Curvilinear grids have been widely used in scientific
computation [1] [2]
since the Finite Element Method (FEM) was
invented in the early 50s. It is possible to
use regular grids instead of curvilinear grids.
However, there are many reasons for using curvilinear
grids instead of regular grids.
First, we are not concerned equally with every portion of a structure
in scientific computation.
Every part is different in its shape and loading conditions.
Sometimes more attention is paid to some parts than to others,
because they are more critical, or because some damage was
reported in prototype designs. Large cells (fewer points)
will be defined in some of the less important areas, and small cells
(more points) in important areas.
Second, using regular grids is too
space- and time-consuming. In the finite element method, if one point is added
into the model, that means two unknowns are added into the system equations
in the two-dimensional case,
or at least three more unknowns(up to six depending on
the element types used in modeling)
in the three-dimensional case.
This amounts to adding two or at least three more columns and
rows to the system matrix, respectively. Computing
the corresponding coefficients is significantly time-consuming.
Solving those equations will take much more time.
Finally, the complex boundaries
of a structure can not be properly emulated by using
regular grids. So, regular grids are not in use for modeling structures.
As computer graphics devices become ever more powerful, they are used
ever more widely. The text table format outputs from FEM software are
no longer satisfactory.
Graphical outputs or even animations are the ideal of the future.
It is highly desired that how the structures behave
under known loading conditions be visualized right after the
design is finished.
To reach this goal, very hard work needs to be done in two major fields:
to provide much more powerful FEM software (new types of
elements for man-made materials and optimization abilities, etc.)
and to equip the software with better pre/postprocessors.
Today the key job in postprocessor technology is volume rendering.
Using curvilinear grids saves us a lot of time and gives us a lot of
flexibility in modeling, but
it also brings some problems in volume rendering.
The main problem is that it is very difficult to determine
inside which cell a point is located in curvilinear grid volume.
In regular grid volume, the point can be very easily located according to its
coordinates, just by some very simple computations.
It seems that the problem can be solved, either by projecting curvilinear grid volume
into regular grid volume, or by resampling curvilinear grids into regular
grids. Often, errors are introduced in projecting,
which makes the problem more difficult to solve.
The efficiency of the curvilinear grid volume visualization algorithm
is very important in scientific visualization.
An ideal visualization algorithm should be fast and reliable.
At the same time, it should use reasonable space and
should also generate no artifacts.
A good visualization algorithm should keep the data set precision
as high as possible, because the data are the output of FEM software,
and there already are some relationships assumed inside this data set, i.e.,
the data are not isolated numbers.
On the other hand, a high-speed visualization algorithm will allow us
to view images interactively; otherwise, that is impossible.