Development of a Multidisciplinary

Design Optimization Framework

applied on UAV Design

Athanasios Papageorgiou

Division of Machine Design

Master Thesis

Department of Management and Engineering

LIU-IEI-TEK-A-15/02189-SE

Development of a Multidisciplinary

Design Optimization Framework

applied on UAV Design

Master Thesis in Multidisciplinary Design Optimization

Department of Management and Engineering

Division of Machine Design

Linköping University

by

Athanasios Papageorgiou

LIU-IEI-TEK-A-15/02189-SE

Supervisors: Edris Safavi

IEI, Linköping University

Examiner: Kristian Amadori

IEI, Linköping University

Linköping, 12 June, 2015

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© Athanasios Papageorgiou

i

ABSTRACT

The present thesis deals with the topic of Multidisciplinary Design Optimization (MDO) and in particular with the development of a framework for Unmanned Aerial Vehicle (UAV) design. The orientation of the research and the overall outlook of the case-study have been based on Saab’s proposal regarding the application of MDO in complex products and are in compliance with the guidelines of the Innovative Multidisciplinary Product Optimization (IMPOz) project initiative. For this initial stage of the project, five principle engineering disciplines related to aircraft design were considered and those were namely the geometric design, the aerodynamics, the antenna analysis, the Radar Cross Section (RCS) signature, and the mission simulation. The aforementioned disciplines were expressed within the framework by developing computational models which were further based on a relevant set of engineering tools. In the present case-study, the primary focus was on using the indicated engineering tools which are available to both Linköping University and Saab but also to investigate viable and more efficient alternatives. A simple optimization strategy was implemented as a guide for the integration of the models and the core framework configuration was evaluated by using the Design Of Experiments (DOE) method. Finally, the use of metamodels as a tool that can increase the computational efficiency of the framework was analyzed and a preliminary optimization of the product was performed as an example of the framework’s capabilities.

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iii

ACKNOWLEDGMENTS

To begin with, I would like to thank my examiner Kristian Amadori and the head of the Machine Design division Johan Ölvander for introducing me to the world of MDO and for giving me the opportunity to work on a very interesting project. Your work has been a source of insightful information as well as inspiration and your support was without a doubt instrumental for the completion of this thesis.

Furthermore, a very special “thank you” goes to my supervisor Edris Safavi for all his help throughout the project and especially for his very useful advice and suggestions. I am really grateful for all the time you spent with me and for your guidance in all aspects of the project.

Moreover, I would like to express my appreciation towards the discipline experts from Saab: Carina Marcus, Jakob Bjerkemo, Natalia Gardberg. Thank you for the time you dedicated to this project despite your already busy schedules.

Undoubtedly, a person that deserves my unconditional gratitude is Raghu Chaitanya M.V. of FluMes division. You have been very helpful throughout those six months and your contribution in many technical parts of my project was of vital importance for its completion.

Last but not least, I would like to thank my friends and peers from the M.Sc. program in aeronautics: Aevan, Alejandro and Sharath. You have been very supportive and our discussions as well as exchange of ideas have proven to be far better than any scientific paper or book.

Linköping, June, 2015 Athanasios Papageorgiou

iv

v

ABBREVIATIONS

AC Aerodynamic Center AKR Anisotropic Kriging AOA Angle Of Attack BL Boundary Layer CAD Computer Aided Design CFD Computational Fluid Dynamics CG Center of Gravity DA Design Automation DIBA Digital Interactive Basic Aircraftanalysis DOE Design Of Experiments DS Direct Sampling (method) FEM Finite Element Method GA Genetic Algorithm GRECO Graphical Electromagnetic Computing IDS In-Direct Sampling (method) IMPOz Innovative Multidisciplinary Product Optimization IR Infrared IRST Infrared Search and Track KR Kriging MAC Mean Aerodynamic Chord MAV Micro Aerial Vehicle MAW Missile Approach Warning MDO Multidisciplinary Design Optimization ML Multi-level (strategy) MOGA Multi-Objective Genetic Algorithm MTOW Maximum Take Off Weight NP Neutral Point NRMSE Normalized Root Mean Square Error RCS Radar Cross Section RF Radio Frequency RMSE Root Mean Square Error SL Single level (strategy) SM Static Margin SVD Singular Value Decomposition UAV Unmanned Aerial Vehicle UCAV Unmanned Combat Aerial Vehicle ULH Uniform Latin Hypercube VLM Vortex Lattice Method

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vii

CONTENTS

Abstract ................................................................................................................................................................ i

Acknowledgments ..........................................................................................................................................iii

Abbreviations .................................................................................................................................................... v

Contents ........................................................................................................................................................... vii

List Of Figures .................................................................................................................................................. ix

List Of Tables .................................................................................................................................................... xi

Introduction ....................................................................................................................................................... 1

1.1 Background ............................................................................................................................................ 1

1.2 The IMPOz Project ............................................................................................................................... 1

1.2.1 In General ........................................................................................................................................ 1

1.2.2 Saab’s Proposal ............................................................................................................................. 2

1.3 Methodology .......................................................................................................................................... 3

1.3.1 Overview ......................................................................................................................................... 3

1.3.2 Description Of The Tools .......................................................................................................... 4

1.3.3 Challenges And Limitations ..................................................................................................... 4

1.3.4 Research Questions ..................................................................................................................... 5

1.4 Objectives And Goals .......................................................................................................................... 5

1.5 Report Outline ....................................................................................................................................... 6

Frame Of Reference ........................................................................................................................................ 7

2.1 Aircraft Design ...................................................................................................................................... 7

2.1.1 Wing And Tail Sizing ................................................................................................................... 7

2.1.2 Stability And Trim ....................................................................................................................... 8

2.1.3 Airfoil Definition .......................................................................................................................... 9

2.1.4 Radar Cross Section ................................................................................................................. 10

2.1.5 Mission Performance............................................................................................................... 10

2.2 Aerodynamics .................................................................................................................................... 12

2.2.1 Vortex Lattice Method............................................................................................................. 12

2.2.2 Computational Fluid Dynamics ........................................................................................... 12

2.3 Antennas And Sensors .................................................................................................................... 13

2.3.1 Overview ...................................................................................................................................... 13

2.3.2 Basic Principles .......................................................................................................................... 13

2.3.3 Coordinate Transformations ................................................................................................ 15

2.3.4 Considerations For MDO ........................................................................................................ 15

2.4 Multidisciplinary Frameworks .................................................................................................... 16

2.4.1 Overview ...................................................................................................................................... 16

2.4.2 Optimization Strategies .......................................................................................................... 16

2.4.3 Optimization Algorithms ....................................................................................................... 17

2.4.4 Efficient Computing ................................................................................................................. 17

viii

Contributions ................................................................................................................................................. 21

3.1 Framework Evaluation ................................................................................................................... 21

3.1.1 Overview ...................................................................................................................................... 21

3.1.2 Problem Formulation .............................................................................................................. 22

3.1.3 Design Variables ........................................................................................................................ 22

3.1.4 Strategy Definition ................................................................................................................... 23

3.2 Module Description .......................................................................................................................... 24

3.2.1 Sizing Loop .................................................................................................................................. 24

3.2.2 Trim ................................................................................................................................................ 24

3.2.3 Antenna Loop ............................................................................................................................. 25

3.3 Model Specifications ........................................................................................................................ 26

3.3.1 Aircraft Geometry ..................................................................................................................... 26

3.3.2 Aerodynamic Performance By Using VLM ...................................................................... 28

3.3.3 Aerodynamic Performance By Using CFD ....................................................................... 30

3.3.4 Antenna Analysis ...................................................................................................................... 31

3.3.5 Radar Signature ......................................................................................................................... 32

3.3.6 Mission Simulation ................................................................................................................... 33

3.4 Evaluation And Optimization ....................................................................................................... 34

3.4.1 Design Of Experiments ........................................................................................................... 34

3.4.2 Performance ............................................................................................................................... 35

3.4.3 Efficient Computing ................................................................................................................. 36

3.4.4 Optimization ............................................................................................................................... 36

Discussion And Conclusions ..................................................................................................................... 41

4.1 Models ................................................................................................................................................... 41

4.2 Framework .......................................................................................................................................... 43

4.3 Results ................................................................................................................................................... 44

4.4 Summary .............................................................................................................................................. 46

4.5 Future Work ........................................................................................................................................ 47

References ....................................................................................................................................................... 49

Appendix A ...................................................................................................................................................... 53

Appendix B ...................................................................................................................................................... 55

Appendix C ...................................................................................................................................................... 57

Appendix D ...................................................................................................................................................... 59

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LIST OF FIGURES

Figure 1.1 Overview of similar UAV applications ............................................................................... 2

Figure 1.2 The multidisciplinary framework of the Saab-IMPOz case-study .......................... 2

Figure 1.3 Outline of the Thesis ................................................................................................................. 6

Figure 2.1 The tail sizing and trim of the aircraft ............................................................................... 8

Figure 2.2 The geometrical details of the present airfoil definition ............................................ 9

Figure 2.3 The “Demo Gotland” hi-lo-hi mission .............................................................................. 11

Figure 2.4 The spherical coordinate system ...................................................................................... 14

Figure 2.5 The coordinate transformation for the directive gain .............................................. 15

Figure 2.6 The DS method and the IDS method ................................................................................ 18

Figure 3.1 The inter-disciplinary relations of the proposed framework ................................ 23

Figure 3.2 The inter-disciplinary relations and operation of the “sizing loop” .................... 24

Figure 3.3 The inter-disciplinary relations and operation of the “trim module” ................ 25

Figure 3.4 The inter-disciplinary relations and operation of the “antenna loop” ............... 26

Figure 3.5 The aircraft geometry ........................................................................................................... 27

Figure 3.6 Geometry parametrization of the engine intake ......................................................... 28

Figure 3.7 Example of aperture and sensor placement on the geometry .............................. 28

Figure 3.8 Result plots from TORNADO .............................................................................................. 29

Figure 3.9 The methodology for importing the airfoil data in TORNADO .............................. 30

Figure 3.10 The domain shape and size with an overview of the mesh density ................. 30

Figure 3.11 An example of the antenna 3D radiation pattern .................................................... 31

Figure 3.12 An example of polar plots of the directive gain ........................................................ 32

Figure 3.13 The reference ground radar position and a RCS result plot ................................ 32

Figure 3.14 The two levels of geometry detail in DIBA ................................................................. 33

Figure 3.15 Example of the available mission performance results in DIBA ........................ 34

Figure 3.16 The optimization of the “sizing loop” with SIMPLEX ............................................. 37

Figure 3.17 Evaluation of MOGA-II for the optimization of the “antenna loop” .................. 37

Figure 3.18 The modified framework used in the preliminary optimization ....................... 38

Figure 3.19 The convergence progress of SIMPLEX when using metamodels ..................... 38

Figure 3.20 The convergence progress of MOGA-II when using metamodels ...................... 39

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xi

LIST OF TABLES

Table 2.1 Airfoil point coordinates ........................................................................................................... 9

Table 3.1 The global and the local optimization problems .......................................................... 22

Table 3.2 The global and local design variables ............................................................................... 23

Table 3.3 The main input and output of the geometry model .................................................... 27

Table 3.4 The main input and output of the TORNADO model .................................................. 29

Table 3.5 The main input and output of the CFD model ............................................................... 30

Table 3.6 The main input and output of the antenna model ....................................................... 32

Table 3.7 The main input and output of DIBA at a low-fidelity level ....................................... 33

Table 3.8 The main input and output of DIBA at a medium-fidelity level .............................. 34

Table 3.9 The specifications of the baseline configuration .......................................................... 35

Table 3.11 Breakdown of the module performance ....................................................................... 35

Table 3.12 The obtained NRMSE results from the “antenna loop” metamodel ................... 36

Table 3.13 The obtained NRMSE results from the CFD metamodel ......................................... 36

Table 3.14 The obtained results from the optimization ................................................................ 39

xii

1

INTRODUCTION

Part 1 of the thesis introduces briefly the topic of design optimization, the specifications of the IMPOz project, and the details of the present case-study. Furthermore, the applied research methodology is elaborated and the thesis objectives as well as the challenges are presented.

1.1 BACKGROUND

The development of complex products is without a doubt a challenging process that involves the interaction of different engineering disciplines. In addition to this, manufacturing industries are oriented towards increased efficiency in all design phases and they strive for time and cost reduction but without losing product quality [1].

Multidisciplinary Design Optimization (MDO) is a promising method which can drastically improve the iterative design process by adding the much needed efficiency in all aspects of the product development process [1], [2]. According to Tarkian [2], “the implementation of MDO allows the designer to map the interdisciplinary relations that exist in a system and automatically search through the design space for optimal solutions”. Furthermore, the continuous computational advancements in design tools and the increasing availability of computer power have nowadays made the implementation of MDO feasible even for complex products and during advanced design phases [1], [2].

An important precondition for an effective MDO of a complex system is a Design Automation (DA) framework with parametric capabilities and automated operational features. According to Amadori [1], “DA refers to a system that is able to perform a design task in an automatic fashion”. More analytically, the framework is able to receive the design specifications as an input and generate a predefined output which can be subsequently evaluated by the optimizer [1]. As a result, the manual and repetitive tasks which are typically included in the iterative optimization process of a given design are automated and an increased development efficiency as well as a higher design quality can be obtained [2].

1.2 THE IMPOZ PROJECT

1.2.1 IN GENERAL

The main objective of the Innovative Multidisciplinary Product Optimization (IMPOz) project is to apply MDO in the manufacturing industry in order to promote more efficient product development processes [3]. The project is under the management of “Kunskapsförmedlingen” (Result Center) which is a Swedish research initiative with focus on product realization, production and support [3]. The research activities emphasize on case-specific products and for this reason there is an ongoing collaboration between Linköping University and industrial partners such as ABB, Bombardier, EnginSoft, Minesto, and Saab.

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According to [3], the basic research topics of the IMPOz project are primarily in respect to the technical and computational requirements of each organization, the partner-specific product design requirements, the available computational tools, and the overall challenges of introducing MDO in the industry. The expected results of the project include guidelines for efficient employment of MDO, optimal product customization, and investigation of possible methods as well as algorithms for MDO [3].

1.2.2 SAAB’S PROPOSAL

The present thesis deals exclusively with the case-study which was proposed by Saab regarding the application of MDO on the development of an Unmanned Aerial Vehicle (UAV). The principle design guidelines include wings of high Aspect Ratio (AR), a V-shaped stabilizer, and symmetry about the XZ (vertical) plane. Similar aircraft which can be considered as reference for the design can be seen in Figure 1.1. Those are the Northrop Grumman RQ-4 Global Hawk, the General Atomics MQ-9 Reaper, and the General Atomics Avenger.

Figure 1.1 Overview of similar UAV applications showing the Global Hawk (left), the Avenger (centre), and the

Reaper (right). The photos are courtesy of [4] and [5] respectively.

The research focus of this application is only on certain engineering disciplines, while several other aircraft design aspects are omitted. Figure 1.2 briefly summarizes the multidisciplinary framework which has been considered for the Saab-IMPOz case-study. Some of the models in Figure 1.2 are not included in the scope of the present thesis (marked with blue colour) but they will be included in the framework as part of a future enhancement.

Primary focus within the present case-study is given to topics that are related to the implementation of MDO and in particular to the integration of the tools, to the effectiveness of the applied methodology, to the limitations of this approach, and to the optimization strategy. At the same time, the development of the final product and the complexity of the models are of secondary importance.

