Arodynamic Optimization of the Nose Shape of a HS Train

Aerodynamic Optimization of theNose Shape of a High-speed Train

By

Jorge Muoz Paniagua

Director: Javier Garca Garca

A THESISSubmitted to

UNIVERSIDAD POLITCNICA DE MADRIDin partial fulfillment of the requirements

for the degree ofDOCTOR IN PHILOSOPHY

Departamento de Ingeniera Energtica y FluidomecnicaEscuela Tcnica Superior de Ingenieros Industriales

2014

A todos los que me acompaaron en este viaje.

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Contents

List of symbols vii

Abstract ix

Resumen xi

Theoretical background 10.1 Context of the study . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 What is high-speed train (HST). Main models of HST . . . . . . . 20.3 The importance of train aerodynamics . . . . . . . . . . . . . . . . 30.4 Objective of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 30.5 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 4

Optimization methods 70.6 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.7 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

0.7.1 Genetic operators . . . . . . . . . . . . . . . . . . . . . . . . 80.7.2 Implicit parallelism. Building blocks . . . . . . . . . . . . . 100.7.3 Limitations of Genetic Algorithms . . . . . . . . . . . . . . . 12

0.8 Adjoint methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130.8.1 Continuous and Discrete Adjoint solver . . . . . . . . . . . . 130.8.2 Discrete Adjoint solver implementation . . . . . . . . . . . . 130.8.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

0.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

I Methodology 17

Shape Parameterization 190.10 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190.11 Shape parameterization requirements . . . . . . . . . . . . . . . . . 190.12 Review of previous researches and normative geometrical constraints 20

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0.13 Shape parameterization techniques. Bzier curves . . . . . . . . . . 210.14 3D nose parameterization approach . . . . . . . . . . . . . . . . . . 24

0.14.1 Cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . 240.14.2 Lateral-side view . . . . . . . . . . . . . . . . . . . . . . . . 260.14.3 Front view . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330.14.4 Top view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340.14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370.14.6 Parameterization validation . . . . . . . . . . . . . . . . . . 38

0.15 A three-parameters 3D geometry description . . . . . . . . . . . . . 410.16 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Metamodels and Design of Experiments 450.17 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450.18 Metamodel or Surrogated-model definition . . . . . . . . . . . . . . 46

0.18.1 Polynomial Response Surface Model (RSM) . . . . . . . . . 480.18.2 Radial Basis Functions (RBF) . . . . . . . . . . . . . . . . . 490.18.3 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . 510.18.4 Why choosing RSM and RBF? . . . . . . . . . . . . . . . . 52

0.19 Generating data to fit the metamodel . . . . . . . . . . . . . . . . . 520.19.1 Classical vs modern DoE . . . . . . . . . . . . . . . . . . . . 52

0.20 Prediction error assessment . . . . . . . . . . . . . . . . . . . . . . 530.21 Online optimization. Adaptive sampling design . . . . . . . . . . . 540.22 Global sensitivity analysis. ANOVA . . . . . . . . . . . . . . . . . . 57

II Results 63

Two-dimensional nose shape optimization 650.23 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650.24 Optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . 65

0.24.1 Genetic algorithms (GA) . . . . . . . . . . . . . . . . . . . . 660.24.2 Adjoint methods . . . . . . . . . . . . . . . . . . . . . . . . 67

0.25 Analysis of sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . 680.26 Numerical set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Three-dimensional nose shape optimization 710.27 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710.28 Shape optimization using the adjoint method . . . . . . . . . . . . . 71

0.28.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 720.28.2 Numerical set-up . . . . . . . . . . . . . . . . . . . . . . . . 730.28.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Table of contents v

0.29 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Shape optimization with crosswind 830.30 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

0.30.1 Genetic algorithm work-flow . . . . . . . . . . . . . . . . . . 850.30.2 Adjoint method work-flow . . . . . . . . . . . . . . . . . . . 86

0.31 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 860.31.1 Numerical set-up . . . . . . . . . . . . . . . . . . . . . . . . 860.31.2 Geometric definition . . . . . . . . . . . . . . . . . . . . . . 870.31.3 Initial design of experiments . . . . . . . . . . . . . . . . . . 870.31.4 Metamodel definition for GA . . . . . . . . . . . . . . . . . 89

0.32 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . 890.32.1 GA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890.32.2 Adjoint method . . . . . . . . . . . . . . . . . . . . . . . . . 930.32.3 Comparison of GA and adjoint method . . . . . . . . . . . . 94

0.33 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Shape optimization in tunnel 990.34 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990.35 Pressure gradient in the compression wave . . . . . . . . . . . . . . 990.36 Geometry description . . . . . . . . . . . . . . . . . . . . . . . . . . 1010.37 Numerical set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

0.37.1 Computational domain . . . . . . . . . . . . . . . . . . . . . 1030.37.2 Initial and boundary conditions . . . . . . . . . . . . . . . . 103

0.38 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . 1040.38.1 Initial design of experiments . . . . . . . . . . . . . . . . . . 1040.38.2 Minimization of the maximum pressure gradient . . . . . . . 1060.38.3 Aerodynamic aspects . . . . . . . . . . . . . . . . . . . . . . 1100.38.4 Metamodel and global sensitivity analysis . . . . . . . . . . 1110.38.5 Parametrical analysis of the design variables . . . . . . . . . 1140.38.6 Estimation of coefficient k . . . . . . . . . . . . . . . . . . . 116

0.39 Multi-objective optimization . . . . . . . . . . . . . . . . . . . . . . 1180.40 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Concluding remarks 1210.41 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . 1210.42 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

References 125

List of symbols

l, L length my height mA train cross-sectional area m2p pressure Pa p pressure drop Paps static pressure PaRe Reynolds number -v train velocity m s1D aerodynamic drag NCD drag coefficient -Q mass flow rate kg s1

Greek boundary layer thickness m boundary layer displacement thickness m boundary layer momentum thickness m dynamic viscosity kg (ms)1 kinematic viscosity m2 s1 density kg m3 standard deviation - yaw angle - pressure drag coefficient -T friction along the train -log(Re) skin-friction drag coefficient -

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viii List of symbols

Sub- and Superscriptst tunnelL leading carB base

Abstract

Aerodynamic design of trains influences several aspects of high-speed trainsperformance in a very significant level. In this situation, considering also thatnew aerodynamic problems have arisen due to the increase of the cruise speed andlightness of the vehicle, it is evident the necessity of proposing an optimizationstudy concerning the train aerodynamics. Thus, the aerodynamic optimization ofthe nose shape of a high-speed train is presented in this thesis. This optimiza-tion is based on advanced optimization methods. Among these methods, geneticalgorithms and the adjoint method have been selected.

A theoretical description of their bases, the characteristics and the implemen-tation of each method is detailed in this thesis. This introduction permits un-derstanding the causes of their selection, and the advantages and drawbacks oftheir application. The genetic algorithms requirethe geometrical parameterizationof any optimal candidate and the generation of a metamodel or surrogate modelthat complete the optimization process. These points are addressed with a specialattention in the first block of the thesis, focused on the methodology considered inthis study. The second block is referred to the use of these methods with the pur-pose of optimizing the aerodynamic performance of a high-speed train in severalscenarios. These scenarios englobe the most representative operating conditions ofhigh-speed trains, and also some of the most exigent train aerodynamic problems:front wind and cross-wind situations in open air, and the entrance of a high-speedtrain in a tunnel.

The genetic algorithms and the adjoint method have been applied in the min-imization of the aerodynamic drag on the train with front wind in open air. Thecomparison of these methods allows to evaluate the methdology and computationalcost of each one, as well as the resulting minimization of the aerodynamic drag.Simplicity and robustness, the straightforward realization of a multi-objective opti-mization, and the capability of searching a global optimum are the main attributesof genetic algorithm. However, the requirement of geometrically parameterize anyoptimal candidate is a significant drawback that is avoided with the use of theadjoint method. This independence of the number of design variables leads to arelevant reduction of the pre-processing and computational cost.

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Considering the cross-wind stability, both methods are used again for the mini-mization of the side force. In this case, a simplification of the geometric parameter-ization of the train nose is adopted, what dramatically reduces the computationalcost of the optimization process. Nevertheless, some of the most important ge-ometrical characteristics are still described with this simplified parameterization.This analysis identifies and quantifies the influence of each design variable on theside force on the train. It is observed that the A-pillar roundness is the mostdemanding design parameter, with a more important effect than the nose lengthor the train cross-section area.

