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Thesis presented to the Instituto Tecnológico de Aeronáutica, in partial fulfillment ofthe 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 São José dos Campos, SP – Brazil 2011 Cataloging-in-Publication Data Documentation and Information Division Ando, Victor Fujii Genetic Algorithm for Preliminary Design Optimisation of High-Performance Axial-Flow Compressors / Victor Fujii Ando. São José dos Campos, 2011. 162f. Thesis of master in science – Program of Aeronautics and Mechanical Engineering. Field of Aerodynamics, Propulsion and Energy – Aeronautical Institute of Technology, 2011. Advisor: Prof. Dr. João Roberto Barbosa. 1. Genetic Algorithm. 2. Axial-flow compressor. 3. Preliminary design. I. Aeronautics Institute of Technology. II. Title BIBLIOGRAPHIC REFERENCE ANDO, Victor Fujii. Genetic Algorithm for Preliminary Design Optimisation of HighPerformance Axial-Flow Compressors. 2011. 162f. Thesis of master of sciences in Aerodynamics, Propulsion and Energy – Aeronautics Institute of Technology, São José dos 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 São José dos Campos, SP. iii Genetic Algorithm for Preliminary Design Optimisation of HighPerformance Axial-Flow Compressors Victor Fujii Ando Thesis Committee Composition: Prof. Dr. Rodrigo Arnaldo Scarpel Chairperson – ITA Prof. Dr. João Roberto Barbosa Advisor – ITA Prof. Dr. Nelson Manzanares Filho Universidade Federal de Itajubá Prof. Dr. Márcio Teixeira de Mendonça ITA ITA the author conveys his thankfulness for the inestimable support of his family. Under this programme. . Prof. 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. Barbosa.iv Acknowledgements This work was executed in the context of the “Programa Integrado GraduaçãoMestrado” – PIGM. The author is also indebted to Prof. Nelson Manzanares Filho. from UNIFEI. Finally. for the guidance and the invaluable assistance. The author acknowledges the support of “Fundação de Amparo à Pesquisa do Estado de São Paulo” (FAPESP) to conduct this study. especially with regard to the axial-flow compressor design program. ITA Bachelor students from the last year undertake disciplines from the post-graduate programme and are encouraged to develop the Bachelor Thesis as an intermediate step towards the research to be conducted during the Masters. The author would like to express his gratitude to his advisor. compressor axial. projeto preliminar. com codificação real e elitismo. um programa de otimização. dezenas de milhares de projetos puderam ser rapidamente avaliados. foram variados os ângulos de saída do escoamento dos estatores. Desse modo. chamado de REMOGA. Contudo. quatro soluções foram tomadas para análise. Palavras-chave: Algoritmo genético. para fácil integração com os programas desenvolvidos pelo Grupo de Turbinas. Em seguida. turbomáquinas . esse processo exige um trabalho longo e exaustivo de tentativas e erros. a distribuição de temperatura nos estágios e a relação de raios. O programa baseia-se em um algoritmo genético multi-objetivo. a fim de auxiliar o projetista na escolha de alguns parâmetros. mas controlando-se o número de De Haller e o ângulo de arqueamento. Finalmente. por meio de um critério de escolha. No contexto do Grupo de Turbinas do ITA. revelando que o programa desenvolvido conseguiu encontrar soluções mais eficientes e plausíveis do que a originalmente proposta. foi desenvolvido em linguagem FORTRAN. empregando-se correlações da literatura para o cômputo das perdas. o REMOGA e o programa de projeto preliminar foram integrados para o projeto de um compressor axial de cinco estágios. Para isso. Graças ao REMOGA. A escolha de diversos parâmetros do ciclo termodinâmico e de geometrias depende da longa experiência acumulada pelos membros do Grupo. visando a maiores eficiências e maiores razões de pressão.v Resumo Este trabalho apresenta uma abordagem para a otimização de projeto preliminar de compressores axiais de alto desempenho. o projeto preliminar é feito utilizando-se um programa computacional baseado no método da curvatura da linha de corrente. vi Abstract This work presents an approach to optimise the preliminary design of highperformance axial-flow compressors. The preliminary design within the Gas Turbine Group at ITA is carried on with an in-house computational program based upon the streamline curvature method, using correlations from the literature to assess the losses. The choice of many parameters of the thermodynamic cycle and of geometries relies upon the expertise from the members of the Group. Nevertheless, it is still a laborious and time-consuming task, requiring successive trial and errors. Therefore, to support the compressor designer in the choice of some parameters, an optimisation program, named REMOGA, was written in FORTRAN language, allowing an easy integration with the programs developed by the Gas Turbine Group. The program is based upon a multi-objective genetic algorithm, with real codification and elitism. Then the REMOGA and the preliminary design program were integrated to design a 5-stage axial-flow compressor. Therefore, the stator air outlet angles, the temperature 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, thousands of designs could be quickly evaluated. Finally, using a choice criterion, four solutions were selected for further analysis, revealing that the developed program was 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: <grc.nasa.gov/WWW/5810/w8.htm>. ........................... 26 Figure 2 – Flow chart of multidisciplinary design optimisation of Luo et al. [30]. ................. 32 Figure 3 – Evolution of domestic processors from 1998 to 2011. ........................................... 33 Figure 4 – Junkers Jumo 004 axial jet engine and Me 262. Source: <www.luftarchiv.de>..... 36 Figure 5 – Rolls-Royce Trent 1000. Source: <www.rolls-royce.com> ................................... 37 Figure 6 – Classification of compressors. ................................................................................ 38 Figure 7 – Centrifugal compressor. Source <history.nasa.gov> .............................................. 39 Figure 8 – Comparison of some compressor types. ................................................................. 39 Figure 9 – Schematic figure of the main components in a gas turbine and the Brayton cycle. ...................................................................................................................... 40 Figure 10 – Scheme of an axial-compressor stage. .................................................................. 41 Figure 11 – Visual aid to the common plane scheme of an axial-flow compressor stage........ 42 Figure 12 – Details of a gas turbine detailing a compressor rotor row. ................................... 42 Figure 13 – Nomenclature according to Saravanamutto [37]................................................... 43 Figure 14 – Generic velocity triangles. .................................................................................... 44 Figure 15 – Hub to tip ratio and tip clearance. ......................................................................... 46 Figure 16 – Divergent isobaric lines and the increased compression difficulty in the last stages...................................................................................................................... 47 Figure 17 – Polytropic or small-stage efficiency...................................................................... 48 Figure 18 – A schematic real gas turbine cycle. ....................................................................... 49 Figure 19 – Axial-flow compressor stage in a T-s diagram. .................................................... 50 viii Figure 20 – Rotor row and stator row with velocity triangles in an axial-flow compressor stage.................................................................................................... 50 Figure 21 – Inlet and outlet relative velocity ratio is reduced with the increase of fluid deflection. .............................................................................................................. 54 Figure 22 – Streamline-blade leading edge coordinate system (s-m). [42] .............................. 58 Figure 23 – Streamlines, stage rows and calculation nodes. Adapted from [42]. .................... 58 Figure 24 – Overview of the SLC program algorithm. ............................................................ 59 Figure 25 – Mapping between the decision space and the objective space. ............................. 61 Figure 26 – Representation of dominance and indifference between solutions in a twoobjective minimisation problem. Solution a dominates b, but is indifferent to c. ........................................................................................................................ 63 Figure 27 – A convex function illustration............................................................................... 64 Figure 28 – Illustrative region where a gradient-based algorithm can get stuck onto a suboptimal solution................................................................................................ 66 Figure 29 – Simple GA algorithm [48]. ................................................................................... 67 Figure 30 – Chromosomal representation of decision variables. ............................................. 68 Figure 31 – Tournament selection illustration.......................................................................... 69 Figure 32 – Biological crossover illustration. .......................................................................... 70 Figure 33 – Bit-wise crossover representation. ........................................................................ 70 Figure 34 – Single-point crossover representation. .................................................................. 71 Figure 35 – Two-point crossover representation. ..................................................................... 71 Figure 36 – Mutation operator. ................................................................................................. 72 Figure 37 – Algorithm of the REMOGA program. .................................................................. 72 ................................................................................................ 81 Figure 46 – Effect of mutation parameter ηm for x=0 and Δmax=1......................... 75 Figure 40 – Optimisation (a) without niche penalty and (b) with niche penalty ................. . ......................................................................... 90 Figure 53 – SLC program acts as blackbox............. 79 Figure 45 – SBX [54] operator and influence of parameter ηc................................................................................................ 97 Figure 58 – Camber angle distribution of the original design.......... Population in the 1st...... 91 Figure 54 – SLCP output data to work together with REMOGA program..................................... 10th and 100th generations............ 78 Figure 43 – Crowded tournament selection operator............................. 83 Figure 47 – Solutions behaviour after each of the implemented operators.................. ........................................................ ................................... 98 ........................ ......................... 97 Figure 57 – Pressure and temperature distributions of the original compressor design....................... ... 75 Figure 41 – Dependence of the sharing function with α............... ........ 96 Figure 56 – Distribution of temperature rise weights along the stages........ ........................... [57] test problem after 500 generations.............. ..... .. ...................... ........... ........................................ 86 Figure 49 – Simple convex test function after 100 generations.......................................... 77 Figure 42 – Visual interpretation of the used value of σshare........ 84 Figure 48 – Testing a simple MOOP.......... 93 Figure 55 – Streamlines and nodes of the original compressor design...ix Figure 38 – Rank assignment algorithm... ............. ................... .................... 74 Figure 39 – Bubble sort pseudocode ............. 88 Figure 51 – Poloni et al. ........................................ 87 Figure 50 – Non-convex test function from Fonseca and Fleming [56].................. 79 Figure 44 – Multiple selections........................... ................. 89 Figure 52 – SLCP and REMOGA coupling........................................................ ..... 100 Figure 63 – Euler diagram representing the sets of feasible and unique solutions........ 111 Figure 78 – Camber angle distribution of solution 2............. camber penalty and last stage stator outlet angle for the limited subset of solutions.................. 102 Figure 65 – History of the hub to tip ratio..................................... 99 Figure 60 – Stage loading distribution of the original design.......... ....... 100 Figure 62 – Blade chord of each row................................................................................... ...... 110 Figure 76 – Number of blades and blade chord of each row for solution 2........... 108 Figure 73 – Nodes and streamlines of solution 2................ .......................................... ...................................................... ........................................................................................... ... 113 ................................................................................... 102 Figure 66 – History of temperature weights distribution................................................ 105 Figure 69 – The initial design is comparatively poor in satisfying de Haller number.......... ......................... 108 Figure 74 – Pressure and temperature rise per row of solutions 1 and 2.............................................. ............ 111 Figure 79 – De Haller number distribution of solution 1......... .. 101 Figure 64 – History of target efficiency ........................... ................................................. .. 110 Figure 77 – Camber angle distribution of solution 1........................................ 103 Figure 67 – History of stator outlet angles distribution............................................... 107 Figure 72 – Nodes and streamlines of solution 1........................................... ............... ...... 106 Figure 71 – Input conditions for solutions 1 and 2................... 106 Figure 70 – Solution 1.....x Figure 59 – De Haller number distribution of the original design............................ ............................. 99 Figure 61 – Number of blades per row ...................................... ......................... ........ 104 Figure 68 – Pressure ratio vs.......................................................... ... .................................... .............................................................................. 109 Figure 75 – Number of blades and blade chord of each row for solution 1..................... ........ 133 ................................................. ................................................. 121 Figure 91 – Camber angle distribution of solution 4....................... .......................... camber angle penalty from the refinement run......... .................... ...................................................................... ...... 118 Figure 87 – Pressure and temperature distribution of solutions 3 and 4............. 123 Figure 95 – Velocity triangles.......... 117 Figure 86 – Streamlines of solutions 3 and 4..... 119 Figure 88 – Number of blades and blade chord of each row for solution 3................................................................................ 119 Figure 89 – Number of blades and blade chord of each row for solution 4....... 121 Figure 92 – De Haller number distribution of solution 3 ............................ 122 Figure 93 – De Haller number distribution of solution 4.......................... . ............. 115 Figure 83 – De Haller numbers do also concentrate close to zero.......................... 114 Figure 82 – Pressure ratio vs............. 122 Figure 94 – Stage loading distribution of solutions 3 and 4.... .... ........... .................. ...................xi Figure 80 – De Haller number distribution of solution 2........ ........ 113 Figure 81 – Stage loading distribution of solutions 1 and 2..... ................................................................................ 120 Figure 90 – Camber angle distribution of solution 3................................................................. 116 Figure 85 – Stages temperature weight and stator air outlet angles........................................ ........... ................................... 116 Figure 84 – Choice of solution 3..................................................... ...... . ................................................................. ..............xii LIST OF TABLES Table 1 – Summary of recent works presented at ASME Turbo Expo on compressor optimisation................ .................... 42 Table 4 – Compressor rows................. ............................................................................................................... 28 Table 2 – Comparison between Junkers Jumo 004 and Rolls-Royce Trent 1000.......................................................... 37 Table 3 – Thermodynamic processes at the rotor and stator............ 92 .................................. 91 Table 5 – Configuration of the computers used in the performance evaluation of the modified SLCP.............................. xiii LIST OF SYMBOLS LATIN SYMBOLS C absolute velocity c blade chord dij normalised distance between solutions i and j f vector of objectives F objective space g vector of inequalities constraints h enthalpy h vector of equalities constraints htr hub-to-tip ratio m mass flow N rotational speed in rpm nc niche count npop population size P pressure r radius  set of real numbers rank (.) sharing function T temperature U tangential velocity V relative velocity W power x vector of decision variables (also referred to as “solution”) X decision space .) rank of a solution rp pressure ratio s pitch or spacing Sh (. xiv GREEK SYMBOLS α angle between the absolute velocity and the axial direction β angle between the relative velocity and the axial direction γ specific heat ratio ζ stagger or settting angle Λ degree of reaction η isentropic efficiency η∞ polytropic efficiency ηc polynomial crossover control parameter ηm polynomial mutation control parameter θ camber angle φ flow coefficient ψ temperature or stage loading coefficient ω angular velocity SUBSCRIPTS 0 total property 1 rotor inlet 2 stator inlet 3 stator outlet a axial component m meridional component w whirl or tangential component . xv LIST OF ACRONYMS AND ABBREVIATIONS ANN Artificial Neural Network DOE Design of Experiments EA Evolutionary Algorithm GA Genetic Algorithm IGV Inlet Guide Vane LHS Latin Hypercube Sampling MOEA Multi-Objective Evolutionary Algorithm MOGA Multi-Objective Genetic Algorithm MOOP Multi-Objective Optimisation Problem N-S Navier-Stokes NSGA Non-dominated Sorting Genetic Algorithm OGV Outlet Guide Vane RANS Reynolds-Averaged Navier-Stokes REMOGA Real-Coded Elitist Multi-objective Genetic Algorithm RSM Response Surface Method SBX Simulated Binary Crossover SLCM Streamline Curvature Method SLCP Streamline Curvature Program SOOP Single-Objective Optimisation Problem . ...............2...................... 41 Nomenclature ..........2 2...............................1 3..........1 3............................................. 25 2..................................2 Objective ...................................................................................................3 History ............................................................................. 21 1............................. 45 Isentropic and polytropic efficiencies ......................................................... 25 Reference stage ........................................................ 19 1..................................................................................2................................. 44 3............4 4 Solvers .. 57 4........... 56 4........................................................................... 26 Optimisation methods......................................................1............................... 22 1...................... 40 Basic operation ..............................................1 Introduction... 52 Camber angle and de Haller number ............3................... 36 3....1...........................................2 3.............................................................................3.................................. 53 Compressor surge .......................................... 45 Hub to tip ratio.......................................................................................... 46 Overview of axial-flow compressor performance ........... 48 3......................................................1......................... 36 Classification .1 Introduction.3 Computational Program ....4 3....3 2.....3......................1........................................................ 55 THE STREAMLINE CURVATURE COMPUTATIONAL PROGRAM ........ 27 Tip speed ...............2 3........................................................................................................................................................3 3..2 3.........................3 3..1 Motivation....................................................................... 59 ...................................................................... 20 1..........1......................5 3............ 36 3............................. 44 Temperature or stage loading coefficient .....................................................3 Methodology .....................................1 3..........................3......................................................................2 3................ 27 AXIAL-FLOW COMPRESSOR OVERVIEW ................................................................................................................ 24 LITERATURE REVIEW................................... 25 2.............................. 56 4...1 2.......................................................1 Introduction.................................................1............................................ 45 Degree of reaction .........................5 Flow coefficient .................. 43 Dimensionless parameters ..........................................................6 Organization of the Thesis ..............................3 3...............5 Research on gas turbine within DCTA ...............2 The Streamline Curvature Method .....................2................................................................................................................4 Context................................................................................................ 54 Compressor choke ..1..........................4 3...................... 20 1.............xvi CONTENTS 1 2 INTRODUCTION ........................................2 3 Review of axial-flow compressor optimisation ..................................1............................ 38 Gas turbine...................2.................................................................. 19 1..............2................................. .... 92 SLCP input data or REMOGA output data ...................................................2 5......................3........................ 82 Test functions.........3........................................................................................................................................................................................4.............................................................5......... 