The expanded 3rd edition of this established textbook offers an updated overview and review of the computational physics techniques used in materials modelling over different length and time scales. It describes in detail the theory and application of some of the most important methods used to simulate materials across the various levels of spatial and temporal resolution. Quantum mechanical methods such as the Hartree-Fock approximation for solving the Schrödinger equation at the smallest spatial resolution are discussed as well as the Molecular Dynamics and Monte-Carlo methods on the micro- and meso-scale up to macroscopic methods used predominantly in the Engineering world such as Finite Elements (FE) or Smoothed Particle Hydrodynamics (SPH).
Extensively updated throughout, this new edition includes additional sections on polymer theory, statistical physics and continuum theory, the latter being the basis of FE methods and SPH. Each chapter now first provides an overview of the key topics covered, with a new “key points” section at the end. The book is aimed at beginning or advanced graduate students who want to enter the field of computational science on multi-scales. It provides an in-depth overview of the basic physical, mathematical and numerical principles for modelling solids and fluids on the micro-, meso-, and macro-scale. With a set of exercises, selected solutions and several case studies, it is a suitable book for students in physics, engineering, and materials science, and a practical reference resource for those already using materials modelling and computational methods in their research.
Author(s): Martin Oliver Steinhauser
Series: Graduate Texts in Physics
Edition: 3
Publisher: Springer Nature
Year: 2022
Language: English
Pages: 432
City: Cham
Tags: Numerical Simulation, Multiscale Modeling, Hartree-Fock Approximation, Molecular Dynamics Simulations, Tensor Analysis, Continuum Mechanics
