Contents

Lithosphere Applications of the Finite Element Method (LAFEM 2009)
An introductory course to practical solutions of static continuum mechanics and
heat conduction, with a specific emphasis on the finite element code GTECTON,
and related software. The course revolves around lab exercises in which
participants explore simplified finite element models of typical lithospheric
problems.
Teachers: Rob Govers (govers@geo.uu.nl) and Wienand Drenth (wdrenth@geo.uu.nl)
Entry level requirements: infinitesimal calculus, partial differential equations,
Green functions, linear algebra, continuum mechanics including tensor analysis,
rheology of rocks. Prepare for this course by studying Part 1 of the textbook of
Ranalli, Rheology of the Earth (1995). I also expect you to be able and work with
LINUX and FORTRAN at an intermediate level.
Course requirements: A handout with relevant GTECTON equations, and labs
questions, will be made available to all participants. Participants are expected to
take notes on when, why, what software is used to tackle steps in the modeling.
Lab reports should document all relevant stages of the modeling.
Detailed class schedule here
Resources Participants 2009
GTECTON reference manual
UNIX manual part 1, part 2a, part 2b
awk manual
sed manual
make manual
GMT home page
triangle home page
gmsh home page
Data Explorer (dx) home page
More info on software tools here
Ward Zineddin (UU)
Sandrine Quéré (UU)
Taco Broerse (TUD)
Ward Stolk (VU)
Andy Hooper (TUD)
GertJan van Zwieten (TUD)
Alí Ozbakir (UU)
Taylor Schildgen (UPotsdam)
Walter Austmann (UU)
Klara Elwenspoek (UU)
Day 1, Monday September 14
General description of theory (handout pdf)
Mechanical equilibrium equation, finite element approaches.
Elasticity equations, going from 3D to 2D, shape functions and element
equations.
Shape functions, element equations, stiffness matrix

Nodal boundary conditions
Analytical solution to the engineering flexure equation, clamped on one
end, force on other end.
Lab 1: flexure of a 2D thin elastic beam (detailed description)
Creating and checking a tessellation in a rectangular domain (triangle)
Creating a GTECTON input file (gedit)
Applying and verifying boundary conditions (plnplt and GMT)
Computing a finite element solution (GTECTON)
Plot vertical surface deformation (plnplt and GMT)
Day 2, Tuesday September 15
A converged solution to 2D elastic flexure in engineering (handout pdf)
Quantifying results, convergence and solution quality
Tesselation, boundary conditions, and other issues affecting the
numerical solution.
Direct and iterative solvers (PETSc).
Stiffness matrix topology.
Lab 2: continuing with previous exercise
Compare Lab 1 results with analytical solution; where did things go
wrong?
Tesselation convergence test for Lab 1 problem.
Day 3, Wednesday September 16
Two-dimensional flexure of an elastic lithosphere (handout pdf)
Numerical integration.
Boundary tractions and Winkler pressures.
Euler angles.
Coloring special nodes, element faces and elements.
Scalar, vector and tensor quantities.
Lab 3: Elastic lithosphere on an inviscid asthenosphere
New rectangular model domain (triangle).
Creating a GTECTON input file (gedit)

Automatic selection elements for applying boundary conditions (picknps
and elmside)
Verifying boundary conditions (plnplt and GMT)
Compute a finite element solution (GTECTON)
Plot displacements and stresses (plnplt and GMT)
Day 4 Thursday September 17
"large" deformation models and gradient BCs (handout pdf)
Symmetry, anti-symmetry, periodic, and no-tilt BC's
Compatibility and completeness
Non-representable stress states
Force balance and residual force update
Lab 4: Buckling of a 2D elastic plate
Rectangular domain, subject to in-plane traction and bifurcation-forcing
Apply theoretical limit traction magnitude, with and without residual
force update
Square domain with corner force
Day 5, Friday September 18
Incorporating fault slip (earthquakes) (handout pdf)
Finite element equations for split nodes
Analytical solution
Local mesh and slip tapering
Block coloring
Lab 5: Fault slip through split nodes
Fault slip with few triangle elements
Investigate model response to correct, and incorrect block coloring.
Fault tip stress field, residual update (non-balance), and mesh density.
Day 6, Monday September 21
Frictional fault surfaces (handout pdf)
Finite element equations for slippery nodes

Fault deformation and fault-parallel slip
Fault intersections; triple junctions
Condition number
Factors affecting stiffness matrix condition number
Lab 6: Dynamic fault slip through slippery nodes
Frictional slip for far field traction boundary conditions
Fault deformation and strain compatibility
Euler angles and fault parallel slip
Day 7, Tuesday September 22
Gravity and restart (handout pdf)
Self-gravitation
Density Stripping method
Pre-stresses
Ridge push and slab pull forcing
Lab 7: incorporating gravity and isostasy
Self-gravitation or the jelly pudding exercise
Partitioning body forces into forcing and non-forcing components
Day 8, Wednesday September 23
Stress flow in a Newtonian body (handout pdf)
General and GTECTON constitutive equations
Time marching basics
Courant limit, stability, oscillation, and accuracy
Fluid limit, Maxwell time
Lab 8: post-seismic relaxation
Time step size examples.
Day 9, Thursday September 24
Heat conduction (handout pdf)
Governing partial differential equations

Finite element implementation
Initializing the temperature field
Source terms and boundary conditions
Coupled thermal-mechanical solution
Lab 9: thermal evolution (McKenzie, 1978 paper)
Creating a GTECTON thermal input file
Stretching model (post-tectonic phase); comparison with analytical
solution
Day 10, Friday September 25
3D (handout pdf)
The philosophy, the tools (gmsh, GTECTON, plt3dt, Data Explorer)
Lab 10: Three-dimensional continuum mechanics
Starting from existing gmsh input file, create tessellation.
Verify boundary conditions (plt3dt and dx).
Solve and visualize, expanding base mesh.
Last updated: September 17, 2009