Welcome to the documentation for Animate, Movement, Goalie and UM2N!

Animate is a package which provides anisotropic mesh adaptation functionality for the Python-based finite element library Firedrake. Given an initial mesh, the user defines a Riemannian metric field (or simply ‘metric’), which provides the ‘optimal mesh concept’, i.e. a ‘blueprint’ for what adapted meshes will look like. Animate can then be used to transform the initial mesh into an adapted mesh, making use of the metric to guide the transformations.

Movement focuses on alternative mesh adaptation strategies to the metric-based approach of Animate. Where Animate allows degrees of freedom to be added and removed from the input mesh in order to obtain an adapted mesh, Movement is concerned only with mesh movement methods, which simply move degrees of freedom around. In some cases the first approach is more suitable; in some cases the latter. Several different strategies are implemented within Movement.

Goalie is more of an ‘outer-loop’ package and is focused on a goal-oriented mesh adaptation pipeline for solving partial differential equations (PDEs) using adapted meshes. Goalie supports solving time-dependent PDEs and their adjoints on sequences of meshes, performing goal-oriented error estimation, and adapting the meshes in the sequence with a user-provided adaptor function. A key example of such an adaptor function is one which uses Animate and the metric-based approach. Goalie uses a ‘fixed-point iteration’ approach, meaning that the PDE (and its adjoint) is solved over the whole space-time domain whenever mesh adaptation is applied.

UM2N is an end-to-end machine-learning based implementation of mesh movement.

Mathematical background

We refer to the Firedrake documentation for an introduction to the finite element method - the discretisation approach assumed throughout. The dolfin-adjoint package (which Goalie uses to solve adjoint problems) contains some excellent documentation on the mathematical background of adjoint problems. The metric-based mesh adaptation, mesh movement, and goal-oriented error estimation functionalities provided by Animate, Movement, Goalie and UM2N are described in the respective manuals.

Source code

The source codes for all four packages are hosted on GitHub: Animate, Movement, Goalie. UM2N.