Thesis presented to the Instituto Tecnolgico de Aeronutica, in partial fulfillment of the requirements for the Degree of Master in Science in the Program of Aeronautics and Mechanical Engineering, Field of Aerodynamics, Propulsion and Energy. Victor Fujii Ando GENETIC ALGORITHM FOR PRELIMINARY DESIGN OPTIMISATION OF HIGH-PERFORMANCE AXIAL-FLOW COMPRESSORS Thesis approved in its final version the signatories below Celso Massaki Hirata Prorector of Graduate Studies and Research Campo Montenegro So J os dos Campos, SP Brazil 2011 Cataloging-in-Publication Data Documentation and Information Division Ando, Victor Fujii GeneticAlgorithmforPreliminaryDesignOptimisationofHigh-PerformanceAxial-Flow Compressors / Victor Fujii Ando.So J os dos Campos, 2011. 162f. ThesisofmasterinscienceProgramofAeronauticsandMechanicalEngineering.Fieldof Aerodynamics, Propulsion and Energy Aeronautical Institute of Technology, 2011. Advisor: Prof. Dr. J oo Roberto Barbosa. 1. Genetic Algorithm.2. Axial-flow compressor. 3. Preliminary design. I. Aeronautics Institute of Technology. II. TitleBIBLIOGRAPHIC REFERENCE ANDO,VictorFujii.GeneticAlgorithmforPreliminaryDesignOptimisationofHigh-PerformanceAxial-FlowCompressors.2011.162f.Thesisofmasterofsciencesin Aerodynamics,PropulsionandEnergyAeronauticsInstituteofTechnology,SoJ osdos Campos. CESSION OF RIGHTS AUTOR NAME: Victor Fujii Ando PUBLICATION TITLE: Genetic Algorithm for Preliminary Design Optimisation of High-Performance Axial-Flow Compressors PUBLICATION KIND/YEAR: Thesis / 2011 It is granted to Aeronautics Institute of Technology permission to reproduce copies of this thesis to only loan or sell copies for academic and scientific purposes. The author reserves other publication rights and no part of this thesis can be reproduced without his authorization. Victor Fujii Ando DCTA, ITA, IEM, Grupo de Turbinas So J os dos Campos, SP. iii Genetic Algorithm for Preliminary Design Optimisation of High-Performance Axial-Flow Compressors Victor Fujii Ando Thesis Committee Composition: Prof. Dr. Rodrigo Arnaldo ScarpelChairperson ITA Prof. Dr. J oo Roberto BarbosaAdvisor ITA Prof. Dr. Nelson Manzanares FilhoUniversidade Federal de Itajub Prof. Dr. Mrcio Teixeira de MendonaITA ITA iv Acknowledgements ThisworkwasexecutedinthecontextoftheProgramaIntegradoGraduao-MestradoPIGM.Underthisprogramme,ITABachelorstudentsfromthelastyear undertake disciplines from the post-graduate programme and are encouraged to develop the BachelorThesisasanintermediatesteptowardstheresearchtobeconductedduringthe Masters. TheauthoracknowledgesthesupportofFundaodeAmparoPesquisado Estado de So Paulo (FAPESP) to conduct this study. The author would like to express his gratitude to his advisor, Prof. Barbosa, for the guidance and the invaluable assistance, especially with regard to the axial-flow compressor design program. The author is also indebted to Prof. Nelson Manzanares Filho, from UNIFEI, who was very supportive with insightful discussions on Genetic Algorithms. Thanks are also addressed to the colleagues from the Gas Turbine Group at ITA for the amiable companionship. Finally, the author conveys his thankfulness for the inestimable support of his family.

v Resumo Este trabalho apresenta uma abordagem para a otimizao de projeto preliminar de compressoresaxiaisdealtodesempenho.NocontextodoGrupodeTurbinasdoITA,o projeto preliminar feito utilizando-se um programa computacional baseado no mtodo da curvatura da linha de corrente, empregando-se correlaes da literatura para o cmputo das perdas. A escolha de diversos parmetros do ciclo termodinmico e de geometrias depende da longaexperinciaacumuladapelosmembrosdoGrupo.Contudo,esseprocessoexigeum trabalho longo e exaustivo de tentativas e erros. Desse modo, a fim de auxiliar o projetista na escolhadealgunsparmetros,umprogramadeotimizao,chamadodeREMOGA,foi desenvolvidoemlinguagemFORTRAN,parafcilintegraocomosprogramas desenvolvidospeloGrupodeTurbinas.Oprogramabaseia-seemumalgoritmogentico multi-objetivo, com codificao real e elitismo. Em seguida, o REMOGA e o programa de projeto preliminar foram integrados para o projeto de um compressor axial de cinco estgios. Para isso, foram variados os ngulos de sada do escoamento dos estatores, a distribuio de temperatura nos estgios e a relao de raios,visandoamaioreseficinciasemaioresrazesdepresso,mascontrolando-seo nmero de De Haller e o ngulo de arqueamento. Graas ao REMOGA, dezenas de milhares deprojetospuderamserrapidamenteavaliados.Finalmente,pormeiodeumcritriode escolha, quatro solues foram tomadas para anlise, revelando que o programa desenvolvido conseguiu encontrar solues mais eficientes e plausveis do que a originalmente proposta. Palavras-chave: Algoritmo gentico, projeto preliminar, compressor axial, turbomquinas vi Abstract Thisworkpresentsanapproachtooptimisethepreliminarydesignofhigh-performance axial-flow compressors. The preliminary design within the Gas Turbine Group at ITAiscarriedonwithanin-housecomputationalprogrambaseduponthestreamline curvature method, using correlations from the literature to assess the losses. The choice of manyparametersofthethermodynamiccycleandofgeometriesreliesupontheexpertise from the members of the Group. Nevertheless, it is still a laborious and time-consuming task, requiringsuccessivetrialanderrors.Therefore,tosupportthecompressordesignerinthe choiceofsomeparameters,anoptimisationprogram,namedREMOGA,waswrittenin FORTRAN language, allowing an easy integration with the programs developed by the Gas TurbineGroup.Theprogramisbaseduponamulti-objectivegeneticalgorithm,withreal codification and elitism. Then the REMOGA and the preliminary design program were integrated to design a 5-stageaxial-flowcompressor.Therefore,thestatorairoutletangles,thetemperature distribution and the hub-tip ratio were varied aiming at higher efficiencies and higher pressure ratios, but controlling the de Haller number and the camber angle. Thanks to the REMOGA, thousandsofdesignscouldbequicklyevaluated.Finally,usingachoicecriterion,four solutionswereselectedforfurtheranalysis,revealingthatthedevelopedprogramwas successful in finding more efficient and feasible compressor designs. Key words: Genetic algorithm, preliminary design, axial-flow compressor, turbomachinery vii LIST OF FIGURES Figure 1 NASA Rotor 37. Source: . ........................... 26Figure 2 Flow chart of multidisciplinary design optimisation of Luo et al. [30]. ................. 32Figure 3 Evolution of domestic processors from 1998 to 2011. ........................................... 33Figure 4 J unkers J umo 004 axial jet engine and Me 262. Source: ..... 36Figure 5 Rolls-Royce Trent 1000. Source: ................................... 37Figure 6 Classification of compressors. ................................................................................ 38Figure 7 Centrifugal compressor. Source .............................................. 39Figure 8 Comparison of some compressor types. ................................................................. 39Figure 9 Schematic figure of the main components in a gas turbine and the Brayton cycle. ...................................................................................................................... 40Figure 10 Scheme of an axial-compressor stage. .................................................................. 41Figure 11 Visual aid to the common plane scheme of an axial-flow compressor stage. ....... 42Figure 12 Details of a gas turbine detailing a compressor rotor row. ................................... 42Figure 13 Nomenclature according to Saravanamutto [37]. .................................................. 43Figure 14 Generic velocity triangles. .................................................................................... 44Figure 15 Hub to tip ratio and tip clearance. ......................................................................... 46Figure 16 Divergent isobaric lines and the increased compression difficulty in the last stages. ..................................................................................................................... 47Figure 17 Polytropic or small-stage efficiency. ..................................................................... 48Figure 18 A schematic real gas turbine cycle. ....................................................................... 49Figure 19 Axial-flow compressor stage in a T-s diagram. .................................................... 50viii Figure20Rotorrowandstatorrowwithvelocitytrianglesinanaxial-flow compressor stage. ................................................................................................... 50Figure 21 Inlet and outlet relative velocity ratio is reduced with the increase of fluid deflection. .............................................................................................................. 54Figure 22 Streamline-blade leading edge coordinate system (s-m). [42] .............................. 58Figure 23 Streamlines, stage rows and calculation nodes. Adapted from [42]. .................... 58Figure 24 Overview of the SLC program algorithm. ............................................................ 59Figure 25 Mapping between the decision space and the objective space. ............................. 61Figure 26 Representation of dominance and indifference between solutions in a two-objective minimisation problem. Solution a dominates b, but is indifferent to c. ........................................................................................................................ 63Figure 27 A convex function illustration. .............................................................................. 64Figure 28 Illustrative region where a gradient-based algorithm can get stuck onto a suboptimal solution. ............................................................................................... 66Figure 29 Simple GA algorithm [48]. ................................................................................... 67Figure 30 Chromosomal representation of decision variables. ............................................. 68Figure 31 Tournament selection illustration. ......................................................................... 69Figure 32 Biological crossover illustration. .......................................................................... 70Figure 33 Bit-wise crossover representation. ........................................................................ 70Figure 34 Single-point crossover representation. .................................................................. 71Figure 35 Two-point crossover representation. ..................................................................... 71Figure 36 Mutation operator. ................................................................................................. 72Figure 37 Algorithm of the REMOGA program. .................................................................. 72ix Figure 38 Rank assignment algorithm. .................................................................................. 74Figure 39 Bubble sort pseudocode ........................................................................................ 