Ebook Variational analysis and aerospace engineering: Part 1

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Ebook Variational analysis and aerospace engineering: Part 1 presents the following content: Algorithm Issues and Challenges Associated with the Development of Robust CFD Codes; Flight Path Optimization at Constant Altitude; A Survey on the Newton Problem of Optimal Profiles;...Please refer to the documentation for more details.

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For More Visit :www.LearnEngineering.in VARIATIONAL ANALYSIS AND AEROSPACE ENGINEERING For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in Springer Optimization and Its Applications VOLUME 33 Managing Editor Panos M. Pardalos (University of Florida) Editor–Combinatorial Optimization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (University of Chicago) C.A. Floudas (Princeton University) F. Giannessi (University of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (McMaster University) Y. Ye (Stanford University) Aims and Scope Optimization has been expanding in all directions at an astonishing rate dur- ing the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the ﬁeld has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the ﬁeld. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences. The series Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network ﬂow problems, stochastic optimization, optimal control, discrete optimization, multiobjec- tive programming, description of software packages, approximation tech- niques and heuristic approaches. For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in VARIATIONAL ANALYSIS AND AEROSPACE ENGINEERING Edited By GIUSEPPE BUTTAZZO University of Pisa, Italy ALDO FREDIANI University of Pisa, Italy 123 For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in Editors Giuseppe Buttazzo Aldo Frediani Universit` di Pisa a Universit` di Pisa a Dipto. Matematica Dipto. Ingegneria Aerospaziale Largo B. Pontecorvo, 5 Via Diotisalvi, 2 56127 Pisa, Italy 56126 Pisa, Italy buttazzo@dm.unipi.it a.frediani@ing.unipi.it ISSN 1931-6828 ISBN 978-0-387-95856-9 e-ISBN 978-0-387-95857-6 DOI 10.1007/978-0-387-95857-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009929316 AMS Subject Classiﬁcations (2000): 76-06, 79-06 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in This book is dedicated to Angelo Miele on the occasion of his 85th birthday. For More Visit :www.LearnEngineering.in
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For More Visit :www.LearnEngineering.in Preface In recent years, new mathematical methods and tools have been developed and ap- plied extensively in the ﬁeld of aerospace engineering, for example, ﬁnite element method, computational ﬂuid dynamics, optimization, control, eigenvalues problems. The interaction between aerospace engineering and mathematics has been signiﬁ- cant in the past for both engineers and mathematicians and will be even stronger in the future. The School of Mathematics “Guido Stampacchia” of the “Ettore Majorana” Foundation and Centre of Scientiﬁc Culture is the most appropriate site for aerospace engineers and mathematicians to meet. The present volume collects the papers pre- sented at the Erice Workshop held on September 8–16, 2007, which was organized in order to allow aerospace engineers and mathematicians from Universities, Re- search Centres, and Industry to debate advanced problems in aerospace engineering requiring extensive mathematical applications. The editors are conﬁdent to capture the interest of people from both academia and industry, particularly, young researchers working on new frontiers of mathematical applications to engineering. The workshop was dedicated to Angelo Miele, Professor at Rice University in Houston, on the occasion of his 85th birthday. Angelo Miele is both an eminent mathematician and a famous engineer, among other activities, able to conceive new scenarios for space exploration. He has been the advisor of many PhD students at Houston, who became well-known professors in universities worldwide and are speakers at this workshop. Pisa, Giuseppe Buttazzo, Pisa (Italy) July 2008 Aldo Frediani, Pisa (Italy) vii For More Visit :www.LearnEngineering.in
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For More Visit :www.LearnEngineering.in Acknowledgments This volume collects the contributions presented in the workshop on “Variational Analysis and Aerospace Engineering,” held in Erice on September 8–16, 2007. The workshop as well as the preparation of this volume have been possible thanks to the contributions of the following organisations and individuals: – Universit` di Pisa, Pisa, Italy a – GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilit` e le loro a Applicazioni), Italy – Dipartimento di Ingegneria Aerospaziale di Pisa, Pisa, Italy – Technical University of Technology, Delft, Holland – Contessa Maria Fede Caproni, Roma, Italy – IDS, Pisa, Italy – AgustaWestland, Cascina Costa, Italy – Fondazione Cassa di Risparmio di Pisa, Pisa, Italy – Consorzio Etruria SCArl, Montelupo, Firenze, Italy We gratefully acknowledge the E. Majorana Centre and Foundation for Scientiﬁc Culture and the precious help by Franco Giannessi and Emanuele Rizzo. ix For More Visit :www.LearnEngineering.in
