Discrete Mechanics

Preface xi Introduction xiii List of Symbols xxi Chapter 1. Fundamental Principles of Discrete Mechanics 1 1.1. Definitions of discrete mechanics 1 1.1.1. Notion of discrete space-time 1 1.1.2. Notion of a discrete medium 4 1.2. Properties of discrete operators 6 1.3. Invariance under translation and rotation 9 1.4. Weak equivalence principle 11 1.5. Principle of accumulation of stresses 13 1.6. Duality-of-action principle 14 1.7. Physical characteristics of a medium 16 1.8. Composition of velocities and accelerations 20 1.9. Discrete curvature 23 1.10. Axioms of discrete mechanics 28 Chapter 2. Conservation of Acceleration 31 2.1. General principles 31 2.2. Continuous memory 34 2.3. Modeling the compression stress 37 2.3.1. Compression experiment 37 2.3.2. Modeling the stress in a solid 39 2.3.3. Modeling the stress in a fluid 39 2.3.4. Compression with small time constants 41 2.3.5. Modeling the accumulation of the normal stress 42 2.3.6. The energy formula, e = mc2 43 2.4. Modeling the rotation stress 44 2.4.1. Couette's experiment 44 2.4.2. Behavior over time 45 2.4.3. Rotation stress in solids 46 2.4.4. Rotation stress in fluids 47 2.4.5. Stresses in a porous medium, Darcy's law 47 2.4.6. Modeling the accumulation of the rotation stress 48 2.4.7. Rotation in Couette and Poiseuille flows 49 2.5. Modeling other effects 49 2.5.1. Gravitational effects 50 2.5.2. Inertial effects 52 2.6. Discrete equations of motion 56 2.6.1. Geometric description 56 2.6.2. Derivation of the equations of motion 58 2.6.3. Dissipation of energy 60 2.7. Coupling conditions 63 2.8. Formulation of the equations of motion at a discontinuity 65 2.9. Other forms of the equations of motion 66 2.9.1. Curl and vector potential formulation 67 2.9.2. Conservative form of the equations of motion 69 2.10. Incompressible models derived from the discrete formulation 70 2.10.1. Kinematic projection methods 70 2.10.2. Incompressibility in discrete mechanics 74 2.11. Consequences on the dynamics of the vorticity 74 Chapter 3. Conservation of Mass, Flux and Energy 77 3.1. Conservation of mass in a homogeneous medium 77 3.1.1. In continuum mechanics 78 3.1.2. In discrete mechanics 80 3.2. Transport within multicomponent mixtures 81 3.2.1. Classical approach 81 3.2.2. Discrete model for the transport of chemical species 84 3.2.3. Equilibrium in a binary mixture 86 3.3. Advection 88 3.4. Conservation of flux 89 3.4.1. General remarks 89 3.4.2. Model 90 3.5. Conservation of energy 93 3.5.1. Conservation of total energy 93 3.5.2. Conservation of kinetic energy 94 3.5.3. Conservation of internal energy 95 3.5.4. Monotonically decreasing kinetic energy 97 3.6. A complete system of equations 98 3.7. A simple heat conduction problem 99 3.7.1. Case of anisotropic materials 101 3.8. Phase change 102 3.8.1. The Stefan problem 103 3.8.2. Condensation 108 Chapter 4. Properties of the Discrete Formulation 115 4.1. Fundamental properties 115 4.1.1. Limitations on the velocity 115 4.1.2. Inverting the formulas V = and V = x 118 4.1.3. Material frame-indifference 121 4.1.4. Fundamental invariants 123 4.2. System of equations 124 4.3. Differences from continuum mechanics 126 4.3.1. Differences from the Navier-Lame equations 126 4.3.2. Differences from the Navier-Stokes equations 127 4.3.3. Dissipation 131 4.3.4. Compatibility conditions for the Navier-Stokes equations 133 4.4. Examples of analytic solutions of the equations of motion 136 4.4.1. Rigid rotational motion 136 4.4.2. Planar Couette flow 138 4.4.3. Poiseuille flow 140 4.4.4. Radial flow 144 4.5. Incompressible motion 145 4.5.1. The Green-Taylor vortex 145 4.5.2. Lid-driven cavity 148 4.6. Compressible fluids and perfect fluids 150 4.6.1. Generalized Bernoulli equation 151 4.6.2. Propagation of linear waves 152 4.6.3. Sod shock tube 155 4.7. Statics of fluids and solids 158 4.8. Conditions for modeling a rigid solid 159 4.9. Flows in a porous medium 160 4.10. Stretching of space-time and Hugoniot's theorem 165 Chapter 5. Two-Phase Flows, Capillarity and Wetting 169 5.1. Formulation of the equations at the interfaces 169 5.1.1. Modeling the curvature 170 5.1.2. Formulation of the equations of motion 174 5.2. Two-phase flows 179 5.2.1. Two-phase Poiseuille flow 179 5.2.2. Sloshing of two immiscible fluids 181 5.3. Capillarity-dominated flows 187 5.3.1. The Laplace problem 187 5.3.2. Oscillating ellipse 188 5.3.3. Marangoni-type flow in a droplet 190 5.3.4. Interacting bubbles 192 5.3.5. Simulating foam in equilibrium 194 5.4. Partial wetting 195 5.4.1. Droplet in equilibrium on a plane 198 5.4.2. Spreading of a droplet 200 5.4.3. Droplet acted upon by gravity 204 5.4.4. Flows within a lens 205 5.4.5. Capillary ascension in a tube 206 Chapter 6. Stresses and Strains in Solids 209 6.1. Discrete solid medium 209 6.2. Stresses in solids 211 6.2.1. Discrete equations 212 6.2.2. Material frame-indifference 214 6.2.3. Solid statics equations 215 6.2.4. Calculating the displacement 217 6.3. Properties of solid media 219 6.3.1. In continuum mechanics 220 6.3.2. In discrete mechanics 223 6.4. Boundary conditions 225 6.5. Rigid motion 228 6.6. Validation of the model on examples 230 6.6.1. Simple example of a monolithic fluid-structure interaction 230 6.6.2. Mechanical equilibrium of sloshing 233 6.6.3. Beam under extension 235 6.6.4. Multimaterial compression 237 6.6.5. Planar shearing 238 6.6.6. Flexing beam 239 6.6.7. Settling of a block under gravity 240 6.6.8. Mechanical equilibrium of a solid object 242 6.6.9. Extension to other constitutive laws 243 6.7. Toward a unification of solid and fluid mechanics 246 Chapter 7. Multiphysical Extensions 249 7.1. Deflection of light 249 7.1.1. Description of the physical phenomenon 250 7.1.2. Deflection of light by the Sun in Newtonian mechanics 252 7.1.3. Deflection of light by the Sun according to the duality-of-action principle 256 7.1.4. Deflection of light by the Sun in a one-dimensional approach 257 7.2. On a discrete approach to turbulence 262 7.2.1. General remarks about the approach 262 7.2.2. Dynamics of the vorticity in two spatial dimensions 264 7.2.3. Analysis of a turbulent flow in a planar channel 266 7.2.4. Model of the turbulence in discrete mechanics 271 7.2.5. Application to a flow in a channel with Re = 590 272 7.3. The lid-driven cavity problem with Re = 5,000 280 7.4. Natural convection into the non-Boussinesq approximation 283 7.5. Fluid-structure interaction 286 References 289 Index 295

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