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Computational Modelling and Simulation of Aircraft and the Environment Computational Modelling and Simulation of Aircraft and the Environment: Platform Kinematics and Synthetic Environment Volume 1 D. J. Diston © 2009 John Wiley & Sons, Ltd. ISBN: 978-0-470-01840-8 Computational Modelling and Simulation of Aircraft and the Environment Volume 1: Platform Kinematics and Synthetic Environment Dominic J. Diston University of Manchester, UK A John Wiley and Sons, Ltd. , Publication This edition first published 2009 O 2009 John Wiley & Sons, Ltd.

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MATLABO MATLAB and any associated trademarks used in this book are the registered trademarks of The MathWorks, Inc. Library of Congress Cataloguing-in-Publication Data Diston, Dominic. Computational modelling of aircraft and the environment / Dominic Diston. v. cm. Includes bibliographical references and index. Contents: Volume 1. Platform kinematics and synthetic environment ISBN 978-0-470-01840-8 (cloth) (v. 1) 1. Airplanes—Computer simulation. 2. Airplanes—Performance—Mathematical models. 3. Aeronautics—Systems engineering—Data processing. 4. Navigation (Aeronautics)— Computer simulation. . Atmosphere—Computer simulation. 6. Gravitational fields— Computer simulation. 7. Spherical astronomy—Data processing. I. Title. TL671. 4. D57 2009 629. 13010 13—dc22 2009008333 A catalogue record for this book is available from the British Library. ISBN: 978-0-470-01840-8 Set in 9/11pt Photina by Integra Software Services Pvt. Ltd. , Pondicherry, India Printed in the UK by Antony Rowe Ltd, Chippenham, Wiltshire. IN MEMORY OF John Joseph Diston Master Mariner 1925–2000 Contents Preface Acknowledgements List of Abbreviations How To Use This Book Series Preface Chapter 1: Introduction 1. 1 1. 1. 3 1. 4 1. 5 1. 6 Computational Modelling Modelling and Simulation (M&S) Development Processes Models Meta-models Aerospace Applications 1. 6. 1 Synthetic Environment 1. 6. 2 Aerospace Vehicles Integration and Interoperability The End of the Beginning xiii xv xvii xxi xxv 1 1 2 4 8 12 14 14 17 21 22 1. 7 1. 8 Chapter 2: Platform Kinematics 2. 1 Axis Systems 2. 1. 1 Platform Axis System 2. 1. 2 Local Axis Systems 2. 1. 3 Earth-Centred Axis Systems 2. 1. 4 Orientation 2. 1. 5 Flight Axis System Changing Position and Orientation Rotating Axis Systems 2. 3. 1 Inertial and Non-inertial Frames 2. . 2 Vector Differentiation 2. 3. 3 Poisson’s Equation 25 25 25 25 27 29 39 42 45 45 45 47 vii 2. 2 2. 3 viii Contents 2. 4 2. 5 Quaternions 2. 4. 1 Method of Construction 2. 4. 2 Frame Rotation via Quaternions 2. 4. 3 Relationship between Quaternions and Euler Angles Line of Sight 49 49 52 54 56 Chapter 3: Geospatial Reference Model 3. 1 3. 2 3. 3 3. 4 3. 5 Spherical Earth Spherical Trigonometry Great Circle Navigation Rhumb Line Navigation Reference Ellipsoids 3. 5. 1 World Geodetic System (WGS84) 3. 5. 2 Geoid Approximations Coordinate Systems 3. 6. 1 Geocentric and Geodetic Latitude 3. 6. Parametric or Reduced Latitude 3. 6. 3 Cartesian Coordinates 3. 6. 4 Approximate Cartesian Coordinates 3. 6. 5 Latitude, Longitude and Altitude Navigation on an Ellipsoidal Earth 3. 7. 1 Differential Geometry 3. 7. 2 Geodesics 3. 7. 3 Geodesic Trajectory 3. 7. 4 Geodesic Length 3. 7. 5 Meridian Distances 3. 7. 6 Rhumb Lines Mapping General Principles of Map Projection 63 63 66 72 78 81 81 82 86 86 87 89 91 92 93 93 96 99 100 101 103 104 104 109 113 113 116 119 120 123 124 125 125 126 127 131 3. 6 3. 7 3. 8 3. 9 3. 10 3. 11 Mercator Projection Transverse Mercator Projection 3. 11. 1 Forward Projection 3. 1. 2 National Grid of Great Britain 3. 11. 3 Universal Transverse Mercator (UTM) Grid 3. 11. 4 Projection Geometry 3. 11. 5 Inverse Projection 3. 12 Conformal Latitude 3. 13 Polar Stereographic Projection 3. 13. 1 Basic Formulation 3. 13. 2 Universal Polar Stereographic (UPS) Projection 3. 14 Three-Dimensional Mapping 3. 15 Actual Latitudes, Longitudes and Altitudes Contents ix Chapter 4: Positional Astronomy 4. 1 4. 2 Earth and Sun Observational Reference Frames 4. 2. 1 Horizontal Frame 4. 2. 2 First Equatorial Frame 4. 2. 3 Second Equatorial Frame 4. 2. 4 Frame Transformations Measurement of Time 4. . 1 Mean Time 4. 3. 2 Diurnal Cycle 4. 3. 3 Universal Time 4. 3. 4 Time Zones 4. 3. 5 Sidereal Time 4. 3. 6 Terrestrial Time Calendars and the J2000 Reference Epoch Chronological Scale Astrometric Reference Frames 4. 6. 1 Inertial Frame 4. 6. 2 Rotating Frame 4. 6. 3 Precession 4. 6. 4 Coordinate Transformations Orbital Mechanics 4. 7. 1 Kepler’s Laws 4. 7. 2 Orbital Energy and Velocity 4. 7. 3 Anomalies and Kepler’s Equation 4. 7. 4 Orbital Elements Solar System Orbit Models 4. 8. 1 Planetary Data 4. 8. 2 Planetary Trajectories 4. 8. 3 Mean Sun and the Equation of Time GPS Orbit Models 4. . 1 GPS Almanac Model 4. 9. 2 GPS Ephemeris Model 4. 9. 3 YUMA Almanac 4. 9. 4 Two-Line Elements 4. 9. 5 GPS Constellation Night Sky 137 137 139 140 140 142 142 143 143 144 147 149 151 152 153 156 156 156 157 158 160 161 161 166 166 169 171 172 175 180 183 184 185 187 187 189 190 4. 3 4. 4 4. 5 4. 6 4. 7 4. 8 4. 9 4. 10 Chapter 5: Geopotential Fields 5. 1 Potential Fields 5. 1. 1 Gauss’s Theorem 5. 1. 2 Applications of Gauss’s Theorem 5. 1. 3 Poisson’s Equation and Laplace’s Equation 197 197 197 198 199 x Contents 5. 1. 4 5. 2 Generic Solution of Laplace’s Equation 00 202 202 202 204 205 206 208 211 213 214 215 215 215 216 218 222 224 225 226 229 235 5. 3 5. 4 Gravitation 5. 2. 1 Gravitation Attraction 5. 2. 2 Apparent Gravity: Spherical Earth 5. 2. 3 Apparent Gravity: WGS84 Ellipsoid 5. 2. 4 Gravitational Moments 5. 2. 5 Earth Gravitational Model (EGM96) 5. 2. 6 MacCullagh’s Formula 5. 2. 7 Earth Flattening or ‘Oblateness’ 5. 2. 8 Cartesian Components of Gravity 5. 2. 9 WGS84 Gravity Formula 5. 2. 10 Geoid Geomagnetism 5. 3. 1 Earth’s Magnetic Field 5. 3. 2 Magnetic Attraction 5. 3. 3 World Magnetic Model (WMM2005) 5. 3. 4 Approximate Dipole Geopotential Computation 5. . 1 EGM96 and WMM2005 Spherical Harmonics 5. 4. 2 Recurrence Formulae 5. 4. 3 Cunningham’s Method Final Comment on Geopotential Models 5. 5 Chapter 6: Atmosphere 6. 1 6. 2 6. 3 6. 4 Overview Standard Atmosphere Models ISA Constants and Relationships Geopotential Altitude 6. 4. 1 Standard Definition 6. 4. 2 Generalised Definition Vertical Structure of the Atmosphere Pressure Altitude Reference Atmospheres Seasonal Variation Climatic Regions 6. 9. 1 MIL-HDBK-310 Classification 6. 9. 2 Koppen-Geiger Classification ? Air Density Water Vapour 6. 11. 1 Gas Constant 6. 11. 2 Humidity Weather Systems 237 37 239 239 242 242 243 245 251 253 255 255 256 256 260 261 261 263 272 6. 5 6. 6 6. 7 6. 8 6. 9 6. 10 6. 11 6. 12 Contents xi Appendix A: Introduction to MATLAB A. 1 A. 2 A. 3 A. 4 A. 5 A. 6 A. 7 A. 8 A. 9 MATLAB The MATLAB Product Family Getting Started Getting Help Where? Numbers: Variables and Literals Arithmetic Logic M-Files and Functions 279 279 280 280 281 284 286 290 293 295 298 298 298 300 A. 10 Built-in Functions A. 11 Constants A. 12 Creating Graphs A. 13 Summary of Appendix A Appendix B: Data and Functions B. 1 B. 2 Types of Data Data Type Descriptions B. 2. 1 ‘double’ B. 2. 2 ‘logical’ B. . 3 ‘char’ B. 2. 4 ‘cell’ B. 2. 5 ‘struct’ B. 2. 6 ‘function_handle’ Program Structure B. 3. 1 Syntax B. 3. 2 Conditional Execution B. 3. 3 Iterative Execution B. 3. 4 Exception Handling B. 3. 5 Omissions User-defined Functions B. 4. 1 Interfacing B. 4. 2 Generic Functions B. 4. 3 Recursive Functions B. 4. 4 Private Functions User-defined Classes Practical Implementation B. 6. 1 Naming Convention B. 6. 2 Program Architecture 303 303 305 305 306 306 307 308 310 313 313 317 319 321 322 322 322 325 326 327 327 331 331 333 B. 3 B. 4 B. 5 B. 6 xii Contents B. 6. 3 B. 6. 4 B. 7 Precedence Preferences 36 338 338 Summary of Appendix B Appendix C: Organisations C. 1 C. 2 C. 3 C. 4 C. 5 C. 6 Specialist Agencies of the United Nations International Organisations US Government Organisations UK Government Organisations European Organisations Open Projects and Consortia 339 339 340 343 345 346 347 Bibliography Index 349 353 Preface The growth in computational processing over the past thirty years has been nothing short of revolutionary. What is available now was unthinkable in 1979 when I began my career. Back then, computing was a long-winded activity involving cards, punch-tape and teleprinters.

