[Gmsh] Are curved and high order gmsh meshes really high order ?
Pierre.Saramito at imag.fr
Pierre.Saramito at imag.fr
Thu Jan 12 22:52:46 CET 2012
Dear gmsh developers and users,
It seems that there is a major problem with the computation of
internal node for curved 2d and 3d elements : an isoparametric Pk
finite element computation of a basic PDE problem does not converge as
expected with such meshes.
I have tested with the following problem:
- Laplace u = f in the domain
u = 0 on the boundary
The domain is the unit circle. The right-hand side f is computed such
that the exact solution is u(x) = cos(pi*r) with r=sqrt(x1^2 + x2^2).
I put a small .pdf in attachment: Fig. 2, left column shows the
convergence curves |uh - pi_h(u)| for various norms (L2, L^infty, H1),
where pi_h(u) denotes the Lagrange interpolation of the exact solution
u, and uh is the isoparametric finite element solution on the mesh, as
generated by gmsh.
I do not known how internal nodes are computed in gmsh.
Nevertheless, by using a blending formulae, such as eqn (27) in:
S. Dey, M. S. Shephard and J. E. Flaherty.
Geometry representation issues associated with p-version finite
element computations.
Comput. Meth. Appl. Mech. Engrg., 150:39-55, 1997.
for the computation of boundary and internal nodes, then the
isoparametric finite element method converges as expected. See Fig. 2,
right column. Fig. 1 represents the difference between gmsh nodes and
those obtained by this procedure: this difference is subtle ;
nevertheless it has a dramatic effect on convergence properties. All
finite element computations has been performed with the development
version of Rheolef :
http://www-ljk.imag.fr/membres/Pierre.Saramito/rheolef
Please, could you check the convergence ?
If it is confirmed, could you consider in the future a boundary and
internal node placement procedure in such way that isoparametric FEM
converges as expected ?
I take this opportunity to congratulate all the developers for the
wonderful features in gmsh : the only one to my knowledge that
consider high-order and curved elements.
Pierre
--
Pierre.Saramito at imag.fr
Directeur de Recherche CNRS
Laboratoire Jean Kuntzmann, Grenoble, France
http://www-ljk.imag.fr/membres/Pierre.Saramito
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