This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Previous experimental studies of convection in fluids with temperature‐dependent viscosity reached viscosity contrasts of the order of 105. Although this value seems large, it still might not be large enough for understanding convection in the interiors of Earth and other planets whose viscosity is a much stronger function of temperature. The reason is that, according to theory, above 104–105 viscosity contrasts, convection must undergo a major transition—to stagnant lid convection. This is an asymptotic regime in which a stagnant lid is formed on the top of the layer and convection is driven by the intrinsic, rheological, temperature scale, rather than by the entire temperature drop in the layer. A finite element multigrid scheme appropriate for large viscosity variations is employed and convection with up to 1014 viscosity contrasts has been systematically investigated in a 2D square cell with free‐slip boundaries. We reached the asymptotic regime in the limit of large viscosity contrasts and obtained s...

Numerical investigation of 2D convection with extremely large viscosity variations

https://hal.archives-ouvertes.fr/hal-00333883/document

A Lagrangian integration point finite element method for large deformation modeling of viscoelastic geomaterials

http://jupiter.ethz.ch/~pjt/papers/Tackley2008PEPI_yinyang.pdf

Modelling compressible mantle convection with large viscosity contrasts in a three-dimensional spherical shell using the yin-yang grid

SUMMARY Calculating surface topography in geodynamic models is a common numerical problem. Besides other approaches, the so-called ‘sticky air’ approach has gained interest as a free-surface proxy at the top boundary. The often used free slip condition is thereby vertically extended by introducing a low density, low viscosityfluid layer. This allows the air/crust interface to behave in a similar manner to a true free surface. We present here a theoretical analysis that provides the physical conditions under which the sticky air approach is a valid approximation of a true free surface. Two cases are evaluated that characterize the evolution of topography on different timescales: (1) isostatic relaxation of a cosine perturbation and (2) topography changes above a rising plume. We quantitatively compare topographies calculated by six different numerical codes (using finite difference and finite element techniques) using three different topography calculation methods: (i) direct calculation of topography from normal stress, (ii) body-fitting methods allowing for meshing the topography and (iii) Lagrangian tracking of the topography on an Eulerian grid. It is found that the sticky air approach works well as long as the term (ηst/ηch)/(hst/L) 3 is sufficiently small, where ηst and hst are the viscosity and thickness of the sticky air layer, and ηch and L are the characteristic viscosity and length scale of the model, respectively. Spurious lateral fluctuations of topography, as observed in some marker-based sticky air approaches, may effectively be damped by an anisotropic distribution of markers with a higher number of markers per element in the vertical than in the horizontal direction.

http://jupiter.ethz.ch/~pjt/papers/Crameri2012GJI_StickyAirBenchmark.pdf

A comparison of numerical surface topography calculations in geodynamic modelling: an evaluation of the ‘sticky air’ method

Summary
We have carried out a comparison study for a set of benchmark problems which are relevant for convection in the Earth's mantle The cases comprise steady isoviscous convection, variable viscosity convection and time-dependent convection with internal heating We compare Nusselt numbers, velocity, temperature, heat-flow, topography and geoid data Among the applied codes are finite-difference, finite-element and spectral methods In a synthesis we give best estimates of the ‘true’ solutions and ranges of uncertainty We recommend these data for the validation of convection codes in the future

/pdf/a-benchmark-comparison-for-mantle-convection-codes-59nbvxglo3.pdf

G. Jarvis

Papers

A benchmark comparison for mantle convection codes