75Figure 40 Optimisation (a) without niche penalty and (b) with niche penalty ...................... 75Figure 41 Dependence of the sharing function with . ......................................................... 77Figure 42 Visual interpretation of the used value of share. ................................................... 78Figure 43 Crowded tournament selection operator. .............................................................. 79Figure 44 Multiple selections. ............................................................................................... 79Figure 45 SBX [54] operator and influence of parameter c. ................................................ 81Figure 46 Effect of mutation parameter m for x=0 and max=1. .......................................... 83Figure 47 Solutions behaviour after each of the implemented operators. ............................. 84Figure 48 Testing a simple MOOP. Population in the 1st, 10th and 100th generations. ......... 86Figure 49 Simple convex test function after 100 generations. .............................................. 87Figure 50 Non-convex test function from Fonseca and Fleming [56]. ................................. 88Figure 51 Poloni et al. [57] test problem after 500 generations. ........................................... 89Figure 52 SLCP and REMOGA coupling. ............................................................................ 90Figure 53 SLC program acts as blackbox. ............................................................................. 91Figure 54 SLCP output data to work together with REMOGA program. ............................. 93Figure 55 Streamlines and nodes of the original compressor design. ................................... 96Figure 56 Distribution of temperature rise weights along the stages. ................................... 97Figure 57 Pressure and temperature distributions of the original compressor design. .......... 97Figure 58 Camber angle distribution of the original design. ................................................. 98x Figure 59 De Hallernumber distribution of the original design. ......................................... 99Figure 60 Stage loading distribution of the original design. ................................................. 99Figure 61 Number of blades per row ................................................................................... 100Figure 62 Blade chord of each row. .................................................................................... 100Figure 63 Euler diagram representing the sets of feasible and unique solutions. ............... 101Figure 64 History of target efficiency ................................................................................. 102Figure 65 History of the hub to tip ratio. ............................................................................. 102Figure 66 History of temperature weights distribution. ...................................................... 103Figure 67 History of stator outlet angles distribution. ......................................................... 104Figure 68 Pressure ratio vs. camber penalty and last stage stator outlet angle for the limited subset of solutions. .................................................................................. 105Figure 69 The initial design is comparatively poor in satisfying de Haller number. .......... 106Figure 70 Solution 1. ........................................................................................................... 106Figure 71 Input conditions for solutions 1 and 2. ................................................................ 107Figure 72 Nodes and streamlines of solution 1. .................................................................. 108Figure 73 Nodes and streamlines of solution 2. .................................................................. 108Figure 74 Pressure and temperature rise per row of solutions 1 and 2. ............................... 109Figure 75 Number of blades and blade chord of each row for solution 1. .......................... 110Figure 76 Number of blades and blade chord of each row for solution 2. .......................... 110Figure 77 Camber angle distribution of solution 1. ............................................................. 111Figure 78 Camber angle distribution of solution 2. ............................................................. 111Figure 79 De Haller number distribution of solution 1. ...................................................... 113xi Figure 80 De Haller number distribution of solution 2. ...................................................... 113Figure 81 Stage loading distribution of solutions 1 and 2. .................................................. 114Figure 82 Pressure ratio vs. camber angle penalty from the refinement run. ...................... 115Figure 83 De Haller numbers do also concentrate close to zero. ........................................ 116Figure 84 Choice of solution 3. ........................................................................................... 116Figure 85 Stages temperature weight and stator air outlet angles. ...................................... 117Figure 86 Streamlines of solutions 3 and 4. ........................................................................ 118Figure 87 Pressure and temperature distribution of solutions 3 and 4. ............................... 119Figure 88 Number of blades and blade chord of each row for solution 3. .......................... 119Figure 89 Number of blades and blade chord of each row for solution 4. .......................... 120Figure 90 Camber angle distribution of solution 3. ............................................................. 121Figure 91 Camber angle distribution of solution 4. ............................................................. 121Figure 92 De Haller number distribution of solution 3 ....................................................... 122Figure 93 De Haller number distribution of solution 4. ...................................................... 122Figure 94 Stage loading distribution of solutions 3 and 4. .................................................. 123Figure 95 Velocity triangles. ............................................................................................... 133xii LIST OF TABLES Table 1 Summary of recent works presented at ASME Turbo Expo on compressor optimisation. .......................................................................................................... 28Table 2 Comparison between J unkers J umo 004 and Rolls-Royce Trent 1000. ................... 37Table 3 Thermodynamic processes at the rotor and stator. ................................................... 42Table 4 Compressor rows. ..................................................................................................... 91Table 5 Configuration of the computers used in the performance evaluation of the modified SLCP. ..................................................................................................... 92 xiii LIST OF SYMBOLS LATIN SYMBOLS Cabsolute velocity cblade chord ijd normalised distance between solutions i and j fvector of objectives Fobjective space gvector of inequalities constraints henthalpy hvector of equalities constraints htrhub-to-tip ratio m mass flow Nrotational speed in rpm ncniche count npoppopulation size Ppressure rradius set of real numbers ( ) . rank rank of a solution rppressure ratio spitch or spacing ( ) . Sh sharing function Ttemperature Utangential velocity Vrelative velocity Wpower xvector of decision variables (also referred to as solution) Xdecision space xiv GREEK SYMBOLS angle between the absolute velocity and the axial direction angle between the relative velocity and the axial direction specific heat ratio stagger or settting angle degree of reaction isentropic efficiency polytropic efficiency c polynomial crossover control parameter m polynomial mutation control parameter camber angle flow coefficient temperature or stage loading coefficient angular velocity SUBSCRIPTS 0total property 1rotor inlet 2stator inlet 3stator outletaaxial component mmeridional component wwhirl or tangential componentxv LIST OF ACRONYMS AND ABBREVIATIONS ANNArtificial Neural Network DOEDesign of Experiments EAEvolutionary Algorithm GAGenetic Algorithm IGVInlet Guide Vane LHSLatin Hypercube Sampling MOEAMulti-Objective Evolutionary Algorithm MOGAMulti-Objective Genetic Algorithm MOOPMulti-Objective Optimisation Problem N-SNavier-Stokes NSGA Non-dominated Sorting Genetic Algorithm OGVOutlet Guide Vane RANS Reynolds-Averaged Navier-Stokes REMOGAReal-Coded Elitist Multi-objective Genetic AlgorithmRSMResponse Surface Method SBXSimulated Binary Crossover SLCM Streamline Curvature Method SLCPStreamline Curvature Program SOOP Single-Objective Optimisation Problem xvi CONTENTS 1INTRODUCTION ......................................................................................................... 191.1 Motivation .............................................................................................................. 191.2 Objective ................................................................................................................ 201.3 Methodology .......................................................................................................... 201.4 Context ................................................................................................................... 211.5 Research on gas turbine within DCTA .................................................................. 221.6 Organization of the Thesis ..................................................................................... 242LITERATURE REVIEW ............................................................................................. 252.1 Introduction ............................................................................................................ 252.1.1 Solvers ..................................................................................................... 252.1.2 Reference stage ........................................................................................ 262.1.3 Optimisation methods .............................................................................. 272.2 Review of axial-flow compressor optimisation ..................................................... 273AXIAL-FLOW COMPRESSOR OVERVIEW ......................................................... 363.1 Introduction ............................................................................................................ 363.1.1 History ..................................................................................................... 363.1.2 Classification ........................................................................................... 383.1.3 Gas turbine ............................................................................................... 403.1.4 Basic operation ........................................................................................ 