Processing was usually performed overnight on a mainframe computer and chargeable by CPU time and memory usage. Now, computers are ubiquitous, tiny and very powerful. Internet access, portable computers, mobile phones and personal music players are taken for granted. At the same time Modelling and Simulation has found its way into popular culture, most recognisably in the form of computer games (in particular through animation and visual scene generation). This too is taken for granted, to the extent that there is far greater interest in the consumption of gaming scenarios than in the production of the underlying models and simulations.

In the professional engineering world, computational modelling has enabled a radical approach to product development, via digital prototypes and dynamic simulation. Complete systems can be represented with a small computer, providing the basis for extensive analysis and experimentation (within the limitations of the underlying models). Gradually, industry has changed its approach to this multi-discipline, through integrated development processes underpinned by databases and design tools.

The educational opportunities are widespread as this multi-disciplinary subject is assimilated into corporate culture and expectations become enshrined in corporate doctrine. At a time when major aerospace programmes are investing heavily in the skills and technologies for modelling and simulation, there is a need for a broad-based text covering the applicable mathematics and science in the key domains that are needed in order to build the new generation of simulators, i. e. incorporating environments and vehicles.

This is what motivates this particular text; it is in two volumes that cover environments and vehicles, respectively. In a very real sense, this is the text that I would have liked to have had seventeen years ago when I began to get heavily involved in diverse simulation projects. I have extensive experience across many aircraft projects, as a control engineer and then as a modelling and simulation specialist. I have written a lot of simulation code for aircraft, gas turbines, onboard systems (especially fuel systems and hydraulic systems), control systems, signal processing and various aspects of the natural environment.

All of this was undertaken for real applications, to allow synthetic aircraft to fly with synthetic systems inside a synthetic environment. The underlying physics can be as sophisticated as anyone wishes it to be. However, the art of simulation is to draw upon expert knowledge in order to form a combined perspective on a system of interest. To this end, I have concentrated on the framework of computational models that are required in order to establish a reasonably compete operational space, containing operational platforms.

