文件名称：Variational Methods with Applications in Science and Engineering
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更新时间：2019-02-25 03:42:27
Variational Applications
1 Preliminaries 3 1.1 A Bit of History 4 1.2 Introduction 7 1.3 Motivation 8 1.3.1 Optics 8 1.3.2 Shape of a Liquid Drop 10 1.3.3 Optimization of a River-Crossing Trajectory 12 1.3.4 Summary 14 1.4 Extrema of Functions 14 1.5 Constrained Extrema and Lagrange Multipliers 17 1.6 Integration by Parts 20 1.7 Fundamental Lemma of the Calculus of Variations 21 1.8 Adjoint and Self-Adjoint Differential Operators 22 Exercises 26 2 Calculus of Variations 28 2.1 Functionals of One Independent Variable 29 2.1.1 Functional Derivative 30 2.1.2 Derivation of Euler’s Equation 31 2.1.3 Variational Notation 33 2.1.4 Special Cases of Euler’s Equation 37 2.2 Natural Boundary Conditions 44 2.3 Variable End Points 53 2.4 Higher-Order Derivatives 56 2.5 Functionals of Two Independent Variables 56 2.5.1 Euler’s Equation 57 2.5.2 Minimal Surfaces 61 2.5.3 Dirichlet Problem 62 vii viii Contents 2.6 Functionals of Two Dependent Variables 64 2.7 Constrained Functionals 66 2.7.1 Integral Constraints 66 2.7.2 Sturm-Liouville Problems 74 2.7.3 Algebraic and Differential Constraints 76 2.8 Summary of Euler Equations 80 Exercises 81 3 Rayleigh-Ritz, Galerkin, and Finite-Element Methods 90 3.1 Rayleigh-Ritz Method 91 3.1.1 Basic Procedure 91 3.1.2 Self-Adjoint Differential Operators 94 3.1.3 Estimating Eigenvalues of Differential Operators 96 3.2 Galerkin Method 100 3.3 Finite-Element Methods 103 3.3.1 Rayleigh-Ritz–Based Finite-Element Method 104 3.3.2 Finite-Element Methods in Multidimensions 109 Exercises 110 P A R T I I P H Y S I C A L A P P L I C A T I O N S 115 4 Hamilton’s Principle 117 4.1 Hamilton’s Principle for Discrete Systems 118 4.2 Hamilton’s Principle for Continuous Systems 128 4.3 Euler-Lagrange Equations 131 4.4 Invariance of the Euler-Lagrange Equations 136 4.5 Derivation of Hamilton’s Principle from the First Law of Thermodynamics 137 4.6 Conservation of Mechanical Energy and the Hamiltonian 141 4.7 Noether’s Theorem – Connection Between Conservation Laws and Symmetries in Hamilton’s Principle 143 4.8 Summary 146 4.9 Brief Remarks on the Philosophy of Science 148 Exercises 152 5 Classical Mechanics 160 5.1 Dynamics of Nondeformable Bodies 161 5.1.1 Applications of Hamilton’s Principle 161 5.1.2 Dynamics Problems with Constraints 174 5.2 Statics of Nondeformable Bodies 178 5.2.1 Vectorial Approach 179 5.2.2 Virtual Work Approach 181 5.3 Statics of Deformable Bodies 184 5.3.1 Static Deflection of an Elastic Membrane – Poisson Equation 184 5.3.2 Static Deflection of a Beam 185 Contents ix 5.3.3 Governing Equations of Elasticity 189 5.3.4 Principle of Virtual Work 196 5.4 Dynamics of Deformable Bodies 197 5.4.1 Longitudinal Vibration of a Rod – Wave Equation 197 5.4.2 Lateral Vibration of a String – Wave Equation 199 6 Stability of Dynamical Systems 202 6.1 Introduction 202 6.2 Simple Pendulum 203 6.3 Linear, Second-Order, Autonomous Systems 207 6.4 Nonautonomous Systems – Forced Pendulum 212 6.5 Non-Normal Systems – Transient Growth 215 6.6 Continuous Systems – Beam-Column Buckling 222 7 Optics and Electromagnetics 225 7.1 Optics 225 7.2 Maxwell’s Equations of Electromagnetics 229 7.3 Electromagnetic Wave Equations 232 7.4 Discrete Charged Particles in an Electromagnetic Field 233 7.5 Continuous Charges in an Electromagnetic Field 237 8 Modern Physics 240 8.1 Relativistic Mechanics 241 8.1.1 Special Relativity 241 8.1.2 General Relativity 247 8.2 Quantum Mechanics 251 8.2.1 Schrödinger’s Equation 252 8.2.2 Density-Functional Theory 256 8.2.3 Feynman Path-Integral Formulation of Quantum Mechanics 257 9 Fluid Mechanics 259 9.1 Introduction 260 9.2 Inviscid Flow 262 9.2.1 Fluid Particles as Nondeformable Bodies – Bernoulli Equation 262 9.2.2 Fluid Particles as Deformable Bodies – Euler Equations 263 9.2.3 Potential Flow 266 9.3 Viscous Flow – Navier-Stokes Equations 269 9.4 Multiphase and Multicomponent Flows 275 9.4.1 Level-Set Methods 276 9.4.2 Phase-Field Models 278 9.5 Hydrodynamic Stability Analysis 281 9.5.1 Introduction 281 9.5.2 Linear Stability of the Navier-Stokes Equations 282 x Contents 9.5.3 Modal Analysis 283 9.5.4 Nonmodal Transient Growth (Optimal Perturbation) Analysis 289 9.5.5 Energy Methods 296 9.6 Flow Control 297 P A R T I I I O P T I M I Z A T I O N 301 10 Optimization and Control 303 10.1 Optimization and Control Examples 305 10.2 Shape Optimization 306 10.3 Financial Optimization 310 10.4 Optimal Control of Discrete Systems 312 10.4.1 Example: Control of an Undamped Harmonic Oscillator 312 10.4.2 Riccati Equation for the LQ Problem 320 10.4.3 Properties of Systems for Control 330 10.4.4 Pontryagin’s Principle 331 10.4.5 Time-Optimal Control 337 10.5 Optimal Control of Continuous Systems 342 10.5.1 Variational Approach 342 10.5.2 Adjoint Approach 346 10.5.3 Example: Control of Plane-Poiseuille Flow 347 10.6 Control of Real Systems 351 10.6.1 Open-Loop and Closed-Loop Control 352 10.6.2 Model Predictive Control 353 10.6.3 State Estimation and Data Assimilation 354 10.6.4 Robust Control 356 10.7 Postscript 356 Exercises 357 11 Image Processing and Data Analysis 361 11.1 Variational Image Processing 362 11.1.1 Denoising 367 11.1.2 Deblurring 368 11.1.3 Inpainting 368 11.1.4 Segmentation 370 11.2 Curve and Surface Optimization Using Splines 371 11.2.1 B-Splines 372 11.2.2 Spline Functionals 373 11.3 Proper-Orthogonal Decomposition 374 12 Numerical Grid Generation 379 12.1 Fundamentals 379 12.2 Algebraic Grid Generation 381 12.3 Elliptic Grid Generation 385 12.4 Variational Grid Adaptation 389 Contents xi 12.4.1 One-Dimensional Case 390 12.4.2 Two-Dimensional Case 398 Bibliography 403 Index 409