On Mon, 6 Jul 2009, Peter Brune wrote:
I'm working on some problems with sub/super/isoparametric elements and have very quickly implemented it
entirely using UFL and a transformation of the form to include a geometric coefficient. This is done using
the transformation framework to append the geometric information to the parts of the form that require
transformation.
I had previously written calculations to include the coefficient space for geometry by hand, but I talked
with Anders at ENUMATH and he told me more about the Transformer machinery. Right now, a Poisson form with
this looks like:
cell = triangle
iso_element = VectorElement("Lagrange", cell, 4)
aff_element = VectorElement("Lagrange", cell, 1)
element = FiniteElement("Lagrange", cell, 4)
iso_func = Function(iso_element)
affine_func = Function(aff_element)
u = TrialFunction(element)
v = TestFunction(element)
f = Function(element)
b = inner(grad(u), grad(v))
K = v*f
J = dot(inv(grad(iso_func)), grad(affine_func))
detJ = det(J)
a = apply_geometry(b, J)
L = apply_geometry(K, J)
a = a*dx
L = L*dx
Where apply_geometry applies a Transformer to the form in order to include the geometric coefficients.
The benefits of this approach are:
1. General function spaces for the geometry -- no reliance on the somewhat contradictory concept of "extra
vertices." on a simplex.
2. Uses all the already existing mechanisms for compilation and optimization
How exactly do you represent the geometry as a function space? We had
thought about this before and there was a problem (because you need a
mesh to create a function space in the furst place).
Right now I'm stuck transforming the affinely-transformed components back to the reference and applying the
map. The ideal would be appending something to the measure, which is then appended to the form. This might
look like:
J = grad(iso_func)
a = inner(grad(u), grad(v))*dx(0) + inner(grad(u), grad(v))*dx(1, jacobian = J)
L = v*f*dx(0) + v*f*dx(1, jacobian = J)
With the form compiler then omitting the generation of the affine Jacobian. Like this we can easily have the
higher-order geometry only defined on, say, boundary cells where we have a higher-order geometry defined.
Otherwise the affine form can be used.
This is great. There is some support for reading in higher order meshes
in DOLFIN now. There is even a boolean parameter for saying which
triangles are curved and which are affine.
Please keep in mind that you may want to have triangles that are NOT on
the boundary to also be curved. This is necessary if the mesh is highly
anisotropic (picture a wing with a very anisotropic curved mesh at the
boundary).
Thoughts on how I should go about this? I'm still generalizing my transformer, but have run a demos of
simple forms (Poisson, Stokes) with no real problems; now I'm moving onto what I actually want to do like
this, but improving the interface would be nice eventually.
- Peter Brune
So, you have done this in UFL? Does that mean just the notation is
setup? I had done a hand modification of a poisson demo that reads in a
higher order mesh and computes the stiffness matrix on two triangles
(only one is curved) and compare it to a stiffness matrix computed by
other means. I don't know if this will be useful for you; what I did was
a little hacky. Have you thought about how the higher order mesh would be
stored in an XML file? I can resend the example I made (that shows
this), but it should be in the archive on DOLFIN.
I am traveling right now, so I won't be able to say much on this,
unfortunately.
