Loosely coupled, modular framework for linear static aeroelastic analyses

JUNE 2019

Loosely coupled, modular framework for linear static aeroelastic analyses

Master Thesis Aaron Dettmann

Supervisor: Prof. Arthur Rizzi Examiner: Prof. Stefan Hallström

KTH Royal Institute of Technology

Aerospace Engineering

Aaron Dettmann (dettmann@kth.se)

Master’s programme in Aerospace Engineering KTH Royal Institute of Technology

100 44 Stockholm, Sweden Place for Project Airinnova AB

Drottning Kristinas väg 53 114 28 Stockholm, Sweden Dr. Mengmeng Zhang, CEO Examiner

Prof. Stefan Hallström

KTH Royal Institute of Technology 100 44 Stockholm, Sweden

Supervisor Prof. Arthur Rizzi

KTH Royal Institute of Technology

100 44 Stockholm, Sweden

A computational framework for linear static aeroelastic analyses is pre- sented. The overall aeroelasticity model is applicable to conceptual air- craft design studies and other low-fidelity aero-structural analyses. A partitioned approach is used, i. e. separate solvers for aerodynamics and structure analyses are coupled in a suitable way, together forming a model for aeroelastic simulations. Aerodynamics are modelled using the vortex- lattice method (VLM), a simple computational fluid dynamics (CFD) model based on potential flow. The structure is represented by a three-dimen- sional (3D) Euler-Bernoulli beam model in a finite element method (FEM) formulation. A particular focus was put on the modularity and loose coup- ling of aforementioned models. The core of the aeroelastic framework was abstracted, such that it does not depend on any specific details of the under- lying aerodynamics and structure modules. The final aeroelasticity model constitutes independent software tools for the VLM and the beam FEM, as well as a framework enabling the aeroelastic coupling. These different tools have been developed as part of this thesis work. A wind tunnel exper- iment with a simple wing model is presented as a validation test case. An aero-structural analysis of a fully elastic unmanned aerial vehicle (UAV) (OptiMale) is described and results are compared with an existing higher- fidelity study.

Keywords

Aeroelasticity, Simulation, CFD, FEM, VLM, Euler-Bernoulli beam theory

III

Rapporten beskriver en beräkningsmodell för linjära, statisk aeroelas- tiska analyser. Modellen kan användas för konceptuella designstudier av flygplan. En partitionerad metod används, d v s separata lösare för aerodynamik- och strukturanalyser kopplas på ett lämpligt sätt, och bil- dar tillsammans en modell för aeroelastiska simulationer. Aerodynamik modelleras med hjälp av en så kallad vortex-lattice method (VLM), en en- kel modell för beräkningsströmningsdynamik (CFD) som är baserad på friktionsfri strömning. Strukturen representeras av en tredimensionell (3D) Euler-Bernoulli-balkmodell implementerad med hjälp av en finita ele- mentmetod (FEM). Ovannämnda modeller har utvecklats med fokus på modularitet och lös koppling. Kärnan i den aeroelastiska modellen har abstraherats så att den inte beror på specifika detaljer i de underliggande aerodynamik- och strukturmodulerna. Aeroelasticitetsmodellen i sin hel- het består av separata mjukvaruprogram för VLM och balk-FEM, såväl som ett ramverk som möjliggör den aeroelastiska kopplingen. Dessa olika program har utvecklats som en del av examensarbetet. Ett vindtunnelför- sök med en enkel vingmodell presenteras som ett valideringstest. Des- sutom beskrivs en analys av ett elastiskt obemannad flygplan (OptiMale) och resultaten jämförs med en befintlig studie som har genomförts med modeller av högre trovärdighet.

Keywords

Aeroelasticitet, Simulation, CFD, FEM, VLM, Euler-Bernoulli-balkmodell

IV

The presented work was carried out at Airinnova AB and the Depart- ment of Aeronautical and Vehicle Engineering at the KTH Royal Institute of Technology in Stockholm in the period from January to June 2019.

I would like to thank Dr. Mengmeng Zhang for giving me the opportunity to work on this thesis at Airinnova and her support throughout the project.

I am very grateful to Prof. Arthur Rizzi for his valuable feedback, guidance and concerns. Many thanks also to Prof. Stefan Hallström for his help and always having time for a discussion. Also warm thanks to Prof. Jesper Oppelstrup for his patience, support and for sharing his expertise with many of the small but important details in this work. I want to thank Aidan Jungo, Jan-Niclas Walther and Reinhold Maierl for their continuous support with CPACS and the OptiMale.