Figure 1.2 The multidisciplinary framework of the Saab-IMPOz case-study

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1.3 METHODOLOGY

1.3.1 OVERVIEW

In order to effectively apply MDO in the development process of the given product, an automated multidisciplinary framework had to be developed. The proposed framework is comprised of several models which correspond to the considered engineering disciplines. Some of the models of the framework are related to only one computational tool but there are also models that require additional support-tools in order to perform the necessary calculations.

The ultimate choice of the computational tool which will represent a model in the framework was based on three basic criteria which are namely the compatibility, the availability, and the fidelity of the software. The term compatibility refers to the ability of the tool to seamlessly communicate with the other components of the framework. By the term availability one refers to whether or not the software can be obtained and used for the current case-study. Finally, the term fidelity is what defines the overall computational capabilities of the software. In a nutshell, low-fidelity tools will give a fast but simplified result, whereas high-fidelity tools are more accurate but come at a higher computational expense.

The models of the proposed framework use tools that are initially suggested by Saab but are also available to Linköping University. In this way, the framework is developed according to the specific technical requirements of the partner-organization. The compatibility between the various tools is not always ideal and therefore, several modifications are required in order to facilitate the interaction between the models. Additional modifications regarding the support of DA features have also been implemented so that the framework can enable a MDO. The overall fidelity of the framework is defined by the preselected tools and it is aimed towards implementing accurate solutions. However, in the case of aircraft aerodynamics, it was decided to test three different options which are different in fidelity and compatibility but are all available to both Linköping University and Saab.

A summary of the research methodology which was implemented in the thesis is presented in the following workflow, while more specific details regarding the development methodology of the individual models and the framework are further elaborated in sections 3.2 and 3.3:

STEP-1:-Identification of the given and suggested computational tools. Evaluation of aspects such as availability, compatibility and fidelity.

STEP-2:-Development of the models. Evaluation of different tool alternatives and possible integration solutions. Implementation of modifications so that the models can handle the expected input and deliver the expected output.

STEP-3:-Definition of the optimization problem, the design variables, and the optimization strategy. Evaluation of the models with respect to handling the selected optimization scheme and implementation of changes in order to effectively handle the inter-disciplinary relations.

STEP-4:-Integration of the models in the common framework and evaluation of the framework’s performance. Identification of the computationally expensive segments and application of corrective changes.

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STEP-5:-Further improvement of the framework’s performance by implementing efficient computing solutions. Optimization of the product and evaluation of the optimization scheme and the MDO method.

1.3.2 DESCRIPTION OF THE TOOLS

The basis for the framework of the present case-study was developed by using the commercial software modeFRONTIER by ESTECO [6]. According to the developers, the software is aimed towards MDO and enables the automation of the simulation process as well as the coupling between third-party computer applications.

In the center of the framework is the geometrical model of the UAV which was created in Dassault Systemes CATIA V5R21 design tool suite [7]. The software offers the standard Computer Aided Design (CAD) functions but it is also possible to include knowledge-based features which are essential in a DA framework.

A computational tool that was used in the development of several of the models which are included in the present framework and in many other instances in order to perform support calculations is Mathworks MATLAB R2014b [8]. In particular, the mission simulation tool DIBA, the aerodynamic performance tool TORNADO, and the sensor analysis tool are all MATLAB-based codes. Furthermore, MATLAB is also used for sizing and trimming the aircraft as well as for pre- and post-processing of the input and output of several models.

The model that analyzes the aerodynamic performance of the aircraft has been developed by using two different approaches. As a medium-fidelity approach, the Vortex Lattice Method (VLM) was considered, whereas as a high-fidelity approach two Computational Fluid Dynamic (CFD) solvers were implemented. For the VLM analysis, the MATLAB-based code TORNADO developed by Melin [9] was selected. The CFD codes which are included and tested within the framework are ANSYS Fluent and CFX [10].

The preliminary design toolbox DIBA (Digital Interactive Basic Aircraftanalysis) of version FOI-FFA was used to predict the behavior of the aircraft under specific mission requirements. The tool is provided by Saab and includes several in-house MATLAB codes that can be used to predict the weight, the aerodynamic and the performance characteristics of an aircraft during the conceptual design phase.

For the Radar Cross Section (RCS) signature analysis, the Graphical Electromagnetic Computing (GRECO) method was initially considered. Due to time restrictions and software availability issues the RCS computation was performed by DIBA at a low fidelity level so that the design space can be effectively filled.

1.3.3 CHALLENGES AND LIMITATIONS

A number of challenges were identified throughout the thesis and can be generally placed in two major categories. The first category is about the development challenges of the individual models:

Although the complexity is of secondary importance, the models should have sufficient fidelity in order to be implemented in a MDO framework.

The models should be robust and error-free when operating with inputs from the predefined design space of the present case-study.

The input and output data to and from each model should be manipulated in a compatible form that will allow a smooth flow of information inside the framework.

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Especially for the geometry model, a proper set of aircraft design rules should be followed and the parametrization should be done in a realistic way that it also corresponds to the specified design variables.

The second category is about the integration of the models in the common framework and the technical requirements that arise from that:

The framework should be able to clearly depict the chosen optimization strategy but also be flexible to future changes.

The integration of the models should be robust and at the same time oriented towards increased efficiency.

The software compatibility issues should be identified and different solutions within the technical limits of the organization should be investigated.

Computationally expensive segments of the framework must be isolated and possible solutions that can decrease the iteration times should be implemented.

Further challenges include software availability issues and contradictory or unfeasible design demands on behalf of the discipline experts. The latter point is particularly sensitive as it can induce large scale changes at a model- and a framework-level. Good communication, compromise and ingenuity were required throughout the project in order to find an acceptable solution.

1.3.4 RESEARCH QUESTIONS

Apart from the manual work that is related to the development of the individual models and their integration in a common framework, the research also focuses on certain topics which are presented below in the form of research questions:

RQ1:-Are the suggested tools adequate in terms of availability, compatibility, and fidelity?

RQ2:-How effective are the models with respect to handling the inter-disciplinary relations?

RQ3:-Is the integration of the models in a common framework under the current optimization strategy feasible?

RQ4:-Are there any alternative solutions that can increase the efficiency of the framework?

RQ5:-Is MDO a viable solution for the proposed case-study and what should be the direction of the future work?

1.4 OBJECTIVES AND GOALS

The primary objective of the present thesis is to develop a multidisciplinary framework which will enable a MDO on the design of a UAV. This is further analyzed into two main components which are namely the development of the individual models and the integration of the models into a common DA framework. The models should follow the specifications set in Saab’s proposal and should have adequate complexity in order to support the MDO method. The framework must be above all functional and capable of supporting an optimization strategy which can yield realistic results.

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The primary goal set by the author was to develop the basic models which will comprise the initial framework (see in Figure 1.2) and to integrate them in a common platform. The secondary goal is to test the functionality of the framework through several Design Of Experiments (DOEs), to evaluate the obtained results, and to identify the required iteration times. The third goal is to investigate different methods that can increase the efficiency of the framework and then apply them in order to run a preliminary optimization of the final product.

1.5 REPORT OUTLINE

The present thesis is organized into four main chapters with the introduction being the first. The outline of the report is graphically summarized in Figure 1.3.

Figure 1.3 Outline of the report

The second chapter includes an overview of the guidelines which were used during the development of the framework models. The scope is to give the reader an overview of the methodology but not to delve into each individual discipline in depth. Throughout the chapter, several relevant case-studies are evaluated and compared to the present application in respect to the most common and practical aspects of MDO.

The third chapter is about the contributions of the present thesis towards the Saab-IMPOz case-study. In this chapter, the proposed framework is elaborated, the methodology and the challenges are described in detail, and the specifications of the individual models are defined.

The fourth chapter is a discussion on the obtained results and an evaluation of the overall project development. Additionally, a summary of the work is offered at the end and the research questions are answered briefly. Finally, the thesis concludes with some suggestions for future work and a description of further research possibilities.

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FRAME OF REFERENCE

Part 2 of the thesis gives a brief overview of some theoretical aspects which are considered relevant to the development of the models and also presents a review of similar design optimization applications.

2.1 AIRCRAFT DESIGN

2.1.1 WING AND TAIL SIZING

It is a common approach during the definition of the aircraft geometry to use a set of suitable conventions [11], [12], [13], [14]. Formulas (2.1) to (2.3) illustrate how the aspect ratio AR, the taper ratio λ, the span b, the surface area S, the root chord Cr, and the tip chord Ct are related.

𝐴𝑅 =

𝑏2

𝑆 (2.1)

𝜆 =

𝐶𝑡

𝐶𝑟 (2.2)

𝐶𝑟 =2

1 + 𝜆√

𝑆

𝐴𝑅 (2.3)

In the conceptual design phase, empirical formulas are often used in order to size the tail control surfaces [1], [15]. Therefore, the surfaces Sh and Sv of the horizontal and vertical stabilizer can be approximated by using formulas (2.4) and (2.5) according to both Raymer [11] and Kroo [12]. In order to do that, the volume coefficients Vvt and Vht must first be defined. In addition to this, the distances lh and lv which are measured from the Aerodynamic Center (AC) of each stabilizer to the Center of Gravity (CG) of the aircraft (see Figure 2.1) should also be known. However, if the CG is not given, then a good approximation is to consider the distance between the AC of each stabilizer and the AC of the wing [13] [14].

𝑆ℎ =

𝑉ℎ𝑡 �̅� 𝑆

𝑙ℎ (2.4)

𝑆𝑣 =

𝑉𝑣𝑡 𝑏 𝑆

𝑙𝑣 (2.5)

For a V-shaped stabilizer the distances lh and lv are clearly equal. Furthermore, the total surface area S is given by formula (2.6) and the “opening angle” v by formula (2.7).

1

1 𝑆𝑡 = 𝑆ℎ + 𝑆𝑣 (2.6)

𝑣 = 180 − 2 ∗ tan−1 √𝑆𝑣

𝑆ℎ⁄ (2.7)

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2.1.2 STABILITY AND TRIM

The hypothetical longitudinal position of the CG where the aircraft stability is neutral is called the Neutral Point (NP) [13]. The distance from the actual position of the CG xcg to the position of the NP xnp is called the Static Margin (SM) and it is expressed as a percentage of the Mean Aerodynamic Chord (MAC) c as seen in formula (2.8).

𝑆𝑀 =

𝑋𝑛𝑝 − 𝑋𝑐𝑔

𝑐 (2.8)

Among other things, the SM defines the pitch stability of the aircraft and hence, it is clear that the design should be within a predefined range [11], [12]. Provided that there is no dynamic model in a MDO framework for predicting the flight characteristics, the SM can be used simply as a constraint like in the case-studies of Choi et al. [16] and Haar et al. [17]. Amadori et al. [18] included the SM in the objective function as a penalty, whereas Lundström et al. [15] approached the stability issue by including an internal loop which balances the aircraft for every optimization evaluation.

Figure 2.1 The basic lengths which are used in the tail sizing and trim of the aircraft, adapted from [13]

An aircraft is considered to be trimmed if there is a balance of forces and moments [11]. For the case of trimming about the pitch axis, a simplified expression for the moments acting about the CG can be seen in formula (2.9). In this formula, Lw and Lht are the lift forces, while mw and mht are the aerodynamic moments of the wing and the horizontal stabilizer respectively. The expression neglects thrust terms and assumes that the AC of the wing and tail as well as the CG are all on the same plane [13].

1

1 𝑚𝑤 + 𝑚ℎ𝑡 + 𝐿𝑤𝑥𝑐𝑔 + 𝐿ℎ𝑡𝑙ℎ = 0 (2.9)

During a straight and level cruise the aircraft should be trimmed. For standard aircraft configurations this is achieved by changing the incidence of the horizontal tail stabilizer which in turn affects the drag and hence, the total performance. This issue becomes more complicated if a versatile flight mission is considered where different cruise conditions are expected [19].

For a mission-oriented optimization it is important to calculate the performance based on the correct trim. Lundström et al. [15] addressed this issue in a flying-wing concept by running a panel code aerodynamic analysis at three different Angles Of Attack (AoA) within the linear range of the lift coefficient. In this way, it was possible to get the curves of the lift and the induced drag coefficients as a function of the AoA and therefore, they were able to make performance calculations at various flight conditions. A similar approach but with CFD tools was followed by Hitzel et al. [19] on a standard-tail configuration UAV design and by Haar et al. [17] on a generic aircraft with rear-

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mounted engines. The difference of [19] is that they isolated the horizontal stabilizer from the rest of the aircraft so that they could evaluate it at different AoA. Thus, it was possible to trim the aircraft in the linear region without having to run computationally expensive CFD simulations of the whole aircraft.

2.1.3 AIRFOIL DEFINITION

The airfoil definition and parametrization which is used as reference in the present design is based on the work of Melin et al. [20]. According to the authors, this type of parametric description gives an infinite coordinate resolution that is able to represent an extremely large design space which includes even degenerate profiles. Furthermore, this approach defines the wing profile with only 15 parameters which enables the modeling of many existing airfoils with good tolerance and allows the continuous optimization of the airfoil profile without resulting in discrete curvature changes [20].

Coordinates Points X Y

1 1 A

2 x2 y2 3 x3 B 4 x4 B 5 x5 B 6 0 y6

7 0 0 8 0 y8 9 x9 C

10 x10 C 11 x11 C 12 x12 y12 13 1 A

Table 2.1 Airfoil point coordinates

Figure 2.2 The geometrical details and the control points of the present airfoil definition, adapted by [20]

The airfoil representation is based on four cubic Bezier curves which are defined by 13 control points. The control points are described by their x- and y-coordinates which are subsequently reduced to 14 independent parameters if certain geometrical assumptions are made. The numbering, the order, and the position of the control points can be seen in Figure 2.2, whereas Table 2.1 shows the reduced input parameters. The coordinates for the control points of many known airfoils can be found in the two-part airfoil catalog published by Melin [21], [22].

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2.1.4 RADAR CROSS SECTION

The term Radar Cross Section (RCS) refers to the detectability of an object such as an aircraft with respect to one or more defined radar positions [23]. According to Berry [23], the designer should first define the expected mission and thereafter, it is possible to make several geometrical or material choices that can give a lower RCS. Although there is no absolute solution to the problem, some notable stealth geometrical considerations that can be applied in the critical radar directions can be seen below [23]:

Avoid three- or two-surface reflectors and large vertical flat plates Minimize cavities, discontinuities, and straight edges Use radar-absorbing materials

The RCS of any target can be described by the term σ which is measured in m2 and it is expressed in spherical directions (see section 2.3.2 for details). A simple definition of the RCS is given in formula (2.10) where r is the examined range, St is the power density which is intercepted by the target, and Sr is the scattered power density in a distance equal to r [24]. If the directive gain of an antenna transmitter is known, then a simplification can be made where the RCS can be assumed to be 0.1 m2 at its peak [25].

𝜎 =

4𝜋𝑟2𝑆𝑟

𝑆𝑡 (2.10)

Hitzel et al. [19] did not have a dedicated model for RCS calculations but instead they included the aforementioned design guidelines as geometrical constraints. In particular, they applied an equal minimum sweep to both the wing and tail leading and trailing edges together with some engine placement constraints.

Tianyuan et al. [26] noted the importance of stealth features in military UAV applications and developed a MDO framework that included a model for RCS signature calculations which were used as design constraints. Their RCS analysis tool was based on the “physical optics” theory and it was implemented through a FORTRAN code which required a triangulated surface mesh of the 3D CAD geometry similar to the GRECO method. The authors managed to reduce the computational time by using the same mesh file for both the RCS and the aerodynamic calculations.

A similar need for RCS calculations in military aircraft applications is also expressed in the work of Allison et al. [27]. In this case-study, the authors considered the stealth features early on during the conceptual design as configuration and geometry constraints but they also included a RCS analysis tool in their MDO framework. For the RCS calculations, they used the POFACETS MATLAB-based code [28] together with a meshed 3D geometry. The obtained RCS signature results were used as a comparison between the various designs and were a primary optimization objective.

