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== An Overview of the Method of Moments ==

== Numerical Solution of Integral Equations ==

:<math>J(r) = \sum_{n=1}^N I_n^{(J)} f_n^{(J)} (r)</math>

The volume polarization currents in 2.5-D MoM have a vertical direction along the Z-axis. These are discretized using prismatic basis functions that have either a rectangular or triangular base with a constant profile along the Z-axis.

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== The Rectangular Mesh Advantage ==

:<math> Z_{ij}^{(\mu \nu)} = \iiint_{V_i} d\nu f_i^{(\mu)}(r) \cdot \iiint_{V_j}d\nu ' \overline{\overline{G}}_{\mu \nu}(r|r') \cdot f_j^{(v)}(r') </math>

where the spatial-domain dyadic Green's functions are a function of the observation and source coordinates, '''r'''and '''r' '''. The MoM matrix elements can indeed be interpreted as interactions between two elementary basis functions '''f_i(r)''' and '''f_j(r')''' on that particular background structure. The spatial-domain dyadic Green's functions can themselves be expressed in terms of the spectral-domain dyadic Green's functions as follows:

:<math> \overline{\overline{G}}_{\mu \nu}(r|r') = \frac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \tilde{\overline{\overline{G}}}_{\mu \nu} (k_p, z|z') e^{-j[k_x(x-x')+k_y(y-y')]} \, dk_x \, dk_y , \quad {k_p}^2 = {k_x}^2 + {k_y}^2 </math>

where the doubly infinite integration is performed with respect to the spectral variables k_x and k_y. As can be seen from the above expression, the spatial-domain dyadic Green's functions are functions of z, z', as well as (x-x') and (y-y'). The MoM matrix elements can now be transformed into the spectral domain as

:<math> Z_{ij}^{(\mu \nu)} = \dfrac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \tilde{f}_i^{(\mu)} (k_x, k_y) \cdot \tilde{\overline{\overline{G}}}_{\mu \nu} (k_{\rho}, z|z') \cdot \tilde{f}_j^{(\nu)} (k_x, k_y) \, dk_x \, dk_y </math>

where the tilde symbol signifies the Fourier transform of a function defined as

:<math> \tilde{f}(k_x, k_y) = \dfrac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} f(x,y) e^{j(k_x x + k_y y)} \, dx \, dy </math>

Rectangular cells have simple Fourier transforms. The rooftop basis functions are triangular functions in the direction of current flow and constant in the perpendicular direction. This means that their Fourier transform is a product of a sinc-squared function along one spectral direction and a sinc function along the other. You can see from the figure below that if one deals with a rectangular mesh of identical cells (all equal and parallel), then the interactions among the rooftop basis functions become a functions of the index differences and not the absolute indices:

:<math> Z_{(i,k)|(j,l)} = Z \Big\langle f_{i,k}(x,y)| f_{j,l}(x', y') \Big\rangle = Z_{(i-j)|(k-l)} </math>

In the above equation, the vectorial rooftop basis functions have explicit, double indices: i and k along the local X and Y directions, respectively, for the test (observation) basis function, and j and l along the local X and Y directions, respectively, for the expansion (source) basis function. Thus, uniform rectangular cells, i.e. structured rectangular cells of identical size aligned in the same direction, can speed up the planar MoM simulation significantly due to these symmetry and the invariance properties. For example, all the self-interactions are identical regardless of the location of a rooftop basis function. This reduces the matrix fill process for a total of N rooftop basis functions from an N2 process to one of order N.

== Computing The Near Fields in Planar MoM ==