Some of the mathematics is quite involved, as befits a text that can support teaching on university programmes. I hope that you find this material interesting and informative and that it will make a difference to how you see the world around you. Happy Reading! xiii Acknowledgements Terry Hillyer, who taught me maths at the Marist College in Hull from 1974 to 1976. This was the start of my love of all things mathematical. Allan Seabridge (formerly head of Flight Systems at BAE SYSTEMS), who is the editor of this book series and the instigator of this particular book. The Mathworks Inc. for their generous support via the MATLABO Book Program. Software and graphics that appear in this book were developed using MATLAB and parts of the MATLAB product family that were supplied for this purpose. Many former colleagues and friends at BAE SYSTEMS (formerly British Aerospace), for help, advice and inspiration over two and half decades. My family . . . an island of turmoil in a sea of calm. xv List of Abbreviations 2-D 3-D 6-DOF ADC ADDS ALT AOA AOS ASCC ASTM ATM AZ BIPM BGS BSC BST CAD CAE CCIR CGPM CIA CIPM COESA DCW DEC DEM DMA DMSO DTED EASA EATCHIP ECAC ECEF ECI EDC EGM96 EME2000 ENU EROS

Two-Dimensional Three-Dimensional Six Degrees-Of-Freedom Astronomical Data Center Aviation Digital Data Service Altitude Angle of Attack Angle of Sideslip Air Standardization Coordinating Committee American Society for Testing and Materials Air Traffic Management Azimuth Bureau International des Poids et Mesures British Geological Survey Bright Star Catalogue British Summer Time Computer-Aided Design Computer-Aided Engineering International Radio Consultative Committee General Conference on Weights and Measures Central Intelligence Agency International Conference on Weights and Measures Committee on Extension to the Standard Atmosphere Digital Chart of the World Declination Digital Elevation Model Defense Mapping Agency Defense Modeling and Simulation Office Digital Terrain Elevation Data European Aviation Safety Agency European Air Traffic Control Harmonisation and Integration Programme European Civil Aviation Conference Earth-Centred Earth-Fixed (Axis System) Earth-Centred Inertial (Axis System) Earth Resources Observation Systems (EROS) Data Center Earth Gravitational Model (1996) Earth’s Mean Equator and Equinox for Julian year 2000 East-North-Up (Axis System) Earth Resources Observation Systems xvii xviii List of Abbreviations ESDI ESDU ET EU EUREF FGDC FK5 GHA GIS GMC GMT GPS GRS67 GRS80 GSDI GSHHS GST GTOPO30 HA HMNAO IAG IAU IATA ICAO ICD ICRF ICSU IEC IERS Inf IP ISA ISO ITRF ITU ITU-D ITU-R ITU-T IUGG JAA JD JPL JSD KBO LAT LHA LON LST

European Spatial Data Infrastructure Engineering Sciences Data Unit Ephemeris Time European Union Reference Frame Sub-Commission for Europe, Federal Geographic Data Committee Fifth Catalogue of Fundamental Stars Greenwich Hour Angle Geographical information Systems Geometric Mean Chord Greenwich Mean Time Global Positioning System Geodetic Reference System (1967) Geodetic Reference System (1980) Global Spatial Data Infrastructure Global Self-consistent Hierarchical High-resolution Shoreline Greenwich Sidereal Time Global Topography (30-arc-minutes resolution) Hour Angle HM Nautical Almanac Office International Association of Geodesy International Astronomical Union International Air Transport Association International Civil Aviation Organisation Interface Control Document International Celestial Reference Frame International Council for Science (formerly International Council of Scientific Unions) International Electrotechnical Commission International Earth Rotation and Reference Systems Service (formerly International Earth Rotation Service) Infinity Intermediate Pressure International Standard Atmosphere International Standardisation Organisation International Terrestrial Reference Frame International Telecommunication Union International Telecommunication Union – Development International Telecommunication Union – Radiocommunication International Telecommunication Union – Telecommunications International Union of Geodesy and Geophysics Joint Aviation Authorities Julian Date Jet Propulsion Laboratory Joint Services Designation Kuyper Belt Object Latitude Local Hour Angle Longitude Local Sidereal Time List of Abbreviations xix

MJD MSL NaN NACA NASA NGA NGDC NGS NIMA NMCAs NOAA NSDI OGC OS PA PLF PR RA SEDRIS SI TAI TCB TCG TLE TMY TNO TR TT UK UN US USA USGS USNO UT UTC UTM VMAP VTP WDB-II WGS84 WMM2005 WMO WN WVS Modified Julian Date Mean Sea Level Not a Number National Advisory Committee on Aeronautics National Aeronautics and Space Administration National Geospatial-Intelligence Agency National Geophysical Data Center National Geodetic Survey National Imagery and Mapping Agency National Mapping and Cadastral Agencies National Oceanic and Atmospheric Administration National Spatial Data Infrastructure Open Geospatial Consortium (OGC) Ordnance Survey Parallactic Angle Pressure Loss Factor Pressure Ratio Right Ascension Synthetic Environment Data Representation and Interchange Standardisation ` ?

Systeme International d’Unite International Atomic Time Barycentric Coordinate Time Geocentric Coordinate Time Two-Line Element Typical Meteorological Year Trans-Neptunian Object Temperature Ratio Terrestrial Time United Kingdom United Nations United States (of America) United States of America US Geological Survey US Naval Observatory Universal Time Coordinated Universal Time [commonly thought of as GMT] Universal Transverse Mercator Vector Map Virtual Terrain Project World Data Bank II World Geodetic System (1984) World Magnetic Model (2005) World Meteorological Organisation Week Number (GPS) World Vector Shoreline How To Use This Book This is a book about applied mathematics and the physics that underpin the creation of aerospace simulations. It is a textbook.

It is the first volume of a two-volume set, which aim to provide a working manual for students and practitioners that is focused on relevant theory and its translation into computational algorithms. Volume 1 deals with the geometry and geophysics that defines the operational environment for flight around the Earth. This defines the synthetic environment and the viewpoint from a mobile platform. Volume 2 will deal with flying vehicles and their embedded systems, with a strong emphasis on equations of motion, force generation and energy transfer. It will conclude with an integrated approach to dynamic simulation. Primarily what is provided in the two volumes is a package of explanations and derivations so that, if anyone were to required to programme a simulation from scratch, they would be able to do so.