V

1 Introduction 1

1.1 Background and motivation . . . . 1

1.2 Objectives and scope . . . . 3

2 Theoretical background 4 2.1 Aeroelasticity . . . . 4

2.1.1 Static deformations and divergence . . . . 4

2.1.2 Control efficiency and reversal . . . . 4

2.2 Approach and requirements for the aeroelastic framework . . . . 5

2.3 Aerodynamic modelling (vortex-lattice method) . . . . 6

2.4 Structural modelling (Euler-Bernoulli beam theory) . . . . 8

2.4.1 Model-induced mesh distortions . . . 10

2.5 Aeroelastic coupling . . . 12

3 Implementation 14 3.1 Technical requirements . . . 14

3.1.1 Loose coupling . . . 14

3.1.2 Modularity . . . 14

3.1.3 Programming language and paradigm . . . 14

3.1.4 Serialisation and data formats . . . 15

3.2 Implementation of the aerodynamics model . . . 15

3.2.1 General approach . . . 15

3.2.2 Lattice generation . . . 16

3.2.3 Aerofoil and control surface modelling . . . 16

3.2.4 API for wing deformations . . . 17

3.2.5 Input data formats . . . 18

3.2.6 Verification . . . 18

3.3 Implementation of the structural model . . . 18

3.3.1 General approach . . . 19

3.3.2 API for loads with an offset to the elastic axis . . . 19

3.3.3 Verification . . . 20

3.4 Implementation of the aero-structural framework . . . 20

3.4.1 Interface with underlying tools . . . 20

3.4.2 Aeroelastic loop for static equilibrium . . . 21

3.4.3 Data exchange within the framework . . . 21

3.4.4 Load mapping . . . 22

3.4.5 Convergence criterion . . . 23

3.4.6 Verification . . . 23

3.5 Publication of developed code . . . 23

4 Case studies 24 4.1 Experimental validation . . . 24

4.1.1 Wind tunnel model, experimental setup and measurements . . . 24

4.1.2 Computational model . . . 25

4.1.3 Comparison of experiment and simulation results . . . 26

4.1.4 Discussion and conclusions . . . 28

VI

4.2.2 Convergence of the FE mesh . . . 30

4.2.3 Convergence of the VLM mesh . . . 31

4.2.4 Conclusions and comments . . . 33

4.3 OptiMale UAV . . . 33

4.3.1 The OptiMale aircraft . . . 33

4.3.2 Flight manoeuvres and analyses . . . 34

4.3.3 Model generation . . . 34

4.3.4 Comparison with high-fidelity models . . . 36

4.4 Conclusions . . . 38

5 Summary 40 6 Outlook 40 6.1 Tool verification and validation . . . 40

6.2 Extended beam models . . . 41

6.3 Aeroelastic sizing and beam model generation . . . 41

6.4 Dynamic aeroelastic simulations . . . 42

7 References 43 A Appendix 47 A.1 Finite element modelling . . . 47

A.1.1 Galerkin weighted residual method . . . 47

A.1.2 The beam element . . . 49

A.2 Tool verification . . . 54

A.2.1 PyTornado verification . . . 54

A.2.2 FramAT verification . . . 58

A.3 JSON files . . . 60

A.3.1 PyTornado – Example aircraft definition . . . 60

A.3.2 FramAT – Example structure definition . . . 62

A.4 Comments on beam load mapping schemes . . . 65

A.4.1 Elastic work . . . 65

List of Figures 1.1 Glider with flexible wings . . . . 1

1.2 OptiMale: Aeroelastic analysis in the AGILE project . . . . 2

2.1 Basic aeroelastic coupling concept . . . . 5

2.2 Concept of the VLM discretisation . . . . 6

2.3 Horseshoe vortex geometry . . . . 7

2.4 Beam element in equilibrium . . . . 9

2.5 Cantilever beam elongation . . . 11

2.6 Concept of joining the VLM with the FE beam model . . . 12

3.1 Object hierarchy of the aircraft model . . . 16

3.2 Basic meshing algorithm . . . 16

3.3 Aerofoil and control surface modelling . . . 17

3.4 Mesh deformations . . . 17

VII

3.7 Conceptual implementation of the aeroelastic loop . . . 21

3.8 Data exchange within the framework . . . 22

3.9 Mapping of aerodynamic forces onto the FE beam . . . 23

4.1 Wind tunnel model: Illustration . . . 24

4.2 Wind tunnel model: Wing mounted in the wind tunnel . . . 25

4.3 Wind tunnel model: Divergence and control reversal . . . 26

4.4 Wind tunnel model: Loss of control efficiency with airspeed . . . 27

4.5 Wind tunnel model: Comparison of beam and plate deformations . . . . 29

4.6 Wind tunnel model: Convergence at reversal . . . 29

4.7 Convergence of the FE mesh . . . 30

4.8 Load mapping and nodal loads for a beam with two elements . . . 31

4.9 Convergence of the VLM mesh . . . 32

4.10 OptiMale: General arrangement . . . 33

4.11 OptiMale: FE beam model . . . 36

4.12 OptiMale: The aeroelastic loop (loads) . . . 36

4.13 OptiMale: The aeroelastic loop (deformations) . . . 37

4.14 OptiMale: Deformation during a pull-up and a push-down (AGILE result) 38 4.15 OptiMale: Tip deflection with an additional wing tip mass . . . 38

A.1 FEM bar element . . . 48

A.2 FEM beam element . . . 50

A.3 Definition of the direction cosines . . . 52

A.4 Assembly of global matrices and vectors . . . 53

A.5 Convergence of lift, drag and moment coefficients for a rectangular wing 55 A.6 Pressure distribution for a D150 analysis . . . 56

A.7 Pressure distribution for a Boxwing analysis . . . 57

A.8 Beam orientation and applied load in a FramAT verification test case . . 58

A.9 Undeformed and deformed structure in a FramAT verification test case . 59 A.10 Comparison of load mapping schemes . . . 65

A.11 Projection of off-axis loads onto an inclined cantilever beam . . . 66

A.12 Elastic work for different mapping schemes and beam inclinations . . . 68

A.13 Beam deformation for different mapping schemes and beam inclinations 69 List of Tables 2.1 Linear Euler-Bernoulli beam theory: Estimated area distortion . . . 11

4.1 Wind tunnel model: Material parameters . . . 24

4.2 Wind tunnel model: Simulated and measured critical airspeeds . . . 27

4.3 OptiMale: Deformations from low- and high-fidelity analyses . . . 37

A.1 PyTornado verification: Comparison of global coefficients . . . 57

VIII

1D One-dimensional. 48

3D Three-dimensional. III, IV, X, 5, 6, 8, 32, 40, 49, 59

AGILE Aircraft 3rd Generation MDO for Innovative Collaboration of Heterogeneous Teams of Experts. VII, VIII, 2, 15, 33–35, 37, 38, 40, 56, 57

API Application Programming Interface. 14, 15, 17, 20–23, 67, Glossary: API

CFD Computational Fluid Dynamics. III, IV, X, 2, 5, 12–14, 21, 22, 33, 40, 42, 65 CFRP Carbon Fibre Reinforced Polymer. 35

CPACS Common Parametric Aircraft Configuration Schema. V, 2, 3, 15, 18, 34–36, 38, 40, 56

d.o.f. Degrees of freedom. 8, 18, 24, 26, 27, 36, 37, 41, 49, 50, 54, 68 DLR German Aerospace Center. 2, 15, 43

EASA European Aviation Safety Agency. 1 EU European Union. 2

FE Finite Element. VII, VIII, 8, 10, 12, 13, 17–19, 22–24, 27, 28, 30, 31, 33, 34, 36, 37, 39–42, 47, 49, 58

FEM Finite Element Method. III, IV, VIII, X–XII, 19, 22, 32–35, 37, 47–49, 67

GFRP Glass Fibre Reinforced Polymer. 24

JSON JavaScript Object Notation. XI, XIII, 15, 18–20, 35, 58, 60, 62 MALE Medium Altitude Long Endurance. 33

MDO Multi-disciplinary Design Optimization. IX, 2, 15, 40 MTOM Maximum Take-Off Mass. 33, 38

OOP Object-Oriented Programming. 14, 15, 19, Glossary: OOP UAV Unmanned Aerial Vehicle. III, 2, 33