Finally, a third scenario is considered for the validation of these methods inthe aerodynamic optimization of a high-speed train. The entrance of a train in atunnel is one of the most exigent train aerodynamic problems. The aerodynamicconsequences of high-speed trains running in a tunnel are basically resumed intwo correlated phenomena, the generation of pressure waves and an increase inaerodynamic drag. This multi-objective optimization problem is solved with ge-netic algorithms. The result is a Pareto front where a set of optimal solutions thatminimize both objectives.

Resumen

La influencia de la aerodinmica en el diseo de los trenes de alta velocidad,unida a la necesidad de resolver nuevos problemas surgidos con el aumento dela velocidad de circulacin y la reduccin de peso del vehculo, hace evidente elinters de plantear un estudio de optimizacin que aborde tales puntos. En estecontexto, se presenta en esta tesis la optimizacin aerodinmica del testero de untren de alta velocidad, llevada a cabo mediante el uso de mtodos de optimizacinavanzados. Entre estos mtodos, se ha elegido aqu a los algoritmos genticos y almtodo adjunto como las herramientas para llevar a cabo dicha optimizacin.

La base conceptual, las caractersticas y la implementacin de los mismos sedetalla a lo largo de la tesis, permitiendo entender los motivos de su eleccin, ylas consecuencias, en trminos de ventajas y desventajas que cada uno de ellosimplican. El uso de los algorimos genticos implica a su vez la necesidad de unaparametrizacin geomtrica de los candidatos a ptimo y la generacin de unmodelo aproximado que complementa al mtodo de optimizacin. Estos puntosse describen de modo particular en el primer bloque de la tesis, enfocada a lametodologa seguida en este estudio. El segundo bloque se centra en la aplicacinde los mtodos a fin de optimizar el comportamiento aerodinmico del tren endistintos escenarios. Estos escenarios engloban los casos ms comunes y tambinalgunos de los ms exigentes a los que hace frente un tren de alta velocidad:circulacin en campo abierto con viento frontal o viento lateral, y entrada entnel.

Considerando el caso de viento frontal en campo abierto, los dos mtodos hansido aplicados, permitiendo una comparacin de las diferentes metodologas, ascomo el coste computacional asociado a cada uno, y la minimizacin de la re-sistencia aerodinmica conseguida en esa optimizacin. La posibilidad de evitarparametrizar la geometra y, por tanto, reducir el coste computacional del pro-ceso de optimizacin es la caracterstica ms significativa de los mtodos adjuntos,mientras que en el caso de los algoritmos genticos se destaca la simplicidad ycapacidad de encontrar un ptimo global en un espacio de diseo multi-modal ode resolver problemas multi-objetivo.

El caso de viento lateral en campo abierto considera nuevamente los dos mto-

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dos de optimizacin anteriores. La parametrizacin se ha simplificado en esteestudio, lo que notablemente reduce el coste numrico de todo el estudio de opti-mizacin, a la vez que an recoge las caractersticas geomtricas ms relevantes enun tren de alta velocidad. Este anlisis ha permitido identificar y cuantificar la in-fluencia de cada uno de los parmetros geomtricos includos en la parametrizacin,y se ha observado que el diseo de la arista superior a barlovento es fundamental,siendo su influencia mayor que la longitud del testero o que la seccin frontal delmismo.

Finalmente, se ha considerado un escenario ms a fin de validar estos mtodosy su capacidad de encontrar un ptimo global. La entrada de un tren de altavelocidad en un tnel es uno de los casos ms exigentes para un tren por el pico desobrepresin generado, el cual afecta a la confortabilidad del pasajero, as como a laestabilidad del vehculo y al entorno prximo a la salida del tnel. Adems de esteproblema, otro objetivo a minimizar es la resistencia aerodinmica, notablementesuperior al caso de campo abierto. Este problema se resuelve usando algoritmosgenticos. Dicho mtodo permite obtener un frente de Pareto donde se incluyen elconjunto de ptimos que minimizan ambos objetivos.

Theoretical background

0.1 Context of the study

People have never traveled as much as over the last two decades, and accordingto the experts, even with the present economical crisis, this vital trend is notgoing to be reversed in the coming years. Result of this is an increasing saturationof the communication highways and an irreparable damage to an already fragileenvironment. More mobility is a problem to which the high-speed train can bea proper answer. Indeed, the reduced traveling times, higher levels of passengercomfort and safety, and low environmental impact enable high-speed trains tocompete with and complement road and air travel. In this context, the EuropeanUnion was committed to making the mobility of people, and also the transportof goods, more efficient, more secure and more environmentally friendly, withpriority given to social and territorial cohesion, [?]. Big cities such as London,Paris, Brussels, Frankfurt, Milan or Madrid among others are now connected bytrains traveling at speeds of 300 kmh1, so that high-speed trains do clearly helpto implement viable mobility at European level.

The year which is considered the beginning of the modern high-speed trainsera is 1964, when the first passenger service was launched in Japan with trainsrunning at 210 kmh1, [?]. In Europe, it was the 1974 petrol crisis that madeseveral European countries to develop a new, fast mode of transport which wouldnot dramatically consume fossil fuels. Italy, France, Germany and Spain were thecountries that most invested in this new mode. At the end of 2009, Europe had6214 km of high-speed lines on which trains could run at speeds over 250 kmh1,[?]. Nowadays, trains running on the most recently installed lines can reach speedsas high as 300 kmh1 not only in Western Europe, but also in densely populatedareas such as Japan and South Korea.

This situation is particularly relevant in Spain, where there were plans to laysome 10000 km of high-speed line between 2007 and 2020. [...]

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2 Abstract

0.2 What is high-speed train (HST). Main mod-els of HST

There is no single definition for high speed in the context of railway services,although the reference is always to passenger services and not to freight, [?]. Ahigh-speed train (HST) is a train capable of reaching speeds of over 200 kmh1 onupgraded conventional lines, and over 250 kmh1 on new lines designed specificallyfor high speeds, according to the Directive 96/48 from the European Union, [?].Although this operational definition does not indicate anything about the traindesign itself, the particular characteristics of the countries aforementioned mayintroduce different configurations of the train models.

In Japan, a high demand for travel between Tokyo and other big cities likeOsaka or Nagoya (populations of more than 8 million people), up to 500 km faraway from the capital city, leads to a dedicated line just for Shinkansen high-speed service. The geographic features of Japan, together with the requirementof avoiding tight curves and steep gradients, resulted in many tunnels and bridgesalong the route. This implies the need of tunnel aerodynamics optimized trains.

The main difference between the French TGV (Train Grande Vitesse) andthe Japanese Shinkansen is related to the ability of the former to operate on con-ventional tracks as well, leading to significant cost savings. The differences are alsoattributable to the distinct characteristics of France and Japan, [?], so that tunnelaerodynamics may not be so demanding, but the crosswind stability or trains pass-ing by each other might be the most critical situations. The AVE (Alta VelocidadEspaola), from Spain, and the German ICE (Inter-City Express) are similar tothe French one, with the particularity of the latter that the high-speed line is usedfor both passenger and freight transport, [?]. Nevertheless, in all these cases theHST use a newly built track on the sections where high speed is achieved, whichtranslates into high construction costs that are not always worthwhile. The Ital-ian Pendolino (ETR-450) and the Swedish X-2000 are examples of HST runningon conventional rail using the tilting mechanism, what permits the bogie to beattached to the track while the car-body tilts in tight radius curves. In this way, itis avoided the excessive cost of new tracks, although not reaching so high speeds.In fact, as the train speed increases, the centrifugal forces in tight curves increase,leading to discomfort to passengers. Thus, the tilting mechanism results into alimitation of speed and may introduce differences in the aerodynamic stability incurves and also in crosswind situations. Related to the Swedish case, it is evidentthat the operating conditions are notably different to those of Southern Europe,basically because of the special climate of Scandinavia. Therefore, crosswind sta-bility or the design of elements like the spoiler, used to remove the snow from thetrack, could be critical in that case.

Abstract 3

0.3 The importance of train aerodynamics

The differences indicated before mostly refer to operating conditions or specialcharacteristics or climate of the country. However, probably it is the progressiveincrease of the cruise speed and the reduction of weight that take the most im-portant role on the design of a HST. In fact, a consequence of the evolution tofaster and lighter trains is the introduction of new aerodynamic problems that wereneglected before but are notably significant nowadays. The main technical chal-lenges are related to avoid a severe increase in energy consumption (aerodynamicdrag), maintain stability and the comfort of passengers (crosswind stability, trainspassing by each other or trains entering a tunnel), avoid track maintenance costs(uprise of ballast) or avoid an increase in noise and vibration to areas adjacentto the line (micro-pressure waves at the exit of a tunnel or aerodynamic noise).Other matters, like safety for maintenance workers or passengers on platforms orquality of comfort for the latter are related to the winds induced by trains and theventilation in underground stations or tunnels respectively, [75]. Consequently,the aerodynamic design plays a key role on the design of a HST and, although itis not the only discipline that requires attention, .

