Preparation of Geometry Models for Mesh Generation and CFD
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T-Splines and U-Splines
A T-Spline (Sederberg & Sederberg [18]) is a spline with a partially empty parametric space; the spline's control points need
not be defined at each parametric (u,v) coordinate. A row of control points ends at a so-called T-junction in the middle of the
parametric space, hence the name T-Spline. This non-rectangular parametric space affords a T-Spline surface a great deal of
flexibility relative to a NURBS surface when modeling a complex shape. In general, a T-Spline model can consist of fewer
surfaces than a NURBS model on the same shape (Figure 2).
Figure 2- A single T-Spline surface (right) can be used to model this personal watercraft instead of 13 NURBS (left). Image from Reference [18].
Used with the permission of the author.
Thomas et al. [19] introduced U-Splines (Unstructured Splines), a further generalization of NURBS that are defined on trian-
gular unstructured meshes versus the structured grid that is the rectangular parametric space of a NURBS or T-Spline.
Subdivision Surfaces
Subdivision (Sub-D) is a method for modeling freeform surfaces that start with a coarse mesh. Through recursive point
insertion achieves a limit surface that either interpolates or approximates the points in the original coarse mesh. Subdivision
is equally credited to Catmull & Clark [20] and Doo & Sabin [21]. The nature of the Sub-D method, as described here, may seem
more like a discrete representation than an analytic one. However, the original development of Sub-D modeling was based on
uniform B-Splines, while more recent work by Cashman [22] demonstrates compatibility with NURBS.
Figure 3: Example of the recursive point insertion in a subdivision that evolves a coarse grid (left) into a smooth, limit surface (right). Image
from Ben-Chen & Lai Lin [23].