71 Real-coded elitist multi-objective genetic algorithm program (REMOGA) .....................1 Search: REMOGA history and filtering of solutions ......... 112 Refinement of the search space ... 90 6...............................................3 5............3 5...............1 Definitions ......... 105 7............................................................. 60 5...............................1 6..4 Overview ...............................................3 7................................................ 88 Summary of the chapter .... 63 Convexity ........3................3 REMOGA settings ............................ 110 De Haller number ..................4 5......................................... 64 SLCP output data or REMOGA input data ..................1 5...........................................4 5................................................5.............1............3 5................................................2 7.................................1....... 114 ...............1................................................. 107 Camber angle ......................................... 85 Non-convex test function .........4..............................................4....... 80 Real-coded Polynomial Mutation Operator............................................ 112 Stage loading ........................................ 90 6..................... 72 5...................................................4 7...............1 Modifications in the SLC program ...........................................................2 5........................................3.................................. 60 5........3 Genetic Algorithm Fundamentals ................................. 67 5. 65 5....................................................................................................................... 79 Real-coded Polynomial and Elitist Crossover Operator ...............................1 5................3.. 61 Domination ..3 Analysis of search step solutions .............................................................................................. 107 7.....2 Traditional methods and the Genetic Algorithm ........................................... 62 Pareto-optimal set ............................2 Looking for solutions..xvii 5 REAL-CODED ELITIST MULTI-OBJECTIVE GENETIC ALGORITHM PROGRAM.......................... 87 Non-convex domain and disconnected Pareto set test function .............................4....................................... 95 6...............................................................................................................5 Convex 2-variable 2-objective test function........ 89 METHODOLOGY..............1.............6 Selection or reproduction operator ..........................................................1......... 101 7........4 Human design start point ............2 5..................................................1 5........................... 69 Crossover operator.........................2 5..........................1 7........3 5..............1........................................................... 96 RESULTS AND DISCUSSION .................. 101 7....................2 7 Multi-objective formulation .................... 73 Crowded Tournament Selection ........................1 5....................................... 95 6.......... 94 6...................................................5.......3.......................................2 Formulation of the MOOP ...3....................................... 85 5.................................4 6 Multi-objective optimisation problem ...................... 70 Mutation operator ..................... .xviii 7..........6 Stator incidence angle .........2....3 Reading initial population and program parameters .......4 Evaluating objectives .....................1 Improvements ........................................................2 Global variables ..................5 Analysis of refinement step solutions ................. 117 Camber angle .. 140 B................... 161 D.....................................................3 7................ 149 B. 117 7.......................................................................... 139 B.....1 Niche count subroutine .................................................................................................................1 Main program .............................................................................. 127 9......................5............................................................. 153 ADDITIONAL INFORMATION FROM THE OBTAINED APPENDIX D SOLUTIONS ...............................1 Rotor inlet Mach number ....................... 128 APPENDIX A SLC SUMMARY ...................................... 126 9............... 120 Stage loading ....................................................................................................................... 159 D......................5 Fitness subroutine .................... 123 8 CONCLUSIONS ....................................................... 160 D............... 147 B.........................................5................................ 163 .......................................................................................5................................................... 145 B......... 142 B.............................................................................. 138 B.................................... 126 9...........................................................2..................................................................................................................................................................................7 Real-coded elitist crossover subroutine .5................................................................................................. 120 De Haller number ..................................3 Rotor total loss ............................................. 157 D..........5 Rotor incidence angle ......................................................................................... 127 Robust optimisation ...6 Crowded tournament selection subroutine .... 127 REFERENCES ........................... 162 D.....4 Stator total loss .................................................................. 150 B........................................................................................................................................................5..................................................... 138 B......................................................2 Detailed project ..............................................................................................2 Suggestion of works ...........................................4 Overview .2 7........................................................................... 124 9 FURTHER WORK ....................... 158 D............................................................................... 133 APPENDIX B OPTIMISATION PROGRAM ................1 7................................................2 Stator inlet Mach number ....................................8 Real polynomial mutation..............................................................................................................1 9...................................... 151 APPENDIX C ORIGINAL SLCP INPUT FILE ............. are spreading quickly as design tool assistant. they are particularly suited for MOOP and computational parallelisation. Classical Methods. high efficiency. high pressure ratio. and numerous constraints.g. encompassing several conflicting ones. Therefore.. Conversely.19 1 1. . thereby reducing the design evaluation time. Its design involves a very large amount of design parameters. e. They require numerical differentiation. etc. such as Multi-Objective Genetic Algorithm (MOGA). such as Gradient-based methods are deterministic and mathematically demanding. low weight. Moreover. To support the designer in choosing the most effective design parameters.. modern Evolutionary Algorithms (EAs) are robust and mathematically simple. and risk being stuck onto suboptimal solutions. Hence. EAs.1 INTRODUCTION MOTIVATION The axial-flow compressor is one of the most challenging components to be designed in a gas turbine. it is virtually impossible to find an optimal compressor design by successive trial and error. it is demanding and time consuming to properly decide on design parameters. even to an experienced compressor designer. a plethora of design requirements. low number of stages. Thus. Furthermore. which tends to be a source of numerical errors. wide surge margin. tools for Multi-Objective Optimisation Problems (MOOP) have been developed and are constantly being improved. etc. Non-dominated Sorting Genetic Algorithm (NSGA). those parameters influence differently many distinct and competing design objectives. Therefore the SLCM is very useful in the preliminary design. b. Optimisation: a. as it combines good accuracy and quick evaluation. Literature review on the use of optimisation procedures in the design of axial-flow compressors. the work was divided in two parts: 1. as the SLCM bypasses the time consuming and demanding viscous-related calculations. instead.2 OBJECTIVE The objective of this work is to develop a procedure to optimise the preliminary design of a high-performance axial-flow compressor by coupling an existing in-house developed preliminary design computational program and a multi-objective genetic algorithm. This coordinate system is preferred due the easily-derived calculation grid. 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. 1. hence providing reasonable predictions.20 The Streamline Curvature Method (SLCM) consists of writing the non-viscous equations of continuity. but also offers a better understanding of the impact of the design parameters. The losses are. it is very fast. motion and energy along a coordinate system laying on the streamlines and on the tangent to the blade edges.3 METHODOLOGY Aiming at the proposed objective. assessed by empirical correlations derived from several tests carried on laboratory facilities. Furthermore. Study of multi-objective genetic algorithms. . 1. Thus. Development of a FORTRAN program to compute a real-coded elitist multi-objective genetic algorithm. Understanding of the fundamentals of the program b. The streamline curvature program a.21 c.4 CONTEXT This present work was executed under the “Programa Integrado GraduaçãoMestrado” – PIGM. Diffusion factors and camber angles were controlled by means of a penalty factor treated as objectives to be minimised. entitled “Project Optimisation of High-Performance Axial-Flow Compressors” was executed under the supervision of Prof. the design variables were the efficiency. which used four control points. Review of functions and main algorithm (carried on by the advisor) c. the Bachelor Thesis (“Trabalho de Graduação” – TG) was supervised to provide a well-developed start point to the Master Thesis. hub-to-tip ratio and the stator air outlet angles via a multivariate interpolation. Dr. In that work. João Roberto Barbosa (the same supervisor of this work). namely hub and tip at the first row and hub and tip at the last row. In this context. . The TG of the author. Modifications to couple with the GA program 1. This Program aims at the integration of the Undergraduate and the Masters Programs by allowing the student from the last year of the undergraduate course at Instituto Tecnológico de Aeronáutica (ITA) to undertake courses from the post-graduate programs. 2. shortening the necessary time to fulfil the requirements to the title of Master in Science. It preliminarily validated the design optimisation procedure by coupling the Streamline Curvature Method to a Multi-Objective Genetic Algorithm. including the supervisor of this work. and Kongsberg (Norway). the research only flourished in the 1970s. mainly in performance . graduated. who obtained his PhD degree in Cranfield in 1987. Plans to develop gas turbines in Brazil are found in the Plans of Foundation [2] of the “Centro Técnico de Aeronáutica” – CTA (Aeronautical Technical Centre). the project was seriously hindered due to lack of experienced professionals.22 The Master Thesis was developed under the scholarship from “Fundação de Amparo à Pesquisa do Estado de São Paulo” – FAPESP (São Paulo State Research Foundation) at the Centre for Reference on Gas Turbine at ITA. 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. working in research related to Gas Turbines. thereby many opportunities of partnerships with important manufacturers. At the time a new turbine project was developed. Thus. From this Institute. the necessity of specialists in those machines. an ambitious program of training the CTA personnel commenced with Cranfield Institute of Technology (currently Cranfield University). Pratt & Whitney (USA and Canada). with the establishment of a Research Program at CTA. engineers from CTA and ITA.5 RESEARCH ON GAS TURBINE WITHIN DCTA Tomita [1] describes the research on gas turbine within DCTA. Lucas Aerospace (UK). Even with the present practice of importing gas turbines rather than designing and manufacturing in Brazil. like Rolls-Royce (UK). succeeded and were valuable. However. Garret (USA). Thereafter. 1. A summary of this history is presented hereafter. in 1947. transient performance. Programs of design point performance. which belongs to the Mechanical Engineering Department of ITA.000 rpm) and a combustion chamber test bed (for hot gases up to 1500 K. a turbine test bed (2000 kW brake power and rotation speed up to 60. The Centre. CTA was renamed DCTA – “Comando-Geral de Tecnologia Aeroespacial” (Brazilian General Command for Aerospace Technology). as well. 1.000 rpm). two companies should be highlighted: Vale Soluções em Energia (VSE). noise prediction have been written and are fully operational. which launched a massive investment program in Brazil in 2010. and General Electric. The process of choosing the correct turbine is vital.23 analysis and applications is evident. or even big companies moving to the energy sector. it should include a compressor test bed (1500 kW shaft power and up to 60. Observing the current actions of the major players from the energy sector in Brazil. According to Barbosa [3]. combustion chamber performance. The research on gas turbine at ITA is conducted by the Centre for Reference in Gas Turbine (CRTG – “Centro de Referência em Turbinas a Gás”). . computational fluid dynamics. since it undoubtedly allows a significant reduction in operation and maintenance costs. The development of a small gas turbine for research should be carried on. but the efforts to implement a modern Turbine Laboratory persist. which is preparing to design and manufacture its own gas turbines. one might again note a real requirement for specialists in turbines and compressors. off-design performance. relies its research upon information of public domain and upon many years of experience from its members. to secure its high energy demand in mining operations.0 MPa). In this context. The centrepiece of the research developed at CRTG is on numerical simulation. Chapters 3 and 4 provide the basic theory on axial-flow compressors and on the streamline curvature method. which was developed as part of this work. A review of ASME Turbo Expo congresses since 2000 in this particular field is also shortly conducted. suggesting future works as well. . Chapter 2 contains a review of studies published in axial-flow compressor optimisation. objective and methodology are presented. In chapter 5. algorithms and models used in the REMOGA program. The first contains a basic derivation of the SLC method.6 ORGANIZATION OF THE THESIS In chapter 1 the reader finds the introduction.24 1. which were selected among thousands of solutions proposed by the REMOGA. A brief history of the research on gas turbine within DCTA is also presented. The second contains the FORTRAN code of the developed optimisation program. Chapters 8 and 9 conclude this work. The third appendix offers the design parameters of the start-point axial-flow compressor. the author starts with the basic ideas behind Genetic Algorithms and then he details features. where the motivation. Chapter 6 describes how the integration of the SLC program and the REMOGA program took place. Four appendixes are provided. And the last appendix provides further graphical information from the compressors analysed in this work. Chapter 7 shows the results obtained through the aforementioned integration and analyse four solutions. that describes the motion of fluids.1 INTRODUCTION Before proceeding with the comparative table. CFD is concerned with numerical solutions of the set of governing equations of fluid dynamics and heat transfer. and has been taking place every year since 1956.25 2 LITERATURE REVIEW Among several turbomachinery conferences. ASME Turbo Expo is recognised as one of the most important events. a summary of Turbo Expo papers from 2000 to 2011 that are tied to the theme is presented in Table 1. some preliminary concepts are presented. Therefore. It is the use of numerical methods and algorithms to obtain approximate solutions. In turbomachinery. N-S equations lead to mathematically complicated problems. The N-S equations are a set of nonlinear partial differential equations. in order to present the recent progress of the studies on compressor optimisation. 2.1 Solvers Solvers can be defined as computational programs that solve a given mathematical problem.1. most flow-field-related solvers rely upon a computational tool called CFD. 2. The fundamental governing equations of interest for CFD are the Navier-Stokes equations (N-S). which stands for Computational Fluid Dynamics. which are . the transport of mass and of energy. The simplified equations are called Euler equations. depending on the desired accuracy and on the computational resources available.htm>. As its flow field was used by the American Society of Mechanical Engineers in 1994 in a CFD blind-test exercise. 2. Euler equations can be used accurately if losses are assessed by correlations derived from experiments. According to the problem. The resulting equations are called Reynolds-Averaged Navier-Stokes equations. the user may choose a 2D or 3D solver. . instantaneous quantities of the N-S equations are timeaveraged to provide an approximation.1. which are not of real-world interest. To describe turbulent flows. as well. except for very simple cases.26 virtually impossible to solve. Source: <grc. Nevertheless. or RANS. as viscosity plays an important role. Figure 1 – NASA Rotor 37. see Figure 1. which is easier to calculate. Therefore numerical methods and algorithms are employed to obtain approximate solutions. plenty of studies on the flow field in the aforementioned rotor were derived [4].nasa.gov/WWW/5810/w8. If used per se it provides very rough approximations in turbomachinery calculation. A further simplification of the N-S equations can be carried out by ignoring viscosity and heat conduction.2 Reference stage The most frequent reference stage used for academic purposes is the NASA Rotor 37. 27 NASA Rotor 37 was designed and tested at NASA Lewis Research Center (renamed NASA Glenn Research Center) in the late 1970s. Rotor 37 has a pressure ratio of 2. MOOPs which were solved with a single objective function (weighted average) were considered SOOP. NASA rotor 37. as well as its evolution.3 Optimisation methods A brief introduction to optimisation methods is provided in chapter 5. design variables and objectives. when applicable: • Problem: whether single-objective or multi-objective.2 REVIEW OF AXIAL-FLOW COMPRESSOR OPTIMISATION A comparative table of works presented at ASME Turbo Expo from 2000 to 2011 regarding optimisation in axial-flow compressors is drawn to provide a panorama of the theme. • Solver: which method was used to obtain quantitative results from the design. • Optimisation method. 2. the following information was taken. Therefore. The works were primarily taken from the topic “Design Methods and CFD Modelling for Turbomachinery”.. It is a low aspect ratio inlet with 36 multiple-circular-arc (MCA) blades. 2.g.19 kg/s. • Reference stage: many optimisation studies are carried on long-timeestablished open-data stages.1. e. .106 at a mass flow of 20. Method design variables objective [5] 2000 The combined use of Navier-Stokes solvers and optimization methods for decelerating cascade design C4 airfoil gradient-based inlet Pt. deviation angle [12] 2003 Numerical optimization of turbomachinery bladings SOOP Quasi-3D N-S and 3D N-S CONMIN (gradient-based) blade deformation max. flow angle. efficiency [11] 2003 Advanced high turning compressor airfoils for low Reynolds number condition. adiabatic efficiency [9] 2002 Towards a reduction of compressor blade dynamic loading by means of rotor-stator interaction optimization MOOP CFD code. adiabatic efficiency [8] 2001 Shape optimization of high-speed axial compressor blades using 3D NavierStokes flow physics SOOP 3D NavierStokes NASA rotor 37 modified feasible directions algorithm blade section geometry max. Part 1: design and optimization MOOP Quasi-3D N-S Evolution Strategies and MOGA blade spline control points min. camber. dynamic loading and max. inlet mech. angle. total pressure loss and min. efficiency [10] 2002 Aerodynamic design optimization of an axial flow compressor rotor SOOP 3D NavierStokes NASA rotor 37 RSM stack line profile max. Ref. [6] 2000 Design optimization of axial flow compressor blades with threedimensional Navier-Stokes solver SOOP 3D NavierStokes four-stage ATKOM NPT steepest decent and conjugate direction stacking lines max. title problem opt. M1. time-avg. efficiency [13] 2003 Automated design optimization of compressor blades for stationary. sliding mesh and time dependent NACA 65-12-10 Multi-objective Evolutionary Algorithm axial distance between rows and circumferential clocking min. solidity. largescale turbomachinery 3D blade geometry weigted sum: aerodynamic losses. t/c SOOP Navier-Stokes min. efficiency [7] 2000 Shape optimization of transonic compressor blades usign quasi-3D flow physics SOOP Quasi-3D N-S NASA rotor 37 gradient-based and sensitivity analysis 8 blade section geometry variables max.28 Table 1 – Summary of recent works presented at ASME Turbo Expo on compressor optimisation. total-to-total pressure loss coef. Tt. maximum Mach. etc MOOP solver Mises (Euler Q3D) reference stage Covariance Matrix Adaption (CMA) . chord. adiabatic efficiency [17] 2006 Modern compressor aerodynamic blading process using multi-objective optimization MOOP 3D-CFD Rolls-Royce datum design DOE.29 Ref. NSGAII blade section geometry min. constrained augmented functional DOE (CCD). static pressure and min. Monte-Carlo Simulation. Method design variables objective [14] 2004 Application of multipoint optimization to the design of turbomachinery blades SOOP 3D NavierStokes NASA rotor 37 ANN. 65 control points min. leaned and skewed blades in a transonic axial compressor SOOP 3D NavierStokes NASA rotor 37 DOE. working range [18] 2006 Optimal design of swept. lean and skew max. Simulated Annealing blade parameters efficiency and weighted sum of penalties [15] 2005 Multiobjective optimization approach to turbomachinery blades design MOOP Reynoldsaveraged 2D NS real-coded MOEA blade geometry: Bezier control points max. ANN 3D blade geometry total pressure loss (DP) and total pressure (ODP) [20] 2006 Compressor blade optimization using a continuous adjoint formulation SOOP 3D NavierStokes steepest decent and adjoint method blade geometry: 3D NURBS. RSM (second-order polynomial) and GA leading edge line: sweep. adiabatic efficiency [19] 2006 Automated Multiobjective optimisation in axial compressor blade design MOOP 3D NavierStokes (DLRcode TRACE) asynchronous MOEA. RSM (second-order polynomial) sweep. title problem solver reference stage opt. GA. which embraces efficiency and pressure ratio IOSO blade geometry efficiency for operation mode [21] 2007 A first-principles based methodology for design of axial compressor configurations SOOP CFD code SWIFT [22] 2007 Optimization of the gas turbine engine parts using methods of numerical simulation SOOP CFD NUMECA NASA stage 35 . total pressure loss [16] 2006 Design optimization of transonic compressor rotor using CFD and Genetic Algorithm SOOP 3D NavierStokes NASA rotor 37 DOE. design point loss and max. RSM and LSM blade parameters: CCGEOM desirability function. bow max. DOE 48 blade parameters 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 MOOP 2D MISES DOE (Latin Hypercube or SOBOL). 