Since train aerodynamic problems are related to the flow around the vehicle, ithas been observed over the last decade an extensive research in the analysis of theflow characteristics in many different scenarios. Examples of the flow situationswhere trains are influenced by crosswind situations include both trains influencedby steady winds [] and wind gusts [], trains on the open terrain [] and trains partlyshielded from winds while leaving tunnels [], trains on the flat ground [], on anembankment [] or on a bridge [].

Once the flow structures are pictured, it is possible to propose a geometricmodification that can improve the aerodynamic performance of trains. Dealingwith these optimization problems has traditionally been done by a trial-and-errorprocedure, which is very expensive in terms of computer and designer time. Fur-thermore, as Krajnovic points out, [45], several aerodynamic objectives are knownto be in conflict, what makes optimization of aerodynamic properties of trains atrue multiobjective optimization problem.

0.4 Objective of this thesis

It has been explained that designing an aerodynamically efficient HST is ofparamount importance in the railway industry. In general, it is very easy to seethe benefits of improved aerodynamic efficiency, but it is not that easy to achieve.Considering as an initial milestone the experience of the research group of AppliedFluid Mechanics, the objective is to go one step beyond and set up the basics of

4 Abstract

the aerodynamic optimization of a HST using advanced optimization algorithms.This report attempts to (I) introduce the two of the most popular advanced

optimization algorithms in the aerodynamics field, (II) describe the methodologyassociated to these, and (III) present the optimal designs obtained in the resolutionof several single-objective and multi-objective optimization problems concerningto train aerodynamics. This work is believed to contribute to the understandingof two optimization algorithms in the aerodynamics field, namely the genetic algo-rithms and the adjoint method. The basis and requirements of them are detailedin the second section of the report, with special emphasis to the geometric pa-rameterization and the construction of the metamodel or surrogate model. Thethird block is devoted to the practical application of the genetic algorithms andthe adjoint method. Different problems are considered during the thesis

0.5 Overview of this thesisHere the outline of the thesis report is described. Chapter 0.5, Optimization

methods explains the theoretical basis of the optimization methods in general,and particularly of the genetic algorithms and the adjoint method. The originof the former, the genetic operators, similar to those that govern the naturalevolution, the concept of Implicit Parallelism in which the performance of thegenetic algorithms is based and the limitations of them are introduced in thischapter. The adjoint method is also presented here, where a more mathematicaldeduction of the method is described, with special emphasis of the continuousformulation. The implementation of the method and its limitations are also givenin this chapter.

Once the description of the two considered optimization methods is done, thedistinctive features and innovative aspects of this thesis are resumed into twoblocks. The first block (I) is referred to the methodology implemented in the suc-cessive optimization problems faced in this thesis. The second block (II) presentsthe main results obtained within the previous methodology. Block I compiles

Chapter I Shape parameterization, where the technique used to para-metrically define the shape of the nose of a high-speed train. Two geometricdescriptions are presented. First, a more complex parameterization basedon twenty-five design variables is adopted. This parameterization permitsdescribing a realistic nose shape, and it is also validated in two dimensionswith other realistic trains. The second geometric description is a very plainone, based on three design variables. This parameterization is simple enoughto reduce the dimensionality of the design space and thus to facilitate thesearch of the global optimum, while at the same time is robust enough toinclude some characteristic features of realistic train models.

Abstract 5

Chapter 0.16 Metamodels and Design of Experiments, where theobjective of using a metamodel or surrogate-model in the optimization work-flow is indicated.

Optimization methods

0.6 Introduction

Because of the development of numerical methods and resources, the processof aerodynamic design optimization has been revolutionized.

to see how the GA processes schemata, an not the individual strings, withinthe population

In [?], a equivalent classification of the optimization methods is proposed. Herethey are gathered into calculus-based, enumerative and random search methods.

A possible classification is based on the order of derivatives of the objectivefunction used. While zero order methods use only the function values in theirsearch for the minimum, the first and second order methods use respectively thefirst and second order derivatives of the objective function. This simple divisionmay englobe many other subclassifications, and terms as calculus-based, enumer-ative, random search methods, evolutive methods or adjoint methods would beincluded into any of the former.

If a new design is based on a previous one, [?],

xi+1 = xi + iSi (1)

where at the ith iteration,

0.7 Genetic Algorithm

Genetic algorithms occupy a gap in the range of optimization techniques, placedbetween random methods and gradient-based methods, [?]. They are less efficientthan gradient-based methods on problems that are amenable to solution by calcu-lus techniques, but they are able to deal with constraints and multi-modal prob-lems, not smooth (even discontinuous) topologies and domains in which the dataare noisy, [?]. In fluid mechanics, GA have been applied to the parametric designof aircraft [?] and design of rockets and missiles [?]. Tong has used GA in thepreliminary design of turbines, [?].

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8 Abstract

A GA is a zero-order method that search in a stochastic fashion, capable offinding the global optimum and dealing with multi-objective optimization prob-lems. Firstly introduced by Holland [?] and developed by Goldberg, [21], it isbased on darwinian natural evolution. A GA repeatedly modifies a set of individ-uals (population) considered as optimal candidates by means of three operators,selection, crossover and mutation. At each iteration the algorithm selects indi-viduals at random from the current population to be parents, and use them toproduce the children for the next generation. Although such operation works ran-domly it is driven in such a way that benefits the selection of those individuals whoresult in the fittest candidates. Children are produced either by making randomchanges to a single parent (mutation operator) or by combining a couple of parents(crossover). There are a large number of different definitions of each operator1.Both operations are performed with an specific probability, mutation probabilityPm and crossover probability Pc respectively. Once new individuals are obtained,the algorithm replaces the current population with the children to form the nextgeneration, and the population size remains constant. The optimal values is alwayssearched for within a group of possible solutions, which is an important differencefrom other one-by-one basis search methods.

Genetic algorithms process populations of strings.In the simple GA, operators are applied to an entire population at each gener-

ation, as shown in Figure A. The population size is kept as constant, although itis possible also to change it as it happens in reality. However, for simplicity hereit is fixed to a constant value popSize. The first population is generated randomlyor according to a certain criteria if a previous information is available.

0.7.1 Genetic operators

In this subsection a brief introduction of the most characteristic operatorsapplied in Genetic algorithms is included. The operators are usually known asgenetic operators, and among the most common are the selection, crossover andmutation operators. These operators drive the search process by selecting thefittest individuals in the current population, combine [...]

The selection of the parents is a critical point in the GA procedure. Onemight expect, as it happens in the nature, that by mating two individuals withfavorable characteristics, also the offspring will have favorable characteristics evennot better. Then, the way the fittest individuals are recognized and selected isvital to obtain a next generation with better skills. As in MATLAB and Minamodifferent selection operators are implemented, it is considered necessary to presentthe ones considered in the following sections. These are the tournament and theroulette wheel selection methods.

Abstract 9

In a tournament selection, s individuals are chosen randomly from the popu-lation and the best among them is selected as the first parent. The second parentis selected in the same way. The parameter s is called the tournament size. Largevalues of s result in a more elitist selection while low values allow less fit parentsto be selected and result in a more diverse population. Harinck et al [56] proposeto use s = 2. Note that the entire optimization strategy rests only on retainingthe best out of two randomly selected individuals. Although this is a very simpleselection rule, it has proven to be very effective.

As an alternative, a roulette wheel selection can be used, see Figure. In thisselection mechanism, the probability of an individual to be selected is proportionalto its fitness. The higher its fitness is, the more portion of the roulette wheel isassigned to this individual and, therefore, a higher probability to be selected. [82]indicates that, in spite of being more ellaborated than the tournament selection, itconverges slower to the optimum in some cases. Where in a tournament selectionmore diversity is present under the individuals, as the less fit individuals also havea chance to be reproduced, the population of the roulette wheel selection will beless diverse due to the more elistist selection. The diversity of the population oftenallows the exploration of new domains of the search space, which is not possible ifthe binary strings of the individuals are too similar.