More generally, the explanations and derivations have an inherent educational value. They provide a technical overview of the multidisciplinary science that is involved in simulation and provide reference points for university courses in aerospace engineering, as well as for continuing professional development. The content herein is based on teaching materials used at The University of Manchester from 2005. Specifically, this book contains six chapters (1-6) and three appendices (A-C): 1. 2. 3. 4. 5. 6. A. B. C. Computational Modelling Platform Kinematics Geospatial Reference Model Positional Astronomy Geopotential Fields Atmosphere Introduction to MATLAB Data and Functions Organisations

Chapter 1 introduces the context and motivation for computational modelling, and establishes the underlying definitions and philosophies. It presents an overview of development processes and the role played by through-lifecycle modelling and simulation. It summarises the scope of a synthetic environment that could be assembled in order to provide a virtual Earth and identifies the main sources of relevant information and (importantly) mathematical models. There then follows a discussion of aerospace vehicles and the appropriate way of interpreting them as platforms with associated systems. Final comments relate to integration and interoperability. Collectively this sets the scene for Volume 1 and provides the link with the companion volume.

Chapter 2 develops the subject of vehicle kinematics, which defines position and orientation and their rates of change without any concern for forces and moments. Most textbooks would xxi xxii How To Use This Book bundle this together with equations of motion and with flight dynamics. However, for current purposes, it is important to have the mathematical definition of a mobile platform from which to observe the synthetic environment. Thus, if the platform trajectory and velocity profile are known, then a range of external parameters (e. g. atmospheric pressure, magnetic variation) can be calculated as the platform traverses the environment. Also, an outside-world visual scene can be generated at any instant for the observer’s eye position and the associated field of view.

Accordingly this chapter also considers the issues of establishing lines of sight and projecting them on to displays (as would be required in a flight simulator). [Note: The key elements of this chapter will be reviewed in Volume 2, as part of the Equations of Motion. ] Chapter 3 gives the geospatial reference model. This is a lengthy and wide-ranging discussion of technical principles that enable positions and trajectories to be determined around the Earth. This starts with a spherical earth model and its reference frame and then considers coordinate the methods for navigation using great circles and rhumb lines. It continues with an ellipsoidal earth model, specialised for various reference ellipsoids and datum definitions, and re-develops the navigational methods based on differential geometry.

Two-dimensional map projections are introduced, with the aid of examples, and lead into the development of the global mapping systems based on Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) grids. In addition, the National Grid of Great Britain (so-called OSGB36) is defined as an examplar for regional grid reference systems. The chapter concludes with a brief overview of three-dimensional mapping and finally, for practical purposes, some airport locations around Europe. Chapter 4 provides the main principles of positional astronomy. Again, this is quite lengthy and is needed for navigation. Thus, it follows on from the previous chapter and sets the Earth in the context of the solar system and the celestial background.

It defines reference frames for observations and it examines various concepts of ‘‘time’’ . Specifically, it considers mean and actual solar time, the diurnal cycle and the overarching chronological scale. There is a concise mathematical summary of orbital mechanics and the related geometrical transformations. Keplerian orbital elements are defined in several ways that cover solar system dynamics and Earth satellite orbits. Particular emphasis is placed on the Global Positioning System (GPS) satellite constellation because of its importance in airborne navigation. Finally there is a very brief discussion of star catalogues and the construction of maps of the night sky.

Chapter 5 deals with potential field models that describe gravitation and geomagnetism. Both are based on Gauss’ Theorem and are expressed using Laplace’s Equation. This gives rise to mathematical models that are constructed from a series of spherical harmonics, which can be extended in order to improve accuracy. While aerospace applications may vary in their requirements for accuracy, it is important to explain the basis for calculation, especially as the sophistication of this subject area is not necessarily appreciated amongst the wider engineering community. The discussion is grounded in the common concepts of gravitational and magnetic potential, with relevant simple explanations.

The chapter closes with a heavily mathematical treatment of computational methods based on Cunningham’s Method. Chapter 6 is an amalgamation of many interesting and useful aspects of atmospheric modelling, without delving into too much detail in any one aspect. The focus is on bulk properties, as these will provide the basis for calculations related to atmospheric flight (in Volume 2). The aim is to develop the relevant mathematics for standard atmosphere models and then to characterise the typical regional, seasonal and diurnal variations. There is reinforced by a reasonably detailed treatment of humidity. The closing section presents a brief discussion of weather systems. How To Use This Book xiii The technical development within this book proceeds sequentially up to Chapter 3 and splits into parallel streams thereafter, i. e. Chapter 1 Platform Kinematics Chapter 2 Computational Modelling Chapter 3 Geospatial Reference Model Chapter 4 Positional Astronomy Chapter 5 Geopotential Fields Chapter 6 Atmosphere Appendix A gives an introduction to MATLAB and the general capabilities of its programming language, with an explanation of how it accesses program files and performs arithmetic. A number of basic examples are presented in order to show an end-to-end process from data definition, through calculation and finishing with data plotting.

Appendix B opens up some of the detail of the MATLAB programming language, examining data and functions. The main data types are defined and described, as applicable to general computational tasks. Likewise, program/function structure is defined and described, with various small-scale examples, in order to illustrate the range of possibilities that available. The final sections discuss general prinicples of program architecture and practical aspects of programming. This does not do justice to the product; inevitably there is much more that could be written on this subject but a condensed overview is sufficient to underpin the technical contents of this book.

Appendix C is a collection of useful information sources, arranged by organisational category, namely Specailist Agancies of the United Nations, International Organisations, US Government Organisations, UK Government Organisations, European Organisations and finally Open Projects and Consortia. This is by no means an exhaustive list but it does provide a set of starting points for detailed research, making the best of what the internet has to offer. Each organisation is named and described by a short paragraph. The wording is essentially a ? precis of what is presented on the respective web sites. Series Preface The field of aerospace is wide ranging and covers a variety of products, disciplines and domains, not merely in engineering but in many related supporting activities.

These combine to enable the aerospace industry to produce exciting and technologically challenging products. A wealth of knowledge is contained by practitioners and professionals in the aerospace fields that is of benefit to other practitioners in the industry, and to those entering the industry from University. The Aerospace Series aims to be a practical and topical series of books aimed at engineering professionals, operators, users and allied professions such as commercial and legal executives in the aerospace industry. The range of topics is intended to be wide ranging covering design and development, manufacture, operation and support of aircraft as well as topics such as infrastructure operations, and developments in research and technology.

The intention is to provide a source of relevant information that will be of interest and benefit to all those people working in aerospace. Modelling of systems is a valuable aid to understanding system behaviours and to assisting with trade-off and selection of candidate solutions. Using models and simulations to replace physical mock-ups and test rigs also makes a significant contribution to reducing project costs and time-scales. It is now second nature for engineers to develop quite sophisticated models and simulations of their systems and to use them as the basis of debate and agreement of system behaviours, and to develop an understanding of system interactions.

