The Waveguide Mesh in General Curvilinear Coordinates

Here we may assume any number of physical spatial coordinates , so that . and are both assumed to be -dimensional column vectors. , , and are all positive functions of ( and are strictly positive), and and are the source terms. If is 1 or 2, then we have the transmission line or parallel-plate transmission line system respectively, and if , we have the system describing linear acoustic phenomena (assuming that the material parameters are constant).

Consider the mapping

where are the transformed coordinates. A rectilinear grid in the coordinates can be mapped to a curvilinear grid in the physical coordinates. We can then define the matrix of partial derivatives by

where is the th component of . We assume to be nonsingular everywhere in the problem domain (though this assumption may be relaxed as will mention later in this section) . Defining the differential operator by , it then follows [69] that

for any column vector . Here, is the so-called

or

where

and

System (4.98) is similar to (4.97), except that we now have ``vector'' inductance and resistance coefficients (note that both and are positive definite matrices, if is non-singular). In particular, it is still symmetric hyperbolic (see §3.2), so we may expect that it is possible to derive a waveguide structure.

Consider now the transformed system (4.98) in (2+1)D. If
is diagonal, then
and
will be as well; in this case, system (4.98) is in the same form^{} as the parallel-plate system in radial coordinates (4.82), so we need not discuss this case further here. Indeed, the radial system is a special case of (4.98) with
and

On the other hand, if
is not diagonal (so that we are working in non-orthogonal or oblique coordinates), then the situation is more complex. Due to the cross-coupling between the components of
through the matrices
and
, it will no longer be possible to stagger all the components of the solution; in particular, it will be necessary to use *vector* scattering junctions. Let us look at the case , so that (4.98) are the equations of the parallel-plate system in the curvilinear coordinates . Furthermore, we will set and . A centered difference approximation to (4.98), over grid points with coordinates and , and at times for , and half-integer is

Here, we have the vector grid function , which is a two-vector with components and as well as the scalar grid function . and are second-order approximations to and at the indicated grid points. The scheme above has been written so that it is clear that it can operate for integer, and for and such that is integer; notice that and are calculated at alternating time instants and grid locations, but the components of can not, in general, be calculated at separate locations. , again, is equal to , and (4.99) will be a second-order accurate approximation to (4.98).

We will skip the tedious procedure of deriving a waveguide mesh, and simply present the resulting structure in Figure 4.34.

The junction impedance is defined to be the sum of these four matrices. The admittances at the parallel junctions are defined in a manner similar to those of the DWN in rectilinear coordinates. Also, we have the source voltage waves at the parallel junctions and vector source current waves at the series junctions.

This DWN can be identified with the difference system (4.99) if we set

at the grid points for which such quantities are defined.

There are, of course, various realizations, depending on how the self-loop and connecting immittances are chosen. First, note that because is not diagonal, it will not be possible to distribute it equally among the two connecting impedances and , which are constrained to be diagonal from (4.100). Thus a type II (current-centered) realization analogous to that which was discussed in the case of the rectilinear mesh will not be possible, even in the absence of losses and sources. A type I realization is certainly possible, but for brevity sake, we will only provide the settings for the type III DWN. Here all the connecting impedances are all set to be some constant value . This then implies that

where is the identity matrix. Requiring the positivity of and the positive definiteness of gives the constraint

for the minimum of over parallel junction locations and the minimum of the eigenvalues of over series junction locations, and where we have made the choice

In general, this bound will depend on the choice of coordinates.

FDTD in general curvilinear coordinates has developed in a similar way; most formulations are slightly different in that they are based on a tensor density formulation [91,209] and employ a double set of variables (covariant and contravariant) in the non-orthogonal case; differencing involves interleaving these two sets of components at alternating time steps. They have also been used as a starting point for developing FDTD methods in ``local'' coordinates defined with respect to an automatically generated grid [72,73]. Curvilinear coordinate systems have been touched upon in the MDWD framework as well; An approach similar to the above is discussed in [69], and a tensor density formulation is given in [131].