Hermite elements for FEniCS

This is a repo to collect the notes and notebooks on my progress in the implementation of Cubic Hermite and Discrete Kirchhoff Triangle elements for FEniCS, as well as their application to some models (Euler-Bernoulli, Poisson, Biharmonic, non-linear Kirchhoff...) .

Contents

• A bunch of Jupyter notebooks with implementation steps, simple tests and some models for deeper testing. See index.ipynb for details.
• hermite.tm, hermite.bib, img/: Document with the math for Hermite elements. In progress.
• discrete_gradient.tm: A document explaining how these operators are built. Contains descriptions of the linear Kirchhoff and nonlinear Kirchhoff models. In progress.
• test.py: Use this to run tests with pdb (or realgud:pdb in emacs)
• checkout_hermite.sh: A script useful after first pulling the docker image with the official sources. It checks out the right commits and branches and configures the remotes with my private branches.

To do

• More tests.
• Finish implementing Dirichlet boundary conditions: Check the effect that meddling with the stiffness matrix has over the solution of the system. In particular check that symmetry is not broken, cf §6.3 of the FEniCS book.
• Finish essential Neumann BCs for 4th order problems (beyond the current hack).
• Implement arbitrary (d > 3) polynomial orders in Hermite elements using hierarchic bases (cf. [Solin, §6.3.2]).
• Disable vertex_to_dof_map() and dof_to_vertex_map() for spaces using Hermite elements.
• Implement evaluate_dof() for PointDerivative to enable nodal interpolation. See interpolation.ipynb.
• Implement Hermite transformation for optimisedquadraturetransformer.py too.
• ???

Dependencies

Most of the code in the notebooks and models depends on my branches on ffc, fiat and ufl, where Hermite and DKT elements are implemented (although a description of the required modifications is contained in some of the notebooks).

As to the other subprojects (dolfin, dijitso, instant and mshr), this code should be compatible with tags/2017.1.0.

The script checkout-hermite.sh automates the checkouts inside the container.

License

Licensed under the GPL v.3.