This book, Computational Modelling and Simulation of Aircraft and the Environment: Volume 1 – Platform Kinematics and Synthetic Environment provides an insight into the use of models and simulations, as well as the system environment they inhabit, to provide analysis and information which is used to support the design and certification of aircraft systems. The first stage in this process is to understand how to develop a consistent understanding of the outside world in which models and simulations exist. This is followed by the development of specific system models to integrate with this synthetic world to be published as a companion volume – Aerospace Vehicles and Flight Dynamics. The result of this combination is a set of tools and techniques which produce robust and trustworthy design information. Ian Moir, Allan Seabridge and Roy Langton xxv Chapter 1 Introduction 1. 1 Computational Modelling

Computational modelling is an increasingly important activity in the development of new aerospace systems. Its importance reflects the growth in system complexity as well as the emphasis on integration and interoperability. This approach can encompass virtually every aspect of system behaviour within an operational environment. The philosophy is to apply mathematical methods and relevant sciences in order to build synthetic systems, which are expressed as equations and algorithms and supported by relevant data. These can be exercised inside a computer for the purposes of analysis, optimisation, simulation and visualisation. The term ‘system’ is ubiquitous.

It is applied to an array or network of components that perform integrated functions and exhibit dynamic behaviour. By definition, a component is viewed as the lowest unit of decomposition for a particular system and, as such, it is treated as indivisible. Components are assembled in order to form a system and systems can be assembled into bigger system, i. e. a system of systems. Boundaries and partitions can be drawn anywhere that makes sense in the context of what a system is intended to do and how it is organised. System representations are commonly called ‘models’ and can be given any form or content that is relevant to a specific purpose.

Computational models are based on constitutive equations and relationships that describe the interconnection of system components and the interdependency of system parameters. What a model contains is dictated by what the model is intended to do and how it will be used. The level of detail establishes the capability of a complete model and, ultimately, how close it comes to being a replication of a real system. The scale and complexity of systems can vary considerably, as can the capacity for information processing. Most systems involve some level of human interaction, such that human beings become part of the system when in use. Other systems are designed to be automatic or autonomous, which raises a whole set of issues regarding their safety and dependability in the absence of human involvement.

All systems have an external environment that defines the extent of the operational space and the variability of operational parameters. It is often appropriate to think of many environments coexisting in the same operational space, each supporting a different mode of transaction (e. g. information transfer, heat transfer). Thus, within all prescribed environments, a system must operate under ambient conditions and it must interoperate with other systems, Computational Modelling and Simulation of Aircraft and the Environment: Platform Kinematics and Synthetic Environment Volume 1 D. J. Diston © 2009 John Wiley & Sons, Ltd. ISBN: 978-0-470-01840-8 2 Computational Modelling and Simulation of Aircraft and the Environment artifacts or agents.

This implies a real-world environment (where physical contact and geophysical processes take place) in conjunction with a network environment (whereby information exchange takes place). Thus, a system can be just about anything, provided that it is an assembly of parts, it performs a function and it exhibits time-varying behaviour. Moreover, a model of a system can be just anything, provided that it represents the system in some credible manner (regardless of its format or level of detail). As a general expectation, the term ‘system’ conveys the dual assumption of complexity and (in the age of software-controlled systems) adaptability. The real challenge is to understand what is important in the construction of a system and how best to represent it.

In simple terms, computational modelling is usually undertaken with one or more of the following objectives: • • • • • • • to confirm the concept of system operation; to understand the inherent functional mechanisms; to predict system behaviour; to investigate parametric uncertainty and physical constraints; to exercise control and monitoring functions; to propose and refine design requirements; to assess aspects of safety and interoperability. The challenge is to combine objectives in order to build an appropriate facility for investigating system functionality, operability and interoperability. In the context of Computational Modelling and Simulation of Aircraft and Environment, the emphasis is placed on the principles that support the simulation of airborne systems, namely the technical descriptions of air vehicles and flight physics (which treat the system platform as a system in its own right) and the theoretical foundations for spatial reference models and geophysical processes (which treat the external environment as a system in its own right).

Thus, an aircraft can be considered as a system of systems that comprises a structural platform with external geometry that determines aerodynamic parameters and internal systems for vehicle management and operational tasks. Taking a wider perspective, the aircraft can be considered as one system in a larger system of systems such that it belongs to a network of vehicles and supporting infrastructure that collectively operates as a collaborative enterprise. Inherently, this idea underpins the concept of network-centric operations or network-enabled capability, in which the ability to acquire and distribute information is the critical factor in ensuring the optimum mix of resources and the maximum effectiveness of all participants.

Looking further outwards, there is the big environment within which all the operational platforms and services reside. This defines the universe or, at least, the small part of it that contains the Earth and a projection of major objects that are visible from Earth. This too could be considered as a system of systems, constructed as a set of interlocking physical processes, but it makes better sense to model this as a synthetic environment. This provides a single operational space for all participants, with a common definition of physical processes and interactions. In principle this can be extended in order to encompass a Virtual Earth within which activities and missions can be undertaken . . . and all by simulation. 1. 2 Modelling and Simulation (M&S)

Modelling and Simulation encompasses a wide range of techniques for building representations of systems and using them in order to predict/replicate system behaviour over time. Often the distinction between these terms has been blurred and this is probably because of the interdependent Introduction 3 nature of the two activities. This is reflected in the combined designation ‘Modelling and Simulation’ (or M&S) that has entered the common vocabulary of system developers. ‘Modelling’ is a process that collects and collates information about a system and then constructs representations of that system, based on mathematical methods and applicable sciences.

Computational modelling has become the standard approach to building system prototypes because of the flexibility that is afforded by software in the building diverse representations (e. g. functional, physical, operational models) of any system that is amenable to mathematics or informatics, i. e. any system at all. A computational model is an artifact that is constructed from equations, functions and data (regardless of its presentation). It is expressed in a form that allows computational experiments to be performed in order to answer questions about a system of interest. As with the term ‘system’, the term ‘model’ is ubiquitous and its meaning has become ambiguous, lying somewhere between the ideas of structure, content, purpose and mechanisation.

Generically, ‘simulation’ is a process that generates dynamic performance and operational features that are appropriate to one or more systems of interest, together with modes of interaction between systems, operators and environment. It uses models that are mechanised in software (i. e. machine-readable or machine-executable) and that are exercised via numerical algorithms that are capable of evaluating cause-and-effect relationships and aggregating the resultant changes over time, with updates associated with discrete time increments or discrete events. Models must be fit for their intended purpose and, inevitably, all will have limitations.

