MDKC Modeling of Boundaries

Suppose that the problem of interest is D, and defined with respect to coordinates , or equivalently, to transformed coordinates , with . We will assume that the problem has one spatial boundary, namely the hyperplane , and is defined over a time interval . As such, the problem domain is then

or its equivalent in the coordinates, obtained under a transformation of the form (3.21). We restate the energy balance for an -port defined over , which is

where is the instantaneous applied power at the ports, is an internal source power, is dissipated power, and is a column -vector representing stored energy flux; is the differential volume element. As mentioned previously, the -port is

for some , all of whose components in the coordinates are positive everywhere in . Here, is the boundary of , is the unit outward normal, and is a differential surface element on the boundary.

Note that consists of the union of three sets of points, i.e.,

where, in terms of the physical coordinates

We can thus rewrite (6.3) as

For a *closed network*--that is, an -port with no free terminals (corresponding to a complete system of PDEs)--the instantaneous applied power is zero, so we are left with

In other words, the stored power flux leaving the boundary must be negative (the -port is passive).

Suppose, now, that there is an D -port defined on the spatial boundary of . Renaming this region , we have another energy balance

over coordinates derived from physical coordinates on . The quantities , , and are the applied power, source power, dissipated power, and stored energy flux in the boundary network. Again, if the boundary network is passive, we have

where is the boundary of the region , and consists of the union of the two regions

so that we have, finally,

The boundary network is intended to model a passive distributed termination to the problem defined over the region . It should be clear that if both networks are passive, then if the transfer of energy between them is passive, then the terminated system as a whole will be passive. See Figure 6.3 for a representation of the relevant regions.

We can ensure this by requiring that the power applied through the ports of the boundary network over the region is equal to the stored energy flux of the interior network leaving through its spatial boundary (recall that we have set ). In other words, we require

Inequality (6.4) can then be rewritten as

or, by employing (6.6), as

From (3.30), the quantities in (6.7) have the following interpretation:

The negative signs in the definitions of the initial energies result from the fact that the outward normal to and points in the negative time direction. As such, (6.7) can be restated simply as

or, in other words: the total energy stored in the interior and boundary networks must not increase as time progresses.

It is straightforward to extend this idea to more complex boundaries. For example, if the region were to be defined by

so that there is a corner at , we could model passive boundary conditions using four networks: an D network for the interior of , two D networks for the two ``faces,'' and a D network for the corner itself; an energy inequality similar to (6.7) results.

Here, we have said absolutely nothing about discretization (and indeed, we have not investigated this problem in any detail). We have, however, indicated the possibility for arbitrary *distributed* passive boundary termination of a given MDKC; only lumped conditions have been examined so far in the literature.