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Doubled Grid Density Across an Interface

Figure 4.35: Interface between mesh and mesh with doubled grid density-- (a) grid arrangement, where boundary junctions are labelled $ B$, and (b) scattering junction at such a boundary junction.
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\end{picture} \end{center} \vspace{0.35in}

In Figure 4.35(a) is shown an interface between a regular rectilinear grid with spacing $ \Delta$ (region $ I$) and a grid with spacing $ \Delta/\sqrt{2}$ (region $ II$) whose orientation is rotated by 45 degrees with respect to that of region $ I$. Clearly then, the density of grid points in region $ II$ is double that of region $ I$. At any point in the interior of either regions $ I$ or $ II$, if we are interested in solving the parallel-plate transmission line equations, we can use the rectilinear mesh described in §4.4. We indicate waveguide connections between junctions located at the gridpoints by black lines. At points lying on the interface between the two regions (labelled $ B$) however, we need to develop special scattering junctions. The most straightforward arrangement requires a six-port junction at a boundary point (waves enter the junction from five irregularly spaced directions, as well as through a self-loop). Such a junction is shown in Figure 4.35(b). The problem, then, is in finding the correct admittance settings for the waveguides connected to such boundary points. If these admittances can be chosen positive and in such a way that the resulting scheme is consistent with the parallel-plate system, then we are assured convergence over the entire problem domain. We note that such interfaces bear a resemblance to the very early work of MacNeal [123], who developed asymmetric resistive networks as a means of solving elliptic problems via relaxation.

We assume that the boundary is aligned with the $ y$-axis, so that the boundary junctions are located at coordinates $ (0,j\Delta)$, for $ j$ integer. We also assume, for the moment, that all delays in the network will be unit sample delays (we will return to interfaces between grids with differing delay lengths in the next section). As before, we will set $ v_{0} = \Delta/T$. At a boundary junction at coordinates $ (0,j\Delta)$, we will have six port admittances: $ Y_{x^{-},0,j}$, $ Y_{y^{-},0,j}$ and $ Y_{y^{+},0,j}$ corresponding to waveguide connections with junctions to the west, south and north respectively, $ Y_{x^{+}y^{+},0,j}$ and $ Y_{x^{+}y^{-},0,j}$ for connections to junctions to the northeast and southeast, respectively in region $ II$ and a self-loop admittance $ Y_{c,0,j}$. The junction admittance at a boundary point $ B$ is then

$\displaystyle Y_{J,0,j} \triangleq Y_{x^{-},0,j}+Y_{y^{-},0,j}+Y_{y^{+},0,j}+Y_{x^{+}y^{+},0,j}+Y_{x^{+}y^{-},0,j}+Y_{c,0,j}$    

Because this waveguide mesh is an extension of the type II mesh described in §4.4.2, we might expect that the waveguide admittances will be related to values of the material parameters $ l$ and $ c$ at the midpoints of the waveguides. This is, in fact, true, even at the boundary junctions, though because of the asymmetric nature of these junctions with respect to the coordinate axes, we must perform a judicious scaling of some of these admittances. In fact, we must only scale the admittances of the waveguides which lie along the boundary itself, and that of the self-loop. The admittances of waveguides connected to interior points in region $ I$ or $ II$ should be treated as ``interior,'' so that the scattering will be correct at junctions neighboring the boundary).

The difference scheme operating at a junction on the boundary will be

$\displaystyle \hspace{-0.2in}\frac{Y_{J,0,j}}{2}\big(U_{J,0,j}(n+1)+U_{J,0,j}(n-1)\big)$ $\displaystyle =\hspace{0.1in}$   $\displaystyle Y_{x^{+}y^{+},0,j}U_{J,\frac{1}{2},j+\frac{1}{2}}(n) +Y_{x^{+}y^{-},0,j}U_{J,\frac{1}{2},j-\frac{1}{2}}(n)\notag$    
      $\displaystyle \hspace{-0.05in}+Y_{x^{-},0,j}U_{J,-1,j}(n)+Y_{y^{+},0,j}U_{J,0,j+1}(n)\notag$    
      $\displaystyle \hspace{-0.05in}+Y_{y^{-},0,j}U_{J,0,j-1}(n)+Y_{c,0,j}U_{J,0,j}(n)$    

If we now treat the junction voltages as samples of a continuous function $ u$, then the difference scheme above can be expanded in a Taylor series about $ (0,j\Delta,nT)$ to give

