Chapter 4 Vortex detection - Computer Graphics and Visualization

Chapter 3. Particle Tracing
a decrease in accuracy and efficiency, as was investigated in detail in [Sadarjoen et al.,
1994; Sadarjoen et al., 1997].
In brief, it turned out that the transformation of the vector field, when done in a
simple way, introduced large errors, but when done in an accurate way, caused an
enormous increase of storage. For an accurate transformation, each vector defined
at a grid node in È-space has to be transformed to eight different vectors in -space,
depending on which cell the node is considered to be part of. It also turned out that
the transformations from È-space to -space, and back to È-space again, were more
costly than point location directly in È-space.
Another strategy works by calculating the particle path directly in the curvilinear
grid in the normal domain, È-space. This avoids transformations between the two
domains, although at the expense of more difficult point location. This is the case
because there is no longer a direct relation between the coordinates of a point and the
cell indices. Instead, a search must be performed in several cells, to check which of
them contains the point. In addition, in curvilinear grids it is harder to determine the
local offsets, i.e. the local position inside a cell.
3.2.2 Tetrahedral 5-decomposition
One way to cope with curved cells works by decomposing the hexahedral cells into
tetrahedra. The advantages of tetrahedra is that they are convex and planar, which
facilitates containment tests and face intersection tests.
The simplest and most efficient scheme is to decompose the hexahedral cells into
five tetrahedra, henceforth called the 5-decomposition [Sadarjoen et al., 1994; van Walsum,
1995]. This method was later adopted in [Kenwright & Lane, 1995]. Figure 3.2a
shows a cube which is decomposed into one central tetrahedron and four corner tetrahedra.
In a structured grid, the decomposition can be done in two orientations. To
guarantee connection of cell faces and to avoid overlapping cells, these two orientations
should be alternated in adjacent cells, as shown in Figure 3.2b.
In tetrahedra, interpolation is typically performed using linear interpolation. Figure
3.3 shows a tetrahedron ABCD, where « ¬ ­ denote the local offsets in the tetrahedron,
with the restriction that « ¬ ­ .IfÚ is the data value in node A, Ú
the data value in node B, etc., then the interpolated value ÚÈ in some position È in the
tetrahedron is:
ÚÈ Ú « Ú Ú ¬ Ú Ú ­ Ú Ú
The local offsets « ¬ ­ may be found by inverting the interpolation of the known
position of P in the tetrahedron:
È « ¬ ­ (3.5)
« ¬ ­ È (3.6)
Point location is done as follows: a line is drawn between the previous, known
position and the new, unknown position. Along this line, intersections are calculated
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