dolfin team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Wed, Jan 14, 2009 at 09:38:23AM -0500, Shawn Walker wrote:
> 
> On Wed, 14 Jan 2009, Garth N. Wells wrote:
> 
> >
> >
> > Anders Logg wrote:
> >> On Wed, Jan 14, 2009 at 08:34:08AM +0000, Garth N. Wells wrote:
> >>>
> >>> Anders Logg wrote:
> >>>> On Wed, Jan 14, 2009 at 08:10:13AM +0000, Garth N. Wells wrote:
> >>>>> Shawn Walker wrote:
> >>>>>> On Tue, 13 Jan 2009, Garth N. Wells wrote:
> >>>>>>
> >>>>>>> Shawn Walker wrote:
> >>>>>>>> I cleared out the old email some because the discussion had changed a
> >>>>>>>> little.  See below for a recap of higher order mesh data stuff:
> >>>>>>>>
> >>>>>>>> -------------
> >>>>>>>>
> >>>>>>>>>>> It will if we want to be able to store a higher-order function space
> >>>>>>>>>>> as a function space with a regular mesh and an additional function
> >>>>>>>>>>> that stores the layout of the coordinates.
> >>>>>>>>>>>
> >>>>>>>>>> Perhaps that is not the best way to do the higher order mesh
> >>>>>>>> coordinates.
> >>>>>>>>>> If we want the higher order mesh data to be a general Function
> >>>>>>>> (requiring
> >>>>>>>>>> a FunctionSpace), then I do not see how you can get away from needing
> >>>>>>>> the
> >>>>>>>>>> FiniteElement signature associated with it, and possibly other things.
> >>>>>>>>>>
> >>>>>>>>>> Even if you have the vector of data and the DoFmap, that info must
> >>>>>>>> still
> >>>>>>>>>> be used to create a Function/FunctionSpace in the code.  And in order
> >>>>>>>> for
> >>>>>>>>>> that to work the DoFmap must be `compatible' with the particular
> >>>>>>>>>> FiniteElement you will be using.  I probably have this wrong, sorry
> >>>>>>>>>> for
> >>>>>>>>>> my confusion.
> >>>>>>>>>>
> >>>>>>>>>> Another way to do the higher order mesh data would be to keep a little
> >>>>>>>>>> simpler.  Have a vector of data, a DoFmap, and an indicator about the
> >>>>>>>>>> degree of polynomial used.  This would be less general but not
> >>>>>>>>>> bad.  In
> >>>>>>>>>> case of higher-order mesh data, you will ALWAYS use a continuous
> >>>>>>>> lagrange
> >>>>>>>>>> finite element.  At least I cannot think of a situation where you
> >>>>>>>>>> would
> >>>>>>>>>> use something else.  Would this not be desirable?
> >>>>>>>>> If we decide to remove input/output for Functions and FunctionSpaces
> >>>>>>>>> (as I've understood is desirable since we then we don't need to rely
> >>>>>>>>> on precompiled elements and dofmaps) then how should we read in a
> >>>>>>>>> higher-order mesh from file?
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>> Anders wrote:
> >>>>>>>>> Here's one option:
> >>>>>>>>>
> >>>>>>>>>   Mesh mesh("mesh");
> >>>>>>>>>   LagrangeFunctionSpace V(mesh);
> >>>>>>>>>   File file("mesh_coordinate_vector.xml");
> >>>>>>>>>   Vector x;
> >>>>>>>>>   file >> x;
> >>>>>>>>>   V.set_coordinates(x);
> >>>>>>>>>
> >>>>>>>>> That might work, but it's a bit long. There should be room for
> >>>>>>>>> improvement.
> >>>>>>>> The discussion on higher-order meshes got a bit confusing for me a
> >>>>>>>> little while back. In summary, exactly what information intended to be
> >>>>>>>> in the mesh file for a high-order mesh?
> >>>>>>>>
> >>>>>>>> Garth
> >>>>>>>>
> >>>>>>>> -------------------------------------------
> >>>>>>>>
> >>>>>>>> Ok, I will try to recap the higher order mesh stuff.
> >>>>>>>>
> >>>>>>>> Currently, in a triangulation, there is an implicit assumption on the
> >>>>>>>> form of the map that takes you from the `unit' reference triangle (or
> >>>>>>>> tetrahedron).  The assumption is that the local map is linear.  As
> >>>>>>>> you well know, this makes for various simplifications which can be
> >>>>>>>> used during matrix assembly.
> >>>>>>>>
> >>>>>>>> But, for various reasons, it can be more useful (or possibly required
> >>>>>>>> depending on the nature of the FEM method) to have a curved triangle
> >>>>>>>> to better approximate domain boundaries or to better compute higher
> >>>>>>>> order geometric motion!
> >>>>>>>>
> >>>>>>>> In this case, one could use a vector quadratic polynomial map and
> >>>>>>>> have a triangle with edges given by a quadratic parametrization.  The
> >>>>>>>> implementation of this only requires a local Lagrange finite element
> >>>>>>>> basis, whose DoFs are just the coordinates of the nodes (for a
> >>>>>>>> quadratic polynomial on a 2-D triangle, this would be 6 nodes per
> >>>>>>>> triangle).  Of course, you will have this for every triangle, and it
> >>>>>>>> makes sense to take the finite element basis to be continuous
> >>>>>>>> lagrange over the whole domain. This continuity is especially
> >>>>>>>> important when deforming the mesh!
> >>>>>>>>
> >>>>>>>> So, way back we thought it would be a good idea to have a separate
> >>>>>>>> functionspace to store this `higher order' mesh data.  But that
> >>>>>>>> seemed problematic.
> >>>>>>>>
> >>>>>>> Sounds complicated.
> >>>>>>>
> >>>>>>>> However, in principle, all you need is a DoFmap and a vector of data
> >>>>>>>> containing the node coordinate positions.
> >>>>>>> This is what I thought. Will we add a field the Mesh xml file to store
> >>>>>>> this extra data?
