Variational Analysis and Aerospace Engineering (eBook)

Preface 7
Acknowledgments 8
Contents 9
Contributors 18
Algorithm Issues and Challenges Associated with the Development of Robust CFD Codes 23
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 23
Introduction 23
Algorithm Issues Related to the Solution of the Navier--Stokes Equations 24
Grid Adaption and Error Estimation 25
Discretization Issues 28
Higher Order Elements 34
Domain Decomposition and Linear Solver 38
Conclusions 41
References 41
Flight Path Optimization at Constant Altitude 42
Mark D. Ardema and Bryan C. Asuncion 42
Introduction 42
Singular Optimal Control 44
The Cruise Problem 45
Fanjet Specific Fuel Consumption 47
An Example 49
Conclusions and Discussion 52
References 53
A Survey on the Newton Problem of Optimal Profiles 54
Giuseppe Buttazzo 54
Introduction 54
Radially Symmetric Profiles 58
The Existence Result 61
References 68
Innovative Rotor Blade Design Code 
70 
Vittorio Caramaschi and Claudio Monteggia 
70 
Introduction 
96 
Helicopter’s Aeromechanics Outlines 
97 
Helicopter’s Rotor Mathematical Model Features 
 
99 
AW Aeromechanics Code GYROX II 
101 
Applications 
105 
Conclusion 109
Fields of Extremals and Sufficient Conditions for the Simplest Problem of the Calculus of Variations in n-Variables 96
Dean A. Carlson and George Leitmann 96
Introduction 96
Notations and the Problem Definition 97
Leitmann's Direct Method 99
Fields of Extremals 101
Sufficient Conditions for Optimality 105
Conclusion 109
References 109
A Framework for Aerodynamic Shape Optimization 111
Giampiero Carpentieri and Michel J.L. van Tooren 111
Introduction 111
Adjoint-Based Sensitivity Analysis 112
Optimization Framework 113
Flow Solver 114
Adjoint Solver 116
Shape Parameterization 118
Geometric Sensitivities 119
Optimization Algorithm 119
Optimization Test Cases 120
RAE2822 at M=0.73 and = 2 120
NACA64A410 at M=0.75 and = 0 121
NACA0012 at M=1.5 and = 2 122
ONERA-M6 wing at M=0.84 and = 3.06 123
Conclusions 125
References 126
Optimal Motions of Multibody Systems in Resistive Media 127
Felix L. Chernousko 127
Introduction 127
Basic Equations 128
Linear Resistance 130
Relative Motions 130
Piecewise Linear Resistance 132
Quadratic Resistance 133
Dry Friction: Velocity-Control Motion 134
Dry Friction: Acceleration-Control Motion 140
Generalizations 144
Experiments 144
Conclusions 145
References 145
Instationary Heat-Constrained Trajectory Optimization of a Hypersonic Space Vehicle by ODE--PDE-Constrained Optimal Control 147
Kurt Chudej, Hans Josef Pesch, Markus Wächter, Gottfried Sachs and Florent Le Bras 147
Introduction 148
Trajectory Optimization Problems with Active Cooling 150
Trajectory Optimization Problem with an Instationary Heat Constraint 154
Conclusions 160
References 162
Variational Approaches to Fracture 165
Gianpietro Del Piero 165
Fracture as a Minimum Problem 165
The Numerical Solution 167
Energy Barriers and Local Minima 168
Barenblatt's Regularization 171
Two Solution Strategies 173
The Dissipative Model 174
From Surface to Bulk Regularization 177
References 181
On the Problem of Synchronization of Identical Dynamical Systems: The Huygens's Clocks 183
Rui Dilão 183
Introduction 183
A Model for the Synchronization of the Two Pendulum Clocks 186
A Simple Clock Model 188
Synchronization of Two Pendulum Clocks with Equal Parameters 189
Conclusions 199
References 200
Best Wing System: An Exact Solution of the Prandtl's Problem 202
Aldo Frediani and Guido Montanari 202
Introduction 202
The Induced Drag for Lifting Multiwing Systems 203
The Problem of Minimum Induced Drag in a Box Wing 206
Case A: Elliptical Circulations on the Horizontal Wings and Zero on the Vertical Ones 210
Case B: Constant Circulations on the Horizontal Wings and Unknown on the Vertical Ones 211
Final Equations 213
The Optimum Lift Distribution Along the Vertical Wings 215
Results and Conclusions 216