413.1.5 Nomenclature .......................................................................................... 433.2 Dimensionless parameters ..................................................................................... 443.2.1 Flow coefficient ....................................................................................... 443.2.2 Temperature or stage loading coefficient ................................................ 453.2.3 Degree of reaction ................................................................................... 453.2.4 Hub to tip ratio ......................................................................................... 453.2.5 Isentropic and polytropic efficiencies ..................................................... 463.3 Overview of axial-flow compressor performance ................................................. 483.3.1 Tip speed ................................................................................................. 523.3.2 Camber angle and de Haller number ....................................................... 533.3.3 Compressor surge .................................................................................... 543.3.4 Compressor choke ................................................................................... 554THE STREAMLINE CURVATURE COMPUTATIONAL PROGRAM .............. 564.1 Introduction ............................................................................................................ 564.2 The Streamline Curvature Method ........................................................................ 574.3 Computational Program ......................................................................................... 59xvii 5REAL-CODEDELITISTMULTI-OBJECTIVEGENETICALGORITHM PROGRAM .................................................................................................................... 605.1 Definitions ............................................................................................................. 605.1.1 Multi-objective optimisation problem ..................................................... 615.1.2 Domination .............................................................................................. 625.1.3 Pareto-optimal set .................................................................................... 635.1.4 Convexity ................................................................................................ 645.2 Traditional methods and the Genetic Algorithm ................................................... 655.3 Genetic Algorithm Fundamentals .......................................................................... 675.3.1 Selection or reproduction operator .......................................................... 695.3.2 Crossover operator ................................................................................... 705.3.3 Mutation operator .................................................................................... 715.4 Real-coded elitist multi-objective genetic algorithm program (REMOGA) ......... 725.4.1 Multi-objective formulation .................................................................... 735.4.2 Crowded Tournament Selection .............................................................. 795.4.3 Real-coded Polynomial and Elitist Crossover Operator .......................... 805.4.4 Real-coded Polynomial Mutation Operator ............................................. 825.5 Test functions ......................................................................................................... 855.5.1 Convex 2-variable 2-objective test function ............................................ 855.5.2 Non-convex test function ........................................................................ 875.5.3 Non-convex domain and disconnected Pareto set test function .............. 885.6 Summary of the chapter ......................................................................................... 896METHODOLOGY ........................................................................................................ 906.1 Modifications in the SLC program ........................................................................ 906.1.1 SLCP output data or REMOGA input data ............................................. 926.1.2 SLCP input data or REMOGA output data ............................................. 946.2 Formulation of the MOOP ..................................................................................... 956.3 REMOGA settings ................................................................................................. 956.4 Human design start point ....................................................................................... 967RESULTS AND DISCUSSION ................................................................................. 1017.1 Search: REMOGA history and filtering of solutions .......................................... 1017.2 Looking for solutions ........................................................................................... 1057.3 Analysis of search step solutions ......................................................................... 1077.3.1 Overview ............................................................................................... 1077.3.2 Camber angle ......................................................................................... 1107.3.3 De Haller number .................................................................................. 1127.3.4 Stage loading ......................................................................................... 1127.4 Refinement of the search space ........................................................................... 114xviii 7.5 Analysis of refinement step solutions .................................................................. 1177.5.1 Overview ............................................................................................... 1177.5.2 Camber angle ......................................................................................... 1207.5.3 De Haller number .................................................................................. 1207.5.4 Stage loading ......................................................................................... 1238CONCLUSIONS ......................................................................................................... 1249FURTHER WORK ..................................................................................................... 1269.1 Improvements ...................................................................................................... 1269.2 Suggestion of works ............................................................................................ 1279.2.1 Detailed project ..................................................................................... 1279.2.2 Robust optimisation ............................................................................... 127REFERENCES ..................................................................................................................... 128APPENDIX ASLC SUMMARY ..................................................................................... 133APPENDIX BOPTIMISATION PROGRAM .............................................................. 138B.1Main program ...................................................................................................... 138B.2Global variables ................................................................................................... 139B.3Reading initial population and program parameters ............................................ 140B.4Evaluating objectives ........................................................................................... 142B.5Fitness subroutine ................................................................................................ 145B.5.1Niche count subroutine .......................................................................... 147B.6Crowded tournament selection subroutine .......................................................... 149B.7Real-coded elitist crossover subroutine ............................................................... 150B.8Real polynomial mutation .................................................................................... 151APPENDIX CORIGINAL SLCP INPUT FILE ........................................................... 153APPENDIX DADDITIONALINFORMATIONFROMTHEOBTAINED SOLUTIONS ............................................................................................................... 157D.1 Rotor inlet Mach number ..................................................................................... 158D.2 Stator inlet Mach number .................................................................................... 159D.3 Rotor total loss ..................................................................................................... 160D.4 Stator total loss .................................................................................................... 161D.5 Rotor incidence angle .......................................................................................... 162D.6 Stator incidence angle .......................................................................................... 163 19 1INTRODUCTION 1.1MOTIVATION The axial-flow compressor is one of the most challenging components to be designed in a gas turbine. Its design involves a very large amount of design parameters, a plethora of designrequirements,encompassingseveralconflictingones,andnumerousconstraints. Therefore, even to an experienced compressor designer, it is demanding and time consuming to properly decide on design parameters. Moreover, those parameters influence differently many distinct and competing design objectives, e.g., high efficiency, high pressure ratio, low number of stages, wide surge margin, low weight, etc. Thus, it is virtually impossible to find an optimal compressor design by successive trial and error. To support the designer in choosing the most effective design parameters, tools for Multi-ObjectiveOptimisationProblems(MOOP)havebeendevelopedandareconstantly beingimproved,therebyreducingthedesignevaluationtime.ClassicalMethods,suchas Gradient-basedmethodsaredeterministicandmathematicallydemanding.Theyrequire numerical differentiation, which tends to be a source of numerical errors, and risk being stuck onto suboptimal solutions. Conversely, modern Evolutionary Algorithms (EAs) are robust and mathematicallysimple.Furthermore,theyareparticularlysuitedforMOOPand computationalparallelisation.Hence,EAs,suchasMulti-ObjectiveGeneticAlgorithm (MOGA), Non-dominated Sorting Genetic Algorithm (NSGA), etc., are spreading quickly as design tool assistant. 