2.1.5 MISSION PERFORMANCE

Two of the most commonly referred performance characteristics are range and endurance. According to Gudmundsson [14], “range is the distance an airplane can fly in a given time”, whereas “endurance is the length of time an airplane can stay aloft while consuming a specific amount of fuel”. The mathematical expressions for the range R and the endurance E can be seen in formulas (2.11) and (2.12) respectively where V is the

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airspeed, ct is the thrust specific fuel consumption, CL is the lift coefficient, CD is the drag coefficient, and W is the weight.

𝑅 = ∫

𝑉

𝑐𝑡

𝑊𝑖𝑛𝑖

𝑊𝑓𝑖𝑛

𝐶𝐿

𝐶𝐷

1

𝑊 𝑑𝑊 (2.11)

𝐸 = ∫

1

𝑐𝑡

𝑊𝑖𝑛𝑖

𝑊𝑓𝑖𝑛

𝐶𝐿

𝐶𝐷

1

𝑊 𝑑𝑊 (2.12)

Range and endurance are often encountered in MDO studies as optimization objectives [15], [16], [17], [19], [29], [30] or constraints [18]. The reason for this is that the definitions of both the range and the endurance include aerodynamic and weight parameters but are also integrated for over a mission segment as seen in formulas (2.11) and (2.12). Therefore, it is possible to have a single objective that expresses a design improvement which is related to both structural and aerodynamic performance but also considers the flight conditions [17], [30].

A definition of the term “mission analysis” is given in [14] as “the investigation of an entire flight of an airplane from engine start to shut-down”. For the purpose of this analysis, it is a common practice to break the mission into many different parts which will be addressed and analyzed individually [14]. A simple mission consists of the basic segments “taxi and take-off”, “climb”, “cruise”, “loiter”, “descent” and “landing” but there could also be complex missions where the above segments appear more than once (see the example in Figure 2.3).

Figure 2.3 The “Demo Gotland” hi-lo-hi mission which has been considered in the present case-study

The concepts of aircraft performance and mission simulation have been included in many MDO case-studies [15], [16], [18], [19], [27], [31], [30]. The reason is that a given aircraft configuration has different performance characteristics depending on the chosen flight condition [30]. Therefore, in order to get a fully optimized design, it is important to consider all possible flight conditions (mission) rather than isolate just one which will probably lead to a sub-optimal solution or a non-flyable aircraft [27] [31].

Amadori et al. [18], [31] as well as Lundström et al. [15] addressed the issue of mission performance in aircraft MDO by using simple empirical formulas which were included in a dedicated mission simulation module. The aforementioned modules were developed in either MATLAB [31] or MS EXCEL [15] and used the available aerodynamic, propulsion and weight data in order to evaluate the performance for a given mission. In this way, it was possible to check if the concept was capable of flying the mission but also to execute several calculations related to stability and balance.

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The need for feasibility constraints regarding the mission performance is also expressed in the case-study of Allison et al. [27]. For this reason, they included the medium-fidelity mission analysis tool FLOPS [32] which is able to calculate the performance in complex conditions by utilizing the computed outputs from the rest of the framework models. The authors emphasized that an advanced mission analysis tool can add supplementary design fidelity to the framework, however, it should be noted that the same was also reported by [16] and [30] who achieved very good optimization results by using only statistical and empirical methods.

2.2 AERODYNAMICS

2.2.1 VORTEX LATTICE METHOD

The Vortex Lattice Method (VLM) is a numerical method which is used in the early aircraft design phases [1]. A typical VLM solver can compute the flow around a complex wing geometry and hence, retrieve the force distribution and the aerodynamic coefficients [33]. The main disadvantages of the VLM is that it neglects the viscous, the compressibility and the body-interaction effects [33].

The TORNADO code was developed by Melin [33] and it is a VLM implemented in MATLAB. According to the developer [34], the code can solve for most aerodynamic derivatives and for a wide range of aircraft configurations at a very high computational speed. Over the years, the code has been enhanced with further functions and third-party add-ons which have increased the overall computational fidelity [9]. The features which are most relevant to the present application are the “zero lift drag prediction”, the “consideration of airfoil camber data”, the “Prandtl-Gauert compressibility correction”, and the “availability of data regarding the International Standard Atmosphere”.

A number of MDO frameworks have successfully included TORNADO as the main tool for aerodynamic predictions mostly due to its fast computational speed and its modifiable interface which enables a simple integration [35], [30]. Smith et al. [30] performed an aero-structural wing optimization, while Safavi et al. [35] used TORNADO for initial aircraft design optimization coupled with a dynamics model. It is noted in [30] that there are limitations at high AoA but nevertheless, the linear aerodynamics theory is valid for normal aircraft operating conditions such as those of cruise, climb, decent, landing, and take-off which have also been considered in the present case-study.

2.2.2 COMPUTATIONAL FLUID DYNAMICS

Computational Fluid Dynamics (CFD) tools implement numerical methods and algorithms in order to resolve a given flow problem [36]. Versteeg et al. [36] state that CFD solvers are high-fidelity computational tools which take into account all aspects of the flow and yield accurate results given that a proper simulation setup has been considered. They are used in later design phases where the design space has been reduced due to their high computational requirements [1]. As Amadori et al. [18] pointed out, the use of a CFD code in very early design phases might be unnecessary as the geometry is still not precisely defined. Furthermore, the authors of [18] note that if the main objective is the comparison of different designs, then CFD can be replaced by a tool of lower fidelity in order to speed up the optimization process.

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CFD tools have been implemented in a variety of MDO applications and it is commonly accepted that they are an accurate but computationally expensive option [16], [17], [18], [19]. Choi et al. [16] used ANSYS Fluent to create a metamodel which they coupled with other low- or medium-fidelity tools. They validated the obtained optimization results with low- and high-fidelity methods and resulted that their approach can increase the accuracy of the framework at an acceptable design cost. Haar et al. [17] as well as Hitzel et al. [19] used the CFD solver TAU [37] for optimization of aircraft designs directly and without the use of metamodels. They comment mostly on the challenges of the integrating a LINUX-based software and on the setbacks that arise from the high failure rate (average of 25 %) which they observed due to CFD process errors. Finally, a more detailed description of the challenges as well as the possible integration methodologies regarding MDO and CFD that can increase the overall efficiency of the framework can be found in the work of Breitkopf [38] but will not be repeated here. Among the many topics that are analyzed, the author points out the advantages of metamodeling as an alternative to CFD and gives examples of successful aircraft design optimizations where the results are “quite close” to the real simulations.

2.3 ANTENNAS AND SENSORS

2.3.1 OVERVIEW

In general, an antenna or a sensor is a unit which is comprised of two components, namely the aperture and the electronics [25]. For an antenna that operates in the Radio Frequency (RF), it can be assumed that the electronics and the aperture are separate components. For Infrared (IR) sensors, the electronics and the aperture are usually integrated into one part but there could be designs where the aperture signal is collected in a central processing unit.

In UAV applications the communications are mostly achieved through RF antennas in different frequency bands, whereas the IR sensors are used to increase the situational awareness. For the coverage of the RF bands, it is common to use “aperture antenna” type which can be easily mounted on the aircraft surfaces [24]. The considered IR sensors for the present case are the Infrared Search and Track (IRST) and the Missile Approach Warning (MAW) systems. A small contribution to the overall radar signature of the aircraft is anticipated from the RF antennas but not from the IR sensors which operate as passive systems [25].

2.3.2 BASIC PRINCIPLES

An important parameter that defines the field performance of an antenna or a sensor is the radiation pattern. According to Balanis [24], “the radiation pattern is defined as a mathematical representation of the radiation properties as a function of the space coordinates”. Another fundamental parameter is the directivity which indicates the antenna or sensor capabilities over a certain direction [39]. Balanis [24] defines directivity as “the ratio of the radiation intensity in a given direction to the radiation intensity averaged over all directions”. A proper set of coordinates is necessary for the representation of the antenna and sensor properties. The spherical coordinate system (see Figure 2.4) is the most popular, since the interest is usually on a certain direction and at a certain distance from the source [39].

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Figure 2.4 The spherical coordinate system which is used in the analysis of antennas, adapted from [24]

A simple methodology which is used for the modeling of rectangular apertures is elaborated below. The variables that characterize an aperture are the frequency v as well as the dimensions a and b. First, the wavenumber k is calculated based on formulas (2.13) and (2.14).

𝑣 =

1

𝜆 (2.13)

𝑘 =

2𝜋

𝜆 (2.14)

Secondly, the far zone fields X and Y are defined as a functions of the azimuth and elevation angles φ and θ as shown in formulas (2.15) and (2.16).

𝑋 =

𝑘𝑎

2sin 𝜃 cos 𝜑 (2.15)

𝑌 =

𝑘𝑏

2sin 𝜃 cos 𝜑 (2.16)

Thirdly, the far-zone electric fields Eθ and Eφ are calculated by formulas (2.17) and (2.18) where C is the maximum amplitude.

𝐸𝜃 = 𝐶 sin 𝜑

sin 𝑋

𝑋

sin 𝑌

𝑌 (2.17)

𝐸𝜑 = 𝐶 cos 𝜃 cos 𝜑

sin 𝑋

𝑋

sin 𝑌

𝑌 (2.18)

The radiation intensity U is then given by formula (2.19) where η is the intrinsic impedance of the medium.

𝑈(𝜃, 𝜑) =

1

2𝜂[(𝐸𝜃)2+(𝐸𝜑)

2] (2.19)

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Finally, the directive gain Dg can be determined by formula (2.20) where Prad is the total radiated power in all directions.

𝐷𝑔 =

4𝜋𝑈

𝑃𝑟𝑎𝑑 (2.20)

The characteristics of the IR sensors are computed in a similar way as shown above with the only difference being the definition of the far-zone electric fields. Those can be modelled according to formulas (2.21) and (2.22).

1

1 𝐸𝜃 = 𝐶 cos 𝜃 (2.21)

1

1 𝐸𝜑 = 𝐶 sin 𝜃 (2.22)

2.3.3 COORDINATE TRANSFORMATIONS

The formulas of section 2.3.2 calculate the radiation pattern of each aperture on a set of local spherical coordinates. In order to enable a holistic quantification of the coverage, the directive gain of each aperture must be expressed on a global spherical coordinate system. A simple graph of the transformation methodology which is also applied in the present model can be seen in Figure 2.5.

Figure 2.5 The coordinate transformation for the directive gain of a typical rectangular aperture

2.3.4 CONSIDERATIONS FOR MDO

The coverage and the radar signature of antennas and sensors are important characteristics which can drive the design of UAVs. During the conceptual design phase it is considered acceptable to make several simplifications and assumptions in order to enable a MDO [25]. Thus, the terms “coverage” and the “radar signature” can be substituted by the directive gain, normalized to its maximum amplitude, and multiplied with a suitable amplitude factor.

As far as the power is concerned, one may assume that a sensor requires 1000W of power of which half is used for cooling and the other half is transmitted through the aperture [25]. The aperture is assumed to be a flat-shaped patch that adds little weight (0.3 kg/aperture), while each sensor can be considered to be a box with 1.5x2x3 dm sides and with a mass density of 3 kg/dm3 [25].

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2.4 MULTIDISCIPLINARY FRAMEWORKS

2.4.1 OVERVIEW

The development of an aircraft is a complex process that involves many engineering disciplines with complicated inter-dependencies which means that a holistic MDO can be a quite challenging task. One common element of all the case-studies that are referenced in this thesis is the integration of a CAD geometry in the MDO framework. The other trend that can be observed is that an aerodynamic analysis is included in the majority of the cases. Finally, it can be seen that depending on the application, several disciplines are often omitted or are expressed by very simple (empirical) formulas.

The importance of the CAD geometry in MDO frameworks regarding aircraft design is analyzed by Amadori [1] who notes that the “geometry in-the-loop” approach can be very computationally expensive but at the same time it is extremely powerful with respect to modelling and analysis. Moreover, a CAD geometry can also facilitate the integration of other high-fidelity tools inside the framework like in the case of FEM and CFD.

Aerodynamics are also instrumental in aircraft studies and this is mainly because the obtained coefficients are used to define the structural loads, the trim requirements, the propulsion, and the mission performance [27]. Therefore, it is clear that an aerodynamics model can affect the overall quality of the design. Hence, it should not come as a surprise that there is a tendency to use higher fidelity tools and that authors strive for higher quality results [19].

The rest of the disciplines which will comprise a MDO framework depend mostly on the mission of the aircraft, the targeted design space, and the available computational resources [27]. In many instances, handbook formulas can be used in order to create a simple model just for the purpose of closing the design space and to provide support to the other models [18]. Another characteristic that is emphasized by many authors is the flexibility of the framework which can enable the seamless continuation of the research [15], [16], [17], [18], [26], [27], [29], [31]. In simple terms, a flexible framework should be able to adapt to a future refinement of the models by tools of different fidelity and allow the addition of new models.

2.4.2 OPTIMIZATION STRATEGIES

One of the possible categorizations that can be found in some of relevant literature samples regarding the optimization methods is the Single Level (SL) and Multi Level (ML) strategies which are mainly based on the number of the involved optimization processes [1], [2]. A SL strategy has a single optimizer, while in a ML strategy the optimization process is distributed in many segments of the framework which according to Tarkian [2] can be a more suitable approach for complex engineering products.

A ML optimization strategy is elaborated in the study of Tianyuan [26] regarding the optimization of UCAV. Initially, an optimization is performed on the aerodynamics module in order to improve the aerodynamic coefficients based on local geometry variables and stealth constraints. Then, the results are sent to the structural module so that the weight can be minimized against several strength constraints. Finally, the obtained aerodynamic and structural data are send to the global optimizer which evaluates the performance of the aircraft given several mission objectives. According to the authors, the ML strategy can take advantage of different algorithms for different

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processes and it is easier to tackle with because of the considerably fewer global parameters. However, they note that the multiple optimization processes require an increased number of evaluations which can be an additional computational expense.

A typical SL strategy is applied in the case-study of Hitzel et al. [19] for the optimization of UAV configurations. First, a geometry layout is defined in a CAD model which is then analyzed by an aerodynamics model. The obtained results are then used for structural, stability, and propulsion simulations which finally give the necessary data for performance evaluations.

Lundström et al. [15] as well as Amadori et al. [29] also used a SL strategy but the optimization was performed in two successive phases. In the initial phase the calculations were based on simplified equations which allowed a fast evaluation of the optimum propulsion system. After having narrowed down the possible propulsion options and design variables, the second phase was initiated but this time with higher fidelity tools.

2.4.3 OPTIMIZATION ALGORITHMS

Optimization algorithms are an indispensable part of a MDO framework because they automate the iterative optimization process and hence, they enable the system to search for an optimal solution inside the design space [40]. According to Tarkian [2], the algorithms that are used in numerical methods can be classified into two main groups which are characterized as gradient and non-gradient. Safavi [40] explains that “gradient-based methods are used when the gradient of the function is easily accessible and calculable”, whereas “non-gradient methods are common in non-differentiable, discrete, non-smooth and non-linear engineering problems”.

An evaluation of both gradient and non-gradient algorithms can be found in the UCAV case-study of Amadori et al. [18]. First, they tested the gradient-based algorithms Fmincon which was found to be very fast but unsuccessful for their application because of its inability to escape the local optimum. As a second attempt, they used the non-gradient algorithm Complex together with “forgetting” and “randomization” functions and got significantly better results but the algorithm did not always reach the absolute optimum. Finally, they tested a Genetic Algorithm (GA) and identified that it requires a longer computing time, but it has an improved hit-rate and higher probability of finding the true optimum solution which makes it far more suitable for complex applications.

