Introduction to Mathematical Modeling

John Tinsley Oden, PhD, is Associate Vice President for Research and Director of the Institute for Computational Engineering and Sciences (ICES) at The University of Texas at Austin. He was the founding Director of the Institute, which was created in January of 2003 as an expansion of the Texas Institute for Computational and Applied Mathematics. A member of the U.S. National Academy of Engineering, the National Academies of Engineering of Mexico and of Brazil, and The American Academy of Arts and Sciences, he serves on numerous national and international organizational, scientific, and advisory committees including the NSF Blue Ribbon Panel on Simulation-Based Engineering Science and the Task Force on Cyber Science and Grand Challenge Communities and Virtual Organizations. Dr. Oden has worked extensively on the mathematical theory and implementation of numerical methods applied to problems in solid and fluid mechanics and, particularly, nonlinear continuum mechanics and, in recent years, multi-scale modeling, stochastic systems, and uncertainty quantification.

• Preface xiiiI Nonlinear Continuum Mechanics 11 Kinematics of Deformable Bodies 31.1 Motion 41.2 Strain and Deformation Tensors 71.3 Rates of Motion 101.4 Rates of Deformation 131.5 The Piola Transformation 151.6 The Polar Decomposition Theorem 191.7 Principal Directions and Invariants of Deformation and Strain 201.8 The Reynolds' Transport Theorem 232 Mass and Momentum 252.1 Local Forms of the Principle of Conservation of Mass 262.2 Momentum 283 Force and Stress in Deformable Bodies 294 The Principles of Balance of Linear and Angular Momentum 354.1 Cauchy's Theorem: The Cauchy Stress Tensor 364.2 The Equations of Motion (Linear Momentum) 384.3 The Equations of Motion Referred to the Reference Configuration: The Piola-Kirchhoff Stress Tensors 404.4 Power 425 The Principle of Conservation of Energy 455.1 Energy and the Conservation of Energy 455.2 Local Forms of the Principle of Conservation of Energy 476 Thermodynamics of Continua and the Second Law 497 Constitutive Equations 537.1 Rules and Principles for Constitutive Equations 547.2 Principle of Material Frame Indifference 577.2.1 Solids 577.2.2 Fluids 597.3 The Coleman-Noll Method: Consistency with the Second Law of Thermodynamics 608 Examples and Applications 638.1 The Navier-Stokes Equations for Incompressible Flow 638.2 Flow of Gases and Compressible Fluids: The Compressible Navier-Stokes Equations 668.3 Heat Conduction 678.4 Theory of Elasticity 69II Electromagnetic Field Theory and Quantum Mechanics 739 Electromagnetic Waves 759.1 Introduction 759.2 Electric Fields 759.3 Gauss's Law 799.4 Electric Potential Energy 809.4.1 Atom Models 809.5 Magnetic Fields 819.6 Some Properties of Waves 849.7 Maxwell's Equations 879.8 Electromagnetic Waves 9110 Introduction to Quantum Mechanics 9310.1 Introductory Comments 9310.2 Wave and Particle Mechanics 9410.3 Heisenberg's Uncertainty Principle 9710.4 Schrödinger's Equation 9910.4.1 The Case of a Free Particle 9910.4.2 Superposition in Rn 10110.4.3 Hamiltonian Form 10210.4.4 The Case of Potential Energy 10210.4.5 Relativistic Quantum Mechanics 10210.4.6 General Formulations of Schrödinger's Equation 10310.4.7 The Time-Independent Schrödinger Equation 10410.5 Elementary Properties of the Wave Equation 10410.5.1 Review 10410.5.2 Momentum 10610.5.3 Wave Packets and Fourier Transforms 10910.6 The Wave-Momentum