$\displaystyle Y_{J,0,j}T^{2}\frac{\partial^{2}u}{\partial t^{2}}$ $\displaystyle =\hspace{0.1in}$   $\displaystyle 2\Delta\left(-Y_{x^{-},0,j}+\frac{1}{2}(Y_{x^{+}y^{-},0,j}+Y_{x^{+}y^{+},0,j})\right)\frac{\partial u}{\partial x}$    
  $\displaystyle \hspace{0.2in}$   $\displaystyle \hspace{-0.05in}+ 2\Delta\left(Y_{y^{+},0,j}-Y_{y^{-},0,j}+\frac{1}{2}(Y_{x^{+}y^{+},0,j}-Y_{x^{+}y^{-},0,j})\right)\frac{\partial u}{\partial y}$    
  $\displaystyle \hspace{0.2in}$   $\displaystyle \hspace{-0.05in}+ \Delta^{2}\left(Y_{x^{-},0,j}+\frac{1}{4}(Y_{x^{+}y^{-},0,j}+Y_{x^{+}y^{+},0,j})\right)\frac{\partial^{2}u}{\partial x^{2}}$    
  $\displaystyle \hspace{0.2in}$   $\displaystyle \hspace{-0.05in}+ \Delta^{2}\left(Y_{y^{-},0,j}+Y_{y^{+},0,j}+\fr...
...{x^{+}y^{-},0,j}+Y_{x^{+}y^{+},0,j})\right)\frac{\partial^{2}u}{\partial y^{2}}$    
  $\displaystyle \hspace{0.2in}$   $\displaystyle \hspace{-0.05in}+ \frac{\Delta^{2}}{2}\left(Y_{x^{+}y^{+},0,j}-Y_{x^{+}y^{-},0,j}\right)\frac{\partial^{2}u}{\partial x\partial y}$    
  $\displaystyle \hspace{0.2in}$   $\displaystyle \hspace{-0.05in}+ O(\Delta^{3},T^{4})$ (4.117)

In order to associate this expansion with the reduced form of the parallel-plate equations of (4.77), we may set

$\displaystyle Y_{x^{-},0,j} = \frac{1}{v_{0}l_{-\frac{1}{2},j}}\hspace{0.3in}Y_...
...+\frac{1}{2}}}\hspace{0.3in}Y_{y^{-},0,j} = \frac{1}{2v_{0}l_{0,j-\frac{1}{2}}}$    

$\displaystyle Y_{x^{+}y^{+},0,j} = \frac{1}{v_{0}l_{\frac{1}{4},j+\frac{1}{4}}}\hspace{0.3in}Y_{x^{+}y^{+},0,j} = \frac{1}{v_{0}l_{\frac{1}{4},j-\frac{1}{4}}}$    

and for the self-loop admittance, we set
$\displaystyle Y_{c,0,j}$ $\displaystyle =$ $\displaystyle \frac{3}{16}v_{0}\left(2c_{-\frac{1}{2},j}+2c_{\frac{1}{4},j+\fra...
...\frac{1}{4},j-\frac{1}{4}}+c_{0,j+\frac{1}{2}}+c_{0,j-\frac{1}{2}}\right)\notag$ (4.118)
    $\displaystyle \hspace{0.3in}- Y_{x^{-},0,j} - Y_{y^{-},0,j}- Y_{y^{+},0,j}- Y_{x^{+}y^{+},0,j}- Y_{x^{+}y^{-},0,j}$ (4.119)

These settings yield a difference scheme which is consistent with the transmission line equations, and which is first-order accurate in the grid spacing $ \Delta$. It is important to note that the admittances of the waveguides connecting two junctions on the boundary itself are set to be half what they would be in the interior of region $ I$. Also note that the mixed derivative term in (4.101) becomes $ O(\Delta^{3})$ (because $ Y_{x^{+}y^{+},0,j}$ and $ Y_{x^{+}y^{-},0,j}$ are the same to zeroth-order, and hence their difference, which is the coefficient of the mixed-derivative term, will be $ O(\Delta)$).

The additional stability requirement, from (4.102) is

$\displaystyle v_{0}\geq \max_{\begin{minipage}[t]{1.0in}\begin{center}\tiny {boundary waveguide midpoints}\end{center}\end{minipage}}\sqrt{\frac{8}{3lc}}$    

which is marginally more restrictive than the requirement on the interior of region $ I$ (by a factor of $ \sqrt{\frac{4}{3}}$). This deterioration in the stability bound is offset, however, by the fact in region $ II$, the grid spacing is $ \Delta/\sqrt{2}$ we must have

$\displaystyle \frac{\Delta}{\sqrt{2}T}\geq \max_{\begin{minipage}[t]{1.0in}\beg...
...aveguide midpoints in region II}\end{center}\end{minipage}}2\sqrt{\frac{1}{lc}}$    

because we are by necessity operating away from the CFL bound in this particular multi-grid setting, which incorporates different grid spacings and yet maintains the same time step throughout the mesh.

The choice of $ Y_{x^{-},0,j} = \frac{1}{v_{0}l_{-\frac{1}{2},j}}$ means that all other admittances in region $ I$ may be set as previously discussed in §4.4.2 for a type II mesh.

next up previous
Next: Corners Up: Interfaces Between Grids Previous: Interfaces Between Grids
Stefan Bilbao 2002-01-22