Probabilistic Multidisciplinary Design Optimization on a high …1135651/... · 2017-08-23 · UMEA˚ UNIVERSITY MASTER’S THESIS Probabilistic Multidisciplinary Design Optimization

UMEA UNIVERSITY

MASTER’S THESIS

Probabilistic Multidisciplinary DesignOptimization on a high-pressure sandwich

wall in a rocket engine application

Author:Dennis WAHLSTROM

Supervisors:Soren KNUTSRobert TANOJan OSTLUND

A thesis submitted in fulfillment of the requirementsfor the degree of Master of Science

in Engineering Physics

August 6, 2017

http://www.umu.se

https://www.linkedin.com/in/dennis-wahlstr%C3%B6m-62b563101/




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Declaration of AuthorshipI, Dennis WAHLSTROM, declare that this thesis titled, “Probabilistic MultidisciplinaryDesign Optimization on a high-pressure sandwich wall in a rocket engine applica-tion” and the work presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a master degreeat Umea University for GKN Aerospace.

• Confidential information has been removed or modified for publication.

• Where I have consulted the published work of others, this is always clearlyattributed.

• Where I have quoted from the work of others, the source is always given. Withthe exception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

Signed:

Date:

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“The important thing is not to stop questioning. Curiosity has its own reason for existence.One cannot help but be in awe when he contemplates the mysteries of eternity, of life, of themarvelous structure of reality. It is enough if one tries merely to comprehend a little of thismystery each day.”

Albert Einstein

Quote taken from: Old Man’s Advice to Youth: ’Never Lose a Holy Curiosity.’ LIFEMagazine. 1955: 64.

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Umea University

AbstractDepartment of Physics

Master of Science

Probabilistic Multidisciplinary Design Optimization on a high-pressuresandwich wall in a rocket engine application

by Dennis WAHLSTROM

A need to find better achievement has always been required in the space industrythrough time. Advanced technologies are provided to accomplish goals for human-ity for space explorer and space missions, to apprehend answers and widen knowl-edges. These are the goals of improvement, and in this thesis, is to strive and de-mand to understand and improve the mass of a space nozzle, utilized in an upperstage of space mission, with an expander cycle engine.

The study is carried out by creating design of experiment using Latin HypercubeSampling (LHS) with a consideration to number of design and simulation expense.A surrogate model based optimization with Multidisciplinary Design Optimization(MDO) method for two different approaches, Analytical Target Cascading (ATC)and Multidisciplinary Feasible (MDF) are used for comparison and emend the con-clusion. In the optimization, three different limitations are being investigated, de-sign space limit, industrial limit and industrial limit with tolerance.

Optimized results have shown an incompatibility between two optimization ap-proaches, ATC and MDF which are expected to be similar, but for the two limita-tions, design space limit and industrial limit appear to be less agreeable. The ATCformalist in this case dictates by the main objective, where the children/subproblemsonly focus to find a solution that satisfies the main objective and its constraint. Forthe MDF, the main objective function is described as a single function and solvedsubject to all the constraints. Furthermore, the problem is not divided into subprob-lems as in the ATC.

Surrogate model based optimization, its solution influences by the accuracy ofthe model, and this is being investigated with another DoE. A DoE of the full facto-rial analysis is created and selected to study in a region near the optimal solution.In such region, the result has evidently shown to be quite accurate for almost allthe surrogate models, except for max temperature, damage and strain at the hottestregion, with the largest common impact on inner wall thickness of the space nozzle.

Results of the new structure of the space nozzle have shown an improvement ofmass by ≈ 50%, ≈ 15% and ≈ −4%, for the three different limitations, design spacelimit, industrial limit and industrial limit with tolerance, relative to a reference value,and ≈ 10%, ≈ 35% and ≈ 25% cheaper to manufacture accordingly to the definedproducibility model.

http://www.umu.se

http://www.physics.umu.se/english

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Umea University

SammanfattningDepartment of Physics

Master of Science

Probabilistic Multidisciplinary Design Optimization on a high-pressuresandwich wall in a rocket engine application

by Dennis WAHLSTROM

Ett behov av att oka prestandan har alltid varit eftertraktat inom rymdindustrin.Avancerade teknologier skapar nya mojligheter for manniskan att genomfora rym-dresor for att kunna besvara fragor och forbattra bade kunskap och forstaelse omrymden. I den har uppsatsen handlar om att forsta och minimera massan pa ettrymdmunstycke som anvands i ett ovre raketstegsuppdrag med en expandercykelmotor.

Studien ar genomford genom att skapa forsoksplan med Latin Hypercube Sam-pling (LHS) som tar hansyn till behovet av ett stort antal design och simuleringstid.Surrogat modeller ar framtagna och optimering ar genomford med MultidisciplinarDesign Optimering (MDO) metod for tva olika ansatser, Analytical Target Cascad-ing (ATC) och Multidisciplinary Feasible (MDF) som har anvants for att jamfora ochverifiera slutsatsen. I optimeringen, har tre olika begraningar undersokts, design-srum grans, industriell grans, och industriell grans med tolerans.

Resultat fran optimeringen har visat pa en inkompatibilitet mellan de tva olikaansatserna, dar ATC och MDF forvantades vara lika for de tva begransningarna,designsrum och industriell grans. I det har fallet, styrs ATC formuleringen av hu-vudproblemet, dar underproblemen fokuserar enbart pa att hitta en losning somuppfyller huvudproblemet och dess tvangsvillkor. For MDF formuleringen, ar hu-vudproblemet definierad som ett singel problem som loses med avseende pa allatvangsvillkor, dessutom har problemet inte delats in i underproblemen som i ATCformuleringen.

Valet av surrogatmodell har stor paverkan pa optimeringens exakthet, och robus-theten i optimeringen har undersokts med en annan typ av forsoksplan. I den harstudien skapas en forsoksplan med full faktoriellt metod som ar centrerad nara denoptimala losningen. Inom en sadan region har resultatet visat en overenstammelsefor de flesta surrogatmodellerna forutom max temperatur, livslangd och tojning paden varmaste delen och har visat sig ha storst effekt pa innervaggstjockleken parymdmunstycket.

Med de nya geometrin pa rymdmunstycke har massa forbattrats med ≈ 50%,≈ 15% och ≈ −4%, for de tre begransningarna, designsrum, industriell grans ochindustriell grans med tolerans, relativt till ett referensvarde och ≈ 10%, ≈ 35% och≈ 25% billigare att tillverka med tankt tillverkningsmetoden.

http://www.umu.se

http://www.physics.umu.se/english

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AcknowledgementsTo begin with I would like to thank my supervisors, Soren Knuts, Robert Tano andJan Ostlund, who have been great support to this thesis and given me this uniqueopportunity to work with a very interesting project and what I like to do. Especially,Soren Knuts for his high effort for supporting and giving me a clear figure aboutthis project and how to solve it, this has been an important step to keep solvingthe problems and moving forward. During the thesis at GKN I have learned manythings such as how to cooperate in a group and about space industry, things that Idid not know before.

My advisors, Raja Visakha at GKN and Eddie Wadbro at Umea University forsuggesting and providing me an idea and understanding about MultidisciplinaryDesign Optimization method. Magnus Wir for supporting on how to describe theproduction cost and manufacturing limits. Lars and Ingegerd Ljungkrona for help-ing me with ModeFrontier Software. Arne Boman for his aerodynamics and thermo-dynamics explanations, Martin Carlsson for the help for understanding FEI spacenozzle better, Oscar Linde for helping with linux OS, and many thanks to everyoneat space nozzle’s team that was very supportive and able to provide me a happywork place. Alexander Hall, another thesis workers and trainees for all the enter-tainments.

Time at GKN has been a great pleasure and a memory of life.

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Contents

Declaration of Authorship iii

Abstract vii

Sammanfattning ix

Acknowledgements xi

1 Introduction 11.1 GKN Aerospace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Main component of a space rocket . . . . . . . . . . . . . . . . . . . . . 21.4 Space Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Objective of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theoretical Background 72.1 Physics of Space Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Thermodynamics and Chemical Reaction . . . . . . . . . . . . . 82.1.2 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Fatigue crack initiation . . . . . . . . . . . . . . . . . . . . . . . . 10Ductile Rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4 Physics in a sandwich wall . . . . . . . . . . . . . . . . . . . . . 122.2 Statistic Background and Design of Experiment . . . . . . . . . . . . . . 13

2.2.1 Latin Hypercube Sampling . . . . . . . . . . . . . . . . . . . . . 132.2.2 Full Factorial Analysis . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Multidisciplinary Design Optimization Architecture . . . . . . . . . . . 152.3.1 Single Objective, Multidisciplinary Design Optimization Ar-

chitecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Multi Objective, Multidisciplinary Design Optimization Ar-

chitecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.3 Individual Disciplinary Feasible . . . . . . . . . . . . . . . . . . 18

Analytical Target Cascading . . . . . . . . . . . . . . . . . . . . . 18Augmented Lagrangian Relaxation . . . . . . . . . . . . . . . . 19

2.3.4 Multidisciplinary Feasible . . . . . . . . . . . . . . . . . . . . . . 20

3 Implementation and Multidisciplinary optimization 213.1 Main study methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Decomposition of cooling system . . . . . . . . . . . . . . . . . . . . . . 233.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1 Data Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 Aerodynamics and thermodynamics simulation . . . . . . . . . 26

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3.3.3 Solid mechanics simulation and finite element method . . . . . 273.4 Prediction of surrogate model . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.1 Natural behavior of the surrogate model . . . . . . . . . . . . . 283.5 Multidisciplinary Optimization . . . . . . . . . . . . . . . . . . . . . . . 28

3.5.1 Optimization constraints . . . . . . . . . . . . . . . . . . . . . . 283.5.2 Optimization problem formulations . . . . . . . . . . . . . . . . 303.5.3 Local and global optimization . . . . . . . . . . . . . . . . . . . 333.5.4 Different limitations . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Certainty of Surrogate model . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Data analysis and Results with Discussion 354.1 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Fitting of the surrogate model . . . . . . . . . . . . . . . . . . . . 354.1.2 Quality of response surface and surrogate model . . . . . . . . 364.1.3 The limitations used to optimize . . . . . . . . . . . . . . . . . . 36

4.2 Results of optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Probabilistic analysis of uncertainty . . . . . . . . . . . . . . . . . . . . 41

4.3.1 Accuracy of Surrogate Model . . . . . . . . . . . . . . . . . . . . 414.3.2 Source of uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 434.3.3 Impact of uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Conclusion and Recommendation 475.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Bibliography 49

A Derivation 51A.1 Main jet expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

B Figures and Tables from Surrogate model 53

C Addition figures 61

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List of Figures

1.1 Schematic picture of Ariane 6 main components. (ESA) . . . . . . . . . 31.2 Schemetic system picture over an expander cycle engine. . . . . . . . . 41.3 Picture of ETID (left) and FEI (right) [5]. . . . . . . . . . . . . . . . . . . 51.4 Two identical channels in a sandwich wall, observed from top view

point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Sections of space nozzle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Cooling system in sandwich wall. . . . . . . . . . . . . . . . . . . . . . . 122.3 Full factorial experiment for 22. . . . . . . . . . . . . . . . . . . . . . . . 142.4 Single-Objective, MDO architecture. . . . . . . . . . . . . . . . . . . . . 162.5 An illustration of Hierarchical in the ATC case. . . . . . . . . . . . . . . 182.6 An illustration Nof on-Hierarchical in the ATC case. . . . . . . . . . . . 19

3.1 MDO Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Half of cooling channel. Left side is the flame side and right side is

toward the outside, with meshes. . . . . . . . . . . . . . . . . . . . . . . 233.3 LHS for 7 design factors for two different design spaces 0.5− 1.5 and

0.2− 1.7, with 50 and 200 as number of designs. . . . . . . . . . . . . . 253.4 Condensation operating point with wall temperature toward and in

the flame side, plotted with the lowest temperature different ∆Tcondenseand the saturation line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Channels in cooling system [5]. . . . . . . . . . . . . . . . . . . . . . . . 293.6 System Level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.7 Hieararchical ATC problem formulation. . . . . . . . . . . . . . . . . . 303.8 ATC Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.9 An example concept and illustration on the different limits, Design

