discretize 0.7.0 Release Notes

April 30, 2021

This Minor release has several new features, including support for boundary conditions, quiver vector plotting for TreeMesh.plot_slice(), and a rather large change in removing the properties requirement from discretize.

Removal of properties

Starting with the largest change, we have made the decision to remove the properties backend of discretize. Our main motivation for this was to give ourselves more control over the documentation of classes. Our goal was to recreate the functionality without any adverse effects to our users, and have taken many steps to hopefully address any possible issues with the conversion. That being said, please raise an issue on the github page if you run into any unexpected issues.

The next step here is to finalize the documentation..

Boundary Conditions

We have added an improved support for implementing boundary conditions for the finite volume formulation that underlies discretize on TreeMesh, TensorMesh , and CurvilinearMesh.

In general, discretize often makes use of the following two identities for the weak form of the finite volume:

\[\int_\Omega \nabla u \cdot \vec{v} dV = -\int_\Omega u \nabla \cdot \vec{v} dV + \int_{\partial\Omega} u \vec{v}\cdot \hat{n} dA\]

\[\int_\Omega (\nabla \times \vec{w}) \cdot \vec{v} dV = \int_\Omega \vec{w} \cdot (\nabla \times \vec{v}) dV - \int_{\partial\omega} (\vec{w} \times \hat{n}) \cdot \vec{v} dA\]

Previously we focused on approximating the volume integrals in the above equations, however to implement boundary conditions, we use the boundary face integrals!

As part of this, meshes now have a few new properties:

• boundary_faces

• boundary_face_outward_normals

• boundary_edges

• boundary_nodes

• average_node_to_face

• average_edge_to_face_vector

• project_edge_to_boundary_edge

• project_face_to_boundary_face

• project_node_to_boundary_node

Together these are used in a few items that are correspond to the mass matrices, operators.InnerProducts.get_edge_inner_product() and operators.InnerProducts.get_face_inner_product(). These relate to the item that they are operating on, and return the necessary matrix to integrate that quantity on the default boundaries of the meshes.

• boundary_face_scalar_integral

• boundary_node_vector_integral

• boundary_edge_vector_integral

You can investigate the source code of these functions to see how they are built if you need to design your own customized boundary.

All together we have also implemented two helper operators that can be used to reproduce common types of boundary conditions for a few PDE’s. These are based on Robin type conditions that can flexibly support multiple types of boundary conditions depending on the discrete value’s locations.

• operators.DiffOperators.cell_gradient_weak_form_robin()

• operators.DiffOperators.edge_divergence_weak_form_robin()

Future Work

With the removal of properties we will be updating the documentation to be more explicit, but in the meantime this will make some items look less clear. We will also be pushing out some tutorials on how to use the boundary conditions to solve the boundary value PDE’s. In the meantime though, you can look at the boundary condition test codes where we form a few PDE’s for convergence tests.

Contributors

• @jcapriot

With reviews from:

• @prisae

• @domfournier/@fourndo

• @lheagy

Pull requests

• #232: Exorcise Properties

• #234: Boundary conditions

• #240: Tree quiver plot

• #241: Update azure-pipelines.yml