# [Gmsh] Optimization of triangulation

Christophe Geuzaine cag32 at case.edu
Fri Nov 3 04:23:21 CET 2006

```Jonas Forssell wrote:
> Hello,
>
> When meshing a surface in parametric space, the mesh might be perfect
> Delaunay, but in 3D space the elements may be distorted due to the shape
> of the surface.
>
> There seems to be several ways to handle this, including using ellipses
> in parametric space to get circumcircles in 3D space.
>
> I wonder if you could explain a little about the method you use in Gmsh
> to achieve good mesh quality and perhaps give me a few references to papers.

Hi Jonas - In Gmsh 1.65 we don't mesh in parametric space: we mesh in
some "mean" plane (obtained by least squares from the boundary mesh) and
just do an orthogonal projection back onto the original surface. This is
Really Bad (TM) when the surface has high curvature, or cannot be mapped
1-to-1 to a plane, etc.

In the (forthcoming) version 2.0 of Gmsh, we mesh in parametric space,
and then modify the initial (e.g. Delaunay) grid until the quality of
the elements in real space is OK. To perform the mesh modifications we
follow the techniques proposed by Shephard and his collaborators at RPI
(see e.g. [Li, Shephard, and Beall: “3-D Anisotropic Mesh Adaptation by
Mesh Modifications”, Comp. Meth. Appl. Mech. Eng., 2003]).

Note that resulting mesh is not Delaunay anymore: but this technique
seems to work very well in practice, and is much simpler than the
classical approach you mentioned. (The classical technique uses a metric
derived from the first fundamental form of the surface to transform
vectors and distances in parametric space--the "empty circle" property
in the Delaunay algorithm then effectively becomes an "empty ellipse"
property, which, once mapped back to the 3D surface, leads to isotropic
triangles. See e.g. the book from George and Borouchaki for references.)

Cheers,

Christophe

> Thanks
>
>
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--
Christophe Geuzaine
Assistant Professor, Case Western Reserve University, Mathematics
http://www.case.edu/artsci/math/geuzaine

```