Space (D.S.), Industrial (Ind.) and Industrial with Tolerance, are de-fined with function’s minimum. . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Optimized geometry from different limits and methods. . . . . . . . . 404.2 Plots of ANOVA, Actual vs. Surrogate model with its residual. . . . . 414.3 Plots of ANOVA, Actual vs. Surrogate model with its residual. . . . . 424.4 Effect influence for each design variable. . . . . . . . . . . . . . . . . . 44

B.1 Correlation matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54B.2 Plot of mass over design variables. . . . . . . . . . . . . . . . . . . . . . 54B.3 Plot of ∆Tc over design variables. . . . . . . . . . . . . . . . . . . . . . . 55B.4 Plot of ∆PC in logarithmic scale over design variables. . . . . . . . . . 55B.5 Plot of Tmax over design variables. . . . . . . . . . . . . . . . . . . . . . 56B.6 Plot of damage in logarithmic scale over design variables. . . . . . . . 56B.7 Plot of heat pick-up over design variables. . . . . . . . . . . . . . . . . . 57B.8 Plot of ∆ε1 over design variables. . . . . . . . . . . . . . . . . . . . . . . 57B.9 Plot of ∆ε3 over design variables. . . . . . . . . . . . . . . . . . . . . . . 58

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B.10 Plot of mach number over design variables. . . . . . . . . . . . . . . . . 58B.11 Plot of temperature different, ∆Tcondense over design variables. . . . . . 59B.12 Plot of ∆Pc over design variables. . . . . . . . . . . . . . . . . . . . . . . 59B.13 Plot of Damage over design variables. . . . . . . . . . . . . . . . . . . . 60

C.1 Different strain peaks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61C.2 Optimized with mass as objective and no physical constraints. . . . . 61C.3 Temperature different from Ansys simulation. . . . . . . . . . . . . . . 62

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List of Tables

1.1 Tolerance cost for weight reductions 1 kg or 1% of the structural weight.[4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.1 Description of design variables. . . . . . . . . . . . . . . . . . . . . . . . 233.2 List of constraints and each of its bound with description . . . . . . . . 29

4.1 Number of data left to create surrogate model for 0.5 to 1.5 and 0.2 to1.7 interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Fitting quality to the data for each variable using 200 designs betweena LHS interval of 0.2 to 1.7. . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Different limitations for each design variable. . . . . . . . . . . . . . . . 364.4 Total number of local solver evaluated and converged in the MDF and

ATC for different limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.5 Result from the optimization for MDF and ATC, for different limits.

The results are relative to reference values. . . . . . . . . . . . . . . . . 374.6 Result from the optimization for MDF and ATC, with a comparison

between surrogate models and actual results without industrial lim-its. The results are relative to reference values. . . . . . . . . . . . . . . 38

4.7 Result from the optimization for MDF and ATC, with a comparisonbetween surrogate models and actual results with industrial limits.The results are relative to reference values. . . . . . . . . . . . . . . . . 38

4.8 Result from the optimization for MDF and ATC, with a comparisonbetween surrogate models and actual results with industrial limitsand tolerances. The results are relative to reference values. . . . . . . . 39

B.1 Fitting quality to the data for each variables using 50 designs betweenan interval of 0.5 to 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

B.2 Fitting quality to the data for each variables using 50 designs betweenan interval of 0.2 to 1.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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List of Abbreviations

ATC Analytical Target CascadingASL Airbus Safran LaunchersDS Design SpaceDoE Design of ExperimentESA European Space AgencyETID Expander Technology Integrated DemonstratorFEI Flight Engine ImageFEM Finite Element MethodHCF High Cycle FagtigueIDF Individual Discipline FeasibleLCF Low Cycle FagtigueLHS Latin Hypercube SamplingMDO Multidisciplinary Design OptimizationMDF MultiDisciplinary FeasibleMOMDOA Multi Objective Multidisciplinary Design Optimization ArchitectureNE Nozzle ExtensionSOMDOA Single Objective Multidisciplinary Design Optimization ArchitectureSQP Sequential Quadratic ProgrammingSWAN SandWich Advanced Nozzle

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List of Symbols

Ae nozzle exhaust area m2

E modulus of elasticity Pahg heat transfer coefficient W m−2K−1

H enthalpy JIsp specific impulse m/s2

l area moment of inertia m4

m mass kgm mass change kg/snp particle density mol−1m−3

Nc number of cycle Unit-lessP pressure PaPa ambient pressure PaPc coolant pressure PaPe exhaust pressure Paq heat flux W/m2

qg heat flux of gas W/m2

Q heat WQtot heat pick-up Wr radius of gyration mR ideal gas constant J mol−1 K−1

Rε strain ratio Unit-lessRσ stress ratio Unit-lessT temperature KTc coolant temperature Ku velocity m/sU internal energy JV volume m3

ve exhaust velocity m/sW work J

∆ physical different Unit-lessε strain rate Unit-lessγ heat capacity ratio Unit-lessρ density kg/m3

σ stress Pa

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Dedicated to my parents.

1

Chapter 1

Introduction

Space is a subject that has been curious to us for many centuries and even beforeChristian. Aristotle ideal about cosmology and heliocentric model was grounded byAristarchus of Samos had stunned many individuals, but at that time it was lackedmathematics and unable to prove, and was barely accepted as a fundamental con-tent [1]. These doctrines had become more underwritten many centuries later whentelescope had been invented and able to observe and prove this theory with both ob-servational and mathematical. Copernicus presented a heliocentric model indepen-dently of Aristarchus based on his astronomical observations. Later in seventeenthcentury, Galilei could prove the heliocentric model with a scientific method, duringthe time in his house arrest caused by the heliocentric ideology instead of geocen-tric ideology that was approved by the church. The heliocentric model describesplanetary motion, and in fact was one of the starting point for Kepler. He advancedthis ideology and provided laws that planets unavoidably must obey and known asKepler’s law. The result was essential to physicist at that time, and its derivation ofKepler’s law was utilized by Newton to generalize motion laws and universal gravi-tation, Newton’s law [2] [3].

Curiosity is one of the strongest ”sense” in human nature, and it is a center ofevolution to strive to answer on these inquires, what, why and how. Without theseperspectives, the evolution and development of technology would have been differ-ent from today and this comes to our cause of matter.

In this section, a short introduction about GKN Aerospace will be reviewed aswell for background and objective of this thesis, together with an explanation ofspace rocket mission, and briefly introduce the importance of space nozzle that willbe focusing.

1.1 GKN Aerospace

GKN Aerospace is a sub-division of GKN concern, where GKN is one of the biggestcompany in the world with employee’s number of 55000 worldwide in over 33 coun-tries, and 3500 employees mainly in Sweden, in 2017. GKN contains of four dif-ferent main divisions, aerospace, driveline, land systems and powder metallurgywith a headquarters located in Redditch, England. The company focus is on man-ufacture vehicle and aircraft components. GKN Aerospace Engine Systems is lo-cated in four countries, Sweden, Norway, Mexico and USA, with a headquartersin Trollhattan, Sweden. In Sweden, GKN Aerospace is located in two differentlocations, Trollhattan and Linkoping with focus on aircraft in production of aero-structures, engine and transparencies.

2 Chapter 1. Introduction

1.2 Background

The development of space rocket is essential for us to provide a possibility to travelouter space and explore the outer world. This conception requires a more advancedtechnology that we do not have today, the components for each space rocket partneed to be more effective, which is challenging for engineers and scientists. Spacerocket and space shuttle have been manufactured since middle of twentieth cen-tury, when Yuri Gagarin went outer space and Neil Armstrong went to the moon,and when the first artificial satellite Sputnik in-landed in the Earth’s orbit, and eversince the space industry keep expanding. New methods and technologies providepossibility in opportunity to make software more effective and complex with morefunctions able to interpret and handle issues and bugs cause from the methods. Theissues and causes that today stand as a center is that the solutions are not robust andneed to be investigated. Last but not least, cost of all actions need to be reduced totheir minimum for maintenance. Table 1.1 show costs of weight reduction betweenvehicle, aircraft, large-aircraft and space rocket. An optimized design solution forspace rocket has a greater impact for cost standpoint comparing with aircraft, there-fore this is an essential task for engineers and scientists to examine.

TABLE 1.1: Tolerance cost for weight reductions 1 kg or 1% of thestructural weight. [4]

Te/kg Te/wt.%Vehicle 0.01 0.1Aircraft 0.5 100Large-aircraft 1 1000Space rocket 10 10000

One project that GKN participates in is Ariane 6 program, originally proposed byEuropean Space Agency (ESA) and sub-divided to Airbus Safran Launcher (ASL).GKN is enrolling in and responsible for the development and production of turbineengine for upper stage and first stage, an upgrade engine from Vulcain II use in Ar-iane 5. For the upcoming Ariane 6, the nozzle is called, Sandwich Advanced Nozzle(SWAN). In the future, with great technology GKN is processing and wanting toadvance to design and manufacture even for the upper stage nozzles.

1.3 Main component of a space rocket

Space rocket body can be divided into three parts, where each part has its own func-tion, booster + first stage, upper stage and fairing, an illustration can be seen in figure1.1. The space rocket is containing of main engine, space nozzle, booster, load (e.g.satellite, supply) and space rocket body, where most of the weight is fuel. Boostersare used to give an initial speed for the space rocket and only remain in the first fewminutes of the flight, and at the same time the main engine starts and burns throughthe first stage, until it reaches the calculated altitude. Furthermore, the main engineignites before the lift-off to ensure the system is performing correctly and avoidingthe catastrophe. Where later, the upper stage engine starts to run and leave half ofthe space rocket body behind, at this point the space rocket is at higher altitude andthe air resistance is low. Lastly, the fairing remains and the mission continuous tooperate in space.

1.4. Space Nozzle 3

FIGURE 1.1: Schematic picture of Ariane 6 main components. (ESA)

1.4 Space Nozzle

Space nozzle experiences high load in thermal and pressure produced by the rocketengine, where the gas temperature can reach up to 5000 K and normally exceedthe material melting point. The high thermal load can be handled by creating acooling system to reduce the temperature and still being able to bring out the sameeffectiveness from the engine.


FIGURE 1.2: Schemetic system picture over an expander cycle engine.

There are different types of space nozzles suited to specific engines. In Ariane 6,an upgrade Vulcain II is designed to use a gas generator cycle, differently from theupper stage engine Vinci utilizes another principle. That is a gas expander cycle, forreusing the fuel in a different way. The gas expander cycle is shown in figure 1.2which will be focused on in this thesis.

Gas expander cycle utilizes heat/temperature generated from the engine in away that fuel travels down and absorbs the heat, and later is being reused to drivethe turbine. Another concept is a gas generator cycle, fuel separates and divides intotwo ways, first, to a pre-burner and second to the nozzle acts as coolant. Further-more, the fuel that has cycled around the nozzle will be disposed in the combustionchamber without being reused. The pre-burner is connected to oxidizer pump andto fuel pump, which some of the oxidizer is divided to the pre-burner and the com-bustion chamber for decreasing the temperature. Fuel and oxidizer that accumulatesin the pre-burner is used to drive the turbine, and discarded after being used.

Flight Engine Image (FEI) and Expander Technology Integrated Demonstrator(ETID) are shown in figure 1.3, and they are the space nozzles suit for the gas ex-pander cycle engine. Their structures are similar as well for their physics and forthe biggest noticeable in the figure is the height, that affects to the physics and isnegligible. In this thesis, the space nozzle FEI will be selected to study.

1.4. Space Nozzle 5

FIGURE 1.3: Picture of ETID (left) and FEI (right) [5].

There is another important thing concerning this thesis, is the sandwich wall ofcooling channels, and it is shown in figure 1.4. The observing point is from topview, where the considered system is for two channels with a complete symmetricbetween them when it comes to size and this two channels system is repeated intothe whole size space nozzle. In the figure, an idea is to illustrate the channels thatwill be studied and a fully and more rigorously explanation will be provided in 2.1.4and 3.3.1 sections.

FIGURE 1.4: Two identical channels in a sandwich wall, observedfrom top view point.


