dolfin team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Wed, Jan 14, 2009 at 03:38:39PM -0500, Shawn Walker wrote:
>
>
> On Wed, 14 Jan 2009, Anders Logg wrote:
>
>> On Wed, Jan 14, 2009 at 08:23:04PM +0000, Garth N. Wells wrote:
>>>
>>>
>>> Anders Logg wrote:
>>>> On Wed, Jan 14, 2009 at 08:12:33PM +0000, Garth N. Wells wrote:
>>>>>
>>>>> Anders Logg wrote:
>>>>>> 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.
>>>>>>
>>>>> This is how most codes work and what mesh generators provide. We still
>>>>> need a dof map to deal with cases other than scalar Lagrange.
>>>>>
>>>>> Garth
>>>>
>>>> What do you mean? Why would we need a dofmap? The cell_nodes list
>>>> above is the dofmap for the coordinates.
>>>>
>>>
>>> If each point has more than one dof.
>>>
>>> Garth
>>
>> I still don't understand. There will be no dofs, only coordinates for
>> points in the mesh. For a quadratic mapping of a triangular mesh, we
>> just need to list 6 numbers for each cell, and those numbers indicate
>> which coordinates are associated with the 6 points (three vertices and
>> three edge midpoints). Each coordinate (in 2D) is just two double
>> values.
>>
>
> I think technically, each "higher order vertex" has 3 dofs in 3-D (1 dof  
> for each scalar coordinate position).  But isn't this already taken into  
> account by the above data structure?  I guess it depends on how the  
> coordinate positions are stored.

Yes, we store the coordinates for each node together so we only need
to map one thing.

> The above data structure seems reasonable.  It generalizes for higher  
> order lagrange.
>
> Now, where will this be stored.  This is not a Function.  The coordinates 
> could be stored as a Vector.  Not sure about the DoFmap.

I think the natural thing would be to store it in MeshGeometry. That
was the original thought (to be able to extend it from just storing
vertex coordinates to higher order mappings).

-- 
Anders

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