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. Turbo-Grid. CFXSolver NASA rotor 37 Latin hypercube. Kriging and polynomial surfaces chordwise s-Shift. 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 NavierStokes coupled with BaldwinLomax NASA rotor 37 Simple gradientbased Multi-section blade parameters adiabatic efficiency [29] 2009 Optimization of variable stator's angle for off design compression systems using streamline curvature method SLC method NACA 10-stage subsonic axial compressor VSV and VIGV angles total pressure at surge-marginrelated operating point SOOP Genetic Algorithm . etc. RSMRBF. Method design variables objective Stacking and thickness optimization of a compressor blade using weighted average surrogate model MOOP Blade-Gen. 38 design parameters weigted sum: losses and inlet angle [27] 2008 Multi-objective optimization in axial compressor design using a linked CFDsolver MOOP 3D-RANS and throuflow MAGELAN IDAC3 of RWTH Aachen MOEA.30 Ref. stagger variation. thickness. RSM and gradient-based 6 design variables defined by parametric curves efficiemcy. suction side control points. annulus. title problem solver [23] 2008 reference stage opt. PRESS based averaging. CFX-Pre. Kriging RSM 2D blade profile pressure loss at DP. ANN. total pressure and the combination of both [24] 2008 Design optimization of a HP compressor blade and its hub endwall SOOP CFD code elsA Cenaero GA. NSGAII. Angle between axis of rotation and camber tangent surge margin and peak adiabatic efficiency MOGA and gradient-based improvements 53: inlet Mach. AD).31 Ref. Method design variables objective NASA rotor 37 Multiobjective Differential Evolution (MDE) 3D blade parameters non-uniform B-spline control points isentropic efficiency and min. rotor blade count and stator blade count 3D-RANS MOOP ANSYS-CFX MOOP T-AXI: axyisymmetric solver NASA rotor 37 . efficiency. length. rVθ stator outlets. GA and RSM (polynomial and basis-function) blade sections B-spline parameters. taper. RSM. no. SQP circumferential grooves: width. mass. hub spline control points. NSGA-II circumferential grooves: width. stall margin DOE (LHS). etc. title problem [30] 2009 Multiobjective optimization approach design of a three-dimensional transonic compressor blade [31] 2010 Blade geometry optimization for axial flow compressor SOOP [32] 2010 Design optimization of circumferential casing grooves for a transonic axial compressor to enhance stall margin SOOP [33] 2011 Optimization of a transonic axial compressor considering interaction of blade and casing treatment to improve operating stability [34] 2011 Optimization of a 3-stage booster part1: the axisymmetric multi-disciplinary optimization approach to compressor design MOOP solver reference stage opt. blades. velocity ratios. lean and sweep combination off overall eficiency and pressure ratio 3D-RANS NASA rotor 37 DOE (LHS). depth. depth normalized by tip chord max. Radial Basis Neural Network. maximum stress CFD NUMECA NASA rotor 67 DOE (FCCD. Start Preprocessing Design Parametrisation of 3D blade variable Generation aero. the use of EA was the rule.32 From Table 1 one might notice that in the early 2000s. [30] conducted a study on multi-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. a tendency to MOOPs is observed. however. Similarly. MOOPs were mostly treated as SOOPs by means of encompassing many objectives in a single objective function (weighted average). Before. mesh Generation mech. Later. . aero and mechanical mesh were required and the aero solution had to be calculated to feed the FEM boundary conditions. which is related to the spread of MOEA. In 2000. as may be clear in Figure 2. [30]. mesh Surfaces Parallel MDE CFD solution Aero efficiency pressure Aero perf ormance computation FEM solution Mechanics perf ormance computation Mechanics performance function value End Figure 2 – Flow chart of multidisciplinary design optimisation of Luo et al. Luo et al. most of the optimisation methods were based on gradient. The optimisation aimed not only at higher isentropic efficiencies. Nine years afterwards. To achieve that. but also at the minimisation of the maximum mechanical stress. Evidently the techniques employed are closely related to the computer capabilities. 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. Table 1 also shows that blade profile optimisation has been extensively studied in the context of compressor optimisation. Celeron. but the increase of the number of transistors and of threads is still taking place. It is noticeable that a stabilisation in clock speed was reached close to 4 GHz.0 1. To plot Figure 3.5 1200 4.0 8 threads Clock [GHz] # Transistors (in millions) bubble size: Cache memory [0. the following processor families were taken into account: Pentium III. Celeron D. Core 2 Extreme. The major move from simple SOOP gradient-based strategies to multidisciplinary optimisation involving several design variables and objectives was certainly due to the advances in computer hardware.250] nm 0 1998 2000 2002 2004 2006 year 2008 2010 2012 0.125. a glimpse of the evolution of the processors in a decade can be put into perspective.5 bubble size: Lithography [32. Gathering information from 44 Intel domestic processors.5 3. Core i5. But the main benefit in recent computation for MOGA is the parallelisation capabilities provided by multiple threads. as EAs require considerable amount of computational effort and are particularly suited to parallel computing [35].5 2. Core 2 Duo. Core i7 and Core i7 Extreme Edition.0 200 0.33 These two different approaches to the optimisation of the NASA rotor 37 highlight the evolution of the optimisation capabilities in the 2000s decade. Core i3. Core 2 Quad.0 1 thread 2 threads 1000 4 threads 800 12 threads 3.5 400 1. Transistors and Cache memmory over time Clock and die lithography over time 1400 4. Pentium Extreme Edition.0 1998 2000 2002 2004 2006 year 2008 2010 2012 Figure 3 – Evolution of domestic processors from 1998 to 2011. summarised in Figure 3. Pentium 4. Pentium D.12] MB 600 1 2 4 8 12 thread threads threads threads threads 2. Pentium 4 HT. . Similarly. . Choi et al. To achieve that. a single-objective GA was employed. the diffusion factor was constrained. the theme has been thoroughly explored. Furthermore. Response Surface Method. Apart from researches published at Turbo Expo. From blade section geometry through spline control points to blade stacking line and from leading edge line to sweep. 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 avoid flow separation. 24 were centred on the blade profile. one may observe that among 30 selected papers. [33] carried an investigation on circumferential grooves targeting higher stall margin and peak efficiency. From 2000 to 2011. [32] and Kim et al. [29] presented a work on compressor optimisation using the Streamline Curvature Method at Turbo Expo. however. Apart from blade geometry optimisation. and at the latter. Binini and Toffolo [9] studied the axial distance between rows and circumferential clocking on dynamic loading and efficiency. lean and skew. Oyama and Liou [35] developed a multiobjective design optimisation tool based on the SLC method and on a real-coded MOGA aiming at higher efficiencies and pressure ratios of a 4-stage axial flow compressor. Latin Hypercube Sampling and Artificial Neural Network. 3D or quasi-3D Navier-Stokes solvers were employed. the methods ranged from simple gradient-based ones to various Evolutionary Algorithms. At the former. they used design parameters at the rotor trailing edge and at the stator trailing edge. To achieve that. The study revealed hundreds of feasible Pareto-optimal solutions. total pressures and solidities are design variables. The optimisation was conducted via MOGA.34 Recalling Table 1. only Shadaram et al. flow angles and solidities. Predominantly. Koch parameters. . The optimisation goal was overall polytropic efficiency. Additionally.35 Keskin and Bestle [36] presented at the German Aerospace Congress 2005 a procedure to automate a given Rolls-Royce preliminary design process to find Pareto-optimal solutions for design conditions. compressor exit Mach number.11% point keeping the surge margin constant or improve the surge margin by 3. Bézier-spline parameterisation was employed to describe the annulus lines and the stage pressure ratio distribution. overall pressure ratio and surge margin at design point. The constraints were: stage loadings. In this manner. the control points of the Béziersplines were used as decision variables. A meanline prediction process was integrated to sampling methods like Design of Experiments and Monte-Carlo Simulation and to a Multi-island Genetic Algorithm (MIGA).2% points without diminishing efficiency. relative rotor and absolute stator inlet Mach numbers. diffusion numbers and de Haller numbers. a gradient-based Lagrange-Newton type algorithm is used. Keskin and Bestle found that the efficiency could rise by 0. In order to reduce the number of design variables and keep the design freedom to save computational costs. the world first operational jet-powered fighter aircraft. Source: <www. Boyce [40] and Walsh [41] . 3. among them. the famous Messerschmitt Me 262 Schwalbe. The Germans took the lead with the engine Junkers Jumo 004. Figure 4 – Junkers Jumo 004 axial jet engine and Me 262. It was written based primarily on the books of Saravanamuttoo [37].1 INTRODUCTION The purpose of the compressor is to raise the total pressure of the working fluid to a level required by the thermodynamic cycle. which was mounted on many aircrafts. The pressure rise should consume the minimum shaft power.de> . Horlock [39].1.luftarchiv.1 History Axial-flow compressors for aeronautical applications started their development in the 1930s and entered into service at the end of the WW2. as this component absorbs approximately one third of the turbine power. Aungier [38]. 3.36 3 AXIAL-FLOW COMPRESSOR OVERVIEW This chapter aims at providing the basic knowledge about axial-flow compressors. certified in 2007 to show the evolution after a bit more than half of a century. From 1940s to 2010s. there was a considerable technological leap in axial-flow compressor design.rolls-royce.7 – 8. Table 2 provides some illustrative data about the Jumo 004 and the Rolls-Royce Trent 1000 (Figure 5). Metallurgy technology.37 A British axial engine program was also carried (The Metropolitan-Vickers F.298 240 – 330 Figure 5 – Rolls-Royce Trent 1000.147 8. For the sake of comparison.com> . Source: <www. computational resources and test facilities contributed for the increase in efficiency and achievement of higher pressure ratios with fewer stages. but it was unsuccessful to deliver an engine to the war.2 was the first axial British design). multi-spool configurations. Table 2 – Comparison between Junkers Jumo 004 and Rolls-Royce Trent 1000. Type Entry Pressure ratio Spools Number of stages Average pressure ratio per stage Thrust [kN] Junkers Jumo 004 Turbojet 1944 3:1 1 8 1. variable geometries.8 Rolls-Royce Trent 1000 Turbofan 2007 (FAA certified) 50:1 3 1+8+6 = 15 1. new materials. 2 Classification Compressors are classified into two major groups: positive displacement and dynamic. multistage configurations present considerable losses due to the high fluid deflections required to deliver the compressed fluid from one stage to another. but range from small to very large pressure ratio. while the axial-flow compressor achieves lower pressure ratios per stage. Dynamic compressors continuously transfer energy to the fluid. Normally. which does also flow continuously. they handle small flow rate. A basic compressor classification scheme is shown in Figure 6. the air flow along the radius in a centrifugal compressor and along the axial direction in an axial-flow compressor. then the frontal area increases.38 3.1. the centrifugal compressor (see Figure 7) is capable of higher pressure ratios per stage. Although centrifugal compressors achieve higher pressure ratios per stage. but if a high mass-flow is desired. Besides the rotation which is implied by the rotor. Therefore. Positive displacement compressors capture fluid in a certain pressure. Compressor Positive displacement Dynamic Centrif ugal Axial-f low Figure 6 – Classification of compressors. but handles higher mass flow per unit frontal area. The flow in an axial-flow compressor suffers little change in radius compared to a centrifugal compressor. Centrifugal compressors and axial-flow compressors are examples of dynamic compressors. trap it in a hermetic volume and deliver it to a higher pressure end. . 00 1. High-performance axial-flow compressors seek high efficiencies and high pressure ratios.05 1. Schematically. This is almost contradictory. but this normally incurs in higher friction and higher losses.10 Axial-Flow Compressor 1. Thus a tuned . because then high air velocities are required.39 Figure 7 – Centrifugal compressor. Another difference between axial-flow and centrifugal compressors is that the latter has narrower operational range than the former. a small variation of flow rate around the design point results in great pressure ratio variation in comparison with centrifugal compressors. the comparison between centrifugal and axialflow compressors is shown in Figure 8. Source <history.nasa. but with few stages.95 1. In an axial-flow compressor.10 Q / Qdesign Figure 8 – Comparison of some compressor types.00 Centrifugal 0.90 0.gov> Therefore. P / Pdesign Positive displacement Head 1. it was recognised from the beginning of the gas turbine history that axialflow compressors would be capable of higher pressure ratio and higher efficiency than centrifugal compressors[37].90 Axial-Flow Compressor 0.05 1.95 Compressor Centrifugal Compressor 0.85 0.80 Flow 0. . etc. The turbine must extract energy in excess to drive a load (e. the ideal cycle efficiency may be calculated as:  T   T  T3 1 − 4  − T2 1 − 1  wcycle c p (T3 − T4 ) − c p (T2 − T1 )  T2   T3  η= = = cycle q23 c p (T3 − T2 ) T3 − T2 (1) . The working fluid (e.40 temperature distribution along the stages is required. leading to a dramatic increase in temperature and energy of the mixture in a isobaric process. Figure 9 shows a simple gas turbine scheme and its related ideal temperature-entropy diagram.3 Gas turbine A simple and ideal gas turbine basically consists of three components: the compressor. In a simplistic approach. the combustion chamber and the turbine.1. which raises the pressure and the temperature of the fluid in an isentropic process (ideally). air) enters the compressor. the working fluid expands isentropically in the turbine. 3.g. as well as a proper selection of the airfoil. free turbine. transferring energy to its blades. generator.. The compressed fluid is then provided to the combustion chamber.).g. T P2 fuel 3 combustion chamber 2 3 P1 power output 2 air 4 exhaust gas 1 compressor turbine 4 1 s Figure 9 – Schematic figure of the main components in a gas turbine and the Brayton cycle. Finally. propeller. considering constant specific heat at constant pressure cp. The turbine and the compressor are connected by a shaft. wherein fuel is added and burnt. which transfers mechanical energy from the turbine to the compressor. Then. 3. for isentropic compression or expansion:  Ta   Tb   Pa  =    Pb  γ −1 γ . the air with high velocity is delivered to the stationary row. where mechanical energy from the shaft is transferred to the fluid to accelerate it.. .4 Basic operation An axial-flow compressor consists of a series of rotating blades and stationary blades. where it flows through a divergent nozzle and is diffused.41 Using. as shown in Figure 10. The air first enters a row of rotating blades. rotor rotation Mechanical Energy  Fluid kinetic energy Fluid kinetic energy  Static pressure rise stator Figure 10 – Scheme of an axial-compressor stage. (2) Then.1. if rp denotes the pressure ratio P2 / P1 : γ −1 γ −1     γ  P4    P1  γ    T3 1 −   − T 1−   P3   2   P2   1     ηcycle = = 1−  r T3 − T2  p    γ −1 γ (3) From Equation (3) one immediately notices the relevance of the compressor in the overall engine efficiency. i. fluid kinetic energy is converted to static pressure rise.e. Figure 12 – Details of a gas turbine detailing a compressor rotor row. Static pressure Total pressure Static temperature Total temperature Relative velocity Absolute velocity Enthalpy Density Rotor Increase Increase Increase Increase Decrease Increase Increase Increase Stator Increase Small decrease Increase Constant Decrease Constant Increase . Figure 11 illustrates the aforementioned “unfold”. which is as if the cylindrical surface. Figure 11 – Visual aid to the common plane scheme of an axial-flow compressor stage. where the blades are laid on. Walsh and Fletcher [41] present a summary of the thermodynamic processes occurring at the rotor and at the stator in Table 3: Table 3 – Thermodynamic processes at the rotor and stator. Actually the scheme refers to the surface at the mid-line.42 Figure 10 is a recurrent drawing. was unfolded. In this work.α’2 ) Ca2 chord Cw2 Figure 13 – Nomenclature according to Saravanamutto [37]. Thus. Blade angle. Subscripts w and a indicate the whirl (tangential) and the axial components. The meridional direction m is given by the composition of the radial and axial directions of the flow: mˆ = Vw rˆ + Va zˆ Vw2 + Va2 (4) Greek letters α and β indicate absolute air and relative air angle. which is the angle between the chord direction and the axial coordinate.43 3. An overview is presented in Figure 13 α1 α1’ ι θ ζ a c Point of maximum camber s δ α2’ V2 α2 V1 α’1 α’2 θ ζ s ε α1 α2 V1 V2 ι δ c blade inlet angle β1 V1 α1 C1 blade outlet angle C a1 blade camber angle ( α’1 . Letters C.α2 ) U air inlet angle air outlet angle air inlet velocity air outlet velocity V2 β 2 α2 C2 incidence angle ( α1 . V and U are used for absolute. α ′ denotes blade ι α1 − α1′ and deviation angle by δ= α 2 − α 2′ . Subscript 0 denotes total property.α’1 ) deviation angle ( α2 . relative to the rotor and tangential (or peripheral) velocities. 2 and 3 denote respectively rotor inlet. Letter ζ denotes the stagger or setting angle. the nomenclature used by Saravanamuttoo [37] is preferred. incidence angle is given by = camber angle is given by θ= α1′ − α 2′ and the deflection of the air by ε= α1 − α 2 . .1. rotor outlet and stator outlet.α’2 ) setting or stagger angle Cw1 pitch or space deflection ( α1 . respectively. Subscripts 1.5 Nomenclature The literature presents many different nomenclatures for blade and cascade. then the meridional component is the axial component.1 DIMENSIONLESS PARAMETERS Flow coefficient The first dimensionless parameter commonly used in performance calculation is the flow coefficient. The distance from one blade to another measured at constant axial coordinate is the space or pitch s. 3.0] .1. The axial velocity is directly related to the flow coefficient and for advanced aero engines. which is defined as: φ= Ca1 U (5) Saravanamuttoo [37] suggests a range φ ∈ [ 0. Cm1 Cm 2 U1 Vw1 Cw1 Vw 2 U2 Cw 2 Figure 14 – Generic velocity triangles. .4. it can reach up to 200 m/s. as shown in Figure 14. If the row is purely axial.2 3.44 The distance from the leading edge to the trailing edge is the chord c. The inlet velocities and the outlet velocities of a rotor row are usually drawn together in a recurrent scheme named velocities triangles.2. 2. but a decrease in stage loading implies more stages. 0. leading to smaller losses. 3.45 3.4 Hub to tip ratio The hub to tip ratio is defined as the ratio of hub and tip radii: htr = rhub .2 Temperature or stage loading coefficient The temperature or stage loading coefficient indicates the amount of work per stage and is defined as: ψ = c p ∆T0 stage h03 − h01 c p constant ψ ⇒= 2 U U2 (6) For satisfactory operation Walsh and Fletcher [41] suggest ψ ∈ [ 0.25. Efficiency improves as loading is reduced.2.3 Degree of reaction The distribution of the flow diffusion taking place at the rotor and the stator rows is indicated by the degree of reaction. Thus. rtip (8) . a compromise is in question for aero engines.2. as both high efficiency and low weight (fewer stages) are mandatory. 3. The degree of reaction is the ratio between the static enthalpy rises in the rotor and in the stage: = Λ h2 − h1 c p constant T2 − T1 = ⇒ Λ h3 − h1 T3 − T1 (7) Many preliminary compressor designs start with a 50% reaction. due to even distribution of diffusion.5] . more difficult mechanical mounting in the rotor disc. the total enthalpy and total temperature are. A subscript 0 is used to denote total properties. the tip clearance becomes relatively higher. hence more pronounced secondary losses. is the distance between the blade tip and the compressor casing.5 Isentropic and polytropic efficiencies It is noted that total properties refer to the fluid with zero velocity and all of the kinetic energy has been adiabatically converted to internal energy. due to leakage flow through the spacing. as the name suggests. High values of tip clearance lead to lower efficiencies. 2c p (10) where h is the static enthalpy. . Hub to tip ratio Tip clearance rtip rhub Figure 15 – Hub to tip ratio and tip clearance. 2 (9) T0= T + C2 . The tip clearance. hence. Low values of hub to tip ratio yield long blades. respectively: h0= h + C2 . In a given point. Figure 15 shows the hub to tip ratio and the tip clearance in an actual compressor. C is the absolute velocity of the fluid and cp is the specific heat at constant pressure.46 High values of hub to tip ratio usually indicate short blades. 3.2. as well as. ηc = T02 T01 T02 − −1 T01 T01 T01 (13) For later compressor stages. This can be explained by the fact that the isobaric lines in a T-s diagram are divergent (to the right). as the pressure is already high. then: = ηc c p (T02′ − T01 ) T02′ − T01 = . c p (T02 − T01 ) T02 − T01 (12) γ −1 T02′ T01  p02  γ −   −1 T01 T01  p01  = .47 The compressor total-to-total isentropic efficiency is given by: ηc = ′ − h01 h02 . Noticeably. T p4 p2 = p3 p1 Last stage p4 p3 p2 p1 First stage s Figure 16 – Divergent isobaric lines and the increased compression difficulty in the last stages. even for the same pressure ratio. Hence the efficiency of the latter stages tends to be smaller than the initial stages. the compressor requires more energy to compress the fluid in the first stage than in the last stage. h02 − h01 (11) If the variation of cp with the temperature is ignored. as shown in Figure 16. . even when the technological level is the same. it is much more difficult to increase the pressure. Thus. Thus the overall efficiency is smaller.3 OVERVIEW OF AXIAL-FLOW COMPRESSOR PERFORMANCE In section 3. the discussion here will focus on the compressor side. the compression is not isentropic.48 This fact revealed the necessity of another definition of efficiency for multi-stage compressor: the polytropic efficiency or small-stage efficiency.3. . a very simple gas turbine thermodynamic cycle was presented. 