UID Unique Identifier. 19, 36, 58, 59, 62

VLM Vortex-lattice Method. III, IV, VII, VIII, XI, XII, 6–8, 10, 12, 13, 15, 16, 19, 21, 22, 26–37, 39, 40, 42, 55, 56

XML Extensible Markup Language. XI, 2, 15, 40

IX

AeroFrame A software tool that couples a CFD and a structure solver for partitioned aeroelastic analyses. The tool coordinates the solvers and enables a data transfer between them. The tool has been written as part of this thesis work. 14, 20, 26, 27, 33, 37, 38, 40

API An application programming interface is a software interface which enables a simpler interaction with the software. The interaction consists of data input and output (of some kind). 14

Elastic axis The elastic axis is a reference line along a beam’s lengthwise direction. If transverse loads were applied along this line they would cause only bending but no torsion at all positions along the beam. However, such a line does not exist except for a few special load cases, and then its location varies with the applied load [1]. Instead, the elastic axis is usually defined as the line connecting the elastic centres of the beam, which is not equivalent to the first definition but a more practical and often sufficient approximation. The elastic centre is defined as the cross section point midway between shear centre (flexural centre) and tor- sional centre (centre of twist). The shear centre is defined as the cross section point at which a transverse load causes bending but no twist of the section. The computation of this point only requires knowledge about the local cross section geometry (cf. [2]). The torsional centre is the cross section point which remains stationary if a torsional load is applied to a straight beam with this cross section.

The shear centre and torsional centre may be identical which is a typical assump- tion. More elaborate definitions and further comments on the elastic axis, the elastic centre, the shear centre and the torsional centre are given by Kuhn [1] and by Young et al. [3]. See also Megson [2] for the computation of the shear centre and torsional centre (centre of twist) of open and closed cross sections. 4, 9, 11, 19, 23, 41, 42, 51, 65

FramAT A FEM software tool for generic 3D beam analyses. The tool has been writ- ten as part of this thesis work. VIII, 14, 18, 20, 23, 31, 35–37, 40, 47, 58–60, 62, 67–69

Loose coupling A design of a computational model in which individual components of this model (modules) have little or no knowledge of each other. III, IV, 3, 14, 40

Modularity In a modular software design large, complex systems are divided into smaller, self-contained pieces (modules). Each separate, independent module has specific functionality. Modules may be interchangeable if they expose the same external interface. Modularity may be implemented on some top-level, but modules themselves may also be broken down into smaller submodules. III, IV, 3, 14, 21, 40, 42

OOP A programming paradigm based on the core idea that systems, which are to be modelled, can be broken down into smaller logical subsystems (objects). Objects

X

down into smaller components of this system. 14

PyTornado An implementation of the vortex-lattice method (VLM). The software tool has been further developed during this thesis work. VIII, 14, 15, 17, 18, 23, 32, 34, 37, 40, 55–57, 60

Serialisation The process of translating data structures into a format that can be stored, e. g. as a file. JSON and XML are examples for text-based serialisation formats. The opposite process of translating data structures from, for instance, a JSON file is called deserialisation. 2, 15, 60

State Information that a software internally keeps track of during program execu- tion. During execution, the state may be modified as a result of some events.

Software designed in this way is also described as being stateful (opposite to being stateless). 21

Nomenclature

Frequently used symbols and indices are listed below. Other symbols and indices that are less frequently used, or that have different meanings in different contexts are defined where used. Matrices and vectors are generally indicated by boldface type (exceptions in tables). Elements of matrices or vectors are of normal (roman) type (e. g. M = ( M

, M

, M

)

). Lengths of vectors (Euclidean norm) are also indicated by normal type (e. g. ∥

∥

= V

). Indices referring to axes of a coordinate systems are generally denoted by lower case symbols x, y and z (for better readability) independent of whether they refer to a global system (X, Y, Z) or a local system (x, y, z).

Latin Symbols

A Area m

b

Reference span for aerodynamic coefficients m

c

Reference chord for aerodynamic coefficients m

e_X

, e

, e

Standard basis vectors (global system) m

E Young’s modulus N / m

F

Force vector or FEM vector of nodal loads *

g Gravitational acceleration m / s

G Shear modulus N / m

I

Second moment of area about a local y-coordinate m

I

Second moment of area about a local z-coordinate m

XI

K

FEM stiffness matrix *

L Length m

m Mass kg

m

Distributed moment vector N m / m

M

Moment vector or FEM mass matrix *

N Number of elements (FEM) or number of panels (VLM) 1

N

FEM shape function matrix *

P

Point in space m

q

Distributed force vector N / m

r

Position vector m

S

Reference area for aerodynamic coefficients m

t Time s

T

FEM transformation matrix 1

U

FEM vector of nodal deformations *

V_∞

Free stream velocity vector m / s

x, y, z Cartesian coordinates (local system) m

X, Y, Z Cartesian coordinates (global system) m

Greek Symbols

α Angle of attack rad

β Sideslip angle rad

Γ Circulation m

/ s

ε Elastic strain m / m

η Local, relative coordinate 1

Θ Twist or rotation angle rad

ξ Local, relative coordinate 1

ϱ Density kg / m

XII

Subscripts

accel Acceleration

air Air

conc Concentrated

def Deformed

dist Distributed

e Element

glob Global system i, j Index variables loc Local system nn Nearest node proj Projected ref Reference

rot Rotated

tip Beam tip

Other Symbols

f

(... )

Internal loads (internal forces

e

e function ( u ˜ ≈ u) in the Galerkin method

(... )T

Transpose

(... )′

=

Derivative by x

(... )

¨ =

Derivative by t

*) Unit is not uniquely defined.

Other conventions

• If not stated otherwise, the reference air density for airspeeds is assumed to be 1 . 225 kg / m

(standard density at sea level [4]).

• JSON files have been printed without quotation marks for better readability.

XIII

1 Introduction

1.1 Background and motivation

Aircraft structures are built upon lightweight design principles resulting in not only light but usually quite flexible structures. Flexibility may become particularly appar- ent in case of long and slender components such as wings. When subjected to aero- dynamic loads, such structures are prone to exhibit significant deformations during flight (fig. 1.1). Since the aerodynamic loads themselves depend on structural geo- metry, these loads will change due to the deformations. The interaction between the fluid and structure is also known as aeroelasticity [5].

Figure 1.1: The Nimeta glider has a span of 30.9 m and a wing aspect ratio of 51.5 [6]. The wings are particularly flexible and deform visibly during flight. (Photo by J. Lehle / CC BY 3.0)

Aeroelasticity needs to be taken into account when designing and certifying aircraft.

