Thomas,
I should not have given Mesquite such short shrift in my reply
yesterday. With regard to the MappingQEulerian approach, I would
imagine Mesquite would take the mapped points as the mesh points
(basically, what my figures show as the mesh), then smooth those out a
bit, and return new vertex locations. Then I would need to update the
euler_vector to reflect the new vertex positions - this seems to me like
a difficult step since the euler_vector describes displacements of
degrees of freedom rather than vertices. My difficulty comes from not
truly understanding how to think about degrees of freedom in that
euler_vector in relation to my naive view of the descriptors of a mesh -
i.e. vertices.
The way you ought to see Eulerian mappings is that the geometry is
simply described by the graph of a piecewise polynomial function. It
happens that you represent this piecewise polynomial function by a
vector-valued finite element field, but this field need not have
anything to do with the solution of the PDE you're trying to solve and
may in fact use completely different elements. At least in principle,
the two also don't need to be defined on the same mesh, though that is
convenient in practice.
What mesh relaxation would do is replace the Eulerian (geometric) field
described by one solution vector by another solution vector that leads
to less distortion of cells. That the solution vector may correspond to
an interpolating finite element that also has degrees of freedom located
at vertices is really not all that important. Think simply of the
Eulerian field as a vector field with arrows from every point of the
reference domain to a corresponding point in the domain you're trying to
describe, without thinking of the reference domain being subdivided into
cells.
I hope this helps a bit.
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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