3. Nevertheless. From now on.c dT ′ = constant dT (14) The idea of the polytropic efficiency can be visualised in Figure 17. it is reasonable in preliminary design to consider constant polytropic efficiency for all stages. T ∆T ∆T ′ dT ′ dT Elemental stage s Figure 17 – Polytropic or small-stage efficiency. which is defined as a constant isentropic efficiency of an elemental stage throughout the whole compression stage: η= ∞ . The polytropic efficiency represents the particular technological level for a particular design. the combustion is not isobaric and the expansion is not isentropic.1. The stator. A compressor stage with its velocity triangles is shown in Figure 20. To proceed with the blade preliminary geometry.49 T P2 P3 3’ 3 4 2 P4 P1 4’ 2’ 1 s Figure 18 – A schematic real gas turbine cycle. transforms kinetic energy to static pressure rise. its angles are written together with aerodynamic and thermodynamic equations. which delivers air at high speed to the stator. Assuming that C = C= Ca 2 . Ca (17) . Assuming adiabatic process. consider the compressor stage in a temperatureentropy diagram in Figure 19. to support the derivation. through diffusion. simple trigonometry yields: a a1 U = tan α1 + tan β1 . Firstly. one immediately finds that the power input to the compressor rotor is given by:  p (T02 − T01 ) = W mc (15) The adiabatic assumption in the stator yields: T02 = T03 (16) The power is solely transferred to the rotor. Ca T p02 02 T02 =T03 T03’ C32 2cp 03’ 3 C22 2cp p2 1 p01 T01 p03 03 p3 T2 (18) 2 3 2 p1 01 T1 S R C12 2cp 1 s Figure 19 – Axial-flow compressor stage in a T-s diagram. . α1 C1 β1 V1 rotor U α1 C1 Ca1 Cw1 C2 U α2 V2 β2 stator α2 C2 Ca2 α3 C3 Cw2 Figure 20 – Rotor row and stator row with velocity triangles in an axial-flow compressor stage.50 U = tan α 2 + tan β 2 . the required torque for a mass flow rate m is = T m ( r2Cw 2 − r1Cw1 ) . (21) W m ω ( r2Cw 2 − r1Cw1 ) . .51 From (17) and (18): tan α 2 − tan α1 = tan β1 − tan β 2 (19) tan α1 + tan β1 = tan α 2 + tan β 2 . otherwise. Thus. (23) yield  a ( tan α 2 − tan α1 ) . thus more power is transferred to the fluid in this condition. β1 → 90° and β 2 = 0° would be the undoubted design. = W mUC (24)  a ( tan β1 − tan β 2 ) . however. Later. the flow enters with tangential velocity Cw1 at radius r1 and leaves with tangential velocity Cw 2 at radius r2 . = (22) and the power to drive it: For the special case in which r1 = r2 :  ( Cw 2 − Cw1 ) . = W mUC (25) Equation (25) shows that more power is used by the compressor if the blade has higher camber angle. (20) In the compressor. it will be shown that there is a limit for this camber. = W m ω r1 ( Cw 2 − Cw= mU 1) (23) The velocity triangles from Figure 20 and Eq. 3. Advanced aero engines can handle axial velocities up to 200 m/s. This happens due to elevated levels of centrifugal tensile stress under which the blades are submitted. if the compressor power input is transferred to the fluid to raise its pressure. cp (26) Then Eqs. the designer has no control on the working fluid and ambient conditions. cp and γ provide elevated pressure ratios per stage. and its maximum value. The rotational velocity is limited by material technology and the compressor efficiency is given by the technological level at disposal. which are related to the temperature rise.52 Continuing with the derivation. Usually. 3. the whole power input contributes to the total pressure rise: T03 =T02 ∆T0 stage = T03 − T01 = T02 − T01 = UCa ( tan β1 − tan β 2 ) . High values of compressor efficiency. γ and T01 are not case of study here. rotational speed. thus. occurring at the blade root is given by: .1 Tip speed The rotational velocity of a gas turbine is limited by material technology. axial velocity and camber angle and low values of inlet total temperature. major analysis is focused on angles. The axial velocity does play an important role. Thus. (13) and (26) yield: γ −1 η= c rp γ − 1  T02  − 1   T01  ⇒ ηc T01 (T02 − T01=) γ −1 γ rp γ −1 UCa − 1 ⇒ ηc ( tan β1 − tan β 2=) rp γ − 1 ⇒ c pT01 (27) γ  UCa  γ −1 = ⇒ rp ηc ( tan β1 − tan β 2 ) + 1  c pT01  Equation (27) provides wise advices on how to obtain higher pressure ratios per stage. changes in c p . but is limited due to high losses. r ⇒ ω = 3. however. rt radius at blade tip. . r rr radius. rt rr (28) where: ρb ω angular velocity. then for a 5 cm radius microturbine and a 1.5 m radius high bypass-ratio turbofan: U = ω.05 s (29) U 350 rad = ⇒ ω = 233 ⇒ N = 2228 rpm r 1. blade material density.53 (σ ct )max = ρbω 2 Sr ∫ S ( r ) . as depicted in Figure 21. (27).2 Camber angle and de Haller number As prescribed by Eq.3. radius at blade root. Present technology imposes a 400 m/s limit at the blade tip. Sr area at blade root. let the tip speed limit be 350 m/s. high fluid deflection contributes to high pressure ratios per stage. S ( r ) blade cross-section at any radius. To evaluate the angular velocity.r dr . Consider the case. at which material limitation is critical. In fans. this figure reaches 450 m/s. which the relative inlet angle β1 is kept constant and the angle β 2 is diminished to provide higher fluid deflection.5 s (30) U = ω.r ⇒ ω = U 350 rad = ⇒ ω = 7000 ⇒ N = 66845 rpm r 0. Usually it is related to excessive vibrations and a particular noise. defined as: deHaller = V2 > 0. entails a high rate of diffusion.69 . The phenomenon yet very harmful to the engine is still not fully understood.3 Compressor surge The surge is an unstable operation of the compressor. Due to excessive losses.72. beyond which reversal of the flow is expected. V1 (31) Originally. the limit was 0.54 β2 V2 α2 β1 β2 α 2 V2 C2 V1 α1 C2 C1 Ca U Figure 21 – Inlet and outlet relative velocity ratio is reduced with the increase of fluid deflection. It is clear that high fluid deflection results in lower outlet relative velocity. which propagates from its origin to the whole engine. this means that more kinetic energy is converted to static pressure. The surge is seen as the lower limit of stable operation. 3. hence camber angle. characterised by a sudden drop of delivery pressure and by intense aerodynamic pulsation. a limit of diffusion exists and in preliminary design it is quantified by the de Haller number. high fluid deflections.69. but accumulated experience pushed this figure to 0. . In other words.3. No matter what is done to increase the pressure ratio. is bounded by surge and choke. Thus. it is known that the maximum mass flow rate through a nozzle is reached when the throat is at Mach 1.55 3.4 Compressor choke From the gas dynamics.3. for each rotational speed. no extra flow pass through this nozzle. the operational range of a compressor. As the space between the blades forms a nozzle. the compressor choke happens when the blade throats choke. . refer to those works. . To start with. much of the preliminary design relies upon an experienced designer and information gathered from costly and time-consuming experimental studies. but some are found in the open literature. which are mostly proprietary. In this particular case of study. it involves a careful and wise choice of a plethora of design parameters. the National Advisory Committee for Aeronautics (NACA). hundreds of parameters have to be properly set. Among the main open data resources. Thus.56 4 THE STREAMLINE CURVATURE COMPUTATIONAL 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]. the publications from the National Aeronautics and Space Administration (NASA) and from its predecessor. For a careful and in-depth analysis.1 INTRODUCTION The design of a multi-stage axial-flow compressor is a laborious task for various reasons. Montsarrat. are yet the basis. 4. Keenan and Tramm (NASA CR-72562) [45] and Schwenck. Lewis and Hartmann (NACA RM E57A30) [46] contain the fundamentals of axial-flow compressors. The publications from Johnsen and Bullock (NASA SP-36) [44]. Dummy stages represent inlet and outlet ducts. which are solved iteratively by finite differences in a meridional plane grid. whose blades do not disturb the flow (i. the set of differential equations are integrated at the nodes along the streamline.). in which the losses due to viscosity effects are computed empirically.57 Due to the complex flow field observed in axial-flow compressors. as well as. an initial positioning guess is required and later it is varied iteratively to satisfy the continuity equations.2 THE STREAMLINE CURVATURE METHOD The actual flow in an axial compressor is highly complex. etc. The Streamline Curvature Method (SLCM) consists of writing the non-viscous equations of continuity. Nevertheless. high computational costs. The nodes of this grid are the intersection of the streamlines and the blade edges. a bladed stage. as shown in [42] by comparing its results with a real three-stage transonic compressor and with commercial codes [47]. turbulent and viscous.e. To assess the flow according to detailed mathematical models. motion and energy along a coordinate system based upon the flow streamline and the tangent to the blade edges. from inlet. The outcome is a set of non-linear partial differential equations. an axisymmetric and inviscid model. as depicted in Figure 23. many early computational models failed to accurately predict performance characteristics. 4. Therefore. illustrated in Figure 22. no deviation. potentially offers adequate accuracy and velocity. Finally. or computationally. Nevertheless. As the streamlines are not known previously. to outlet. no diffusion. quick answers are constantly demanded in the preliminary design phase. the SLCP demonstrated to be a fast and reliable performance prediction tool. three-dimensional. .. long evaluation times are required. at the trailing edge. stage rows and calculation nodes. the flow is divided into concentric streamtubes. In the SLCM. [42] dummy dummy IGV rotor stator dummy dummy dummy nodes streamlines Figure 23 – Streamlines. by means of empirical correlations. Adapted from [42]. pressure decrease.58 r s γ casing blade trailing edge m ε streamline z blade leading edge hub Figure 22 – Streamline-blade leading edge coordinate system (s-m). wherein the flow is axisymmetric. The flow is calculated according to inviscid equations and the losses due to viscosity are incorporated as entropy increase. etc. The basic derivation of the method can be found in Appendix A. . start Read reference curves Read input Stage loading distribution Channel inlet Channel outlet Axial channel design Channel intake Vortex Inlet Radial equilibrium Outlet Incidence Boundary layer effect Cutaway Velocity triangles Blading Boundary layer New grid Efficiency calculation N Efficiency converged? Y Print tables Write complete geometry Write optimisation input file •Calculated efficiency •De Haller number •Camber •Pressure ratio end Figure 24 – Overview of the SLC program algorithm.3 COMPUTATIONAL PROGRAM The design mode of the SLC program is based on reference [42]. a simplification of the main structure of the program is illustrated in Figure 24. The SLC program is interactive. At the present time. It is written in FORTRAN language and it has been continuously updated.59 4. Many convergence loops and subroutines were omitted for the sake of clarity. Losses DCA . fully modular and its high flexibility allows new features to be easily integrated. To compare two solutions in a MOOP. Optimisation may be defined as the search for solution(s) which correspond to minimum (or maximum) of one or more objectives. 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. In this case. the specific vocabulary and definitions are made clear. a real-coded elitist multiobjective genetic algorithm program (REMOGA) developed in FORTRAN is detailed.60 5 REAL-CODED ELITIST MULTI-OBJECTIVE GENETIC ALGORITHM PROGRAM This chapter aims at providing the reader. normally one obtains not one. The definitions presented hereinafter are extracted or adapted from Deb [48] and Büche . which may be conflicting. with the basis to proceed without loss of understanding. 5. the fundamentals of Genetic Algorithms (GAs) are explained. important definitions about Multiple Objective Optimisation Problem (MOOP) are presented.1 DEFINITIONS To accurately describe a MOOP. who is not familiar with Genetic Algorithms. A singleobjective optimisation problem (SOOP) usually has one single optimal solution. A MOOP accounts for multiple objectives. satisfying given constraints. Finally. but a set of optimal solutions named Pareto optimal solutions. Next. To start with. totalling p+q constraints and 2n variable bounds is called a feasible solution. . which minimises/maximises: = min/max f ( x) ( f ( x ) . a MOOP can be described by a vector of decision variables x and the corresponding vector of objectives. . Decision space Objective space x3 f2  x  z x2 x1 f1 Figure 25 – Mapping between the decision space and the objective space. The functions g ( x ) and h ( x ) are the constraint functions. otherwise. 2. infeasible solution. x2  . . g p ( x ) ) ≥ 0 = h ( x ) (= h1 ( x ) . .1 Multi-objective optimisation problem Definition 1 The multi-objective optimisation problem is defined as the search for the set of solutions x . 5. An illustrative mapping from a 3-dimensional decision space to a 2-dimensional objective space is shown in Figure 25. (32) i= 1.61 [49]. hq ( x ) ) 0 with = x xi( L ) ≤ xi ≤ xi(U ) . g 2 ( x ) . ( where X ⊆  n is the n-dimensional decision space bounded by xi L) ( ) and xi . f ( x ) . f ( x ) ) ∈ F 1 2 m ( x1 . Any solution which satisfies every constraint. n. To start with. and F ⊆  m is U the m-dimensional objective space.1. h2 ( x ) . f = f ( x ) . xn ) ∈ X = subject to g ( x ) ( g1 ( x ) . . or. . Analogously.2 Domination To compare different solutions from (32). Definition 2 A solution a ∈ X dominates a solution b ∈ X .62 Let a and b ∈ X . 5. m} : f j (a)  f j (b ) (33) If solution a dominates solution b. and f i ( a ) / f i ( b ) denotes that a is a not worse solution than b. than f i ( a )  f i ( b ) means that fi ( a ) < fi ( b ) . f i ( a )  f i ( b ) denotes that a is a worse solution than b. then f i ( a )  f i ( b ) denotes that a is a better solution than b with respect to the i-th objective. a / b . . . than f i ( a )  f i ( b ) implies that fi ( a ) > fi ( b ) . If the i-th objective is a maximisation one. or. This statement can be expressed as: ab⇔ ∀ i ∈ {1. 2. m} : fi ( a ) / fi ( b ) ∧ ∃ j ∈ {1. then it is said that a does not dominate b. f i ( a ) / f i ( b ) denotes that a is a not better solution than b.1. an ordering among different solutions is obtained by the dominance criterion. • a is non-dominated by b. which is expressed by a  b . 2. if and only if a is no worse in all objectives and at least superior in one objective then b . If Definition 2 does not apply. If the i-th objective is a minimisation one. it is also usual to write any of the following: • b is dominated by a. This can be expressed as: .63 Definition 3 The solution a ∈ X is indifferent to a solution c ∈ X . thus. if and only if neither solution is dominating the other one. in both objectives). Among a set of solutions P ⊆ X . f2 b a ab a / c c f1 Figure 26 – Representation of dominance and indifference between solutions in a two-objective minimisation problem. Figure 26 illustrates the dominance and indifference as defined in a two-objective minimisation problem. but is indifferent to c. named Non-dominated set: Definition 4 (Non-dominated set). This leads to an important set.3 Pareto-optimal set If in a given set of solutions. solution a is indifferent to solution c. Solution a dominates solution b. one eventually finds which solution dominates which and which solutions are not dominated with respect to each other. all possible pairwise comparisons are performed. as it is no worse than b in both f1 and f2 and is better in at least one of those objectives (in this case. i. 5.1. one notices that solution a does not dominate solution c and solution c does not dominate solution a.e. Furthermore. Solution a dominates b. the non-dominated set of solutions P′ ⊆ P are those that are not dominated by any member of the set P .. a / c and c / a . 1.. i. b x .1] ⊂  (35) A convex f :  →  function is illustrated in Figure 27. ∀λ ∈ [ 0. Note that the line segment joining a and b is always greater or equal the function evaluated between those values.4 Convexity Definition 6 A function f :  n →  is said to be a convex function if for any pair of solutions a.e. P = X .64 P′ ⊆ P is a non-dominated set ⇔ ∀a ∈ P′. ∀b ∈ P : a  / b (34) Definition 5 (Globally Pareto-Optimal set). 5. the globally Pareto-optimal set is often referred to as Pareto set. then the non-dominated set P′ is called globally Pareto-optimal set. When the set P is the entire search space. the following condition is true: f ( λa + (1 − λ ) b ) ≤ λ f ( a ) + (1 − λ ) f ( b ) . f (x) λ f ( a ) + (1 − λ ) f ( b ) f (b ) f (a) f ( λa + (1 − λ ) b ) a λa + (1 − λ ) b Figure 27 – A convex function illustration. For the sake of concision. b ∈ X . Definition 8 A multi-objective optimisation problem is convex if all objective functions are convex and the feasible region is convex [48]. Indirect methods rely on solving the set of equations provided by equalling the gradient of the objective functions to zero. • Numerical differentiation is prone to severe errors.2 TRADITIONAL METHODS AND THE GENETIC ALGORITHM According to Goldberg [50]. optimisation and search techniques fall onto three main methods: calculus-based. • It does not work properly with discontinuous functions.65 m Definition 7 Let F ⊂  .e. 5. The main disadvantages of the method are: • The objective function has to be differentiable. A brief description of each one is provided to elucidate the reader the reason of the success of GAs in the turbomachinery context. F is said to be a convex set if. • Almost implies that the objective function surface has to be known a priori. enumerative and random.. given any two points members of F . Direct methods are based on the iterative hill-climbing concept. a local optimum is found eventually. . • The algorithm is likely to be stuck onto a suboptimal solution when dealing with surfaces like Figure 28. i. starting from a given point the gradient is calculated to provide the climb-direction of the next point. Calculus-based methods are divided into two categories: indirect and direct. Successively. the line segment joining these points is entirely contained in F . 66 Figure 28 – Illustrative region where a gradient-based algorithm can get stuck onto a suboptimal solution. It relies on evaluating the objective function at every point. calculus-based methods most commonly convert the MOOP into a SOOP summing all objectives with certain weights. g 2 ( x ) . Nevertheless. which full enumeration is practicable. The problem that arises is with the choice of the weights and the fact that the optimisation will result in a single solution. ∑ wi 1 ∑= =j 1 =i 1 ( x1 . as the whole search space is covered. given a finite search space. To treat MOOPs. it only fits problems. . as there is no reason to explore unnecessary regions. . however. it is computationally inefficient and expensive. unquestionably. rather than a compromise Pareto set. (36) 1. 2. . g p ( x ) ) ≥ 0 h1 ( x ) . xn ) ∈ X subject to = g ( x ) ( g1 ( x ) . Goldberg [50] distinguishes random walks and randomized techniques. n. the maximum is found. hq ( x ) ) 0 = h ( x ) (= with = x xi( L ) ≤ xi ≤ xi(U ) . x2  . h2 ( x ) . Enumerative methods are definitely infeasible for real-world multi-dimensional engineering problems. min/max f (x) = m m w j f j ( x ). Random search algorithms are gaining popularity in Turbomachinery design as shown in chapter 2. i= The enumerative method is the “least intelligent” algorithm and applicable to few a simple cases. . 67 The former implies a random scheme to search and save the best.3 GENETIC ALGORITHM FUNDAMENTALS Evolutionary techniques are being extensively used by researchers in various fields due to their effectiveness and robustness in searching for a set of global trade-off solutions [51]. Yes Stop . where the survival of the fittest. In that sense Genetic Algorithms are completely different from classical techniques. uses random choices as guiding tool. where GAs are included. while the latter. start Initialise population gen=0 Evaluation gen = gen+1 Assign fitness Condition satisfied ? No Reproduction Crossover Mutation Figure 29 – Simple GA algorithm [48]. reproduction and mutation are the basis of the heuristic counterpart. 5. The working principle is motivated by natural genetics and Darwinian natural selection. but a pool of solutions. Next. in the “evaluation” block. the first step after the algorithm is started is “Initialise population”. a simple GA works with three operators: reproduction. crossover and mutation as shown in Figure 29. Fitness is the biological correspondent to “objective” and in SOOP is the same as “objective”. A triangle with height 9 cm and base 12 cm is then represented by the x1 = 9 cm 8-bit string (1001 1100). the objective functions are calculated and the fitness is assigned to each solution. the solution must be represented in a special manner. let ignore that an area of zero might result from this unconstrained example. . If a refined step is desired. let an SOOP be the minimisation of the area of a right triangle with the decision variables being the catheti length. Chromosome 10011100 A = 54 cm² x1 = 9 x2 = 12 x2 = 12 cm Figure 30 – Chromosomal representation of decision variables. recalling Figure 29. Thus. This means that a certain amount of solutions in the design space is chosen by a given criteria (which can follow a certain rule or be purely random). To perform those operations.68 Briefly. GAs strategies usually evaluate not a single solution per time. Then the GA operators are performed to search optimal solutions. which allows a biological parallel. crossover and mutation operations will be more transparent in the following sections. but the concept goes beyond when dealing with MOOP as will be treated afterwards. With the binary string representation in mind. This is carried out by writing the decision variables in a binary string. than more bits should be used. For the sake of simplicity. Using the example from [52]. the understanding of the selection. 4. fit. Each individual #1... the average fitness is improved. 4. It can be performed via tournament selection [50]. Then. Tournament 1 # x1 x2 fit. fit.5 Figure 31 – Tournament selection illustration. keeping the population size constant. Recalling the triangle area minimisation example. after the selection operator. the best solutions are copied and the worst solutions are eliminated. 8 and 7 are winners. # x1 x2 fit.#8 has initially assigned values of x1 and x2. 8. the best solution among the initial pool of candidates would be . after selec. solutions 1. prior selec. The two tables in the centre of Figure 31 illustrate the matches of each tournament. 5. 6. Tournament 2 # x1 x2 fit. in the second tournament solutions 6. the population size is kept. 5 and 8 are winners. #2. 14 av.3. Moreover. In other words.1 Selection or reproduction operator The selection or reproduction operator identifies and multiplies good solutions in a population and eliminates bad solutions. 7 and 8 survive and solutions 2 and 3 are eliminated. In the first tournament. solutions number 1. Therefore. # x1 x2 fit. 4.69 5. let the initial population be of eight individuals and given as shown in the leftmost table from Figure 31. If each individual from the initial population plays the tournament twice (against different individuals). the matches are shuffled. Prior to the second tournament. in which each solution “plays” against another solution and the fittest wins and the other is eliminated. Were the selection the unique operator. 