dolfin team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Wed, Jan 14, 2009 at 04:17:19PM -0500, Shawn Walker wrote:
>
>
> On Wed, 14 Jan 2009, Anders Logg wrote:
>
>> On Wed, Jan 14, 2009 at 03:53:30PM -0500, Shawn Walker wrote:
>>>
>>>
>>> On Wed, 14 Jan 2009, Anders Logg wrote:
>>>
>>>> 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).
>>>>
>>>>
>>>
>>> Yes, I remember that.  And this would require a modification of ufc.h
>>> and/or UFCCELL.h.  I guess that when the mesh cell information is
>>> originally read (at the beginning of code), each cell can store it's own
>>> DoFmap, which is ONLY used for the higher order mapping.  Unless you want
>>> to store the DoFmap more globally; that might be more desirable.
>>>
>>> - Shawn
>>
>> It would be enough to have a function in MeshGeometry that returned
>> the coordinates for a given cell, something like
>>
>>  const double* cell_coordinates(uint cell) const;
>>
>> or
>>
>>  const double * const * cell_coordinates(uint cell) const;
>>
>
> And, obviously, this set of coordinates would be ordered (0,1,2,3,4,5) 
> for P2 map on a triangle.  Will there need to be a flag or something to 
> say if the map takes the triangle to 2-D or 3-D?  This higher order 
> mapping could certainly be used to map 2-D triangles into 3-D space.
>
> - Shawn

2D or 3D will be decided by mesh.geometry().dim() and triangle/tet
will be decided by mesh.topology().dim().

-- 
Anders

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