The effectiveness of GA in optimization of complex products is also expressed in the case-study of MAV by Lundström et al. [15] as well as Amadori et al. [29]. In this case, the Multi-Objective Genetic Algorithm (MOGA-II) was selected because of its capability to process multiple objectives and because it was provided by modeFRONTIER which the authors used to build their framework. Overall, the authors concluded that the algorithm was successful in identifying the “pareto front” of dominant designs but they also noted that a high computing time was required. Several tests were also performed in order to analyze how the algorithm operates in respect to the different weighted objectives and constraints. Although a set of optimum control parameters was identified, it was proven that a correct problem formulation was far more important for guiding the algorithm to the region of interest.

2.4.4 EFFICIENT COMPUTING

The overall computational time of a MDO process depends on the number of designs which will be evaluated and the required analysis time per design. High-fidelity models

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can give a better design quality but will also require several hours for a single simulation which might render many MDO problems impractical [2].

The concept of metamodels or surrogate models or Response Surface Models (RSM) is to develop a mathematical approximation which will replace a computationally expensive model [26]. Clearly, a metamodel cannot always be a perfect approximation of the system and therefore, it should always be validated in order to ensure that it is sufficiently accurate for the intended application [26], [40]. According to Tianyuan et al. [26] a simple process of developing a metamodel begins with the setup of a set of cleverly chosen sample designs (input), then the system is simulated in order to get the response (output), and finally the input and the output data are combined by a suitable approximation method.

Since metamodels are only mathematical approximations it is possible that their output might differ from the real model. In order to measure the deviation between the real model and the metamodel, the Root Mean Square Error (RMSE) approach can be used [40]. If each of the estimated variables from the metamodels is Xmeta,i and the respective output from the real model is Xreal,i, then for n designs the RMSE is defined as seen in formula (2.23). For applications where different units have to be compared, then the normalized form of the RMSE can be used as shown in formula (2.24).

𝑅𝑀𝑆𝐸 = √∑ (𝑋𝑟𝑒𝑎𝑙,𝑖 − 𝑋𝑚𝑒𝑡𝑎,𝑖)2𝑛

𝑖=1

𝑛 (2.23)

𝑁𝑅𝑀𝑆𝐸 =

𝑅𝑀𝑆𝐸

𝑋𝑟𝑒𝑎𝑙̅̅ ̅̅ ̅̅ ̅

(2.24)

Tarkian [2] notes that sampling can be a very complicated task especially if the inputs depend on the outputs which are generated by the other models. According to the same author, a common solution to this problem is to use the Direct Sampling (DS) and Indirect Sampling (IDS) methods which can be seen in Figure 2.6. The author concludes that for specific tasks the IDS can be a more effective method as it provides a better mapping of the design space but on the downside, it requires further post-processing in order to remove samples that are too close to each other.

Figure 2.6 The DS method (left) and the IDS method (right), adapted from Tarkian [2]

Metamodels are often encountered in MDO frameworks of complex engineering products like aircraft as an alternative to computationally expensive models such as CFD or CAD. The work of Tianyuan et al. [26] as well as Choi et al. [16] illustrated that reasonable results can be obtained if metamodels are implemented in the MDO process. In the former case [26], the authors decided to use the “Kriging” model to replace the computationally expensive models which were used for the RCS and CFD calculations.

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The obtained results from the real- and meta-models were evaluated and the output in both cases was found to be in good agreement. A more thorough investigation and implementation of metamodels is presented in the case-study of Safavi et al. [35]. In this case, the authors used modeFRONTIER to create metamodels of all the simulation models of an aircraft MDO framework by using the “Anisotropic Kriging” algorithm and the “Uniform Latin Hypercube (ULH)” sampling method. The deviation from the real models was found to be very low and the authors reported that metamodels can be an efficient and viable alternative for MDO applications of complex nature.

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21

CONTRIBUTIONS

Part 3 of the thesis includes a detailed analysis of the main framework as well as a description of the module integration and development. Furthermore, the individual model specifications are elaborated and the miscellaneous evaluation results are presented.

3.1 FRAMEWORK EVALUATION

3.1.1 OVERVIEW

The proposed multidisciplinary framework is aimed towards the development of a UAV aircraft by considering several engineering disciplines such as geometry, aerodynamics, antenna placement, radar signature and mission performance. In order to effectively integrate the aforementioned disciplines, the framework has been further divided into different modules which utilize specific computational tools and execute predefined calculation routines.

The overall development approach is to use low- to medium-fidelity tools at the initial phase of the optimization process so that some of the calculations can be performed relatively fast at a local level and therefore, reduce the number of design variables that are included in the global optimization scheme. The aforementioned local processes are in principle iterative and they are able to define several of the design characteristics which are needed for later calculations in the framework. Hence, it can be said that this approach aims at reducing the design space at a minimum computational cost as it was also shown in the work of Amadori [29] and Lundström [15]. High-fidelity tools are also included in the basic framework structure but they are engaged at a later phase of the optimization process so that a more accurate result can be obtained. A graphical illustration of the inter-disciplinary relations between the modules can be seen in Figure 3.1 and it is analyzed in detail in section 3.2.

The integration methodology which was applied in the present case-study was to prepare the framework in successive steps by developing one module at a time. The division of the framework into modules was further deemed necessary for the following reasons:

The different tools which are required for a specific operation are grouped together in a local integration platform and hence, the complexity of the main framework is reduced.

A future refinement within the modules can be done in an efficient way since this will only induce limited local changes instead of the complex and sometimes radical global changes.

It is possible to optimize each module locally depending on the chosen optimization strategy or the available computational resources.

The modules can be simulated individually in order to create metamodels and they also provide a platform for simple metamodel implementation.

Every module can be tested alone without the need of running time consuming iterations of the whole framework.

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3.1.2 PROBLEM FORMULATION

The performance of the aircraft for a given mission is closely related to the weight, the aerodynamic, the propulsion as well as the geometric characteristics of the design and therefore, it constitutes a useful objective for a MDO (see section 2.1.5). Among the many outputs of the mission simulation model, a simple but yet indicative performance variable that can be used as an objective is the weight of the fuel that is required to fly the mission. A further design consideration, especially for military applications, is the radar signature of the aircraft (see section 2.1.4). In the present case-study, the obtained maximum RCS area as well as the RCS area on a hypothetical radar direction are used as design constraints (more details on section 3.3.5). Since there were no available data to base this design constraint, it was assumed that the limit was the value of RCS in the baseline configuration increased by 10 percent.

As far as antennas are concerned, it is important to be able to quantify and optimize the placement of the apertures on the aircraft surfaces so that the coverage over some specified angular sectors can be maximized. At the same time, it is equally important to keep the apertures close to the sensor units which is highly beneficial in terms of performance, cost, and reduced manufacturing complexity.

The longitudinal stability characteristics of the aircraft are defined by the SM and hence, it is desirable to strive towards an optimum value (see section 2.1.2). Furthermore, a constant value of the SM throughout the different designs will ensure that each configuration is equally stable.

In the present application the optimization process has been divided into a global, a local (nested) optimization scheme, and an iterative process (similar to a local optimization) which will be further elaborated in sections 3.2.1 and 3.2.3. The formulation of the global optimization problem is given in Table 3.1 (left), while the local “antenna” optimization problem in Table 3.1 (right). The aforementioned iterative process is with respect to aircraft sizing and the goal is to optimize the design against a predefined target value of SM=0.4 which will enable good handling characteristics.

Optimization problem formulation obj. = min (Wfuel)

s.t. RCS<RCSlim obj. = max (DG-XA-S)

Table 3.1 The global (left), the local “antenna” (centre) and the local “sizing” optimization problems

3.1.3 DESIGN VARIABLES

The global and local design variables which have been included in the evaluation scheme of the framework can be seen in Table 3.2. The chosen sample is comprised of representative variables that are usually considered in aircraft design. It should be noted that the models and the framework structure can support a much larger design space but such complexity was not considered relevant at the present stage of the development.

Two examples of available parametrization that has been omitted are the fuselage shape as well as the wing and tail partitions. In the case of the former, it is possible to make changes in width and height as well as in the shape of each cross-section but it was decided to use only the total fuselage length. In the case of the latter, it is possible to select a different airfoil type and thickness in each partition but a simplification was made where a single airfoil type with constant thickness applies to all partitions.

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Global Design Variables

Total fuselage length (L)

Wing surface area (SW)

Wing aspect ratio (ARW)

Tail taper ratio (TRW)

Wing sweep (ΛW)

Wing Dihedral (Y)

Wing airfoil (AIRW) x14

Wing airfoil thickness (TW)

Tail horizontal vol. coef. (VH)

Tail vertical vol. coef. (VV)

Tail aspect ratio (ART)

Tail taper ratio (TRT)

Tail sweep (ΛT)

Tail airfoil (AIRT) x14

Tail airfoil thickness (TT)

Local Sizing Variables

Wing relative position (XW) Tail relative position (XT)

Local Antenna Performance Variables

Aperture relative position (Xi) x5

Aperture dimensions (dxi, dyi) x5

Aperture frequency (fi) x5

Table 3.2 The global and local design variables which were used in the evaluation of the framework

3.1.4 STRATEGY DEFINITION

The present case study was modelled by using a multi-level optimization strategy which is comprised of a global as well as two local optimization processes. In this way it is possible to reduce the number of the global design variables and therefore, enable the optimizer to converge much faster as shown in the work of Tianyuan [26]. Furthermore, the local optimization modules are comprised of low-fidelity tools which means that the local iterations are performed relatively fast and at a much lower computational cost.

A simple graphical illustration of the connectivity between the modules and the important data generation of each model can be seen in Figure 3.1. The “sizing loop” and the “antenna loop” nodes symbolize the local optimization processes. The CFD model node is marked with a star because it is possible to be omitted, if TORNADO (which is included in the “trim module”) is selected as the primary aerodynamics tool.

Figure 3.1 The basic inter-disciplinary relations of the proposed framework. The CFD model is marked with a star because it can be omitted if TORNADO is selected to be used as the primary aerodynamics tool

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As far as the RCS model is concerned, it should be noted that due to time limitations and availability issues, the RCS calculations have been performed by the mission simulation tool DIBA. In principle, the operation of DIBA regarding the RCS computations is the same to that of the GRECO method in respect to the integration and the inter-disciplinary relations, with the only difference being that it offers calculations of a lower fidelity. Therefore, it can be safely assumed that the present approach is a good alternative that can fill the design space until a more suitable tool becomes available.

3.2 MODULE DESCRIPTION

3.2.1 SIZING LOOP

The objective of the “sizing loop” is to optimize the position of the wing and the tail so that a predefined value of SM can be achieved. It is comprised of low fidelity tools and the aim is to reduce the design space by adding a very small penalty in terms of computational cost. The inter-disciplinary relations between the different tools of the module can be seen in Figure 3.2, while a brief description of its operation is given below:

The input to the module is the global and the local (sizing) design variables (see section 3.1.3).

Several geometric manipulations are performed in order to convert the input data to a compatible form.

The mission simulation tool DIBA is used in a low-fidelity version (level 1) in order to predict the weight and the CG position.

Two support MATLAB codes calculate the NP and the SM and the results are evaluated by the local optimization algorithm.

The output of the module is the converged values of MTOW, the CG position, and the relative placement of the wing and tail on the fuselage.

Figure 3.2 The inter-disciplinary relations and operation of the “sizing loop”

3.2.2 TRIM

The “trim module” uses the medium-fidelity aerodynamics tool TORNADO and a designated support MATLAB code in order to identify the correct trim angles for the wing and tail as well as the trimmed lift and drag coefficients. The inter-disciplinary relations of the “trim module” can be seen in Figure 3.3, whereas its operation is described below:

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The input to the module is the position of the wing and tail, the prediction of the MTOW, the position of the CG and the global design variables (see section 3.1.3).

Several geometric manipulations are performed so that the input data can be converted to a compatible form.

The airfoil data are converted from control points into a cloud of coordinates and they are then loaded to TORNADO (see section 3.3.2).

A description of the cruise flight conditions is given as reference for the simulations.

The wing is tested at three different AoA with a constant tail trim angle and then the tail is tested at the same AoA with the wing at zero incidence.

A simple trim model that uses basic equilibrium equations (see section 2.1.2) is used to calculate the lift and drag curves.

The output of the module is the incidence angle of the wing and the trim angle of the tail for steady and level cruise flight.

Furthermore, the trimmed lift and drag coefficients are computed and can be used as the primary aerodynamic data if there is no other high-fidelity aerodynamic analysis (CFD) included in the framework.

Figure 3.3 The inter-disciplinary relations and operation of the “trim module”

3.2.3 ANTENNA LOOP

The “antenna loop” is a local optimization process that uses the CAD geometry and the antenna analysis model in order to evaluate the coverage of the apertures and their distance from the sensor units. The inter-disciplinary relations within the module are presented in Figure 3.4 and the operation is as follows:

The inputs of the module are the CAD geometry and the local (antenna) design variables (see section 3.1.3).

Every aperture and sensor is placed by using relative measures on a realistic position on the fuselage.

The global Cartesian coordinates of each component and the rotation of the local axis system based on the surface curvature are send to the antenna model.

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A calculation routine is run in the antenna model (see section 3.3.4) and the coverage as well as the aperture-sensor distance is computed.

After the optimizer has converged, the weight and the power output are sent to the mission simulation model, while the antenna RCS contribution is added to the total RCS value.

Figure 3.4 The inter-disciplinary relations and operation of the “antenna loop”

3.3 MODEL SPECIFICATIONS

3.3.1 AIRCRAFT GEOMETRY

The CAD geometric model was developed in Dassault Systemes CATIA V5R21. The software allows the user to create designs of high fidelity and implement various types of knowledge-based features. In this way, it is possible to have a model with DA characteristics which in turn will enable an effective integration in the MDO framework.

In general, the baseline configuration of the aircraft is based on a combination of characteristics that can be seen in similar UAV applications (see section 1.2.2). Overall, the CAD model is comprised of three main parts which are namely the fuselage, the wing and the tail (see Figure 3.5). In addition to this, the model includes a set of support geometry features that are intended to be used as reference for the sensor and aperture placement (see section 3.3.4).

As far as the possible transformations of the model are concerned, it should be noted that the geometry is able to effectively map the expected design space of the present case-study. However, it is important to mention that the model is oriented towards a particular configuration which means that there is only a finite number of possible topological and morphological changes. Nevertheless, the model is highly robust within the predefined design space and in fact, it has been identified that it is error-free for all the tested (realistic) configurations. A summary of the most important characteristics of the model is summarized below:

Fuselage: Distinct side edge and hexagonal-looking cross-sections for low RCS signature, smooth surface definition through cubic Bezier curves, uniform elongation in length and width by using relative measures for parametrization.

Wing: Flexible planform based on the desired design variables, airfoil definition based on [20] with thickness manipulation, division of the wing

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in four partitions, streamline fairing in the fuselage junction for improved aerodynamic and RCS performance (see Figure 3.6), not possible to generate a crescent wing type.

Tail: V-shape configuration, flexible planform based on the desired design variables, airfoil definition based on [20] with thickness manipulation.

Engine nacelle: Optional feature that is controlled by a rule, flexible surfaces made with Bezier curves, two types of intake shapes, one blended intake for faster aerodynamic calculations, one intake with a “lip” and a plate cover for accurate RCS calculations (see Figure 3.6).

Sensors: Only possible placement is the area in the front part of the fuselage, positioning is defined by relative measures (see Figure 3.7).