In crossover, two individuals swap part of the string so two new individu-als are formed as combinations of parts of the old strings. An example of theeffect of crossover is shown in Figure A. The operation shown is a single-pointcrossover, meaning that each parent string is disrupted at only one point. Multi-point crossover can also be used, and has been found to be useful in some appli-cations,

The population size is an important factor that determines the performance ofthe GA. Increasing the population size enables the GA to search more points andthereby obtain a better result. It is obvious that the population size should belarge enough to guarantee a satisfactory genetic diversity, which is essential to theGA because it allows the algorithm to explore more regions in the design space.However, the larger the population size, the longer the GA takes to computeeach generation. As it is clear that the total number of evaluations is directlyrelated to the population size (and the number of generations), without taking aprohibitive amount of time to run. To cut down this time cost without affectingsignificatively the population size Population diversity is another parameter relatedto the population concept which also affects the GA performance.

One of the most important factors that determines the performance of the GAis the diversity of the population.

10 Abstract

0.7.2 Implicit parallelism. Building blocks

Once it is understood the main characteristics of GA, and the genetic operatorsresponsible of its work, here we

In some sense, GA are not interested in strings as strings alone, but in what in-formation is contained in a set of strings and their objective function values to helpguide a directed search for improvement. To drive this search in a more efficientfashion, similarities among highly fit strings need to be observed. Holland definesa schema as a similarity template describing a subset of strings with similaritiesat certain string positions. A schema is represented by the binary alphabet {0,1}and a special metasymbol * which is a dont care or wild card symbol. In thisway, any individual in the population can be expressed by means of this ternaryalphabet. A schema acts as a pattern matching device, this is if, as an example,strings and schemata of length 8 are considered, the schema *00*0000 matchesfour strings or optimal candidates, namely {00000000}, {00010000}, {10000000}and {10010000}. It is observed that a schema gives a powerful and compact wayto talk about all the well-defined similarities among finite length strings over afinite alphabet, [?].

For alphabets of cardinality (number of alphabet characters) p, there are (p+1)lschemata, where l is the length of the string. Within the previous string lengthand alphabet, the total number of schemata is 38. However the number of differentstrings in the design space is only pl (28), so it seems that schemata, which aresupposed to help guide the search, enlarge the space and, therefore, complicatethe operation. What at first glance seems to be a drawback is in fact the keystoneof Genetic algorithms. The objective is to admit as much information as possibleby the similarities, and this amount of information does depend on the number ofunique schemata englobed in the population. A single string of length l is memberof 2l schemata, so a population of size n contains somewhere between 2l and n2ldifferent schemata, depending on the diversity of the population. Thus, a GA thatmanipulates relatively few strings (n) actually samples a vastly larger number ofregions (up to n2l). This behavior is called the implicit parallelism, and is whatgives the GA the central advantage over other problem-solving schemes, [?].

The way this information is exploited and transmitted into successive genera-tions is crucial for understanding the efficiency of Genetic algorithms. First, notall the schemata are equal, in terms that some are more specific than others, orare more or less compact. The order of a schema H in particular, is simply thenumber of fixed positions present in the template. This is, the order of the schema0*001*** is 4 whereas the order of schema *0*0011010 is 2. The higher the orderis, the more specific the schema results. The defining length of a schema is thedistance between the first and last specific string position, [?]. For example, theschema 0******1* has a defining length of 6, and the schema **1*01** has a

Abstract 11

defining length equal to 3. The larger portion of the string the schema span, themore likely to be disrupt in following operations. These properties are interestingfor analyzing the net effect of reproduction and genetic operators (crossover andmutation) on the implicit information contained within the population.

During reproduction (selection), a particular schema grows as the ratio of theaverage fitness of the schema to the average fitness of the population. This meansthat schemata with fitness values above the population average will receive anincreasing number of samples in the next generation, while schemata with fitnessvalues below the population average will receive a decreasing number of samples.This increasing or decreasing is exponential as given by [21]

m(H, t) = m(H, 0) (1 + c)l (2)

where m(H, t) refers to the number of samples of schema H at time (or gener-ation) t, 0 to the initial population and l the string length. Coefficient c indicatesthe amount above (or below) population average fitness a particular schema is,namely

f(H) = f + cf (3)

being f(H) the fitness of schema H and f the average fitness of the entirepopulation.

Crossover clearly complicates the effects of implicit parallelism. A particularschema is more or less likely to survive depending on the defining length. In fact, ifthe defining length is large, on average the cut point is more likely to fall betweenthe extreme fixed points and disrupt the schema. If the probability to be destroyedis pd = (H)/(l 1) where (H) is the defining length and considering that thecrossover is applied with a certain probability pc

m(H, t) = m(H, 0) (1 + c)l[1 pc (H)

l 1]

(4)

where the term in brackets refers to the probability of the schema to survive.Consequently, a schema is transmitted to the following generation and its presenceincrease or decrease depending whether the schema is above or below the popula-tion average and whether the schema has relatively short or long defining length.Mutation affects the growth or decay of the schema by the order of it o(H), givenby

m(H, t) = m(H, 0) (1 + c)l[1 pc (H)

l 1 o(H)pm]

(5)

with pm the mutation probability. It is clear that the higher the order is, theless likely to survive the schema is. Equation 5 is called the Schema Theorem

12 Abstract

or the Fundamental Theorem of Genetic Algorithms. Despite the disruption oflong1, high-order schemata by crossover and mutation, GA inherently process alarge quantity of schemata while processing a relatively small quantity of strings(individuals). The amount of schemata processed is of the order O(n3), where nwas the population size. These short, low-order and highly fit schemata whichare sampled, recombined and resampled to form potentially highly fit schemataare called building blocks. This term is crucial for understanding the GA per-formance. Instead of building high-performance strings (i.e. finding new optimalcandidates) by trying every conceivable combination, the GA constructs betterand better strings from the best partial solutions of past samplings.

0.7.3 Limitations of Genetic Algorithms

In the previous subsection it has been explained why GA work and we un-derstood that its performance does rely on the survival of the building blocks.The effectiveness of GA depends on useful correlations between parts of the ge-netic string (genotype) and the performance or fitness of the individuals it repre-sents (phenotype). High-performance, short-defining length, low-order schemata,called building blocks, are likely to survive into the next generation even if the geno-type is broken up by the action of crossover or mutation, but in general terms, itis the population size and the string definition itself, with the crossover and muta-tion probabilities, that play a relevant role in the survival of them. In [?], DeJongsuggested that good GA performance requires the choice of a high crossover prob-ability, a low mutation probability (inversely proportional to the population size)and a moderate population size.

Indeed, these choices are crucial, as problems may be observed in domainswhere building blocks are not recognized or they are not retained. This is directlyrelated to the population size and the genetic operators. The population is a smallsample of all possible cases in the design space. The idea is that the individualsinvolved in the current population contain as many building blocks as possible. Ifsome helpful building blocks are not present, the population does not represent soaccurately the design space, and a sampling error is assumed. Population size hasa strong effect on sampling error, with the error reduced in large populations wherebuilding blocks are more likely to be represented, [?]. Crossover operator is likelyto disrupt useful schemata when the components of those lie far apart in the string,because the probability of the cut point falling between the components is high.This is even more likely if a multi-points crossover mechanism is considered in theGA and if a high crossover probability is adopted. Therefore, the variables whichare expected to be coupled should be consecutive in order to form a useful building

1Here terms long and short refer to the defining length of the schema H.

Abstract 13

block which is more difficult to be broken by the crossover operator. In summary,GA parameters need to be conscientiously selected in order to both satisfactorilylead and speed up the search. A compromise between exploitation (helping thesurvival of the building blocks) and exploration (breaking them in order to createnew strings or, in other words, to move individuals to new unexplored regions inthe design space) has to be reached. As there is no prior knowledge about thedesign space landscape, different tests should be run to determine which are thebest conditions in which to find the optimum. Nevertheless,

0.8 Adjoint methods

0.8.1 Continuous and Discrete Adjoint solver

There are two different strategies to use the adjoints in CFD. These two ap-proaches are the continuous or the discrete one. The continuous way formulatesan adjoint partial differential equation system and the adjoint boundary condi-tions derivation. After that, both get discretized and numerically solved. Thediscrete approach starts from the linearization of the discretized primal equationsand transforms them into the corresponding adjoint equations. Thus, this strategyis not based on the form of the differential equations governing the flow but theactual discretized form of the equations used in the flow solver, [?]. In [?], adjointmethods based on continuous approach are considered, as Othmer proposes in [?]for ducted flows. In the ANSYS FLUENT Adjoint Solver user guide, a clarifyingexplanation of both methods is included. The continuous approach presents a po-tential flexibility of discritizing and solving the partial differential equations whichmay become into a pitfall as inconsistencies in modelling, discretization and solu-tion approaches can affect the sensitivity information. For this reason, the discreteadjoint approach is applied in ANSYS FLUENT Adjoint Solver and in this project.Othmer also uses discrete approach for automotive aerodynamic optimization, [?].