Specifically, a ‘simulation’ is a computational task or experiment that is initialised in a known condition and then is steered by stimuli (i. e. commands or environmental conditions) that are applied at system boundaries (which necessarily match the boundaries that delineate the associated models). The nature of simulation will dictate whether stimuli are routed via physical interfaces (which implies real-time simulation) or are predefined via time-based records (which implies non-real-time simulation). A ‘simulator’ is an assemblage of computers, interfaces and related equipment that support the execution of specific types of simulation. Perhaps the example that is brought to mind most readily is a flight simulator.

In its full configuration, this comprises a six degree-of-freedom motion platform, carrying a fully functional replica of a cockpit or flight deck surrounded by a high-fidelity display system for projecting images of a synthetic outside world. All airborne systems are simulated, together with navigational and atmospheric environments. For training and operational purposes, external operators provide air traffic management and induce faults and failures. In its minimal configuration, this comprises software inside a computer with a limited array of control and display functions. At this level, ‘software only’ simulators are applicable to all types of system. The context in which simulators are used leads to a number of designations for the type of simulation being undertaken.

There is some degree of variability and preference in the choice of designation but typical distinctions might include: • Constructive M&S involving software-based simulations only, representing functional principles of the various systems of interest; • Hybrid M&S involving software-based simulations plus equipment components and, representing a mix of functional principles and physical embodiment; • Live M&S involving software-based simulations plus equipment components, with so-called Human In The Loop (HITL) operation. By definition, Live and Hybrid M&S necessitate real-time simulation, while Constructive M&S can be real-time or non-real-time. Note that non-real-time simulation can be set at an arbitrary update rate (i. e. slower or faster than real time). 4 Computational Modelling and Simulation of Aircraft and the Environment 1. 3 Development Processes On major development programmes, M&S is an integral feature of lifecycle planning, as typified by Figure 1. 1.

This is based on the conventional ‘V-Diagram’ for system development, starting with requirements at top-left. Time runs from left to right. The left side of the ‘V’ follows the design path down to detailed specification and acquisition of system components; the right side of the ‘V’ follows the integration of the end-product and the related test and verification activities. This example considers the creation of a hypothetical air system (i. e. a complete airborne capability), through the progressive design of an air vehicle, its integrated systems and the constituent system components. The complete system is assembled to completion and reaches operational readiness.

In this case, the requirements are implemented in an operational simulation, which implies that the operational capability of the end-product is completely predicted in advance and that performance can be confirmed and refined during development. However, Figure 1. 1 reinterprets the conventional lifecycle by subdividing it into mini-lifecycles that are applied in a hierarchy. Each ‘V’ starts with requirements at top-left, proceeds to the solution components at bottom-centre and concludes with an assembled product at top-right. It must be demonstrated that the product satisfies the requirements and any noncompliance would have to be addressed by either changing requirements or repeating development.

So it is highly preferable that problems with requirements and/or solutions and/ or products are rectified as soon as possible. In this setting, models and simulations have the unrivalled potential of being able to represent architectures and predict behaviour during any level of system development. Stacking up the hierarchy, the complete set of mini-lifecycles can be tracked in a comprehensive and coordinated manner. Crucially, each mini-lifecycle contains integration and verification, performed via M&S as a prediction supported by analysis. Thus, components can be demonstrated in advance of their construction and assembly; they can be placed in the context of the full system before the full system exists.

Subject to the quality and validity of computational models, M&S will predict the outcomes of physical integration and verification activities towards the end of the lifecycle. Early prediction will reduce the amount of physical testing that is required and will allow that testing to be targeted effectively. The underlying philosophy is that design will be validated by M&S early in the lifecycle and M&S will be validated by physical test later in the lifecycle. Design Validated by M&S Models System Concepts and Component Definitions Operation Operational Readiness Operational Simulation Model-based Prediction Air Vehicle gr at ion Implementation Allocated Design Requirements Actual Interfaces Real-time Control

System Product Definition Figure 1. 1 Model and simulation within the system development lifecycle. In Implementation Concepts and Allocated Performance Function Integrated Systems Individual Systems te & M&S Validated by Physical Test Ve rif ica tio Air System n Introduction 5 The ‘depth’ of development is revealed in different class of models, as suggested in Figure 1. 1. There are no rules for classification but it is reasonable to suppose the high-level development will focus on capabilities and outcomes, the mid-level development will focus on functionality and performance and low-level development will focus on implementation and build standards.

Thus, three broad groupings are designated as Operational Models, Functional Models and Implementation Models. All contribution to the overall System Product Definition. Simplistically, the V-Diagram can be seen as a linear process in two parts, namely design followed by integration. Realistically this is better characterised as a V-shaped bucket, containing numerous iterations that confirm design properties as they emerge in order to ensure that development is contributing progressively to a viable end-product. This is shown in Figure 1. 2, as a combination of Design and Build activities, conducted at levels that are labelled as Product, Systems and Units.

In addition, the development path continues to the right as the product enters service and is subsequently maintained. As time runs from left to right, the problem with this traditional view of lifecycles is the large gap between top-left (where the requirements start) and the top right (where the product arrives). This is bridged by validation activities and, if problems are found here, they will be expensive to fix. The application of Verification and Validation (V&V) at each level of development is shown schematically in Figure 1. 3. This involves testing on the upward path but, still, it is not desirable to find too many problems during the Build phase. Product Systems Design Units Build Maintenance Figure 1. 2 Simplified lifecycle.

Product Validation System Verification Unit Verification Figure 1. 3 Verification and validation. 6 Computational Modelling and Simulation of Aircraft and the Environment The principal benefit and attraction of modelling is the ability to predict how a product will work and what it will look like well in advance of actually building it. This requires a set of models at different levels of detail, each having its own lifecycle, as shown in Figure 1. 4. These are superimposed on the product lifecycle and suggest that significant time can be saved if good models can be produced early enough. Clearly there is a trade-off between the content of a model and the time required to create it.

Top-level concept models can help confirm requirements before committing to full development. They provide a rehearsal for detailed design and testing, as well as allowing assessment of many options prior to down-select. System models look at implementation issues and detailed design. Importantly they can identify problem hotspots that require further modelling and analysis. The aim of a modelling framework to support the product lifecycle is to build up a family of models, as shown in Figure 1. 5, each representing different aspects of the product. In principle, the bulk of product validation can be performed against models of what the product will be, leaving a set of acceptance test procedures to be applied to the final product.