Inadequate design may cause static or dynamic stability problems, potentially caus- ing structural failure or severe reduction of controllability. Certification specifications such as the CS-25 issued by the EASA [7] therefore define strict requirements to pre- vent undesired aeroelastic phenomena in order to ensure safe flight within the flight envelope.

It is of importance to take aeroelastic considerations into account in an early stage of

the aircraft design process, as late changes and retrofits may become very costly [8,

9, 10]. Computational models can provide tools to perform aeroelastic analyses and

thereby help to identify and improve problematic designs in an early development

stage. If aeroelastic analysis tools are built from separate aerodynamic and structural

models, which is a fairly common approach, there is an inherent necessity to exchange

deformation and load data between these models. One practical issue that can arise in

such situations is a mismatch of data formats and tool interfaces. Different software

tools often require different input data formats and may have different interfaces.

To reduce this data-transfer-and-translation barrier, the German Aerospace Center (DLR) has since 2005 developed a common file format for the inherently multidiscip- linary aircraft design process. This open-standard format, called Common Parametric Aircraft Configuration Schema (CPACS), is specifically aimed to be used in conceptual and preliminary aircraft design studies [11, 12, 13]. The CPACS format itself is based on XML, a widely used textual serialisation format. CPACS can store, for instance, the aircraft geometry, detailed structural definitions or engine performance data. The idea is that this highly elaborate format can be used by all tools involved in the air- craft design process which significantly reduces the complexity of tool interfaces and translation of different data formats. Using a standardised format may also help to ensure overall consistency and reduce the risk of misinterpreting data.

CPACS was extensively used in the EU funded research project AGILE (Aircraft 3rd Generation MDO for Innovative Collaboration of Heterogeneous Teams of Experts) from 2015 to 2018 [14]. A main objective of the AGILE project was the development of new and improved multi-disciplinary design optimization (MDO) processes, which can contribute to reductions of time and costs in aircraft development, especially for novel, more efficient aircraft concepts. The AGILE project included numerous aircraft- related MDO case studies, in particular several aero-structural studies [15, 10, 16, 9]. Overall, 19 industry, research and academia partners from Europe, Canada and Russia were involved in various collaborative MDO design tasks [14, 15], making a commonly understood data format even more important. The general viability of the CPACS format was successfully demonstrated during the AGILE project.

Figure 1.2: Aerodynamic loads and CFD mesh deformation of the OptiMale [15]

One MDO study conducted in the AGILE project was a structure mass optimisation of

the OptiMale unmanned aerial vehicle (UAV) [17, 15]. The main objective of this study

was a minimisation of structural weight while still complying with strength and stabil-

ity constraints, as well as top-level mission requirements. This optimisation challenge

was solved using a collaborative aeroelastic analysis framework involving multiple

companies, namely Airbus, Airinnova, CFSE and DLR. Starting from an undeformed

state, aerodynamic loads were computed and subsequently mapped onto a structure

model from which deformations could be obtained. The aerodynamic model was up-

dated to reflect these deformations (fig. 1.2), which in turn changed loads computed

with the CFD analysis. The exchange of loads and deformations was implemented

in an aeroelastic loop which was run until a structural equilibrium was found. This

report will make use of a very similar concept, though based on significantly simpler

low-fidelity models. In section 4.3 of this report, the OptiMale will be studied with the

aero-structural framework developed in this thesis project, and results will be com-

pared with higher-fidelity results from AGILE.

1.2 Objectives and scope

The main objective of this work is to describe and demonstrate a basic implement- ation of a loosely coupled, modular framework for aeroelastic analyses. The overall model is based on a so-called partitioned approach which means that separate solvers for the aerodynamic and structural analyses are being used. This particular model- ling approach, however, does involve practical challenges concerning to the exchange and interpolation of data between the separate solvers. Based on the chosen physical models some of these issues will be highlighted and discussed. A secondary object- ive related to the issue of consistent data exchange is to demonstrate the use of the CPACS format.

The presented model is restricted to analyses of static aeroelastic phenomena, such as static wing deformation, divergence and control reversal. The framework is intended to be used for computationally inexpensive low-fidelity analyses in conceptual aircraft design. The framework allows to produce quick estimates of wing loads and deform- ations, and is thus a good starting point for more detailed design work. While this report deals with lower-fidelity models and quasi-static analyses, the general concepts discussed can be extended to higher-fidelity models and dynamic analyses too.

The theoretical background for the components of the aeroelastic framework will be

summarised in section 2. General assumptions and limitations inherent to the chosen

models will be outlined. Section 3 describes the software implementation of the frame-

work and its modules. Case studies and numerical experiments will be presented in

section 4.

2 Theoretical background

2.1 Aeroelasticity

Aeroelastic phenomena can classified to be either of dynamic or static nature, depend- ing on whether they are time-dependent or not. In either case, the structural response to the aerodynamic loads can be stable or unstable [2]. Dynamic aeroelastic phenom- ena, such as classical flutter, involve unsteady aerodynamic loads and structural dy- namics. The unsteady loads continuously feed energy into a structure – a wing for instance – causing the mass distributed along the wing span to be accelerated in an oscillating motion. Hence, not only elastic properties but also the mass distribution influences the response of the wing. This report will, however, only focus on static aer- oelasticity. Model extensions which allow dynamic analyses will be briefly discussed in section 6.

2.1.1 Static deformations and divergence

Even in quasi-steady flight situations aircraft structures, only having a finite stiffness, will deform. In case of sufficiently stiff structures the deformation will converge and form a quasi-static equilibrium with the aerodynamic forces. This situation is usually found during steady level flight but also during quasi-steady flight manoeuvres such as steady climbs and descents, pull-up manoeuvres or steady turns in which the aero- dynamic loads can be considered to be steady [2, 18]. In a quasi-static equilibrium, the aircraft will take on a so-called flight shape, the deformed shape during flight. As a result of the changed aircraft geometry, aerodynamic parameters, such as lift or drag, do not only depend on the incoming flow but also on the flight shape [8, 5]. In fact, wings of large transport aircraft may be designed, optimised and built in such a way, that, at cruise, when deformed by aerodynamic and gravitational loads, they take on a minimum drag configuration [5, 19].