0.8.2 Discrete Adjoint solver implementation

As the discrete approach is selected in this project, the implementation startsfrom the discretized Navier-Stokes equations. A compact fashion of representingthe Navier-Stokes equations is proposed in [?]. Let us denote

Ri(u0,u1 . . .uM1; c

)= 0 (6)

the set of Conservation of Mass and Momentum equations, where i = 0 . . .M1refers to the cell j of the computational mesh, and M is the number of cells in theflow domain. So, the flow is defined by the Navier-Stokes equations resumed in R

14 Abstract

at the cell centroid for the whole flow domain. It is observed that the equationsinvolve a set of vectors ui evaluated at each and every cell, and a vector c. Theformer refer to the flow state and the vector ui compiles any fluid-related variable,as pressure or velocity components, in the cell ith. The latter is known as theinput variables vector, and involves all the user-specified inputs, as the modelgeometry, the computational mesh, any model parameters for turbulence modelsor the fluid properties.

The objective function is assumed to be a scalar, denoted as J , which is alsofunction of the vectors ui and c, i.e. the flow state and the input variables

J(u0,u1 . . .uM1; c

)(7)

The objective is to determine the sensitivity of the objective function withrespect to the input variables or, in particular, to the model geometry. If thegeometry is changed, the flow is also altered, so the objective function is affectedfor both the initial change of the input variables and the flow state modification,complicating the calculation of the aforementioned sensitivity.

If a variation of the input variables cj is considered, from equation 6 it isobtained that the variations in the flow state, namely ukj , with k = 1 . . .M , mustsatisfy

Ri

ukjukj +

Rijcj

u

cj = 0 (8)

where |u denotes that the flow state is held constant, and there is an impliedsummation based on indices j and k, where k = 1 . . .M . From equation 7, theobjective function changes as

J =J

ukjukj +

J

cj

u

cj (9)

Before, difficulties of determining the sensitivity of the objective function withrespect to the geometry because of the flow state have been introduced. Therefore,the aim is to remove the variations of the flow solution in 9 in order to leave theobservation as just an explicitly function of the input variables (geometry). If aset of adjoint variables ui is considered,(

uiRi

ukj

)ukj + u

iRijcj

u

cj = 0 (10)

a similarity between equations 7 and 10 may be proposed, concluding that

uiRi

ukj=

J

ukj. (11)

Abstract 15

It results into a system of equations which are the discrete adjoint equations.Solving this system allows determining the value of the adjoint variables qi, whichare present in

J =J

ukjukj+

J

cj

u

cj = uiR

i

ukjukj+

J

cj

u

cj =

(J

cj

u

ui Rij

cj

u

)cj (12)

after using equation 10, what permits determine the sensitivity of the objectivefunction J with respect to the input variables. It is remarked that the adjointsolution is specific to the flow state, what is pointed out in 12 by |u. It is out ofscope the way to solve this equations, as the methods considered to compute theJacobian of the system Ri/ukj . Details of the involved operations may be foundat [?]. The objective of this deduction is to show how only two simulations, one toobtain the flow solution, and one to compute the adjoint variables, are required.

0.8.3 Limitations

Gradient-based optimization can also be impractical in domains which havemany local minima. These algorithms use only local information to improve thedesign, so they cannot detect the global optimum if there is no path of continuousimprovement between their starting point and the optimum, [?].

0.9 ConclusionsThe operation of Genetic algorithms is remarkably straightforward. Starting

from a (random) population of n individuals, treated as strings, the algorithmcopies strings with some bias toward the best, mate and partially swap sub-strings, and mutate an occasional bit value for good measure. This explicit pro-cessing of strings really causes the implicit processing of many schemata duringeach generation. The fundamental concept of a schema or similarity templatewas introduced. Quantitatively, we found that there are indeed a large number ofsimilarities to exploit in a population of strings. Intuitively, we saw how Geneticalgorithms exploit in parallel the many similarities contained in building blocks orshort, high-performance schemata.

Natural selection eliminates one of the greatest hurdles in software design:specifying in advance all the features of a problem and the actions a program shouldtake to deal with them. Moreover, genetic operators allow optimal candidates toevolve much more rapidly than they would if each new design simply

Part I

Methodology

17

Shape Parameterization

0.10 Introduction

The need of considering any optimal candidate as a design vector introducesshape parameterization as an unavoidable step in shape optimization when GAare applied, see figure F. Actually, its role is of major importance in the successof the optimization procedure. By means of a certain number of design variables2,shape parameterization provides a representation of the design space, defining theshape of any geometry. Thus, a complete, simple and precise selection of thesedesign variables (and its range of variation) is critical for the parameterization ef-fectiveness. Such requirements are detailed below. First, to achieve a satisfactoryrepresentation of the nose of a high-speed train, a short review of the most relevantconclusions from previous train aerodynamics studies is presented. This is com-pleted with a summary of the main constraints for nose shape design. Once thegeometrical requirements are listed, different shape paremeterization techniquesare considered, among which Bzier curves are selected. Precision of the indicatedmethod is analyzed afterwards. Finally, robustness is tested by approximating theproposed parameterization to different nose shapes.

0.11 Shape parameterization requirements

The lack of completeness may preclude the possibility of finding optimal de-signs, [?], as these are not represented by the parameterization and therefore arenot included in the search space. If a not simplified choice of the design variablesis carried out, an excessive number of parameters may result in a baste designspace that difficulties the optimization search. The number of parameters need tobe kept as low as possible, excluding any parameter which can be determined bycorrelations. The range of variation of each design variable has also to be keptin a coherent order of magnitude. Such compromise between completeness and

2References also indicate that some boundary conditions to be optimized may be included inthe parameterization, [?].

19

20 Abstract

simplicity makes shape parameterization a crucial aspect in shape optimization.While leaving a relative grade of freedom to find new creative optimal geometries,simplifications have to be included by omitting parameters that are assumed tohave little or no impact in the performance of the geometry, [?]. As simplificationsare imposed, non-smooth features on the shape may appear. This behaviour hasa larger impact in mesh-morphing, as the algorithm is incapable of deformate themesh to this new non-smooth shape, rather than when remeshing is applied.

Precision is determined by approximating a specific shape, within a predefinedlevel of accuracy, to a shape parameterized geometry. The difference between theoriginal and the parametric geometries gives an estimation of the precision of theproposed parameterization.

0.12 Review of previous researches and normativegeometrical constraints

Before proposing the geometrical parameterization of the nose of a high-speedtrain, a review of the most relevant conclusions from previous works is carriedout. Restrictions included in the European regulations on Technical Specificationsfor Interoperability (TSI) are also taken into account to determine the range ofvariation of the design variables in the parameterization.

Some of these investigators base theirs studies in axially symmetric geometriesor a simplified inter-city express (ICE) 2 train. The first ones are very usefulwhen a general description of the flowfield is required, and is extensively usedin tunnel aerodynamics, but are far away from actual train geometries. Whilstit can be proposed a geometrical transformation from it to a more realistic one,in this project axially symmetric noses are not considered as a reference. Thelatter is a smooth model of the leading control unit of the ICE 2 and is known asthe Aerodynamic Train Model (ATM), which is widely accepted among the trainaerodynamics community as a reference geometry. The leading car is followed by asimplified end-car which is the same shape as the nose. No details as pantograph,bogies, partial bogies skirts, plough underneath the front-end or inter-car gap areincluded. Figure A represents the top, side and front view of the ATM geometry.This geometry is chosen as the base for developing the parameterization. In spite ofrepresenting an important fraction of the total external aerodynamic drag of a train(up to 47% and 20% the bogies and pantograph or roof equipment respectively,[?]), such simplifications are maintained in the nose description.

Orellano, [?] introduces several possible parameters to be optimized when im-proving the aerodynamic behavior of a high-speed train. These design variables area guideline to define our parameterization. Nose shrinking related to nose length,

Abstract 21

height of the nose tip, upper and lower curvature of the nose, bluntness or A-pillarroundness are some of these parameters. In [74] different nose train configurationsare depicted, where the influence of the length of the change in the cross-sectionalarea of the model train and the three-dimensionality on aerodynamic drag andtrain induced flows are analyzed. Noses close to ICE or Shinkansen can be ob-tained depending on these two parameters. In this paper, Raghunathan observedthat the nose length of the fore-body of the train (head) has no influence anymorewhen it is larger than two times the model width. Meanwhile, nose length of theafter-body (tail) still have influence even at four times the model width. This givesan idea of the maximum value of this parameter.