20 TheStreamlineCurvatureMethod(SLCM)consistsofwritingthenon-viscous equationsofcontinuity,motionandenergyalongacoordinatesystemlayingonthe streamlines and on the tangent to the blade edges. This coordinate system is preferred due the easily-derived calculation grid. Furthermore, as the SLCM bypasses the time consuming and demanding viscous-related calculations, it is very fast. The losses are, instead, assessed by empiricalcorrelationsderivedfromseveraltestscarriedonlaboratoryfacilities,hence providingreasonablepredictions.ThereforetheSLCMisveryusefulinthepreliminary design, as it combines good accuracy and quick evaluation.Thus, the blend of an axial-flow compressor performance program which uses the SLCM and an evolutionary algorithm not only does quickly provide an optimised component, but also offers a better understanding of the impact of the design parameters. 1.2OBJECTIVE Theobjectiveofthisworkistodevelopaproceduretooptimisethepreliminary designofahigh-performanceaxial-flowcompressorbycouplinganexistingin-house developed preliminary design computational program and a multi-objective genetic algorithm. 1.3METHODOLOGY Aiming at the proposed objective, the work was divided in two parts: 1.Optimisation: a.Literature review on the use of optimisation procedures in the design of axial-flow compressors; b.Study of multi-objective genetic algorithms; 21 c.Development of a FORTRAN program to compute a real-coded elitist multi-objective genetic algorithm; 2.The streamline curvature program a.Understanding of the fundamentals of the program b.Review of functions and main algorithm (carried on by the advisor) c.Modifications to couple with the GA program 1.4CONTEXT ThispresentworkwasexecutedundertheProgramaIntegradoGraduao-MestradoPIGM.ThisProgramaimsattheintegrationoftheUndergraduateandthe Masters Programs by allowing the student from the last year of the undergraduate course at InstitutoTecnolgicodeAeronutica(ITA)toundertakecoursesfromthepost-graduate programs, shortening the necessary time to fulfil the requirements to the title of Master in Science. Inthiscontext,theBachelorThesis(TrabalhodeGraduaoTG)was supervised to provide a well-developed start point to the Master Thesis. The TG of the author, entitled Project Optimisation of High-Performance Axial-Flow Compressors was executed under the supervision of Prof. Dr. J oo Roberto Barbosa (the same supervisor of this work). It preliminarilyvalidatedthedesignoptimisationprocedurebycouplingtheStreamline Curvature Method to a Multi-Objective Genetic Algorithm. In that work, the design variables weretheefficiency,hub-to-tipratioandthestatorairoutletanglesviaamultivariate interpolation, which used four control points, namely hub and tip at the first row and hub and tip at the last row. Diffusion factors and camber angles were controlled by means of a penalty factor treated as objectives to be minimised. 22 The Master Thesis was developed under the scholarship from Fundao de Amparo Pesquisa do Estado de So Paulo FAPESP (So Paulo State Research Foundation) at the Centre for Reference on Gas Turbine at ITA. 1.5RESEARCH ON GAS TURBINE WITHIN DCTA Tomita [1] describes the research on gas turbine within DCTA. A summary of this history is presented hereafter. Plans to develop gas turbines in Brazil are found in the Plans of Foundation [2] of the CentroTcnicodeAeronuticaCTA(AeronauticalTechnicalCentre),in1947. However,theresearchonlyflourishedinthe1970s,withtheestablishmentofaResearch ProgramatCTA.Atthetimeanewturbineprojectwasdeveloped,therebymany opportunitiesofpartnershipswithimportantmanufacturers,likeRolls-Royce(UK),Garret (USA),Pratt&Whitney(USAandCanada),LucasAerospace(UK),andKongsberg (Norway), succeeded and were valuable. Thereafter, the project was seriously hindered due to lack of experienced professionals. A joint project with Rolls-Royce to design and manufacture of a 300 kW turboprop to be mounted on aircrafts from Bandeirante class was halted as a result of lack of personnel. Thus,anambitiousprogramoftrainingtheCTApersonnelcommencedwith CranfieldInstituteofTechnology(currentlyCranfieldUniversity).FromthisInstitute, engineersfromCTAandITA,workinginresearchrelatedtoGasTurbines,graduated, including the supervisor of this work, who obtained his PhD degree in Cranfield in 1987. Even with the present practice of importing gas turbines rather than designing and manufacturing in Brazil, the necessity of specialists in those machines, mainly in performance 23 analysis and applications is evident. The process of choosing the correct turbine is vital, since it undoubtedly allows a significant reduction in operation and maintenance costs. Observing the current actions of the major players from the energy sector in Brazil, or even big companies moving to the energy sector, one might again note a real requirement forspecialistsinturbinesandcompressors.Inthiscontext,twocompaniesshouldbe highlighted: Vale Solues em Energia (VSE), which is preparing to design and manufacture its own gas turbines, to secure its highenergydemand in mining operations; and General Electric, which launched a massive investment program in Brazil in 2010. CTAwasrenamedDCTAComando-GeraldeTecnologiaAeroespacial (BrazilianGeneralCommandforAerospaceTechnology),buttheeffortstoimplementa modern Turbine Laboratory persist. According to Barbosa [3], it should include a compressor test bed (1500 kW shaft power and up to 60,000 rpm); a turbine test bed (2000 kW brake power and rotation speed up to 60,000 rpm) and a combustion chamber test bed (for hot gases upto1500K;1.0MPa).Thedevelopmentofasmallgasturbineforresearchshouldbe carried on, as well. The research on gas turbine at ITA is conducted by the Centre for Reference in Gas Turbine (CRTG Centro de Referncia em Turbinas a Gs). The Centre, which belongs to the Mechanical Engineering Department of ITA, relies its research upon information of public domain and upon many years of experience from its members. ThecentrepieceoftheresearchdevelopedatCRTGisonnumericalsimulation. Programsofdesignpointperformance,off-designperformance,computationalfluid dynamics,transientperformance,combustionchamberperformance,noisepredictionhave been written and are fully operational.24 1.6ORGANIZATION OF THE THESIS In chapter 1 the reader finds the introduction, where the motivation, objective and methodology are presented. A brief history of the research on gas turbine within DCTA is also presented. Chapter2containsareviewofstudiespublishedinaxial-flowcompressor optimisation. A review of ASME Turbo Expo congresses since 2000 in this particular field is also shortly conducted. Chapters3and4providethebasictheoryonaxial-flowcompressorsandonthe streamline curvature method. In chapter 5, the author starts with the basic ideas behind Genetic Algorithms and then he details features, algorithms and models used in the REMOGA program, which was developed as part of this work. Chapter6describeshowtheintegrationoftheSLCprogramandtheREMOGA program took place. Chapter7showstheresultsobtainedthroughtheaforementionedintegrationand analyse four solutions, which were selected among thousands of solutions proposed by the REMOGA. Chapters 8 and 9 conclude this work, suggesting future works as well.Fourappendixesareprovided.ThefirstcontainsabasicderivationoftheSLC method. The second contains the FORTRAN code of the developed optimisation program. The third appendix offers the design parameters of the start-point axial-flow compressor. And the last appendix provides further graphical information from the compressors analysed in this work.25 2LITERATUREREVIEW Amongseveralturbomachineryconferences,ASMETurboExpoisrecognisedas one of the most important events, and has been taking place every year since 1956. Therefore, in order to present the recent progress of the studies on compressor optimisation, a summary of Turbo Expo papers from 2000 to 2011 that are tied to the theme is presented in Table 1.2.1INTRODUCTION Beforeproceedingwiththecomparativetable,somepreliminaryconceptsare presented. 2.1.1Solvers Solvers can be defined as computational programs that solve a given mathematical problem. In turbomachinery, most flow-field-related solvers rely upon a computational tool calledCFD,whichstandsforComputationalFluidDynamics.CFDisconcernedwith numerical solutions of the set of governing equations of fluid dynamics and heat transfer. It is theuseofnumericalmethodsandalgorithmstoobtainapproximatesolutions.The fundamental governing equations of interest for CFD are the Navier-Stokes equations (N-S), the transport of mass and of energy. The N-S equations are a set of nonlinear partial differential equations, that describes the motion of fluids. N-S equations lead to mathematically complicated problems, which are 26 virtuallyimpossibletosolve,exceptforverysimplecases,whicharenotofreal-world interest. Therefore numerical methods andalgorithms are employed to obtain approximate solutions. According to the problem, the user may choose a 2D or 3D solver, depending on the desired accuracy and on the computational resources available, as well. To describe turbulent flows, instantaneous quantities of the N-S equations are time-averaged to provide an approximation, which is easier to calculate. The resulting equations are called Reynolds-Averaged Navier-Stokes equations, or RANS. A further simplification of the N-S equations can be carried out by ignoring viscosity and heat conduction. The simplified equations are called Euler equations. If used perse it providesveryroughapproximationsinturbomachinerycalculation,asviscosityplaysan important role. Nevertheless, Euler equations can be used accurately if losses are assessed by correlations derived from experiments.2.1.2Reference stage The most frequent reference stage used for academic purposes is the NASA Rotor 37, see Figure 1. As its flow field was used by the American Society of Mechanical Engineers in 1994 in a CFD blind-test exercise, plenty of studies on the flow field in the aforementioned rotor were derived [4]. Figure 1 NASA Rotor 37. Source: . 27 NASA Rotor 37 was designed and tested at NASA Lewis Research Center (renamed NASAGlennResearchCenter)inthelate1970s.Itisalowaspectratioinletwith36 multiple-circular-arc (MCA) blades. Rotor 37 has a pressure ratio of 2.106 at a mass flow of 20.19 kg/s. 2.1.3Optimisation methods A brief introduction to optimisation methods is provided in chapter 5. 2.2REVIEW OF AXIAL-FLOW COMPRESSOR OPTIMISATION A comparative table of works presented at ASME Turbo Expo from 2000 to 2011 regardingoptimisationinaxial-flowcompressorsisdrawntoprovideapanoramaofthe theme,aswellasitsevolution.TheworkswereprimarilytakenfromthetopicDesign Methods and CFD Modelling for Turbomachinery. Therefore, the following information was taken, when applicable: Problem:whethersingle-objectiveormulti-objective.MOOPswhichwere solved with a single objective function (weighted average) were considered SOOP; Solver: which method was used to obtain quantitative results from the design; Referencestage:manyoptimisationstudiesarecarriedonlong-time-established open-data stages, e.g., NASA rotor 37; Optimisation method; design variables and objectives. 