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition in 2008
Contents
Acronyms
List of Algorithms
List of Boxes
blackPart I Fundamentals-1pt
1 Introduction to Multiscale Modeling
1.1 Physics on Different Length- and Timescales
1.1.1 Electronic/Atomic Scale
1.1.2 Atomic/Microscopic Scale
1.1.3 Microscopic/Mesoscopic Scale
1.1.4 Mesoscopic/Macroscopic Scale
1.2 What are Fluids and Solids?
1.3 The Objective of Experimental and Theoretical Physics
1.4 Computer Simulations—A Review
1.4.1 A Brief History of Computer Simulation
1.4.2 Computational Materials Science
1.5 Suggested Reading
2 Multiscale Computational Materials Science
2.1 Some Terminology
2.2 What is Computational Material Science on Multiscales?
2.2.1 Experimental Investigations on Different Length Scales
2.3 What is a Model?
2.3.1 The Scientific Method
2.4 Hierarchical Modeling Concepts Above the Atomic Scale
2.4.1 Example: Principle Model Hierarchies in Classical Mechanics
2.4.2 Structure-Property Paradigm
2.4.3 Physical and Mathematical Modeling
2.4.4 Numerical Modeling and Simulation
2.5 Unifications and Reductionism in Physical Theories
2.5.1 The Four Fundamental Interactions
2.5.2 The Standard Model
2.5.3 Symmetries, Fields, Particles and the Vacuum
2.5.4 Relativistic Wave Equations
2.5.5 Suggested Reading
2.6 Computer Science, Algorithms, Computability and Turing Machines
2.6.1 Recursion
2.6.2 Divide-and-Conquer
2.6.3 Local Search
2.6.4 Simulated Annealing and Stochastic Algorithms
2.6.5 Computability, Decidability and Turing Machines
2.6.6 Efficiency of Algorithms
3 Mathematical and Physical Prerequisites
3.1 Introduction
3.2 Sets and Set Operations
3.2.1 Cartesian Product, Product Set
3.2.2 Functions and Linear Spaces
3.3 Topological Spaces
3.3.1 Charts
3.3.2 Atlas
3.3.3 Manifolds
3.3.4 Tangent Vectors and Tangent Space
3.3.5 Covectors, Cotangent Space and One-Forms
3.3.6 Dual Spaces
3.3.7 Tensors and Tensor Spaces
3.3.8 Affine Connections and Covariant Derivative
3.4 Metric Spaces and Metric Connection
3.5 Riemannian Manifolds
3.5.1 Riemannian Curvature
3.6 The Problem of Inertia and Motion: Coordinate Systems in Physics
3.6.1 The Special and General Principle of Relativity
3.6.2 The Structure of Spacetime
3.7 Relativistic Field Equations
3.7.1 Relativistic Hydrodynamics
3.8 Suggested Reading
4 Fundamentals of Numerical Simulation
4.1 Basics of Ordinary and Partial Differential Equations in Physics
4.1.1 Elliptic Type
4.1.2 Parabolic Type
4.1.3 Hyperbolic Type
4.2 Numerical Solution of Differential Equations
4.2.1 Mesh-Based and Mesh-Free Methods
4.2.2 Finite Difference Methods
4.2.3 Finite Volume Method
4.2.4 Finite Element Methods
4.3 Elements of Software Design
4.3.1 Software Design
4.3.2 Writing a Routine
4.3.3 Code-Tuning Strategies
4.3.4 Suggested Reading
blackPart II Computational Methods on Multiscales-1pt
5 Computational Methods on Electronic/Atomistic Scale
5.1 Introduction
5.1.1 Scale Separation
5.2 Ab-Initio Methods
5.3 Physical Foundations of Quantum Theory
5.3.1 A Short Description of Quantum Theory
5.3.2 A Hamiltonian for a Condensed Matter System
5.3.3 The Born-Oppenheimer Approximation
5.4 Density Functional Theory
5.5 Car-Parinello Molecular Dynamics
5.5.1 Force Calculations: The Hellmann-Feynman Theorem
5.5.2 Calculating the Ground State
5.6 Solving Schrödinger's Equation for Many-Particle Systems: …
5.6.1 The Hartree-Fock Approximation
5.7 What Holds a Solid Together?
5.7.1 Homonuclear Diatomic Molecules
5.8 Semi-empirical Methods
5.8.1 Tight-Binding Method
5.9 Bridging Scales: Quantum Mechanics (QM) - Molecular Mechanics (MM)
5.10 Concluding Remarks
6 Computational Methods on Atomistic/Microscopic Scale
6.1 Introduction
6.1.1 Thermodynamics and Statistical Ensembles
6.2 Fundamentals of Statistical Physics and Thermodynamics
6.2.1 Probabilities
6.2.2 Measurements and the Ergodic Hypotheses
6.2.3 Statistics in Phase Space and Statistical Ensembles
6.2.4 Virtual Ensembles
6.2.5 Entropy and Temperature
6.3 Classical Interatomic and Intermolecular Potentials
6.3.1 Charged Systems
6.3.2 Ewald Summation
6.3.3 The P3M Algorithm
6.3.4 Van der Waals Potential
6.3.5 Covalent Bonds
6.3.6 Embedded Atom Potentials
6.3.7 Pair Potentials
6.4 Classical Molecular Dynamics Simulations
6.4.1 Numerical Ingredients of MD Simulations
6.4.2 Integrating the Equations of Motion
6.4.3 Periodic Boundary Conditions
6.4.4 The Minimum Image Convention
6.4.5 Efficient Search Strategies for Interacting Particles
6.4.6 Making Measurements
6.5 Liquids, Soft Matter and Polymers
6.5.1 Bonded Interactions
6.5.2 Scaling and Universality of Polymers
6.6 Monte Carlo Simulations
7 Computational Methods on Mesoscopic/Macroscopic Scale
7.1 Example: Meso- and Macroscale Shock-Wave Experiments
7.2 Statistical Methods: Voronoi Tesselations and Power Diagrams for Modeling …
7.2.1 Reverse Monte Carlo Optimization
7.3 Dissipative Particle Dynamics
7.4 Ginzburg–Landau/Cahn–Hiliard Field Theoretic Mesoscale Simulation Method
7.5 Bridging Scales: Soft Particle Discrete Elements for Shock Wave Applications
7.6 Bridging Scales: Energetic Links Between MD and FEM
7.6.1 Bridging Scales: Work-Hardening
7.7 Physical Theories for Macroscopic Phenomena: The Continuum Approach
7.7.1 The Description of Fluid Motion
7.8 Continuum Theory
7.8.1 The Continuum Hypothesis
7.9 Theory of Elasticity
7.9.1 Kinematic Equations
7.9.2 The Stress Tensor
7.9.3 Equations of Motion of the Theory of Elasticity
7.9.4 Constitutive Equations
7.10 Bridging Scale Application: Crack Propagation
8 Perspectives in Multiscale Materials Modeling
A Further Reading
A.1 Foundations of Physics
A.2 Programming Techniques
A.3 Journals and Conferences on Multiscale Materials Modeling and Simulation
B Mathematical Definitions
C Sample Code for the Main Routine in a MD Program
D A Sample Makefile
E Tables of Physical Constants
E.1 International System of Units (SI or mksA System)
E.2 Conversion Factors of Energy
References
Index
