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cubes algorithm, which has been extensively re-
searched in the scientific visualization community,
for which (Lorensen and Cline, 1987), (Nielson and
Hamann, 1991), (Nielson, 2003), (Lopes and Brodlie,
2003) are some of the representative papers, (New-
man and Yi, 2006) is a comprehensive survey of
the variants of the algorithm. Additionally, a great
deal of work has been done for both software- and
hardware-based approaches in volume rendering, for
example, (Williams et al., 1998), (Muigg et al., 2007),
etc.
(Gallagher and Nagtegaal, 1989) extended the
marching cubes algorithm to hybrid grids con-
taining tetrahedral, prismatic and hexahedral ele-
ments. (Takahashi et al., 2004) elaborated the look-up
table approach for octahedral elements.
In the visualization software GMV, isosurface ex-
traction from hybrid grids is implemented by deter-
mining intersection points of isosurfaces on edges
of cells and triangulating these points to generate a
2-manifold surface (Ortega, 2008). However, this
method discards the information provided by the grid
during the triangulation stage, which can lead to a dif-
ferent solution from the one obtained when using an
interpolation function for each cell.
(Bhaniramka et al., 2004) presented an algorithm
for automatically generating case tables for isosur-
faces in cells containing hypercubes, cells with 2
k
ver-
tices in k-dimensional space. Their algorithm creates
look-up tables similar to that of (Montani et al., 1994).
This algorithm can be extended to pentahedral cells,
as it is applicable to all topological homotopes of hy-
percubes in the k-dimensional space.
(Weber et al., 2003) described a crack-free isosur-
face extraction algorithm using “stitch cells”, specifi-
cally for grids subjected to adaptive mesh refinement
(AMR). These stitch cells could be tetrahedra, penta-
hedra, or hexahedra. We have used the interpolation
functions that (Weber et al., 2001) use.
While methods by (Gallagher and Nagtegaal,
1989), (Weber et al., 2003) and (Bhaniramka et al.,
2004) work for our application, we are going a step
further towards resolving the topological configura-
tions for pentahedral cells by using similar patterns
that are found in hexahedra. We follow the index-
ing for the configurations used by (Nielson, 2003)
for hexahedra. In relation to our argument against
subdivision of elements, (Carr et al., 2006) have dis-
cussed various artifacts that can be introduced while
performing simplicial subdivision of a hexahedral el-
ement.
3 DISADVANTAGES OF
APPROACHES BASED ON
TETRAHEDRALIZATION OF
HYBRID GRIDS
The marching tetrahedra algorithm is one of the most
convenient isosurface extraction algorithms, devoid
of the ambiguous cases which occur in the case of
the marching cubes algorithm. In the case of hy-
brid unstructured grids, extracting isosurfaces using
the marching tetrahedra algorithm requires an extra
processing step of subdividing the non-tetrahedral el-
ements into tetrahedral ones ensuring continuity of
isosurface across faces. Since tetrahedra are basic
building blocks, and all complex geometric shapes
can be broken down into tetrahedra, it is one of most
commonly used finite elements. However, one has to
be aware of the differences in interpolants that occur
when decomposing a hybrid grid to a tetrahedral grid.
A straightforward tetrahedralization can be done
without inserting new vertices in the grid. However,
this is not possible in certain cases. The first step
in tetrahedralization of cells is the subdivision of its
polygonal faces to triangles. In a standard Lagrange
finite element, a bilinear interpolation function is used
in the quadrilateral face. Considering the parametric
representation of a function F on a bilinear surface, at
any point P(x,y,z), we get, F(x,y,z) = f (s,t) =
(1 − s)(1 −t)F
00
+ s(1 − t)F
10
+ stF
11
+ (1 − s)tF
01
,
where (s,t) is the parametric representation of the
point P(x, y,z) in the quadrilateral P
00
P
10
P
11
P
01
, as
shown in Figure 1(B)(a); and F
i j
is the function value
at vertex P
i j
for {i, j} = {0,1}. On the diagonals
(where s = t or s + t = 1), the interpolating function
is quadratic in either s or t, different from the linear
interpolation function used on triangulating the face.
The different interpolation models used for comput-
ing and visualizing the solution can lead to artifacts
in the isosurface as shown in (Carr et al., 2006).
Subdividing a quadrilateral face can lead to two
solutions as either of its two diagonals can be used to
triangulate the surface. Thus, different possible tetra-
hedralizations can lead to different results for isosur-
faces. There will be differences in the isolines gener-
ated for a quadrilateral face, depending on the choice
of triangulation, as shown in Figure 1(B).
Additionally, in the case of large datasets, the
computational and storage overhead induced by gen-
erating and using the additional elements may cancel
the gain of eliminating ambiguities and using linear
elements. Minimally, a pyramid can be decomposed
to two tetrahedra, a prism to three and a hexahedron to
five. For example, in the missile dataset we have used,
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