References 218
Numerical Simulation of the Dynamics of Boats by a Variational Inequality Approach 231
Luca Formaggia, Edie Miglio, Andrea Mola and Anna Scotti 231
Introduction 231
A Variational Approach to the Floating Body Problem 232
Characteristic Treatment of the Time Derivative 236
Enforcing the Constraint in the Hydrostatic Step 237
The Model for the Dynamics of a Rowing Scull 238
More Realistic Boundary Conditions 241
The Interaction Between the Boat and the Water 241
Numerical Results 242
Sinking and Pitching Motions 242
Reproducing Mean Motion Wave Pattern 243
An Example with the Full Dynamics 244
A Final Detail 244
References 245
Concepts of Active Noise Reduction Employed in High Noise Level Aircraft Cockpits 246
Hatem Foudhaili and Eduard Reithmeier 246
Passive Versus Active Noise Reduction 247
Active Noise Cancellation 247
Active Structural/Acoustic Control (ASAC) 251
Active Aviation Headsets 254
An Aviation Communication Headset Prototype with Digital Adaptive Noise Reduction 255
Conclusions 257
References 257
Lekhnitskii's Formalism for Stress Concentrations Around Irregularities in Anisotropic Plates: Solutions for Arbitrary Boundary Conditions 259
Sotiris Koussios and Adriaan Beukers 259
Introduction 259
Governing Equations 261
General Solution 262
Stress, Strain, and Displacements Formulation 263
Formulation of Boundary Conditions 264
Forces 264
Displacements 265
Solution Strategy 266
Series Representation of the Boundary Conditions 266
Transformation into a Single Variable 267
Boundary Conditions Evaluation 269
Homogeneous Part 269
Logarithmic Part 270
Disturbance Field 272
Evaluation of Stresses and Displacements 275
Example 277
Conclusions 280
References 281
Best Initial Conditions for the Rendezvous Maneuver 282
Angelo Miele and Marco Ciarcià 282
Introduction 283
Algorithm 284
System Description 286
Multiple-Subarc Equations 287
Inequality Constraint 288
Particular Cases 289
Boundary Conditions 289
Performance Index 290
Approaches 291
Minimum Fuel, Time Free 291
Results 292
Minimum Fuel, Time Given 295
Results 297
Conclusions 302
References 303
Commercial Aircraft Design for Reduced Noise and Environmental Impact 305
S. Mistry, Howard Smith, and John P. Fielding 305
Introduction 306
Simple Emission Trade-Off Study 306
Global Warming Costs 306
Noise Costs 307
Local Air Quality Cost (LAQ) 307
Annual Fuel Costs Fro Baseline Aircraft 308
Baseline Aircraft Environmental Costs 308
Summary of Trade-Offs 309
Aircraft Designs for Reduced Noise 309
Background 309
Baseline Aircraft Design and Noise Prediction 310
Low Airframe Noise Design Methodology 311
Low-Noise Aircraft Concept Brainstorming Process 311
Broad Delta Concepts 313
Airframe Approach Noise Prediction 316
Performance Comparison 317
The Cranfield A-6 Greenliner Project 318
Group Design Project Activities 318
Greenliner Description 319
Predicted Performance for the Greenliner 323
Conclusions 325
References 326
Variational Approach to the Problem of the Minimum Induced Drag of Wings 327
Maria Teresa Panaro, Aldo Frediani, Franco Giannessi and Emanuele Rizzo 327
Introduction 328
Finite Span Wings 328
Problem of Minimum Induced Drag of a Straight Wing: An optimality condition 330
Duality: A New Approach to the Design of Wings 333
Direct Methods 339
Elliptic Distribution 339
Ritz Method 341
References 356
Plastic Hinges in a Beam 357
Danilo Percivale and Franco Tomarelli 357
Elastic--Plastic Beam 357
Skew-Symmetric Load 361
References 362
Problems of Minimal and Maximal Aerodynamic Resistance 363
Alexander Plakhov 363
Introduction 363
Translational Motion 364
Translational Motion with Rotation: Two-Dimensional Case 369
Definition of Rough Body and Main Theorems 369
Problems of Minimal and Maximal Resistance for a Slowly Rotating Body 372
Mathematical Retroreflector 374
Effect of Magnus 375
Shock Optimization for Airfoil Design Problems 380