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For More Visit :www.LearnEngineering.in Contents 1 Algorithm Issues and Challenges Associated with the Development of Robust CFD Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Steven R. Allmaras, John E. Bussoletti, Craig L. Hilmes, Forrester T. Johnson, Robin G. Melvin, Edward N. Tinoco, Venkat Venkatakrishnan, Laurence B. Wigton and David P. Young 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Algorithm Issues Related to the Solution of the Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Grid Adaption and Error Estimation . . . . . . . . . . . . . . . . . . 3 1.2.2 Discretization Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Higher Order Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.4 Domain Decomposition and Linear Solver . . . . . . . . . . . . . 16 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Flight Path Optimization at Constant Altitude . . . . . . . . . . . . . . . . . . . 21 Mark D. Ardema and Bryan C. Asuncion 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Singular Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 The Cruise Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Fanjet Speciﬁc Fuel Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 A Survey on the Newton Problem of Optimal Proﬁles . . . . . . . . . . . . . 33 Giuseppe Buttazzo 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Radially Symmetric Proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 The Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 xi For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in xii Contents 4 Innovative Rotor Blade Design Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Vittorio Caramaschi and Claudio Monteggia 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Helicopter’s Aeromechanics Outlines . . . . . . . . . . . . . . . . . . . . . . . . 51 4.3 Helicopter’s Rotor Mathematical Model Features and Aeromechanics Codes Worldwide Status . . . . . . . . . . . . . . . . . . 56 4.4 AW Aeromechanics Code GYROX II . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4.1 General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4.2 Rotor Hub Modelling Features . . . . . . . . . . . . . . . . . . . . . . 59 4.4.3 Pylon Modelling Features . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.4 Rotor Blade Structural Modelling Features . . . . . . . . . . . . 62 4.4.5 Rotor Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4.6 Solution Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4.7 Operational Main Features and Output Data . . . . . . . . . . . 67 4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6.1 Short Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6.2 Long Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5 Fields of Extremals and Sufﬁcient Conditions for the Simplest Problem of the Calculus of Variations in n-Variables . . . . . . . . . . . . . 75 Dean A. Carlson and George Leitmann 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Notations and the Problem Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3 Leitmann’s Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4 Fields of Extremals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5 Sufﬁcient Conditions for Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6 A Framework for Aerodynamic Shape Optimization . . . . . . . . . . . . . 91 Giampiero Carpentieri and Michel J.L. van Tooren 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 Adjoint-Based Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.3 Optimization Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.3.1 Flow Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3.2 Adjoint Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.3 Shape Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3.4 Geometric Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.3.5 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4 Optimization Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.4.1 RAE2822 at M∞ = 0.73 and α = 2◦ . . . . . . . . . . . . . . . . . . 100 6.4.2 NACA64A410 at M∞ = 0.75 and α = 0◦ . . . . . . . . . . . . . . 101 6.4.3 NACA0012 at M∞ = 1.5 and α = 2◦ . . . . . . . . . . . . . . . . . 102 6.4.4 ONERA-M6 wing at M∞ = 0.84 and α = 3.06◦ . . . . . . . . 103 For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in Contents xiii 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7 Optimal Motions of Multibody Systems in Resistive Media . . . . . . . . 107 Felix L. Chernousko 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.3 Linear Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.4 Relative Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.5 Piecewise Linear Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.6 Quadratic Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.7 Dry Friction: Velocity-Control Motion . . . . . . . . . . . . . . . . . . . . . . . . 114 7.8 Dry Friction: Acceleration-Control Motion . . . . . . . . . . . . . . . . . . . . 120 7.9 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.10 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8 Instationary Heat-Constrained Trajectory Optimization of a Hypersonic Space Vehicle by ODE–PDE-Constrained Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Kurt Chudej, Hans Josef Pesch, Markus W¨ chter, Gottfried Sachs a and Florent Le Bras 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.2 Trajectory Optimization Problems with Active Cooling . . . . . . . . . . 130 8.3 Trajectory Optimization Problem with an Instationary Heat Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9 Variational Approaches to Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Gianpietro Del Piero 9.1 Fracture as a Minimum Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 9.2 The Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.3 Energy Barriers and Local Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.4 Barenblatt’s Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.5 Two Solution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.6 The Dissipative Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.7 From Surface to Bulk Regularization . . . . . . . . . . . . . . . . . . . . . . . . . 