Smooth Particle Applied Mechanics: The State of the Art

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Overview

This book takes readers through all the steps necessary for solving hard problems in continuum mechanics with smooth particle methods. Pedagogical problems clarify the generation of initial conditions, the treatment of boundary conditions, the integration of the equations of motion, and the analysis of the results. Particular attention is paid to the parallel computing necessary for large problems and to the graphic displays, including debugging software, required for the efficient completion of computational projects.

The book is self-contained, with summaries of classical particle mechanics and continuum mechanics for both fluids and solids, computer languages, the stability of numerical methods, Lyapunov spectra, and message-passing parallel computing. The main difficulties faced by meshless particle methods are discussed and the means of overcoming them are illustrated with worked examples.

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Product Details

  • ISBN-13: 9789812700025
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 3/28/2007
  • Series: Advanced Series in Nonlinear Dynamics
  • Pages: 316
  • Product dimensions: 6.20 (w) x 9.00 (h) x 0.80 (d)

Table of Contents


Dedication and Motivation     v
Preface     vii
Physical Ideas Underlying SPAM     1
Motivation and Summary     1
Particles versus Continua     3
Newton's Particle Mechanics     4
Eulerian and Lagrangian Continuum Mechanics     9
Computer Simulation of Microscopic Particle Motion     14
Liouville's Theorem; Statistical Mechanics     16
Simulating Continua with Particles     20
SPAM [ Smooth Particle Applied Mechanics ]     24
Example: A Molecular Dynamics Simulation     26
References     31
Continuum Mechanics     33
Summary and Scope of Continuum Mechanics     33
Evolution Equations for Fluids and Solids     35
Initial and Boundary Conditions     39
Constitutive Equations for Equilibrium Fluids     42
Constitutive Relations for Nonequilibrium Fluids     45
Artificial Viscosity and Conductivity     46
Constitutive Relations for Elastic Solids     48
Constitutive Relation for Nonequilibrium Plasticity     52
Plasticity Algorithm     55
Example: Heat Conduction in One Dimension     58
Example: Sound Propagation in OneDimension     59
Example: Rayleigh-Benard Flow in Two Dimensions     60
References     64
Smooth Particle Methods     65
Summary     65
Motivation     66
Basic Equations     67
Interpolation on an Irregular Grid     68
Alternative Averages: [ f0, f1, f2,...]     71
Weight Functions     74
Continuity Equation from [nabla middot] v with SPAM     80
Evaluating the Spatial Derivatives {lcub}[nabla rho], [nabla middot] P, [nabla middot] Q{rcub}     82
SPAM Equation of Motion and Energy Equation     83
Rezoning; Does Particle Size Matter?     84
Ideal-Gas Isomorphism with SPAM     85
Evaluating the Spatial Derivatives {lcub}[nabla]v, [nabla]T{rcub}     87
von Neumann-Richtmyer Artificial SPAM Viscosity     89
Example: Adiabatic Atmospheric Equilibrium     91
Example: Isothermal Atmospheric Equilibrium     94
References     97
Computer Programming     99
Summary     99
FORmula TRANslation languages     100
Designing a SPAM program     105
Runge-Kutta Integration with Fortran and C     112
A Useful Random Number Generator     117
Graphic Displays and Analysis     119
"Debugging" Tools-Finding Errors     125
Parallel Computing     128
Mesh Partitioning     131
Message Passing Techniques     133
Material Interfaces in Parallel Computing     136
Concentric Annuli Undergoing Rotation     137
Free Expansion Problem     138
Crushing of an Elastic-Plastic Sheet     139
Caricature of a Billiard Table     140
References     142
Initial and Boundary Conditions, Interpolation     143
Summary     143
Initial Coordinates     144
Mesh Generation for SPAM with Free Boundaries     147
Implementing Periodic and Mirror Boundaries     150
Alternative Meshes-Regular Lattices     156
Elastic Stability of Embedded-Atom Lattices     157
Invariant Curvature Crystal Stabilization     162
Example: Heat Transfer in One Dimension with SPAM     164
Example: Periodic Shear Flow with SPAM     167
Example: Rayleigh-Benard Flow with SPAM     171
References     175
Convergence and Stability     177
Summary     177
Existence and Uniqueness in Continuum Mechanics     178
Accuracy and Precision in Numerical Solutions     180
Convergence of Numerical Methods     180
Runge-Kutta Integration of Linear Problems     181
Stability     184
Lyapunov Instability     186
Stability Analysis for a Chaotic Problem     188
Size Dependence: Lessons from Molecular Dynamics     190
Smooth-Particle Spatial Integration Errors     191
Lattice Instability     192
Even-Odd Instability     195
Example: Shear-Flow Convergence     196
References     199
Lucy and Embedded-Atom Fluids     201
Summary     201
Trajectory Isomorphism for the Lucy Fluid     202
Statistical Thermodynamics for the Lucy Potential     203
Trajectory Isomorphism for the Embedded-Atom Fluid     205
Embedded-Atom Approach to Structural Relaxation     207
Example: Embedded-Atom Gravitational Relaxation     208
Example: Embedded-Atom Model of Falling Water     211
Example: Free Expansion of a [gamma]-law Gas     213
Example: Lucy-Fluid Shockwave Structure     217
References     225
SPAM: Limitations and Difficulties     227
Summary     227
Surface Tension     227
Tensile Instability     231
Monaghan's Motion Equations     233
Continuum Mechanics: Stress; Rigid-Body Rotation     236
Dynamic and Static Constitutive Relations     237
Example Deformations with Stress and Strain Rates     241
Dynamics with Jaumann's Stress Rotation Rate     244
Conservation of Angular Momentum     246
Artificial Transport Coefficients     248
Residual Stress-Artificial Plasticity in SPAM     249
References     251
SPAM: Sample Applications to Solids     253
Summary     253
The Tension Test     254
Tension Test via Standard Molecular Dynamics     256
Boundary Conditions for Tension     257
Initial Conditions for Tension Using SPAM     260
Tension Test via SPAM-like Molecular Dynamics     261
Tension Test via SPAM     263
Failure Algorithms     266
Penetration Mechanics     266
Penetration via Continuum Mechanics     267
Penetration via Standard Molecular Dynamics     270
Penetration via SPAM-like Molecular Dynamics      271
Penetration via SPAM     272
A Research Suggestion     275
References     276
Summary, Literature, and Outlook     277
Introduction     277
Current State of the Art     278
Cutting and Machining     279
Structural Response to Waves     280
Dynamics of Sea Ice     281
Astrophysics     283
The Near Future of Parallel Computing     285
An Afterword     286
References     287
Alphabetical Bibliography     289
Index     295
Example Problem List     299
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