Book contents
- Frontmatter
- Contents
- Acknowledgements
- Introduction
- 1 The continuity equation
- 2 Density and gravity
- 3 Numerical solutions of partial differential equations
- 4 Stress and strain
- 5 The momentum equation
- 6 Viscous rheology of rocks
- 7 Numerical solutions of the momentum and continuity equations
- 8 The advection equation and marker-in-cell method
- 9 The heat conservation equation
- 10 Numerical solution of the heat conservation equation
- 11 2D thermomechanical code structure
- 12 Elasticity and plasticity
- 13 2D implementation of visco-elasto-plastic rheology
- 14 The multigrid method
- 15 Programming of 3D problems
- 16 Numerical benchmarks
- 17 Design of 2D numerical geodynamic models
- Epilogue: outlook
- Appendix: MATLAB program examples
- References
- Index
15 - Programming of 3D problems
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Acknowledgements
- Introduction
- 1 The continuity equation
- 2 Density and gravity
- 3 Numerical solutions of partial differential equations
- 4 Stress and strain
- 5 The momentum equation
- 6 Viscous rheology of rocks
- 7 Numerical solutions of the momentum and continuity equations
- 8 The advection equation and marker-in-cell method
- 9 The heat conservation equation
- 10 Numerical solution of the heat conservation equation
- 11 2D thermomechanical code structure
- 12 Elasticity and plasticity
- 13 2D implementation of visco-elasto-plastic rheology
- 14 The multigrid method
- 15 Programming of 3D problems
- 16 Numerical benchmarks
- 17 Design of 2D numerical geodynamic models
- Epilogue: outlook
- Appendix: MATLAB program examples
- References
- Index
Summary
Theory: Formulation of thermomechanical problems in 3D and its numerical implementation. Numerical methods for solving temperature, Poisson, momentum and continuity equations in 3D.
Exercises: Programming of numerical methods for temperature and Poisson equations and coupled solving of momentum and continuity equations in 3D.
Why simply not always 3D?
We know very well that the Earth is a 3D, nearly spherical object and, therefore, all the dynamic processes inside our planet are inherently three dimensional. Therefore, it is very logical to assume that realistic geodynamic modelling should always be done in 3D. Also, if you talk to geoscientists studying various natural geological objects, you are frequently told that such objects can only be modelled in 3D. This is a normal expectation since they are perfectly aware of the spatial 2D variability of geological structures on the Earth's surface and they thus know that a similar variability also exists in depth. Therefore, 3D modelling appears to be the natural choice for ‘observers’. What about ‘modellers’? Why don't they always use 3D modelling? What's wrong with it? The ‘uncensored’ truth about 3D modelling is the following:
3D thermomechanical modelling is quite easy from a methodological point of view – it is fairly straightforward to formulate and discretise the governing equations in a 3D Cartesian geometry for both simple viscous and more realistic visco-elasto-plastic rheologies (especially by using the same relatively simple finite-differences and marker-in-cell techniques that we extensively discussed in this book).
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- Introduction to Numerical Geodynamic Modelling , pp. 221 - 240Publisher: Cambridge University PressPrint publication year: 2009