MDKC Modeling of Boundaries

One of the big hurdles yet to be overcome in the MDWD simulation method is the implementation of boundary conditions. As we mentioned briefly in §3.11, this is a very tricky business, and the approaches in the literature for simple model problems do not generalize to more complex systems. Setting boundary conditions for systems such as beams and plates was a time-consuming, and ultimately fruitless venture. We were forced to turn to DWNs, for which appropriate boundary conditions are much easier to find, because the DWN can be interpreted as a lumped network. The problem is that there is not, as yet, a general theory of boundary conditions for MDWD simulation methods [142]. In this section, we briefly mention a possible foundation for such a theory which is based on the ideas presented initially in [48,85,131] and outlined in §3.4.

Suppose that the problem of interest is $(n+1)$D, and defined with respect to coordinates ${\bf u} = [x_{1},\hdots,x_{n},t]^{T}$, or equivalently, to $k$ transformed coordinates ${\bf t} = [t_{1},\hdots, t_{k}]^{T}$, with $k\geq n+1$. We will assume that the problem has one spatial boundary, namely the hyperplane $x_{1} = 0$, and is defined over a time interval $[0,t_{f}]$. As such, the problem domain $G$ is then

$$G = \{{\bf u}\vert\hspace{0.05in} 0\leq t \leq t_{f}, x_{1}\geq 0\}$$

or its equivalent in the ${\bf t}$ coordinates, obtained under a transformation of the form (3.21). We restate the energy balance for an $N$-port defined over $G$, which is

$$\int_{G} \left(w_{inst} +w_{s}\right)dV_{{\bf t}} = \int_{G} \left(w_{d} +\nabla_{{\bf t}}\cdot {\bf E}\right)dV_{{\bf t}}$$

where $w_{inst}$ is the instantaneous applied power at the ports, $w_{s}$ is an internal source power, $w_{d}$ is dissipated power, and ${\bf E}$ is a column $k$-vector representing stored energy flux; $dV_{{\bf t}}$ is the differential volume element. As mentioned previously, the $N$-port is integrally MD-passive over $G$ if

$$\int_{G} w_{inst}dV_{{\bf t}} \geq \int_{G} \nabla_{{\bf t}}\cdot {\bf E}dV_{{\bf t}} = \int_{\partial G} {\bf n}_{G}\cdot {\bf E} d\sigma_{G}$$ (6.3)

for some ${\bf E}$, all of whose components in the ${\bf t}$ coordinates are positive everywhere in $G$. Here, $\partial G$ is the boundary of $G$, ${\bf n}_{G}$ is the unit outward normal, and $d\sigma_{G}$ is a differential surface element on the boundary.

Note that $\partial G$ consists of the union of three sets of points, i.e.,

$$\partial G = \partial G_{0} \cup \partial G_{f} \cup \partial G_{b}$$

where, in terms of the physical ${\bf u}$ coordinates

$$\partial G_{0} = \{{\bf u}\vert\hspace{0.05in} t=0, x_{1}\geq 0\}$$

$$\partial G_{f} = \{{\bf u}\vert\hspace{0.05in} t=t_{f}, x_{1}\geq 0\}$$

$$\partial G_{b} = \{{\bf u}\vert\hspace{0.05in} 0\leq t \leq t_{f}, x_{1} = 0 \}$$

We can thus rewrite (6.3) as

$$\int_{G} w_{inst}dV_{{\bf t}} \geq \int_{\partial G_{0}} {\bf n}_... ...\bf E} d\sigma_{G} + \int_{\partial G_{b}} {\bf n}_{G}\cdot {\bf E} d\sigma_{G}$$

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

$$0 \geq \int_{\partial G_{0}} {\bf n}_{G}\cdot {\bf E} d\sigma_{G}... ...\bf E} d\sigma_{G} + \int_{\partial G_{b}} {\bf n}_{G}\cdot {\bf E} d\sigma_{G}$$ (6.4)