28 Table 1 Summary of recent works presented at ASME Turbo Expo on compressor optimisation. Ref.titleproblemsolverreference stageopt. Methoddesign variablesobjective [5] 2000 The combined use of Navier-Stokes solvers and optimization methods for decelerating cascade design SOOPNavier-StokesC4 airfoilgradient-based inlet Pt, Tt, M1, flow angle; chord; inlet mech. angle, solidity, camber, t/c min. total-to-total pressure loss coef. [6] 2000 Design optimization of axial flow compressor blades with three-dimensional Navier-Stokes solver SOOP 3D Navier-Stokes four-stage ATKOM NPT steepest decent and conjugate direction stacking linesmax. efficiency [7] 2000 Shape optimization of transonic compressor blades usign quasi-3D flow physics SOOPQuasi-3D N-SNASA rotor 37 gradient-based and sensitivity analysis 8 blade section geometry variables max. adiabatic efficiency [8] 2001 Shape optimization of high-speed axial compressor blades using 3D Navier-Stokes flow physics SOOP 3D Navier-Stokes NASA rotor 37 modified feasible directions algorithmblade section geometry max. adiabatic efficiency [9] 2002 Towards a reduction of compressor blade dynamic loading by means of rotor-stator interaction optimization MOOP CFD code; sliding mesh and time dependent NACA 65-12-10 Multi-objective Evolutionary Algorithmaxial distance between rows and circumferential clocking min. dynamic loading and max. time-avg. efficiency [10] 2002 Aerodynamic design optimization of an axial flow compressor rotor SOOP 3D Navier-Stokes NASA rotor 37RSMstack line profilemax. efficiency [11] 2003 Advanced high turning compressor airfoils for low Reynolds number condition. Part 1: design and optimization MOOPQuasi-3D N-S Evolution Strategies and MOGA blade spline control points min. total pressure loss and min. deviation angle [12] 2003 Numerical optimization of turbomachinery bladings SOOP Quasi-3D N-S and 3D N-S CONMIN (gradient-based) blade deformationmax. efficiency [13] 2003 Automated design optimization of compressor blades for stationary, large-scale turbomachinery MOOP Mises (Euler Q3D) Covariance Matrix Adaption (CMA) 3D blade geometry weigted sum: aerodynamic losses, maximumMach, etc 29 Ref.titleproblemsolverreference stageopt. Methoddesign variablesobjective [14] 2004 Application of multipoint optimization to the design of turbomachinery blades SOOP 3D Navier-Stokes NASA rotor 37 ANN, GA, Simulated Annealing blade parameters efficiency and weighted sumof penalties [15] 2005 Multiobjective optimization approach to turbomachinery blades design MOOP Reynolds-averaged 2D N-S real-coded MOEA blade geometry: Bezier control points max. static pressure and min. total pressure loss [16] 2006 Design optimization of transonic compressor rotor using CFD and Genetic AlgorithmSOOP 3D Navier-Stokes NASA rotor 37 DOE, RSM (second-order polynomial) and GA leading edge line: sweep, bow max. adiabatic efficiency [17] 2006 Modern compressor aerodynamic blading process using multi-objective optimization MOOP3D-CFD Rolls-Royce datumdesign DOE, Monte-Carlo Simulation, NSGA-II blade section geometry min. design point loss and max. working range [18] 2006 Optimal design of swept, leaned and skewed blades in a transonic axial compressor SOOP 3D Navier-Stokes NASA rotor 37 DOE, RSM (second-order polynomial) sweep, lean and skew max. adiabatic efficiency [19] 2006 Automated Multiobjective optimisation in axial compressor blade design MOOP 3D Navier-Stokes (DLR-code TRACE) asynchronous MOEA, ANN 3D blade geometry total pressure loss (DP) and total pressure (ODP) [20] 2006 Compressor blade optimization using a continuous adjoint formulation SOOP 3D Navier-Stokes steepest decent and adjoint method blade geometry: 3D NURBS, 65 control points min. constrained augmented functional [21] 2007 A first-principles based methodology for design of axial compressor configurations SOOP CFD code SWIFT NASA stage 35 DOE (CCD), RSM and LSM blade parameters: CCGEOM desirability function, which embraces efficiency and pressure ratio [22] 2007 Optimization of the gas turbine engine parts using methods of numerical simulation SOOPCFD NUMECA IOSOblade geometry efficiency for operation mode 30 Ref.titleproblemsolverreference stageopt. Methoddesign variablesobjective [23] 2008 Stacking and thickness optimization of a compressor blade using weighted average surrogate model MOOP Blade-Gen, Turbo-Grid, CFX-Pre, CFX-Solver NASA rotor 37 Latin hypercube, PRESS based averaging, RSM and gradient-based 6 design variables defined by parametric curves efficiemcy, total pressure and the combination of both [24] 2008 Design optimization of a HP compressor blade and its hub endwall SOOPCFD code elsA Cenaero GA, RSM-RBF, DOE 48 bladeparameters and 16 hub surface parameters isentropic efficiency at two operating points [25] 2008 Accelerated industrial blade design based on multi-objective optimization using surrogate model methodology MOOP2D MISES DOE (Latin Hypercube or SOBOL); NSGA-II; Kriging RSM 2D blade profile pressure loss at DP, stall and choke [26] 2008 A NURBS-based optimization tool for axial compressor cascades at design and off-design conditions SOOP blade-to-blade MISES (Q3D) UKS-31 vane and E/CO-4 stator GA (developed by Carroll) airfoil geometry: LE and TE dimensions, thickness, etc. 38 design parameters weigted sum: losses and inlet angle [27] 2008 Multi-objective optimization in axial compressor design using a linked CFD-solver MOOP 3D-RANS and throuflow MAGELAN IDAC3 of RWTH Aachen MOEA, ANN, Kriging and polynomial surfaces chordwise s-Shift, stagger variation, suction side control points, annulus. 23 parameters efficiency improvement and diffusion factor in stator 3 [28] 2009 Application of simple gradient-based method and multi-section blade parametrization technique to aerodynamic design optimization of a 3D transonic single rotor compressor SOOP 3D Navier-Stokes coupled with Baldwin-Lomax NASA rotor 37 Simple gradient-based Multi-section blade parameters adiabatic efficiency [29] 2009 Optimization of variable stator's angle for off design compression systems using streamline curvature method SOOPSLC method NACA 10-stage subsonic axial compressor Genetic Algorithm VSV and VIGV angles total pressure at surge-margin-related operating point 31 Ref.titleproblemsolverreference stageopt. Methoddesign variablesobjective [30] 2009 Multiobjective optimization approach design of a three-dimensional transonic compressor bladeMOOP3D-RANSNASA rotor 37 Multiobjective Differential Evolution (MDE) 3D blade parameters - non-uniformB-spline control points isentropic efficiency and min. maximumstress [31] 2010 Blade geometry optimization for axial flow compressor SOOPCFD NUMECANASA rotor 67 DOE (FCCD, AD), GA and RSM (polynomial and basis-function) blade sections B-spline parameters, lean and sweep combination off overall eficiency and pressure ratio [32] 2010 Design optimization of circumferential casing grooves for a transonic axial compressor to enhance stall margin SOOP3D-RANSNASA rotor 37 DOE (LHS), Radial Basis Neural Network, SQP circumferential grooves: width, depth normalized by tip chord max. stall margin [33] 2011 Optimization of a transonic axial compressor considering interaction of blade and casing treatment to improve operating stability MOOPANSYS-CFXNASA rotor 37 DOE (LHS), RSM, NSGA-II circumferential grooves: width, depth. Angle between axis of rotation and camber tangent surge margin and peak adiabatic efficiency [34] 2011 Optimization of a 3-stage booster part1: the axisymmetric multi-disciplinary optimization approach to compressor design MOOP T-AXI: axyisymmetric solver MOGA and gradient-based improvements 53: inlet Mach, velocity ratios, rV stator outlets, hub spline control points, taper, no. blades, etc. efficiency, mass, length, rotor blade count and stator blade count 32 FromTable1onemightnoticethatintheearly2000s,mostoftheoptimisation methods were based on gradient. Later, however, the use of EA was the rule. Similarly, a tendency to MOOPs is observed, which is related to the spread of MOEA. Before, MOOPs weremostlytreatedasSOOPsbymeansofencompassingmanyobjectivesinasingle objective function (weighted average). Table 1 also shows that blade profile optimisation has been extensively studied in the context of compressor optimisation. Evidently the techniques employed are closely related to the computer capabilities. In 2000, Chung and Lee [7] used a quasi-3D Navier-Stokes solver and a gradient-based method in a SOOP to optimise the NASA rotor 37 with eight design variables.Nineyearsafterwards,Luoetal.[30]conductedastudyonmulti-disciplinary optimisation of the same NASA rotor 37 using a 3D-RANS solver to the aero domain and FEM to the mechanical domain using 19 design variables related to the blade suction surface geometry. The optimisation aimed not only at higher isentropic efficiencies, but also at the minimisation of the maximum mechanical stress. To achieve that, aero and mechanical mesh wererequiredandtheaerosolutionhadtobecalculatedtofeedtheFEMboundary conditions, as may be clear in Figure 2. Figure 2 Flow chart of multidisciplinary design optimisation of Luo et al. [30]. StartPreprocessingParallelMDEEndParametrisation of 3D bladeGenerationaero. meshGenerationmech. meshCFD solution FEM solutionAero performance computationMechanicsperformance computationDesign variableSurfacespressureAeroefficiencyMechanicsperformance functionvalue33 These two different approaches to the optimisation of the NASA rotor 37 highlight theevolutionoftheoptimisationcapabilitiesinthe2000sdecade.Themajormovefrom simpleSOOPgradient-basedstrategiestomultidisciplinaryoptimisationinvolvingseveral design variables and objectives was certainly due to the advances in computer hardware, as EAsrequireconsiderableamountofcomputationaleffortandareparticularlysuitedto parallel computing [35]. Gathering information from 44 Intel domestic processors, summarised in Figure 3, a glimpse of the evolution of the processors in a decade can be put into perspective. To plot Figure 3, the following processor families were taken into account: Pentium III, Pentium 4, Pentium 4 HT, Celeron, Celeron D, Pentium D, Pentium Extreme Edition, Core 2 Duo, Core 2 Quad, Core 2 Extreme, Core i3, Core i5, Core i7 and Core i7 Extreme Edition. Figure 3 Evolution of domestic processors from 1998 to 2011. It is noticeable that a stabilisation in clock speed was reached close to 4 GHz, but the increase of the number of transistors and of threads is still taking place. But the main benefit inrecentcomputationforMOGAistheparallelisationcapabilitiesprovidedbymultiple threads. 02004006008001000120014001998 2000 2002 2004 2006 2008 2010 2012# Transistors (in millions)yearTransistors and Cache memmory over time1t hr ead2t hr eads4t hr eads8t hr eads12t hr eadsbubbl e si ze: Cachememor y[ 0. 125; 12] MB0. 00. 51. 01. 52. 02. 53. 03. 54. 04. 51998 2000 2002 2004 2006 2008 2010 2012Clock [GHz]yearClock and die lithography over time1t hr ead2t hr eads4t hr eads8t hr eads12t hr eadsbubbl e si ze: Li t hogr aphy[ 32; 250] nm34 Recalling Table 1, one may observe that among 30 selected papers, 24 were centred onthebladeprofile.Frombladesectiongeometrythroughsplinecontrolpointstoblade stackinglineandfromleadingedgelinetosweep,leanandskew,thethemehasbeen thoroughlyexplored.Similarly,themethodsrangedfromsimplegradient-basedonesto various Evolutionary Algorithms, Response Surface Method, Latin Hypercube Sampling and Artificial Neural Network. Predominantly, however, 3D or quasi-3D Navier-Stokes solvers were employed. Apartfrombladegeometryoptimisation,BininiandToffolo[9]studiedtheaxial distance between rows and circumferential clocking on dynamic loading and efficiency. The optimisation was conducted via MOGA. Furthermore, Choi et al. [32] and Kim et al. [33] carriedaninvestigationoncircumferentialgroovestargetinghigherstallmarginandpeak efficiency. From2000to2011,onlyShadarametal.