The expression (2.18,2.19) result from solving the implicit expression
The previous equation can be written in matrix notation
can be expressed as
The explicit components of each vector in (3.3) must be replaced by the
current projections, because this is the only available information that any
discretization procedure provides. So,
is transformed to
where the subindex
is the index of a simplex, a
grid point which includes its neighboring points (See Figure 3.2).
has all the current projections from a
simplex, from the
point
to the neighboring points
.
Because a simplex could have any number of neighboring points,
must hold the same number of current projections too. So, (3.3)
must be modified accordingly. Expression (3.5) gives a 3
3
matrix unless an operation is introduced in such a way that it gives
a square matrix with the same number of current projections of a
simplex. This is done by introducing a transformation matrix
composed of the unitary vectors which are computed from the
point
to every neighboring point
.
Introducing the transformation matrix
in (3.5), and
replacing in (3.6) one gets
The above expression determines a matrix which combines current
projections with the components of a magnetic field. However, the
neighboring points
of the point
for every simplex
can
be arbitrarily placed in the simulation domain. Care must be taken
in this step to obtain the proper weight of current projections
with the magnetic field. This is done by means of a matrix
multiplication with the
vector whose components are the inverse
of the distances between the point
and the neighboring points
.
Finally, the discretization procedure taking into account the magnetic field reads
Rodrigo Torres 2003-03-26