Apertures: All the aircraft surfaces can be used for placement, positioning on different parts is controlled by a rule and by relative measures (see Figure 3.7).

Figure 3.5 Isometric, top, front and side views of the aircraft geometry

Input

From the global optimizer

Design variables (see section 3.1.3)

From the sizing module

Wing and tail relative position (Xw, XT)

From the trim module

Wing incidence and tail trim angle (iW, iT)

From the local antenna optimizer

Aperture position

in relative measures

Sensor position

in relative measures

Output

Antenna model Sensor

position (X, Y, Z)

Aperture

position

(x, y, z)

Aperture orientation

(Xx, Xy, Xz, Yz, Yy, Yz, Zx,

Zy, Zz)

Available volume

for placement

CFD model *.CATProduct file

Mission simulation *.igs file

Table 3.3 The main input and output of the CAD geometry model

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The main input to the model is in respect to the overall geometry layout and it is controlled by the global optimization process and the design variables. Additionally, the model receives an input from the sensor and aperture model regarding the local (antenna) optimization process. The output of the model can be either individual parameters or CAD geometry files and it is used by most of the tools that are included in the framework (see Table 3.3).

Figure 3.6 Geometry parametrization showing the variant with covered engine intake and no wing fairing

(left) and the variant with realistic engine intake and wing fairing (right)

Figure 3.7 Example of aperture and sensor placement on the CAD geometry with visualization of the axis

system rotation and translation

3.3.2 AERODYNAMIC PERFORMANCE BY USING VLM

The medium-fidelity aerodynamic analysis model was developed by using TORNADO which is a VLM implemented in MATLAB (see section 2.2.1). The model is based on a “batch” file of the code which was properly modified so that it can fit the requirements of a DA framework. The main modifications were in respect to the input and output parameters which had to be converted in a form that is compatible with the rest of the models. Further changes include some “bug” fixes, some alterations in the functions and an implementation of various methods for importing airfoil data. The basic input of TORNADO is the aircraft geometry and the flight state, while the main output is the aerodynamic coefficients (see Table 3.4).

Some further comments regarding the integration and operation of TORNADO are given below:

TORNADO has an inbuilt geometry function (see Figure 3.8) which means that it can be directly connected to the global optimizer instead of having to interact with the CAD model. This approach can increase the efficiency

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of the framework due to the significant reduction of inter-disciplinary dependencies.

The flight state is a predefined and constant user-input which is based on the cruise conditions of the selected mission.

An independence study was performed on the number of panels that are used in the solution. The lowest number of elements that does not significantly affect the accuracy of the solution was chosen in order to increase the efficiency of the computations (see Figure 3.8).

Input

From the global optimizer

Design variables (see section 3.1.3)

From the sizing loop

Wing and tail relative position

Flight state Altitude and speed (H, V) AoA and tail trim angle

Output

Trim function Wing and tail lift

coefficients (CLW, CLT) Wing and tail lift induced

drag coefficient (CDiW, CDiT)

Mission simulation *(if no CFD)

Total lift coefficient (CL)

Total lift induced drag

coefficient (CDi)

Total parasitic drag

coefficient (CD0)

Table 3.4 The main input and output of the TORNADO model

Figure 3.8 Result plots from TORNADO showing the geometry (left) and the lattice panels (right)

The import of the airfoil data in the TORNADO poses an additional challenge that requires several modifications. The problem is that the airfoil definition which is included in the optimization is based on control points (see section 2.1.3), whereas TORNADO can only handle a cloud of point coordinates. Several solutions were investigated and it was found that the best approach is to use a support MATLAB file which can convert the control points and export a file with coordinate data (see Figure 3.9). In this way, it was possible to map all the changes that the optimizer made on the airfoil control points without having to include the computationally expensive CAD geometry in the loop.

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Figure 3.9 The methodology which is used for importing the airfoil data in TORNADO

3.3.3 AERODYNAMIC PERFORMANCE BY USING CFD

The high-fidelity aerodynamics model of the framework was developed by using the commercial CFD solvers ANSYS Fluent and CFX which are capable of enabling DA features and can be integrated in a relatively simple way in modeFRONTIER. Both solvers were able to capture the flow phenomena with the same accuracy but at a very high computational cost since an average simulation required around two hours. As a general observation it can be said that Fluent proved to be better in terms of the available problem setup possibilities, whereas CFX was superior in terms of DA features and simplicity.

The operation of the CFD model within the framework depends mostly on the CAD geometry. The main output is the lift and drag forces on the aircraft’s surfaces which are send to a support MATLAB program in order for the aerodynamic coefficients to be calculated. A summary of the input and output to the model is given in Table 3.5.

Input

From the CAD geometry

*.CATProduct file

Flight state Density, viscosity, temperature

and speed (ρ, μ, T, V)

Output

Mission simulation Total lift coefficient

(CL)

Total induced drag

coefficient (CDi)

Total parasitic drag

coefficient (CD0)

Table 3.5 The main input and output of the CFD model

Figure 3.10 The domain shape and size with an overview of the mesh density (left) and a detailed close-up of

the mesh near the engine intake where the Boundary Layer (BL) can be seen (right)

An important aspect of every CFD simulation is the correct setup of the problem in terms of domain, mesh, boundary conditions and solver controls. Although the in depth analysis of CFD procedures is not in the scope of the thesis, a few details regarding the simulations are given below:

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The domain was created sufficiently large to limit the mass imbalances (see Figure 3.10).

A high-quality mesh of 15 million elements was generated so that the code can be utilized in its full potential. Surface inflation and edge sizing was used in the critical regions (see Figure 3.10).

The simulation was steady state with constant velocity and the shear stress transport turbulence model of 5 % intensity was selected.

The solution was controlled by the residuals of the flow equations and it was considered converged at 10-3 in order to speed up the procedure.

3.3.4 ANTENNA ANALYSIS

The antenna analysis model was developed in MATLAB by using simple electromagnetic principles (see section 2.3.2), basic coordinate transformation techniques (see section 2.3.3), and statistical estimation formulas. The scope was to include a tool which can handle a representative selection of input parameters and can generate reasonable output values regarding the antenna coverage and placement. A notable feature of the code is that the CAD geometry is included in the computations. In this way, it is possible to execute the placement function of sensors and apertures with high accuracy but also to export measurements regarding the absolute position and orientation of each component.

Figure 3.11 An example of the 3D radiation pattern plots for a test case of 15 RF and 4 IR sensors

At the current development state, the model is able to predict the coverage and RCS signature of the apertures in one or more specified directions. Furthermore, it is able to perform several calculations regarding the placement of the sensor units. One example is the calculation of the distance between apertures and sensors, while it is also possible to optimize the sensor placement against a predefined CG position. The input and output connections of the model are given in Table 3.6.

Visualization of the results is also possible through several plot functions as seen in Figure 3.11 and Figure 3.12. In a typical example case, it can be clearly seen that the IR

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sensor number 2 is pointing at φ=180o and covers an arc of ±45o in the azimuth plane, whereas it is oriented at θ=90o and covers an arc of ±45o in the elevation plane. Furthermore, by analyzing Figure 3.12 one can observe that the IR sensor covers a larger angular sector but with less radiation intensity, while the RF sensor covers a narrower section but directs all the power in one direction.

Figure 3.12 An example of polar plots of the directive gain against the elevation and azimuth angles θ and φ

Input

From the local antenna optimizer

Local optimization variables (see section 3.1.3)

From the CAD geometry

Aperture global

Position (x, y, z)

Aperture orientation

(Xx, Xy, Xz, Yz, Yy, Yz, Zx, Zy, Zz)

Sensor global

Position (X, Y, Z)

Output

Local optimizer evaluation

Coverage at specified directions θ, φ (DG)

Distance between apertures and sensors (L)

Mission simulation Sensor weight and power requirements (WS, PS)

RCS model RCS contribution in the radar direction (RCS)

Table 3.6 The main input and output of the antenna model

3.3.5 RADAR SIGNATURE

The RCS model that is included in the present framework was developed by using DIBA. According to the program’s documentation, the RCS is calculated on simple empirical formulas by considering the faceting method of the surfaces.

Figure 3.13 The reference ground radar position (left) and a plot of the radar waves scattering map in

spherical coordinates with the angular sector of the radar being highlighted (right)

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The input for the calculations is the CAD geometry of the aircraft, while the output is the equivalent RCS area of the target in spherical coordinates. From the obtained data, the maximum RCS value and the RCS value in the direction of a hypothetical ground radar are extracted and used as constraints. Figure 3.13 shows a plot of the RCS area map over all spherical directions and the assumed position of the radar for the present mission. The marked position on the RCS plot indicates the RCS area that the hypothetical ground radar detects in the present scenario.

3.3.6 MISSION SIMULATION

The simulation of the aircraft performance characteristics for a given mission is performed by the MATLAB-based toolbox DIBA. According to the developers, the data structure is formulated in a general way and it handle nearly all aircraft. Furthermore, the geometry definition, the aerodynamic performance, and the weight analysis can be executed in two levels of detail depending on the available data and the required output fidelity.

In the present application it was deemed necessary to use DIBA twice within the framework. First, DIBA is used as a low-fidelity tool within the “sizing loop” so that a prediction of the MTOW and the CG position can be obtained. The inputs for his preliminary operation are a coarse geometry definition (see Figure 3.14 left) and some available weight data, while the propulsion and the aerodynamic parameters are predicted by the software. Secondly, DIBA is used as a medium-fidelity tool so that a more refined estimation of the mission parameters can be obtained and subsequently evaluated by the global optimizer. During this second run, the input to the program is more detailed in terms of the geometry definition (see Figure 3.14 right) as well as in terms of aerodynamic and weight data which have been computed by the other models of the framework.

Figure 3.14 The two levels of detail for the geometry definition in DIBA

Input (level 1)

From the global optimizer

Global optimization variables (see section 3.1.3)

From the local optimizer

Wing and tail relative position

Output (level 1)

NP and SM calculation models

MTOW (W) CG position (XCG)

Table 3.7 The main input and output of DIBA within the sizing module at a low-fidelity level of detail

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The input and output to the program for the present framework application is presented in Table 3.7 and Table 3.8, while several performance plots which are relevant to the present mission specifications can be seen in Figure 3.15. It should be noted, that DIBA offers a large selection of output performance calculations but for this preliminary evaluation of the framework it was decided to objectify the required fuel weight (see section 3.1.2) As far as the input to the model is concerned, the user is able to manipulate a large number of design parameters. Some of them have been linked to the global design variables, some to the other modules of the framework and finally some have been set to a predefined constant value based on an estimation. An example of the latter is the weight of some components as well as the propulsion specifications.

Input (level 2)

From the global optimizer

Global optimization variables (see section 3.1.3) except the fuselage data

From the CAD geometry

*.igs file

From the antenna model

Aperture RCS in the radar direction (RCS), Sensor weight and power

requirements (WS, PS)

From aerodynamics (CFD or TORNADO)

Total lift coefficient (CL)

Total lift induced drag

coefficient (CDi)

Total parasitic drag

coefficient (CD0)

Output (level 2)

Global optimizer evaluation

Weight of the required mission fuel (WF)

Global optimizer evaluation

Maximum RCS, RCS in the radar direction

Table 3.8 The main input and output of DIBA within the mission simulation module at a medium-fidelity level

Figure 3.15 Example of the results from the analysis of the baseline design configuration in DIBA

3.4 EVALUATION AND OPTIMIZATION

3.4.1 DESIGN OF EXPERIMENTS

A baseline configuration was selected as reference for the optimization process but also as an input for the first-level evaluation of the modules and models that are included in the framework. The baseline design specifications are summarized in Table 3.9 and a good visual example can be seen in Figure 3.5 and Figure 3.7. Due to space limitations the airfoil input details as well as the output design characteristics of the baseline configuration are not given here and the interested reader can refer to Appendix A.

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The second-level evaluation of the framework was performed by using the Design Of Experiments (DOE) methodology with a sequence of 30 random design samples. The range of both the global and local input variables was selected to be within a realistic limit which corresponds to the expected design space of the product for the present application. The interested reader can find the details of the input range in Appendix A, while the obtained output results of the DOEs are given in Appendix B.

Global Baseline Input

Fuselage Length

11 [m] Wing Area

15 [m2] Wing

Aspect Ratio 15 [-]

Wing Taper Ratio

0.5 [-] Wing

Dihedral 2 [deg]

Wing Sweep

3 [deg]

Wing Airfoil Thickness

100 [%] Wing

Airfoil Type Eppler E374

Tail Hor. Volume Coef.

0.55 [-]

Tail Ver. Volume Coef.

0.04 [-] Tail

Aspect Ratio 6 [-]

Tail Taper Ratio

0.8 [-]

Tail Sweep

8 [deg] Tail Airfoil Thickness

100 [%] Tail

Airfoil Type NACA 0010

Local Baseline Input

Wing Relative Position

0.5 [-] Tail Relative

Position 0.9 [-]

Aperture Relative X Position (i=1:5)

(0.01 0.3 0.9 0.01 0.5) Aperture Relative Y

Position (i=1:5) (0.5 0.5 0 0 0)

Aperture Frequency (x5)

11 [GHz] Aperture Opening

(x5) 5x5 [cm]

Table 3.9 The specifications of the baseline configuration

A relatively high failure rate (23 %) was observed during the evaluation of the DOEs and the investigation of the cause showed that it was related to the mission simulation model. By analyzing the failed cases (see Appendix B), it can be seen that 28 % of the times DIBA gave an error during its preliminary use in the “sizing loop”, while the rest of the times the error occurred at the final mission simulation module. Furthermore, it was identified that the reason DIBA failed during the “sizing loop” was that the code could not converge to a final MTOW value. The aforementioned problem poses a big performance challenge as this can stall the whole optimization process if the design is not discarded manually.

3.4.2 PERFORMANCE

A quantification of the framework’s performance was carried out in parallel with the main evaluation. The hardware configuration for this test consisted of a Fujitsu Celsius R940 workstation with 8 physical Intel Xeon® cores at 3.00 GHz and 126 GB of RAM. The observations regarding the simulation time of each model and for different types of schedulers are presented in Table 3.10.

Framework Performance

Module Sizing Trim Antenna CFD RCS and Mission* Time/Iteration <10 sec <30 sec <10 sec ~120 min <30 sec

Table 3.10 Breakdown of the module performance (*the RCS calculations are performed by DIBA)

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It should be noted that the “sizing loop” as well as the “antenna loop” require several evaluations since they comprise a local optimization process. For the “sizing loop” the SIMPLEX algorithm can provide a more than adequate solution with less than 30 iterations which is equivalent to a total time of around 3 min (see section 3.4.4). For the “antenna loop” the MOGA-II algorithm can easily tackle with the problem, but for an initial population of 50 individuals that run for 50 generations the loop required 2500 iterations which took roughly 7 hours (see section 3.4.4).

3.4.3 EFFICIENT COMPUTING

In order to enable a preliminary optimization of the product at a reasonable time span, it was considered relevant to implement metamodels as an alternative to the computationally expensive sub-systems of the framework. The sub-systems of the framework where the application of metamodels was deemed necessary are the “antenna loop” as well as the CFD model. The main reason was that in the former, the optimizer requires thousands of evaluations, whereas in the latter the computational time per iteration is considerably high.

