Boundary Conditions

Boundary terminations have been discussed in [61,107,131,211]. We have not done significant work on this problem, but would like to mention the several disparate approaches which have been proposed. The problem of general passive termination of a MD network is very involved, and would probably merit a long treatment in a separate work; termination of a (1+1)D MD circuit, which is all we will be able to discuss here, is a simple matter, and the ideas can be extended to cover certain important cases in higher dimensions. The most straightforward method was put forth in [107]. We will refer here to Figure 3.22, the MDWD network for the source-free transmission line equations. This network represents the signal behavior at any grid point in the domain. In particular, the signals $x_{2}$ and $x_{3}$ are obtained at each time step from signals input into the shift registers at grid locations immediately to the left and right, respectively. Suppose now that we have a left boundary termination at $x=0$, and that the domain has been sampled such that a grid point coincides with this point. Then at this location, $x_{2}$ cannot be directly obtained because there is no grid point to the left to which the shift register refers.

Let us now examine some simple lossless boundary conditions of the form of (3.8). Suppose that we would like the boundary condition for the transmission line to be that of an open-circuit at the left termination, so that we have

$$i(0,t) = 0$$

The wave digital approximation to the physical current at any grid point is calculated from the wave variables incident on the left series adaptor, so that we have

$$i = \frac{1}{R_{0}+R_{er}+R_{1}}\left(a_{1}+a_{2}\right) = \frac{1}{R_{0}+R_{er}+R_{1}}\left(-x_{1}-\frac{1}{2}(x_{2}+x_{3})\right)$$

If we set, at the left-most grid point,

$$x_{2} = -x_{3}-2x_{1}$$

then the calculated current will be identically zero, and $x_{2}$ is easily obtained from $x_{3}$ and $x_{1}$, both of which are available. A short-circuited termination, i.e.,

$$u(0,t) = 0$$

can be accomplished by treating the right-hand adaptor in a similar manner. It is possible to mix these conditions, and to introduce loss and a lumped terminating source as well [107].

This idea is also easily extended to multiple dimensions for rectilinearly sampled grids, if the boundary is parallel to one of the grid axes. We would like to add, however, that such a termination has never been shown to be passive, and that there is no general theory applicable to boundary termination of MDWD networks [142](though we will provide a possible foundation for such an approach in §6.2.3). While it is easy to prove that the simple cases above correspond to passive lumped terminations, there are situations in higher dimensions when this approach becomes difficult to apply reliably; in several instances, (see Chapter 5 for some added discussion), this approach has failed in simulation. The difficulty with approaching boundary termination in this way is that the physics of the problem (in particular the passivity at the boundary) is not being taken into account; this method, though easy to apply, is essentially no different from what is done using conventional finite difference methods. Fettweis and Nitsche [61] provided an alternative method which is more satisfying from a physical point of view; in this case, the region beyond the boundary is modeled as a material with extreme parameter values (typically $r = \infty$ or $g = \infty$, for the transmission line or parallel-plate problem). These regions are still passive, though it may now be necessary to employ a ``layer'' of this material, which will incur extra calculation costs.

Other recent work has involved more general lumped boundary terminations [5,211,212], as well as the termination of the (2+1)D parallel-plate problem in hexagonal coordinates; we mention that these approaches are unwieldy in the extreme; in at least one case [210], the proposed modelling of a passive boundary condition requires active elements!

The problem with the termination of MDWD networks is that when spatial dependence is expanded out to get a signal flow graph, we do not end up with a lumped network of portwise-connected elements; see, for example, the flow graph for the simple advective system, shown in Figure 3.7. Such is not the case for digital waveguide networks, which are in fact formulated from the outset as large lumped networks. For this reason, boundary termination is much simpler in a DWN. In Chapter 4, which is devoted to digital waveguide networks, we will discuss boundary termination for the (1+1)D transmission line problem in §4.3.9, and for the parallel-plate problem in §4.4.4. Boundary termination for vibrating beam and plate systems is discussed in detail in Chapter 5.