1 7 6 21 1 7 6 21 6 8 4 16 1 7 6 21 2 4 15 30 2 4 15 30 3 6 7 21 6 8 4 16 3 6 7 21 3 6 7 21 1 7 6 21 4 8 1 4 4 8 1 4 4 8 1 4 4 8 1 4 4 8 1 4 5 9 2 9 5 9 2 9 2 4 15 30 5 9 2 9 6 8 4 16 6 8 4 16 8 1 6 3 8 1 6 3 7 2 8 8 7 2 8 8 7 2 8 8 8 1 6 3 8 1 6 3 8 1 6 3 5 9 2 9 7 2 8 8 av.. The selection aims at passing to the next generation only the fittest. Therefore.70 found after some iterations. But the optimum may not be any of the initial candidates. is an exchange of genetic material between chromosomes. Differently from the selection operator. the crossover and mutation operators come to provide further exploration of the design space. contributing to the exploration of the design space. Figure 32 – Biological crossover illustration. illustrated in Figure 32. h = 8 b = 12 48 h = 10 b = 14 70 . A bit-wise crossover is illustrated in Figure 33. The crossover results in a new arrangement and more diversity. The crossover can be a simple exchange of one or some bits in a pair of chromosomes. the crossover generates new solutions. h = 9 b = 12 1001 1100 1001 0100 54 h = 10 b = 6 30 1010 0110 1010 1110 Figure 33 – Bit-wise crossover representation. The computational parallel is performed thanks to the binary string concept. which occurs during the meiosis.3. 5.2 Crossover operator The biological crossover. . as depicted in Figure 35. the operator changes a bit from 1 to 0.3. Bits in position 4. 5. the two-point crossover is operated by exchanging the bits between two selected bits.3 Mutation operator To secure diversity. resulting in two different solutions. Its working principle is simple: with a certain probability pmut . additionally to the crossover. 5 and 6 are exchanged. or vice versa. Lastly. as shown in Figure 34. h = 9 b = 12 1001 1100 1000 0100 54 h = 10 b = 6 30 1010 0110 1011 1110 h = 8 b = 4 16 h = 11 b = 14 77 Figure 35 – Two-point crossover representation. This probability affects the convergence and the exploration of the design space. Note that all bits from the fourth bit (inclusive) are exchanged. resulting in triangles with areas 16 and 77. h = 9 b = 12 1001 1100 1001 0110 54 h = 10 b = 6 30 1010 0110 1010 1100 h = 9 b = 6 27 h = 10 b = 12 60 Figure 34 – Single-point crossover representation. the mutation operator is used.71 The single-point crossover is operated by exchanging all bits “to the right” of a chosen bit. 72 54 h = 9 b = 12 Chromosome 1 0 0 1 1 1 00 Chromosome 1 0 1 1 1 1 00 h = 11 b = 12 66 Figure 36 – Mutation operator. Elitist Crossover Assess niche count Real Polynomial Mutation Assess shared fitness parent  offspring Stop Yes Gen > gen_max? No Figure 37 – Algorithm of the REMOGA program. it will be easier to the reader. which was developed as part of this work.4 REAL-CODED ELITIST MULTI-OBJECTIVE GENETIC ALGORITHM PROGRAM (REMOGA) The real-coded elitist multi-objective genetic algorithm program (REMOGA) written in FORTRAN 90 language is based on the works from Deb [48] and Fonseca and Fleming [53]. start gen=1 gen = gen+1 Read initial population Sort according to fitness Evaluate objectives Write parent population Calculate rank Crowded Tournament Selection Sort according to rank Real Poly. who is not familiar with GAs to follow the mechanisms used in the real-coded multi-objective genetic algorithm. . 5. Now that the basic concepts of GAs were presented. the fitness is the equivalent to “objective” when dealing with a SOOP. i.e. rank ( i ) = 1 + ni . Then. fit R= ( i ) npop − rank ( i ) (38) Computationally. 5. or its raw fitness. The obtained algorithm. Proceeding in this way. is depicted in Figure 37.. as if they refer to the i-th or j-th solutions among a total of npop solutions. indices i and j are preferred. instead of referring to generic solutions a or b.1. fit R . but its features are expanded due to the multi-objective and real-coded approaches. As the selection operator uses the fitness to identify the best solutions and to eliminate the bad ones. the algorithm to assign rank was conducted by pairwise comparisons using flags to determine whether the two domination criteria are satisfied. 5. the rank of the ith solution is equal to one plus the number of solutions that dominate solution i. see Equation (38). which will be detailed hereafter.1 Multi-objective formulation As aforementioned. but its concept is changed when the problem is a MOOP.4. better solutions are assigned with greater fitness values.4. the fitness must encompass the concept of domination for MOOP. Let .73 The basic algorithm is inevitably similar to the one presented in Figure 29. ni.1 Assessing multiple objective in a single function: rank-based fitness Fonseca and Fleming [53] introduced the idea of a rank-based fitness assignment. is defined as the population size minus its rank. In this way an initial MOOP fitness of an individual. (37) Thus the non-dominated solutions are assigned a rank equal to 1 and worse solutions are assigned values greater than 1. Hereafter. which hardly exceeds 500. The pseudocode of the Bubble sort algorithm is shown in Figure 39.4.1. then the rank assignment algorithm is depicted in Figure 38.g. Quicksort. 5. i=1 n ≠ i rank=1 n=1 start N Y flag1=0 flag2=0 j=1 Y fi(j) > f n(j) flag1=1 N Y fi(j) < f n(j) flag2=1 N j=j+1 N j > nobj Y flag1=0 AND flag2=1 Y rank=rank+1 N i=i+1 N Y i>npop N rank(fn)=rank N n>npop n=n+1 Y end Figure 38 – Rank assignment algorithm. The next step in the algorithm is the sorting of the population according to the calculated rank. The choice of this simple algorithm.74 f m( j ) be the m-th objective of the j-th individual of a certain population of size npop. . e. was due to the computationally small population per generation.2 Sorting A simple Bubble sort algorithm was implemented for the sorting of the population. rather than more efficient ones. 3 Preserving diversity: niche count and shared fitness A little thought about the rank-based fitness reveals that diversity of solutions is not guaranteed. 5. To avoid concentration of solutions in the objective space and encourage a broad exploration of the codomain. f2 Without niche penalty f2 f1 With niche penalty f1 Figure 40 – Optimisation (a) without niche penalty and (b) with niche penalty .4. The niche control can be visually understood in Figure 40.1. solutions that are too close to other solutions should have its fitness penalised.75 subroutine BubbleSort( A : list of sortable items ) swapped = true do while (swapped == true) swapped = false for i = 1 to length(A) – 1 do: if A[i] > A[i+1] then memo = A[i] A[i] = A[i+1] A[i+1] = memo swapped = true end if end for end do end subroutine Figure 39 – Bubble sort pseudocode. Fonseca and Fleming [53] have also come up with clever ideas about niche of solutions. This penalty must encompass both the distance to other solutions and the number of solutions that are nearby. the niche count nci is computed as the sum of the sharing functions: nci = µ ( ri ) ∑ Sh ( d ) . Thereafter. dij = =  max ∑ min   k =1  f k − f k     (39) To each pair of solution i and j. which is defined as:   dij α  Sh= ( dij ) max 1 −  σ  . Anything in between will assume a value within the range [0. f ( i ) is to f ( j ) . high values of nci should be avoided in order to maintain diversity. The niche count indicates the crowding around a solution.76 The first step towards niche avoidance is the calculation of the normalised distance between any two solutions i and j in a rank r: 1  m  f (i ) − f ( j ) 2  2 k k  . the sharing function is 1. 0 . there is a corresponding sharing function. j =1 ij (41) where µ ( ri ) is the number of solutions in rank i. but if j is outer the m-hypersphere of radius σ share centred in f ( i ) . than the sharing function is null. . in the objective space.1] according to the choice of parameter α .   share   (40) which can be understood as an indicator of how close. If f ( i ) = f ( j ) . rank ( i ) rank ( j ) . as depicted in Figure 41. for each solution i. 77 Sharing function 1.0 Sharing function alpha 0.8 0.5 1.0 1.5 2.0 0.6 0.4 0.2 0.0 0 0.2 0.4 0.6 0.8 dij / σshare 1 1.2 Figure 41 – Dependence of the sharing function with α. The next step is to use the niche count in the fitness. This can be done by dividing the raw fitness by the niche count. This new fitness should be referred to as shared fitness. An isolated solution has a niche count of 1, thus its shared fitness is not penalised, but another solution in a crowded area has a niche count greater than 1, thus, the shared fitness is lower than the raw fitness. It was not revealed hitherto which value of σ share to use. A simple approach is to define a fixed value based on a rough knowledge of the Pareto optimal region. Nevertheless, rarely it is known a priori. Fonseca and Fleming [53] suggest a value based on the smallest m-hypercube (m is the number of objectives) which contains the objective space and in its minimum subdivisions in smaller m-hypercubes of edge σ share . Eventually it is suggested the solution of the following polynomial equation: m m ( Ωi − ωi + σ share ) − ∏ ( Ωi − ωi ) ∏ m −1 = i 1 =i 1 npop= .σ share = 0, σ share > 0 , σ share where npop is the population size in each generation and: (42) 78 = ω Ω= min f1 , , min f m ) (ω1 , , ωm ) , (= ( max f1 , , max f m ) = ( Ω1 , , Ωm ) . (43) In the developed program, a simplification was used and the σ share was calculated as: 1 σ share = npop 1 m 1 m   ∑  max ( fi ) − min ( fi )   .  m i =1  (44) In Equation (44), the σ share was taken as if a m-hypercube with edge equal to the average of the length of each objective was divided in such a way that npop m-hypercubes of edge σ share would fit in this larger m-hypercube. This idea may be easily understood with a 2-dimensional objective space, as shown in Figure 42. Let 16 solutions be scattered in such a way that the difference of the maximum value and minimum value for objectives 1 and 2 are, respectively, 6 and 4. The average edge length is 5. Now, consider that all 16 individuals shown in Figure 42 are to be distributed in a 2-dimensional “square” set of edge 5. It can be achieved if the “square” can be divided into 16 smaller squares, what means a small square of edge 1.25, whose value will be used as σ share . Naturally, this square may not be so orderly divided if the number of solutions is not a square of a natural number, thus the mathematical formulation given by (44) expands the explained idea not only for this case, but also for hyper-dimensional sets. f2 5 6 16 indv. 5 4 f1 σshare Figure 42 – Visual interpretation of the used value of σshare. 79 5.4.2 Crowded Tournament Selection Following the algorithm shown in Figure 37, the next important step is the Crowded Tournament Selection (CTS). Differently from the GA Tournament Selection, the CTS uses both the rank and the shared fitness to perform the selection of the fittest and elimination of the bad solutions. The CTS operator works as follows: a solution i wins a tournament against solution j, if: rank ( i ) < rank ( j ) ∨ fit sh ( i ) > fit sh ( j ) , if rank ( i ) = rank ( j ) (45) f2 4 1 56 2 1 vs. 4 : 1 4 vs. 5 : 4 3 vs. 1 : 3 3 f1 Figure 43 – Crowded tournament selection operator. Figure 43 may be used to visualise the CTS operator. In a tournament between solutions 1 and 4, solution 1 wins because it has a lower rank. In a tournament between 1 and 3, solution 3 wins because it has the same rank as solution 1, but its shared fitness is greater, as it is more isolated. An additional feature in the selection operator is the possibility to call it multiple times in a row in the same generation. It revealed an interesting solution when dealing with high-error functions (actually, the SLC program). Parameter number of selection nsel can be set in the program input file. It simply acts as follows: DO i=1,nsel CALL selection END DO Figure 44 – Multiple selections. requiring more computational effort. a real-coded genetic algorithm is necessary and a different crossover and mutation mechanisms are required.4.  2 (47) xi( 1 (1 − βi ) xi(1. 1=) 1 (1 + βi ) xi(1.t ) + (1 + βi ) xi( 2.t ) = 1 and xi( 2. even with a considerable refinement.5. u 2 ( )  i  1 βi =   1  ηc −1   . the real space is not fully explored.  2 (48) t+ 2.80 5. the simulated binary crossover (SBX) from Deb and Agrawal [54] is used. the parameter βi is calculated as follows: 1  ηc +1 .3 Real-coded Polynomial and Elitist Crossover Operator Dealing with continuous search space is a limitation of the binary string chromosome approach.t ) represents the i-th parameter of solution 1 of generation t. To assess the real-coded crossover as similar to the binary string. equations (47) and (48) result in an offspring probability density distributions as shown in the histogram of Figure 45. if xi(1. (46) otherwise.t +1) = Considering the parent solutions xi(1.t ) = −1 . the string has to be very long.t ) + (1 − βi ) xi( 2.t )  . If a small step is desired. which will be explained afterwards. . Hence.1) is a random number and ηc is a control parameter. Moreover.   2 (1 − ui )   if ui ≤ 0. where ui ∈ ( 0.t )  . First. the offspring is calculated as follows: xi(1. Then. 0 0.0 5% -2.0 -2.4 0. but incurred in “divergence” or even loss of optimum regions.0 eta_c = 2 2.0 3.8 random number value 1.0 eta_c = 1 2.0 0.0 0.0 -3.0 -3. then with a certain user-defined probability (preferably high). solution i undergoes the crossover with another solution.0 Figure 45 – SBX [54] operator and influence of parameter ηc.0 0. but one of the offspring will receive the value from i without change and the other one will receive a value given by the SBX.0 0.0 0% -3.0 0.0 -1. Offspring histogram Offspring per rand. one notes that the parameter ηc is responsible for the concentration of the probability near the parent solutions.0 2.0 -2.0 0. 20% 3.2 0. High values of ηc showed slower convergence but more stability. This was carried as follows: if solution i has rank ( i ) ≤ relit .0 0% -3.0 5% -2.0 -1. The implemented crossover received an elite-preserving operator.2 0.0 offspring solution 3.0 eta_c = 1 15% 1.6 0.0 0.0 0% -3.8 1.0 5% -2. . good solutions should bypass the crossover.0 0. small values of ηc granted faster convergence for “simple” tests.0 10% -1.0 1.8 1.0 0.0 -2. no. To avoid the loss of good solutions due to disadvantageous crossover.0 20% eta_c = 2 15% 1.0 0.0 1.0 eta_c = 4 2.2 0.0 -1.0 2.4 3.0 3.4 3.81 with step of 0.0 0.6 0.0 2.0 10% -1.6 0.0 1.0 -3.1.0 20% eta_c = 4 15% 1.0 0.0 0. From it. where relit is given by the user.0 10% -1. 82 Differently from what is commonly found. The histogram step is 0. The histogram was illustrates the effect of ηm with a solution xi(1. Large value of ηm concentrates the result close to the initial value and small value of ηm spreads the outcome. Deb and Goyal [55] proposed the following polynomial function: ) yi(= xi( 1. but it is an additional feature.1) is calculated using the following equation: 1  ηm +1 − 1. the decision to use small values of ηm should be . (51) if ui ≥ 0. and: ( ) ( ) = ∆ max max xi( n .t +1 + ∆ maxδ i . (49) where yi(1. 5.5. i    if ui < 0.t +1) after the mutation. Parameter ηm affects the strength of the mutation.t +1) is the solution xi(1.t +1) 1.t +1) . the elite-preserving operator used is not on-off. n n (50) The parameter δ i is a perturbance factor corresponding to a random number ui ∈ ( 0.02. As can be seen.4.998 with 0. No study was conducted by the author to analyse if it is any better.t +1) = 0 and ∆ max = obtained by testing the mutation function for ui from 0.002 to 0.t +1) − min xi( n .4 Real-coded Polynomial Mutation Operator The real mutation operator used in the program follows the same idea as the SBX operator. u 2 ( )  i δi =  1 1 −  2 (1 − u )  ηm +1 . Figure 46 1 . but it allows for a probability of occurrence of elitism.001 step.5. 0 4.0 1.2 0.0 step: 0.4 0.0% -0.0 step: 0.4 2. the code generates “dat” files which are imported by an Excel sheet.0% step: 0.0 1.0 4.0 x1 = 0 delta = 1 0.4 0.2 x1 = 0 delta = 1 0.0% -0.0 -0.6 0.0% -1.83 accompanied by small probabilities of mutation.02 eta_m = 1 6.0 Figure 46 – Effect of mutation parameter ηm for x=0 and Δmax=1.8 1.0% -1.0% -1.0 4.0% -0.6 0.0 -0. To visualise the results from the program.02 6.2 0.5 1.8 0.8 0.0 eta_m = 4 0.2 0.8 1.0% -0.0% -0.0 -1.0% 0.4 0.4 0. which was prepared to provide a graphical interface. To exemplify the selection.2 0. mutation result histogram mutation per random number 1. .0% eta_m = 1 0.0% eta_m = 4 6.0 eta_m = 2 0.0% -1.02 -1.5 8.0% eta_m = 2 0. a simple 2-minimisation objective test function being “operated” is show in Figure 47.2 0.0% -0.4 2.8 0.4 0.5 mutated solutions 1.6 0. otherwise a high and undesired dispersion is observed.4 2.4 0.8 0.2 8. crossover and mutation operators using the graphical interface.2 0.0 0.0 -0.5 0.2 x1 = 0 delta = 1 0.0 0.8 random number 1. The Excel sheet created for visualisation is also capable of displaying an animation of the evolution of the population along the generations and the result after each operator for each generation.8 0.5 1.5 8.8 0.0 0. the REMOGA parameters were chosen to provide a quick convergence. new solutions are created as a result of pair-wise combination. the second row represents the fifth generation and the third row shows the sixth generation. the mutation takes place. The first row shows the objective space in the fourth generation. the selection operator effect is observed. but actually a duplication of good solutions takes place. Lastly. From the first column to the second column. From the second to the third column. from the third to the fourth column. Its effect is similar to the effect of the crossover in the sense that it further explores the objective space. as one notices the rapid migration of the solutions towards the Pareto optimal set.84 selection Objective space crossover Objective space mutation Objective space Objective space 10 Gen 8 4 f2 6 4 2 0 10 Gen 5 8 f2 6 4 2 0 10 Gen 6 8 f2 6 4 2 0 0 2 4 6 f1 8 10 0 2 4 6 f1 8 10 0 2 4 6 8 10 0 2 f1 4 6 8 10 f1 Figure 47 – Solutions behaviour after each of the implemented operators. One should note that this operator does not create any new solution. Visually it does only remove some nonoptimal solutions. the crossover operator effect is observed. . In this case. as a result of the tournament. Being an “easy” and known a priori optimisation problem. 2 (52) −5 ≤ xi ≤ 5 The convergence. .05.0 (from Eq. α 1. x2= ) x12 + x22 . a very simple test of a convex problem was carried only to check the functionality of the selection. as well as a discontinuous set. This test function was the same used to illustrate the steps of the algorithm in Figure 47. which shows the decision and objective spaces at the first.1 Convex 2-variable 2-objective test function The first MOOP tested in the program was the simple and convex problem: minimise  minimise   f1 ( x1 . Then a 2-objective optimisation problem with a non-convex Pareto. = = pelit 0. (53) is relatively fast and its evolution per generation is illustrated in Figure 48. 8) and= = pmut 0. using 120 individuals per generation and ηc 30. 5.5. was successfully tested.5 TEST FUNCTIONS The REMOGA program was verified using test functions found in the literature. two dials labelled “f1” and “f2” show the average of each objective function.= ηm 0.5.8. x2 ) = ( x1 + 2 ) + x22 . 10th and 100th generations. in Figure 48 one can also follow the behaviour of the decision space at the left. First. Nevertheless. At the right. f 2 ( x1 . crossover and mutation operators.85 5. : 10 Gen.86 Decision space Objective space 6 10 Gen.: 1 4 8 2 f2 x2 6 0 4 10 10 8 8 6 6 4 4 2 2 -2 -4 2 -6 0 0 -6 -4 -2 0 x1 2 4 0 6 2 4 6 8 f2 10 f1 Decision space Objective space 6 10 Gen. a total of 12.: 100 Gen.: 1 Gen.: 100 4 8 2 f2 6 x2 0 f1 0 4 10 10 8 8 6 6 4 4 2 2 -2 2 -4 0 0 -6 -6 -4 -2 0 x1 2 4 6 0 f1 0 2 4 6 8 f2 10 f1 Figure 48 – Testing a simple MOOP. Population in the 1st. Figure 49 shows this MOOP after 100 generations displaying every individual of each generation.: 10 4 8 2 f2 6 x2 0 f1 0 4 10 10 8 8 6 6 4 4 2 2 -2 -4 2 -6 0 0 -6 -4 -2 0 x1 2 4 0 6 2 4 6 8 f2 10 f1 Decision space Objective space 6 10 Gen. .000 individuals were evaluated. 10th and 100th generations. As 120 generations were used. x2 ) =− 1 exp  − ( x1 − 1) − ( x2 + 1)  .   2 2 f 2 ( x1 .05.: 100 10 8 6 f2 4 2 0 0 2 4 6 8 10 f1 Figure 49 – Simple convex test function after 100 generations.= pmut 0.0. = ηm 0. a non-convex Pareto is successfully obtained as shown in Figure 50. .5.= pelit 0. x2 ) =− 1 exp  − ( x1 + 1) − ( x2 − 1)  . 5. (55) and using a population of npop = 120 and 200 generations.: 120.2 Non-convex test function The second test was carried out with the non-convex 2-objective and 2-variable problem proposed by Fonseca and Fleming [56]: minimise   minimise    2 2 f1 ( x1 .5.   −4 ≤ xi ≤ 4 (54) Setting the program with the following parameters: = ηc 2.75. gen. = α 1.87 Simple test function pop. 5  = a = .  where  A1 = a (1.5  (56) Using the following GA settings for a population of npop = 120 and 500 generations .0 0.   A2 = a ( 2. 2 ) sin x2 + b ( 2.1) sin1 + b (1.8 0.1) cos x1 + a (1. b   .6 f2 0. 2 ) cos x2 .0 f1 Figure 50 – Non-convex test function from Fonseca and Fleming [56].1) sin x1 + b ( 2. but here it was multiplied by -1 to convert it into a minimisation problem): 2 2 minimise f1 ( x1 . gen.0   −1.5.0 −0. It has a non-convex and disconnected Pareto-optimal set and it is defined as follows (the actual problem is a maximisation one.3 Non-convex domain and disconnected Pareto set test function The test problem proposed by Poloni et al. 2 ) sin x2 + b (1. x2 ) =1 + ( A1 − B1 ) + ( A2 − B2 )  .   0.6 0. 2 ) cos x2 .0 0.: 120.1) sin x1 + b (1.1) cos x1 + a ( 2.: 200 1.   B1 = a (1.8 1.  B2 = a ( 2. 2 ) sin 2 + b (1.4 0.4 0.0 0. 2. [57] has been used by many researchers [48].5 2.5 1.2 0.1) sin1 + b ( 2. 2 ) cos 2. 2 ) sin 2 + b ( 2.    2 2  f 2 ( x1 . i = 1. 2 ) cos 2.0 −1.1) cos1 + a ( 2.88 non-convex test function pop. x2 ) = ( x1 + 3) + ( x2 + 1)  .  1.2 0. minimise    −π ≤ xi ≤ π . 5.1) cos1 + a (1.0   −2. The basic structure is maintained. but the fitness is associated with dominance.0. the reader was prepared to a more complex real-coded multiobjective program.6 SUMMARY OF THE CHAPTER This chapter provided the reader with the basic concepts of GAs and details of the program.= pmut 0. . Starting from definitions and proceeding with the essential ideas behind GA operators.8. = α 1.10.89 = ηc 200. which can be observed in Figure 51. The crossover and mutation operators are substituted for equations instead of bitwise operations. (57) results in two disconnected Pareto sets. = ηm 0. crossover and mutation.5. namely: selection. [57] test problem after 500 generations.= pelit 0. Poloni test function 30 objective 2 25 20 Pareto optimal regions 15 10 5 0 0 2 4 6 objective 1 8 10 Figure 51 – Poloni et al. 5. which was written to perform real-coded multi-objective optimisation. a generic grid node will be defined by the tangent to the blade edge (i) and the streamline (j). Illustratively. 6.90 6 METHODOLOGY The objective of this chapter is to provide the reader with the methods through which an initial compressor was optimised by coupling the SLC program and the real-coded elitist multi-objective genetic algorithm program (REMOGA) described in the previous chapter. i varies from 1 to 16 and j from 1 to 5. the preliminary design was chosen to be carried on with 5 streamlines and 15 rows. where 3 rows were dummy to simulate the inlet channel. The innermost streamline is j=1 and the outermost streamline is j=5.1 MODIFICATIONS IN THE SLC PROGRAM Initially. Modified SLCP File 1 SLCP input REMOGA output File 2 SLCP output REMOGA input REMOGA Figure 52 – SLCP and REMOGA coupling. The tangent to the blade edges i defines the rows given by Table 4. rotors and stators rows after one another to shape 5 stages and 2 final dummies to simulate the compressor outlet. . When recalling the SLCP. where the REMOGA acts reading the SLCP output and proposing inputs. the integration depicted in Figure 52. So. . calculation of square root of a negative number. several routines were reviewed to receive stop criteria and argument tests to avoid crashes. . nor infinite loops should exist in the program. In case of unfeasible solution. the integration should run without full human supervision. e.e. Target isentropic efficiency Calculated isentropic efficiency Temperature distribution De Haller number penalty Stator gas outlet angles Modified SLCP Camber angle penalty Hub to tip ratio Pressure ratio . i 1–2 2–3 3–4 4–5 5–6 6–7 7–8 8–9 Row Dummy inlet channel Dummy inlet channel Dummy inlet channel Rotor 1 Stator 1 Rotor 2 Stator 2 Rotor 3 i 9 – 10 10 – 11 11 – 12 12 – 13 13 – 14 14 – 15 15 – 16 Row Stator 3 Rotor 4 Stator 4 Rotor 5 Stator 5 Dummy compressor outlet Dummy compressor outlet To conduct the integration of the SLC program with the REMOGA program.. some specific modifications were required. Therefore. which receives a couple of input information and returns another group of output information. Figure 53 – SLC program acts as blackbox. Neither message boxes. but “bad” solution with regard to the objectives. As the REMOGA calls the SLCP many times. but never ‘crashes’ or stops without an output file.. if a stop criteria like the square root one is reached. i.. for any input.91 Table 4 – Compressor rows. requiring long hours of problem-seeking simulations followed by laborious debugging.g. This was a relatively time-consuming task. This was conducted so that the “bad” solution is killed in the optimisation process. the program actually returns a feasible. The SLCP was adapted to work as a blackbox. 