1.5 Objective of the thesis

The objective of this thesis is to find the ”optimal” structural design of sandwich wallof FEI space nozzle with mass as the main optimization objective, using Multidis-ciplinary Design Optimization (MDO) method. By creating Design of Experimentand forming a surrogate model of aerodynamics, thermodynamics and solid me-chanics, and analyze their interactions and addiction on the design variables. Thisis for widening knowledge and understanding of the space nozzle. A compari-son between two different MDO problem formulations, Analytical Target Cascading(ATC) and Multidisciplinary Feasible (MDF) will be investigated and concluded forstrengthening the reliability of the solutions. The optimal solutions of the space noz-zle need to be considered to the manufacture and production perspective, that GKNhas high value and expectation.

In this study, simulations will be performed by using GKN’s develop methodcalculating aerodynamics, thermodynamics and solid mechanics. This aerodynam-ics and part of the thermodynamics, their results are received from using a Tccoolscript, and later apply into the Ansys software for solving solid mechanics and ther-modynamics in two dimensional. A post process script CUMFAT is utilized to ex-tract the solid mechanics information from the Ansys simulations. Thus, the resultsfrom the simulations are used to construct surrogate models requiring in the study,for the later matter use in optimization. Herein, the MATLAB software program isselected to use to imply the MDO method for a non-linear optimization.

7

Chapter 2

Theoretical Background

In chapter 1, a short introduction about space nozzle and objective of this thesis havebeen introduced, where the space nozzle that will be used in this study is FEI whichis an upper stage engine with an expander cycle system. The need to understandsuch system is important, with a goal in chapter 2 to give a concise and wide intro-duction about physics theory and statistic theory that will be needed to understandthis thesis.

The study can be done in many ways, with a conception to find surrogate modelsthat describes physics in term of geometry. Besides, the analysis is consisting of fi-nite element method (FEM), with that cause the simulations are time expensive andneeded to construct and formulate the problem in such way that reduces the num-ber of simulations. Therefore, the decomposition of the problem becomes essential.One of the ways to engage the problem is by creating a Design of Experiment (DoE)scheme and benefiting its feasibility of probabilistic spread of the data and simu-late the physics, aerodynamics, thermodynamics and solid mechanics, that will bepreviewed in this chapter.

2.1 Physics of Space Nozzle

Space nozzle experiences high thermal and pressure loads created from turbine andchemical reaction between fuel and oxidizer. This energy production is used to ac-celerate space rocket, higher velocity results in higher thrust force, and at the sametime the material needs to sustain such extreme conditions. A need to understandthe energy travels and acts in the space nozzle is a key to visualize the thermal andpressure loads affecting the material, and with that knowledge, the cooling systemcan be designed designate to these physical behaviors and properties.

A space nozzle is divided into different sections and each section has its ownphysical meaning for bringing out the possible capacity. An illustrated picture of thiscan be seen in figure 2.1. Four different regions, combustion chamber, convergingsection, throat and diverging section.

8 Chapter 2. Theoretical Background

FIGURE 2.1: Sections of space nozzle.

2.1.1 Thermodynamics and Chemical Reaction

Space nozzle regions can be seen in figure 2.1. In the first region, combustion cham-ber, an interaction between fuel and oxidizer occurs, that is exchanging and releasingenergy in form of thermal energy. This phenomenon is called exothermic reaction:

reactant→ product + energy (2.1)

For a reaction between liquid hydrogen and liquid oxygen:

2H2 +O2 → 2H2O (2.2)

Resulting in bonding energy described as an enthalpy ∆H and has its physical for-mula:

∆H = HCreation −HAnnihilation (2.3)

whereHCreation is the enthalpy used to create a system andHAnnihilation is the energyuse to destroy a system, with the definition of enthalpy:

H ≡ U + PV (2.4)

An amount of thermal energy U created from the substances obeys the first law ofthermodynamics and the ideal gas law to the first order. The first law of thermody-namics is defined as:

∆U = Q−W (2.5)

2.1. Physics of Space Nozzle 9

where the change of internal energy ∆U is equivalent to the difference between heatQ and work W in the system. Besides, the pressure P , that produces in the combus-tion chamber in the first order can be illustrated by the ideal gas law:

P = npkbT (2.6)

with np as particle density, kb Boltzmann constant and T temperature. This concep-tion has not yet violated any physical law, but in an experimental concept, the idealgas law is not the fundamental way of describing the pressure in a space nozzle, ex-cept it contains more terms to equation (2.6), by using of the perturbation theory andgetting the correction terms. This also applies to the diverging section. In additionto the diverging section, its physical properties of thermodynamics and aerodynam-ics, their character is behaving as an isentropic, meaning it is both adiabatic andreversible [6].

When the wall of a space nozzle has low heat resistance the temperature tendsto decrease and causes the gas to change its property due to the phase transition.The condensation is determined by temperature and pressure of the hot gas, with adifferent in density between the two phases, liquid and gas. This phase transitionis decided by a saturation line, which is a line that separates different states andphases of the gas, solid and liquid. A well-known theory of condensation on a solidinterface is resolved by Nusselt, which in this thesis will be referred to the originalNusselt’s work [7].

2.1.2 Aerodynamics

In the combustion chamber the temperature and chemical reaction are dominating,but in converging section, throat and diverging section the aerodynamics is the focalproblem. In addition to the earlier statement, the ideology of the space nozzle’s sec-tions is to convert the thermal energy into kinetic energy, with the thrust force of thespace rocket depends on the exit velocity. This can be illustrated as the momentumlaw. This also implies that each section has its specific velocity, and in the convergingsection the fluid has a subsonic pattern, while at the throat the velocity increases andbecomes a sonic. Conceptional, the desired outcome can be regulated by decreasingarea and control the mass flow of the fluid. The mass flow over a surface Ω1 is givenby:

dm

dt=

∫∫Ω1

ρ1u1dA1 (2.7)

Imaginably, the entry of the converging section as 1 and at the throat as 2, by usingthe mass flow ratio between two sub-systems assumed to be equal, a relation can bedetermined:

ρ1u1A1 = ρ2u2A2 (2.8)

In such system, the velocity u2 can be controlled by varying area A2 while considersection 1 as initial, this principle is similar to the venturi principle. For the divergingsection, fluid’s velocity keeps increasing, while temperature, pressure and densitydecreasing, which is reversible from the converging section and the throat. Withoutthis process, the fluid’s velocity in the space nozzle unable to exceed one Mach [8].


The exit velocity from a main jet expansion is:

ve =

√√√√2γ

γ − 1

RTcM

(1−

(PePc

) γ−1γ

)(2.9)

where the derivation of this can be seen in Appendix A. The exit velocity ve is essen-tial to the space rocket’s thrust force and is given by:

F = (mve + PeAe)− PaAe (2.10)

where m is mass change, Pe is pressure at the exit, Ae is area of the nozzle at the exitand Pa, ambient pressure. Same thrust force can be derived for impulse:

F = mIsp (2.11)

with Isp as the specific impulse. The thrust force is depending on the change of massm. When a fluid undergoes phase transition its density is then changed affectingthe specific impulse and the momentum of the space rocket rocket. A well-knownequation of motion for a space rocket is described by Tsiolkovsky’s rocket equation,with the conservation of momentum derived:

∆v = veln(m0

m1) (2.12)

where m0 is initial mass and m1 is final mass.

2.1.3 Solid Mechanics

The material that is used to manufacture a space nozzle is important whereas it mustbe able to keep its properties due to loads from aerodynamics and thermodynamics,such as pressure and heat. Pursuant to earlier sections 2.1.1 and 2.1.2, the largesttemperature production is held in the combustion chamber and can cause materialto break or fail, due to fatigue or ductile rupture.

Fatigue crack initiation

Fatigue is a way to describe metal properties, and the limit is studied by testingmany cycles. There are two different types of fatigue, Low Cycle Fatigue (LCF) andHigh Cycle Fatigue (HCF), these are depending on various factors.

LCF is used to describe life under high temperature and low number of cycles,which the cause of this is predominant of thermal loads and cause the material tofail [9]. But for HCF, the material can withstand many numbers of cycles and theloading is caused mainly from the elasticity [10]. The stress ratio is given by:

Rσ = σmin/σmax (2.13)

where σmin and σmax are the minimum stress and maximum stress respectively. Thestrain ratio is:

Rε = εmin/εmax (2.14)

where ε denoted strain. These ratios have importance of describing behavior of thestress, which is also related to life of the material. There are plenty methods for thecrack initiation analysis, an empirical way to describe the strain range is through

2.1. Physics of Space Nozzle 11

Basquin-Coffin-Manson equation:

∆ε

2=σ′fE

(2Nc)b + ε′f (2Nc)

c (2.15)

This equation has excluded the mean stress, which in some case the solution canbe better approximated by taking account of its mean value, with the mean stresscorrection gives by the Morrow hypothesis [11]:

∆ε

2=

(σ′f − σmean)

E(2Nc)

b + ε′f (2Nc)c (2.16)

where E is the modulus of elasticity. The value of b and c can be fitted by creatinga curve (Strain range versus number of cycles to failure). The mean stress may havea significant effect on life, especially for HCF and for large strain ranges the effectof the mean stress is negligible, independently on LCF or HCF. Another importantphysical property is Damage, and the damage caused by a cycle is defined as:

Damage ≡ 1

Nc(2.17)

For several cycles, the damage is added linearly:

Damagetot = D1 +D2 + ... (2.18)

Observing equation (2.18), the Damage is defined as an inverse proportional to thenumber of cycles, with few numbers of cycles the Damage is high, vice versa formany numbers of cycles the Damage is low.

Ductile Rupture

The ductile rupture is a failure of the material, where the plastic strain is large andexceeded its ductile limitation cause the material to break. The failure conditionis depending on many factors, such as temperature and stress. Ductile rupture istypically evaluated for increasing loads, but the failure strain might be affected byproceeding cyclic loads. The equivalent plastic strain denoted as εmax herein is usu-ally compared to the failure strain.


2.1.4 Physics in a sandwich wall

FIGURE 2.2: Cooling system in sandwich wall.

In sections 2.1.1, 2.1.2 and 2.1.3 the main source of heat and pressure from the rocketengine has been presented. In this section, is to utilize previous knowledge and fo-cus on the sandwich wall channels. For a bi-channel system in the sandwich walla 2D illustration can be seen in figure 2.2, where the inner wall to the gas flame ispositioned toward us and opposite for the outer wall. Moreover, this bi-channel isrepeated into the fully size space nozzle, figure 1.3. Starting from top, fuel is dividedinto each channel equivalently and under its way down, it experiences pressure lossdue to frictions from the wall surfaces. Channel split is used to separate fuel into twoevenly channels, which has an importance for the space nozzle when the area andthe number of channels have increased, shown in figure 2.2. Such structural designis necessary to the inner wall’s prospect of damage and temperature. Continuously,this fuel travels to the second channel and out to the outlet manifold with a hydro-dynamics equilibrium as the physical requirement at the outlet manifold. Whereas,this requirement is the source for the velocity at the inlet of the second channel to behigher than the first channel.

The decisive distant of fuel is between the inlet manifold and outlet manifold,which has a significant physics to the turbine. Energy the fuel successfully pickingup during the transportation can directly be contributed to the turbine. Pressuredifference of the fluid/coolant is given by:

∆Pc = Pcin − Pcout (2.19)

Temperature difference:∆Tc = Tcin − Tcout (2.20)

2.2. Statistic Background and Design of Experiment 13

During the path, the thermal energy that fuel accumulates between the inlet mani-fold and the outlet manifold is called heat pick-up:

Qtot =

∫∫qdA (2.21)

Where q is the heat flux produces from the thermodynamics reactions in the com-bustion chamber. Observing that the temperature of the fuel increases due to theheat, equation (2.20) both with the heat pick-up equation (2.21) are having a com-mon relation. The heat transfer from hot flame side to a solid wall is given by:

qg = hg(Twall − Tgas) (2.22)

This description shows the heat from the gas transfers to the inner wall with a heattransfer factor hg.

2.2 Statistic Background and Design of Experiment

There are many available methods for designing an experiment, all having advan-tages and disadvantages. Factorial analysis is one of the common methods usedto design an experiment, for two levels the number of design grows exponential interm of factors, 2k. Unavoidable, the designs become large for many factors, butthis can be useful when the problem has decomposed and reduced into fewer pa-rameters. A way to decompose the problem is by the Plackett-Burman screening,where this method objectively finds the relations and the main effects between fac-tors, and reduce the number of variables by neglecting those with low correlationand main effect. In this study, one wants to collect information even those with lowmain effect, therefore, a more suitable method for many factors problem is the LatinHypercube Sampling (LHS).