[29]presentedaworkoncompressor optimisation using the Streamline Curvature Method at Turbo Expo. The study aimed at the maximisation of the total pressure ratio at off-design condition of a 10-stage compressor by means of changing the stagger angles of the inlet guide vane (IGV) and two rows of stator vanes. To achieve that, a single-objective GA was employed. Apart from researches published at Turbo Expo, Oyama and Liou [35] developed a multiobjective design optimisation tool based on the SLC method and on a real-coded MOGA aimingathigherefficienciesandpressureratiosofa4-stageaxialflowcompressor.To achieve that, they used design parameters at the rotor trailing edge and at the stator trailing edge. At the former, total pressures and solidities are design variables, and at the latter, flow anglesandsolidities.Toavoidflowseparation,thediffusionfactorwasconstrained.The study revealed hundreds of feasible Pareto-optimal solutions. 35 KeskinandBestle[36]presentedattheGermanAerospaceCongress2005a procedure to automate a given Rolls-Royce preliminary design process to find Pareto-optimal solutions for design conditions. A meanline prediction processwas integrated to sampling methodslikeDesignofExperimentsandMonte-CarloSimulationandtoaMulti-island Genetic Algorithm (MIGA). Additionally, a gradient-based Lagrange-Newton type algorithm is used. In order to reduce the number of design variables and keep the design freedom to save computationalcosts,Bzier-splineparameterisationwasemployedtodescribetheannulus lines and the stage pressure ratio distribution. In this manner, the control points of the Bzier-splineswereusedasdecisionvariables.Theoptimisationgoalwasoverallpolytropic efficiency, overall pressure ratio and surge margin at design point. The constraints were: stage loadings,relativerotorandabsolutestatorinletMachnumbers,compressorexitMach number,Kochparameters,diffusionnumbersanddeHallernumbers.KeskinandBestle foundthattheefficiencycouldriseby0.11%pointkeepingthesurgemarginconstantor improve the surge margin by 3.2% points without diminishing efficiency. 36 3AXIAL-FLOWCOMPRESSOROVERVIEW This chapter aims at providing the basic knowledge about axial-flow compressors. It waswrittenbasedprimarilyonthebooksofSaravanamuttoo[37],Aungier[38],Horlock [39], Boyce [40] and Walsh [41] . 3.1INTRODUCTION The purpose of the compressor is to raise the total pressure of the working fluid to a level required by the thermodynamic cycle. The pressure rise should consume the minimum shaft power, as this component absorbs approximately one third of the turbine power. 3.1.1History Axial-flow compressors for aeronautical applications started their development in the 1930s and entered into service at the end of the WW2. The Germans took the lead with the engine J unkers J umo 004, which was mounted on many aircrafts, among them, the famous Messerschmitt Me 262 Schwalbe, the world first operational jet-powered fighter aircraft. Figure 4 Junkers Jumo 004 axial jet engine and Me 262. Source: 37 A British axial engine program was also carried (The Metropolitan-Vickers F.2 was the first axial British design), but it was unsuccessful to deliver an engine to the war. From1940sto2010s,therewasaconsiderabletechnologicalleapinaxial-flow compressordesign.Metallurgytechnology,newmaterials,multi-spoolconfigurations, variable geometries, computational resources and test facilities contributed for the increase in efficiencyandachievementofhigherpressureratioswithfewerstages.Forthesakeof comparison, Table 2 provides some illustrative data about the J umo 004 and the Rolls-Royce Trent 1000 (Figure 5), certified in 2007 to show the evolution after a bit more than half of a century. Table 2 Comparison between Junkers Jumo 004 and Rolls-Royce Trent 1000. J unkers J umo 004Rolls-Royce Trent 1000 TypeTurbojetTurbofan Entry19442007 (FAA certified) Pressure ratio3:150:1 Spools13 Number of stages81+8+6 =15 Average pressure ratio per stage1.1471.298 Thrust [kN]8.7 8.8 240 330

Figure 5 Rolls-Royce Trent 1000. Source: 38 3.1.2Classification Compressorsareclassifiedintotwomajorgroups:positivedisplacementand dynamic. Positive displacement compressors capture fluid in a certain pressure, trap it in a hermetic volume and deliver it to a higher pressure end. Normally, they handle small flow rate, but range from small to very large pressure ratio. Dynamic compressors continuously transfer energy to the fluid, which does also flow continuously. Centrifugal compressors and axial-flowcompressorsareexamplesofdynamiccompressors.Abasiccompressor classification scheme is shown in Figure 6. Figure 6 Classification of compressors. The flow in an axial-flow compressor suffers little change in radius compared to a centrifugal compressor. Besides the rotation which is implied by the rotor, the air flow along theradiusinacentrifugalcompressorandalongtheaxialdirectioninanaxial-flow compressor. Therefore, the centrifugal compressor (see Figure 7) is capable of higher pressure ratios per stage, but if a high mass-flow is desired, then the frontal area increases, while the axial-flow compressor achieves lower pressure ratios per stage, but handles higher mass flow per unit frontal area.Althoughcentrifugalcompressorsachievehigherpressureratiosperstage,multi-stage configurations present considerable losses due to the high fluid deflections required to deliver the compressed fluid from one stage to another. CompressorPositive displacement DynamicCentrifugal Axial-flow39 Figure 7 Centrifugal compressor. Source Therefore, it was recognised from the beginning of the gas turbine history that axial-flowcompressorswouldbecapableofhigherpressureratioandhigherefficiencythan centrifugal compressors[37]. Figure 8 Comparison of some compressor types.Another difference between axial-flow and centrifugal compressors is that the latter has narrower operational range than the former. In an axial-flow compressor, a small variation of flow rate around the design point results in great pressure ratio variation in comparison with centrifugal compressors. Schematically, the comparison between centrifugal and axial-flow compressors is shown in Figure 8. High-performance axial-flow compressors seek high efficiencies and high pressure ratios, but with few stages. This is almost contradictory, because then high air velocities are required,butthisnormallyincursinhigherfrictionandhigherlosses.Thusatuned P / PdesignQ / Qdesign1.00 0.95 0.90 1.05 1.100.850.800.900.951.001.051.10Axial-FlowCompressorCentrifugalCompressorFlowHeadPositive displacementCentrifugalCompressorAxial-FlowCompressor40 temperaturedistributionalongthestagesisrequired,aswellasaproperselectionofthe airfoil. 3.1.3Gas turbine Asimpleandidealgasturbinebasicallyconsistsofthreecomponents:the compressor, the combustion chamber and the turbine. The working fluid (e.g., air) enters the compressor, which raises the pressure and the temperature of the fluid in an isentropic process (ideally). The compressed fluid is then provided to the combustion chamber, wherein fuel is added and burnt, leading to a dramatic increase in temperature and energy of the mixture in a isobaric process. Finally, the working fluid expands isentropically in the turbine, transferring energy to its blades. The turbine and the compressor are connected by a shaft, which transfers mechanical energy from the turbine to the compressor. The turbine must extract energy in excess to drive a load (e.g., propeller, generator, free turbine, etc.). Figure 9 shows a simple gas turbine scheme and its related ideal temperature-entropy diagram. Figure 9 Schematic figure of the main components in a gas turbine and the Brayton cycle. In a simplistic approach, considering constant specific heat at constant pressure cp, the ideal cycle efficiency may be calculated as: ( ) ( )( )4 13 23 4 2 1 3 223 3 2 3 21 1cycle p pcyclepT TT Tw c T T c T T T Tq c T T T T| | | | || \ . \ .= = = (1) aircombustionchamberturbine compressorfuelexhaustgaspoweroutput12 34sTP1P2123441 Using, for isentropic compression or expansion: 1a ab bT PT P| | | |= ||\ . \ ..(2) Then, if rp denotes the pressure ratio2 1/ P P: 1 14 13 2 13 23 21 111cyclepP PT TP PT T r ((| | | | (( || ((\ . \ .| | (( = = | |\ .(3) From Equation (3) one immediately notices the relevance of the compressor in the overall engine efficiency.3.1.4Basic operation Anaxial-flowcompressorconsistsofaseriesofrotatingbladesandstationary blades, as shown in Figure 10. The air first enters a row of rotating blades, where mechanical energyfromtheshaftistransferredtothefluidtoaccelerateit.Then,theairwithhigh velocity is delivered to the stationary row, where it flows through a divergent nozzle and is diffused, i.e., fluid kinetic energy is converted to static pressure rise. Figure 10 Scheme of an axial-compressor stage. rotorstatorrotationMechanical Energy Fluid kinetic energyFluid kinetic energy Static pressure rise42 Figure 10 is a recurrent drawing, which is as if the cylindrical surface, where the blades are laid on, was unfolded. Actually the scheme refers to the surface at the mid-line. Figure 11 illustrates the aforementioned unfold. Figure 11 Visual aid to the common plane scheme of an axial-flow compressor stage. Figure 12 Details of a gas turbine detailing a compressor rotor row. WalshandFletcher[41]presentasummaryofthethermodynamicprocesses occurring at the rotor and at the stator in Table 3: Table 3 Thermodynamic processes at the rotor and stator. RotorStator Static pressureIncreaseIncrease Total pressureIncreaseSmall decrease Static temperatureIncreaseIncrease Total temperatureIncreaseConstant Relative velocityDecrease- Absolute velocityIncreaseDecrease EnthalpyIncreaseConstant DensityIncreaseIncrease 43 3.1.5Nomenclature The literature presents many different nomenclatures for blade and cascade. In this work, the nomenclature used by Saravanamuttoo [37] is preferred. An overview is presented in Figure 13 Figure 13 Nomenclature according to Saravanamutto [37]. LettersC,VandUareusedforabsolute,relativetotherotorandtangential(or peripheral) velocities, respectively. Subscripts 1, 2 and 3 denote respectively rotor inlet, rotor outlet and stator outlet. Subscript 0 denotes total property. Subscripts wand a indicate the whirl(tangential)andtheaxialcomponents.Themeridionaldirectionmisgivenbythe composition of the radial and axial directions of the flow: 2 2 w aw aVr V zmV V+=+(4) Greek letters andindicate absolute air and relative air angle; denotes blade angle. Thus, incidence angle is given by 1 1 = and deviation angle by2 2 = . Blade camberangle is given by 1 2 = and the deflection of the air by 1 2 = . Letterdenotes the stagger or setting angle, which is the angle between the chord direction and the axial coordinate. a11V1csV22Pointofmaximumcamber21bladeinletangle2bladeoutletangle bladecamberangle( 1- 2) settingorstaggerangles pitchorspace deflection( 1- 2)1airinletangle2airoutletangleV1airinletvelocityV2airoutletvelocity incidenceangle( 1- 1) deviationangle( 2- 2)c chord11C1V1Cw1Ca1UC2V2Ca2Cw22244 The distance from the leading edge to the trailing edge is the chord c. The distance from one blade to another measured at constant axial coordinate is the space or pitch s. The inlet velocities and the outlet velocities of a rotor row are usually drawn together in a recurrent scheme named velocities triangles, as shown in Figure 14. If the row is purely axial, then the meridional component is the axial component. Figure 14 Generic velocity triangles. 