Hello Pierre, Indeed. We are currently working on improving the algorithm to correct this.
Thanks for the feedback! Christophe On 12 Jan 2012, at 22:52, [email protected] wrote: > 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 > <curved-convergence.pdf>_______________________________________________ > gmsh mailing list > [email protected] > http://www.geuz.org/mailman/listinfo/gmsh -- Prof. Christophe Geuzaine University of Liege, Electrical Engineering and Computer Science http://www.montefiore.ulg.ac.be/~geuzaine _______________________________________________ gmsh mailing list [email protected] http://www.geuz.org/mailman/listinfo/gmsh