Aeroelastic divergence is a static instability that may occur if a lifting surface does not have the required stiffness to sustain the aerodynamic loads. At divergence the de- formation of the wing is continuously amplified as the aerodynamic loading increases which contributes to even further structural deformation. When restoring elastic forces are overcome, a point of structural failure is reached. Main factors influencing this phenomenon are the dynamic pressure and wing torsional stiffness. Forward swept wings are particularly sensitive to divergence as the forward sweep introduces a disadvantageous coupling between bending and torsion deformations [5].

2.1.2 Control efficiency and reversal

When flight control surfaces such as ailerons are being deflected, they usually intro- duce forces into the wing at some distance away from its elastic axis. These torsional loads can cause the wing to twist which decreases the effectiveness of the control sur- face deflection. For instance, ailerons mounted on a real, elastic wing will produce smaller roll moments than ailerons mounted on an idealised, perfectly stiff structure.

This adverse effect of fluid-structure interaction, also known as loss of control effi-

ciency, typically becomes more pronounced with increasing airspeeds due to accompa-

nying higher torsional loads. Insufficiently stiff structures can encounter an instabil-

ity called control reversal, in which the lifting surfaces have deformed so much that

control surface deflections produce reversed effects, i. e. moments opposite to their in- tended direction.

2.2 Approach and requirements for the aeroelastic framework

The aircraft design process may be divided into three major phases, namely concep- tual design, preliminary design and detail design [20]. In conceptual design the main focus is to determine the general configuration, size, weight and performance based on some top-level requirements. The aero-structural model presented in this work is aimed to be used for conceptual aircraft design. The overall objective is to set up a computational framework which allows to easily obtain estimates of aircraft loads and deformations. It is of special interest to analyse novel aircraft configurations, for instance boxwing configurations, for which there exist little or no empirical data. This does require a high generality for modelling aircraft structures extending in 3D space.

This work will be restricted to static aeroelastic analyses in low subsonic airflows. Un- steady aerodynamic loads, effects of compressibility and structural dynamics are not taken into account. In the conceptual design stage, characterised by many unknowns, low-fidelity models which indicate trends correctly with reasonable accuracy are usu- ally sufficient. A quick setup and short computation times are desirable in order to easily compare and benchmark various design concepts [21].

Generally, one can distinguish between monolithic and partitioned solution methods for aeroelastic simulations. The so-called monolithic models formulate the governing equations including both fluid- and structural dynamics in a single system of equa- tions, while partitioned methods are based on separate models for the structure and fluid domains. Felippa et al. [22] give an extensive overview about the terminology, general concepts, techniques, merits and examples for the partitioned treatment of coupled mechanical systems. The terms monolithic and partitioned are used accord- ing to the definitions in [22].

CFD Loads Deformations Structure

Figure 2.1: Basic aeroelastic coupling concept in a partitioned approach. Separate solvers for the aerodynamic and structural analyses operate on a common aircraft model. A staggered solution procedure is used in which loads and deformations are updated alternatingly.

One major advantage of the partitioned approach is that for each domain (fluid and

structure) well established modelling-, discretisation- and solution methods can be

applied [8, 23, 19]. The model presented in this work will be based on a partitioned

solution method. In this approach, the separated fluid and structure domains have to

be coupled in a suitable way, and a data exchange for loads and deformations has to be

established. The general coupling concept is illustrated in fig. 2.1. The fluid and the

structure domains are updated with an alternating scheme based on new estimates

of deformations and loads, respectively. This general form of coupling is also known

as staggering [22], though time is considered to be fixed in this particular case. If a static equilibrium position exists, the load and displacement fields may converge in this iterative procedure.

The presented aero-structural model is based on the vortex-lattice method (VLM) for the fluid domain and Euler-Bernoulli beam theory for the structure domain. The gen- eral suitability, theoretical background, model limitations and the conceptual coup- ling method will be described in the following sections. Further requirements pertin- ent to the actual implementation of the software will be discussed in section 3.1.

2.3 Aerodynamic modelling (vortex-lattice method)

The vortex-lattice method (VLM) is the simplest general 3D potential flow model [24].

Since the model is relatively easy to setup and has low computational cost, it is a widely used tool in conceptual aircraft design [24, 25]. The theory is well covered in literat- ure, for instance by Drela [24], Katz and Plotkin [26] and Bertin and Cummings [27].

Only a basic summary will be given here. The VLM makes use of many assumptions and simplifications. Most significantly, quasi-steady, potential flow is assumed. A po- tential flow implies an inviscid and irrotational flow field. Viscous, dissipative effects such as turbulence or the boundary layer cannot be resolved. Only induced drag can be computed. The flow field is further assumed to incompressible, which is a reason- able simplification in low subsonic airflow [24]. The lifting surfaces are assumed to be thin. Small angles of attack and sideslip angles are assumed. In general, large-angle effects like stall cannot be modelled.

Z

Y

X Control points

V(r) Field point r

Lifting surface

Horseshoe vortices V_∞

Figure 2.2: Concept of the VLM discretisation (adapted from [24]). The shown lifting surface is divided into nine panels. A three-legged horseshoe vortex is placed on each panel.

The VLM is a numerical solution method for the general 3D lifting surface problem [24].

The lifting surface problem is a model in which lifting surfaces and the wake are rep- resented by infinitely thin vortex sheets with a sheet strength distribution [24]. The potential flow field around the lifting surfaces is influenced by these vortex sheets.

The basic approach applied in the VLM is to discretise the continuous vortex sheet

strength distribution. To achieve this, each lifting surface (modelled as a flat sheet)

is divided into small quadrilateral panels (fig. 2.2), together constituting a mesh (or

lattice). The distributed vortex distribution is then lumped into so-called horseshoe vortices. On each panel a horseshoe vortex with three legs is placed. A bound leg, modelling the lifting properties, lies at the panel quarter chord line. Two trailing legs, which model the influence of the wake, extend from the ends of the bound leg parallel to the freestream direction to downstream infinity.

Each leg of the same horseshoe vortex i has the same circulation strength Γ

(fig. 2.3).

Z

Y

X

Fi

Vi

si

Γi

Γi

Γi

Panel rb

ra V_∞

Trailing leg

Bound leg

Figure 2.3: Horseshoe vortex geometry and variables involved in the Kutta-Joukowski theorem (adapted from [24])

Every horseshoe vortex will induce some velocity contribution at any field point r.