> >>>>>> Yes.  I don't see why that would be a problem.  And if you don't want to
> >>>>>> use the higher order mesh data (that happens to be in a file), then that
> >>>>>> should also be fine.
> >>>>>>
> >>>>> OK, so we won't have the issue that Anders outlined above with respect
> >>>>> to reading in meshes.
> >>>>>
> >>>>>>> And you need a method for
> >>>>>>>> updating the positions (for a deforming mesh) but that isn't a big
> >>>>>>>> deal. Once this information is properly stored, and accessible to the
> >>>>>>>> matrix assembler, THEN...
> >>>>>>>>
> >>>>>>>> Then the next step would be to modify FFC to use this higher order
> >>>>>>>> (locally defined) map to compute the local matrices, INSTEAD of the
> >>>>>>>> linear map that is implicitly assumed now.
> >>>>>>>>
> >>>>>>>> I realize this will take some time, but we at least need to get a
> >>>>>>>> storage scheme for the higher order mesh data to even proceed!
> >>>>>>>>
> >>>>>>> Kristian is looking at the UFL transition for the FFC quadrature
> >>>>>>> representation at the moment which will be needed for non-affine maps.
> >>>>>>>
> >>>>>>> Perhaps a smaller first step in the non-affine direction would be to
> >>>>>>> support quadrilateral elements.
> >>>>>>>
> >>>>>>> Garth
> >>>>>> Did you mean quadratic elements?  Quadrilaterals are just deformed squares.
> >>>>>>
> >>>>> I meant quadrilaterals (with just 4 nodes) as a first step in having FFC
> >>>>> generate code for non-affine maps. I expect that quads would require
> >>>>> less initial work on the DOLFIN side, perhaps just an extension of
> >>>>> ufc::cell.
> >>>>>
> >>>>>> Yes, I agree.  In reality, I cannot forsee the potential difficulties
> >>>>>> this will cause.  So, trying to have the full implementation ironed out
> >>>>>> before we even put it in may not be helpful.  So, maybe just assuming a
> >>>>>> 2nd order vector polynomial for the local map may suffice.  This is very
> >>>>>> much in line with the current philosophy of implicitly assuming a linear
> >>>>>> map.
> >>>>>>
> >>>>>> So, where would the data be stored in the code?  In FunctionSpace by
> >>>>>> some extra variable field that contains the vector of coordinate data
> >>>>>> and the DoFmap?
> >>>>>>
> >>>>> Using a FunctionSpace sounds complicated to me. What about letting the
> >>>>> mesh carry this data?
> >>>>>
> >>>>> Garth
> >>>> How would it be represented? We already know how to represent such
> >>>> fields (by Functions). We would need to reinvent and reimplement
> >>>> Lagrange elements as part of the Mesh class.
> >>>>
> >>> If we're happy with using a Lagrange basis for the map (at least for
> >>> now), all the form compiler needs is the locations of all the nodes. I
> >>> don't see the need for the complication of a FunctionSpace.
> >>>
> >>> Garth
> >>
> >> That would work if we stored the coordinates as a list of
> >> coordinates for each cell:
> >>
> >>   cell 0:
> >>      node 0: x y z
> >>      node 1: x y z
> >>      node 2: x y z
> >>      node 3: x y z
> >>      node 4: x y z
> >>      node 5: x y z
> >>   cell 1:
> >>      node 0: x y z
> >>      node 1: x y z
> >>      node 2: x y z
> >>      node 3: x y z
> >>      node 4: x y z
> >>      node 5: x y z
> >>    etc.
> >>
> >>
> >> During assembly, we need to retrieve the coordinates for all nodes of
> >> the current cell.
> >>
> >> The problem is if we want to store a single list of coordinates
> >> (without duplication) and then be able to map ourselves from the local
> >> nodes on each cell to the global coordinate list. For that we need
> >> some kind of dofmap which would be exactly the dofmap that the form
> >> compilers generate for Lagrange elements.
> >>
> >
> > Yes I agree that the dof map needs to be aware of this. What I don't see
> > is why it's any more complicated than the mesh carrying some extra data
> > and an appropriate dof map being generated.
> >
> > Garth
> 
> You are all making good points here.  I understand the desire to want to 
> make this as repeatable and extendable as possible.  But at the same time, 
> it would be nice to experiment at first and see how it works in a simple 
> case first.
> 
> Perhaps in the long run, it would be better to have a FunctionSpace.  You 
> may want to interpolate the higher order mesh coordinates in case of a 
> re-mesh.  Incidentally, whenever you interpolate any function on the 
> higher order mesh, you will need to account for the higher order mapping. 
> I don't know how serious that is.  Of course, for now, it may be ok to 
> cheat a little on this point and just leave the interpolation as it is.
> 
> Sorry, I don't know what to say here.  I would just really like this to be 
> in the code in some way.  :)
> 
> - Shawn

How about storing the coordinates in a list and a sort of explicit
dofmap. Something like this:

  coordinates
  -----------
  node 0: x y z
  node 1: x y z
  etc

  cell_nodes
  ----------
  cell 0: 1 3 5 6 7 8
  cell 1: 1 6 7 12 15 16
  etc

So we would eseentially have a dofmap but we store it explicitly so
there's no need to write a function that computes the dofmap.

-- 
Anders

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