Transformations by J, invJ, detJ in the frontend AST

[kynan]

To treat the transformation from physical to reference space in a more natural way, global derivatives and the measures dx/ds are replaced by their local equivalents transformed through J/invJ/detJ.

This process involves the following steps:

1. The Transformation class

   • Transformation is a class with a singleton instance T defined in the transformation module
   • T has the attributes J, invJ, detJ, after it has been initialised with the coordinate field (via init)
   • transform(coordinates) returns T initialised with coordinates

   • UFL code is prepended by

     x = state.vector_field['Coordinates']
     T = transform(x)

     in a preprocessing step

2. UFL derivation and measure operators are overridden in the namespace the UFL code is exectued:

   grad = lambda u: inner(T.invJ, ufl.grad(u))
   div = lambda u: ufl.div(inner(T.invJ, u))
   dx = T.detJ * ufl.dx

This results in a natural integration of all transformations by simply executing the UFL code with these operators brought into the namespace.

[gmarkall]

This (or a similar solution) was implemented in the generate-transforms branch before it was merged (e739ee4bf).