This induced velocity is computed using the Biot-Savart law, analogous to an electric current inducing a magnetic field. The total flow field velocity V (

) relative to the aircraft’s body-fixed axes at a position r is a result of the freestream velocity, the air- craft’s rotation rate and the contribution of all horseshoe vortices. This concept is used to set up a system of linear equations by applying a flow tangency boundary condition.

At control points (also known as collocation points) located at r

on a panel i (centre of the panel three-quarter chord line), the flow in the direction of the panel’s normal vector

is prescribed to be zero, V (

)

= 0, where V (

) is the velocity vector at the control point given as a results of induced and freestream velocity as well as the aircraft’s rotation rate. The boundary condition implies that there cannot be airflow through the wing surface at this point.

The equations enforcing the flow tangency boundary conditions on N panels can be assembled into a linear system of equations for the unknown vortex strength Γ

. The matrix formulation of the N × N system has the form



 

 



a

a

. . . a

a

. .. .. .

.. . . .. .. . a

a

. . . a



 

 





 

  Γ

Γ

.. . Γ



 

  =



 

  b

b

.. . b



 

  ^(2.1)

1 The horseshoe vortex geometry is not consistently defined in literature and in different VLM imple- mentations. Katz and Plotkin [26] include the most elaborate explanation and mention three different variations. Differently defined horseshoe vortex geometries can cause differences in between results provided by different VLM implementations.

2 The normal vector does not have to be the geometric normal of the panel. It may be rotated in order to model control surface deflections or aerofoils [24] (see also section 3.2.3).

where the a

denote the elements of a so-called aerodynamic influence coefficient mat- rix, and Γ

is the circulation of a panel i. The b

on the right-hand side are functions of the freestream velocity, the aircraft rotation rate and the panel normal vectors.

The a

and b

only depend on the geometry and the flight state and are hence known beforehand. Equation (2.1) can be solved to obtain the horseshoe vortex strength Γ

(circulation) for each panel i. With known circulation, the aerodynamic force F

acting at the quarter chord midpoint of a panel can be computed using the Kutta-Joukowski theorem,

F_i

= ϱ

×

Γ

(2.2)

where ϱ

is the air density, V

the velocity relative to the horseshoe vortex bound leg midpoint, and s

=

−

is the vector of the bound leg (see fig. 2.3). Based on the computed force distribution, aerodynamic parameters such as lift and (induced) drag coefficients can be obtained. For all further aeroelastic analyses, the panel forces are considered to be the main result of the VLM. Practically, these forces represent a discretised formulation of the continuous pressure distribution over the lifting sur- faces.

2.4 Structural modelling (Euler-Bernoulli beam theory)

Aircraft are often made up of long and slender structures such as the wings and the fuselage. In the scope of low-fidelity analyses such structures may reasonably be ap- proximated using beam theory, which is the approach taken here. Practically, the beam model is solved with a finite element (FE) formulation. As beams can be de- scribed with relatively few degrees of freedom (d.o.f.), they are computationally fairly inexpensive. Beam theory allows to model 3D geometries. It is a also a widely used tool for low- to medium-fidelity aeroelastic analyses [21, 19, 23, 5, 9], also in combina- tion with the VLM [28, 25, 10]. For the presented work, Euler-Bernoulli beam theory is used. Dorbath et al. [21] performed an extensive study comparing an Euler-Bernoulli FE beam model with a FE shell model in the scope of preliminary wing design stud- ies. Deformations of wings with realistic geometries did generally not differ more than ± ⁵ % . Observed differences were mainly attributed to shear deformations as well as to effects of restrained warping. Both effects are reflected by the FE shell model but not by the standard Euler-Bernoulli model.

Euler-Bernoulli beam theory, also known as classical beam theory, makes use of signi-

ficant simplifications and assumptions. Most notably, it imposes several constraints

on representable deformations of the beam structure. It is assumed that the cross sec-

tion is infinitely rigid in its own plane, and that – after deformation – the cross section

remains plane and normal to the deformed beam axis (fundamental Euler-Bernoulli

assumptions) [29]. As a consequence, shear deformations cannot be resolved. In ad-

dition, warping deformations are neglected. Furthermore, small deformations are as-

sumed. This allows the linearisation of the governing differential equations. Isotropic

material is assumed, and the relation between stresses and strains is modelled using

Hooke’s law, i. e. using a linear-elastic relation. For materials relevant in aircraft man-

ufacturing such as fibre composites or aluminium this assumption implies a restric-

tion to small strains. The governing equations applied in this work have been derived

from an equilibrium consideration in the undeformed state. In particular, this implies

that normal forces do not contribute to any bending loads. Thus, global buckling phe- nomena cannot be represented with these equations. A more extensive overview and discussion about explicit and implicit assumptions can be found in [30, 29].

z

y x

Fe+d_Fe Me +d_Me m

dx

− e^M

−eF

q

r

Figure 2.4: Undeformed beam element in equilibrium. All loads are thought to act on the elastic axis (dashed line). The tilde symbols indicates internal loads rather than externally applied loads.

The Euler-Bernoulli beam model is well covered in literature (see for instance [30, 29, 2, 3]), thus only a brief overview will be given here. The governing equations can be derived from the equilibrium state of an infinitesimal beam element with length dx in its undeformed state (fig. 2.4). The internal loads, the force

e

e may differ between the left and right cut. Externally applied distributed forces q (di- mension force per length) and distributed moments m (dimension moment per length) are assumed to be constant over the beam element length dx. The force equilibrium stated in the local coordinate system yields

−e

+

(

e + d e

)

+

=

⇒ _dx ^d



  e F

e F

F e



  +



 ^q q

q



 =

(2.3)

where F e

is the axial force component, F e

and F e

are shear force components. The moment equilibrium yields

− e

+

(

e + d

e )

+

−

× e

+

2 ×

=

(2.4)

where r = ( − ^dx , 0 , 0 )

is the lever arm for the force contributions. As the term r dx is small of higher order, the moment contribution from the distributed force is henceforth neglected. The shear forces F e

and F e

in eq. (2.4) can be eliminated after a derivation of eq. (2.4) by x and a substitution of the shear loads using eq. (2.3).

d dx



  M e

M e

M e



  +



 ^m m

m



 +



 −e ^{0 F}

e F



 =

⇒ ^d

dx



  M e

M e

M e



  + _dx ^d



 ^m m

m



 +



 q ⁰

− ^q



 =

(2.5)

The internal loads

e

e

cross section area. Using kinematic relations based on the geometry of the deformed

beam, normal and shear strains can be derived. As mentioned above, stresses and strains are connected using Hooke’s law. Together, these equations can be written as a system of linear fourth order differential equations.

d dx

(

E A du

dx

)

+ q

= 0 (2.6)

d

dx

(

E I

d

u

dx

) + ^dm

dx − ^q

= 0 (2.7)

d

dx

(

E I

d

u

dx

)

− ^dm

dx − ^q

= 0 (2.8)

d dx

(

G J dΘ

dx

)

+ m

= 0 (2.9)

The beam stiffness is described by both geometry and material parameters. The beam cross section area A, the second moments of area about the local y- and z-axes, de- noted I

and I

, as well as the torsion constant J are defined by the local cross section geometry. The Young’s modulus E and the shear modulus G are material properties.