157 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 10 On the Problem of Synchronization of Identical Dynamical Systems: The Huygens’s Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Rui Dil˜ o a 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.2 A Model for the Synchronization of the Two Pendulum Clocks . . . 166 For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in xiv Contents 10.3 A Simple Clock Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 10.4 Synchronization of Two Pendulum Clocks with Equal Parameters . 169 10.5 Synchronization of Two Pendulum Clocks with Different Parameters: Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 11 Best Wing System: An Exact Solution of the Prandtl’s Problem . . . . 183 Aldo Frediani and Guido Montanari 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 11.2 The Induced Drag for Lifting Multiwing Systems . . . . . . . . . . . . . . 184 11.3 The Problem of Minimum Induced Drag in a Box Wing . . . . . . . . . 187 11.3.1 Case A: Elliptical Circulations on the Horizontal Wings and Zero on the Vertical Ones . . . . . . . . . . . . . . . . . . . . . . . 191 11.3.2 Case B: Constant Circulations on the Horizontal Wings and Unknown on the Vertical Ones . . . . . . . . . . . . . . . . . . . 192 11.3.3 Final Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 11.4 The Optimum Lift Distribution Along the Vertical Wings . . . . . . . . 196 11.5 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 12 Numerical Simulation of the Dynamics of Boats by a Variational Inequality Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Luca Formaggia, Edie Miglio, Andrea Mola and Anna Scotti 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.2 A Variational Approach to the Floating Body Problem . . . . . . . . . . 214 12.2.1 Characteristic Treatment of the Time Derivative . . . . . . . . 218 12.2.2 Enforcing the Constraint in the Hydrostatic Step . . . . . . . . 219 12.2.3 The Model for the Dynamics of a Rowing Scull . . . . . . . . 220 12.2.4 More Realistic Boundary Conditions . . . . . . . . . . . . . . . . . 223 12.3 The Interaction Between the Boat and the Water . . . . . . . . . . . . . . . 223 12.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 12.4.1 Sinking and Pitching Motions . . . . . . . . . . . . . . . . . . . . . . . 224 12.4.2 Reproducing Mean Motion Wave Pattern . . . . . . . . . . . . . . 225 12.4.3 An Example with the Full Dynamics . . . . . . . . . . . . . . . . . 226 12.4.4 A Final Detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 13 Concepts of Active Noise Reduction Employed in High Noise Level Aircraft Cockpits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Hatem Foudhaili and Eduard Reithmeier 13.1 Passive Versus Active Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . 230 13.2 Active Noise Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 13.3 Active Structural/Acoustic Control (ASAC) . . . . . . . . . . . . . . . . . . . 234 13.4 Active Aviation Headsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in Contents xv 13.5 An Aviation Communication Headset Prototype with Digital Adaptive Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 13.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 14 Lekhnitskii’s Formalism for Stress Concentrations Around Irregularities in Anisotropic Plates: Solutions for Arbitrary Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Sotiris Koussios and Adriaan Beukers 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 14.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 14.3 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 14.4 Stress, Strain, and Displacements Formulation . . . . . . . . . . . . . . . . . 247 14.5 Formulation of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 248 14.5.1 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 14.5.2 Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 14.6 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 14.6.1 Series Representation of the Boundary Conditions . . . . . . 250 14.6.2 Transformation into a Single Variable . . . . . . . . . . . . . . . . . 251 14.7 Boundary Conditions Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 14.7.1 Homogeneous Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 14.7.2 Logarithmic Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 14.7.3 Disturbance Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 14.8 Evaluation of Stresses and Displacements . . . . . . . . . . . . . . . . . . . . . 259 14.9 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 14.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 15 Best Initial Conditions for the Rendezvous Maneuver . . . . . . . . . . . . . 267 Angelo Miele and Marco Ciarci` a 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 15.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 15.3 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 15.3.1 Multiple-Subarc Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 272 15.3.2 Inequality Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 15.3.3 Particular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 15.3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 15.3.5 Performance Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 15.3.6 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 15.4 Minimum Fuel, Time Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 15.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 15.6 Minimum Fuel, Time Given . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 15.6.