2.2.1 Latin Hypercube Sampling

LHS assembles and generates the needed values for each variable Xj by statisticalspan the hyperspace/hypercube. Each variable is generated by multidimensionaldistribution Dj with equal probable/uniformly and disjoint. For N numbers, theprobability for each sample is 1/N , this repeats for each variable and stratifies into amatrix Xij , where i = 1, ..., N and j = 1, ...,M . To illustrate this, assume that X iscontinuous with a cumulative distribution function FX(x).

X ∼ P (X ≤ x) ≡ FX(x) (2.23)

Let also φi be sampling probability with equation (2.23) describes random variable,X . The sample xi can be described by the inverse transformation of the sampling φi:

xi = F−1X (φi) (2.24)

Formulation of equation (2.24) fulfills The inversion principle[12].One of the interesting point in LHS is the variability of the model’s mean value.

An ability of the LHS has shown a superior way in reduction of standard error fromthe mean sampling, comparing with the stratify sample and random sample [13].


For Yi = g(Xi), its mean value is given by:

Y =1

N

N∑i=1

Yi (2.25)

and variance for the random sample:

VRS(YRS) =1

NV (Y ) (2.26)

For stratify case:

VS(YS) = VRS(YRS)− 1

N2

N∑i=1

(µi − µ)2 (2.27)

The variance of LHS:

VLS(YLS) = VRS(YRS) +1

(NM+1(N − 1)M−1)

∑RS

(µi − µ)(µj − µ) (2.28)

with µi, µj as the expectation values in cell i or cell j in a stratify or LHS case, andµ = E(Y ). This has evidently shown a reduction of the variance of the mean valueof a factor 1/N2, equations (2.27). However, in case of LHS, equation (2.28), thevariance is a combination between the random sampling and stratify sampling, insuch a way the method gives higher stability to the probabilistic scheme [14].

2.2.2 Full Factorial Analysis

Full factorial analysis is one of the DoE methods used to study the variations be-tween factors, for two levels (low and high) the number of studies increase by 2k fork factors mentioned earlier. The conception of this is to create points for every edge(low and high) in a k number of variable, and for the 2 dimensional its full factorialcan be illustrated in figure 2.3 [15].

FIGURE 2.3: Full factorial experiment for 22.

2.3. Multidisciplinary Design Optimization Architecture 15

Effect of the full factorial is an average of all the observations between low andhigh value with the main effect of A expressed as [16] :

MA =2∑i=1

(yA=i − y)2 (2.29)

where y is average value of the responses and yA=i is response of the study A.

2.2.3 Analysis of Variance

Analysis of Variance (ANOVA) is a way to investigate variations between variablesand determine quality of the data. The coefficient of determine is given by:

R2 = 1− SSESST

(2.30)

This coefficient is used to evaluate the deviation between data and predicted func-tion, where SSE and SST are error sum square and total sum of square.

SSE =n∑i=1

(yi − yi)2 =n∑i=1

e2i (2.31)

SST =n∑i=1

(yi − y)2 (2.32)

Sum square of error is used to describe the residual between estimated points anddata. Differently from the total sum of square, which is a sum of total error that haspossibility to occur and the sum square of error.

There is other type of coefficient of determine with a consideration to the datapoints used to interpolate and predict a function, this is expressed as R2

adj :

R2adj = 1− SSE/dfe

SST /dft(2.33)

with degree of freedom dft = n− 1 and dfe = n− p− 1 for n number of designs andp number of variables used.

2.3 Multidisciplinary Design Optimization Architecture

Multidisciplinary Design Optimization (MDO) method is based on a single goal,which is to minimize function of design variables. There are two groups in the MDOproblem formulation, monolithic and distributed. The difference between them isthat monolithic subjectively optimize to a single objective, but this is not the case fordistributed that allows the same objective to divide into subproblems and optimizeto the objective [17]. Introduce the general problem formulation for a MDO problem


and its mathematics expression:

Minimize : F (X) (2.34)with respect to : X

subject to : g(X) ≤ 0

ψ(X) = 0

R(X) = 0

with g = g1, g2, ..., gm represents unequal constraints, ψ = ψ1, ψ2, ..., ψl equal con-straint/consistency constraints andR residual constraints, where X = X1, X2, ..., Xn

are the design variables.

2.3.1 Single Objective, Multidisciplinary Design Optimization Architec-ture

Single Objective, Multidisciplinary Design Optimization Architecture (SOMDOA)can be seen in figure 2.4.

FIGURE 2.4: Single-Objective, MDO architecture.

SOMDOA methodology is based on one single objective and controlled by de-sign variables and constraints. A common objective can be e.g. manufacture cost,and mass, with design variables and constraints of time, material, geometry and an-other physical requirement. The SOMDOA is classificated as a monolithic problemformulation. In some case, the objective problem can be partitioned into subprob-lems that contains their own constraints and residuals. By defining in such, the mainproblem is decreased into many subproblems which are often easier to solve, and theadvantage of this can increase the feasibility to the solved problem and the solution[17] [18].


2.3.2 Multi Objective, Multidisciplinary Design Optimization Architec-ture

In Multi Objective, Multidisciplinary Design Optimization architecture (MOMDOA)its environment is based on the SOMDOA except with multiple objectives. Theseobjectives have possibility to link and depend on each other and its formulation isshown below, equation (2.35).

Minimize F(X) (2.35)with respect to X


ψ(X) = 0

R(X) = 0

where F = F1, F2, ..., Fp is a vector of p objectives.There are four different recipes in the MDO method for the monolithic formula-

tion. All-at-Once (AaO) recipe is the general problem formulation of a MDO prob-lem, whereas no assumption has applied, equation (2.35). In turn, this can createproblem in simulation expense due to its difficulty in converging depend on theoriginal problem matrix, and therefore no guarantee on a feasible solution. Nor-mally, the AaO formulation is rare to use because of in a real-world problem, con-sistency of the constraints or residual of the model is often needed to be assumedand in some occasion negligible [17]. This is for keeping the problem in a rangethat is solvable and estimable for understanding the study outcomes. The secondrecipe is Simultaneous Analysis and Design (SAND), which is a simplification ofAaO problem by eliminating the equal consistency constraints ψ of equation (2.35).This assumption can result in a less problematic in convergence comparing with theAaO [18].

For distributed formulation, the main problem is divided into subproblems andreduced the size of the main matrix problem, in turn creates smaller subproblems.This can either way described by the third and fourth recipes of the MDO prob-lem formulation, Individual Discipline Feasible (IDF) and Multidisciplinary Feasible(MDF). These two methods can also be categorized as monolithic formulation whentheir main problem formulation is refined to a single objective [17].


2.3.3 Individual Disciplinary Feasible

Elimination of residual constraints R(X) from the general MDO formulation equa-tion (2.35) results in an IDF formulation.

Minimize : F(X) (2.36)with respect to : X


ψ(X) = 0

IDF solves problems individually and for a large problem the method decomposes insuch way, that increases the feasibility of the problem and easier to find the conver-gence, which is a great benefit for a large problem [17]. There are plenty of methodsin the IDF formulation, but in this the focus will be on Analytical Target Cascading(ATC).

Analytical Target Cascading

ATC is one of the methods that divides problem into subproblems and increases theconvergence ability in comparison with other MDO methods[19] [20]. This allowsthe user to optimize subproblems individually, and keep consistency between allthe subproblems [21]. The feature of this is to utilize penalty function that eachoptimize loop condition is violated, will result in an addition penalty value in termof consistency constraint differences between subproblems and objective.

There are two different ways to solve and describe the problem in the ATC for-mulation, hierarchical and non-hierarchical shown in figures 2.5 and 2.6.

FIGURE 2.5: An illustration of Hierarchical in the ATC case.


FIGURE 2.6: An illustration Nof on-Hierarchical in the ATC case.

Independently on how the main objective problem is described, in hierarchicalATC, the problems level i − 1 share variables with level i, inconsiderate to the sub-problems i− 1 that might have the same dependent variables. Instead, they strictlyneed to share their variables through the upper level problem i. Non-hierarchicaldespites the earlier formulation and allows all the problems and subproblems toshare variables with each other freely, figure 2.6. The number of sharing variablesincrease with the number of problem and subproblem, which can be difficult to keeptracking of all shared variables when the problems and the systems are large.

Augmented Lagrangian Relaxation

Individual Discipline Feasible operates by keeping track on sharing variables in eachoptimize function, holding consistency within the defined system. This consistencyis formulated by Lagrange Multiplier and one conventional penalty function is theAugmented Lagrangian (AL) [23] [22].

The AL relaxation function is described:

π(c) = vT c+ wT (c c) (2.37)

where c is constraint, and in this case is the response and the target value from sys-tem i, ci = Ti − Ri. The AL can be applied to the ATC formulation function for anobjective F :

Minimize: F (xj , gh(xj)) + πk(cj) (2.38)subject to: xj , gh(xj)

where k is number of nearby objective and j is number of discipline constraint and his number of dependent variable of the objective F . In minimizing, for each iterationconstraints c is violated to the consistency condition, the minimized function willgain a penalty of πk, until the condition is fulfilled.


The update of v and w in equation (2.37) is formulated as:

vk+1 = vk + 2wk wk ck (2.39)

wk+1i =

wki if|cki | ≤ γ|c

k−1i |

βwki if|cki | > γ|ck−1i |

For constants β > 1 and 0 < γ < 1.

2.3.4 Multidisciplinary Feasible

Another way to decompose the AaO is Multidisciplinary Feasible (MDF) by elim-inating consistency constraints ψ(X) and R(X) residual constraint from equation(2.35). The MDF problem formulation becomes:

Minimize : F(X) (2.40)with respect to : X


This problem formulation is easier to construct comparing with the ATC, for a smallproblem the method is quite effective and powerful, but for a large problem the con-vergence problem can be encountered[17]. Instead of solving problem individuallyas in the ATC, this requires a full information about all the constraints and the re-sponse variables in the system for each optimization iteration, which is not requiredin the ATC.

21

Chapter 3

Implementation andMultidisciplinary optimization

The theories to understand this thesis have been presented in the previous chapterand now we are able to apply these theories and deepen into our main objective.

In this chapter, the goal is to give readers a clear figure about implementationand optimization, including assumptions that are needed to be made with a consid-eration to the simulation expense. The implementation will carefully be introducedfrom construction of DoE scheme and to surrogate model. In the optimization part,the theory of focus to our problem will be formulated as well for the definition ofdifferent study optimize limitations.

3.1 Main study methodology

Work frame in this thesis is divided into six main steps and are presented in figure3.1.

First step: This is to find and define what is it that needs to be studied in thecooling system. Importantly, understand the physics of how the engine contributesthe main jet and the pressure inside the cooling channels and how these affect thematerial.

Second step: When the physical properties are known, the design variables thatinfluence the physics can be created by making a DoE scheme for probabilistic vari-ation.

Third step: The created DoE is putting into black boxes to get physical relationsinfluence by the design variables. The black boxes in this case are Tccool and Ansys,which will be discussed more in section 3.3.1.

Fourth step: Interpret the outputs from the black boxes, including understand ifthe received results are logic and using to create surrogate models that describes thephysical properties, such as, aerodynamics, thermodynamics and solid mechanics.

Fifth step: Optimization, using the MDO method includes formulation of objec-tive problem and creation of surrogate models application to the previous step. TheMDO method will be used with two different approaches for two distinguish solv-ing ways. Before optimizing the problem, three different limitations will be definedare implied to the MDO method, Design Space limit, industrial limit and industriallimit with tolerance.

Sixth step: The final step is to study the accuracy of the optimized result. Thiswill be performed by creating a DoE scheme with the full factorial in an interval nearto the industrial optimal result and analyzing its uncertainty with the ANOVA.

22 Chapter 3. Implementation and Multidisciplinary optimization

FIGURE 3.1: MDO Methodology.

