There are several varieties of auto meshing
methods [2] [8] used in FEM.
Every structure is built up of basic elements,
such as triangles, quadrilaterals, hexahedrons, pentahedrons, quadrihedrons
and segments of a circle/sphere.
A triangle is considered to be a special case of
quadrilateral, where two vertices are
at the same position.
Likewise, pentahedrons and quadrihedrons are special hexahedrons.
This phenomenon is called degeneration.
So it is only necessary to work on general elements:
quadrilaterals and hexahedrons. If they can be meshed automatically,
then all other basic elements and any other structures will
also be meshed automatically.
For the two-dimensional example shown in Figure 1, the complex figure is first partitioned into triangles, quadrilaterals and/or pentagons.
Figure 1: A two-dimensional example
Usually those blocks are called big elements. Next, every big element is projected
into a 
square in the 
plane (regular plane), and
then this square can be meshed by different algorithms.
A very simple method is to divide every pair of
opposite edges of the square evenly into m,
n segments respectively, so in total there will be
elements generated.
Then, the square from the 
plane is projected back to the
original XY plane to get the coordinates 
of each vertex, because
in FEM the analysis is done on real space, and
the regular space is only a reference.
The projection functions are:
¯where
is the corner point position in irregular space.
N is called the shape function. Usually, it is a set of
Lagrange interpolation
functions. In the bilinear interpolation
case, 
,
= 
is the corner position in regular space corresponding to each 
,
and i= 1,2,3,4. It is obvious that
whenever there is a 
in regular space,
there will be a corresponding 
in curvilinear
(real world) space. It is a one to one mapping.
The reason is widely discussed in
all FEM books (e.g., [2] [8]).
This process is shown in Figure 2. 
Figure 2: Illustration of the auto meshing process