Teaching: 540 Numerical Geodynamics

This advanced geodynamics class discusses the numerical solution of continuum mechanics problems as applied to geological processes in the solid Earth. Quantitative skills to be learned are useful for all Earth scientists, but the focus of the class is on tectonophysics and mantle convection problems. We briefly review math and continuum mechanics fundamentals, then discuss ordinary differential equations, and spend the majority of the class discussing finite difference and element solutions to partial differential equations. Every subject is accompanied by hands-on Matlab programming exrcises.

Geodynamics students should take the sequence of GEOL534 (Intro Geodynamics), 540, and 599 (Deep Earth Seminar).

Online material

• Complete set of GEOL540 Numerical Geodynamics lecture notes as PDF (alternatively, download all GEOL540 notes Latex and EPS input files and my Bibtex references
• All Matlab problem sets (gzipped tar) (solutions for instructors available from me by email)
• GEOL540 Material split up into weeks

Cource structure

• Term: Fall 2008
• Format: The class consists of lectures, computer exercises, student presentations, and discussion of readings.
• Instructor: Thorsten Becker
• Texts:
  • Required
    • Your own and my GEOL540 lecture notes, handouts
      
    • "Myths and Methods in Modeling" M. Spiegelman lecture notes
    • Spencer and Ware Matlab Intro
  • Recommended
    • "Numerical Recipes in C. The art of scientific computing." Press et al., Cambridge U Press, 2nd edition, 1992.
      This is the place to start before doing any numerical work and a standard reference. The current, 3rd 2007 edition is only available in the object-oriented C++ language; it is updated and more complete, but if you get a used 2nd edition, that will do. Also, PDFs of the 2nd edition may still be available freely as PDF online.
    • "Geodynamics", D. Turcotte and G. Schubert, 2nd Edition, Cambridge U Press, 2001.
      A standard reference for geodynamics. While the notation could be more elegant, still a highly useful book that defined the field; available as paperback.
  • For finite element details
    • "The finite element method. Linear static and dynamic finite element analysis." T. J. R. Hughes, Dover, 2000.
      The class, and common mantle dynamics codes such as ConMan and Citcom, follow the notation of this nice book which is also not too expensive as paperback.
• Meeting times: Tuesdays 3:30-6:30pm, ZHS264 and ZHS130 (computer lab).
• Evaluation: The course work will include a mix of regular problem sets, reading and programming/modeling exercises. Students are expected to work toward solving a specific research problem as a final project. There will be no exam, and evaluation will be based on class participation (10%), problem sets (50%), and final project (40%).
• Prerequisites: MATH 126; corequisite: PHYS 135bL or PHYS 152L, GEOL440; GEOL534 or similar recommended (math reqs can be waived, inquire with instructor). The class assumes basic familiarity with the internal structure of the Earth (440), geophysical methods (440), and plate tectonics (440, 534).

Syllabus

• Orientational meeting - August 22
• Week 1 (Aug 25)
  Introduction: Examples for numerical methods in Earth Sciences; Overview of numerical methods; Computer hardware; Computer Language; Principles of Programming. Exercise: Matlab programming
  Notes: Introduction Handout: Math Problem set: Matlab
• Week 2 (Sep 1)
  Ordinary differential equations: Definition of ODEs. Initial value problems. Euler method. Taylor expansions. Accuracy of numerical methods. Midpoint method, 4th order Runge Kutta. Exercise: Program and solve Lorentz equations.
  Notes: ODEs Problem set: ODEs
• Week 3 (Sep 8, SCEC: no lecture, but homework assignment)
  Scaling analysis; Non-dimensionalization; Non-dimensional numbers (Rayleigh, Prandtl, Peclet, Reynolds, Deborah). Stokes velocities for Newtonian and non-Newtonian rheology; shear layers
  Notes/problem set: Scaling
• Week 4 (Sep 15)
  Finite differences I: 1-D heat equation. Explicit solution of diffusion problems. Stability.
  Notes/problem set: Explicit FD
• Week 5 (Sep 22)
  Finite differences II: Implicit methods. Crank-Nicolson method. Order of spatial and temporal accuracy. Stability conditions. Neumann and Dirichlet boundary conditions. Sparse matrices, triangularity. Linear systems of equations. Heat equation in 1-D.
  Notes/problem set: Implicit FD methods
• Week 6 (Sep 29)
  Finite differences III: Non-linear equations. Darcy flow equation for pressure-dependent diffusivity. Two-dimensional heat equation, solution with fully explicit and fully implicit methods. Comparison with analytical solutions.
  Notes/problem set: Non-linear and 2-D FD methods
• Week 7 (Oct 6)
  Finite differences IV: Advection equation for heat transport. FTCS method and stability. Lax method, Courant criterion. Upwind schemes. Staggered leapfrog. Semi-Lagrangian methods. Advection-diffusion combos in 2-D, operator splitting.
  Notes/problem set: Advection equations and combos
• Week 8 (Oct 13)
  Finite elements I: Introduction to the finite element method. Strong and weak forms of PDEs. Discretization of domains into finite elements. Shape functions. Bilinear forms. Variational approaches, virtual work. Galerkin method. One-dimensional heat equation example.
  Notes: FE Intro
• Week 9 (Oct 20)
  Finite elements II: Local and global coordinate systems. Change of variables during integration. Matrix assembly. Solution of linear systems of equations, direct and iterative methods. LU decomposition, Cholesky. Jacobi, Gauss-Seidel, Conjugate gradient, and multigrid methods.
  Notes: FE Implementation Problem set: 1-D FE implementation and matrix inversion
• Week 10 (Oct 27)
  Finite elements III: 2D boundary value problems. Isoparametric elements. Jacobian; global and element-local coordinates. Numerical integration using Gauss quadrature. Triangular and quadrilateral shape functions. Meshing using triangles. Solution of 2-D heat equation.
  Notes: FE 2D, time dependent solution Problem set: 2-D FE heat equation
• Week 11 (Nov 3)
  Finite elements IV: Compressible elastic problems. Elastic moduli, plane stress, plane strain. Gradient operator, elasticity matrix, engineering strain convection. Visualization of stress states, eigensystems.
  Problem set: 2-D FE elastic
• Week 12 (Nov 9)
  Finite elements V: Compressible and incompressible elasticity and Stokes flow. Mixed formulation with discontinuous pressure. Powell-Hestenen iterations.
  Notes: Incompressible elastic/fluid problem Problem set: 2-D FE incompressible Stokes
• Weeks 13, 14 and 15 (Nov 17, Nov 25 and Dec 01)
  Joint project work in computer lab.

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