Note that this is not the same as model validation, which Concept Model System Model Product Development Figure 1. 4 Model lifecycles. Product Validation Family of Models Acceptance Testing uc t od Pr Int Figure 1. 5 Model-based Integration. eg rat i on Introduction 7 Product Validation Family of Models Acceptance Testing od Pr uc t Target Delivery Late Delivery from Early Fix Late Delivery from Late Fix Figure 1. 6 Model-based evaluation. would have to be performed in order to demonstrate the model is an adequate and accurate representation of the product. Having acquired models, they can be deployed for product evaluation, as shown in Figure 1. 6.

The models themselves can be used to generate results to be confirmed on a physical test stand. This allows extensive analysis and simulation, as well as definition of tasks and test scripts. Armed with this preparation, product build will progressively replace modelled units and systems with real units and systems. Note that the aim is to provide a product-wide context for testing such that each unit and system will experience realistic (dynamic) boundary conditions. Increasingly it is recognised that stand-alone testing is not adequate in itself for integrated systems. The whole point of organising a product lifecycle is to be able to achieve a delivery target with minimal risk.

The earlier problems are found and fixed, the smaller the risk of delay and the smaller the actual delay if the risk were realised. This is what Figure 1. 7 attempts to show. Modelling can certainly facilitate early product validation. What it can also do is allow suppliers to be fully engaged at the concept stage, as in Figure 1. 8. This results in a more complete product model and a more effective flow-down of product requirements through the Figure 1. 7 Possible late delivery. alu Ev ati on 8 Computational Modelling and Simulation of Aircraft and the Environment Early Supplier Engagement Early Delivery with Early Validation Contractor Product Model Target Delivery Supplier Unit Models Experience Supplier Lifecycle Figure 1. 8 Possible early delivery. value chain.

There is an opportunity to shift unit development to the left, based on early supplier engagement and early access to supplier experience. In effect, the contractor sees a ‘thin’ and ‘short’ unit development compared with the standard lifecycle. Cost and time can be reduced and quality can be improved in this way. However, this sort of enrichment is based on rapid design iterations and shared design information. Many complex products are developed by several risk-sharing partners and these are supported, in turn, by a large number of supplier organisations. Products evolve through many versions and variants before the final product family emerges. 1. 4

Models Quite simply, a model is defined simply as a representation of a system. It does not have to be an exact representation but it must contain the correct configuration and deliver the correct operational capability. Thus, it is an approximation but it must exhibit credible behaviour and performance. It can also be thought of as a framework that allows a system to be analysed or simulated. As such, the level of detail must be appropriate to the analysis or simulation but note that the level of detail does not have to be uniform across the model. Some technical features may be of greater interest than others and these may warrant greater detail.

A system can have many different representations and therefore a family of models may be required in order to cover the full range of technical features and to reflect the evolution of those features through the development lifecycle. Perhaps the two most important observations are as follows: • All models serve a purpose and that purpose must be made explicit. • All models have limitations and those limitations must be made explicit. What is at issue here is that a model is applicable to the context within which it was created and are not necessarily applicable anywhere else; if it is applicable then either it is based on generic principles or it is a fluke. Thus, it is essential that the authors specify (a) intended usage, (b) known limitations and (c) test coverage.

In critical applications, independent review should establish the adequacy of Introduction 9 this information, as well as the extent to which a model is a true representation of a system. What is needed is a qualification statement that, quite literally, states that a model is qualified to be used for a given purpose. A representation invokes a set of classifications, associations and explanations. These are expressed using a language, which can be thought of as a descriptive framework of concepts, definitions and attributes. The process of converting between representations will be called a ‘transformation’; the process of converting between languages will be called a ‘translation’ (Gawthrop & Ballance, 1998). These nterrelationships are shown generically in Figure 1. 9. It is widely accepted that a model is either a physical replica or a mathematical abstraction. For an abstract model, there are many options for deciding on its content and for adopting suitable methods of construction (cf. Bennett, 1995). These might include: • • • • • stochastic vs deterministic; event-driven vs time-driven; equation-based vs algorithm-based; theoretical vs empirical; physics-based vs effects-based. Having obtained a model of a system, a number of activities can be performed. For instance, it can be used to predict steady-state performance, to establish analytical models (e. g. ia linearisation) and to develop simulation codes. It should be noted that the terms ‘modelling’ and ‘simulation’ are notoriously interchangeable. Under its correct interpretation, modelling is the process of mapping the structure of a physical system into a mathematical form and checking that it is correct and fit for purpose. By contrast, simulation is the process of experimentation whereby behaviour is predicted/reproduced by a computational algorithm that integrates rates of change in continuous states and induces transitions in discrete states, all along an advancing timeline. It is intuitive to think of model development based on objects (cf. Rumbaugh et al. , 1991).

Physical systems are built using physical objects, linked via physical connectors of various types. Systems can contain subsystems (i. e. systems in their own right), giving the property of hierarchy. Connections allow objects to interact and thus provide the interface medium between objects. The content of an object can be defined independently of its interfaces, giving the property of encapsulation. Many different object types could match a single interface definition, giving the property of polymorphism. A hierarchy of subsystem objects would access Language 1 Representation 1 Representation 2 Language 2 Language M Representation N Figure 1. 9 Matrix of model representations vs languages. 10

Computational Modelling and Simulation of Aircraft and the Environment the top-level system interface, giving the property of inheritance. The adoption of this development paradigm confirms and formalises what modellers and simulationists have taken for granted over many decades. The basic concept of object-oriented modelling is shown in Figure 1. 10. This shows an arrangement of component objects within a defined architecture, with interfaces to an external environment. Associations between components are created by means of links which plug into component icons via sockets that are usually referred to as ports. Each port is defined as part of the component definition.

The decomposition reveals that one subsystem happens to be a supertype of various component entities. One is a constitutive relationship (in this case, a mathematical expression involving states x, port variables p, internal variables y and time t). Another is a composite component definition (which is a model in its own right). The question mark indicates that potentially any other entity could satisfy the same interface or, perhaps, the interface is left open (such that a model is only partially defined). The issues and motivations behind polymorphic modelling are discussed by de Vries (1994). In his definition, this is ‘the combined application of modularisation and subtyping during model building’.

The first concept introduces an abstraction principle that focuses on the separation between essential and incidental properties of a subsystem. Essential properties are those that are necessary in order to classify the subsystem: incidental properties serve to add descriptive detail to a particular subsystem and, as such, they may differ depending on context. The second concept makes it possible to refine or specialise a generic type in various forms. Critical aspects of modelling are verification and validation. These have a strong distinction, as recommended by the Society for Computer Simulation (Technical Committee for Model Credibility) (SCS 1979). With a number of specific points of clarification (e. g.