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 15.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in xvi Contents 16 Commercial Aircraft Design for Reduced Noise and Environmental Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 S. Mistry, Howard Smith, and John P. Fielding 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 16.2 Simple Emission Trade-Off Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 16.2.1 Global Warming Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 16.2.2 Noise Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 16.2.3 Local Air Quality Cost (LAQ) . . . . . . . . . . . . . . . . . . . . . . . 293 16.2.4 Annual Fuel Costs Fro Baseline Aircraft . . . . . . . . . . . . . . 294 16.2.5 Baseline Aircraft Environmental Costs . . . . . . . . . . . . . . . . 294 16.2.6 Summary of Trade-Offs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 16.3 Aircraft Designs for Reduced Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 295 16.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 16.3.2 Baseline Aircraft Design and Noise Prediction . . . . . . . . . 296 16.3.3 Low Airframe Noise Design Methodology . . . . . . . . . . . . 297 16.3.4 Low-Noise Aircraft Concept Brainstorming Process . . . . 297 16.3.5 Broad Delta Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 16.3.6 Airframe Approach Noise Prediction . . . . . . . . . . . . . . . . . 302 16.3.7 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 16.4 The Cranﬁeld A-6 Greenliner Project . . . . . . . . . . . . . . . . . . . . . . . . . 304 16.4.1 Group Design Project Activities . . . . . . . . . . . . . . . . . . . . . 304 16.4.2 Greenliner Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 16.4.3 Predicted Performance for the Greenliner . . . . . . . . . . . . . 309 16.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 17 Variational Approach to the Problem of the Minimum Induced Drag of Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Maria Teresa Panaro, Aldo Frediani, Franco Giannessi and Emanuele Rizzo 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 17.2 Finite Span Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 17.3 Problem of Minimum Induced Drag of a Straight Wing: An optimality condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 17.4 Duality: A New Approach to the Design of Wings . . . . . . . . . . . . . . 319 17.5 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 17.5.1 Elliptic Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 17.5.2 Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 18 Plastic Hinges in a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Danilo Percivale and Franco Tomarelli 18.1 Elastic–Plastic Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 18.2 Skew-Symmetric Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in Contents xvii 19 Problems of Minimal and Maximal Aerodynamic Resistance . . . . . . 349 Alexander Plakhov 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 19.2 Translational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 19.3 Translational Motion with Rotation: Two-Dimensional Case . . . . . 355 19.3.1 Deﬁnition of Rough Body and Main Theorems . . . . . . . . . 355 19.3.2 Problems of Minimal and Maximal Resistance for a Slowly Rotating Body . . . . . . . . . . . . . . . . . . . . . . . . . 358 19.3.3 Mathematical Retroreﬂector . . . . . . . . . . . . . . . . . . . . . . . . . 360 19.3.4 Effect of Magnus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 20 Shock Optimization for Airfoil Design Problems . . . . . . . . . . . . . . . . . 367 Olivier Pironneau 20.1 Numerical Optimal Shape Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 20.1.1 An Academic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 20.1.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 20.1.3 Conceptual Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 20.2 Automatic Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 20.2.1 Principle of Automatic Differentiation . . . . . . . . . . . . . . . . 370 20.2.2 Example of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 20.3 Differentiability Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 20.3.1 Extended Calculus of Variation . . . . . . . . . . . . . . . . . . . . . . 372 20.3.2 Sensitivity Analysis for Burgers’ Equation . . . . . . . . . . . . 373 20.3.3 Application to Optimal Control . . . . . . . . . . . . . . . . . . . . . . 373 20.3.4 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 20.3.5 Right and Wrong Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 374 20.4 Small Disturbances and Automatic Differentiations . . . . . . . . . . . . . 376 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 21 Differential Games Treated by a Gradient-Restoration Approach . . . 379 Mauro Pontani 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 21.2 Zero-Sum Differential Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 21.3 Numerical Solution of Two-Sided Optimization Problems . . . . . . . 382 21.3.1 Transformation into Single-Objective Problem . . . . . . . . . 382 21.3.2 Sequential Gradient-Restoration Algorithm . . . . . . . . . . . . 384 21.4 Homicidal Chauffeur Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 21.4.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 385 21.4.2 Method of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 21.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 21.5 Orbital Pursuit-Evasion Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 21.5.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 389 21.5.2 Method of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 21.5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 21.