M)

SOM neuron

minus04

minus02

0

02

04

06

(a) (b)




Range of variation




44 45 4647 48 49


Full tip51 52 5354 55 5657 58







1003817100381710038171003817 (119896 = 1 119872) (9)


Best matching unit

X



m119896 (119905 + 1) = m119896 (119905) + ℎ119895 (119873119895 (119905) 119905) (x minusm119896)119896 = 1 2 119872 (10)



minus0514

00531


(a) 120576119905design

minus0913

minus0337


(b) 120578isdesign

minus022

0465

115 Surge_flow

(c) surge

minus0936

minus00912

0753 Choke_flow

(d) choke








minus0938

minus00552

0827 Variable_21


minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33


000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44


minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58


00894

0813 Variable_68

(h) Variable 68











Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Weight vector

Input layer

Output layer

mi = [mi1 mi2 min] (i = 1 M)

SOM neuron

minus04

minus02

0

02

04

06

(a) (b)




Range of variation




44 45 4647 48 49


Full tip51 52 5354 55 5657 58







1003817100381710038171003817 (119896 = 1 119872) (9)


Best matching unit

X



m119896 (119905 + 1) = m119896 (119905) + ℎ119895 (119873119895 (119905) 119905) (x minusm119896)119896 = 1 2 119872 (10)



minus0514

00531


(a) 120576119905design

minus0913

minus0337


(b) 120578isdesign

minus022

0465

115 Surge_flow

(c) surge

minus0936

minus00912

0753 Choke_flow

(d) choke








minus0938

minus00552

0827 Variable_21


minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33


000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44


minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58


00894

0813 Variable_68

(h) Variable 68











Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





minus0514

00531


(a) 120576119905design

minus0913

minus0337


(b) 120578isdesign

minus022

0465

115 Surge_flow

(c) surge

minus0936

minus00912

0753 Choke_flow

(d) choke








minus0938

minus00552

0827 Variable_21


minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33


000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44


minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58


00894

0813 Variable_68

(h) Variable 68











Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





minus0938

minus00552

0827 Variable_21


minus00301

0468 Variable_22

(b) Variable 22

minus0638

0131

09Variable_33


000473

0319 Variable_40

(d) Variable 40

minus0983

minus00913

0801 Variable_44


minus0284

0327 Variable_52

(f) Variable 52

minus0866

minus00439

0778 Variable_58


00894

0813 Variable_68

(h) Variable 68











Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of













Range of variation












LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





LE

TE

002

004

006

008

01

012

R (m

)


(a) Endwalls


minus1

minus05

0

05

1

15

2

25

3

Thet

a

8 85 9 9575dmr

(b) Root section


16 18 2 2214dmr

minus05

0

05

1

15

2Th

eta

(c) Tip section








Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Start

Preprocessing

3D parameterization





algorithm (NSGA-II)


Optimum geometry

End







30K rpm

40K rpm

50K rpm

2

4

6

Tota

l pre

ssur

e rat

io



BaselineOptimal


45

50

55

60

65

70

75

80

85

Isen

tropi

c effi

cien

cy (

)





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0

002

004

006

008

01

012

014

016

018

02

022

R (m

)


(a) Endwalls

BaselineOptimal

minus05

0

05

1

15

Thet

a

76 78 8 82 84 86 88 9 92 9474dmr


BaselineOptimal

0

05

1

15Th

eta

16 1814dmr







I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





I

IV

III

II

0 50 100 150 200 250 300Entropy (J(kg K))

(a) Baseline

0 50 100 150 200 250 300Entropy (J(kg K))

(b) Optimal



















(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








(b) Optimal 60 span




(d) Optimal 95 span


Nomenclature






BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





BaselineOptimal

0

02

04

06

08

1N

orm

aliz

ed sp

anw

ise



BaselineOptimal


0

02

04

06

08

1

Nor

mal

ized

span

wise




Greek Symbols






Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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