1.05 seconds per compressor design and the PC.92 Additionally.68 seconds. To evaluate the compressor design with the original SLCP. tests were carried using a regular laptop computer and a modern personal computer. To evaluate the modified SLCP. time – original SLCP Av.1 Notebook Intel ® Core ™ 2 Duo P7450 2009 – 1st Quarter 2. .00 GB Windows 7 Professional 12. a dramatic reduction in computational time was achieved. To evaluate how much time could be saved by the aforementioned modifications in the SLCP. the PC required 12. whose configurations are show in Table 5. to gain computational performance.9 at 8. To conduct this performance test. Processor Processor release date Clock Cores / Threads Memory Operating System Av. the laptop required an average of 35. Hence. meanwhile.22 s SLCP output data or REMOGA input data The main output data of the SLC program when working together with the REMOGA program is a file named optimisation. the laptop required an average of 3.68 s 1.00 GB Windows 7 Home Premium 35. a test was conducted with ten thousand solutions and none required manual intervention.dat and is as simple as shown in Figure 54.34 seconds.40 GHz 4/8 16. the screen output was dramatically reduced. It is a 5-stage axial flow compressor with 85% isentropic efficiency and pressure ratio of 4. 1.05 s Modern Personal Computer Intel ® Core ™ i7-2600 2011 – 1st Quarter 3.13 GHz 2/2 4. the program was created in Console mode to avoid program termination boxes. Finally. Table 5 – Configuration of the computers used in the performance evaluation of the modified SLCP.2 kg/s mass flow rate.22 seconds. The original and the modified SLCP were executed 10 times in a row using a simple batch code. Then. time – modified SLCP 6.34 s 3. the original starting-point compressor design was chosen. as well as files writing. j =meanline. This file contains the objectives that the REMOGA has to minimise.1. 6. This limitation is carried on for every streamline. 0.1.1.69 − deHallerij ).93 Figure 54 – SLCP output data to work together with REMOGA program. the camber angle penalty. the parameter de Haller number penalty was defined as: iend pendH =∑ max ( 0.1. Thus.2 Camber angle penalty Similarly to the de Haller number. the four objectives were: 6. the camber angle should also be limited. the parameter to be minimised. In this work. was defined as: . To operate maximisation objectives they were converted to minimisation of the opposite value.1 • Maximise the calculated isentropic efficiency • Minimise the de Haller number penalty • Minimise the camber angle penalty • Maximise the pressure ratio De Haller number penalty To assess the compressor feasibility with regard to de Haller number. (58) i =ibegin where ibegin = 4 and iend = 13. 6. tip speed limit.2 SLCP input data or REMOGA output data The SLCP has two main input data. space to chord ratios. and • Hub-to-tip ratio. • Temperature weights (Tw ) distribution along the stages. The SLCP actually first reads the original input file and then immediately reads the second one. which contains more than 100 parameters to be chosen by the designer.1. mass flow rate. One can find the complete input file in Appendix C. • Stator air outlet angle for each stator. . The first step was a preliminary search and was executed with large ranges for the following 12 variables: • Target isentropic efficiency. The preliminary design optimisation was carried in two steps: search and refinement.94 penθ = iend jend ∑ ∑ =i ibegin =j jbegin min ( 0. θij − 40 ) (59) where ibegin = 4. The first one is the original parameters input. stator air outlet angles. Information such as ambient conditions. number of stages. modifications can be done only in the second file. what eases the handling of parameters. assuming no radial variation of this angle. The second input file is a selection of parameters from the original input file to be used in the REMOGA. So. blade profiles. aspect ratios. iend = 13. etc. jbegin = 1 and jend = 5. tip clearance. hub-to-tip ratio of the first row. have to be properly set. Moreover. Tw5 ∈ [ 0. 40°] (α 3 )5 ∈ [ 0°. but is clearly a not simple one.80.95 The refinement step used the history information from the search step to focus on regions.35] (α 3 )1 . Preliminary tests revealed that the SLC is very sensitive with regard to some parameters and incur in large number of unfeasible solutions.40. Tw4 . multiple selections help eliminating unfeasible solutions and rank-based elitism avoids the loss of converged solutions.1. 0. Tw2 .60] REMOGA SETTINGS The REMOGA program was set with conservative parameters.2 FORMULATION OF THE MOOP The MOOP formulation for the axial-flow compressor search is. the stator air outlet angle was allowed to vary linearly from hub to tip.3 ηcalculated pr pendH penθ 0. The concept for multiple selections and rank-based elitism in the REMOGA was idealised because of tests in which more than 60% of solutions were unfeasible. Tw3 .85 ≤ ηinput ≤ 0. (α 3 )4 ∈ [10°. 6. . as the objective space is unknown. (α 3 )2 . where promising solutions are located. Hence.90 (60) Tw1 . 20°] htr ∈ [ 0. then:  maximise maximise  minimise  minimise  subject to        6. totalling 17 design variables. (α 3 )3 . all stator air outlet angles were set as 25°.00 -0.55 and stagewise temperature rise weights distribution as shown Figure 56.4 HUMAN DESIGN START POINT The start point is a preliminary configuration of an engine for aeronautical application.05 0.96 6.10 0. Compressor nodes and streamlines . . Further details can be found in Appendix C.15 0.25 0.05 0. Firstly. the target isentropic efficiency was set as 85%.2 kg/s.10 -0. It is designed to produce 5 kN with a mass flow rate of 8. In summary.00 0.20 radius [m] 0. requiring a pressure ratio of 5:1.original design 0. Notice that it is a constant outer diameter project. hub to tip ratio of 0. The axial-flow compressor consists of five stages with no inlet guide vane (IGV) nor outlet guide vane (OGV).05 axial coordinate [m] 0.30 Figure 55 – Streamlines and nodes of the original compressor design.20 0.15 0. an overview of the streamlines and nodes distribution can be viewed in Figure 55.10 0. 00 20.05 25.95 15. from where a pressure ratio of 4.00 0. thereby meeting all remaining restrictions. the de Haller number . it is potentially feasible.20 200 0. On one hand.40 400 0. it could proceed to a thorough performance analysis and further detailing. Thus. the overall camber angle distribution is reasonable.00 Temperature [K] 0.00 0.00 1.85 5. but the 4th stator presents some high values.60 0 4 6 8 10 compressor row 12 Figure 57 – Pressure and temperature distributions of the original compressor design. The camber angles and the de Haller number distribution along the nodes are shown in Figure 58 and Figure 59.30 300 0.10 30.00 0. On the other hand. Pressure and temperature rise 600 pt1(i) Pressure [MPa] 0.00 1.50 500 tt1(i) 0.901 is observed. while the grey region indicates unsatisfactory regions.80 s1 s2 s3 s4 s5 S1 S2 S3 S4 S5 Figure 56 – Distribution of temperature rise weights along the stages. The white region indicates satisfactory regions. The pressure and temperature distribution of the human design is show in Figure 57.00 0.90 10. Although the resulting preliminary design does not satisfy all conditions.97 Stage temperature weights Stator outlet air angles 1.00 0. but not dramatic. As the de Haller number is considered only at the meanline.10 100 0. the light grey region indicates an indifferent region. camber angle . original solution .69.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0 5 10 15 20 25 30 camber angle [ deg ] 35 40 45 50 original solution . but also the first stator has an abnormally high values of de Haller number. The stage loading distribution shown in Figure 60 reveals that the majority of the nodes are within the recommended interval. Not only are there five rows in the meanline with de Haller numbers below 0. Moreover. 50 .camber angle . the 4th stator has a steep spanwise de Haller number distribution. but the outermost streamline concentrates low values of ψ and the innermost streamline of the first rotor has an abnormal high value.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0 5 10 15 20 25 30 camber angle [ deg ] 35 40 45 Figure 58 – Camber angle distribution of the original design.98 distribution is quite problematic. the number of blades in each row is shown in Figure 61.50 0.loading coefficient .40 loading coefficient [ .50 0.] 0.80 de Haller number [ .stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0. .70 de Haller number [ .90 1.60 0.30 0.60 0.10 original solution .00 1. original solution .10 0.99 original solution .rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.00 1.50 0.20 0.de Haller number .70 0.40 0.80 0.de Haller number .90 1. Finally.70 Figure 60 – Stage loading distribution of the original design.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.] 0.10 Figure 59 – De Haller number distribution of the original design.60 0. The total number of blades is 529.40 0.] 0. 53 1. Blade chord per row [cm] 3.0 3.5 0.77 1.0 4 5 6 7 8 9 10 11 12 Figure 62 – Blade chord of each row.60 1.0 2.5 2.0 1.5 1.5 3.35 0.25 2.100 Number of blades per row 100 80 60 40 20 0 4 5 6 7 8 9 10 11 12 13 # blades 20 29 40 47 54 59 64 67 66 83 Figure 61 – Number of blades per row The blade chord of each row is show in Figure 62. The values are all reasonable.69 2.97 1. 13 .56 1.18 1.65 1. 162 (30.437 (92. unique and limited solutions with regard to the penalties. Figure 63 illustrates the distribution of solutions in an Euler diagram.625 solutions (23.162 (30.15 ) ∧ ( penC ≤ 50 ) (61) leads to a final subset that contains feasible.8%) solutions contains the feasible solutions and another intersecting subset with 18.1 RESULTS AND DISCUSSION SEARCH: REMOGA HISTORY AND FILTERING OF SOLUTIONS The last REMOGA run aiming at optima compressors counted with 20.e.625 (23.064 solutions.1%) Figure 63 – Euler diagram representing the sets of feasible and unique solutions.2%) solutions contains the unique solutions.8%) Unique 18. The intersection of those sets. i. The history of those solutions can be seen in Figure 64 to Figure 65. Feasible 6. unique and feasible solutions delivers 4. From this set.1%). that are subject to ( pendH ≤ 0. in this smaller subset.101 7 7. . Looking for solutions.000 designs.437 (92. a subset with 6. This subset contains 3..2%) Intersection 4. 860 0 20 40 60 generation 80 100 Figure 64 – History of target efficiency History: hub to tip ratio 0.55 0.40 0.880 0.890 0.50 0.30 0 20 40 60 generation 80 100 Figure 65 – History of the hub to tip ratio.45 0.60 0.875 0.35 0.102 History: target efficiency 0.870 0. .885 0.900 0.895 0.865 0.905 0. 000 1.500 1.000 0.200 1.100 1.200 1.400 1.100 1.500 1.200 1.900 0.300 1.700 0 20 40 60 generation 80 100 History: temperature weight 3 0 20 40 60 generation 80 100 History: temperature weight 4 1.100 1.400 1.400 1.103 History: temperature weight 1 History: temperature weight 2 1.300 1.900 0.800 0.200 1.300 1.700 0.700 0 20 40 60 generation 80 100 Figure 66 – History of temperature weights distribution.000 1.400 1.800 0.900 0.500 1.100 1.300 1.500 1.000 0. 80 100 .800 0.700 0 20 40 60 generation 80 0 100 20 40 60 generation History: temperature weight 5 1.500 1.900 0.300 1.000 0.800 0.900 0.400 1.200 1.700 0.100 1.800 0. 0 25.0 15.0 15.0 10.0 25.0 10.104 History: angle S1 History: angle S2 30.0 5.0 20.0 15.0 20.0 0 20 40 60 generation 80 100 0 History: angle S3 20 40 60 generation 80 100 80 100 History: angle S4 30.0 30.0 25.0 5. .0 20.0 25.0 5.0 15.0 15.0 5.0 10.0 20.0 0 20 40 60 generation 80 100 0 20 40 60 generation History: angle S5 30.0 10.0 20.0 30.0 0 20 40 60 generation 80 100 Figure 67 – History of stator outlet angles distribution.0 25.0 10.0 5. indicated by the de Haller number penalty.3 4. Figure 69 indicates that the optimisation is capable of finding better solutions. A high stator air outlet angle hinders the proper functioning of the combustion chamber.5 0 10 20 30 40 Camber angle penalty 50 0 10 20 5th stage α3 [deg] 30 Figure 68 – Pressure ratio vs.1 3. Figure 69 shows that REMOGA designs present less potential problems with excess of diffusion.1 Pressure ratio 5. with a high angle of 25°. hence the choice to limit it in 20°.7 3.5 4. However.7 4.7 4.3 4. due to elevated swirl velocities.9 4. differently from the initial design.5 5.9 4. .3 5.7 3.105 7. Observing Figure 68 may lead to precipitated conclusions that the human design is undoubtedly better than the REMOGA solutions.9 3.1 4.3 4. Moreover. REMOGA designs REMOGA designs 5.1 Pressure ratio Human design 5.2 LOOKING FOR SOLUTIONS To look for better solutions.5 4.9 3. camber penalty and last stage stator outlet angle for the limited subset of solutions.5 3. an initial approach is to plot the objectives and check where the REMOGA results are located in comparison to the initial design.5 Human design 5. attention shall be conveyed to the fact that the air outlet angle of the last stator row is limited in 20° in the REMOGA designs. whose analysis is of interest.7 3.7 3.000 0. Let this solution be referred to as solution 1. REMOGA designs 5.100 0.9 3. Solution 2 has isentropic efficiency of 90% and pressure ratio of 4.3 4.3 Solution 1 Pressure ratio 5.3 4. “solution 2”).9 3.5 0 10 20 30 40 Camber angle penalty 50 Figure 70 – Solution 1.9 4.0.106 REMOGA designs REMOGA designs 5.9 4.1 Pressure ratio Human design Camber angle penalty 5.100 0. A solution.1 3.300 Figure 69 – The initial design is comparatively poor in satisfying de Haller number.1 3.5 4.5 4.7 4.5 0.3 4.300 0. is the one that is “at the right” of the human design solution (Figure 70).1 4. Another solution of interest it the one with the highest pressure ratio. .200 de Haller penalty 0.000 0. nor camber angle penalty (hereinafter.5 50 Human design 5. It has a similar pressure ratio and camber angle penalty and has a lower stator air outlet angle.7 4. but with no de Haller number penalty.200 de Haller number penalty 40 30 20 10 0 0.5 Human design 5. 00 5.00 0.00 0. Figure 72 shows the streamlines and nodes of solution 1.00 1. It is noticeable that the kink in the later stages does not exist anymore.60 0.00 10.60 10. but a kink in the initial stages is now present.20 1.00 0.00 1.80 0.60 1.3 ANALYSIS OF SEARCH STEP SOLUTIONS Solution 1 and solution 2 were obtained with the following temperature weights and stator outlet air angles: Solution 1 Stage temperature weights 1.00 d1 d2 d3 d4 d5 S1 S2 S3 S4 S5 Figure 71 – Input conditions for solutions 1 and 2.20 0. Similarly.00 Stator outlet air angles 20.00 0. stator outlet air angles of initial stages are lower and increase in later stages.40 0.107 7.40 1.00 0.40 Stator outlet air angles 25.20 0. 7. shown in grey.1 Overview To provide a first visual idea of the optima compressors geometries.3.00 15. the streamlines and nodes are displayed in red together with the original compressor.20 20.40 5. .00 0.80 15.00 0. Both solutions present low temperature weights in the last two stages.00 d1 d2 d3 d4 d5 S1 S2 S3 S4 S5 Solution 2 Stage temperature weights 1. 20 0.15 axial coordinate [m] 0.solution 2 0. neither in the inlet.10 0.30 Figure 73 – Nodes and streamlines of solution 2. Another advantageous aspect of both optima compressors is the smaller outer diameter than the original.108 Compressor nodes and streamlines . the optima solutions do also perform well in this aspect.05 0.25 0. Even though the frontal area was not an objective.05 0.10 0. Solution 2.30 Figure 72 – Nodes and streamlines of solution 1.05 0.00 -0.10 -0. nor in the outlet.20 radius [m] 0.20 radius [m] 0.05 0. Compressor nodes and streamlines .05 0.20 0. has no kinks.10 0. . whose streamlines are shown in Figure 73.00 -0.solution 1 0.10 0.00 0.05 0. since a reduced frontal area reduces the drag in aero engines.15 0.00 0.10 -0.15 0.25 0.15 axial coordinate [m] 0. 00 Temperature [K] 0.0 pt1(i) Pressure [MPa] 0.0 4 6 10 8 compressor row 12 Figure 74 – Pressure and temperature rise per row of solutions 1 and 2. both have less than 529 from the original design.0 0.50 500.0 0.40 400.0 tt1(i) 0.10 100.30 300.0 pt1(i) Pressure [MPa] 0.60 0.00 Temperature [K] 0.0 0. .0 0.20 200. This positive result came without setting the number of blades as an objective.40 400.109 The pressure and temperature rise along the stages obtained by the solutions in study are shown in Figure 74.solution 1 600.30 300. The number of blades per row and the blade chord per row are depicted in Figure 75 and Figure 76.0 4 6 8 10 compressor row 12 Pressure and temperature rise .0 tt1(i) 0. Solution 1 has a total of 507 blades and solution 2.0 0. a total of 515 blades.20 200.10 100.50 500.solution 2 600.0 0.60 0.0 0.0 0. Pressure and temperature rise . 49 1.00 2. with angles of ca.00 0.3.50 1.00 3. 45°.46 2.110 Blade chord per row [cm] Number of blades per row 80 60 40 20 0 4 5 6 7 8 9 10 11 12 13 # blades 18 29 32 41 46 53 60 71 78 79 4.68 9 1. solutions 1 and 2 perform better in this aspect.45 2.53 1.18 1.91 4 5 6 7 1.2 Camber angle While the original design presents high camber angles in the 4th stator.22 1.93 2.24 10 11 12 13 Figure 76 – Number of blades and blade chord of each row for solution 2. 7. but also has a smoother spanwise distribution.16 1.00 3.00 1. .50 0.00 0.35 1.40 1.94 2. Blade chord per row [cm] Number of blades per row 100 80 60 40 20 0 4 5 6 7 8 9 10 11 12 13 # blades 18 29 32 41 46 53 62 71 80 83 4.50 2.30 9 10 11 12 13 Figure 75 – Number of blades and blade chord of each row for solution 1.00 3. Solution 2 not only does have all camber angles inferior to 40°.50 2.00 2.00 3.00 1.50 3. as shown in Figure 77 and Figure 78.50 2.72 8 1.50 1.28 1.50 0.88 4 5 6 7 8 1.50 3. Solution 1 works slightly beyond the established limit of 40° in outermost streamlines of stators 2 and 5.47 2. camber angle .stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0 5 10 15 20 25 30 camber angle [ deg ] 35 40 Figure 78 – Camber angle distribution of solution 2.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0 5 10 15 20 25 30 camber angle [ deg ] 35 40 45 50 Figure 77 – Camber angle distribution of solution 1. 45 50 .camber angle .camber angle .rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0 5 10 15 30 25 20 camber angle [ deg ] 35 40 45 50 solution 1 . solution 2 .111 solution 1 .rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0 5 10 15 30 25 20 camber angle [ deg ] 35 40 45 50 solution 2 .camber angle . Solution 1 has an abnormal value for the last stator row at streamline 5.3 De Haller number The de Haller number was an issue in the original design. 7. Imposing the de Haller penalty as objective. Both solutions 1 and 2 have acceptable figures. both solutions face low values at streamline 5 and rotor 1 has a strange behaviour at streamline 1. meanline and values higher than 0. Solution 2 has a very well-behaved de Haller number distribution. However.e.4 Stage loading The stage loading distribution is better in solution 1.3. as there are values as low as 0.3..69. i. . as shown in Figure 81. Further investigation should be carried on to check whether this high loading is real or is the result of some numerical problem. the de Haller number at the meanline was successfully controlled. as shown in Figure 79 and Figure 80. The yellow rectangle highlights the region of interest for the de Haller number. as more nodes are located within the recommended range (white region).64 (meanline).112 7. 80 de Haller number [ .40 0.40 0. solution 2 .70 0.60 0.70 0.90 Figure 80 – De Haller number distribution of solution 2.50 0.40 0.de Haller number . 1.de Haller number .00 Figure 79 – De Haller number distribution of solution 1.90 1.00 .de Haller number .rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.50 0.] 0.] 0.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.] 0.40 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.80 de Haller number [ .70 0.60 0.00 solution 2 .90 1.80 de Haller number [ .60 0.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.50 0.00 solution 1 .90 1.de Haller number .113 solution 1 .60 0.70 0.] 0.80 de Haller number [ .50 0. 10 0.20 0.114 solution 1 .loading coefficient .60 0.loading coefficient .40 loading coefficient [ .] 0.4 REFINEMENT OF THE SEARCH SPACE As detailed in the previous chapter. the refinement formulation is given by (62): . the first run was a search procedure.30 0.30 0.50 loading coefficient [ .70 solution 2 .50 0. through the history of feasible solutions.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.20 0. Hence.] 0. This step revealed two interesting solutions. the ranges of promising design variable for a second optimisation run were determined. but more importantly.10 0. 7.70 Figure 81 – Stage loading distribution of solutions 1 and 2.40 0.60 0. 90 Tw1 . camber angle penalty from the refinement run. 20°] (α 3 )5h .9 4.1 3.10. resulting in 4372 (14%) feasible and unique solutions.7 Human design 3. Tw5 ∈ [ 0.1 4. 0. REMOGA designs 5.48. There is a good concentration of solutions with very little de Haller number penalty.115  maximise maximise  minimise  minimise subject to            ηcalculated pr pendH penθ 0.3 4. Filtering according to the same criteria from (61) 2857 (9. (α 3 )4t ∈ [15°.85] (α 3 )1h .15°] (α 3 )3h . (α 3 )4 h . as shown in Figure 83. (α 3 )1t .30] (62) Tw 4 .2%) solutions were analysed.87 ≤ ηinput ≤ 0. (α 3 )5t ∈ [15°.9 3. 0.5 4. Again. (α 3 )3t .3 Pressure ratio 5. the more relevant conflicting objectives are the pressure ratio and the camber angle penalty. The de Haller number is not a major issue in the solutions obtained. as shown in Figure 82.7 4.1. (α 3 )2 h . . Tw2 .5 0 10 20 30 40 Camber angle penalty 50 Figure 82 – Pressure ratio vs.52] In the refinement step. (α 3 )2t ∈ [10°.75. 31200 solutions were tested in 156 generations.18°] htr ∈ [ 0. Tw3 ∈ [1. 116 REMOGA designs REMOGA designs 5.5 50 Human design Camber angle penalty 5.3 Pressure ratio 5.1 4.9 4.7 4.5 4.3 4.1 3.9 3.7 40 30 20 10 Human design 3.5 0 0 0.1 0.2 de Haller number penalty 0.3 0 0.1 0.2 de Haller penalty 0.3 Figure 83 – De Haller numbers do also concentrate close to zero. Proceeding similarly to the search step, two solutions from the refinement step are going to be detailed. Solution 3 was chosen due to its high pressure ratio and proximity to the human design. It has a pressure ratio of 4.836 and isentropic efficiency of 87.2%. Solution 4 was chosen as the solution with the highest pressure ratio and no de Haller penalty nor camber angle penalty. This filter yields to a solution with 4.399:1 pressure ratio and 87.1% isentropic efficiency. REMOGA designs 5.3 Pressure ratio 5.1 Solution 3 4.9 4.7 4.5 4.3 4.1 3.9 3.7 Human design 3.5 0 10 20 30 40 Camber angle penalty Figure 84 – Choice of solution 3. 