3.2. Decomposition of cooling system 23

3.2 Decomposition of cooling system

Sandwich wall of the space nozzle is viewed in figure 2.2 and shown a bi-channelsystem, with the coolant travels from the left channel and up to the right channelas explained in the theory part. Further decomposition of the problem is to neglectmanifold at the bottom, with an assumption of low impact to the results.

3.3 Implementation

This section is about the main set-up of the study, with further and specific assump-tions will be presented.

FIGURE 3.2: Half of cooling channel. Left side is the flame side andright side is toward the outside, with meshes.

TABLE 3.1: Description of design variables.

Design variable Descriptionx1 Number of channelx2 Inner wall thicknessx3 Channel heightx4 Mid wall thicknessx5 Outer wall thicknessx6 Mill radiusx7 Weld radius


3.3.1 Data Implementation

The methodology uses to solve the problem is shown in figure 3.1. Firstly, in theimplementation step a DoE scheme is created using LHS method, which is necessityneeded to create surrogate models that describes and suits to the physical problems.LHS scheme is built on seven design variables (mainly are geometries) for fifty andtwo hundred designs, for getting a wide range of study as possible by taking simu-lation time into account.

These design variables are shown in figure 3.2 and table 3.1. Observing, thedesign variables x4 and x1 are divided by a factor 2, because the presented piece isonly half of the original channel and it is positioned at the top of the space nozzlein the combustion chamber. The notation a2 represents a constant, preventing the”buckling” phenomenon in the space nozzle for different load of pressure betweeninside and outside [25], this constant only exist at the top of the space nozzle. Thestress acts on the space nozzle’s wall is given by equation (3.1) which is originallyderived by Euler. Differently, at bottom of the space nozzle the channel height istwice larger than the inlet, and the constant a2 which is shown in figure 3.2 goes tozero.

σ =F

A=π2E(lr

)2 (3.1)

where F is force, A area, E modulus of elasticity, l area moment of inertia and rradius of gyration. This also shows for a constant stress σ, then F ∝ A.

There is one more important thing that must be considered when creating a newdesign variable for the channel height x3. When the design space has large varia-tions, this can result channel height to be smaller than the sum of the radii, x6 andx7, and cause problem when creating mesh elements and geometry for the finiteelement analysis. The problem is manipulated by defining a constant a1 as:

a1 ≡ x3ref − (x6ref + x7ref ) (3.2)

and make variations on this constant a1, instead of the channel height x3. Thereby anew channel height is taking form:

x3i = αi3a1 + αi6x6ref + αi7x7ref (3.3)

where αi is a constant factor for i design number. This concept has similar ideologyas varying x3, and moreover, it has placed a consideration to the variation of theradii, x6 and x7.

3.3. Implementation 25

(A) Design space 0.5− 1.5, with 50 designs. (B) Design space 0.2− 1.7, with 50 designs.

(C) Design space 0.2− 1.7, with 200 designs.

FIGURE 3.3: LHS for 7 design factors for two different design spaces0.5− 1.5 and 0.2− 1.7, with 50 and 200 as number of designs.

The first study is to contribute LHS scheme value between 0.5 to 1.5, figure 3.3with a probability density 1/N described in section 2.2.1. Wide range of the LHScan cause trouble in simulation, e.g. thin walls and large channel give a high hydro-dynamics pressure load on the wall and thin walls cannot withstand such pressure.In some exceptional case, if the walls are thin and the mill and the weld radii arelarge, this could compensate the fact that the walls are thin. These are depending onseveral factors and difficult to determine and face the problem directly. Therefore,by creating a DoE scheme with wide range is important to the study for finding thelimitation of the material, and possibly designing space nozzles that today unable tomanufacture. The factors generated from the LHS 0.5 to 1.5 will be multiplied into areference value.

In the second study, the set-up has a similar procedure as the first study, but theLHS range is expanding, 0.5 to 0.2 and 1.5 to 1.7 with 50 designs. For the third study,the same study range is applied, but with 200 designs. Knowing from the theoreticalpart that by constructing large design variations can also cause a large variation inphysics, therefore, many designs are needed to be able to create well behaved sur-rogate models. These LHS between different intervals 0.5 to 1.5 and 0.2 to 1.7 aredepending on the probability density function and the number of designs. There-fore, by increasing the number of designs a more rigorous solution to the ”actual”behavior can be determined. Results from the three studies, only one will be selectedusing to optimize for reducing the number of optimizations.


3.3.2 Aerodynamics and thermodynamics simulation

For given designs, the varies of the geometries are putting in black boxes, such asTccool and Ansys for physics study, aerodynamics, thermodynamics and solid me-chanics, ∆Tc, ∆Pc, Qtot, Mass, Tmax etc. these are mentioned in section 2.1.4. Theaerodynamics and thermodynamics are the results from the Tccool script, which isused and developed by GKN with the goal to simulate the space nozzle’s channelsfor given geometries of cooling system, figure 2.2. It also neglects effects from thebottom manifold and the study can be chosen between one dimensional or two di-mensional along axial axis. In this study one dimensional will be selected. Thephysical result returns from the Tccool is a mean value of aerodynamics and ther-modynamics, where the bi-channel is assumed to be repeated into the whole spacenozzle. Conceptional, is to minimize the study problem without losing any impor-tant physical properties. The Mach number uses in this study is the largest value inthe first channel, figure 2.1, where another physical has been selected depending ontheir maximum or significant for their specific position in the bi-channel.

There are two operating points that needs to be considered, hot operating pointand cold operating point, which is also called condensation point. The differences ofthese points are physical initial values of the fuel and the engine. The condensationphenomenon has been mentioned in section 2.1.1 and this ∆Tcondense in figure 3.4 isrepresenting the minimum of difference between the wall temperature toward theflame and the saturation temperature.

FIGURE 3.4: Condensation operating point with wall temperature to-ward and in the flame side, plotted with the lowest temperature dif-

ferent ∆Tcondense and the saturation line.

3.4. Prediction of surrogate model 27

3.3.3 Solid mechanics simulation and finite element method

The Tccool script is not functional to provide any information concerning to LCFwhich is necessary for this study, whereas a finite element method simulation is usedand simulated in Ansys, to get the LCF for different geometrical designs. A three-dimensional simulation is time expensive, and the study problem formulation isdecomposed even further into a two dimensional by giving a simulation position inaxial axis, specific for the LCF study. This selection is analyzed from the Tccool scriptof the aerodynamics and thermodynamics with the theoretical support. Accordingly,LCF is expected to have the largest impact influences by the thermal load in thecombustion chamber due to the thermodynamics reactions, and such axial positionwill be selected in this study. At the bottom of the space nozzle, pressure drop offuel is low relative to the inlet. Besides, the pressure force acts on the wall is twotimes larger than at the inlet due to equation (3.1), and can results in an increasing ofεmax for cycles Nc. The study will first be simulated in the hottest region and secondat the bottom, where the data will be analyzed and selected to create a safety factorthat each geometry is forbidden to exceed.

A script CUMFAT is developed by GKN and used to evaluate the solid mechan-ical properties, such as strain and damage with the theoretical support described inthe section 2.1.3. Normally, the stress is described in a three-dimensional tensor withthe CUMFAT script specializes in search and select the highest value in a specific axisand return as result.

3.4 Prediction of surrogate model

A CAE program ModeFrontier is used to create response surfaces and predict surro-gate models, and to analyze behavior of the data. In this study, the surrogate modelshave chosen to be approximated and described as a second order polynomial in R2.The model Y of these for n dimensional is given by:

Y = β0 +n∑i=1

βixi +n∑i=1

K∑k≥i

βikxkxi +n∑i=1

βiix2i (3.4)

For a well-defined polynomial in an interval a and b, the function says to be contin-uous if there exist a value |f(a)− f(b)| ≥ L|a− b| for an arbitrary Lipschitz constantL. The proof of continuity of a polynomial will not be performed, except the conceptof understanding the surrogate model is essential for usage in the optimization [26].If discrete data points can be interpolated into a surrogate model and able to forma polynomial that does not contradict the Lipschitz condition, then the discrete datapoints can be transformed into an unobservable continuous function and is differ-entiable.

The number of design is essential when the computational time is limited, withthe need to reduce the simulation time and still be able to create a well-behavedsurrogate model suits to the physics. An approximate number of design that needsfor creating a polynomial is given by [27]:

mc =

(n+ k

k

)=

(n+ k)!

n!k!(3.5)


where mc is the number of coefficients, n is the number of variables and k, degree ofpolynomial. For 7 design variables and third order polynomial, number of designsrequired is 120 and for a second order polynomial, 36 designs.

In some case, the data is unable to describe as a second order polynomial, anothersurrogate model approximation will be applied. This approximation is experimentaldescribed the data as an exponential function, which is objectively linearized thedata by natural logarithm before fitting the new data into a second order polynomial.

3.4.1 Natural behavior of the surrogate model

The difficulty in this study is to decide if the surrogate models are well behaved fordescribing the physical properties. Taking mass as an example, by describing as afunction of geometries x, it is expected to have the third order polynomial, but insome case in a specific interval the function can vaguely be assumed and describedas a second order polynomial. This is inconsistent with the theory, but as far thefunction has revealed the ”original” behavior of its physical, then this violation canbe ignored. For instant, the Mass,

Mass ∝ x3[m3] (3.6)

This can be discussed even further, at this design space range it is not necessaryfor such physical function to fulfil the theoretical approach, which often is a globaldefinition for the mass, x ∈ [0, inf). But if one has more points, then it is possibleto create such surrogate model that fits to the third order polynomial and fulfills thetheoretical point of view.

3.5 Multidisciplinary Optimization

The optimization problem will be engaged with two different multidisciplinary ap-proaches, ATC and MDF, for comparison between two different methods. A surro-gate model based optimization will be optimized using a non-linear optimizationmethod and in this case the Sequential Quadratic Programming (SQP), with an im-plementation in MATLAB. Following from earlier, when optimizing, a restriction ofsearch interval is needed with lower and upper bound which is this case definedwith three different limitations considerable to the manufacturing aspect.

3.5.1 Optimization constraints

In this study, the surrogate model has been selected to describe the physical functionentirely by the design variables, independently to other physical. This is somehowone of the ways to describe the problem, in such way, it has less influence from theaccuracy of the other surrogate models. Alternatively, a surrogate model can be se-lected to include some physical function. If this is the case, then the uncertainties ofthe other surrogate model are also included into the new function. By taking Dam-age as the example, according to the theory this depends on the temperature, andtherefore, suited to describe the surrogate model as a function of temperature. How-ever, the temperature depends on the geometries, thickness of inner wall, channelsize etc. which contains errors when creating the surrogate model. By adding thetemperature into the Damage is the same as adding the addition errors into itself. Inconclusion, the study is imposed by describing the surrogate model independently

3.5. Multidisciplinary Optimization 29

to the other physical properties and using these functions as constraints to completethe set-up for an optimization problem.

Before formulating the MDO study, the first focus necessary needed is the con-straints on the cooling system, listed in table 3.2.

TABLE 3.2: List of constraints and each of its bound with description

Constraint Descriptiong1 = ∆Pc − C1 ≤ 0 Pressure dropg2 = −∆Tc + C2 ≤ 0 Temperature dropg3 = Tmax − C3 ≤ 0 Max temperatureg4 = Damage− C4 ≤ 0 Damageg5 = M − C5 ≤ 0 Mach numberg6 = Producible− C6 ≤ 0 Producibleg7 = Qtot − C7 ≤ 0 Heat pick-upg8 = ∆ε1 − C8 ≤ 0 Strain range top edgeg9 = ∆ε3 − C9 ≤ 0 Strain range bottom edge

For the strain range ∆ε1 and ∆ε3, their values are based on the strain for eachcycle and not the max strain, a picture of them can be seen in figure C.1 where themax strain εmax is more complicated to create a surrogate model. The notations 1and 3 are the positions shown in figure 3.2 and max temperature, Tmax is usually thetemperature at point 1 where the temperature is the largest.

