A Survey of Unstructured Mesh Generation Technology
4. Surface Meshing
SURFACE
MESHING
4.1 Parametric Space
Parametric space algorithms will form elements in the two-dimensional parametric space of the surface. Since all NURBS surfaces have an underlying u-v representation, it can often be efficient to mesh in two dimensions and as a final step, map the u-v coordinates back to world space, x-y-z coordinates. The drawback to this method is that the elements formed in parametric space may not always form well-shaped elements in three dimensions once mapped back to the surface. To resolve this, parametric surface meshers can do one of two things: 1) modify or reparamaterize the underlying parametric representation so there is a reasonable mapping from parametric space to world space; or 2) modify the mesh generation algorithm so that stretched or anisotropic elements meshed in 2D will map back to well-shaped, isotropic elements in 3D.The first method requires that in order to have a good paramaterization, the surface derivatives, (Du, Dv), should not vary widely over the domain. Some exact arc-length reparamaterizations have been defined in the literature[80], but can be excessively costly. An approximate arc-length paramaterization or "warped parametric space" can be defined by selectively evaluating surface derivatives over the domain and adjusting u-v values to hold the magnitude of Du, Dv roughly constant. For many cases, a warped parametric space can generate reasonable surface meshes, but there are many problems that the reparamaterization cannot adequately resolve. For this reason, much of the literature on surface meshing focuses on the second option of forming anisotropic elements in 2D that will map back to isotropic elements in 3D.
A common method used in practice is to take advantage of surface
derivatives, Du, Dv, easily computed from a NURBS surface.
George and Borouchaki[81]
propose the use of a metric derived from
the first fundamental form of the surface. The metric is in the
form of a 2X2 matrix and is used to transform vectors and distances
in parametric space. With their Delaunay approach, the "empty circle"
property, effectively becomes an "empty ellipse" property. Also
included with the metric is the option to incorporate element
sizing and stretching properties. A similar approach to parametric
Delaunay surface meshing is presented by Chen and Bishop
[82] and available
in MARC's Mentat[67] software.
Equivalent advancing front surface
mesh generation algorithms, which utilize a metric derived from
the first fundamental form of the surface are presented independently
by Cuilliere[83] and
Tristano[84]. Tristano's
implementation is available
in a recent release of the ANSYS[31]
mesh generation tools.
4.2 Direct 3D
Direct 3D surface mesh generators form elements directly on the
geometry without regard to the parametric representation of the
underlying geometry. In some cases where a parametric
representation is not available or where the surface paramaterization
is very poor, direct 3D surface mesh generators can be useful.
Lau and Lo[85][80]
present an advancing front approach for arbitrary 3D
surfaces. In this method surface normals and tangents must be
computed in order to compute the direction of the advancing front.
In addition, a significant number of surface projections are
required to ensure that new nodes remain on the surface. Also
of significance is the increased complexity of the intersection
calculations required to ensure that triangles on the surface
do not overlap.
A direct 3D implementation[59] of the paving[44] algorithm is also available in the CUBIT[37] software. Similar issues regarding additional projection and evaluations are also of significance to 3D paving. Cass[59] defines a heuristic "sticky space" in order to detect intersecting or overlapping quadrilaterals.
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