In [11], an aerodynamic optimization study of EMUV250 train is proposed.Modifications of roof (reducing the roof curvature), hood and lower part of the nosetip are introduced, changing the slope of the hood and the window, and altering thespoiler design and position with respect to the tip. The most interesting feature ofthe presented geometry in [11] is the non-differentiability at the connection betweenthe windshield and the hood (i.e., different slope for each part), close to the duck-type noses. This effect is considered to be represented by our parameterizationwhere, starting from ATM, it might be possible to arrive to a duck-nose, as AVETALGO-Bombardier, the Spanish high-speed train, see figure F1. Maeda et al.concluded that a blunt nose is effective on the micro-pressure wave, (for moreinformation about micro-pressure wave and tunnel aerodynamics read ). Theparabolic nose is the reference geometry in most of tunnel aerodynamics studies asa paraboloid introduces the minimum variation of cross-sectional area change, andthe blunter the nose is, the more reduction of pressure gradient (related with micro-pressure waves at the exit or compression waves inside the tunnel) is obtained.However, this leads to a larger drag coefficient. In [?] an optimal axially symmetricnose shape for reducing micro-pressure and drag is proposed. The slope of thecross-sectional area in the middle zone of the nose changes steeply from a positivegradient to a negative one. Their conclusions stresses the need of including in theparameterization the capability of generating a duck-nose.

0.13 Shape parameterization techniques. Bziercurves

In adjoint methods, optimization starts with an existing design, and the ob-jective is to improve its aerodynamic behavior by using numerical optimization.When GA is considered, a population of optimal candidates has to be defined andthere is no initial design from which start the optimization process. Optimizationaims to find the best design among all the possible ones. Thus, it is no necessary

22 Abstract

hcabin

hcabfloor

Heel-point

Offsetbetweenmaxim

um

andminim

um

cabin

size

Eyes

tall

driver

Endcoupler

Crash-structure

Coupleraxis

d min

d max

Driver

desk

Sightline

ha

Eyes

smalldriver

Figure 1: Lateral-side view (XZ) of the nose of ICE 2 train. Red dots refer to sampling pointsand red line is the generated spline of the lateral-side view. It is divided into four section boxes(represented in dash-dot line), numerated from 1 to 4: 1. Roof, 2. Windshield, 3. Hood and 4.Underbody. Dimensions are normalized with respect to bt, half of the train width. Nose lengthis limited to four times bt.

Abstract 23

to reproduce perturbations of an initial geometry but submit a flexible automatictechnique to represent any optimal candidate. If a geometry is taken as an initialreference (baseline shape), parameterization has to be able to accurately modeland include it in the design space. Definition of the ranges of each design variableis restricted to this condition.

Depending on which optimization method is applied, different shape parame-terization approaches can be deemed. In [?], an excellent survey of shape param-eterization techniques is presented. Discrete representation, analytical, ComputerAided Design (CAD) based or polynomial and spline approaches are included.

Discrete representation is based on using the grid-point coordinates as designvariables, what becomes into an unaffordable task when a 3D complex geometryis analyzed. [51] and [49] use Hicks-Henne functions superposition to obtain aaxially symmetric nose shape representation. Hicks-Henne formulation is basedon adding analytical shape functions linearly to the baseline shape. Design pa-rameters are the weights of each function contribution to the global expression.Low level of complexity for geometry changes is observed in such approach, [?].In [44], a 3D representation of the train nose by implicit functions as simple el-liptical and parabolic equations is applied, based on Chiu and Squire, [?], nosedefinition. [99] uses five control variables to define a very simple two-dimensionalgeometry, and [?] defines a common two-dimensional train nose shape by a set ofspline curves, and obtain a 3D representation of a high-speed train using OpenCAS-CADE CAD tool. CAD systems entail the incapability of calculating sensitivityderivatives analytically, [?]. Apart from it, some drawbacks may be pointed outwhen considering these approaches. First, at some cases the design variables haveno physical meaning. This implies a more complicated understanding of the ge-ometrical parametrization. Second, by using so few design variables most of therepresentations are just simplified approximations of real high-speed trains, andnot a large range of different nose shapes (i.e. more possibilities of optimal de-signs) is available. Other proposals used for different ground vehicles can be foundin and [?]. Rho, [?] presents a vehicle-modeling function in the form of an ex-ponential function to express the two-dimensional and 3D curved shapes of anautomobile. The subsectional parts of the vehicle-modeling funciton are definedas section functions by classifying each subsection of teh automobile as a sectionbox model. The resulting function for each section is given as Here we propose theapplication of Bzier curves, [?], to define the geometry of a 3-D high-speed trainnose in combination with Rho section boxes.

The Bzier curve of degree n equation is given as

C(t) =ni=0

(n

i

)(1 t)niti Pi (13)

24 Abstract

where 0 t 1 is a parameter control, and Pi are the control points to beweighted. A Bzier curve is defined by a set of control points P0 through Pn. Thefirst and last control points are always the end points of the curve; however, theintermediate control points (if any) generally do not lie on the curve. A quadraticcurve is defined by three control points, and a cubic one by four control points.Once the control points coordinates are defined, the Bzier curve is easily depicted.Typically, the control points coordinates are used as design variables.

Bzier curves are highly suited for shape optimization, as they can describe acurve in a very compact form with a small set of design variables, [?]. Comparingthem with previous Hicks-Henne function, they have a simpler formulation bymeans of polynomial functions. Moreover, the characteristics of the curve arestrongly coupled with the underlying polygon of control points, simplifying thelink between parameters and real design variables. Compared with Rho powerfunctions it is avoided to deal with coefficients which physical meaning is notexplicit. The convex hull of the Bzier control polygon contains the curve. Thisproperty is very useful, especially in defining the geometric constraints, [?].

0.14 3D nose parameterization approach

The baseline shape considered is ATM model. The parameterization is flexibleenough to be adapted to any other commercial train geometry.

To parametrically define the volume of the nose of a high speed train, first a2D description of the longitudinal profile of the nose is performed. Once this isobtained, cross-section profiles are generated. The is based on an original noseshape definition.

0.14.1 Cross-section

Since all the optimal candidates will be attached to the same train coach, thecross-section of the train is constant for all the geometries. The optimization isfocused just on defining the shape of the nose, while keeping unaltered the rest ofthe train. Thus, the cross-section is obtained by sampling the actual cross-sectionof ICE 2 train model. Figure 2 plots the original and the created spline throughall the sampled points. Its height Ht and width bt are considered as referencelengths for the following geometry definition. bt is half the width of the wholetrain. This data is taken to define the length of the nose Lt. From [?], as it wasexplained in section 0.12, maximum nose length where to observe notable changesin drag coefficient when front wind is affecting the train is two times the trainwidth. Then, as bt 1500 mm, Lt is fixed to 6000 mm.

Abstract 25

Height at which maximum width (bt) is obtained is hb. This value is usedlaterly to define the widths of frontal views at different transversal planes (Y Z).

13

23

1 43

73

2

53

43

1

23

13

0

83

ybt

zbt

sample

(a) Sampled ICE2 cross-section

13

23

1 43

73

2

53

43

1

23

13

0

83

ybt

zbt

bt

hb

Ht

(b) Generated cross-section after sampling andcharacteristic dimensions

Figure 2: Original and generated cross-section. Red dots in left picture refer to sampling points.On the right, in black line the generated profile is represented. Characteristic lengths, which areconsidered constants for the rest of the parameterization are depicted in red. Dimensions arenormalized with respect to bt, half of the train width.

26 Abstract

0.14.2 Lateral-side view

Two-dimensional description of the nose shape in the longitudinal symmetryplane (XZ) is based on four Bzier curves. These curves are related with fourcharacteristic sections of the nose. Such sections are roof, windshield, hood andlower part of nose tip. Underbody is supposed as horizontal. Curvature changesin a range of 10 mm are neglected. Bogies, bogies skirts and plough underneath(spoiler) are not considered. Resulting two-dimensional profile from simplifiedICE 2 is plotted in figure F. Red line does not involves window curvature, whichis supposed negligible.

In table 0.14.2 degrees of each curve is indicated. Eleven control points arenecessary to define the four curves. Below, a more detailed account of each sectionis included.

Section Order Control pointsRoof Quadratic P1, P2, P3Windshield Quadratic P3, P4, P5Hood Cubic P5, P6, P7, P8Underbody Cubic P8, P9, P10, P11

Table 1: Control points and order of the Bzier curve used to draw each section.In red fixed points are indicated. Shared control points are given in bold.