In this, a consideration to the manufacture cost will be considered with an esti-mation of the cost, Producible describes as:

Producible = vp1x1

x1ref

+ vp2x2 + x3

x2ref + x3ref

+ vp3x3

x3ref

+ vp4x5

x5ref

(3.7)

where∑

i vpi = 1 are weighted coefficients describe the weight of each variable’sproperty in the producible, equation (3.7). In this case the weighted coefficients areassumed to be vp1 = 0.60, vp2 = 0.20, vp3 = 0.15 and vp4 = 0.05, where the numberof channels x1 has the highest impact comparing with the rest of the factors. Thevp2 and vp4 represent the weight of the sheets for both inner and outer sheet, wherethe last vp3 is for the channel height. A concept of how to manufacture a sandwichwall is shown in figure 3.5 together with equation (3.7) has shown an incompletemodel of the producible, where the times that requires and the fact that it becomesmore difficult to manufacture when the size of geometries are small is pretermitted.Except the concept provides a relation of how the producible can be estimated.

FIGURE 3.5: Channels in cooling system [5].


3.5.2 Optimization problem formulations

The main goal is to optimize mass of the space nozzle with respect to design vari-ables and subject to constraints, with the problem formulations for both ATC andMDF are shown below.

FIGURE 3.6: System Level.

Hierarchical ATC formulation is used to formulate the problem with constraintsg1, g2 and g3 reserve in the Mass main objective function, as for the remaining con-straints are selected to include in the penalty functions πk(c). This selection lies withan argument that the cooling system is based on the ∆Pc, ∆Tc and Tmax, that arethe main constraints of the fuel and the inner wall toward the flame side, thereby,reduce number of the subproblems.

FIGURE 3.7: Hieararchical ATC problem formulation.


Minimize: Mass(X) + πk(c) (3.8)with respect to : X

subject to: g1(X) ≤ 0

g2(X) ≤ 0

g3(X) ≤ 0

where k = 4, 5, ..., 9 are equivalent to the number of constraints and c is a vector ofsharing variables. An additional need is to minimize the penalty function AL, withthe requirement to satisfy the constraints gk:

Minimize: πk(c) (3.9)with respect to : X

subject to: gk(X) ≤ 0

Optimization tolerance between the design variables for each system is numericallychosen: Tol ≤ 0.001, where Tol = |XTarget −XResponse| with update factors γ = 0.8and β = 2.1 [22]. These factors are presented in the theory section.

In figure 3.8, the algorithm uses to solve the ATC hierarchical problem is showedwith the objective mentioned earlier and the problem formulations are described ac-cording to equations (3.8) and (3.9). The optimizer starts to solve the main objectiveMass, and sends responses to π4. When the π4 function is solved and the constraintg4 is satisfied, the responses from the Mass and the π4 are forwarded to π5. Thisprocedure is repeated to π9. Lastly, if the consistency constraints c between all sub-problems and main objective problem are unsatisfied, the factors v and w will beupdated and repeated the optimizing scheme until the consistency constraints c aresatisfied.


FIGURE 3.8: ATC Algorithm.

The number of solvers become large for many constraints defined for the ATCformulation. On contrary, the MDF formulation its way of describing the problemis easier to formulate independent to the number of constraints which are gatheredinto a single solver, equation (3.10).

Minimize Mass(X) (3.10)with respect to : X

subject to : g1(X) ≤ 0

g2(X) ≤ 0

...g9(X) ≤ 0

MDF formulation allows a single optimizer to solve the objective function subject tothe constraints using shared design variables, differently from the ATC that solvesthe problems separate.


3.5.3 Local and global optimization

Surrogate model based optimizations are implemented and solved in MATLAB,with two varieties of solutions, local and global. Generally, the global solution is dif-ficult to receive, when the solver is programmed only for finding a solution regard-less to the global solution. One of the ways to handle this is to evaluate many initialpoints, but this can be problematic for many design variables and constraints. Al-ternatively, such feasibility of global evaluation exists in the MATLAB’s Global Op-timization toolbox and will be utilized in this study. The MATLAB’s ”GlobalSearch”constructs the problem and initialize search for the best objective value.

Sequential Quadratic Programming (SQP) method will be applied into the opti-mization, where this requires the objective function and the constraints to be twicedifferentiable and continuous, and in our case the functions are described in secondorder polynomial or exponential function, which both are differentiable and contin-uous.

3.5.4 Different limitations

Different limitations imply to the optimizations with their define concept shows infigure 3.9. The idea behind this is to restrict the search region when optimizing,where each region has its own importance. In consideration to the producibility andmanufacture two limitations are defined, industrial limit and industrial limit withmanufacture tolerance, provided by GKN Aerospace. Manufacturing tolerance is anuncertainty when manufacture such geometry and the uncertainty factor is chosento define as 3σ. By placing 3σ value into the industrial limit, a new restricted regionis then shifted and can be illustrated in figure 3.9, with an ideology to decrease thesearch region for satisfying the industrial requirement. In some case the 3σ value isnot applied into the industrial limit, which is often concerned to the upper boundwhere there is no strictly defined limitation of manufacturing, except in the opti-mization. When these are applied, the optimizer will have to find other optimal inthe defined region. The implement of this is easy to evaluate for multi-dimensional,multi design variables and many numbers of constraints, instead of probabilisticanalyze with the Monte Carlo method to find the density of the function for eachfunction. This can be complicated for many design variables and constraints thatconnects to the main objective and to each other.


FIGURE 3.9: An example concept and illustration on the different lim-its, Design Space (D.S.), Industrial (Ind.) and Industrial with Toler-

ance, are defined with function’s minimum.

3.6 Certainty of Surrogate model

The motivation of this is to analyze and find the variation of the surrogate modelscomparing with the actual value from the black boxes. This is done by creating aDoE scheme with given lower and upper bounds near to the industrial optimumlimit. To find the reliability in a smaller restricted region, instead of analyzing thewhole design space that the result of variants and effects will be averaged of thewhole model. DoE of full factorial is used with the reason, in a small region it hashigher feasibility and easier to construct and evaluate the results comparing withthe LHS. The idea is to analyze its sensitivity in this restricted region and providean understanding of the impact of each design variables, as well for the physicaloutcomes.

35

Chapter 4

Data analysis and Results withDiscussion

Based on chapter 2 and implemented according to chapter 3, the main result of thisthesis is revealed. This chapter contains three parts, data analysis, main optimizationresult and probabilistic analysis of uncertainty.

The results that are presented in the thesis have been normalized with referencevalues and modified in such a way that they are able to publish.

4.1 Data analysis

Results from creating the surrogate models using the ModeFrontier program arepresented in this section with different optimization limitations.

4.1.1 Fitting of the surrogate model

The surrogate model described in equation (3.4) was created in ModeFrontier, wherethe original data contained ”bad data/design”. To improve the accuracy of the surro-gate model, designs had been manually excluded where the selection was depend-ing on the main behavior of the data points. The number of useful results accordingto the three different studies with different LHS design spaces and design numbers,after manual exclusion are shown in table 4.1.

TABLE 4.1: Number of data left to create surrogate model for 0.5 to1.5 and 0.2 to 1.7 interval.

Nr. of design Nr. of removed data Nr. of remaining data Design range50 3 47 0.5− 1.550 8 42 0.2− 1.7200 90 110 0.2− 1.7

Table 4.1 shows the studied range of different defined design space limitationsand the number of designs impact on the surrogate models. For 0.5 − 1.5 designrange with 50 designs, the number of the remaining data is high compared with awider design space 0.2 − 1.7. For the 0.2 − 1.7 design space range, the number ofremoved data is high which can be explained from the spread of data or its distribu-tion, 1/N as described in the section 2.2.1. The spread of the data has an impact tothe physics inside the space nozzle, where e.g., the designs that are based on loweror upper limits will show a large deviation from the mean behavior. This createsdifficulty in creating a well-behaved surrogate model. Analyzing from the numberof the remaining data, the 0.2 − 1.7 range with 200 designs has been selected forcreating the surrogate models and using to optimize.

36 Chapter 4. Data analysis and Results with Discussion

4.1.2 Quality of response surface and surrogate model

TABLE 4.2: Fitting quality to the data for each variable using 200 de-signs between a LHS interval of 0.2 to 1.7.

Property Polynomial order R2 R2adjust

Tmax 2 0.9975 0.9963∆Tc 2 0.9989 0.9905∆Pc 2 0.9869 0.9808ln(∆Pc) 2 0.9977 0.9966Mass 2 0.9996 0.9994ln(Damage) 2 0.9869 0.9808∆ε1 2 0.9941 0.9914∆ε3 2 0.9941 0.9914Heat pick-up 2 0.9996 0.9994Mach number 2 0.9982 0.9974

The fitting quality for 50 designs for 0.5 − 1.5 and 0.2 − 1.7 intervals are shown intables B.1 and B.2, where all the surrogate models have high number fitting qual-ity. With a good fitting quality, the effect on the physical outputs from the designvariables are better known than with poor fitting quality. Pressure drop ∆Pc andDamage are described in a natural logarithmic scale due to their complexity. For the∆Pc the improvement of R2 for defining as the natural logarithm ≈ 1%, table 4.2.The choice to logarithm the data points before creating the polynomial is based onthe data’s character. One reason is to avoid negative value which is unphysical andnot to be expected when defining the function as a second order polynomial.

4.1.3 The limitations used to optimize

TABLE 4.3: Different limitations for each design variable.

Design variable Total D.S. limits D.S. limits Industrial limit Industrial limit with Tol.x1 0.20− 1.61 0.20− 1.61 0.54− 1.61 0.55− 1.60x2 0.20− 1.68 0.67− 1.68 0.67− 1.00 0.83− 1.00x3 0.62− 1.56 0.62− 1.56 0.63− 1.56 0.74− 1.45x4 0.20− 1.66 0.20− 1.66 1.00− 1.38 1.06− 1.31x5 0.20− 1.69 0.62− 1.69 1.23− 1.69 1.37− 1.55x6 0.20− 1.68 0.20− 1.68 0.54− 1.62 0.68− 1.49x7 0.20− 1.70 0.20− 1.70 0.80− 1.70 1.40− 1.60

In table 4.3 the four different limits have been constructed where three of them werementioned in the section 3.5.4. The unmentioned is total Design Space (D.S) lim-its that is the total design range when creating surrogate model and able to fit anddecide reliability of the results. When optimizing with this limit, large strain valueoccurred which point at failure of the structure. A result from the mass optimiza-tion for the total D.S. limit is shown in figure C.2 where the inner wall is unable tosustain the pressure. This can be caused by the uncertainty of the surrogate modeland/or the information of constraint uses in the optimization is insufficient, whichis directed to the max strain εmax, showing in figure C.1. The difference between

4.2. Results of optimization 37

∆εs2 and εmaxs2 , is that the strain keeps increasing cycle by cycle instead of becom-ing stable like ∆ε1 in the same figure. A way to prevent this from happening is todefine a new range which in this case is called for Design Space limit, figure 3.9.

4.2 Results of optimization

The number of local solvers evaluated and converged are shown in table 4.4 andare increasing as the study range decreasing for the ATC approach. The explanationis that when the design range is decreasing, there is less number of local minimato converge on, and by using the same number of points in the optimizer, morepoints will then converge at the same local minimum. For the MDF, the number oflocal solvers have shown to be compatible between the limitations, where most ofthe results successfully converge at the same solution. But for the industrial limitand industrial limit with tolerance the outcomes have shown, that the number oflocal solver evaluated is inconsistent with the number of local solver converged tothe solution, by the cause of difficulty of finding physical allowance in the particu-lar region. Adding into the statement, all the functions are described in one singleproblem and by suffocating its search interval the solver has also trouble to find thesolution, due to the size of information the optimizer needs to handle.

An essential remark to the ATC solutions are that the results have converged inthe first iteration which means that there is no penalty update applied to the solu-tion. This kind of result is not unexpected, since the optimizer evaluates many pointsand succeeds to find the ”best” solution with respect to all the conditions. If oneneeds to understand how the convergence behaves for the two MDO approaches,the ”GlobalSearch” can be removed when trying to find the solution. In other way isto suffocate the constraints even more while solving, in order to make the optimizerworks harder to find the common solution.

TABLE 4.4: Total number of local solver evaluated and converged inthe MDF and ATC for different limits.

D.S. limit Ind. limit Ind. limit with Tol.ATC MDF ATC MDF ATC MDF

LocalSolverConverged

53 39/39 258 51/53 307 31/43

TABLE 4.5: Result from the optimization for MDF and ATC, for dif-ferent limits. The results are relative to reference values.