50 117 7.5 ANALYSIS OF REFINEMENT STEP SOLUTIONS The design parameters of solutions 3 and 4 are given in Figure 85. Solution 3 Stage temperature weights 1.40 Stator outlet air angles 20.00 1.20 hub 15.00 1.00 tip 0.80 10.00 0.60 0.40 5.00 0.20 0.00 0.00 d1 d2 d3 d4 d5 1 2 3 4 5 Solution 4 Stage temperature weights 1.40 Stator outlet air angles 20.00 1.20 15.00 1.00 hub tip 0.80 10.00 0.60 0.40 5.00 0.20 0.00 0.00 d1 d2 d3 d4 d5 1 2 3 4 5 Figure 85 – Stages temperature weight and stator air outlet angles. Solution 3 has a hub-to-tip ratio of 0.481 and solution 4, 0.489. 7.5.1 Overview The streamlines distribution is shown in Figure 86. Little visual difference is noticed between them. Similarly to solutions 1 and 2, the outer diameters are also smaller. A small kink is observed at approximately 0.02 m of axial distance, but no kink at the outlet. 118 Compressor nodes and streamlines - solution 3 0.20 radius [m] 0.15 0.10 0.05 0.00 -0.10 -0.05 0.00 0.05 0.10 0.15 axial coordinate [m] 0.20 0.25 0.30 0.25 0.30 Compressor nodes and streamlines - solution 4 0.20 radius [m] 0.15 0.10 0.05 0.00 -0.10 -0.05 0.00 0.05 0.10 0.15 axial coordinate [m] 0.20 Figure 86 – Streamlines of solutions 3 and 4. Pressure and temperature distributions along the rows are shown in Figure 87. Notice that they have a similar pattern, but solution 3 is slightly superior in every rotor row; this has a considerable impact in the overall pressure ratio of 4.836 vs. 4.399. 00 0.50 Pressure [MPa] Pressure ratio .3 Number of blades per row .0 0.0 0.solution 3 0.90 40 2.0 tt1(i) 0.0 0.00 1.40 1.30 1.0 Temperature [K] 1.50 2.0 tt1(i) 0.40 400.10 1.3 R4 S4 R5 S5 Figure 88 – Number of blades and blade chord of each row for solution 3. R2 S2 R3 S3 1.56 Blade chord per row [cm] . solution 3 totals 493 blades and solution 4 requires 511 blades.00 0.52 0 1.18 60 2.0 0.30 300.119 Pressure and temperature rise .50 3.71 20 1.20 200.10 100.00 0.40 400. 4 600.sol.60 1.31 S1 1.20 1.30 1.00 0.50 pt1(i) 0.47 80 2.0 0.50 1.0 0.50 pt1(i) 0.20 1.0 0.10 100.40 R1 S1 R2 S2 R3 S3 R4 S4 R5 S5 # blades 18 29 32 37 44 51 60 67 76 79 1.0 0.solution 4 500.0 0.24 cm.40 1.sol.98 100 4.00 2. rotor of the 5th stage – has a blade chord of 1.90 4 4 6 8 10 compressor row 5 6 7 8 9 10 11 12 13 12 Figure 87 – Pressure and temperature distribution of solutions 3 and 4.20 200. 3 600. The smallest one – solution 4.30 300.00 0.28 R1 1.50 Pressure [MPa] Pressure ratio .00 3. As Figure 88 and Figure 89 show.0 1.sol.00 3.90 4 4 6 8 10 compressor row 5 6 7 8 9 10 11 12 13 12 Pressure and temperature rise .0 Temperature [K] 1.60 500. . The blade chords are also within an acceptable range.sol.50 0.10 1. respectively.5.00 2. .86 40 2.50 0.36 R1 S1 R2 S2 R3 S3 R4 S4 R5 S5 # blades 18 29 32 41 46 53 62 71 80 79 1. Solution 4 presents a smoother almost linear variation from hub to tip (except stator 2 and 5).50 2.3 De Haller number De Haller numbers at the meanline are also above 0.94 100 3.27 S1 1.50 1.2 Camber angle Figure 90 and Figure 91 show the distribution of camber angles of solutions 3 and 4.4 R4 S4 R5 S5 Figure 89 – Number of blades and blade chord of each row for solution 4.4 Number of blades per row .sol.43 80 4. They are below the upper limit of 40° (except some nodes in solution 3).52 Blade chord per row [cm] . 7.120 R2 S2 R3 S3 1.00 0.00 3.67 20 1.13 60 2.5.69 (Figure 92 and Figure 93) Solution 3 has an abnormal value for the streamline 5 of stator 5. 7.50 3. which certainly is not real.00 2.00 1.24 R1 1.sol.48 0 1. camber angle .121 solution 3 . solution 4 .rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0 5 10 15 30 25 20 camber angle [ deg ] 35 40 45 50 solution 3 .rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0 5 10 15 30 25 20 camber angle [ deg ] 35 40 45 50 solution 4 .stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0 5 10 15 20 25 30 camber angle [ deg ] 35 40 45 50 Figure 90 – Camber angle distribution of solution 3.camber angle .camber angle . 45 50 .stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0 5 10 15 20 25 30 camber angle [ deg ] 35 40 Figure 91 – Camber angle distribution of solution 4.camber angle . 60 0.90 1.80 de Haller number [ .50 0.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0. 1.40 0.50 0.] 0.80 0.50 0.50 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.00 solution 4 .stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.de Haller number .40 0.90 1.80 0.] 0.de Haller number .70 0.40 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.00 solution 3 .de Haller number .80 0.90 Figure 93 – De Haller number distribution of solution 4.122 solution 3 .60 0.70 de Haller number [ .60 0.00 .de Haller number .00 Figure 92 – De Haller number distribution of solution 3 solution 4 .] 0.40 0.70 de Haller number [ .] 0.70 de Haller number [ .90 1.60 0. no noticeable improvement was noticed between the solutions from the search step to the solutions from the refinement step.loading coefficient .4 Stage loading Finally. Appendix D provides further graphical information from the analysed solutions. Eventually.5.40 loading coefficient [ .20 0.10 0. .40 0.50 0. solutions 3 and 4 meet similar results from solutions 1 and 2. but the former is more strict with the stator air outlet angle.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.60 0. This angle is limited due to restrictions in the combustion chamber. No major improvement from solutions 1 and 2 to solutions 3 and 4 is seen. But again.20 0.123 7.] 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.30 0. the stage loading are plotted in Figure 94.30 0. The compressor revealed to be very sensitive to this parameter. whose upper bound is 18° instead of 20°.70 Figure 94 – Stage loading distribution of solutions 3 and 4.70 solution 4 .60 0. solution 3 .loading coefficient .10 0. so that the human design has fewer restrictions to camber angle using the aforementioned angle in 25°. the outermost nodes are not very loaded.50 loading coefficient [ . However.] 0. 124 8 CONCLUSIONS A real-coded elitist multi-objective genetic algorithm (REMOGA) was written in FORTRAN to support the designer upon the choice of some axial-flow compressor parameters. The program was first examined with test functions from the literature. It successfully passed through convex, non-convex and discontinuous Pareto-sets. Then it was coupled to the SLCP. Initial tests with the REMOGA working together with the SLCP demanded further modifications in the former to handle a high level of unfeasible solutions. Afterwards, it was noticed that pushing harder with the limit of the air outlet angle of the last stator complicates the search for feasible solutions. In the original design, this angle was 25° and the optimised solutions were obtained with 18°. This new limit is realistic due to requirements of low swirl velocities prior to the combustion chamber. The last version of REMOGA evaluated 20,000 designs in an initial search and found solutions whose indicators of viability were better than those from the initial design proposed through manual trial and error, long experience and intuition. Among a selected subset of solutions, two have been selected to be further analysed. The search step did also reveal the bounds, wherein good solutions are located. Thus a refinement step followed. Not only were the design parameters bounds restricted, but also the angles of each row were allowed to have a linear variation from hub to tip. Moreover, the stator air outlet angle was restricted to maximum 18°, instead of 20° from the search step. Similar results from the search step were found, but meeting tougher restrictions. 125 Eventually, four solutions (two from the search and two from the refinement) were analysed. Although they revealed to be consistent in terms of de geometry, number of blades, de Haller number, camber angle, etc., none reached the pressure ratio of 5. The procedure developed in this work revealed that the preliminary design of an axial-flow compressor can be optimised thanks to the full automation and inherent intelligence of the developed multi-objective genetic algorithm. More than 100,000 designs were evaluated during this work and this took just some weeks, while a non-automated procedure would require some minutes per design and analysis. If the manual process of deciding upon design variables, running the SLCP and analysing the results takes around 30 minutes for each design, then the 100,000 evaluations would require 50,000 hours, or 2083 days of non-stop work. In this sense this work contributed to the preliminary design of axialflow compressors. 126 9 FURTHER WORK Throughout the development of this work, ideas for further development naturally come. Some are related to improvements and others to works that can be derived from the obtained results. 9.1 IMPROVEMENTS The REMOGA, as presented, was not conceived at once. It suffered many changes throughout the integration with the SLCP. The main changes were related to the high error level of the SLCP. Nevertheless, time is limited and some ideas or models to improve the algorithm were not tested. The first one is the normalisation of the fitness, so that the average fitness is kept constant though the generations. This might improve the efficacy of the other selection operators. The second REMOGA proposed improvement is the treatment of unfeasible solutions. At the present moment, unfeasible solutions receive penalised objective values, in a way that they would be killed in the selection operation. However, when working with high level of errors, the tournament selection can create disputes between two identical penalised solutions, passing a penalised solution to the next generation. A suggestion for this aspect is to substitute penalised solutions by existing feasible solutions with low rank. It may have a 127 similar effect of the selection operator. the next natural step is the deep investigation of them. but grants elimination of unfeasible solutions and the crossover is.2.2 9.2. Investigations can also be carried on with commercial programs. 9. A limiting issue of the REMOGA is that it runs one design per time and does not benefit of multiple threads existing in modern processors.2 Robust optimisation Existing cooperation in turbomachinery optimisation research between ITA and Universidade Federal de Itajubá (UNIFEI) can use the procedure developed in this work to conduct further optimisation studies in robust optimisation. 9. Existing in-house design-point and off-design point performance programs can be used to assess the feasibility of the obtained compressors. so that there is no mixing of input and output files.1 SUGGESTION OF WORKS Detailed project After some optimum preliminary design solutions were found. A relatively simple approach would be the execution of a batch program. then more effective for the generation of new solutions. This would be an enriching further study to be conducted in the axial-flow compressor design. which manages the executable files in different folders. like Concepts NREC. in order to understand the impact of input variables that are given by probability distributions. . which are themes of research at UNIFEI. 3.. In: ENCONTRO DE INICIAÇÃO CIENTÍFICA E PÓSGRADUAÇÃO DO ITA. Large eddy simulation of transonic flow field in NASA Rotor 37. LEE. R.-Y. Proceedings. 2000. KIM... Proceedings. Brasília: [s. Um Programa para Desenvolvimento da Capacidade Nacional em Turbinas a Gás. 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K.. and machine learning. 108p. optimization. K. Thesis (Bachelor in Science in Aeronautical-Mechanical Engineering) Instituto Tecnológico de Aeronáutica. 50. COHEN.. Hybidization of a multi-objective genetic algorithm. 1996. V. 53. p. M.. 51. No. London. 3687). Acesso em 3 de nov. FONSECA. (IITK/ME/SMD-94027). 1989. G. Reading: Addison Wesley. Evolutionary Computation Journal.Swiss Federal Institute of Technology Zürich. Gas turbine theory. 2010.ieee. F.. Multi-objective evolutionary optimization of gas turbine components. F. São José dos Campos. 3. 1972. KHOR. DEB. 3. E. & M. POLONI. 2001. n. 55. Seoul.. C. DEB. 59. Zürich: [s. 403420. Genetic algorithms for multiobjective optimization: formulation. v. 133 APPENDIX A SLC SUMMARY The detailed derivation can be found in the PhD thesis of Barbosa [42] and in the Report of Frost [58]. respectively. (65)  U = ω r zˆ . From Equations (63) to (66). (63) Cm1 Cm 2 U1 Vw1 Vw 2 Cw1 U2 Cw 2 Figure 95 – Velocity triangles. Detailing in cylindrical coordinates:  C = Cr rˆ + Cθ θˆ + Cz zˆ . θ and z indicate the radial. . This section aims at providing basic guidance to the reader who is not familiar with the method. (64)  V =Vr rˆ + Vθ θˆ + Vz zˆ . From the velocity triangles:    C= V + U . tangential and axial components. (66) where r . 134 C r = Vr . a mathematical representation of a steady-state nonviscous flow in cylindrical coordinates rotating about a fixed axis is obtained:       ∂V Wθ ∂V ∂V    P Cr − ∇ = + + Cz + ω × (ω × r ) + 2ω × V + F . θ (68) Cz = Vz . Dt ρ 1 where (70) D ( ⋅) is the material derivative. r ρ ∂θ ∂r r ∂θ ∂z r (74) ∂Cz Vθ ∂Cz ∂Cz 1 ∂P . ρ ∂r ∂r ∂z r ∂θ r − (73) ∂V V ∂V ∂V C V 1 ∂P = Cr θ + θ θ + Cz θ + r θ + 2ωCr . (67) C= Vθ + ω r . r ∂θ ∂r ∂z ρ 1 (72) In terms of radial. (69) For a non-inertial coordinate frame rotating with the rotor at a angular velocity ω . = Cr + + Cz ∂r ∂z ρ ∂z r ∂θ (75) − . tangential and axial components and noticing that    ω × (ω × r ) = −ω 2 r rˆ : ∂Cr Vθ ∂Cr ∂Cr Vθ2 1 ∂P − = Cr + + Vz − − ω 2 r − 2ωVθ . whence: Dt     DV ∂V Vθ ∂V ∂V = Cr + + Vz . the inviscid and steady-flow equation of motion is     DV    − ∇P= + ω × (ω × r ) + 2ω × V + F . Dt r ∂θ ∂r ∂z (71) Combining (70) and (71). = Cm Cr2 + Cz2 . ∂m Cm (81) Cz ∂z = cos = ε . (79) The streamline curve can be set as a function of r and z. Besides that.. ∂z Cz (83) Then Equation (80) may be written as following: ∂ ∂ ∂ = sin ε + cos ε . or ∂m ∂r ∂z ∂ ∂ ∂ Cm= Cr + Cz . ∂m ∂m ∂r ∂m ∂z (80) From Figure 22. z ) . (78) Cz = Cm cos ε . i.e. ∂m Cm (82) Cr ∂r = tan = ε . hence: ∂ ∂r ∂ ∂z ∂ = + . ∂m ∂r ∂z (84) (85) . the meridional velocity is defined as:    C= C m r + Cz . (76) (77) As the streamline is tangent to the velocity and with aid of Figure 22: Cr = Cm sin ε . along the streamline it follows: Cr ∂r = sin = ε . m = m ( r .135 The axisymmetry permits to drop differentiation with respect to the tangential coordinate. Now.  Cr ∂m Cm ∂r Cm ∂z ∂m Cm  ∂r ∂z  (87) ∂ ( rCθ ) ∂C ∂r .(75).136 Moreover: ∂Cr Cr ∂Cr Cz ∂Cr ∂Cr ∂C 1  ∂Cr = + ⇒ = + Cz r  Cr ∂m Cm ∂r Cm ∂z ∂m Cm  ∂r ∂z  . Therefore. z ) . ∂r ∂z ∂m ρ ∂r ρ ∂r r r ∂Cθ ∂C C V Cm ∂ ( rCθ ) . so: ∂ ∂r ∂ ∂z ∂ = + ∂s ∂s ∂r ∂s ∂z (93) . an analogous procedure is conducted by noticing that s = s ( r . which is along the blade edge.  Cr Cm  r  ∂m ∂r ∂z (89) Having in mind that Cθ= U + Vθ and from (86) . 0 + Cz θ + r θ ⇒ = ∂r ∂z r r ∂m (91) ∂C z ∂C z ∂C z 1 ∂P 1 ∂P = Cr + Cz ⇒− = Cm ∂r ∂z ∂m ρ ∂z ρ ∂z (92) 0 Cr = − (90) The right-hand side of Equations (90)-(92) are written in terms of the streamline mcoordinate. one gets: ∂C ∂C C 2 ∂C C 2 1 ∂P 1 ∂P − = Cr r + C z r − θ ⇒ − = Cm r − θ .(89) substituted into (73) .  (86) ∂Cz Cr ∂Cz Cz ∂Cz ∂Cr ∂C  1  ∂Cz = + ⇒ = + Cz z  . the left-hand side should be written in terms of the s-coordinate. = r θ + Cθ ∂m ∂m ∂m (88) Substituting (81) and (85) into (88): ∂ ( rCθ ) ∂C C C  r  ∂Cθ = + CZ θ + r θ  .  ∂ε tan ( ε + γ )  − sec ( ε + γ ) + +  C ∂Cm   Rc ∂s Cm = . after laborious mathematical manipulations.137 Recalling Figure 22 and that the angle γ is defined as the angle between the r-axis and s-axis: ∂r = cos γ ∂s (94) ∂z = sin γ ∂s (95) ∂ ∂ ∂ = cos γ + sin γ ∂s ∂r ∂z (96) Eventually.        (97) where. 2  ∂m (1 − M m )  1 ∂S 1  2 + R ∂m − (1 + M m ) r sin ( ε )  2 m (98) Equations (97) and (98) form a system of partial differential equations. Barbosa [42] obtains:   cos ( ε + γ ) tan ( β ) ∂β −   cos 2 ( β ) ∂s R c     2U cos ( β )  + − tan ( β ) ∂Cm ∂S  r   ∂I 2 2  Cm + Cm  Cm= cos ( β )   − T ∂s ∂s    ∂s  − tan 2 ( β ) cos ( γ ) +   r     + sin ( ε + γ ) ∂Cm   ∂m Cm   +    . . The system can be solved if it is previously know that flow properties vary smoothly at the blade edges. nobje.Simulated Binary Crossover (SBX) from DEB and Kumar (1995) ! .138 APPENDIX B B.nvare.Crowded Tournament Selection ! .Elitism probability within SBX ! .Polynomial Mutation.Sharing function based on pseudo-averaged edge hypercube ! .imax. Eng.nrank. DEB and GOYAL (1996) !--------------------------------------------------------------- PROGRAM main INCLUDE "zcommon.nfit nvarb=2 nvare=1+nvar nobjb=nvare+1 nobje=nobjb+nobj-1 nrank=nobje+1 nfit=nrank+1 !Initialise generation gen=1 !'Randomise' the random operator seed_time=TIME()-1318600000 CALL SEED(seed_time) .1 OPTIMISATION PROGRAM MAIN PROGRAM !--------------------------------------------------------------!REAL-CODED ELITIST MULTI-OBJECTIVE GENETIC ALGORITHM !--------------------------------------------------------------!--------------------------------------------------------------!developed by VICTOR FUJII ANDO.Shared fitness evaluation ! .jmax seed_time !population comumns INTEGER :: nvarb.ITA 2011 !18th of October 2011 !--------------------------------------------------------------!--------------------------------------------------------------! Description: ! .nobjb. MSc Aer-Mech. .f90" REAL*8 REAL*8 REAL*8 INTEGER INTEGER(4) :: :: :: :: :: var1 mem1(20) objective i.j. nsel .nvar=12.ngen !Evaluate Objective Functions CALL eval_objective !Assign fitness by rank CALL fitness !Write parent population prior selection CALL write_parent !Selection DO i=1.139 !Read initial design parameter values CALL read_ini !Open and write head of output files CALL open_files DO gen=1.rankel.2 GLOBAL VARIABLES !zcommon.totcol=nvar+nobj+3 INTEGER :: obj_selec.f90 USE IFPORT INTEGER.ngen.nsel CALL selection ENDDO !Write parent population after selection CALL write_parent !Cross-over CALL cross_over !Write offspring population prior mutation CALL write_offspring !Mutation CALL mutation !Write offspring population after mutation CALL write_offspring !Parent receives offspring parent=offspring ENDDO ENDPROGRAM main B.PARAMETER :: npop=200.gen.nobj=4. gen.nvarb+ 6).nobjb.offspring.file="GA_parameters.parent(i.parent(i.f90" INTEGER :: i !population comumns INTEGER :: nvarb.nvarb+ 2).nsel.offspring(npop.totcol) REAL :: upper_var(nvar).nvarb+ 5).parent(i.nfit nvarb=2 nvare=1+nvar nobjb=nvare+1 nobje=nobjb+nobj-1 nrank=nobje+1 nfit=nrank+1 OPEN(3. !18th of October 2011 !-------------------------------------------------------- SUBROUTINE read_ini INCLUDE "zcommon.ngen.nvarb+ 9). the GA parameters and the ! variable limits ! !-------------------------------------------------------!developed by VICTOR FUJII ANDO.parent.nvarb+11) parent(i.& &parent(i.nvarb+ 8).nvarb+ 4).p_mut.npop parent(i.eta_m.1)=REAL(i) READ(3.nrank.parent(i.& &parent(i.140 REAL :: parent(npop.3 & & READING INITIAL POPULATION AND PROGRAM PARAMETERS !-------------------------------------------------------! ! Subroutine to read the initial user-defined ! population. & eta_c.parent(i.90 .alpha_sh.file="ini.nvarb) ENDDO !Define variable limits !effisen_given lower_var(1)=-0.p_elit COMMON obj_selec. MSc Aer-Mech. Eng.& &parent(i.*) parent(i. & upper_var.parent(i.nvarb+ 0).nvarb+ 3).totcol).nobje.dat") OPEN(4.nvare.nvarb)=-parent(i.nvarb+10).eta_m.nvarb+ 7).lower_var(nvar) !Genetic Algorithm parameters REAL :: eta_c.lower_var.rankel.p_elit B.p_mut.parent(i.parent(i.dat") !Read initial population from an external file DO i=1.alpha_sh.nvarb+ 1). 80 upper_var(4)=1.*)p_elit !To simplify calculations: !redefine Crossover parameter "eta_c" eta_c=1/(1+eta_c) !To simplify calculations: !redefine Mutation paramenter "eta_m" eta_m=1/(eta_m+1) ENDSUBROUTINE read_ini .35 !temp dist 4 lower_var(5)=0.35 !temp dist 2 lower_var(3)=0.80 upper_var(6)=1.*)ngen READ(4.S3 lower_var(9)=10 upper_var(9)=40 !alfa2 .80 upper_var(3)=1.141 upper_var(1)=-0.*)rankel READ(4.40 upper_var(12)=0.*)p_mut READ(4.*)eta_m READ(4.S5 lower_var(11)=0 upper_var(11)=20 !hub-tip ratio lower_var(12)=0.S4 lower_var(10)=10 upper_var(10)=40 !alfa2 .S1 lower_var(7)=10 upper_var(7)=40 !alfa2 .35 !alfa2 .*)obj_selec READ(4.85 !temp dist 1 lower_var(2)=0.*)nsel READ(4.*)alpha_sh READ(4.60 !Read GA parameters READ(4.35 !temp dist 3 lower_var(4)=0.35 !temp dist 5 lower_var(6)=0.80 upper_var(5)=1.S2 lower_var(8)=10 upper_var(8)=40 !alfa2 .*)eta_c READ(4.80 upper_var(2)=1. ' dist5' !Pressure ratio !Isentropic !itype dist 1=temp !number of weights !dist1 !dist2 !dist3 !dist4 !dist5 . 2011 ! Use result from axial design program ! !Revision 2: 21st Nov.nvarb+4).count nvarb. !18th of October 2011 !Last update: 21th Nov 2011 !-------------------------------------------------------!Features ! 1.*)'1' 2=pressure WRITE(5.B2 LOGICAL(4) result nvarb=2 nvare=1+nvar nobjb=nvare+1 nobje=nobjb+nobj-1 !Streamline Curvature Program objectives IF (obj_selec.' dist1' WRITE(5.*)parent(i.dat') WRITE(5.nvarb+5). 2011 ! Selection of objectives using obj_selec !-------------------------------------------------------- SUBROUTINE eval_objective INCLUDE "zcommon.' eff' efficiency given WRITE(5. discontinuous front test function ! 4.nvarb+1).nobje int_time char_time x1.' dist3' WRITE(5.EQ.*)parent(i. A 2-objective non-convex front test ! 3.*)'5' WRITE(5.nvarb+3). Simple 2-objective convex front test ! 2.*)parent(i.file='opt_input.nvarb) OPEN(5.142 B.*)'5' WRITE(5.nvare.4 EVALUATING OBJECTIVES !-------------------------------------------------------! ! Subroutine to evaluate the objective functions ! !-------------------------------------------------------!developed by VICTOR FUJII ANDO. Eng.1)THEN DO i=1.*)parent(i.B1.j. from an external program !-------------------------------------------------------!Revision 1: 08th Nov.A2.f90" INTEGER INTEGER INTEGER(4) CHARACTER*8 REAL REAL :: :: :: :: :: :: i.' dist2' WRITE(5.x2 A1. MSc Aer-Mech. User-defined.nvarb+2).nvarb). Poloni et al.npop parent(i.*)parent(i.' dist4' WRITE(5.nvarb)=-parent(i.*)parent(i.nobjb. S3 WRITE(5.S1 WRITE(5.parent(i.00' WRITE(5.*)'1.parent(i.*)'1.*)'2.I3.S4 WRITE(5.*)'2.87' WRITE(5.*)'0.parent(i.*)'0.nvarb+ 8).88' WRITE(5.' time: '.00' WRITE(5.89' WRITE(5.00' WRITE(5.parent(i.*)'0.nvarb+ 9).' !alfa2 hub .59' WRITE(5.91' WRITE(5.' htr' WRITE(5.nvarb+ 9).*)parent(i.parent(i.*)'0.*)'2.00' WRITE(5.79' WRITE(5.nvarb+10).*)'0.81' WRITE(5.parent(i.*)'2.*)'2.*)parent(i.*)'0.95' WRITE(5.*)'3.' !alfa2 hub .*)'1.file='optimisation.*)parent(i.00' WRITE(5.nvarb) .nvarb+10).00' WRITE(5.parent(i.*)parent(i.nvarb+9).parent(i.nvarb+ 7).' !alfa2 hub .S3' alfa2 hub .S5 WRITE(5.85' WRITE(5.*)'1.*)'1.10' WRITE(5.59' WRITE(5.& &parent(i.*)parent(i.nvarb+ 7).*)'0.parent(i.*)parent(i.88' CLOSE(5) alfa2 hub .nvarb+8).10)gen.nvarb+ 6).<nvar>F10.S1' alfa2 hub .nvarb+ 2).93' WRITE(5.143 WRITE(5.00' WRITE(5.*)'2.*)'2.3.83' WRITE(5.*)'0.*)'0.nvarb+ 6).' !alfa2 hub .*)'2.00' WRITE(5.nvarb+ 3).parent(i.*)'2.parent(i.nvarb+ 8).11)parent(i.*)'3.dat') .*)'2.nvarb) result=SYSTEMQQ('axial_dp') OPEN(6.' n = '.00' WRITE(5.59' WRITE(5.nvarb+5).00' WRITE(5.A8) WRITE(*.parent(i.nvarb+ 6).*)'0.00' WRITE(5.parent(i.*)'0.' !alfa2 hub .S5' !hub !s/c !s/c !s/c !s/c !s/c !s/c !s/c !s/c !s/c !s/c !s/c !s/c !s/c !s/c !s/c !s/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c !h/c tip ratio D1 D2 D3 R1 S1 R2 S2 R3 S3 R4 S4 R5 S5 D1 D2 D3 D1 D2 D3 R1 S1 R2 S2 R3 S3 R4 S4 R5 S5 D1 D2 D3 int_time=TIME() CALL TIME(char_time) WRITE(*.*)'1.I3.00' WRITE(5.nvarb+1).5) parent(i.3.S4' alfa2 hub .nvarb+11) 11 FORMAT (' '.S2' alfa2 hub .i.00' WRITE(5.*)'3.nvarb+10).char_time 10 FORMAT (' gen = '.00' WRITE(5.nvarb+ 7).nvarb+11).*)'1.00' WRITE(5.S2 WRITE(5.*)'0.88' WRITE(5.nvarb+4).nvarb)=-parent(i.& &parent(i.parent(i.00' WRITE(5. )+1.j)=(2+x1)**2+x2**2 IF(obj_selec.0*COS(1.nobjb+1).j)=(1+(A1-B1)**2+(A2-B2)**2) ENDIF IF (j.2) THEN parent(i.j)=1-EXP(-(x1+1)**2-(x2-1)**2) IF(obj_selec.2) x2=parent(i.nobjb+2).5*SIN(1.*)' out: --------.0*SIN(2.nobjb)=0 parent(i.0*SIN(x2)-0.fail ---------' ELSE WRITE(*.2)parent(i.AND.3)parent(i.20).nobjb) parent(i.EQ.nobjb+3) IF (count .EQ.LT.5) THEN IF(obj_selec.) B1=0.parent(i.)