Olivier Pironneau 380
Numerical Optimal Shape Design 380
An Academic Problem 380
Sensitivity Analysis 381
Conceptual Algorithm 382
Automatic Differentiation 383
Principle of Automatic Differentiation 383
Example of Application 384
Differentiability Issues 385
Extended Calculus of Variation 385
Sensitivity Analysis for Burgers' Equation 386
Application to Optimal Control 386
A Simple Example 387
Right and Wrong Schemes 387
Small Disturbances and Automatic Differentiations 389
References 390
Differential Games Treated by a Gradient-Restoration Approach 391
Mauro Pontani 391
Introduction 391
Zero-Sum Differential Games 392
Numerical Solution of Two-Sided Optimization Problems 394
Transformation into Single-Objective Problem 394
Sequential Gradient-Restoration Algorithm 396
Homicidal Chauffeur Game 397
Formulation of the Problem 397
Method of Solution 398
Numerical Results 399
Orbital Pursuit-Evasion Game 400
Method of Solution 402
Numerical Results 404
Conclusions 407
References 407
Interval Methods for Optimal Control 409
Andreas Rauh and Eberhard P. Hofer 409
Introduction 410
Optimal and Robust Control of Dynamical Systems 411
Optimal Control of Discrete- and Continuous-Time Processes 412
Specification of Robustness in the Time Domain 413
Optimality Criteria for Systems with Uncertainties 414
Interval Arithmetic Optimization Algorithm 415
Parallelization of the Optimization Algorithm 417
Combination with Classical Controller Design 418
Validated Modeling and Simulation of Dynamical Systems with State-Dependent Switchings 419
Optimization Results 422
Interval Algorithm for Structure Optimization 422
Linear State Controller for Improvement of Robustness 425
Interval Algorithm for Parameter Optimization 427
Conclusions and Outlook on Future Work 428
References 429
Application of Optimisation Algorithms to Aircraft Aerodynamics 431
Emanuele Rizzo and Aldo Frediani 431
Introduction 431
An Algorithm for the Search of Global Minima 436
Test Cases 440
Test Case 1 (Unconstrained): Ackley's Function 440
Test Case 2 (Unconstrained): Rastrigin's Function 443
Test Case 3 (Unconstrained): Rosenbrock's Function 443
Test Case 4 (Unconstrained): Schwefel's Function 444
The AEROSTATE Program: An Application to Aeronautics 446
Minimum Induced Drag of a Wing 447
Minimum Total Drag of a Wing 450
The Trimmed Aircraft 451
The PrandtlPlane 453
Conclusions 457
References 457
Different levels of Optimisation in Aircraft Design 459
Dieter Schmitt 459
Air Transport System 460
Industrial Process of Aircraft Design 461
Different Levels of Aircraft Design vs. Development Phases 464
Tools Used in Different Phases 467
Conclusion 471
References 471
Numerical and Analytical Methods for Global Optimization 472
Paolo Teofilatto and Mauro Pontani 472
Introduction 472
Green's Theorem Approach 474
Morse Theory Approach 480
Final Comments 485
References 485
The Aeroservoelasticity Qualification Process in Alenia 487
Vincenzo Vaccaro 487
Introduction 487
Company Presentation 488
What Is Aeroelasticity 489
Aeroelastic Tradition in Alenia 490
Aeroservoelastic Certification Process 491
Analytical Models 492
Theoretical Background 494
Ground Test 496
Flight Test 496
Research and Future Developments 496
Further Steps Towards Quantitative Conceptual Aircraft Design 500
Michel van Tooren, Gianfranco La Rocca and Teodor Chiciudean 500
Introduction 500
The Systems Engineering Approach 505
Requirements on Computational Systems 505
The Design and Engineering Engine Concept 506
Describing Design Options 506
The Initiator 510
The Multi-model Generator 512
The Life-Cycle Analysis with Expert Tools 513
The Converger/Evaluator 513
The Agent-Based Framework 513
Results and Discussion 514
Conclusions 516
References 517
Some Plebeian Variational Problems 518
Piero Villaggio 518
Introduction 518
Mechanical Plebeian Problems 519
Locomotion 522
Peeling and Cooking 524
Conclusions 527
References 527