Mesh ideas

Currently ClimaCore:

1. Domain: mathematical domain, e.g. RectangularDomain
2. Mesh: how to divide up domain into elements, e.g. EquispacedRectangular
3. Topology: how the elements are connected
4. Space: place quadrature points, compute metric terms

Warping

1. Wrapper around a domain WarpedDomain(domain, (x,y) -> (x, y + sin(x))
2. Mesh: applied to the underlying domain
3. Topology: applied to the underlying mesh
4. Space: place quadrature points on underlying mesh, then transform (and calculate Jacobian, etc)

CubedSphere

1. domain = SphereDomain(radius=...)
2. mesh = EquiangularCubedSphere(domain)

• 6 charts
  • each with their own warping function (SquareDomain -> SphereDomain)
  • each has their own "flat" mesh, which might connect to other meshes?
• implicit underlying cube mesh, which has 6 facets/patches/charts

3. Topology:

• CubeTopology:
  • store # faces per cube
  • has 6 facets
• Unstructured

4. Space: rule to place nodes & compute Jacobians a. directly from warp func b. via polynomial interpolations (a la ClimateMachine) c. Guba (specific to sphere: bilinear interpolate, re-normalize)

Fully unstructured...

Key ideas

• a warped mesh has an underlying "flat" mesh, with a warping function from (flatcoord, facet) -> coord
• placing quadrature points/dofs & grid refinement require a flat mesh
  • alternatively we can place quadrature points like HOMME
• topology/connectivity depends only on the flat mesh (can use standard grid partitioning)
• some notion of facets/charts: a warping function is homeomorphic within a given chart (i.e. we can compute partial derivatives)