The beam deformation is described by displacements u

, u

and u

, and by rotations Θ

, Θ

and Θ

, where the subscripts x, y and z refer to the local coordinate system.

Equations (2.6) to (2.9) are fully uncoupled differential equations which is a result of the simplifying assumptions mentioned above. In the preceding development it was also assumed that the y-and z-axes are principal centroidal axes of the cross section.

This implies that the bending about the y- and z-axes is uncoupled (no skew bending).

It is further noted that rotations Θ

and Θ

, assuming small-angle approximations, are related through basic kinematic relations.

Θ

= − ^du _dx

^{and Θ}

= ^du

dx (2.10)

Practically, the governing equations are solved using a FE formulation which is well suited for computational analyses. Details on the matrix formulation are covered in appendix A.1.

2.4.1 Model-induced mesh distortions

At this point, another limitation of the coupled aero-structural model should be noted, as it is a direct consequence of using the linear Euler-Bernoulli model. Consider a cantilever beam loaded with a point force at its tip (fig. 2.5). In the Euler-Bernoulli model the deformed tip point will always remain in the same plane, independent of the magnitude of the deflection. This implies that the beam length is increased, simply as a result of the mathematical model. When such a stretched beam deformation is transformed to a VLM mesh, the mesh surface area will increase accordingly which implies non-physical changes of the aerodynamic loads.

3 A more elaborate derivation of the governing equations can be found in [30, 29].

s

L

uz(x)

E, Iy

x z

uz(L) Fz

y

Figure 2.5: Undeformed and deformed state of a cantilever beam loaded with a point force Fz

at its tip. The undeformed beam has a length L while the deformed beam has a length s.

Example To get an idea of the relevance of the surface area distortion, it may be illustrative to estimate the error based on the simple cantilever beam. With appropri- ate boundary conditions, an analytical expression for the deflection u

of a cantilever beam loaded with a point force F

at its tip can be derived from eq. (2.8) as

u

= ^F

6 E I

( 3 L x

− ^x

)

= ^δ

2 L

( 3 L x

− ^x

)

with δ

:= ^u

( L )

L (2.11)

The deformed beam shape may be formulated in terms of a relative tip deflection δ

which is a simple measure for the overall deformation. Assuming that the beam (or the corresponding wing surface represented by the beam) has a unit width, the relative area increase is given as

A − ^A

A

= ^s − ^L

L with s =

∫

√ 1 +

( du

dx

)

dx (2.12)

where A

and A are the undeformed and deformed planform areas, respectively, and s is the arc length of the deformed elastic axis defined by u

. The integral in eq. (2.12) can be solved numerically. Table 2.1 summarises the estimated area distortion for different tip deflections.

Table 2.1: Estimated area distortion for different relative tip deflections

δ_tip

0.1 0.2 0.3 0.4 0.5

A−A0

A0

0 . 60 % 2 . 36 % 5 . 21 % 9 . 03 % 13 . 69 %

For instance, a rectangular wing with a 10 m span and a 1 m tip deflection will in-

crease its surface area by approximately 0 . 60 % (given a deformation shape similar

to eq. (2.11)). Larger errors are to be expected for larger deformations. This model-

induced mesh distortion further emphasizes that additional model restrictions can

arise due to the coupling of different models, and, in particular, that the small-defor-

mation restriction is not only pertinent to the Euler-Bernoulli model in isolation but to

the entire coupled aeroelastic model as well. Whether the described model-induced dis-

tortions can be accepted must be decided based on overall objectives of the simulation

and the required accuracy. With this in mind, it may for instance not be very sensible

to try to accurately predict changes of aerodynamic properties (e. g. lift or drag coeffi-

cient) with a change of deformation since there would be an unknown contribution to

this change from the mesh area distortion. The described distortions could be avoided

altogether using non-linear beam models (see also section 6). In the results presented

in this work, the relative tip deflection δ

is rather small, thus the resulting overall error is assumed to be small. One study presented by Seywald [28] comparing linear and non-linear beam model deformations also suggest that the error may not affect overall conclusions too much.

2.5 Aeroelastic coupling

The conceptual idea of connecting the chosen CFD and structure models is illustrated in fig. 2.6. While the VLM uses a surface mesh, the FE beam uses a line mesh. It is obvious that there is a mismatch in the discretisations of these two models. Thus, there has to be a scheme which transfers and interpolates loads computed by the VLM and deformations computed by the FE beam model.

FE mesh

VLM mesh Wing

Tail plane

Fuselage

Figure 2.6: Concept of joining the VLM with the FE beam model. Aircraft components use different meshes for the aerodynamics and the structure analyses. For some components (e. g.

the fuselage) there may not be a corresponding CFD mesh.

The coupling of the separate fluid and structure solvers constitutes the central as- pect of the partitioned solution approach. For a meaningful physical representation of the whole fluid-structure system, coupling boundary conditions in accordance with Newton’s 3rd law have to be fulfilled [8]. The boundary conditions demand that both spatial and temporal deformations and loads must be equal at the fluid-structure in- terface. In other words, a suitable transfer must ensure that loads and deformations are equal at the boundary of the fluid and structure domain within a same time step.

As this work is restricted to static analyses, temporal coupling boundary conditions

do not apply. However, the spatial boundary conditions do apply and require that all

aerodynamic loads must be transferred from the fluid to the structure domain, and

that the work done by the fluid domain must equal the work done by the structure

domain. The implementation of the required load and deformation transfers will be

described in section 3.

Braun [8] gives a more detailed overview about the coupling boundary conditions for the general case of dynamic aeroelastic analyses. It should also be noted that vari- ous practical issues in the implementation of partitioned, modular, multi-fidelity, dy- namic aeroelastic analyses have been covered in-depth by Reimer et al. [23], Braun [8]

and Boucke [19]. Amongst others, issues such as mismatched discretisations, poten- tial problems with load mappings and mesh deformations methods are being covered.