D.S. limit Ind. limit Ind. limit with Tol.DesignVariable

ATC MDF ATC MDF ATC MDF

x1 0.77 1.26 0.71 0.54 0.83 0.82x2 0.67 0.67 0.67 0.67 0.83 0.83x3 0.86 0.88 0.94 0.89 0.96 0.95x4 0.20 0.20 1.00 1.00 1.11 1.11x5 0.62 0.62 1.23 1.23 1.37 1.37x6 0.20 0.20 0.54 0.54 0.68 0.68x7 0.20 0.20 0.80 0.80 1.40 1.40


TABLE 4.6: Result from the optimization for MDF and ATC, with acomparison between surrogate models and actual results without in-

dustrial limits. The results are relative to reference values.

ATC MDFPhysical and Pro-ducibility

Surro.Model

Actual Surro.Model

Actual

Mass 0.48 0.49 0.49 0.51∆Pc 0.86 0.81 0.86 0.86∆Tc 1.01 1.01 1.01 1.01Tmax 0.94 0.95 0.93 0.95∆ε1 0.95 0.95 0.95 1.00∆ε3 1.00 1.00 1.00 1.00Damage 0.25 0.33 0.21 0.34Producibility 0.65 0.65 0.92 0.92Mach number 0.96 0.92 0.90 0.87Heat pick-up 1.00 1.01 1.01 1.01εmax 1.55 1.15∆Tcondense 0.28 0.29

TABLE 4.7: Result from the optimization for MDF and ATC, with acomparison between surrogate models and actual results with indus-

trial limits. The results are relative to reference values.


Surro.Model

Actual Surro.Model

Actual


4.2. Results of optimization 39

TABLE 4.8: Result from the optimization for MDF and ATC, with acomparison between surrogate models and actual results with indus-trial limits and tolerances. The results are relative to reference values.


Surro.Model

Actual Surro.Model

Actual


Optimized designs can be seen in table 4.5. The designs are different depend-ing on the limits defined in the optimization. For the D.S. limit solution, the resulthas shown an incompatibility between ATC and MDF approaches, but both leadto the similar objective function Mass. Using figure B.2, the Mass is less affectedby increasing x1 compared to increasing x5. Differences in the results can be origi-nated from the problem formulation of the ATC and utilization of the ”GlobalSearch”.Combining these two facts, the ATC problem formulation is dictated by the mainobjective function and all initial conditions defined at the beginning. Where thechildren/subproblems equation (3.9), only requires to satisfy the responses from themain objective and its own constraints.

The industrial limit has shown the same behavior of x1 for ATC and MDF asearlier which can be directed to the above argument. In the opposite, the indus-trial limit with tolerance gives similar results between the two different MDO ap-proaches, which is not observed in the D.S and the industrial limit. Meaning that,in the defined lower and upper bounds there are fewer remaining solutions, andtherefore, the two disparate formulations of optimization tend to seek and convergein the same optimum.

The main objective Mass has decreased by≈ 50% for D.S. limit relative to the ref-erence value, see table 4.6. By applying the industrial search regions, the Mass hasproven to decrease ≈ 15% and increase ≈ 4%, for the industrial limit and the indus-trial limit with tolerance relative to the reference value, see tables 4.7 and 4.8. Thevariability is not only affected the Mass but also included to other physical proper-ties, Tmax, Damage, Mach number, Producibility, ∆Pc and all are listed in the showntables. An increase pattern of Tmax, ∆Tc, ∆Pc, ∆ε1, Mach number, Heat pick-upand Damage are observed when applying the industrial search limits in compari-son with the D.S limit result. Anyhow, these factors are selected by the user whenoptimizing and in our case the strive is to increase the performance of the spacenozzle. Remembrance of keeping larger ∆Tc and Heat pick-up and less Mass, ∆Pc,Tmax, ∆ε1, ∆ε3, Damage and Producibility which most of the results are promised.When the industrial limit and industrial limit with tolerance are compared, the innerwall thickness x2 has shown to be thicker which results in higher temperature and


higher damage, see figures in Appendix.B. For high temperature in the inner wallat a constant pressure, ∆Tcondense will increase inversely with the risk for gas phasechange/condensation that can affect the space rocket’s specific impulse.

Another important measure to each solution is the max strain, εmax. The indus-trial limit with tolerance has the lowest εmax value which is also in an agreementwith ∆ε1, meaning that the max strain is independent to the number of cycles, or ithas found its steady state, such solution can be considered as preservable. The man-ufacture tolerance limit itself has limited the solution in the way that it is realizable.On the other hand, the industrial limit and the D.S. limit indicate an increase maxi-mum strain cycle by cycle. For many cycles, this may lead to material failure muchearlier than for the case with a stable strain. Meaning for many cycles, the damageand strain are accumulated for each cycle and when it exceeds the limit it will crack.

(A) ATC without industrial limit. (B) MDF without industrial limit.

(C) ATC with industrial limit. (D) MDF with industrial limit.

(E) ATC with industrial limit and tolerance. (F) MDF with industrial limit and tolerance.

FIGURE 4.1: Optimized geometry from different limits and methods.

The optimized geometries in the combustion chamber of the cooling system areshown in figure 4.1. These geometries have shown an incompatibility between meth-ods, which can indicate to that, there exist many local minima and to the MDO prob-lem formulation.

Regardless to the manufacture requirements the solution for optimization willonly face the physical constraints that needs to be satisfied, and this is by reducingthe size of material as much as possible. Illustrative pictures can be seen in figures

4.3. Probabilistic analysis of uncertainty 41

4.1a and 4.1b. The between wall thickness x4 and the radii x6 and x7 have reducedto their allowed minima which is not the case for the industrial limits.

4.3 Probabilistic analysis of uncertainty

The error from the surrogate model is caused by the deviations from inability to fitthe data when creating the surrogate models, particular when points are includedthat have large deviations from the main data. These points can be caused by factorssuch as noise from the black box, or other numerical error. Indisputable, a study ofthe surrogate model is needed.

4.3.1 Accuracy of Surrogate Model

Accurateness of the surrogate model is being investigated by creating a full facto-rial scheme with a lower and upper bound near the industrial optimum solution.The differences between the results in plots of the actual value from the black boxesagainst the surrogate models are shown below, figures 4.2 and 4.3.

(A) Actual vs. Surrogate model, Mass. (B) Actual vs. Surrogate model, Tmax.

(C) Actual vs. Surrogate model, ∆Pc. (D) Actual vs. Surrogate model, ∆Tc.

FIGURE 4.2: Plots of ANOVA, Actual vs. Surrogate model with itsresidual.


(A) Actual vs. Surrogate model, Damage. (B) Actual vs. Surrogate model, ∆ε1.

(C) Actual vs. Surrogate model, ∆ε3. (D) Actual vs. Surrogate model, Heat pick-up.

(E) Actual vs. Surrogate model, Mach num-ber.

FIGURE 4.3: Plots of ANOVA, Actual vs. Surrogate model with itsresidual.


The results have shown to be undistinguishable between the actual and the sur-rogate model except for Tmax, Damage and ε1, figures 4.2b, 4.3a and 4.3b, where aclear deviation can be seen. The damage is found to has the largest residual andthe R2 value ≈ 84%, which contributes highest impact to the optimization result.Besides, the max temperature, Tmax has inferior R2 quality but in turn its residualis significantly lower in comparison to the Damage. Similarly, ∆ε1 has low residualand low R2 value. Anyhow, these are clearly shown in the optimized results, tables4.6, 4.7 and 4.8, which has ensured the agreement between the uncertainty studyand the main results. In the MDO formulation the goal was to optimize the massand for this the surrogate model has shown to be accurate. The accuracy of the otherconstraints ∆Pc, ∆Tc, ∆ε3, Heat pick-up and Mach number was also found to behigh.

All the ANOVA results, figures 4.2b, 4.3a and 4.3b have shown a statistical cer-tainty between the actual values and the surrogate models in a 99% confidence in-terval, where the p-value ≈ 0.001. This concludes that one is unable to reject the factthat there are differences between the actual result and the surrogate model result.

4.3.2 Source of uncertainties

The main source of uncertainty is produced when creating the surrogate model fromthe black boxes. This has been analyzed in the previous section that results in well-behaved surrogate models with the agreement between the surrogate models andthe actual values at 99% level of confidence interval. Even with this result the relia-bility of the surrogate model can be improved by creating a larger DoE scheme withmore designs. This is unavoidable fact for receiving an extraordinary well-behavesurrogate model, which is time expensive.

In this thesis, the study contains a few assumptions that were made to reduce thesimulation time. The one-dimensional tool Tccool was used to simulate aerodynam-ics and thermodynamics and Ansys two-dimensional model the structural responsewhich may introduce errors since in reality the problem is formed as three dimen-sional. With full information about the aerodynamics, thermodynamics and solidmechanics, the results might have been different.


4.3.3 Impact of uncertainties

The effect for each variable from the design variables are shown below,

(A) Mass. (B) ∆Pc.

(C) ∆Tc. (D) Tmax.

(E) ∆ε1. (F) ∆ε3.

(G) Damage. (H) Heat pick-up.

(I) Mach number.

FIGURE 4.4: Effect influence for each design variable.

The largest residuals from the previous section is Tmax, Damage and ∆ε1 whichhave the highest common impact on x2. This is the expected result since the innerwall thickness x2 is essential for the cooling system and the fuel to reduce the walltemperature effectively. If the wall is thick the cooling effect decreases due to anincreased heat resistance. With such high heat resistance, the temperature increasesresulting in higher damage and strain which at a certain limit will cause a failure ofthe structure.

Studying the tables 4.5 and 4.3, it can be seen that the design variables x2, x4, x5, x6

and x7 have results close to the lower limit, but that x1 and x3 have not. By increas-ing x1 this could result in reduction of temperature Tmax, Damage, and ∆ε1, whichin turn affect to some other properties, i.e. increasing ∆Pc, Mass and Mach number.Producibility is however improved. The plots of these in a two-dimensional relationcan be seen in Appendix B.


According to figure B.1, by increasing Mass, it can result in increasing Damage,∆ε1, ∆ε3, Tmax which is not strange but these properties also depend on other fac-tors. The correlation matrix gives a basic understanding of the physical behaviorand how each behavior is related to each other and design variables. An example ofthis is the inner wall thickness x2 and outer wall thickness x5, these design variableshave the largest effect and impact on the mass, figure 4.4a. In fact, the parametershaving high influence on Tmax, ∆ε1 and ∆ε3 is x2, where e.g. x4 has no impact ontemperature and strains.

Design variable x2, the inner wall thickness has a high effect on Mass, Tmax, ∆ε1,∆ε3 and Damage, figure 4.4. The number of channels, x1, is dominating for ∆Pc,∆Tc, Tmax, Damage, Heat pick-up and Mach number. When the number of channelsincreases the channel width decreases resulting in higher pressure loss and Machnumber decrease. Similarly, for the channel height, x3, where it mostly effects ∆Pc,∆ε1, ∆ε3, Damage and Mach number. The strain ranges ∆ε1 and ∆ε3 increase whenthe height increases since the stress become higher, see section 3.3.1. However, themid wall thickness, x4, is observed to has the highest effect on the Mass, ∆ε1 andTmax. The second largest impact on the Mass is the outer wall thickness, x5, whichhas low effect on the other physical properties. For the radii, x6 and x7, the millradius, x6, is important for the temperature in the wall which is illustrated in figureC.3. In the red and orange regions, the temperature is differing by≈ 10%, dependingon the size of mill radius. By increasing the radius, the strain range ∆ε1, the maxstrain εmax and the Damage will also increase. On the other hand, if the mill radiusis small the fuel can cool the wall more effectively due to less heat resistance, whichgives the opposite effect to the earlier statement. Lastly, the weld radius, x7, has notshown any significant high effect on the physical properties for the piece of spacenozzle positioned in the combustion chamber. It has also shown to contribute a loweffect on the Mass, ∆Pc and Mach number. However, this can be different for thestudy at bottom of the space nozzle where the channel height is larger, the stress onwalls is more intense and there is no extra sheet a2. This can be concluded, that themill radius will contribute and increase the effect of strain range and damage on theouter wall. Which in this study such thought has been putting when defining theoptimization limitations.

