4.11 Transform condition
Some boundary conditions represent 
at the
boundary as a transformation of the cell value 
. They can be expressed
in terms of a general transform condition
|
(4.13) |
where
is the geometric transformation of variable
by a tensor
.
The transformation is calculated
as follows:
|
(4.14) |
When
is a scalar, the transform condition is equivalent to a
zero gradient condition. Otherwise, it is implemented so that terms
in
in Eq. (
4.13
) contribute to coefficients in
.
These contributions are implicit which improves convergence when
solving the matrix equation
.
A factor 
is introduced to specify the contribution
to internal coefficients. It represents a single internal coefficient
for each component of 
so has the same rank
as 
, i.e. it is a
vector when 
is a vector, and a tensor when 
is a tensor.
The multiplication of each coefficient of
by
its respective component of 
is denoted by by
.
In the case of vector 
, this “component multiplication” is
For advection discretisation, the face value is represented as
|
(4.15) |
where
is an explicit boundary value,
calculated from the expression in
Eq. (
4.13
) using the current values
.
The other explicit term uses the current
with “
” denoting
for a vector
and
for a tensor.
Laplacian discretisation requires the face normal
gradient 
. Combining 
with Eq. (4.15
) gives
|
(4.16) |
where the explicit gradient is calculated by
|
(4.17) |
The transform condition is summarised within the table below by the
value and gradient contributions.