Roof shape

Roof shape is defined as the curve from the car body to the start position of thewindshield. A quadratic Bzier curve is used to represent this section as not bigcurvature is expected between these two points. Figure 4 plots the roof section.Three control points (P1, P2 and P3) adjust the curvature. P1 is the connectionpoint between front nose and car body. Coordinates of this point are (l0, Ht),where Ht is already known. l0 is the position at which the nose starts (origin isimposed at 6000 mm from the nosetip in xdirection). This value can be fixedor kept as a variable. Depending on its value, the length of the nose can vary. Asnose shrinking can be controlled by other design variables, it is recommended tokeep this value as a constant. One of these variables that control the length of thenose is l1, which actually defines the horizontal dimension of the roof. Point P3is located at (l0 + l1, h1), being h1 the height at which the windshield starts. Infigure 4, h1 is not given explicitly but as function of s1, where s1 = Ht h1.

A larger value of l1 drives to a longer roof while reducing the rest of the nose.Depending on the position of point P2, this can be translated into a nose shrinking.Vertical position of P2 is restricted to Ht, as tangency condition is expected at the

Abstract 27

x b t

z b t

1 32 3

14 3

7 3 2 5 3 4 3 1 2 3 1 38 3Sideview

sample

1

2

3 4

27 3

5 38 3

30

11 3

10 3

13 3

4

Figure 3: Lateral-side view (XZ) of the nose of ICE 2 train. Red dots refer to sampling pointsand red line is the generated spline of the lateral-side view. It is divided into four section boxes(represented in dash-dot line), numerated from 1 to 4: 1. Roof, 2. Windshield, 3. Hood and 4.Underbody. Dimensions are normalized with respect to bt, half of the train width. Nose lengthis limited to four times bt.

28 Abstract

connection between car body and front nose. This is a characteristic of Bziercurves, [?]. Therefore, three design variables are introduced, l1, h1 (or s1) and1. The latter controls the slope of the roof, defined by the convex hull of thecontrol points (red dashed line in figure 4). Maximum value of 1 is defined so asto generate a separation of the flow as low as possible. Default value considered is1max =

pi2rad. Meanwhile, in figure 4 it is observed that 1 cannot be lower than

a minimum value, as distance P2P3 has to be no larger than l1. The minimumvalue is function of l1 and s1. As roof slope is linked to window slope (see below),it is fixed that at least 1min =

pi9rad.

1min = arctans1l1

(14)

So, design variable 1 is translated into a coefficient k1 that defines the valueof the angle between the minimum and the maximum one. Range of variationof l1 and h1 are obtained from the cabin dimensions, which are, in turn, definedfrom the heel point position. In table 0.14.2, coordinates of each control point aresummarized. xcoordinate is based on the assumption of l0 as a constant.

1 = max(pi

9, arctan

s1l1

) + k1

(pi

2max(pi

9, arctan

s1l1

)

)(15)

Control Point x-coordinate z-coordinateP1 l0 (0) HP2 l1 - s1 cot1 HP3 l1 h1

Table 2: Coordinates definition of roof control points. ycoordinate is fixed tovalue 0 as lateral-side view refers to symmetry plane (XZ).

Windshield shape

Once P3 is already located, windshield is traced. It is sketched in two-dimensionsby a quadratic Bzier curve. The practical difficulty of building a windowpanewith two curvatures (what drives to the existance of an inflexion point) makes aquadratic curve enough for its representation. Three control points are neededto draw the curve, P3, P4 and P5. Then, only P4 and P5 remain to be placed.Dimensions of the windshield section box are l2 and s2, where s2 = h1 h2, beingh1 and h2 the heights of points P3 and P5 respectively. So, coordinates of P5 are[l0 + l1 + l2, h2]. Driver table location fixes the height of the lower edge of thewindow. This is transformed as the maximum value of h2. The minimum value ofthis design variable is obtained from the crash structure dimensions. Apart from

Abstract 29

x

z

P3

s1

l1

1P2P1

l0

Figure 4: Roof parameterization in symmetry (XZ) plane. Red line represents the roof givenby a quadratic Bzier curve. Red dots refer to control points. They are labeled in red color asP1, P2 and P3. Dashed red line represents the control polygon. In blue geometry restrictions ofdriver cabin size are given. Design variables involved in roof parameterization (l0, l1, 1 and s1)are written and depicted in black.

l2 and h2 (implicit in s2), a third design variable is introduced in this part. k2 is aparameter that controls the window curvature and, therefore, the position of P4.In order to satisfy first order (geometric) continuity at the shared control point(P3), the last segment of the roof curve and first segment of the windshield haveto be collinear. Hence, P4 position is function of 1. Depending on windshielddimensions (l2 and s2) and on 1 value, different cases might be observed. But be-fore window slope constraint is introduced. TSI, [63], indicates that the minimumslope (given by parameter w, see figure 5) of the windowpane is around 5pi36 rad.To prevent large flow detachment, the slope is limited to 5pi

18rad. If

tanw =l2s2

(16)

it is obtained that

l2 = (h1 h2) tan(wmin + kw {wmax wmin)} . (17)Consequently l2 is given by the parameter kw which implicitly defines the slope

of the windowpane. kw values range is function of the positional restrictions of thewindow with respect to the eyes of the driver. In [63] normative details which arethe closest and furthest position of the window following the sightline. In figure 5these points are plotted as a blue . Sightlines are represented by a blue dashedline. Also the field of vision is considered in two-dimensions to determine s2.

Depending if w < 1 or not, where w is the complementary angle of w, P4coordinates in x or z direction are limited by l2 or s2 respectively. Window

30 Abstract

curvature parameter k2 controls the position of P4 with respect to these limits.In table 0.14.2, coordinates of each control point are detailed. A more illustrativedescription is given in figure 5.

Control Point x-coordinate z-coordinateP3 l1 h1P4 l1 + k2 min(l2, s2 cot1) h1 k2 min(l2 tan1, s2)P5 l1 + l2 h2

Table 3: Coordinates definition of windshield control points. Again no data fory-coordinate is given as this profile is in symmetry plane (XZ).

x

z

Driver table

P4

P5

Eyes tall driver

Eyes small driver

w

P3

l2

s2

1

Figure 5: Windshield parameterization in symmetry (XZ) plane. Red line represents thewindshield given by a quadratic Bzier curve. Red dots refer to control points. They are labeledin red color as P3, P4 and P5. Dashed red line represents the control polygon. In blue geometryrestrictions of driver cabin size and driver table are given. Design variables involved in roofparameterization (1, l2, s2 and w) are written and depicted in black. It is also indicated theposition of the eyes of a tall and small driver according to TSI, []. Positions are defined respectto the heal point and the floor of the cabin. Blue dashed line refers to the view line of bothcases. Blue stars indicate the minimum and maximum distance from the driver to the front cabinwindow.

Hood shape

Train noses generally are streamlined, as ICE 2 or ICE 3, but there are alsodesigns where a discontinuous gradient point at the connection between windshield

Abstract 31

and hood is observed, see figure ??. To capture this feature, no tangency conditionis applied at P5 between both Bzier curves. The main restrictions of the hoodshape is the must of not intersecting the energy absorption box (crash structure)and the front connection size (end coupler). A cubic Bzier curve has be chosenbecause of its simplicity and flexibility to reproduce an inflection point in a curve.

Four control points are used to create the curve. P5 is already defined fromwindshield shape. P7 and P8 are fixed to be at the same xcoordinate in order toguarantee differentiability at the nosetip (i.e. avoid a cusp at the end of the nose).As point P8 represents the nosetip, such xcoordinate is the nose length itself. Itsvertical position (of P8) is given by height h3, from the ground. Vertical positionof P7 is given by an additional height h4.

Apart from these two new design variables, two more parameters are introducedto trace the cubic spline. These parameters help positioning point P6. In ordernot to intersect the curve with the end coupler, P6 is placed so that the curvepasses within a certain distance far from P , which is the limiting condition. Thus,a critical point in the Bzier curve lets defining the coordinates of P6, particularlythe zcoordinate. The xcoordinate is given as k3 [0, 1] times the hood lengthl3, where l3 = L (l0 + l1 + l2). The zcoordinate is such that the critical point isover P a distance given by parameter kh as z = P (z)+kh(P (z)h2), being z andP (z) the height of the critical point and P respectively. From z, zcoordinate ofP6 is obtained as Bzier curve equation is known.

Control Point x-coordinate z-coordinateP5 l1 + l2 h2P6 l1 + l2 + k3l3 h2 k3l3 tanhP7 L h3 + h4P8 L h3

Table 4: Coordinates definition of hood control points. Variable kh is translatedinto h as it gives the slope of the hood.