47

Chapter 5

Conclusion and Recommendation

5.1 Conclusion

For the surrogate model optimization using a second order polynomial, the resultshave shown to be more stable for the industrial limit and industrial limit with tol-erance compared to the design space limit for both ATC and MDF approaches. Theresults show minor variations almost all the design variables except for the numberof channel and the channel height. The bound of the industrial limit has proven torestrict the optimization outcomes and the physical properties. Besides, most of theresults have shown to be near lower optimization limit, which is the evident that theresults are near to the global solution optimizing toward the mass. The mass reduc-tion of the industrial limit has shown to be≈ 15% and when applying the tolerances− ≈ 4% relative to a reference. If new manufacturing methods are developed thatcan reduce the dimensions even further, then the results have shown a potential todecrease the mass of a space nozzle over 50%. Problematically, this is an expensiveoption, but can result in an investment in the future for newer space nozzle.

The results of optimization with industrial limit have shown to be in a rangeof 0.5 − 1.5 for both ATC and MDF methods. It can also be concluded, that theoptimum we are searching for using the current producibility exists somewhere inbetween 0.5− 1.5 relative to the reference value. In this study range, it has also beenshown that the amount of designs needed to exclude is less than using a 0.2 − 1.7study range.

For a small system, the MDO problem formulation can be considered as MDFformulation. ATC is more suited to larger problem where the MDF formulation hasdifficult to converge, see section 2.3. For the design space limit, the MDF approachwill be chosen due to a more stable result on the εmax and higher ∆Tcondense being2% heavier than the ATC solution. But for the industrial limit, the ATC solution hasshown better values of Damage and ∆Tcondense compared to the MDF. Lastly, for theindustrial limit with tolerance, their solutions are approximately the same, but theMDF solution leads to lower Damage and cheaper to manufacture.

The highest effect on the Mass is from the inner wall thickness x2, by varying thethickness it has shown an impact on Tmax, ∆ε1, ∆ε3 and Damage. Besides the millradius x6, has the lowest effect on the Mass, which is in the opposite for Damage,strains and temperature. The temperature issue is often occurred in the combustionchamber, but at the bottom part, the channel is larger and colder which results in abetter number of temperature, Damage and strains. Anyhow, the information for thebottom section of the space nozzle is still necessary, where the channel size is largerwhich can affect the strain and the Damage. Difficulties have been encounteredwhen creating the surrogate model for the lower part of the space nozzle, since thestress on the wall is large due to the size of the channel. This requires the designspace to decrease which leads to a limited solution when optimizing toward Mass.

48 Chapter 5. Conclusion and Recommendation

5.2 Recommendation

For today’s industrial limitation there is no point to use a design range wider than0.5− 1.5 since the optimized results have shown to not overlap this limit.

The surrogate model for physical properties can also be formulated in anotherway, where e.g. ∆ε is depending on the temperature which depends on the geom-etry. In this case by creating a second order surrogate model of ∆ε ∝ T 2 and sinceT ∝ x2, the surrogate model function of ∆ε will be having a higher order. This prin-ciple works effectively if the simulation time is expensive and the number of datapoints are few, but it can also increase the uncertainty of the surrogate model. Thetemperature is created in terms of design variables which has its own uncertainty,and by adding this function into the other, the uncertainty will then be accumulatedinto the new function.

In the Finite Element method simulations for designs that cannot carry the load,Ansys program has decreased the step size and taken more iterations which resultsin long simulation times. This can be handled by defining a maximum allowableiteration number which exceeds points that the design is unable to use.

The optimal result, where its foundation is based on the surrogate model and canbe analyzed using another way to describe the problem. Instead of using a secondorder polynomial, the Kriging model or the Radial Basis Functions can be used. Thenew result of using a new surrogate model should give a similar result as in thisstudy, due to the defined limitations. Almost all the design variables have shown tobe at their allowed lower optimization bound except for the x1 and x3. These can beused to vary and receive new physical properties differently from this study.

For further investigation, another type of design can be constructed having theoutlet height of the channel twice as big as an inlet. An even higher degree of free-dom is obtained when allowing both the inlet and the outlets height to vary. Also,the inner wall thickness can be allowed to vary at the inlet and the outlet of the noz-zle. Where at the inlet the temperature is high, which is on the contrary at the outlet.This variation can be interesting in the next study and to find out more limitationson the space nozzle manufacturing. A further limitation can also be studied, whichconcerns the robustness of the model. The current robust solution was performed bymanually adjusting the limit in the manufacture tolerance range and seek a solutionby running the optimization. Another approach is to use a probabilistic way and tryto find the density function for each physical constraint and objective.

49

Bibliography

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[4] Peters, M., Jorg, F. K., Charles, W. H., Christoph, L. 2003. Titanium Alloys forAerospace Applications. Advanced Engineering Materials 5 (6): 419-427.

[5] Lindblad, K., Amnell, H., Persson, S., Lindblad, H., Olofsson, H., Palmnas, U.,Brox, L. 2016. Sandwich NE for Vinci evolution. GKN Aerospace internal report.

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[7] Nusselt, W. 1916. Die oberflachenkondensation des wasserdampfes. Zeitschriftdes Vereines Deutscher Ingenieure 60(27): 541– 575 (in German).

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[11] Wetterholm, E. 2016. Crack Initial Analysis. GKN Aerospace internal report,VOLS:10029214.

[12] Minasny, B., McBratney, A.B. 2005. A conditioned Latin hypercube method forsampling in the presence of ancillary information. Computer Oriented GeologicalSociety, Elsevier 32: 1378–1388.

[13] McKay, M. D., Conover, W. J., Beckman, R. J. 1979. A Comparison of ThreeMethods for Selecting Values of Input Variables in the Analysis of Output Froma Computer Code. Technometrics 21: 239-245.

[14] Helton, J. C. and Davis, F. J. 2002. Latin Hypercube Sampling and the Propagation ofUncertainty in Analyses of Complex Systems, SAND2001-0417. Albuquerque, NM:Sandia National Laboratories.

[15] Montgomery, D. C., Runger, G. C., Hubele, N. F. 2012. Engineering Statistics. 5thed. NJ: Wiley & Sons.

50 BIBLIOGRAPHY

[16] Cavazzuti, M. 2013. Design of Experiments. Optimization Methods: From Theoryto Design, Springer-Verlag Berlin Heidelberg: 13-42.

[17] Martins, J. R. R. A., Lambe, A. B. 2013. Multidisciplinary Design Optimization:A Survey of Architectures. American Institute of Aeronautics and Astronautics 51(9): 2049-2075. doi: 10.2514/1.J051895.

[18] Cramer, E. J., Dennis, J. E., Frank, P. D., Lewis, R. M., Shubin, G. R. 1994. Prob-lem formulation for multidisciplinary optimization. Society for Industrial and Ap-plied Mathematics 4 (4): 754-776.

[19] Alexandrov, N. M., Lewis, R. M. 1999. Comparative properties of collaborativeoptimization and other approaches to MDO. Proceeding of ASMO UK /ISSMOCONFERENCE on Engineering Design Optimization. MCB Press.

[20] Allison, J. Kokkolaras, M. Zawislak, M. Papalambros, P. Y. June 2005. On theUse of Analytical Target Cascading and Collaborative Optimization for ComplexSystem Design. Proceeding of 6th World Congress of Structural and MultidisciplinaryOptimization, Rio de Janeiro, Brazil.

[21] Allison, J., Walsh, D., Kokkolaras, M., Papalambros, P. Y., Cartmell, M. Jan-uary 2006. Analytical Target Cascading in Aircraft Design. 44th American Insti-tute of Aeronautics and Astronautics Aerospace Sciences Meeting and Exhibit. doi:10.2514/6.2006-1325.

[22] Tosserams, S., Etman, L. F. P., Papalambros, P. Y., Rooda, J. E. June 2005. AnAugmented Lagrangian Relaxation for Analytical Target Cascading using theAlternating Directions Method of Multipliers. Proceeding of 6th World Congress ofStructural and Multidisciplinary Optimization, Rio de Janeiro, Brazil.

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51

Appendix A

Derivation

A.1 Main jet expansion

Modified Bernoulli’s equation for isentropic gas and neglecting gravitation term fortwo points 1 and 2:

P1 + ρv2

1

2= P2 + ρ

v22

2(A.1)

From ideal gas law, a rewritten in term of specific heat ratio Cp is:

P = ρCpT (A.2)

Putting this back to equation (A.1) and uses that v2 ≡ v22 − v2

1 :

1

2v2 + CpT2 = CpT1 (A.3)

then

∆T = T2 − T1 =v2

2Cp(A.4)

The Mach number can be used in this case instead of using velocity v. The Machnumber M = v/cs, where cs is the speed of sound in v’s medium. Together with theMach number equation (A.4) can be modified:

T2

T1=

(1 +

1

2(γ − 1)M2

)−1

(A.5)

For isentropic relation, T γ2 P1−γ2 = T γ1 P

1−γ1 this results:

P2

P1=

(T2

T1

)γ/(γ−1)

,ρ2

ρ1=

(T2

T1

)1/(γ−1)

(A.6)

Stagnation enthalpy for point 1 is given by:

ht1 = h1 +v2

1

2(A.7)

The exit velocity of the nozzle can be determined by using different in stagnationenthalpy point 1 and point 2. For a case two points are sharing the same stagnationpoint along its axis. Combining isentropic relations, equation A.6 and A.7:

v22 = 2(h1 − h2) + v2

1 = 2γ

(γ − 1)

RT1

M

(1−

(P2

P1

) γ−1γ

)+ v2

1 (A.8)

52 Appendix A. Derivation

Velocity at point 1 is placed in the converging section which is less than the velocityat point 2, then this relation holds v2

1 v22 , and putting point 2 to exit e and 1 to

converging section c:

ve ≈

√√√√2γ

(γ − 1)

RTcM

(1−

(PePc

) γ−1γ

)(A.9)

53

Appendix B

Figures and Tables from Surrogatemodel

TABLE B.1: Fitting quality to the data for each variables using 50 de-signs between an interval of 0.5 to 1.5.

Type Polynomial order R2 R2adjust

Tmax 2 0.9999 0.9997∆Tc 2 0.9999 0.9997∆Pc 2 0.9959 0.9858Mass 1 0.9839 0.9813ln(Damage) 1 0.9892 0.9865∆ε1 2 0.9998 0.9995∆ε3 1 0.9954 0.9942Heat pick-up 1 0.9999 0.9999

TABLE B.2: Fitting quality to the data for each variables using 50 de-signs between an interval of 0.2 to 1.7.

Type Polynomial order R2 R2adjust

Tmax 2 0.9999 0.9987∆Tc 2 0.9999 0.9995∆Pc 2 0.9881 0.9290Mass 2 0.9998 0.9990ln(Damage) 2 0.9979 0.9874∆ε1 2 0.9996 0.9979∆ε3 2 0.9990 0.9940Heat pick-up 2 0.9998 0.9990

54 Appendix B. Figures and Tables from Surrogate model

FIGURE B.1: Correlation matrix.

FIGURE B.2: Plot of mass over design variables.

Appendix B. Figures and Tables from Surrogate model 55

FIGURE B.3: Plot of ∆Tc over design variables.

FIGURE B.4: Plot of ∆PC in logarithmic scale over design variables.


FIGURE B.5: Plot of Tmax over design variables.

FIGURE B.6: Plot of damage in logarithmic scale over design vari-ables.


FIGURE B.7: Plot of heat pick-up over design variables.

FIGURE B.8: Plot of ∆ε1 over design variables.


FIGURE B.9: Plot of ∆ε3 over design variables.

FIGURE B.10: Plot of mach number over design variables.


FIGURE B.11: Plot of temperature different, ∆Tcondense over designvariables.

FIGURE B.12: Plot of ∆Pc over design variables.


FIGURE B.13: Plot of Damage over design variables.

61

Appendix C

Addition figures

FIGURE C.1: Different strain peaks.

(A) Optimized with mass as objective. (B) After Plastic simulation in Ansys.

FIGURE C.2: Optimized with mass as objective and no physical con-straints.

62 Appendix C. Addition figures

FIGURE C.3: Temperature different from Ansys simulation.

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