Many of the issues and solution methods presented by these authors do apply in this

work as well. As their presented model also deals with dynamic analyses, highly mod-

ular systems and CFD volume meshes, the methods are somewhat more elaborate

than needed to couple the models chosen for this work. The stated issues will only be

dealt with on a level appropriate to the VLM and FE beam model.

3 Implementation

This section gives an overview about the practical implementation of the aero-structur- al framework including the CFD and structure modules. The basic theories for these tools have been covered in section 2. The software packages discussed in this section (PyTornado, FramAT and AeroFrame) have been developed or extended as part of this thesis work at Airinnova AB.

3.1 Technical requirements

A few requirements relevant to model choices have already been discussed in sec- tion 2.2. Further requirements pertinent to the practical implementation of the frame- work shall be mentioned here.

3.1.1 Loose coupling

The CFD and the structure tools are developed as stand-alone tools which have no in- terdependencies. In other words, the CFD tool knows nothing about the structure tool and vice versa. The actual core of the aeroelasticity model should just be a framework which couples the two modules, but – it too – knows no implementation details about the underlying CFD and structure modules.

3.1.2 Modularity

The framework shall further have a modular design. This concept, closely related to the idea of loose coupling, implies that the CFD and structure tools should be independ- ent and interchangeable. In other words, these tools are “plugged-in” as a modules, but as long as any other tool exposes the same interface it can be used instead.

3.1.3 Programming language and paradigm

Python was chosen as the main programming language. Some parts of the CFD tool were also written in C++ and integrated using Python’s native application program- ming interface (API) for C++. Python is a widely used high-level programming lan- guage. It is known for being a very readable and flexible language. Python also sup- ports object-oriented programming (OOP) which all developed tools make use of. OOP is a programming paradigm which can help to model complex systems by breaking them down into smaller objects (subsystems), each of which has specific properties and capabilities (see also glossary).

4 Notice that the term loose coupling may not be defined consistently in the context of partitioned aer- oelastic analyses ([19, 8]). In this work, the term is to be understood from a software design perspective (see also glossary). Loose coupling is not thought to be equivalent with partitioning (cf. section 2.2).

Instead, a loosely coupled software implementation can be consequently realised as a results of parti- tioning the aeroelastic analysis.

3.1.4 Serialisation and data formats

Another aim was to make use of the CPACS format developed by DLR. CPACS is an open-standard format aimed specifically to be used in conceptual and preliminary air- craft design [12, 31]. As demonstrated in the AGILE project (see section 1.1), the format can be used for MDO studies of different fidelity levels. The basic concept of CPACS is that it allows to store complex object definitions, such as a complete air- craft, by breaking it down into smaller components using a hierarchical system. For instance, an aircraft has any number of wings. Each of these wings may be further divided into segments, and within such a segment there may be controls surfaces.

CPACS does not restrict itself to only store the aircraft geometry, but it can also store, for instance, engine performance data or flight mission data. This concept is realised using the standardised serialisation format XML which can be easily parsed by com- puters. To simplify the interaction with CPACS and XML, the DLR provides two ad- ditional software libraries. TIXI [32] is a library which facilitates the parsing of XML and the extraction of required data. A second library, namely TiGL [33, 34], provides additional geometric functions which, for instance, allow to interpolate points on a wing.

While CPACS is a proven concept and tailored for conceptual aircraft design, it is also very general and elaborate. The data relevant for a specific application may be very deeply nested in the XML structure. Thus, even to store simpler models, there can be a lot of overhead. This is not a problem from a technical perspective. However, it makes it impractical to write by humans. In order to more easily define models relevant in this work, another format was used, namely JSON. Compared to XML, the standardised JSON format is generally considered to be more human-readable. JSON was used as a tool native format, essentially storing same data as CPACS though in a significantly flatter and simpler structure.

3.2 Implementation of the aerodynamics model

The VLM discussed in section 2.3 was implemented in a tool called PyTornado. This program is loosely based on Tornado by Melin [35], a VLM implementation written in Matlab. The PyTornado tool already provided functionality to perform basic VLM analyses using CPACS [36]. The tool was further extended by the author of this thesis.

Most notably, modelling of control surfaces and of aerofoils was added. In addition, an API for mesh transformations was added, as well as model import from JSON files.

3.2.1 General approach

The VLM is first and foremost relying on a geometric definition of the aircraft. Based

on the OOP paradigm, the aircraft was modelled by breaking it down into smaller

geometric components. An aircraft has any number of lifting surfaces (wings), which

may be divided into separate wing segments with variable aerofoil geometries. On each

wing there can be leading and trailing edge control surfaces. For reasons of consistent

panelling and modelling of wing deformations, each segment was further divided into

so-called segment strips and strip subdivisions as indicated in fig. 3.1.

Segment Segment

strip Strip

subdivisions Wing

Panels

Control surfaces

Figure 3.1: Object hierarchy of the aircraft model (aircraft and aerofoils not shown)

3.2.2 Lattice generation

A wing may have any number of leading and trailing edge control surfaces. In order generate a suitable VLM mesh, a systematic approach is required. Figure 3.2 illus- trates the general meshing algorithm. The planform geometry of an example wing is shown fig. 3.2a. It has three segments as well as one leading edge device and three trailing edge devices. In a first meshing step, spanwise subdivisions are added at each segment or control surface border (fig. 3.2b). This ensures that control surface geo- metries are correctly captured. More spanwise division are added to produce a finer spanwise discretisation (fig. 3.2c). Finally, chordwise subdivisions are added, such that each resulting quadrilateral is a panel used in the VLM analysis (fig. 3.2d).

Leading edge control Segments

Trailing edge controls a

b

d c

Panels

Figure 3.2: Basic meshing algorithm

3.2.3 Aerofoil and control surface modelling

The effect of cambered aerofoils and control surface deflections can be modelled using the same general concept of rotating panel normal vectors. The rotated normals (loc- ated at the panel control points, see section 2.3) change the flow tangency boundary condition, and thus alter the flow field accordingly.

The VLM cannot reflect the real aerofoil geometry which has some thickness with an

upper and lower surface. However, the aerofoil camber can be taken into account by

tilting panel normals according to the local normal of the camber line. This idea is

illustrated in fig. 3.3a. Control surface deflections, effectively changing the wing cam-