In all metamodeling cases, the Direct Sampling (DS) strategy (see section 2.4.4) was used due to the fact that at this evaluation stage it is preferable to strive for efficient results rather than increased accuracy. Furthermore, a brief investigation of the accuracy of each metamodeling approach was carried out by testing different DOE methods against various metamodeling algorithms. The DOE sequence generation methods which were considered are the Uniform Latin Hypercube (ULH) and SOBOL, while the algorithms which were evaluated are the Anisotropic Kriging (AKR), the Kriging (KR) and the Polynomial SVD. For the purpose of evaluating the metamodels, random sequence DOE samples were generated and simulated in both the real- and the meta-model and the NRMSE (see section 2.4.4) was calculated. The whole set of the obtained results is given in Appendix C while Table 3.11 and Table 3.12 present the selected configuration which was used in the preliminary optimization.

Antenna Metamodel

Samples DOE Algorithm Evaluation Output NRMSE [%]

300 ULH AKR 50 Random

Coverage 0.43

Distance 0.02

RCS 0.87

Table 3.11 The obtained NRMSE results from the evaluation of the “antenna loop” metamodel

CFD Metamodel

Samples DOE Algorithm Evaluation Output NRMSE [%]

90 ULH AKR 15 Random

CLtotal 0.05

CDitotal 0.07

CD0total 0.03

Table 3.12 The obtained NRMSE results from the evaluation of the CFD metamodel

3.4.4 OPTIMIZATION

The first local optimization process is the “sizing loop” where the system must evaluate different positions of the wing and tail in order to reach the predefined target value of SM (see sections 3.1.2 and 3.2.1). Overall, it can be said that this loop does not

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constitute a challenge since it is a single-objective optimization and the involved models are simple linear functions. Based on the above, the SIMPLEX algorithm was selected to schedule the optimization routine and the evaluation showed that it was able to give an accurate and the same time fast solution. Figure 3.16 shows the progress of the algorithm in the case of the baseline configuration, where SIMPLEX was able to reach the target value in 29 iterations and with a convergence value of 10-4.

Figure 3.16 The optimization of the “sizing loop” with the SIMPLEX algorithm showing the convergence to the

target value of SM=0.4

The second local optimization process is the “antenna loop” where the goal of the optimizer is to maximize the coverage and minimize the distance of the apertures from the sensors. The multi-objective nature of the problem as well as the dependency of the module on the complex CAD geometry require a more advanced algorithm that can look into all the areas of the design space. Based on the above, the genetic algorithm MOGA-II was considered to be a suitable solution to the problem which is also supported by the findings of both Amadori [29] and Lundström [15].

Like all genetic algorithms, MOGA-II depends on the initial population, the number of generations as well as the cross-over and mutation characteristics. An evaluation test was carried out to investigate the performance of the algorithm under different population sizes and varying number of generations, while the mutation and cross-over probabilities were kept constant and equal to 0.1 and 0.5 respectively (see Figure 3.17). It can be seen that the algorithm was not able to locate the correct pareto front when a small number of generations and population was used. Increasing the number of generations and population size improved the performance of the algorithm but clearly some regions of high coverage were not investigated. Finally, for a very high number of generations and population size the algorithm seemed to be able to explore all the design space including regions of high coverage.

Figure 3.17 Evaluation of the algorithm MOGA-II for the optimization of the “antenna loop”

By evaluating the findings of the local optimization processes and combining them with the identified performance characteristics of the framework (see section 3.4.2), it

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becomes clear that the global optimization under the current settings is not viable given the available time and computational resources. In order to increase the efficiency of the framework and enable a global optimization, it was considered to use metamodels for the computationally expensive sub-systems of the framework. As already described in section 3.4.3, the NRMSE of the metamodels in the present case is very low which indicates that they can be an accurate substitute of the main modules.

The modified framework with metamodels which was used for enabling the optimization of the product can be seen in Figure 3.18. The range of the input variables was chosen to be representative of the application and the interested reader can refer to Appendix B for details. The objectives and the constraints of both the local and global processes have not changed and a good description is given in section 3.1.2.

Figure 3.18 The modified framework used in the preliminary optimization

The first optimization evaluation of the modified framework was performed by using the SIMPLEX algorithm and the progress of its convergence against the evolution of the designs can be seen in Figure 3.19. In total, it can be said that the algorithm was allowed to run for 310 iterations but required only 200 iterations before the objective could reach a value that was very close to the final one. After this stage, only small improvements could be noticed and finally the algorithm entered into a fluctuation state with no significant change over time around the 300th iteration which was an indication that the process could be interrupted. The obtained results from this optimization are presented in Table 3.13 and it can be clearly seen that the objective of minimizing the fuel weight was undoubtedly improved (25.7-%) compared to the baseline configuration. Similarly, the examination of the obtained pareto front of the local antenna optimization process, showed that there was also an improvement to the antenna coverage and placement problem (16.4 % and 48.2 % respectively). From the obtained pareto front, the design with the best coverage has been selected and presented in Table 3.13.

Figure 3.19 The convergence progress of the required fuel design objective for the optimization of the framework with metamodels and by using the SIMPLEX algorithm

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The second optimization evaluation of the framework was performed with MOGA-II and the progress of the fuel minimization objective against the evolution of the designs can be seen in Figure 3.20. The settings which were used include the generation of an initial population of 50 individuals by using the ULH sequence that evolved for 50 generations, while the probabilities of mutation and directional cross-over were 0.1 and 0.5 respectively. On the upside, it can be said that MOGA-II was very successful and managed to improve the optimization objective by 36.1 % which is a far better result than that which was obtained from SIMPLEX, but on the downside the computational time was four times higher. Overall, the algorithm run for 2000 iterations but after the 1500th iteration no significant improvement could be observed, while after the 1800th iteration there was absolutely no change (see Figure 3.20). Furthermore, the local optimization process of antennas was also successful and with similar results to those that were obtained in the SIMPLEX optimization since it depends on the local scheduler which is in both cases MOGA-II. From the obtained pareto front, the antenna design with the maximum coverage has been selected and presented in Table 3.13 together with the rest of the global optimization results. A list of the design specifications and the CAD geometry of the optimum configuration can be found in Appendix D.

Figure 3.20 The convergence progress of the required fuel design objective for the optimization of the

framework with metamodels and by using the MOGA-II algorithm

SIMPLEX MOGA-II

Results Improvement Results Improvement

Maximum coverage 5.97 [-] 3.5 % 5.85 [-] 0.7 %

Minimum Distance 11.41 [m] 40.2 % 11.85 [-] 37.8 %

RCS max 14.89 [m^2] 2.1 % 15.53 [m^2] -1.4 %

RCS radar direction 10.15 [m^2] -6.3 % 7.74 [m^2] 18.9 %

Minimum Fuel Weight 1763 [kg] 25.7 % 1517 [kg] 36.1 %

Table 3.13 The obtained results from the optimization of the framework with metamodels by using the SIMPLEX and the MOGA-II algorithms

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DISCUSSION AND CONCLUSIONS

Part 4 is the epilogue of the thesis where the development of the framework and the obtained results are commented and discussed. Moreover, a summary of the research contribution is given and suggestions for future work are offered.

4.1 MODELS

The empirical formulas and the simple geometry which were used in the “sizing loop” managed to minimize the design space and provided a good starting point for all the other models of the framework at a very low computational cost as it was also reported by Lundström [15] and Amadori [29]. Clearly, the accuracy of the obtained results is not high due to the simplicity of the sizing method which is only valid at the very early aircraft design stages. Nevertheless, it provided instrumental data to the rest of the framework such as an estimation of the CG and the MTOW for trim calculations as well as the wing and tail position which in turned defined the CAD geometry.

The integration of the “trim module” proved to be of particular significance as it enabled the estimation of the correct wing incidence and tail trim angles which in turn limited the available design space even more. Hence, it was possible to generate the correct CAD geometry which further increased the fidelity of the calculations in the rest of the modules that were depended on it.

The “antenna loop” is clearly one of the weak features of this framework and a re-evaluation of the optimization strategy might be necessary. The problem is that this local optimization process may have fast iteration times but requires thousands of evaluations until it reaches the correct pareto front when the MOGA-II algorithm is used (see section 3.4.4). On the other hand, if the local objectives are moved to a global level, then the global problem becomes much more complicated and will inevitably require more iterations which as one might imagine, would be very computationally expensive as also reported by Tianyuan [26]. For the time being, the solution of running the optimization locally was selected due its simplicity and as explained in section 3.4.4, one can obtain a first set of results relatively fast if accuracy is not the main consideration. The viable alternative of using metamodels seems to gain a lot of ground as a candidate strategy since the expected error is very small and the required time for a local optimization is trivial. As far as the sensors model is concerned, it should be noted that the empirical formulas have proven to be adequate for the present application as they were able to effectively map the desired design space and illustrate that all the inter-disciplinary relations can be accurately handled.

The CAD model is a very important part of the framework since it defines the outlook of the aircraft’s geometry but it is also used as input to other models such as the aerodynamic performance, the RCS signature, the antenna analysis, and the mission simulation. In a nutshell, it can be said that the main development methodology was to aim for high-robustness, while at the same time an effort was made to achieve as fast update times as possible. In particular, the model exhibits an error-free behavior throughout the predefined design space and at the same time it is compatible with all the sub-systems and models of the framework. As far as efficiency is concerned, it

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should be noted that the model initially illustrated a very poor update performance. It was identified that the complicated geometric relations between the different parts which comprised the total product as well as the advanced curvature features were the main cause. Regarding the former point, the solution was to minimize the internal geometric relations and to merge several parts together when possible. The latter point proved to be more challenging to tackle with because the changes in the surface fidelity eventually reflected on the compatibility with the CFD solver but also generated an aesthetically inferior result. Furthermore, and in respect to parametrization, it was found through the main framework evaluation that a more than adequate level of DA was achieved. The included parameters and rules were able to map the design space with simple morphological transformations, while the use of topological features was only limited to the aperture and sensor placement.

The development of the CFD model was based on a compromise between accuracy and performance. A fine mesh will most probably guarantee good results but it will also increase the total simulation time. In the present case no mesh independence study was carried out and the Boundary Layer (BL) was simply resolved which means that no particular attention was paid to the “law of the wall”. Nevertheless, since the scope of the thesis is MDO, it was consider acceptable to make the aforementioned simplifications because the importance is on the comparison of different designs and not on the final product [18]. As far as the performance of the model is concerned, it should be noted that high computational times were observed which clearly necessitates for the investigation of viable alternatives in terms of other CFD software as also commented by [16], [17], [18], [19]. Furthermore, another approach would be to use metamodels which can yield satisfactory results at almost no computational cost. The implementation of metamodels in the present CFD application showed very small deviation from the real model and proved that they can be an efficient solution as also reported by Choi [16] and Tianyuan [26]. Finally, it should be noted that a relatively high failure rate (28 %) similar to the remarks of Haar [17] and Hitzel [19] was observed during the CFD simulations for creating the metamodels. It was identified that this was due to a limitation in the DA features of the model which failed to resolve certain geometry configurations. The applied solution in the present case was to limit some of the DA features such as “named selections” and “mesh sizing” which improved the success rate to more than 90 % but inevitably decreased the overall fidelity of the model.

Apart from the initial RCS considerations during the geometry generation similarly to the work of Allison [27] and Hitzel [19], a RCS analysis model based on DIBA was included in order to fill the design space and illustrate the capabilities of the framework in absence of a more advanced software. Despite the low-fidelity of the calculations, it was proven that a RCS model can be included and effectively coupled with the CAD geometry and the output from the antennas model. Furthermore, it was shown that one or more specific radar directions can be used together with the maximum obtained RCS value for the evaluation of the problem. Finally, it can be said that the use of the RCS area as a constraint was verified as an efficient strategy because it does not affect the solution of the optimization problem or the performance of the optimizer as already indicated by Tianyuan [26]. Clearly, if a strictly military design is desired, then it is possible to include the RCS signature as an objective, provided that results from a tool of higher fidelity can be obtained.

The second (level 2) mission simulation analysis was able to map the performance of the aircraft over a wide range of flight conditions and hence, it was possible to obtain a more thorough overview of the design as already pointed out by many authors (see

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section 2.1.5). The medium-fidelity approach which was implemented added supplementary accuracy as also commented by Allison [27] because areas of uncertainty like propulsion and weight were adequately filled with well-estimated data. Moreover, the model enabled a holistic evaluation of the concept regarding whether it is flyable or not which minimized the risk of a having a seemingly optimum design but with poor individual characteristics which is a very common phenomenon as pointed out by Amadori [18] and Allison [27]. As far as the objective of the maximum required fuel is concerned, one can comment that it is a representative characteristic of the design since it is coupled with the propulsion, weight, aerodynamic, geometry as well as mission variables and therefore, it is considered to be an advantageous monitor point for optimization as also commented by Haar [17] and Smith [30]. In the present application, the range, the altitude and the cruising speed were kept constant, but in future applications it is possible to include them as design variables together with other performance characteristics like the turn rate, the fuel consumption, or the landing and takeoff distance.

4.2 FRAMEWORK

The integration of the models in the common MDO framework was performed by using a down-top approach were several models were first grouped together in modules based on their relevance regarding a specific calculation routine. The modules were tested locally and the framework was developed in successive steps that allowed further evaluation at all integration stages which in turn increased the robustness of the system as also reported by Tianyuan [26] and Choi [16]. The assessment of the framework showed that this down-top approach can also be beneficial in terms of simplicity since the inter-disciplinary relations are decreased and there is a reduction of the floating parameters inside the workspace. On the downside, it is clear that this multiple-level approach requires more development effort and it is expected to be slightly slower than the traditional flat-level approach. The latter point was verified during the evaluation of the framework where a small delay was noticeable when data had to be moved between the different levels of simulations.

A preliminary quantification of the framework’s performance was attempted by using the present optimization strategy as reference (see section 3.1.4). According to the findings, one should expect iteration times in the order of seconds for the sizing, trim, antenna and mission simulation models, while the CFD model under the current solver settings required two hours at the very minimum.

The “sizing loop” was proved to be very efficient when used in the standard form of the framework but if metamodels were implemented, then it became a time-consuming segment of the optimization. This problem was clearly identified during the optimization with metamodels and it was concluded that in future applications the sizing loop and possibly the trim model should also be replaced. Apart from that, it was found that DIBA may generate critical errors for certain design configurations since the code can enter a divergent loop and block the entire optimization process. A first analysis of the results did not reveal the exact design factors that can induce this behavior and further investigation is required.

The local optimization process of antennas without metamodels proved to be a non-viable solution which is contradictory to the reported high-efficiency framework in the work of Tianyuan [26]. Nevertheless, a proper comparison could not be made since the

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exact specifications of the hardware the authors used as well as the complexity of their models was not known. Furthermore, one should note that the alternative case of a SL strategy was not tested and hence, no real reference for an accurate comparison between the SL and ML approach was available.

A first attempt to increase the efficiency of the framework was based on the use of metamodels for the “antenna loop” and the CFD model by using the Direct Sampling (DS) method (see section 2.4.4). A brief investigation of the metamodel accuracy was performed by testing several DOE sequence methods and different metamodeling algorithms but the analysis of the obtained NRMSE (see Appendix C) shows that only a very small improvement can be observed between the different approaches. Clearly, this points to the fact that the accuracy of the metamodel might depend more on training samples but it is not so sensitive to the DOE sequence or the metamodeling algorithm. The latter observation is in direct contrast with the findings of Tarkian [2] and Safavi [40] but it can be assumed that this is due to the simplicity of the present framework and that sensitivity will increase as the framework becomes more complex.

Overall, the values of the obtained NRMSE for the individual systems show that a good agreement was reached between the metamodels and the real models. In total, a NRMSE deviation of 0.1 % was identified for the whole system after comparing the output data from the random DOEs that were used for the main framework evaluation (see section 3.4.1) with the respective data from the framework with metamodels. This coincides with the reported error from the CFD model and it is a good indication that the DS strategy for this simple application was effective as it was suggested by Tarkian [2].

