Introduction to Numerical Geodynamic Modelling
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About this book: Have you ever thought that modelling of geological processes is an exciting topic but too difficult to enter because there is no introductory textbook on this subject? Yes? Then come the good news! Here is the textbook written for you to learn numerical geodynamic modelling from scratch. It does not require any preliminary knowledge besides simple linear algebra and derivatives. It provides a consistent basic background in continuum mechanics, partial differential equations, numerical methods and geodynamic modelling. It is illustrated with 47 practical exercises and 67 MATLAB examples as successive and successful stages on your learning path. In addition, several state-of-the-art, well-commented visco-elasto-plastic codes are provided to allow numerical modelling in two-dimensions of several key geodynamic processes such as subduction, lithospheric extension, collision, slab breakoff, intrusion emplacement, mantle convection and planetary core formation. Below are keywords and the book content. |
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Keywords for this book: geodynamic models, models of tectonic plates, models of lithospheric processes, divergent plate model, convergent plate model, model of continental collision, subduction zone model, mantle convection model, igneous intrusion model, model of the core of Earth, rocks properties, Earth mantle properties, numerical modeling, applied numerical methods, finite differences, particle in cell, numerical code, numerical example, numerical analysis, numerical integration, numerical matlab codes, numerical tests |
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Book content
ACKNOWLEDGEMENTS
INTRODUCTION
Theory: What is this book? What this book is not? Get started. Seven
golden rules for learning the subject. Short history of geodynamics and
numerical geodynamic modelling. Few words about programming and
visualization. Nine programming rules.Exercises: Starting with MATLAB. Visualization exercise.
CHAPTER 1: THE CONTINUITY EQUATION
Theory: Definition of a geological media as a continuum. Field
variables used for the representation of a continuum. Methods for
definition of the field variables. Eulerian and Lagrangian points of
view. Continuity equation in Eulerian and Lagrangian forms and their
derivation. Advective transport term. Continuity equation for an
incompressible fluid.Exercises: Computing the divergence of velocity field in 2D.
CHAPTER 2: DENSITY AND GRAVITY
Theory: Density of rocks and minerals. Thermal expansion and
compressibility. Dependence of density on pressure and temperature.
Equations of state. Poisson equation for gravitational potential and
its derivation.Exercises: Computing and visualising density, thermal expansion and compressibility.
CHAPTER 3: NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
Theory: Analytical and numerical methods for solving partial
differential equations. Using finite-differences to compute various
derivatives. Eulerian and Lagrangian approaches. Transition from
partial differential equations to systems of linear equations. Methods
of solving large systems of linear equations: iterative methods (Jacobi
iteration, Gauss-Seidel iteration), direct methods (Gaussian
elimination). Indexing of unknowns in 1D and 2D.Exercises: Numerical solutions of Poisson equation in 1D and 2D.
CHAPTER 4: STRESS AND STRAIN
Theory: Deformation and stresses. Definition of stress, strain and
strain-rate tensors. Deviatoric stresses. Mean stress as a dynamic
(non-lithostatic) pressure. Stress and strain rate invariants.Exercises: Computing the strain rate tensor components in 2D from the material velocity fields.
CHAPTER 5: THE MOMENTUM EQUATION
Theory: Momentum equation. Viscosity and Newtonian law of viscous
friction. Navier-Stokes equation for the motion of a viscous fluid.
Stokes equation of slow laminar flow of highly viscous incompressible
fluid and its application to geodynamics. Simplification of the Stokes
equation in case of constant viscosity and its relation to the Poisson
equation. Analytical example for channel flow. Stream function approach.Exercises: Solving continuity and momentum equations for the case of constant viscosity with a stream function approach.
CHAPTER 6: VISCOUS RHEOLOGY OF ROCKS
Theory: Solid-state creep of minerals and rocks as the major mechanism
of deformation of the Earth’s interior. Dislocation and diffusion creep
mechanisms. Rheological equations for minerals and rocks. Effective
viscosity and it’s dependence on temperature, pressure, and strain
rate. Formulation of the effective viscosity from empirical flow laws.Exercises: Programming viscous rheological equations for computing effective viscosities from empirical flow laws.
CHAPTER 7: NUMERICAL SOLUTION OF THE MOMENTUM AND CONTINUITY EQUATIONS
Theory: Types of numerical grids and their applicability for different
differential equations. Staggered, half-staggered and non-staggered
grids in one, two and three dimensions. Discretisation of the
continuity and Stokes equations on a rectangular grid. Conservative and
non-conservative discretisation schemes for Stokes equations.
Mechanical boundary conditions and their numerical implementation. No
slip and free slip conditions. Exercises: Programming different mechanical boundary conditions. Solving continuity and momentum equations for the case of variable viscosity.