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

Suppose, now, that there is an $n$D $N$-port defined on the spatial boundary $\partial G_{b}$ of $G$. Renaming this region $G^{(b)}$, we have another energy balance

$$\int_{G^{(b)}} \left(w_{inst}^{(b)} +w_{s}^{(b)}\right)dV_{{\bf t... ...w_{d}^{(b)} +\nabla_{{\bf t}^{(b)}}\cdot {\bf E}^{(b)}\right)dV_{{\bf t}^{(b)}}$$

over coordinates ${\bf t}^{(b)}$ derived from physical coordinates ${\bf u} = [x_{2},\hdots,x_{n},t]^{T}$ on $G^{(b)}$. The quantities $w_{inst}^{(b)}$, $w_{s}^{(b)}$, $w_{d}^{(b)}$ and ${\bf E}^{(b)}$ 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

$$\int_{G^{(b)}} w_{inst}^{(b)}dV_{{\bf t}^{(b)}} \geq \int_{G^{(b)... ... \int_{\partial G^{(b)}} {\bf n}_{G^{(b)}}\cdot {\bf E}^{(b)} d\sigma_{G^{(b)}}$$ (6.5)

where $\partial G^{(b)}$ is the boundary of the region $G^{(b)}$, and consists of the union of the two regions

$$\partial G_{0}^{(b)} = \{{\bf t}^{(b)}\vert\hspace{0.05in} t=0 \}$$

$$\partial G_{f}^{(b)} = \{{\bf t}^{(b)}\vert\hspace{0.05in} t=t_{f}\}$$

so that we have, finally,

$$\int_{G^{(b)}} w_{inst}^{(b)}dV_{{\bf t}^{(b)}} \geq \int_{\parti... ...t_{\partial G_{f}^{(b)}} {\bf n}_{G^{(b)}}\cdot {\bf E}^{(b)} d\sigma_{G^{(b)}}$$ (6.6)

The boundary network is intended to model a passive distributed termination to the problem defined over the region $G$. 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.

Figure 6.3: A region with one spatial boundary.

We can ensure this by requiring that the power applied through the ports of the boundary network over the region $G^{(b)}$ is equal to the stored energy flux of the interior network leaving through its spatial boundary (recall that we have set $G^{(b)} = \partial G_{b}$). In other words, we require

$$w_{inst}^{(b)} = {\bf n}_{G}\cdot {\bf E}\hspace{0.5in}{\rm on}\hspace{0.5in} \partial G_{b} = G^{(b)}$$

Inequality (6.4) can then be rewritten as

$$0 \geq \int_{\partial G_{0}}{\bf n}_{G}\cdot {\bf E}d\sigma_{G} +... ...}{\bf n}_{G}\cdot {\bf E}d\sigma_{G} + \int_{G^{(b)}}w_{inst}^{(b)}dV_{G^{(b)}}$$

or, by employing (6.6), as

$$0 \geq \int_{\partial G_{0}}{\bf n}_{G}\cdot {\bf E}d\sigma_{G} +... ...int_{\partial G_{f}^{(b)}}{\bf n}_{G^{(b)}}\cdot {\bf E}^{(b)}d\sigma_{G^{(b)}}$$ (6.7)

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

$$\mathcal{E}(0) \triangleq -\int_{\partial G_{0}}{\bf n}_{G}\cdot {\bf E}d\sigma_{G} = \mbox{{\rm Energy of interior network at time $t=0$}}$$

$$\mathcal{E}^{(b)}(0) \triangleq -\int_{\partial G_{0}^{(b)}}{\bf n}_{G^{(b)}}\cdot {\bf E}^{(b)}d\sigma_{G^{(b)}} = \mbox{{\rm Energy of boundary network at time $t=0$}}$$

$$\mathcal{E}(t_{f}) \triangleq \int_{\partial G_{f}}{\bf n}_{G}\cdot {\bf E}d\sigma_{G} = \mbox{{\rm Energy of interior network at time $t=t_{f}$}}$$

$$\mathcal{E}^{(b)}(t_{f}) \triangleq \int_{\partial G_{f}^{(b)}}{\bf n}_{G^{(b)}}\cdot {\bf E}^{(b)}d\sigma_{G^{(b)}} = \mbox{{\rm Energy of boundary network at time $t=t_{f}$}}$$

The negative signs in the definitions of the initial energies result from the fact that the outward normal to $\partial G_{0}$ and $\partial G^{(b)}_{0}$ points in the negative time direction. As such, (6.7) can be restated simply as

$$\mathcal{E}(t_{f})+\mathcal{E}^{(b)}(t_{f})\leq \mathcal{E}(0)+\mathcal{E}^{(b)}(0)$$

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 $G$ were to be defined by

$$G = \{{\bf u}\vert x_{1}\geq 0, x_{2}\geq 0, 0\leq t\leq t_{f}\}$$

so that there is a corner at $x_{1} = x_{2} = 0$, we could model passive boundary conditions using four networks: an $(n+1)$D network for the interior of $G$, two $n$D networks for the two ``faces,'' and a $(n-1)$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.