Duality 11010.7 Appendix: A Brief Review of Probability Densities 11111 Dynamical Variables and Observables in Quantum Mechanics: The Mathematical Formalism 11511.1 Introductory Remarks 11511.2 The Hilbert Spaces L2(R) (or L2(Rd)) and H1(R) (or H1(Rd)) 11611.3 Dynamical Variables and Hermitian Operators 11811.4 Spectral Theory of Hermitian Operators: The Discrete Spectrum 12111.5 Observables and Statistical Distributions 12511.6 The Continuous Spectrum 12711.7 The Generalized Uncertainty Principle for Dynamical Variables 12811.7.1 Simultaneous Eigenfunctions 13012 Applications: The Harmonic Oscillator and the Hydrogen Atom 13112.1 Introductory Remarks 13112.2 Ground States and Energy Quanta: The Harmonic Oscillator 13112.3 The Hydrogen Atom 13312.3.1 Schrödinger Equation in Spherical Coordinates 13512.3.2 The Radial Equation 13612.3.3 The Angular Equation 13812.3.4 The Orbitals of the Hydrogen Atom 14012.3.5 Spectroscopic States 14013 Spin and Pauli's Principle 14513.1 Angular Momentum and Spin 14513.2 Extrinsic Angular Momentum 14713.2.1 The Ladder Property: Raising and Lowering States 14913.3 Spin 15113.4 Identical Particles and Pauli's Principle 15513.5 The Helium Atom 15813.6 Variational Principle 16114 Atomic and Molecular Structure 16514.1 Introduction 16514.2 Electronic Structure of Atomic Elements 16514.3 The Periodic Table 16914.4 Atomic Bonds and Molecules 17314.5 Examples of Molecular Structures 18015 Ab Initio Methods: Approximate Methods and Density Functional Theory 18915.1 Introduction 18915.2 The Born-Oppenheimer Approximation 19015.3 The Hartree and the Hartree-Fock Methods 19415.3.1 The Hartree Method 19615.3.2 The Hartree-Fock Method 19615.3.3 The Roothaan Equations 19915.4 Density Functional Theory 20015.4.1 Electron Density 20015.4.2 The Hohenberg-Kohn Theorem 20515.4.3 The Kohn-Sham Theory 208III Statistical Mechanics 21316 Basic Concepts: Ensembles, Distribution Functions, and Averages 21516.1 Introductory Remarks 21516.2 Hamiltonian Mechanics 21616.2.1 The Hamiltonian and the Equations of Motion 21816.3 Phase Functions and Time Averages 21916.4 Ensembles, Ensemble Averages, and Ergodic Systems 22016.5 Statistical Mechanics of Isolated Systems 22416.6 The Microcanonical Ensemble 22816.6.1 Composite Systems 23016.7 The Canonical Ensemble 23416.8 The Grand Canonical Ensemble 23916.9 Appendix: A Brief Account of Molecular Dynamics 24016.9.1 Newtonian's Equations of Motion 24116.9.2 Potential Functions 24216.9.3 Numerical Solution of the Dynamical System 24517 Statistical Mechanics Basis of Classical Thermodynamics 24917.1 Introductory Remarks 24917.2 Energy and the First Law of Thermodynamics 25017.3 Statistical Mechanics Interpretation of the Rate of Work in Quasi-Static Processes 25117.4 Statistical Mechanics Interpretation of the First Law of Thermodynamics 25417.4.1 Statistical Interpretation of Q 25617.5 Entropy and the Partition Function 25717.6 Conjugate Hamiltonians 25917.7 The Gibbs Relations 26117.8 Monte Carlo and Metropolis Methods 26217.8.1 The Partition Function for a Canonical Ensemble 26317.8.2 The Metropolis Method 26417.9 Kinetic Theory: Boltzmann's Equation of Nonequilibrium Statistical Mechanics 26517.9.1 Boltzmann's Equation 26517.9.2 Collision Invariants 26817.9.3 The Continuum Mechanics of Compressible Fluids and Gases: The Macroscopic Balance Laws 269Exercises 273Bibliography 317Index 325

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