-2.nobjb)=-parent(i.EQ.5*COS(x2) ENDIF IF (j.nobjb+3)=0 WRITE(*.) A2=1.3) IF(obj_selec.0*SIN(2.nobje x1=parent(i.nobj)) READ(6.5) ENDIF CLOSE(6.5*COS(x2) B2=1.nobjb+count) count=count+1 ENDDO parent(i.j)=x1**2+x2**2 IF(obj_selec.0*COS(x1)+1.)-1.NOT.EQ.nobjb). EOF(6)).parent(i.nobjb+3)=-parent(i.0*COS(x1)+2.5*SIN(x1)-2.EQ.EQ.status='delete') ENDDO ENDIF !------------------------------------------------------------!Test functions !------------------------------------------------------------! Simple ! Fonseca and Fleming ! Poloni !------------------------------------------------------------IF(obj_selec.0*COS(1.5*COS(2.1)THEN DO i=1. 4) THEN IF(obj_selec.LE.)+2.0*SIN(x2)-1.4)THEN A1=0.5*SIN(1.EQ.144 count=0 DO WHILE ((.3)parent(i.2)parent(i.(count.)-0.EQ.4)parent(i.5*SIN(x1)-1.nobjb+2)=600 parent(i.<nobj>F10.npop DO j=nobjb.nobjb+1)=500 parent(i.j)=1-EXP(-(x1-1)**2-(x2+1)**2) IF(obj_selec.parent(i.*)parent(i.)-1.5*COS(2.no bjb+3) 20 FORMAT(' '.j)=((x1+3)**2+(x2+1)**2) ENDIF ENDDO .NE.4)parent(i.parent(i.EQ. SHARED FITNESS and RANK ! !-------------------------------------------------------!developed by VICTOR FUJII ANDO. Eng.nobje.j).(flag2.npop rank=1 DO i=1.j)) flag2=1 ENDDO IF ((flag1.5 FITNESS SUBROUTINE !-------------------------------------------------------! ! Subroutine to perform the calculation of ! FITNESS.swap.0).LT. Assignement of rank of each solution ! 2.n.j).NE. !18th of October 2011 !-------------------------------------------------------!Structure ! 1.1)) rank=rank+1 .EQ.memo_real.memo_int flag1.mu(npop) memo(totcol).nrank.flag2.sum niche(npop) INTEGER :: nvarb.j)) flag1=1 IF (parent(i.f90" INTEGER INTEGER REAL REAL :: :: :: :: i.parent(n.nvare.nfit nvarb=2 nvare=1+nvar nobjb=nvare+1 nobje=nobjb+nobj-1 nrank=nobje+1 nfit=nrank+1 ! Assign rank DO n=1.GT. Definition of array mu(n).rank.npop IF (n . which gives the number ! of solutions with rank n ! 4.145 ENDDO ENDIF !------------------------------------------------------------- ENDSUBROUTINE eval_objective B.nobje IF (parent(i. MSc Aer-Mech. Calculation of Fitness by the fitness-averaging ! method !-------------------------------------------------------- SUBROUTINE fitness INCLUDE "zcommon.AND.EQ.j.parent(n. Sorting of the population using simple BubbleSort ! 3. i) THEN flag1=0 flag2=0 DO j=nobjb.nobjb. 5*(memo_real-1) ELSE memo_int=INT(parent(n.146 ENDIF ENDDO parent(n.npop rank=parent(n.1) THEN parent(n.:)=parent(n. inclusive memo_real=REAL(mu(INT(parent(n.nrank)-1) DO j=1.nfit)/(niche(n)+1) ENDDO .nrank).:) parent(n+1.:)=memo swap=1 ENDIF ENDDO ENDDO !Calculate the array 'mu' of number of solutions for each rank mu=0 DO n=1.nrank)) THEN memo=parent(n+1.nfit)=REAL(npop)-0.nfit)=parent(n.memo_int sum=sum+(REAL(mu(j))-0.EQ.npop parent(n.(npop-1) IF(parent(n+1.LT.npop sum=0 !Memo_real represents the number of solutions with rank !equal to the rank of parent n.parent(n.5*(memo_real-1)) ENDDO parent(n.nrank) mu(rank)=mu(rank)+1 ENDDO ! Assign fitness DO n=1.1) swap=0 DO n=1.nrank)=rank ENDDO !\Assign rank !Sort the parent according to the rank !Bubblesort algorithm swap=1 DO WHILE (swap.niche) ! Calculation of Shared fitness DO n=1.nrank).:) parent(n.nrank)))) IF (parent(n.nfit)=REAL(npop)-sum ENDIF ENDDO !\Assign fitness ! Definition of Niche-Count array CALL niche_count(mu.EQ. :)=memo swap=1 ENDIF ENDDO ENDDO ENDSUBROUTINE fitness B.npop) !Sharing function REAL :: sharing(npop.EQ.:) parent(n.parent(n.:)=parent(n.1 Niche count subroutine !-------------------------------------------------------! ! Subroutine to perform the calculation of ! NICHE COUNT ! !-------------------------------------------------------!developed by VICTOR FUJII ANDO. Eng. Calculation of Sharing function ! 4.nfit)) THEN memo=parent(n+1.i. Dynamic update of Sigma share ! 3. Calculation of nomalised distance ! 2.5.npop) !Niche count REAL :: niche(npop) !Sharing parameter . Calculation of Niche count !-------------------------------------------------------SUBROUTINE niche_count(mu.j.auxi1.:) parent(n+1. !18th of October 2011 !-------------------------------------------------------!Structure ! 1.GT.niche) INCLUDE "zcommon.(npop-1) IF(parent(n+1.nfit).1) swap=0 DO n=1.auxi2 INTEGER :: mu(npop) !Normalized Euclidian distance between solutions REAL :: distance(npop.dynamic update REAL :: sigma_sh .f90" INTEGER :: n. MSc Aer-Mech.147 !Sort the parent according to the fitness !Bubblesort algorithm swap=1 DO WHILE (swap. npop IF ((parent(n.delta_min) delta_min=aux3 IF (aux3.nobje aux1=MAXVAL(parent(:.sum REAL :: delta_min.sigma_sh) THEN sharing(n.0.nrank)).aux3.j)) aux3=aux2-aux1 sum=sum+aux3 IF (aux3.aux2.i)=0 ENDIF ENDDO ENDDO !\Calculation of sharing function .npop DO i=1.npop DO i=1.148 !Auxiliary variables REAL :: aux1.nvare.nobjb)) sum=0 DO j=nobjb.j))-MINVAL(parent(:.nobjb)) delta_max=MAXVAL(parent(:.001) THEN sum=0 DO j=nobjb.LT.LE.j)) aux2=parent(n.j)-parent(i.nobjb))-MINVAL(parent(:.i).nobjb))-MINVAL(parent(:.nobje.i)/sigma_sh)**alpha_sh ELSE sharing(n.LT.nrank.j)) aux2=MAXVAL(parent(:.nobjb.nobje aux1=MINVAL(parent(:.i)=sum ENDIF ENDDO ENDDO !\Calculation of normalised distance ! Dynamic update of Sigma Share delta_min=MAXVAL(parent(:.i)=1-(distance(n.j) sum=sum+(aux2/aux1)**2 ENDDO distance(n.nfit nvarb=2 nvare=1+nvar nobjb=nvare+1 nobje=nobjb+nobj-1 nrank=nobje+1 nfit=nrank+1 ! Calculation of normalised distance distance=0 DO n=1.npop IF (distance(n.nrank)-parent(i.GT.delta_max) delta_max=aux3 ENDDO sum=sum/REAL(nobj) sigma_sh=sum/(npop**(1/REAL(nobj))) !\Dynamic update of Sigma Share ! Calculation of sharing function sharing=0 DO n=1.delta_max INTEGER :: nvarb. :)=parent(2*n.(parent(2*n1.GT.nrank).nvare.npop auxi1=INT(parent(n.:) ELSE memory(2*n-1.nfit).:) ENDIF ENDDO CALL shuffle .:)=parent(2*n-1.nrank)).nvar+nobj+1)) IF (auxi1.parent(2*n. MSc Aer-Mech.npop/2 IF (parent(2*n-1.EQ.parent(2*n.nfit))) THEN memory(2*n-1.LT.n REAL :: memory(npop.parent(2*n.j.nobjb.nrank).nobje.npop auxi2=INT(parent(i.nrank.AND.i) ENDDO ENDDO !\Calculation of niche count ENDSUBROUTINE niche_count B.f90" INTEGER :: i.nfit nvarb=2 nvare=1+nvar nobjb=nvare+1 nobje=nobjb+nobj-1 nrank=nobje+1 nfit=nrank+1 !Shuffle prior to reproduction CALL shuffle DO n=1.nrank)) THEN memory(2*n-1.totcol) INTEGER :: nvarb.149 ! Calculation of niche count niche=0 DO n=1.:) ELSEIF ((parent(2*n-1. !15th of October 2011 !-------------------------------------------------------SUBROUTINE selection INCLUDE "zcommon.EQ.6 CROWDED TOURNAMENT SELECTION SUBROUTINE !-------------------------------------------------------!Subroutine to perform the reproduction of the solutions !using the crowded tournament selection method !-------------------------------------------------------!developed by VICTOR FUJII ANDO.nvar+nobj+1)) DO i=1.:)=parent(2*n-1.auxi2)niche(n)=niche(n)+sharing(n. Eng. :) ENDIF ENDDO parent=memory CALL shuffle ENDSUBROUTINE selection B.nobje.2 DO i=nvarb.nrank).:)=parent(2*n-1.nobjb.npop-1.nfit nvarb=2 nvare=1+nvar nobjb=nvare+1 nobje=nobjb+nobj-1 nrank=nobje+1 nfit=nrank+1 DO n=1.(parent(2*n1.i.nrank)) THEN memory(2*n.GT.nrank.:)=parent(2*n-1.parent(2*n.seed_time REAL :: betaq.:) ELSEIF ((parent(2*n-1.nvare j=i-nvarb+1 ! Random number seed_time=TIME()-1318600000 CALL SEED(seed_time) rand1=RAND() !\Random number . Eng.:) ELSE memory(2*n.7 REAL-CODED ELITIST CROSSOVER SUBROUTINE !-------------------------------------------------------!Subroutine to perform the real polynomial and elitist !cross-over !-------------------------------------------------------!developed by VICTOR FUJII ANDO.AND.nfit).j.f90" INTEGER :: n.nvare.LT.:)=parent(2*n.parent(2*n.rand1.EQ.nrank).npop/2 IF (parent(2*n-1. !15th of October 2011 ! !revised: ! 3rd Nov: elitism implemented !-------------------------------------------------------SUBROUTINE cross_over INCLUDE "zcommon.nfit))) THEN memory(2*n.nrank)).150 DO n=1.etac INTEGER :: nvarb.parent(2*n. MSc Aer-Mech. 5*((1betaq)*parent(n.5*((1+betaq)*parent(n.5) THEN betaq=(2*rand1)**eta_c ELSE betaq=(1/(2*(1-rand1)))**eta_c ENDIF !\Beta_q function calculation ! Offspring calculation offspring( n.LT.i)+(1+betaq)*parent(n+1.LT.1)=parent(n+1.upper_var(j))offspring(n.nrank)-1).i)=0.i)) ELSE offspring( n.upper_var(j))offspring(n+1.LT.1) IF(((parent(n.AND.i).151 ! Beta_q function calculation IF (rand1.i).i)=0.i)=upper_var(j) IF (offspring(n+1.lower_var(j))offspring(n+1.i).GT.GT.01).i)=lower_var(j) IF (offspring(n+1.i)=lower_var(j) IF (offspring(n.i)=0.0.i)=0.i). Eng.i)+(1betaq)*parent(n+1.0.i)) ENDIF !\Offspring calculation ! Check limits IF (offspring(n.i)+(1+betaq)*parent(n+1.i)+(1betaq)*parent(n+1.lower_var(j))offspring(n.i)) ELSEIF(((parent(n+1.1)=parent( n.p_elit))THEN offspring( n.5*((1+betaq)*parent(n.AND.LE. MSc Aer-Mech.i)) offspring(n+1.LT.8 REAL POLYNOMIAL MUTATION !-------------------------------------------------------! !Subroutine to perform the polynomial mutation ! !-------------------------------------------------------!developed by VICTOR FUJII ANDO. !18th of October 2011 !-------------------------------------------------------- SUBROUTINE mutation .(RAND().(RAND().5*((1betaq)*parent(n.i)=upper_var(j) !\Check limits ENDDO ENDDO ENDSUBROUTINE cross_over B.LT.p_elit))THEN offspring(n+1.01).i)=parent(n+1.i)=parent(n.nrank)-1).LT.i) offspring(n+1.0.i) offspring( n.1) offspring(n+1. 5) THEN delta=(2*rand1)**eta_m-1 ELSE delta=1-(2*(1-rand1))**eta_m ENDIF offspring(n.i)=offspring(n.p_mut)THEN DO i=nvarb.nrank.LT.rand2 REAL :: min_var(nvar).i)) max_var(i-nvarb+1)=MAXVAL(offspring(:.i).152 INCLUDE "zcommon.i)=lower_var(invarb+1) IF (offspring(n.nobjb.max_var(nvar).lower_var(i-nvarb+1))offspring(n.i)) delta_var(i-nvarb+1)=max_var(i-nvarb+1)-min_var(i-nvarb+1) ENDDO DO n=1.npop rand2=RAND() IF (rand2.rand1.f90" INTEGER :: n.nvare ! Random number rand1=RAND() !\Random number ! Parameter delta IF (rand1.i).delta_var(nvar) INTEGER :: nvarb.GT.LT.j.nvare min_var(i-nvarb+1)=MINVAL(offspring(:.nobje.nvare.i.LT.i)+delta_var(i-nvarb+1)*delta ! Check limits IF (offspring(n.seed_time REAL :: delta.i)=upper_var(invarb+1) !\Check limits ENDDO ENDIF ENDDO ENDSUBROUTINE mutation .nfit nvarb=2 nvare=1+nvar nobjb=nvare+1 nobje=nobjb+nobj-1 nrank=nobje+1 nfit=nrank+1 DO i=nvarb.upper_var(i-nvarb+1))offspring(n.0. xkb1 .0 alfa2 S 1 jmax streamlines 25.97 .00 distrib of etac stage 1 0.00 ispd (1=U.55 hub-tip ratio 0.95 .stage 5 2 iloss (1=Swan 2=Msarratt) 2 iaxial_chanel (=1 linear rotor-estator.98 .Dummy1 before 1.R 3 0.xkb1 .0 25. 2=RPM) 1 flag 1=temp 2=press 0 iogv 3 ndummyafter 1 itype_dist =1 temperature =2 pressure 5 num_weigths 0.0 400.00 xkb1 .950 temp or press distribution .0 Pa Ta Flight Mach No 3 ndummybefore 6. 288.001 var_effic (precisao) 0.stage 4 0. which is the starting point of this work follows: 0 =0 => no IGV =1 => with IGV =99 sample file 101325. =2 linear rotor-rotor) 0.0 25.0 25.0 alfa2 S 5 jmax streamlines 0.Dummy3 before 1.00 distrib of etac stage 4 0.0 25.85828 0.0 25.0 25.stage 2 1.0 alfa2 S 2 jmax streamlines 25.00 distrib of etac stage 5 0 idistalfa2 =0 linear =1 for each streamline 25.84669 0.S 1 0.0 25.0 dvoutlet 1.0 25.250 temp or press distribution .R 2 0.850 effisen_given 0.94 .0001 check_effic 0.000 dr1p 10.153 APPENDIX C ORIGINAL SLCP INPUT FILE The original SLCP input file as given by the human design.50 0.xkb1 .stage 3 0.0 25.xkb1 .92 .xkb1 .S 2 0.0 25.20 mass flow kg/s 1 400.0 25.01 ac_eff=var_effic (delta effic para acerto) 0.stage 1 1.S 3 .00 distrib of etac stage 3 0.0 25.0 25.R 1 0.Dummy2 before 1.0 25.xkb1 .0 25.000 5 5 5 pr nstage jmax njmax 0.00 xkb1 .00 xkoxamb 0.0 dvinlet 10.0 25.800 temp or press distribution .0 alfa2 S 4 jmax streamlines 25.0 25.30 axial Mach N.00 xkb1 .0 0.00 . at inlet and outlet of the compressor 1.950 temp or press distribution .0 25.60 dstall 0.100 temp or press distribution .0 25.0 alfa2 S 3 jmax streamlines 25.00 distrib of etac stage 2 0.250 f15=lim wtotal 8.0 25. 89 .91 .R2 dca .S1 dca .for convergence purposes only 1 ivortx dca .86 xkb1 .91 .00 .s/c .R 3 0.s/c .R 4 2.h/c .00 .Dummy3 after 0.S 2 2.Dummy2 after 0.S 5 0.h/c Dummy 3 before 2.154 0.00 .S 1 2.R5 dca .h/c Dummy 3 after .R 5 0.86 .86 xkb1 .86 xkb1 .S5 1.86 xkb1 .59 .R 2 0.s/c .xkb1 .R 4 0.00 .h/c Dummy 1 after 1.R 4 0.s/c Dummy 2 before 1.30 0.00 .S 1 0.h/c .88 .59 .00 .00 .S4 dca .30 lslr srlr ssls 0.s/c .s/c .R 1 0.s/c .h/c .00 .xkb1 .s/c Dummy 1 after 0.93 .00 .h/c Dummy 2 after 1.88 .h/c Dummy 1 before 3.S 4 0.R 3 2.s/c .R3 dca .s/c .88 .00 .xkb1 .90 relax relax1 .S 5 1.R1 blade type .s/c Dummy 2 after 0.10 .s/c Dummy 1 before 1.S 4 0.00 .00 .R 5 0.R 5 2.h/c .h/c .s/c .s/c .S 2 0.h/c .s/c Dummy 3 before 1.s/c Dummy 3 after 3.S2 dca .S 5 0.00 .dca/b65s/c4/myblade dca .h/c Dummy 2 before 3.83 .79 .h/c .85 .R 1 2.R 2 2.00 .00 0.h/c .Dummy1 after 0.S 4 2.81 .S 3 2.xkb1 .Dummy4 after cod cod=constant outer diam cid=constant inner diameter cmd=constant mean diameter v1u distrib of Vu 1 2 iradeq1 (1=old 2=new) 1 2 iradeq2 (1=old 2=new) 1 0 ikt =0 single precision =1 double precision 1 0 iio10 =0 single precision =1 double precision 1 0 iin =0 single precision =1 double precision 1 0 ikdelta =0 single precision =1 double precision 1 0 ido10 =0 single precision =1 double precision 1 0 inb =0 single precision =1 double precision 1 0 id0di2d =0 single precision =1 double precision 1 0 ii0ci2d =0 single precision =1 double precision 1 0 id0cd2d =0 single precision =1 double precision 1 secondary loss 1=Griepentropg_s method 2=Howell 1.R4 dca .h/c .87 .90 0.00 .95 .00 .59 .S 3 0.89 .h/c .S3 dca .88 . R 1 0.010 min space between rows .50 6.hub .50 percentage of stage pressure loss attributed to the 0.50 percentage of stage pressure loss attributed to the 10.R 1 .010 min space between rows .50 6.R 5 .S 4 .010 min space between rows .0d0 xacochamb 6.tip thickness-chord ratio % 8.010 min space between rows .Dummy 3 before 0.0 r/t .001 rmin 0.tip 6.S 4 0.tip 10.0 V1a .0 V1a .010 min space between rows .010 min space between rows .R 2 0.tip 10.010 min space between rows .50 6.tolerance for mass flow !era 0.0 8.relaxation factor for boundary layer 0.hub .tip thickness-chord ratio % 8.S 3 0.S 1 .hub .Dummy 1 after 0.S 1 0.hub .01 tip clearance R 1 0.0 t/c .0 r/t .0 r/t .S 1 .S 2 0.R 1 .0 6.R 4 .0 t/c .hub .0 8.R 5 0.0 6.50 6.50 additional incidence R 1 3.01 tip clearance S 1 0.hub .0 6.010 min space between rows .hub .010 min space between rows .R 3 .001 wrdes (streamlines reposition tolerance) 1.010 min space between rows .50 3.01 tip clearance S 5 0.S 2 .50 percentage of stage pressure loss attributed to the 0.R 2 .0 8.0 r/t .5 xkzdummy .01 tip clearance R 5 0.tip thickness-chord ratio % 8.0 8.001 0.010 min space between rows .tip thickness-chord ratio % 8.hub .0 r/t .0 t/c . angbullet .0 8.hub .50 6.010 min space between rows .0 6.0 6.tip 10.hub .spinner 0.50 3.50 percentage of stage pressure loss attributed to the 0.S 5 .50 additional incidence R 2 rotor rotor rotor rotor rotor R R R R R 1 2 3 4 5 .tip 10.010 min space between rows .hub .hub .50 3.0 V1a .0 r/t .tip thickness-chord ratio % 8.50 additional incidence S 1 6.155 0.R 2 166.S 5 0.S 5 .R 3 164.R 4 0.tip leading and trailig edges radius % 6.tip 6.Dummy 1 before 0.0 6.0 6.R 3 .0 8.5 alfakb .% dummy no bullet 0.hub .tip leading and trailig edges radius % 6.00001 wmcn1 .S 2 .01 tip clearance S 4 0.R 4 .R 5 0 idevmod (0=carter deviation correlation) 45.01 tip clearance R 4 0.0 t/c .0 V1a .Dummy 3 after 10.tip leading and trailig edges radius % 6.01 tip clearance R 2 0.Dummy 2 before 0.010 min space between rows .S 3 .0 6.0 8.tip leading and trailig edges radius % 6.0 r/t .0 t/c .50 6.50 6.01 tip clearance S 2 0.0 r/t .0 flare 0.tip 170.50 6.010 min space between rows .R 5 .tip 6.hub .0 8.R 4 160.hub .50 percentage of stage pressure loss attributed to the 0.0 V1a .0 6.0 8.0 6.tip 6.0 t/c .50 3.S 4 .0 t/c .0 t/c .Dummy 2 after 0.R 2 .hub .0 t/c .01 tip clearance R 3 0.R 1 168.0 8.hub .0 t/c .hub .tip leading and trailig edges radius % 6.01 tip clearance S 3 0.R 3 0.tip 6.hub .0 r/t .hub .0 r/t .S 3 .010 min space between rows . 40d0 0.para convergencia do 'va' relax6 .50 6.50 additional incidence S 3 6.50 additional incidence R 5 3.para convergencia do 'va' relax8 .50 3.50 6.50 6.045 0.50 3.50 additional incidence S 5 dhlim drh=0.para convergencia do 'va' no no no no radeq2 radeq2 radeq2 radeq2 .50 additional incidence R 3 3.50 3.070 0.50 3.50 6.50 6.50 3.070 0.50 3.070 0.50 3.50 6.50 additional incidence S 2 6.50 3.72 -0.50 additional incidence R 4 3.25d0 0.070 50 0.50 3.40d0 0.156 3.50 6.50 6.50 0.50 3.para convergencia do 'va' relax7 .50 6.001d0 radius_shaft radius_labirinth( 1) radius_labirinth( 2) radius_labirinth( 3) radius_labirinth( 4) radius_labirinth( 5) n_nradeq2_lim relax5 .005 0.50 3.50 3.50 3.070 0.50 3.003d0 drm=0.50 additional incidence S 4 6.50 3.000d0 drt=0.007 0.50 6.50 6.000 0.25d0 3.50 6. 157 APPENDIX D ADDITIONAL INFORMATION OBTAINED SOLUTIONS FROM THE . Inlet Mach .30 0.10 1.00 1.] 1.30 solution 1 .rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.80 0.70 0.70 0.] 1.50 0.Inlet Mach .158 ROTOR INLET MACH NUMBER original solution .10 1.50 0.30 0.30 solution 2 .30 solution 4 .rotor 5 4 streamline D.] 1.1 R1 R2 R3 R4 R5 3 2 1 0.10 1.90 Inlet Mach [ .40 0.20 1.] 1.00 1.40 0.40 0.30 0.40 0.00 1.Inlet Mach .60 0.60 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.80 0.10 1.70 0.] 1.50 0.30 0.00 1.40 0.20 1.30 solution 3 .70 0.60 0.70 0.80 0.90 Inlet Mach [ .10 1.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.80 0.50 0.Inlet Mach .20 1.90 Inlet Mach [ .00 1.50 0.30 0.20 1.60 0.30 .Inlet Mach .60 0.90 Inlet Mach [ .90 Inlet Mach [ .80 0.20 1. 50 0.stator 5 4 streamline D.2 S1 S2 S3 S4 S5 3 2 1 0.40 0.40 0.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.50 0.Inlet Mach .30 solution 2 .30 0.159 STATOR INLET MACH NUMBER original solution .40 0.60 0.20 1.] 1.70 0.20 1.20 1.10 1.60 0.20 1.] 1.30 solution 1 .70 0.] 1.00 1.80 0.90 Inlet Mach [ .80 0.10 1.90 Inlet Mach [ .30 0.40 0.70 0.90 Inlet Mach [ .00 1.80 0.30 solution 3 .stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.70 0.10 1.Inlet Mach .50 0.30 0.Inlet Mach .30 0.30 .50 0.] 1.30 0.00 1.90 Inlet Mach [ .70 0.20 1.80 0.Inlet Mach .80 0.60 0.50 0.90 Inlet Mach [ .40 0.Inlet Mach .10 1.00 1.60 0.10 1.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.60 0.30 solution 4 .] 1.00 1. 05 0.10 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.15 total loss [ .rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.20 0.10 0.rotor 5 4 streamline D.15 total loss [ .] 0.20 0.25 solution 1 .25 .] 0.] 0.] 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.05 0.20 0.25 solution 3 .20 0.total loss .10 0.00 0.total loss .10 0.00 0.00 0.10 0.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 0.15 total loss [ .05 0.05 0.00 0.15 total loss [ .] 0.05 0.20 0.total loss .00 0.25 solution 2 .total loss .160 ROTOR TOTAL LOSS original solution .15 total loss [ .25 solution 4 .3 R1 R2 R3 R4 R5 3 2 1 0.total loss . 4 S1 S2 S3 S4 S5 3 2 1 0.03 0.02 0.01 0.02 0.05 total loss [ .total loss .] 0.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.03 0.08 solution 3 .06 0.06 0.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.08 solution 2 .] 0.04 0.04 0.total loss .00 0.07 0.total loss .03 0.05 total loss [ .] 0.total loss .07 0.08 solution 4 .04 0.06 0.00 0.06 0.] 0.07 0.00 0.02 0.02 0.07 0.05 total loss [ .stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.08 .05 total loss [ .04 0.03 0.08 solution 1 .stator 5 4 streamline D.00 0.01 0.total loss .] 0.07 0.06 0.01 0.04 0.01 0.161 STATOR TOTAL LOSS original solution .01 0.05 total loss [ .03 0.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 0.00 0.02 0. 40 -7.80 -7.162 ROTOR INCIDENCE ANGLE original solution .80 incid.40 -6. angle .80 -7.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 -8. angle [ deg ] -6.60 -7.00 solution 3 .00 solution 1 .20 -6.00 -6.60 -7.60 -6.20 -7.00 -6.60 -7.00 -7. angle .40 -6. angle [ deg ] -6.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 -8.80 incid. angle .40 -6.incid.00 .60 -7. angle [ deg ] -6.00 -6.20 -6. angle [ deg ] -6.20 -7.60 -6.80 -7.incid.00 solution 4 .20 -6.60 -7.rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 -8.00 -6.80 incid.20 -7.00 -6.00 -7.60 -6.60 -6.20 -7.20 -6.80 -7. angle [ deg ] -6.60 -6.40 -6. angle .rotor 5 streamline 4 R1 R2 R3 R4 R5 3 2 1 -8.00 solution 2 .40 -7. angle .incid.5 R1 R2 R3 R4 R5 3 2 1 -8.00 -7.00 -7.40 -7.rotor 5 4 streamline D.80 -7.20 -6.40 -7.80 incid.incid.40 -6.80 incid.20 -7.00 -7.incid.40 -7. 80 -4.20 -5.00 -4.40 incid.20 -4.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 -5.00 -4.6 S1 S2 S3 S4 S5 3 2 1 -5.80 -4.00 -3.00 -4.20 -4. angle [ deg ] -4.00 -3.60 solution 4 .60 -4.20 -5. angle .40 -4.20 -5.00 -4. angle [ deg ] -4.60 solution 1 .163 STATOR INCIDENCE ANGLE original solution .80 -3. angle .60 solution 3 .20 -5.incid. angle [ deg ] -4.60 -4.incid.00 -3.40 incid.00 -4.60 .incid.80 -3.60 solution 2 .80 -4.80 -3.incid.40 -5. angle .20 -4.40 -5.40 incid.80 -3.80 -4.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 -5. angle [ deg ] -4.80 -3.20 -5. angle .incid.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 -5.00 -3.60 incid. angle [ deg ] -4.40 incid.stator 5 4 streamline D.20 -4.60 -4.60 -4.stator 5 streamline 4 S1 S2 S3 S4 S5 3 2 1 -5.00 -3.40 -5.20 -4.40 -5.80 -4.40 -5. angle . Projeto de máquinas. an optimisation program. Programa de Pós-Graduação em Engenharia Aeronáutica e Mecânica. using correlations from the literature to assess the losses. Orientador: Prof. Curso de Mestrado. PALAVRAS-CHAVE SUGERIDAS PELO AUTOR: Algoritmo genético. it is still a laborious and timeconsuming task. Algoritmos genéticos. Therefore. Propulsão e Energia. four solutions were selected for further analysis. allowing an easy integration with the programs developed by the Gas Turbine Group.FOLHA DE REGISTRO DO DOCUMENTO 1. RESUMO: This work presents an approach to optimise the preliminary design of high-performance axial-flow compressors. Publicada em 2011. APRESENTAÇÃO: X Nacional Internacional ITA. was written in FORTRAN language. 2. The program is based upon a multi-objective genetic algorithm. Turbinas a gás. The preliminary design within the Gas Turbine Group at ITA. Turbomáquinas. Área de Aerodinâmica. DATA 3. Projeto preliminar. Nevertheless. Therefore. The choice of many parameters of the thermodynamic cycle and of geometries relies upon the expertise from the members of the Group. the temperature distribution and the hub-tip ratio were varied aiming at higher efficiencies and higher pressure ratios. with real codification and elitism. thousands of designs could be quickly evaluated.PALAVRAS-CHAVE RESULTANTES DE INDEXAÇÃO: Turbocompressores. 11. the stator air outlet angles. Engenharia mecânica 10. N° DE PÁGINAS 162 TÍTULO E SUBTÍTULO: Genetic algorithm for preliminary design optimisation of high-performance axial-flow compressors 6. Finally. but controlling the de Haller number and the camber angle. GRAU DE SIGILO: (X ) OSTENSIVO ( ) RESERVADO ( ) CONFIDENCIAL ( ) SECRETO . is carried on with an in-house computational program based upon the streamline curvature method. Dr. Compressor axial. INSTITUIÇÃO(ÕES)/ÓRGÃO(S) INTERNO(S)/DIVISÃO(ÕES): Instituto Tecnológico de Aeronáutica . Then the REMOGA and the preliminary design program were integrated to design a 5-stage axialflow compressor. CLASSIFICAÇÃO/TIPO DM 5. requiring successive trial and errors. Turbomáquinas 9.ITA 8. using a choice criterion. named REMOGA. to support the compressor designer in the choice of some parameters. 12. Thanks to the REMOGA. João Roberto Barbosa. São José dos Campos. AUTOR(ES): Victor Fujii Ando 7. REGISTRO N° 13 de janeiro de 2012 DCTA/ITA/DM-081/2011 4. Defesa em 19/12/2011. revealing that the developed program was successful in finding more efficient and feasible compressor designs. Documents Similar To 62031Skip carouselcarousel previouscarousel nextThesisInverse SimulationCFX Introductioncfx_intrAntisurge Control SystemEconomics of Internal Pipe Coating04-AxialTurbines (1)1.Turbo FundamentalsSwirl BrakeSyllabusmodule 15. 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