Underbody shape

In order to guarantee both first order continuities at the connections with thehood and the body of the train, underbody shape is described with a cubic Bziercurve. Nor spoiler neither bogies are considered in the geometry definition, sojust the outlined curve is depicted. This curve is constructed with four controlpoints. P8 is already settle. P11 is fixed although it may be leave free if lSpoiler3 isnot considered constant. If it is, coordinates of P11 are [L lSpoiler, hc], where hc

3Though the spoiler is not exactly defined by this design variable, proximity of P11 and thebeginning of the spoiler allows this similarity.

32 Abstract

x

z

h3

P5

P6

Crash structure

Critical point

End coupler

P

l3

k3l3

h4

P8

P7


is the clearance distance between the car and the top of the rail. Because of theaforementioned first order continuities, P9 and P10 are already partially positioned.The shape of the underbody is controlled by these two points. If length l4, whichplaces P10 is larger, a bluffer underbody profile is traced. The closer P10 to P11, themore slender the shape is. In any case, a restriction is the curve not to intersectwith the end coupler (in the lower side). The same procedure as in the hood is

Abstract 33

applied in the underbody to estimate the zcoordinate of P9, since the xone isfixed to L. Hence, a variable permits control such condition (ku). This variablecan also be translated as the slope of the underbody shape (u), while l4 gives anestimate of the actual dimension of the underbody. In table, coordinates of all thecontrol points are summarized.

Control Point x-coordinate z-coordinateP8 L h3P9 L hc + (lSpoiler l4)tanuP10 L lSpoiler + l4 hcP11 L lSpoiler hc

Table 5: Coordinates definition of underbody control points

x

z

P8

P9

End coupler

P

l4 lSpoiler

P10P11

Critical point


0.14.3 Front view

The above parameterization is enough to draw a two-dimensional profile of thenose of a train. When a three-dimensional construction is required, more base pro-

34 Abstract

files are necessary. If front view is considered, apart from the reference cross-sectiongiven in section 0.14.1 two more frontal profiles are used. Each one is defined atthe beginning of the windshield and hood respectively (i.e. at xcoordinates l0 +l1and l0 + l1 + l2 respectively). In this way, each section box represented in figure3 (but the underbody one) is based on two frontal profiles. In figure 8 the threefrontal curves are depicted in red. To draw these curves, each profile is dividedinto two parts, one referred to the upper side and one to the lower side, being thepoint where the curve is divided the one for the maximum width in each profile.Doing so it is possible to introduce different curvature at the upper and lower sideby just two cubic Bzier curves per profile. As these profiles initiate at the wind-shield and hood start respectively, control points P3 and P5 are included within thefourteen control points required to trace the frontal profiles. Each part of a profileconsist of four control points. No first order continuity condition is imposed atthe connection between the two parts, as it happens in the reference cross-section.However tangency is imposed at the symmetry plane connection to avoid cuspswhen the whole train is represented. To enhance this condition an offset of yoffthe lateral-side profile is introduced when the CAD tool is used to build up thegeometry. This is represented as the vertical red line in figure 8. In this figureICE 2 windshield and hood frontal profiles are plotted in dot-dash black line. Itis stressed that width profile, A-pillar roundness and tumblehome can be alteredby moving the respective control points.

The same philosophy is followed in the construction of the frontal profiles.Thus, here only the upper part of the hood frontal shape is explained. A cubicBzier curve is used for the upper part. Four control points are employed to drawthe spline, namely P5, P18, P19 and P20. P5 is already defined from the lateral-sideview. P20 coordinates are [bh, hbH ], where bh and hbH are two new design variablesthat determine the width of the frontal profile at the hood start and the heightrelated to this width, see figure 9. Points P18 and P19 are used to control thecurvature of the Bzier spline so that the A-pillar roundness can be changed. Toplace these points two more design variables are introduced. These are lAH andAH

4, where H denotes hood.

0.14.4 Top view

Frontal curves are not enough if a three-dimensional body construction is de-sired. To improve the surface control two extra curves are introduced. They areenglobed in the top view group as they consist in a two-dimensional curve laying

4Because of construction limitations in the CAD tool, tangency at the connection betweenupper and lower part at the hood is imposed, reducing the total number of five in the windshieldto four design variables in the hood

Abstract 35

ybt

zbt

13

23

1 43

13

23

1

43

53

2

73

83

0


in a XY -plane. At a height h3 from the ground, a plane is defined. In this planea curve resulting of the combination of a cubic Hermite interpolating polynomialcurve and a cubic Bzier curve is depicted. Using an Hermite interpolating poly-nomial, no extra points are introduced since the curve is based on already definedpoints belonging to the frontal profiles. Nosetip roundness is controlled by thecubic Bzier curve in order to have a more complete representation. Two designvariables are defined to fix the bluntness of the nose. bpeak and peak helps con-

36 Abstract

lAH

AH

P18

P5

P21

P22P23bh

hbH

P20

P19

hb


trolling the bluffness at the nose peak. Two of the four required control pointsare defined a priori as one is P8 and the other results from hood frontal profile (atheight h3). P25 is restricted to be at xcoordinate equal to the nose length in orderto guarantee tangency as the nosetip. Peak width is given by the ycoordinateof P25, so [L, bpeak] are the coordinates of this control point. P24 is placed suchthat tangency is observed between the Hermite interpolating polynomial and the

Abstract 37

x

Hermite polynomial Point at hood frontalprofile and h3

P25

P8

P24

b peak

bpeak


Bzier curve (what gives b) while at the same time satisfies chamfer given by anglepeak.

The second curve lays on a sloped plane to help the control of surface bumpinessbetween the frontal profiles. This curve is also a cubic Hermite interpolatingpolynomial, so no new design variables are demanded. In figure 8 this curve isrepresented in solid black line.

0.14.5 Summary

Table summarizes the meaning and function of the design variables in the shapeparameterization,. Only design variables for one part of one of the frontal profilesare included in the table. While the curve definition for each part is the same forboth the windshield and hood, in order to reduce the number of design variablesand taking advantage of the ICE 2 geometry, here lower part of the windshield issimplified to a geometric transformation of the reference cross-section lower part.It results in a complete three-dimensional shape representation based on 25 designvariables.

38 Abstract

Design variable Physical meaningl1 dimension of the roofh1 height at which starts the windshield (i.e. window)1 curvature at the connection between nose and car bodyl2 horizontal dimension of the windowpaneh2 height at which starts the hood (i.e. end of window)k2 window curvaturew slope of windowh3 height of the nosetiph4 (vertical) bluntness of the nosek3 hood slopekh gap between end coupler and hood surfacel4 relative size of the spoileru inclination of the underbodybh width of the frontal profile defining the hoodhbH height related to bhlAH controls where the A-pillar starts (front view)AH roundness of the A-pillar at the hood frontal profilebpeak (horizontal) bluntness of the nosepeak curvature of the nosetip (top view)

Table 6: Coordinates definition of underbody control points

0.14.6 Parameterization validation

Robustness and precision of the proposed parameterization are tested in thissection. Apart from the simplified ICE 2 train nose, two more commercial high-speed trains are considered to validate the parameterization robustness. These arethe Spanish high-speed train AVE, from Bombardier and Talgo, and the FrenchTGV, from Alstom. Only information of the lateral-side view are available, [] and[], so validation is restricted to check if the set of Bzier curves can reproduce thetrain profile satisfactorily. By changing the control points, and fixing the constantsto the train model in question, a very accurate representation can be obtained. Infigure 12 the results are plotted. The parameterization is flexible enough to fit thegeometry while keeping itself as simple as possible.

In order to validate the parameterization precision in this project, a compari-son of the pressure flow field and velocity at the train surface is performed betweenthe original geometry of ICE 2 train and the one resulting by the parameterizationand Bzier curves aforementioned. Qualitative conclusions are accompanied by aquantitative comparison, focusing on drag force and drag coefficient of the train.Both the original and the generated geometries are meshed within the same reso-lution criteria. y+ is set to 100, and standard wall functions are used to capture

Abstract 39

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 11: Examples of three-dimensional shape parameterization and nose construction.Train bodies are constructed with CATIA. Figure (a) corresponds to ICE 2.

the boundary layer close to the surfaces. k SST turbulence model is used. Asecond order upwind discretization is used for the momentum equation, turbulentkinetic energy and specific dissipation rate. Simulations are run with ANSYS-FLUENT commercial code, version 14.5. Reynolds number based on the train

Arodynamic Optimization of the Nose Shape of a HS Train

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