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in xviii Contents 22 Interval Methods for Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . 397 Andreas Rauh and Eberhard P. Hofer 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 22.2 Optimal and Robust Control of Dynamical Systems . . . . . . . . . . . . . 399 22.2.1 Optimal Control of Discrete- and Continuous-Time Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 22.2.2 Speciﬁcation of Robustness in the Time Domain . . . . . . . 401 22.2.3 Optimality Criteria for Systems with Uncertainties . . . . . . 402 22.3 Interval Arithmetic Optimization Algorithm . . . . . . . . . . . . . . . . . . . 403 22.4 Parallelization of the Optimization Algorithm . . . . . . . . . . . . . . . . . . 405 22.5 Combination with Classical Controller Design . . . . . . . . . . . . . . . . . 406 22.6 Validated Modeling and Simulation of Dynamical Systems with State-Dependent Switchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 22.7 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 22.7.1 Interval Algorithm for Structure Optimization . . . . . . . . . . 410 22.7.2 Linear State Controller for Improvement of Robustness . . 413 22.7.3 Interval Algorithm for Parameter Optimization . . . . . . . . . 415 22.8 Conclusions and Outlook on Future Work . . . . . . . . . . . . . . . . . . . . . 416 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 23 Application of Optimisation Algorithms to Aircraft Aerodynamics . 419 Emanuele Rizzo and Aldo Frediani 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 23.2 An Algorithm for the Search of Global Minima . . . . . . . . . . . . . . . . 424 23.3 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 23.3.1 Test Case 1 (Unconstrained): Ackley’s Function . . . . . . . . 428 23.3.2 Test Case 2 (Unconstrained): Rastrigin’s Function . . . . . . 431 23.3.3 Test Case 3 (Unconstrained): Rosenbrock’s Function . . . . 431 23.3.4 Test Case 4 (Unconstrained): Schwefel’s Function . . . . . . 432 23.4 The AEROSTATE Program: An Application to Aeronautics . . . . . . 434 23.4.1 Minimum Induced Drag of a Wing . . . . . . . . . . . . . . . . . . . 435 23.4.2 Minimum Total Drag of a Wing . . . . . . . . . . . . . . . . . . . . . . 438 23.4.3 The Trimmed Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 23.4.4 The PrandtlPlane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 23.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 24 Different levels of Optimisation in Aircraft Design . . . . . . . . . . . . . . . 447 Dieter Schmitt 24.1 Air Transport System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 24.2 Industrial Process of Aircraft Design . . . . . . . . . . . . . . . . . . . . . . . . . 449 24.3 Different Levels of Aircraft Design vs. Development Phases . . . . . 452 24.4 Tools Used in Different Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 24.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 For More Visit :www.LearnEngineering.in
For More Visit :www.LearnEngineering.in Contents xix 25 Numerical and Analytical Methods for Global Optimization . . . . . . . 461 Paolo Teoﬁlatto and Mauro Pontani 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 25.2 Green’s Theorem Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 25.3 Morse Theory Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 25.4 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 26 The Aeroservoelasticity Qualiﬁcation Process in Alenia . . . . . . . . . . . 477 Vincenzo Vaccaro 26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 26.2 Company Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 26.3 What Is Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 26.4 Aeroelastic Tradition in Alenia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 26.5 Aeroservoelastic Certiﬁcation Process . . . . . . . . . . . . . . . . . . . . . . . . 481 26.5.1 Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 26.5.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 26.5.3 Ground Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 26.5.4 Flight Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 26.5.5 Research and Future Developments . . . . . . . . . . . . . . . . . . 486 27 Further Steps Towards Quantitative Conceptual Aircraft Design . . . 491 Michel van Tooren, Gianfranco La Rocca and Teodor Chiciudean 27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 27.2 The Systems Engineering Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 496 27.3 Requirements on Computational Systems . . . . . . . . . . . . . . . . . . . . . 496 27.4 The Design and Engineering Engine Concept . . . . . . . . . . . . . . . . . . 497 27.4.1 Describing Design Options . . . . . . . . . . . . . . . . . . . . . . . . . 497 27.4.2 The Initiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 27.4.3 The Multi-model Generator . . . . . . . . . . . . . . . . . . . . . . . . . 503 27.4.4 The Life-Cycle Analysis with Expert Tools . . . . . . . . . . . . 504 27.4.5 The Converger/Evaluator . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 27.4.6 The Agent-Based Framework . . . . . . . . . . . . . . . . . . . . . . . 504 27.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 27.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 28 Some Plebeian Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 Piero Villaggio 28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 28.2 Mechanical Plebeian Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 28.3 Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 28.4 Peeling and Cooking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 28.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 For More Visit :www.LearnEngineering.in