Gridap GSoC 2021

Contains some code related to my Google Summer of Code. I mostly fill this up with experiments and to test concepts. Checkout DETAILS.md for more details and also the blog post at the end.

Evaluating function on an arbitrary point.

Extend evaluate to handle arbitrary points. Build kdtree using the coordinates and search the tree for the cell containing the point. Already under development in Gridap. It works:

reffe = ReferenceFE(lagrangian, Float64, 1)
model = CartesianDiscreteModel((0,1,0,1), (20,20))
V1 = FESpace(model, reffe)
fh = interpolate_everywhere(f, V1)

evaluate(fh, Point(rand(2))
fh(Point(rand(2)) # Also works!

Use evaluate to interpolate between two meshes.

Interpolation can be done by using evaluate from before. The simplest one is

phys_point = get_cell_points(get_fe_dof_basis(V2)).cell_phys_point
fh_phys_coords(x) = evaluate(fh, x)
phys_point_fx = lazy_map(fh_phys_coords, phys_point)

One of my mentors came up with this neat solution.

model = CartesianDiscreteModel((0,1,0,1), (40,40))
V2 = FESpace(model, reffe)
gh = interpolate_everywhere(x->fh(x), V2)

But this is a little bit slow since it builds the trees everytime a new point is encountered. There are a couple of ways to do it. Most define a method _cell_vals(b::CellDof, fh) where b is the cell basis of the target space V2 obtained using b = get_fe_dof_basis(V2). Then evaluating them in the new cell points

function _cell_vals(b::CellDof, fh::CellField)
  # Compute cell dof values of fh on the new mesh
  return cell_vals # cell-wise dof values
end

This can be done in many ways and README.md will be soon updated once the optimal way is implemented.

Blog posts

Main Page

1. Google Summer of Code with Gridap.jl
2. Google Summer of Code 2021 - Community Bonding
3. Interpolation for Lagrangian Elements - I
4. Interpolation for Lagrangian Elements - II