CHAPTER 8: THE ADVECTION EQUATION AND MARKER-IN-CELL METHOD
Theory: Advection equation. Solution methods for continuous and
discontinuous variables. Eulerian schemes: upwind differences, higher
order schemes, flux corrected transport (FCT). Lagrangian schemes:
marker-in-cell method. Runge-Kutta advection schemes. Numerical
interpolation schemes between markers and
nodes. Exercises: Programming of various advection schemes and markers
CHAPTER 9: HEAT CONSERVATION EQUATION
Theory: Fourier’s law of heat conduction. Heat conservation equation
and its derivation. Radioactive, viscous and adiabatic heating and
their relative importance. Heat conservation equation for the case of a
constant thermal conductivity and its relation to the Poisson equation.
Analytical examples: steady and non-steady temperature profiles in case
of channel flow.Exercises: Computing shear heating and adiabatic heating distribution for buoyancy driven flow.
CHAPTER 10: NUMERICAL SOLUTION OF THE HEAT CONSERVATION EQUATION
Theory: Discretisation of the heat conservation equation with finite
differences. Conservative and non-conservative discretisation schemes.
Explicit and implicit solution schemes of the heat conservation
equation. Advective terms: upwind differences, numerical diffusion.
Advection of temperature with markers. Subgrid diffusion. Thermal
boundary conditions: constant temperature, constant heat flow, combined
boundary conditions. Numerical implementation of thermal boundary
conditions. Exercises: Programming various thermal boundary conditions. Solving the heat conservation equation in the case of constant and variable thermal conductivity with explicit and implicit solution schemes. Advecting temperature with Eulerian schemes and markers.
CHAPTER 11: 2D THERMO-MECHANICAL CODE STRUCTURE
Theory: Principal steps of a coupled thermo-mechanical solution with
finite differences and marker-in-cell techniques. Organisation of a
thermo-mechanical code for the case of viscous, multi-component flows.
Adding self-gravity. Handling free planetary surfaces with weak layer
approach.Exercises: Building a 2D thermo-mechanical code.
CHAPTER 12: ELASTICITY AND PLASTICITY
Theory: Elastic rheology. Maxwell visco-elastic rheology. Rotation of
stresses during advection. Plastic rheology. Plastic yielding
criterion. Plastic flow potential. Plastic flow rule.Exercises: Stress buildup/relaxation with a visco-elastic Maxwell rheology.
CHAPTER 13: 2-D IMPLEMENTATION OF VISCO‑ELASTO-PLASTICITY
Theory: Numerical implementation of visco-elasto-plastic rheology.
Organisation of a thermo-mechanical code in case of 2D,
visco-elasto-plastic, multi-phase flows.Exercises: Programming a 2D thermo-mechanical code with a visco-elasto-plastic rheology.
CHAPTER 14: THE MULTIGRID METHOD
Theory: Principles of multigrid method. Multigrid method for solving
the Poisson equation in 2D. Coupled solving of momentum and continuity
equations in 2D with multigrid for the cases with constant and variable
viscosity.Exercises: Programming of multigrid methods for solving Poisson equation and coupled solving of momentum and continuity equations in 2D.
CHAPTER 15: PROGRAMMING OF 3-D PROBLEMS
Theory: Formulation of thermo-mechanical problems in 3-D and its
numerical implementation. Multigrid method for solving temperature,
Poisson, momentum and continuity equations in 3-DExercises: Programming of multigrid methods for temperature and Poisson equations and coupled solving of momentum and continuity equations in 3D.
CHAPTER 16: NUMERICAL BENCHMARKS
Theory: Numerical benchmarks: testing of numerical codes for various
problems. Examples of thermo-mechanical benchmarks.Exercises: Programming of models for various numerical benchmarks.
CHAPTER 17: DESIGN OF 2-D NUMERICAL GEODYNAMIC MODELS
Theory: Warning message! What numerical modeling is about? Rock
properties for numerical geodynamic models. Designing of numerical
models for different geodynamic processes: visco-elasto-plastic slab
bending, retreating subduction, lithospheric extension, collision, slab
detachment, intrusion emplacement, core formation. Comparison with
natural examples.Exercises: Designing numerical model for extension of the continental lithosphere.
EPILOGUE: OUTLOOK
Theory: Where are we now? Where to go further? Current and future
directions of numerical geodynamic modelling development: 3D, MPI,
OpenMp, PETSC, AMR, FEM, FVM, GPU/Cell-based computing, interactive
computing, realistic physics, visualization challenges etc.Exercises: No more exercises and home works!
APPENDIX
MATLAB PROGRAM EXAMPLES
REFERENCES
INDEX
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