Moreover, in the case of the antenna metamodel, the maximum error was 0.87 % for the RCS and 0.43 % for the coverage prediction, while the aperture-to-sensor distance deviated by only 0.02 %. This shows clearly that simple linear functions like the prediction of the distance can be effectively replaced by metamodels, whereas more complex calculations such as the RCS and coverage require supplementary attention. Furthermore, in the case of the CFD model, the NRMSE was less than 0.1 % although the training sample was comprised of only 90 samples. Nevertheless, it can be verified that it is in the same order as the findings of Choi [16] and Breitkopf [38] which points out that additional expensive simulations can be possibly avoided.

4.3 RESULTS

Starting with the obtained results from the simulation of the baseline configuration and the evaluation of the DOEs (see section 3.4.1 as well as Appendix A and B), it can be seen that the initial estimations of the “sizing loop” regarding the MTOW and the CG position show a deviation from the final calculated values. As one may expect, this is due to the fact that new aerodynamic and weight data as well as a more refined geometry have been considered in the final mission analysis. Nevertheless, it should be noted that the main aim of this successive strategy was not to get the same results but to limit the design space of the other models as seen in the work of Amadori [29] and Lundström [15]. Having narrowed down the optimum position of the wing and tail it was possible to close the optimization loop and obtain designs that have realistic specifications which are very close to the desired SM characteristics. Furthermore, in order to verify that the method is valid, the SM of the final design was monitored and evaluated as a constraint during the final mission simulation. For all the considered designs, it was found that the

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SM constraint was never violated and in fact, the final obtained values deviated from the initial ones by less than 5 %.

As far as the VLM and the CFD results are concerned, it can be seen that there is a difference of less than 6 % in the lift and more than 60 % in the drag prediction. Clearly, the difference in drag is due to the fact that the VLM does not consider several aspects of the flow like compressibility, viscosity and surface interactions as noted in section 2.2.1. The lift on the other hand, appears to be in good agreement in both methods which indicates that trimming the aircraft with the VLM can be a viable and fast alternative to the computationally expensive multiple-CFD simulations which were used in the case-studies of Haar [17] and Hitzel [19].

A further examination of the obtained DOE results reveals that there is an average difference of 10 % when comparing the calculated lift force from the CFD and VLM simulations to the final aircraft weight. This is attributed to the fact that the first MTOW prediction from DIBA within the “sizing loop” is based on rough weight estimations and empirical aerodynamic calculations. Therefore, it becomes clear that the initial computations of the MTOW neglect to account for several of the additional weight penalties like the one of the sensors and apertures as well as for the more refined set of aerodynamic data that are computed from the CFD and VLM simulations at a later stage. The identified solution to this problem is a more detailed weight prediction at an earlier stage that is as independent as possible from any sort of aerodynamic data, and a compensation of the expected weight penalties in advance. Conclusively, it can be said that with the current tools it was not possible to fully resolve the inter-disciplinary relations between the weight and aerodynamics since they depend on each other through an iterative process. Hence, it was not feasible to compute one without knowing the other which inevitably led to an over- or under-prediction of the required lift. Nevertheless, the identified deviation of 10 % can be seen as an acceptable result especially if one considers that the product is at the preliminary design phase and that it is also a common practice in aircraft design to adjust the trim conditions at later development stages as Gudmundsson [14] explains.

The results from the evaluation of the local optimization processes indicate that the used algorithms were successful in handling the given problems. As far as the “sizing loop” is concerned, SIMPLEX was able to reach the target SM value all the times, while MOGA-II which was used in the “antenna loop” gave different results depending on the chosen settings. In particular, it was found that MOGA-II is sensitive to the number of generations as well as the number of individuals in the initial population as it was also reported by Lundström [15] and Amadori [18]. The obtained results indicate that the ability of the algorithm to identify the true pareto front increases as the number of generations and the population size grow but at the same time this can have a negative effect on the computational cost. Furthermore, it was observed that the algorithm has the tendency to merit one objective over the other which means that even a seemingly acceptable solution can be misleading as shown in section 3.4.4. All the above point to the fact that there is a need for a more detailed definition of the objective function which can guide the algorithm to the desired design space faster. This will in turn allow a reduction in the number of generations and the population size which can further increase the efficiency of the process as already pointed out by Amadori [29] and Lundström [15].

The optimization of the total aircraft was performed by considering a hybrid framework of both real- and meta-models which decreased the total time per iteration from an average of 9 hours to roughly 5 min. In this way it was possible to obtain a first

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set of results which further proved the potential of the framework regarding MDO and allowed the evaluation of two algorithms which were namely SIMPLEX and MOGA-II. Overall, both algorithms were able to optimize the baseline design but MOGA-II was far more successful than SIMPLEX with a total improvement of 36.1 % against 25.7 % respectively. The obvious explanation for this difference is that MOGA-II uses mutation and cross-over functions that can help the algorithm escape from a local optimum and search for the true optimum which was clearly not possible when SIMPLEX was implemented. This finding together with the increased required time that MOGA-II exhibited (5 times compared to SIMPLEX) coincide with the observations of Amadori [29] and Lundtröm [15] and indicates that MOGA-II is far more suitable for the present application provided that computational power is not limited.

4.4 SUMMARY

In this thesis a multidisciplinary framework for UAV design was developed by considering the geometric characteristics, the aerodynamic performance, the antenna efficiency, the RCS signature and the mission parameters. The engineering tools which were considered include high-fidelity solutions like CAD and CFD software, medium-fidelity tools like the VLM, and low-fidelity codes that are based on empirical equations. The development of the individual models was centered on principles such as robustness, increased compatibility, and advanced DA features which enabled the mapping of the expected design space and allowed a seamless MDO.

A preliminary ML strategy which was comprised of successive simulations that can limit the design space and local processes that optimize one module at a time acted as the main guideline for the integration. For the time being, the strategy illustrated that the recommended tools can be used effectively in a MDO framework by considering all the inter-disciplinary relations but revealed certain critical points. The most notable issues which were identified were the questionable efficiency of the proposed ML strategy in respect to the alternative of a SL approach and the challenge of trimming and sizing the aircraft. Regarding the latter, it should be noted that since there is no model to compute the exact weight it is impossible to accurately predict the trimmed aerodynamic performance which can only be closely estimated.

The efficiency of the selected strategy and the performance of the framework were evaluated by several DOEs since it was identified that an optimization at the present phase and the current computational resources would have been impossible. Nevertheless, the important contribution is that the obtained results were verified for their accuracy and showed that the operation of the framework is above all robust with an acceptable low failure rate.

Furthermore, in order to illustrate the capabilities of the framework and generate a first set of optimization results, metamodels were considered as an efficient alternative to the computationally expensive sub-systems. The DS method was used due to the limited inter-disciplinary relations between the models and several DOE sequences against different metamodeling algorithms were tested. Although the research on the metamodel possibilities was brief, it was shown that very good results can be obtained and that they can be a viable alternative to the MDO of complex products.

Moreover, the thesis addressed the topic of optimization by using the modified framework with metamodels and by evaluating two different algorithms which were namely SIMPLEX and MOGA-II. Overall, both algorithms managed to improve the design

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but MOGA-II illustrated a far better hit rate which clearly makes GA more suitable for the present application provided that the increased computational time can be tolerated.

Conclusively, some brief answers to the research questions which were posed at the introduction are presented as a summarization of the thesis:

RQ1:-Are the suggested tools adequate in terms of availability, compatibility, and fidelity?

The tools which were used in the present case-study were available to both Saab and Linköping University. Modifications were required in many cases to ensure their compatibility but this did not pose a critical issue. The used tools were of different fidelity but this did not constitute a problem with respect to their integration in the framework.

RQ2:-How effective are the models with respect to handling the inter-disciplinary relations?

The models which were developed for this case-study were able to handle the expected input and provide a suitable output that is compatible with the other models of the framework. Overall, it can be said that it was feasible to have a proper representation of the inter-disciplinary relations.

RQ3:-Is the integration of the models in a common framework under the current optimization strategy feasible?

The integration of the models was successful and the obtained failure-rate was within reasonable limits. The chosen optimization strategy was able to show the capabilities of the framework but it illustrated a very poor performance in terms of iteration times.

RQ4:-Are there any alternative solutions that can increase the efficiency of the framework?

Metamodels were shown to be an efficient alternative to the computationally expensive models. The investigation of their deviation from the real models was found to be very low which indicates that they can be an effective solution for enabling a fast MDO.

RQ5:-Is MDO a viable solution for the proposed case-study and what should be the direction of the future work?

Given the successful development of the models and their seamless integration in a common framework it is clear that MDO can be a viable option in the design of UAVs. Furthermore, the preliminary optimization results indicate that MDO can improve the specified characteristics and provide engineers with valuable information during the early design phases.

4.5 FUTURE WORK

At a model level, the future work includes mostly refinements in respect to the individual models and evaluation of new tools like the RCS software QGRECO which was omitted from this case-study due to time limitations. In addition to that, it will be of utmost importance to regenerate segments of the CAD model but this time the development objective should be aimed towards performance rather than accuracy.

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Furthermore, it would be interesting to evaluate the RAPID model which is already used in many of the MDO studies that are performed in the Machine Design division of Linköping University. Moreover, it is clearly instrumental to investigate different CFD solvers that can increase the overall efficiency of the framework as well as the CFD codes that are indicated by the partner organization. Regarding the former, it is worth considering the mesh generator SUMO by the Royal Institute of Technology in Stockholm as well as the SU2 solver by Stanford University which have been reported to have shown increased efficiency in terms of computational time. Regarding the latter, it is important to verify that MDO can work with the specific tools that the partner organization uses and therefore, the EDGE code developed by FOI and the mesh generator ICEM should be evaluated since they are commonly used in SAAB’s projects. Finally, it is the author’s personal opinion that the antenna model can and should be replaced by a tool of higher fidelity. At the moment there are various commercial products available and one notable case is that of the CST suite which is also directly compatible with modeFRONTIER.

At a framework level, there are not many improvement possibilities if the current optimization strategy and the models remain unchanged. However, if new models or a different strategy are considered, then it becomes clear that new integration challenges arise. One case that is worth investigating is that of the SL strategy but it would also be interesting to evaluate the performance of the current strategy under multiple optimization objectives. Furthermore, it can be seen that the weight is not properly defined which indicates that there is need for a more accurate weight prediction model at an early stage of the framework. In addition to this, there is clearly a lot of ground to be covered in the field of metamodels and hence, it would be interesting to look into different sampling and training methodologies. Finally, one must not forget that this is an MDO study and therefore, optimization algorithms should be a major part of the future research. This includes a more thorough analysis of the non-gradient solutions but also a feasibility study of gradient algorithms which would be interesting to extend to an ad-joint optimization application.

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APPENDIX A Summary of the input range that the design variables received for the preliminary

evaluation and optimization of the framework.

Global Input Variable Range

Fuselage Length

10 - 12 [m] Wing Area

14 - 16 [m2] Wing

Aspect Ratio 14 - 16 [-]

Wing Taper Ratio

0.4 - 0.6 [-] Wing

Dihedral 0 - 4 [deg]

Wing Sweep

0 - 5 [deg]

Wing Airfoil Thickness

80 - 160 [%] Wing

Airfoil Type Eppler E374

Tail Hor. Volume Coef.

0.5 - 0.6 [-]

Tail Ver. Volume Coef.

0.03 - 0.05 [-] Tail

Aspect Ratio 5 - 7 [-]

Tail Taper Ratio

0.7 - 0.9 [-]

Tail Sweep

5 - 10 [deg] Tail Airfoil Thickness

60 - 120 [%] Tail

Airfoil Type NACA 0010

Local Input Variable Range

Wing Relative Position

0.4 - 0.6 [-] Tail Relative

Position 0.85 - 0.95 [-]

Aperture Relative X Position (i=1:5)

(0.01 0.20 0.50 0.01 0.20) (0.20 0.50 0.99 0.20 0.70)

Aperture Relative Y Position (i=1:5)

(0.3 0.3 -0.2 -0.2 -0.2) (0.7 0.7 0.2 0.2 0.2)

Aperture Frequency (x3)

10 - 12 [GHz] Aperture

Opening (x3) 4x4 - 6x6 [cm]

Eppler E374

A 0.001 B -0.03172 C 0.0776 X2 0.85221 X3 0.63969

X4 0.31145 X5 0.27407 X9 0.11634 X10 0.3894 X11 0.51246

X12 0.677286 Y2 -0.01126 Y6 -0.03172 Y8 0.024399 Y12 0.052693

NACA 0010

A 0.001 B -0.05001 C 0.05006 X2 0.85594 X3 0.53405

X4 0.3 X5 0.165 X9 0.165 X10 0.3 X11 0.53405

X12 0.85594 Y2 -0.01763 Y6 -0.037 Y8 0.037004 Y12 0.017634

54

Baseline Configuration Output

MTOW (level 1)

5118 [kg] CG

(level 1) 5.21 [m] SM 0.54 [-]

Trimmed CL (VLM)

0.3686 [-] Trimmed CDi (VLM)

0.0059 [-] Trimmed

CD0 (VLM) 0.0193 [-]

Tail trim angle

-0.65 [deg] Wing

inc. angle +1.67 [-] Coverage 5.81 [-]

Ap.-sen. distance

19.08 [m] Max RCS 15.31 [m2] RCS (radar

dir.) 9.55 [m2]

CL total (CFD) 0.4128 [-] CDi total (CFD) 0.0041 [-] CD0 total

(CFD) 0.0805 [-]

MTOW (level 2)

5566 [kg] CG

(level 2) 5.21 [m] Max Fuel 2374 [kg]

55

APPENDIX B The random sequence input which was used for the evaluation of the framework

through the DOE method.