3.2DIMENSIONLESS PARAMETERS 3.2.1Flow coefficient The first dimensionless parameter commonly used in performance calculation is the flow coefficient, which is defined as: 1 aCU = (5) Saravanamuttoo [37] suggests a range| |0.4,1.0 . The axial velocity is directly related to the flow coefficient and for advanced aero engines, it can reach up to 200 m/s. 1 wV1 wC2U2 wV2 wC1 mC2 mC1U45 3.2.2Temperature or stage loading coefficient The temperature or stage loading coefficient indicates the amount of work per stage and is defined as: constant003 012 2pcp stagec Th hU U = = (6) For satisfactory operation Walsh and Fletcher [41] suggest| |0.25,0.5 .Efficiency improves as loading is reduced, but a decrease in stage loading implies more stages. Thus, a compromise is in question for aero engines, as both high efficiency and low weight (fewer stages) are mandatory.3.2.3Degree of reaction The distribution of the flow diffusion taking place at the rotor and the stator rows is indicatedbythedegreeofreaction.Thedegreeofreactionistheratiobetweenthestatic enthalpy rises in the rotor and in the stage: constant2 1 2 13 1 3 1pch h T Th h T T = = (7) Manypreliminarycompressordesignsstartwitha50%reaction,duetoeven distribution of diffusion, leading to smaller losses. 3.2.4Hub to tip ratio The hub to tip ratio is defined as the ratio of hub and tip radii: hubtiprhtrr= .(8) 46 High values of hub to tip ratio usually indicate short blades, hence, the tip clearance becomes relatively higher. The tip clearance, as the name suggests, is the distance between the blade tip and the compressor casing. High values of tip clearance lead to lower efficiencies, due to leakage flow through the spacing. Figure 15 shows the hub to tip ratio and the tip clearance in an actual compressor. Figure 15 Hub to tip ratio and tip clearance. Low values of hub to tip ratio yield long blades, hence more pronounced secondary losses, as well as, more difficult mechanical mounting in the rotor disc. 3.2.5Isentropic and polytropic efficiencies It is noted that total properties refer to the fluid with zero velocity andall of the kinetic energy has been adiabatically converted to internal energy. A subscript 0 is used to denotetotalproperties.Inagivenpoint,thetotalenthalpyandtotaltemperatureare, respectively: 202Ch h = + ,(9) 202pCT Tc= + ,(10) where h is the static enthalpy, C is the absolute velocity of the fluid and cp is the specific heat at constant pressure. rhubrtipHub to tip ratio Tip clearance47 The compressor total-to-total isentropic efficiency is given by: 02 0102 01.ch hh h =(11) If the variation of cp with the temperature is ignored, then: ( )( )02 0102 0102 01 02 01,pcpc T TT Tc T T T T = = (12) 102 02 0101 01 0102 01 0201 01 011.1cp T Tp T TT T TT T T| | |\ .= = (13) For later compressor stages, as the pressure is already high, it is much more difficult to increase the pressure. This can be explained by the fact that the isobaric lines in a T-s diagramaredivergent(totheright),asshowninFigure16.Noticeably,thecompressor requires more energy to compress the fluid in the first stage than in the last stage, even for the same pressure ratio. Figure 16 Divergent isobaric lines and the increased compression difficulty in the last stages. Hence the efficiency of the latter stages tends to be smaller than the initial stages, even when the technological level is the same.sTp1p2p3p4FirststageLaststage4 23 1p pp p=48 This fact revealed the necessity of another definition of efficiency formulti-stage compressor: the polytropic efficiency or small-stage efficiency, which is defined as a constant isentropic efficiency of an elemental stage throughout the whole compression stage: ,constantcdTdT= = (14) The idea of the polytropic efficiency can be visualised in Figure 17. The polytropic efficiencyrepresentstheparticulartechnologicallevelforaparticulardesign.Thus,itis reasonable in preliminary design to consider constant polytropic efficiency for all stages. Figure 17 Polytropic or small-stage efficiency. 3.3OVERVIEW OF AXIAL-FLOW COMPRESSOR PERFORMANCE Insection3.1.3,averysimplegasturbinethermodynamiccyclewaspresented. Nevertheless,thecompressionisnotisentropic,thecombustionisnotisobaricandthe expansionisnotisentropic.Thustheoverallefficiencyissmaller.Fromnowon,the discussion here will focus on the compressor side. sTT T dTdTElemental stage49

Figure 18 A schematic real gas turbine cycle. Firstly, to support the derivation, consider the compressor stage in a temperature-entropy diagram in Figure 19. Assuming adiabatic process, one immediately finds that the power input to the compressor rotor is given by: ( )02 01 pW mc T T = (15) The adiabatic assumption in the stator yields: 02 03T T = (16) The power is solely transferred to the rotor, which delivers air at high speed to the stator. The stator, through diffusion, transforms kinetic energy to static pressure rise. To proceed with the blade preliminary geometry, its angles are written together with aerodynamic and thermodynamic equations. A compressor stage with its velocity triangles is shown in Figure 20. Assuming that1 2 a a aC C C = = , simple trigonometry yields: 1 1tan tanaUC = + ,(17) sTP1P21234P3P423450 2 2tan tanaUC = + .(18) Figure 19 Axial-flow compressor stage in a T-s diagram. Figure 20 Rotor row and stator row with velocity triangles in an axial-flow compressor stage. T03T02=T03232pCc212pCc222pCcp1p01p2p3p02p0302033201103T2T01T1TsRS1 2 311C1V1Cw1Ca1UC2V2Ca2Cw222C11rotorstatorC22C3 3U51 From (17) and (18): 2 1 1 2tan tan tan tan = (19) 1 1 2 2tan tan tan tan . + = + (20) Inthecompressor,theflowenterswithtangentialvelocity 1 wC atradius 1r and leaves with tangential velocity 2 wCat radius 2r . Thus, the required torque for a mass flow rate mis ( )2 2 1 1 w wT mr C rC = ,(21) and the power to drive it: ( )2 2 1 1 w wW m r C rC = .(22) For the special case in which1 2r r = : ( ) ( )1 2 1 2 1 w w w wW m r C C mUC C = = .(23) The velocity triangles from Figure 20 and Eq. (23) yield ( )2 1tan tanaW mUC = ,(24) ( )1 2tan tanaW mUC = .(25) Equation (25) shows that more power is used bythecompressor if theblade has highercamberangle,thusmorepoweristransferredtothefluidinthiscondition.Later, however,itwillbeshownthatthereisalimitforthiscamber,otherwise, 190 and 20 = would be the undoubted design. 52 Continuing with the derivation, if the compressor power input is transferred to the fluid to raise its pressure, the whole power input contributes to the total pressure rise: ( )03 020 03 01 02 01 1 2tan tanT TastagepUCT T T T Tc = = = = .(26) Then Eqs. (13) and (26) yield: ( ) ( )( )11 102 01 1 201 01 020111 20111 tan tan 11tan tan 1pc ac p c ppap cprUCT T r rT c T TTUCrc T = = = | | |\ . ( = + ( ( (27) Equation(27)provideswiseadvicesonhowtoobtainhigherpressureratiosper stage. High values of compressor efficiency, rotational speed, axial velocity and camber angle and low values of inlet total temperature, cp and provide elevated pressure ratios per stage.Usually, the designer has no control on the working fluid and ambient conditions, thus, changes in pc , and 01Tare not case of study here. The rotational velocity is limited by materialtechnologyandthecompressorefficiencyisgivenbythetechnologicallevelat disposal. The axial velocity does play an important role, but is limited due to high losses. Advanced aero enginescan handle axial velocities up to 200 m/s. Thus, major analysis is focused on angles, which are related to the temperature rise. 3.3.1Tip speed Therotationalvelocityofagasturbineislimitedbymaterialtechnology.This happensduetoelevatedlevelsofcentrifugaltensilestressunderwhichthebladesare submitted, and its maximum value, occurring at the blade root is given by: 53 ( ) ( )2max.trrbctrrS r r drS =,(28) where: ( )blade material density;angular velocity;area at blade root;blade cross-section at any radius;r radius;radius at blade root;radius at blade tip.brrtSS rrr Present technology imposes a 400 m/s limit at the blade tip. In fans, however, this figurereaches450m/s.Toevaluatetheangularvelocity,atwhichmateriallimitationis critical, let the tip speed limit be 350 m/s, then for a 5 cm radius microturbine and a 1.5 m radius high bypass-ratio turbofan: 350 rad. 7000 66845rpm0.05 sUU r Nr = = = = = (29) 350 rad. 233 2228rpm1.5 sUU r Nr = = = = = (30) 3.3.2Camber angle and de Haller number As prescribed by Eq. (27), high fluid deflection contributes to high pressure ratios per stage. Consider the case, which the relative inlet angle 1is kept constant and the angle 2is diminished to provide higher fluid deflection, as depicted in Figure 21. 54 Figure 21 Inlet and outlet relative velocity ratio is reduced with the increase of fluid deflection. Itisclearthathighfluiddeflectionresultsinloweroutletrelativevelocity,this meansthatmorekineticenergyisconvertedtostaticpressure.Inotherwords,highfluid deflections, hence camber angle, entails a high rate of diffusion. Due to excessive losses, a limit of diffusion exists and in preliminary design it is quantified by the de Haller number, defined as: 210.69VdeHallerV= > .(31) Originally, the limit was 0.72, but accumulated experience pushed this figure to 0.69. 3.3.3Compressor surge The surge is an unstable operation of the compressor, characterised by a sudden drop of delivery pressure and by intense aerodynamic pulsation, which propagates from its origin tothewholeengine.Thephenomenonyetveryharmfultotheengineisstillnotfully understood. Usually it is related to excessive vibrations and a particular noise. The surge is seen as the lower limit of stable operation, beyond which reversal of the flow is expected. UaC1C1V2C2V22112C2V2 255 3.3.4Compressor choke Fromthegasdynamics,itisknownthatthemaximummassflowratethrougha nozzleisreachedwhenthethroatisatMach1.Nomatterwhatisdonetoincreasethe pressure ratio, no extra flow pass through this nozzle. As the space between the blades forms a nozzle, the compressor choke happens when the blade throats choke. Thus, the operational range of a compressor, for each rotational speed, is bounded by surge and choke. 56 4THESTREAMLINECURVATURECOMPUTATIONAL PROGRAM This chapter provides a brief explanation of the Streamline Curvature Computational Program developed by Barbosa [42] and revised and further developed by Figueiredo [43]. For a careful and in-depth analysis, refer to those works. 