Several volume rendering methods for
curvilinear grid volumes have been developed by
Garrity [3], Wilhelms [4], Neeman [5],
Mao [6], Helfrick [7], etc.
Some of those articles process a complete curvilinear
data set first (preprocess) [3] [4] [7],
and some even include sorting.
If the whole data set is preprocessed, there are two problems.
A lot of memory space is needed, and the processing
takes time. As the data sets become large,
both the space and the time get bigger.
For example, a data set with
ten thousand cells is commonly used in FEM. Time
and space needed to preprocess such a large data set are significant.
The fastest sort algorithm is 
. Sorting this kind of huge
three-dimensional data set is monumental task.
Therefore, if there is preprocessing and sorting involved in an algorithm,
the algorithm will be unacceptable.
The splatting method developed by Mao [6] is a good idea,
but it sometimes generates
artifacts, because the cell size in curvilinear grids varies greatly.
It is very difficult to determine a generalized sphere diameter
used in this method, because
it is very common that there is a ten-fold difference
between the sizes of two cells within one data set.
A much more serious problem in the splatting method is that
the value in a voxel may not be correct in some cases,
because the value depends not only on the values at
the vertices of the cell where the voxel is located, but also
on what kind of cell the FEM used. Different kinds of cells use different
interpolation functions. FEM engineers select cell types according to
the nature of the structures (materials used, loading conditions, etc.).
If different interpolation functions are used
in FEM analysis and visualization, the value will not be correct.
This is discussed in more detail in the next section.
So far, there is no paper that has addressed this issue,
while also discussing curvilinear grid volume rendering.