Murray-Smith, 1995), Verification confirms that the internal structure of a model is correct and that its constituent parts are mutually consistent and validation confirms that the external behaviour is credible and that it satisfies the user requirements. Typically, the latter implies the use of test scenarios in order to demonstrate that a model can reproduce known benchmarks and it is important to distinguish between theoretical validation (which considers general principles), Environment System f(x,p,? ,t) ? Figure 1. 10 Polymorphic modelling with encapsulated components. Introduction 11 Model Validation Real System Modelling & System Identification Model Suitability Computer Model Programming Model Verification Numerical Model Figure 1. 1 Modelling of a dynamic system. functional validation (which deals with specific mechanisms contained in actual systems) and empirical validation (which compares model outputs with real measurements). In overall terms, this view is depicted in Figure 1. 11 (Buccholz et al. , 1995) highlighting the modelling and programming activities which lead from a real system through to a computerbased simulation. Validation and verification are shown as comparative exercises, along with a vague reference to model ‘suitability’. Adopting a different perspective, as discussed in the previous section, it is useful to separate the structure of a model from its parametric instantiation.

In this way, the fundamental task is to map the constituents of a real system into an object model. A structural validation can be performed in order to confirm that both the component resolution and interface definition correctly reflect the functional organisation of the system and that the model is capable of delivering the information required of it. Instance data transforms an object model into a parametric model that can support analysis and simulation. A parametric validation is then applied in order to compare model prediction with actual measurement. This revised scheme is shown in Figure 1. 12. In contrast, the numerical and computer models of Figure 1. 1 are merely two parametric representations. The prime distinction is the explicit declaration of model structure which offers an interpretation of ‘suitability’ which can support a strategy for model testing. Parametric Validation Real System Computer Model Structural Verification Internal Verification Numerical Model Figure 1. 12 Validation and verification. 12 Computational Modelling and Simulation of Aircraft and the Environment Test strategies for models fit into six broad categories although the terminology can vary considerably. These involve simulation and other analyses in order to confirm system performance within known tolerances.

The main categories are, as follows: • replication where consistent results are produced by independent means; • substitution where part of a model is replaced by an equivalent model to confirm its behaviour; • approximation where a model is reduced to its bare essentials to confirm dominant properties; • inversion where a model is reconstructed in order to reproduce input stimuli from a known response; • identification where internal model parameters are reconstructed from measured inputs/ outputs; • sensitivity where parametric variation is quantified against predefined design margins. The rationale is to build confidence in the correct operation of a model. Choice of strategy and the level of testing will depend on the criticality of application and the perceived complexity of the system of interest.

This will be reflected in the number of components and component interfaces; it will be influenced by carry-over experience from similar systems and previous application of relevant technologies. With increasing scale and connectivity of systems, less reliance can be placed on testing and more has to be placed on the process of model development and the traceability of development activities. The basic principle is that, when a model is too big to test effectively, verification rests on adherence to a set of standard practices. Auditors will always look for evidence of this before looking at the technical content of any work that has been done.

There is a general recognition that risks associated with a complex system can never be zero but should be as low as reasonably practicable. In pragmatic terms, this says that risk reduction should be pursued until the cost grossly outweighs benefit. In the context of modelling, there is an additional recognition that it is not feasible to fully specify a model in advance of its design and implementation. Invariably, in all but the simplest of systems, there will be significant uncertainty about the detailed characteristics of the system of interest and, depending on the modelling requirements, the full extent of development problems may not be immediately apparent. 1. 5 Meta-models

In order to illustrate some of the formalities that underpin model development, it is appropriate to consider how to specify the information content of a generic model in such a manner as to enable its storage within a database. An example is shown in Figure 1. 13, using graphical notation that is based on ISO-10303. 1 This is a model of a model, which is otherwise known as a meta-model or a schema. The design of meta-models is open to preference and reflects the concepts and interpretations that are important to users and designers. Crucially this establishes a structure within to express information and necessarily it removes the use of ambiguous terminology. With reference to Figure 1. 3, the main objectives are, firstly, to associate model components, model connectivity, constitutive relationships and data items with distinct entities and, secondly, to rigorously separate definitions from instances. In this case, an instance relates to an 1 ISO-10303 Product Data Representation and Exchange, otherwise known as the STandard for Exchange of Product model data (STEP). Introduction 13 Data_Type_Instance definition data_type data_type Connection destination current_data source Data_Type_ Definition Data_Instance Port 1 Formal_Port definition data_type current_data port_of Actual_Port instance port_of Port_Binding definition Model Component_Definition 1 hasParameters Component_Instance contains L[1:? ]

Parameter Primitive_Component L[1:? ] behaviour Composite_Component inter_relationship Constitutive_Relationship 1 usesFunctions S[0: ? ] Function_Instance definition Function_ Definition 1 Mathematical_ Expression Numerical_ relationship contains L[1:? ] Composite_Function Primitive_Function Figure 1. 13 Extract of information model for system ‘modelling’. actual object, while a definition relates to the object type. Note that all entities are given singular names (i. e. never plural), relational links are traced along thin lines and inherited links are traced along thick lines (with the number ‘1’ signifying mutual exclusion between subtypes).

All links are traced towards the line-end that is marked by the symbol ‘o’. A model is a Component_Definition in this particular context (although the term could be given several interpretations). In turn this can be specialised as a Composite_Component or a Primitive_Component, depending on whether the model is reducible or not, respectively. A composite component is decomposed into a list of one or more Component_Instance entities (annotated as ‘L(1:? )’). A primitive component has its behaviour defined by a Constitutive_Relationship. Note that, in this formulation, component definitions are nonrecursive. Models are drawn using components, ports and connections.

A connection has a port at each end, typically designated a source and a destination. The ports of a component definition are defined by the concept of a Formal_Port while those of a component instance are defined by the concept of an Actual_Port. This is akin to the concepts of actual and formal arguments in structured programming languages, which relate respectively to the use and definition of functions. When models are built, each connection to an actual port needs to be referenced to the equivalent formal port in the relevant component definition by means of a Port_Binding. 14 Computational Modelling and Simulation of Aircraft and the Environment A constitutive relationship can be specialised in various forms, as shown.

For illustration, one scenario has been developed in which functions can be defined using a hierarchy. Also, mathematical expressions (in other words, equations) can optionally contain functions. This topic is a major area of research in its own right2 and will not be discussed further. Finally, what about data?

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