4.1INTRODUCTION Thedesignofamulti-stageaxial-flowcompressorisalaborioustaskforvarious reasons. To start with, it involves a careful and wise choice of a plethora of design parameters. In this particular case of study, hundreds of parameters have to be properly set. Thus, much of the preliminary design relies upon an experienced designer and information gathered from costly and time-consuming experimental studies, which are mostly proprietary, but some are found in the open literature.Among the main open data resources, the publications from the National Aeronautics andSpaceAdministration(NASA)andfromitspredecessor,theNationalAdvisory Committee for Aeronautics (NACA), are yet the basis. The publications from J ohnsen and Bullock (NASA SP-36) [44], Montsarrat, Keenan and Tramm (NASA CR-72562) [45] and Schwenck,LewisandHartmann(NACARME57A30)[46]containthefundamentalsof axial-flow compressors. 57 Duetothecomplexflowfieldobservedinaxial-flowcompressors,manyearly computational models failed to accurately predict performance characteristics. Nevertheless, the SLCP demonstrated to be a fast and reliable performance prediction tool, as shown in [42] by comparing its results with a real three-stage transonic compressor and with commercial codes [47]. 4.2THE STREAMLINE CURVATURE METHOD Theactualflowinanaxialcompressorishighlycomplex,three-dimensional, turbulent and viscous. To assess the flow according to detailed mathematical models, high computationalcosts,aswellas,longevaluationtimesarerequired.Nevertheless,quick answers are constantly demanded in the preliminary design phase. Therefore, an axisymmetric and inviscid model, in which the losses due to viscosity effects are computed empirically, potentially offers adequate accuracy and velocity. TheStreamlineCurvatureMethod(SLCM)consistsofwritingthenon-viscous equations of continuity, motion and energy along a coordinate system based upon the flow streamline and the tangent to the blade edges, illustrated in Figure 22. The outcome is a set of non-linear partial differential equations, which are solved iteratively by finite differences in a meridional plane grid. The nodes of this grid are the intersection of the streamlines and the blade edges, as depictedinFigure23.Asthestreamlinesarenotknownpreviously,aninitialpositioning guess is required and later it is varied iteratively to satisfy the continuity equations. Dummy stages represent inlet and outlet ducts, or computationally, a bladed stage, whose blades do notdisturbtheflow(i.e.,nodeviation,nodiffusion,etc.).Finally,thesetofdifferential equations are integrated at the nodes along the streamline, from inlet, to outlet. 58 Figure 22 Streamline-blade leading edge coordinate system (s-m). [42] Figure 23 Streamlines, stage rows and calculation nodes. Adapted from [42]. In the SLCM, the flow is divided into concentric streamtubes, wherein the flow is axisymmetric. The flow is calculated according to inviscid equations and the losses due to viscosity are incorporated as entropy increase, pressure decrease, etc. at the trailing edge, by means of empirical correlations.The basic derivation of the method can be found in Appendix A. r smzstreamlinebladeleadingedgebladetrailingedgecasinghubdummy dummy dummy dummy dummy IGV rotor statornodesstreamlines59 4.3COMPUTATIONAL PROGRAM The design mode of the SLC program is basedon reference [42]. It is written in FORTRAN language and it has been continuously updated. The SLC program is interactive, fully modular and its high flexibility allows new features to be easily integrated. Atthepresenttime,asimplificationofthemainstructureoftheprogramis illustrated in Figure 24. Many convergence loops and subroutines were omitted for the sake of clarity. Figure 24 Overview of the SLC program algorithm. Read r ef er ence cur vesRead i nputSt age l oadi ngdi st r i but i onAxi al channel desi gnVor t exBoundar y l ayer ef f ectCut awayNew gr i dEf f i ci encycal cul at i onEf f i ci encyconver ged?Pr i nt t abl esWr i t e compl et e geomet r yWr i t e opt i mi sat i oni nput f i l eYNst ar tendI nl etOut l etVel oci t y t r i angl esRadi al equi l i br i umI nci denceLossesBl adi ngBoundar y l ayerCal cul at ed ef f i ci encyDeHal l er numberCamberPr essur e r at i oChannel i nl etChannel out l etChannel i nt akeDCA60 5REAL-CODEDELITISTMULTI-OBJECTIVEGENETIC ALGORITHMPROGRAM Thischapteraimsatprovidingthereader,whoisnotfamiliarwithGenetic Algorithms, with the basis to proceed without loss of understanding. To start with, important definitions about Multiple Objective Optimisation Problem (MOOP) are presented. Next, the fundamentals of Genetic Algorithms (GAs) are explained. Finally, a real-coded elitist multi-objective genetic algorithm program (REMOGA) developed in FORTRAN is detailed. Optimisationmaybedefinedasthesearchforsolution(s)whichcorrespondto minimum (or maximum) of one or more objectives, satisfying given constraints. A single-objective optimisation problem (SOOP) usually has one single optimal solution. A MOOP accounts for multiple objectives, which may be conflicting. In this case, normally one obtains notone,butasetofoptimalsolutionsnamedParetooptimalsolutions.Tocomparetwo solutions in a MOOP, the concept of dominance is introduced to encompass the idea that if a certain solution a dominates solution b, then solution a is at least better in one objective and is better or at least equal in all the other objectives. 5.1DEFINITIONS To accurately describe a MOOP, the specific vocabulary and definitions are made clear. The definitions presented hereinafter are extracted or adapted from Deb [48] and Bche 61 [49]. To start with, a MOOP can be described by a vector of decision variablesx and the corresponding vector of objectives, ( ) = f f x . 5.1.1Multi-objective optimisation problem Definition 1The multi-objective optimisation problem is defined as the search for the set of solutionsx , which minimises/maximises: ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )() ( )1 21 21 21 2min/max , , ,with, ,subject to , , ,, , ,, 1,2, , ,mnpqL Ui i if f fx x xg g gh h hx x x i n= = = = = =f x x x xxg x x x x 0h x x x x 0FX(32) where n X is the n-dimensional decision space bounded by () Lixand ( ) Uix , and m F is them-dimensionalobjectivespace.Thefunctions( ) g x and( ) h x aretheconstraint functions.Anysolutionwhichsatisfieseveryconstraint,totallingp+qconstraintsand2n variable bounds is called a feasible solution, otherwise, infeasible solution. Anillustrativemappingfroma3-dimensionaldecisionspacetoa2-dimensional objective space is shown in Figure 25. Figure 25 Mapping between the decision space and the objective space. 1x2x3xxDecision space1f2fzObjective space62 Letand a b X,then( ) ()i if f a b denotes that a is a better solution than b with respect to the i-th objective. If the i-th objective is a minimisation one, than( ) ()i if f a b meansthat ( ) ()i if f < a b .Ifthei-thobjectiveisamaximisationone,than( ) ()i if f a b implies that ( ) ()i if f > a b . Analogously,( ) ()i if f a b denotes that a is a worse solution than b;( ) ()i if f/a b denotes that a is a not better solution than b, and( ) ()i if f/a b denotes that a is a not worse solution than b. 5.1.2Domination To compare different solutions from (32), an ordering among different solutions is obtained by the dominance criterion. Definition 2A solution a X dominates a solution b X, which is expressed bya b , if and only ifais no worse in all objectives and at least superior in one objective thenb. This statement can be expressed as: { } ( ) (){ } ( ) ()1,2, , :1,2, , :i ij ji m f fj m f f / a b a ba b (33) If solution a dominates solution b, it is also usual to write any of the following: b is dominated by a, or; a is non-dominated by b. If Definition 2 does not apply, then it is said that a does not dominate b, or,/ a b . 63 Definition 3Thesolution a Xisindifferenttoasolution c X,ifandonlyifneither solution is dominating the other one, i.e.,and / / a c c a . Figure 26 illustrates the dominance and indifference as defined in a two-objective minimisation problem. Solution a dominates solution b, as it is no worse than b in both f1 and f2andisbetterinatleastoneofthoseobjectives(inthiscase,inbothobjectives). Furthermore, one notices that solution a does not dominate solution c and solution c does not dominate solution a, thus, solution a is indifferent to solution c.

Figure 26 Representation of dominance and indifference between solutions in a two-objective minimisation problem. Solution a dominates b, but is indifferent to c. 5.1.3Pareto-optimal set If in a given set of solutions, all possible pairwise comparisons are performed, one eventually finds which solution dominates which and which solutions are not dominated with respect to each other. This leads to an important set, named Non-dominated set: Definition 4(Non-dominated set). Among a set of solutions P X , the non-dominated set of solutions P P are those that are not dominated by any member of the setP. This can be expressed as: 1f2facba b / a c 64 is a non-dominated set / a b a b P P P, P: (34) Definition 5 (Globally Pareto-Optimal set). When the setPis the entire search space, i.e., = P X, then the non-dominated set Pis called globally Pareto-optimal set. Forthesakeofconcision,thegloballyPareto-optimalsetisoftenreferredtoas Pareto set. 5.1.4Convexity Definition 6Afunction:nf issaidtobeaconvexfunctionifforanypairof solutions , a b X, the following condition is true: ( ) ( ) ( ) ( ) () | |1 1 , 0,1 f f f + + a b a b (35) A convex: f function is illustrated in Figure 27. Note that the line segment joining anda b is always greater or equal the function evaluated between those values. Figure 27 A convex function illustration. x( ) f xab( ) f a() f b( ) ( ) () 1 f f + a b( ) 1 + a b( ) ( )1 f + a b65 Definition 7Let m F .Fis said to be a convex set if, given any two points members of F , the line segment joining these points is entirely contained in F . Definition 8A multi-objective optimisation problem is convex if all objective functions are convex and the feasible region is convex [48]. 5.2TRADITIONAL METHODS AND THE GENETIC ALGORITHM According to Goldberg [50], optimisation and search techniques fall onto three main methods:calculus-based,enumerativeandrandom.Abriefdescriptionofeachoneis providedtoelucidatethereaderthereasonofthesuccessofGAsintheturbomachinery context. Calculus-based methods are divided into two categories: indirect and direct. Indirect methodsrelyonsolvingthesetofequationsprovidedbyequallingthegradientofthe objective functions to zero. Direct methods are based on the iterative hill-climbing concept, i.e., starting from a given point the gradient is calculated to provide the climb-direction of the next point. Successively, a local optimum is found eventually. The main disadvantages of the method are: The objective function has to be differentiable; Numerical differentiation is prone to severe errors; It does not work properly with discontinuous functions; The algorithm is likely to be stuck onto a suboptimal solution when dealing with surfaces like Figure 28; Almost implies that the objective function surface has to be known a priori. 66 Figure 28 Illustrative