dolfin team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Mon, Jul 06, 2009 at 09:34:43AM -0400, Shawn Walker wrote:
>
> 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.

I'm on vacation so I also won't have much to say about this right now,
but the form-based approach is very appealing since (1) it is a simple
layer on top of existing functionality (so we don't need to modify
the form compilers) and (2) it is more general since we can use any
Function to map the mesh from the reference geometry.

We still have the problem of reading/storing the geometry and we
concluded a while back that we didn't want to use Function for this,
since it would create a circular dependency between the Mesh and
Function classes. But if the mapping of the mesh can be stored
separately from the mesh (as a coefficient in the form) then I guess
this is no longer a problem. (?)

-- 
Anders

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