The generation of the grid used for the solution of the physical equations
is separated into two tasks [Wim91b]. The first task is the generation
of a mapping from physical domain 
to computational domain 
and vice versa. This mapping is described by the functions 
and
. To generate the mapping we use an approximately equidistant grid
in the computational domain which is just fine enough to resolve all
geometric details (uv-geo in Figure 3.3-4). This grid is
mapped onto the physical domain (xy-geo in Figure 3.3-4)
using algebraic, elliptic or variational methods such that the points at the
boundaries are approximately equidistributed along the arc length
(cf. Section 3.3.3). This geometry grid establishes the
mapping function , , i.e. it serves as reference
mapping.
The second task is the generation and adaption of the grid for solving the physical equations (physics grid). A gradient resolution criterion and a dose conservation criterion control the adaption of the grid according to the evolving dopant profiles. After each time step during a transient simulation the adaption is achieved by inserting and deleting lines in the computational domain (uv-phys in Figure 3.3-4). The corresponding grid points in the physical domain (xy-phys in Figure 3.3-4) are obtained by interpolation in the geometry grid.
Actually, we could generate the mapping using the physical grid, too. The
reason why we are using an independent geometry grid is twofold: (1) In many
applications the geometry (physical domain) is stationary, i.e. the geometry
does not change with time. Then, the mapping has to be calculated just once
for the whole simulation. (2) For non-algebraic mapping methods the point
position for a given may depend on the 
grid. This
dependency entails that the position for a point can change
each time a grid update for the physical quantities is made. The
application of a separate grid for establishing the mapping function
guarantees an unambiguous mapping from computational domain to physical
domain.