011.4

618

5.8

202

72.4

629

6.6

636

0.8

936

0.5

006

0.0

493

15.8

797

155.7

756

15.8

742

1.5

887

1.7

376

0.4

588

111.0

130

5.2

319

106.2

322

8.2

995

0.7

313

0.5

378

0.0

328

15.3

899

144.4

182

14.0

101

2.0

925

3.7

199

0.4

284

210.9

635

6.0

891

94.6

260

6.0

246

0.8

247

0.5

185

0.0

302

14.3

221

94.2

444

15.0

808

3.8

953

1.2

271

0.4

789

310.4

352

5.8

640

73.9

893

9.4

495

0.7

077

0.5

592

0.0

431

14.2

397

132.1

981

15.9

686

0.8

270

1.8

732

0.4

927

410.6

672

5.8

864

90.2

481

9.9

949

0.8

261

0.5

910

0.0

402

14.9

829

114.3

027

14.6

162

2.8

693

4.8

121

0.4

419

510.3

453

6.0

979

93.3

281

7.9

391

0.8

578

0.5

699

0.0

341

14.5

098

142.2

333

14.4

500

3.9

324

4.0

181

0.5

673

610.3

262

6.2

750

60.5

256

8.1

559

0.7

416

0.5

880

0.0

441

15.4

467

120.7

594

15.9

742

0.6

130

3.5

529

0.5

648

710.2

473

6.2

339

88.9

117

5.4

988

0.8

234

0.5

028

0.0

418

14.1

461

85.9

238

15.3

599

1.1

200

3.0

199

0.5

699

810.7

076

6.1

973

77.5

256

9.3

822

0.8

488

0.5

423

0.0

375

14.5

426

138.3

859

15.7

735

0.1

748

2.2

885

0.5

725

911.8

038

5.1

250

105.8

292

7.1

385

0.8

998

0.5

062

0.0

417

15.1

346

132.9

621

15.8

146

0.9

859

1.2

424

0.5

414

10

10.6

754

6.2

147

78.9

254

5.1

224

0.7

902

0.5

369

0.0

393

14.3

545

134.0

899

15.4

311

2.9

735

4.6

455

0.5

180

11

11.2

880

5.2

210

106.4

762

6.5

324

0.8

761

0.5

630

0.0

486

14.3

934

140.3

717

14.3

153

0.1

687

1.8

931

0.5

573

12

11.4

255

5.2

205

105.7

280

5.1

503

0.7

633

0.5

667

0.0

385

14.8

011

139.5

009

14.7

306

3.0

550

1.6

133

0.5

025

13

10.9

574

5.2

623

61.9

078

7.2

223

0.8

383

0.5

719

0.0

456

14.5

762

142.5

084

15.9

325

3.9

861

1.1

955

0.4

890

14

10.2

604

5.4

689

99.3

649

7.8

611

0.8

112

0.5

250

0.0

484

15.4

475

142.1

593

14.4

846

0.7

028

2.2

240

0.5

659

15

11.2

038

6.2

759

78.2

020

5.3

270

0.8

876

0.5

266

0.0

346

14.6

080

109.7

552

15.7

143

3.2

283

1.9

436

0.4

530

16

10.0

647

6.3

396

64.4

617

8.0

457

0.7

357

0.5

030

0.0

442

14.5

556

89.2

631

15.3

825

1.3

960

1.1

407

0.4

335

17

11.1

893

6.6

198

90.8

957

6.7

449

0.7

106

0.5

606

0.0

436

15.1

675

109.5

802

15.8

920

2.7

457

1.1

229

0.4

889

18

11.4

152

5.0

596

99.5

336

6.5

796

0.8

632

0.5

156

0.0

391

14.6

394

123.9

754

15.0

445

1.2

613

3.4

806

0.4

190

19

11.9

176

5.2

594

61.4

159

5.4

236

0.7

105

0.5

470

0.0

464

15.9

382

107.9

989

14.5

458

0.0

449

3.9

701

0.4

201

20

11.6

379

6.4

957

82.7

918

5.5

385

0.8

532

0.5

209

0.0

489

14.0

551

139.7

539

15.2

727

3.7

542

0.4

693

0.5

919

21

11.6

376

5.4

296

110.0

263

6.2

786

0.8

281

0.5

716

0.0

460

14.9

228

89.5

712

15.0

212

2.9

717

0.2

844

0.5

120

22

10.7

360

6.7

645

112.7

986

5.3

023

0.7

144

0.5

229

0.0

345

15.2

909

103.4

386

15.7

710

2.1

774

3.5

091

0.4

366

23

10.6

947

5.0

796

74.9

571

8.9

834

0.8

416

0.5

466

0.0

461

14.1

642

118.3

684

15.9

224

0.9

525

2.2

659

0.4

962

24

10.1

315

5.2

176

113.5

943

9.3

219

0.8

598

0.5

350

0.0

326

15.7

840

114.5

572

14.2

597

2.6

785

2.5

885

0.5

109

25

11.8

335

5.7

607

70.3

499

5.6

664

0.7

957

0.5

875

0.0

382

15.0

585

130.6

441

14.3

221

0.9

850

4.2

596

0.4

098

26

11.7

797

5.0

553

96.7

169

6.4

504

0.7

163

0.5

771

0.0

372

15.2

063

109.3

690

15.2

637

3.5

379

3.2

874

0.5

287

27

10.1

356

6.2

054

93.1

318

7.6

527

0.8

826

0.5

944

0.0

487

14.3

468

93.2

446

15.8

446

3.9

842

3.6

825

0.5

638

28

11.1

640

5.8

954

84.8

105

7.3

281

0.8

338

0.5

312

0.0

478

15.4

435

137.7

148

15.0

422

3.8

693

3.7

411

0.5

482

29

11.7

007

6.0

661

76.0

641

8.7

322

0.8

388

0.5

066

0.0

404

15.6

226

122.3

700

15.4

427

2.1

159

3.1

449

0.5

424

Win

g

Are

a

[m^

2]

Win

g

Dih

ed

ral

[de

g]

Win

g

Sw

ee

p

[de

g]

Win

g T

R

[-]Id

Fu

sela

ge

Le

ng

th

[m]

Ta

il AR

[-]

Ta

il Airfo

il

Th

ickn

ess

[%]

Ta

il

Sw

ee

p

[de

g]

Ta

il Ho

r.

Vo

l. Co

ef

[-]

Ta

il Ve

r.

Vo

l. Co

ef.

[-]

Win

g A

R

[-]

Win

g A

irfoil

Th

ickn

ess

[%]

Ta

il TR

[-]

56

5275

5.3

053

0.4

022

0.4

778

0.8

984

0.4

200

0.3

900

0.3

588

0.0

046

0.0

191

5.8

569

19.8

605

0.0

079

0.2

822

0.0

018

0.1

247

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

4986

5.0

307

0.3

999

0.4

691

0.8

900

0.8

700

0.6

900

0.3

841

0.0

055

0.0

198

5.8

107

19.1

078

0.0

079

0.3

326

0.0

025

0.1

171

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

5016

5.0

292

0.4

005

0.4

748

0.8

952

1.6

400

0.0

700

0.3

589

0.0

054

0.0

188

5.8

136

19.0

247

0.0

079

0.3

974

0.0

039

0.0

696

14.7

986

8.1

079

5296

5.0

281

2174

5094

4.9

719

0.3

997

0.4

926

0.9

049

0.8

100

0.4

300

0.3

442

0.0

050

0.0

185

5.7

506

18.1

387

0.0

080

0.3

380

0.0

028

0.1

023

15.6

105

8.9

374

6097

4.9

639

2926

5025

5.0

243

0.3

997

0.4

826

0.8

999

1.3

300

0.4

400

0.3

715

0.0

054

0.0

194

5.8

128

18.5

279

0.0

080

0.3

961

0.0

037

0.0

890

14.8

597

10.4

856

5620

5.0

195

2489

4996

4.8

782

0.3

992

0.4

841

0.8

975

0.8

500

0.6

600

0.3

733

0.0

056

0.0

193

5.7

568

17.9

878

0.0

080

0.3

359

0.0

028

0.1

163

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

5233

5.0

038

0.4

004

0.4

968

0.9

151

1.0

100

0.4

200

0.3

538

0.0

048

0.0

186

5.7

563

17.9

558

0.0

080

0.3

727

0.0

032

0.0

923

15.6

222

8.2

502

6006

4.9

961

2732

5041

4.9

121

0.3

993

0.4

972

0.9

221

1.7

600

-0.2

100

0.3

545

0.0

054

0.0

186

5.7

595

17.8

235

0.0

080

0.4

100

0.0

042

0.0

663

15.3

528

7.5

902

5270

4.9

106

2139

5140

5.0

472

0.4

005

0.4

865

0.9

085

0.7

300

0.4

700

0.3

519

0.0

049

0.0

186

5.8

102

18.5

957

0.0

080

0.3

273

0.0

026

0.1

074

15.6

816

10.4

510

6216

5.0

380

2998

5247

5.3

452

0.4

010

0.4

674

0.8

963

0.8

500

0.2

900

0.3

581

0.0

050

0.0

190

5.8

414

20.4

342

0.0

079

0.3

395

0.0

027

0.1

006

15.6

887

7.0

784

6165

5.3

369

2875

5073

5.0

860

0.3

998

0.4

921

0.9

119

0.8

700

0.3

900

0.3

552

0.0

052

0.0

188

5.8

125

18.5

416

0.0

080

0.3

413

0.0

029

0.1

058

14.9

711

8.8

845

6096

5.0

777

2928

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

5087

5.1

570

0.3

985

0.4

631

0.8

925

0.8

700

0.6

700

0.3

728

0.0

054

0.0

196

5.8

587

19.7

997

0.0

079

0.3

391

0.0

027

0.1

091

14.9

628

10.3

292

6052

5.1

488

2868

5150

5.0

940

0.4

011

0.4

792

0.8

955

0.6

400

0.5

700

0.3

491

0.0

050

0.0

188

5.8

135

19.0

145

0.0

079

0.3

153

0.0

024

0.1

096

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

5079

4.8

275

0.3

999

0.4

838

0.9

035

0.8

500

0.3

400

0.3

790

0.0

054

0.0

195

5.7

595

17.8

455

0.0

080

0.3

418

0.0

027

0.1

180

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

5111

5.1

825

0.4

000

0.4

784

0.9

015

1.2

700

0.2

200

0.3

511

0.0

050

0.0

187

5.8

030

19.4

279

0.0

079

0.3

790

0.0

035

0.0

787

15.0

421

10.5

118

5591

5.1

795

2406

5008

4.8

183

0.3

983

0.4

971

0.9

195

1.6

700

-0.1

300

0.3

514

0.0

053

0.0

186

5.6

906

17.5

172

0.0

080

0.4

091

0.0

041

0.0

671

15.3

279

7.1

310

5258

4.8

163

2152

5231

5.2

029

0.4

004

0.4

783

0.8

936

1.2

600

0.2

700

0.3

553

0.0

049

0.0

189

5.8

027

19.4

035

0.0

079

0.3

856

0.0

035

0.0

814

15.3

411

8.0

832

5745

5.1

983

2479

5056

5.2

336

0.3

989

0.4

738

0.8

866

1.1

200

0.2

400

0.3

629

0.0

054

0.0

193

5.8

585

19.7

825

0.0

079

0.3

640

0.0

032

0.0

944

15.4

591

8.3

571

5797

5.2

283

2645

5170

5.3

936

0.4

003

0.4

635

0.8

933

1.4

900

0.2

600

0.3

837

0.0

055

0.0

200

5.8

933

20.6

250

0.0

079

0.4

041

0.0

036

0.0

777

15.5

166

6.6

800

5504

5.3

911

2272

5186

5.3

109

0.3

998

0.4

713

0.8

910

0.8

700

0.3

000

0.3

668

0.0

055

0.0

195

5.8

504

20.1

559

0.0

079

0.3

336

0.0

028

0.1

067

14.9

722

6.6

674

6157

5.3

020

2914

5174

5.2

362

0.4

012

0.4

616

0.8

797

1.7

800

0.0

700

0.3

725

0.0

056

0.0

196

5.8

504

20.1

554

0.0

079

0.4

356

0.0

045

0.0

650

15.0

549

7.6

541

5314

5.2

359

2093

5124

5.0

931

0.3

995

0.4

901

0.9

108

1.3

500

0.1

700

0.3

508

0.0

048

0.0

185

5.8

080

18.6

432

0.0

080

0.3

903

0.0

035

0.0

808

15.3

480

10.5

134

5649

5.0

889

2450

5096

5.0

442

0.3

997

0.4

880

0.8

991

1.0

900

0.1

800

0.3

459

0.0

053

0.0

186

5.8

108

18.5

739

0.0

080

0.3

682

0.0

034

0.0

881

15.5

854

7.9

780

5785

5.0

392

2622

5001

4.7

316

0.4

003

0.4

796

0.8

957

1.3

400

0.3

400

0.3

787

0.0

054

0.0

192

5.7

649

17.6

292

0.0

080

0.3

980

0.0

035

0.0

926

14.9

141

10.2

519

5641

4.7

259

2511

5074

5.3

508

0.3

999

0.4

626

0.8

981

1.1

100

0.6

700

0.3

825

0.0

056

0.0

200

5.8

404

20.4

840

0.0

079

0.3

682

0.0

032

0.0

973

15.4

351

10.3

274

5786

5.3

450

2613

5201

5.3

396

0.3

997

0.4

642

0.8

855

1.3

600

0.3

600

0.3

684

0.0

053

0.0

193

5.8

434

20.3

937

0.0

079

0.3

947

0.0

036

0.0

797

15.0

351

10.4

977

5638

5.3

365

2387

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

#N

AN

5196

5.2

066

0.3

998

0.4

791

0.9

025

0.8

700

0.4

200

0.3

726

0.0

052

0.0

194

5.8

046

19.3

610

0.0

079

0.3

502

0.0

028

0.1

094

14.8

401

6.2

217

6200

5.1

969

2940

5263

5.3

730

0.4

002

0.4

727

0.8

942

1.1

100

0.2

000

0.3

683

0.0

050

0.0

192

5.8

475

20.2

612

0.0

079

0.3

669

0.0

030

0.0

906

15.4

781

7.0

227

5925

5.3

667

2621

Win

g

Po

sition

[-]

To

tal

CD

0

(CF

D) [-]

MT

OW

(initia

l)

[kg

]

CG

(initia

l)

[m]

To

tal

Cd

i

(CF

D) [-]

To

tal

CL

(CF

D) [-]

Re

qu

ired

Fu

el [k

g]

MT

OW

(fina

l)

[kg

]

An

ten

na

RC

S

[m^

2]

RC

S

Ra

da

r

Dir. [m

^2]

Ma

x.

RC

S

[m^

2]

Sta

tic

Ma

rgin

[-]

CG

(fina

l)

[m]

Ap

ertu

re

Co

ve

rag

e

[-]

Ap

er.-

Se

ns.

Dist. [-]

Ta

il

Po

sition

[-]

Ta

il Trim

An

gle

[de

g]

Trim

me

d

CD

0

(VL

M) [-]

Trim

me

d

Cd

i

(VL

M) [-]

Trim

me

d

CL

(VL

M)

[-]

Win

g In

c.

An

gle

[de

g]

57

APPENDIX C The NRSME values which were obtained during the evaluation of different DOE

sequence methods and various metamodeling algorithms.

Antenna Metamodel Samples DOE Algorithm Evaluation Output NRMSE [%]

300

ULH

AKR

50 Random

Coverage 0.43

Distance 0.02

RCS 0.87

KR

Coverage 0.41

Distance 0.03

RCS 0.86

SVD

Coverage 0.40

Distance 0.02

RCS 0.85

SOBOL

AKR

Coverage 0

Distance 0.01

RCS 0

KR

Coverage 0.36

Distance 0

RCS 0.89

SVD

Coverage 0.38

Distance 0.02

RCS 0.84

CFD Metamodel Samples DOE Algorithm Evaluation Output NRMSE [%]

90 ULH

AKR

15 Random

CLtotal 0.05

CDitotal 0.07

CD0total 0.03

KR

CLtotal 0.06

CDitotal 0.13

CD0total 0.03

SVD

CLtotal 0.04

CDitotal 0.09

CD0total 0.04

58

59

APPENDIX D The optimized configuration of the aircraft which was obtained by using the MOGA-

II algorithms on the metamodel-modified framework.

Global Input Variable Range

Fuselage Length

12 [m] Wing Area

14 [m2] Wing

Aspect Ratio 16 [-]

Wing Taper Ratio

0.6 [-] Wing

Dihedral 0 [deg]

Wing Sweep

0 [deg]

Wing Airfoil Thickness

80 [%] Wing

Airfoil Type Eppler E374

Tail Hor. Volume Coef.

0.6 [-]

Tail Ver. Volume Coef.

0.03 [-] Tail

Aspect Ratio 5 [-]

Tail Taper Ratio

0.9 [-]

Tail Sweep

10 [deg] Tail Airfoil Thickness

60 [%] Tail

Airfoil Type NACA 0010

Local Input Variable Range

Wing Relative Position

0.4 - 0.6 [-] Tail Relative

Position 0.85 - 0.95 [-]

Aperture Relative X Position (i=1:5)

(0.09 0.20 0.50 0.04 0.28) Aperture Relative Y Position (i=1:5)

(0.47 0.45 0.11 0.15 0.11)

Aperture Frequency (x3)

(10.57 10.15 10.26) [GHz] Aperture

Opening (x3) (4.2x5.9 4.8x4.1 5.2x4.1)

[cm]

60