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== Free-Space Green’s Function ==

The right choice of the basis functions that are used to represent the elementary currents is very important. It will determine the accuracy and computational efficiency of the resulting numerical solution. Rooftop basis functions are one of the more popular types of basis functions used in a variety of MoM formulations. The simplest rooftop function is the one-dimensional triangular functions defined as in the figure below:

This function provides a linear interpolation of the unknown currents or fields in one dimension. Note that the function vanishes at it two ends. This is a desirable feature for basis functions that represent electric currents on metallic wires as the current must vanish at the two ends of a wire. The total current on the wire can be approximated in a linear fashion by a set of one-dimensional rooftop functions as shown in the figure below:

This can be written as

:<math> I(l) = \sum_{n=1}^N a_n f_n(l) \mathbf{\hat{s}_n} </math>

where l is the length coordinate along the wire with l=0 at its start point. <math>f_n(l)</math> is the scaled and translated version of the linear basis function <math>f(l)</math> shown in the previous figure. <math>\mathbf{\hat{s}_n}</math> is the unit vector along wire.

== Multilayer Green’s Functions ==

The Green’s functions are the solutions of boundary value problems when they are excited by an elementary source. This is usually assumed to be an infinitesimally small vectorial point source. In order for Green’s functions to be computationally useful, they must have analytical closed forms like a mathematical expression, or one should be able to compute them using a recursive process. It turns out that only very few boundary value problems have closed-form Green’s functions. Planar layered structures with laterally infinite extents are one of those few cases, which can be represented by recursive dyadic Green's functions.

In general, a structure may support both electric ('''J''') and magnetic ('''M''') currents. The total electric ('''E''') and magnetic ('''H''') fields can be expressed in terms of the electric and magnetic currents in the following way:

:<math>H = H^{inc} + \iiint\limits_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') \, dv' + \iiint\limits_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') \, dv'</math>

where '''G_EJ''', '''G_EM''', '''G_HJ''', '''GH_M''' are the dyadic Green’s functions for the electric and magnetic currents due to electric and magnetic current source, respectively, and '''Eⁱ''' and '''Hⁱ''' are the incident or impressed electric and magnetic fields, respectively. In these equations, '''r''' is the position vector of the observation point and '''r'''' is the position vector of the source point. V is the volume that contains all the sources and the volume integration is performed with respect to the primed coordinates. The incident or impressed fields provide the excitation of the structure. They may come from an incident plane wave or a gap source on a microstrip line, a short dipole, etc. The complexity of the Green’s functions depends on what is considered as the background structure. If you remove all the unknown currents from the structure, you are left with the background structure.

== Numerical Solution of Integral Equations ==

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

Similar expressions can be derived for the T^(EM), U^(HJ) and Y^(HM)elements of the MoM matrix.

The right choice of the basis functions to represent the elementary currents is very important. It will determine the accuracy and computational efficiency of the resulting numerical solution. Rooftop basis functions are one of the most popular types of basis functions used in a variety of MoM formulations. The surface currents (whether electric or magnetic) are discretized using 2D rooftop basis functions shown in the figure below:

The rooftop basis functions are defined over two adjacent cells with a common edge of length. If the two cells are triangular, then the so-called RWG functions are obtained. It is also possible to define rooftop functions over two adjacent rectangular cells or two adjacent rectangular and triangular cells with a common edge. On a rectangular cell, the function is defined as having a (descending or ascending) linear profile in one direction and a constant profile in the other perpendicular direction.

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.

== 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>

:<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 ==

<!--[[File:PMOM93(1).png]]-->

{{Note|When the observation plane is placed very close to the radiating J and M currents, the Green's functions exhibit singularities, which translate to very slow convergence or divergence of the integrals. You need to be careful to place field sensors at adequate distances from these radiating sources.}}

The above doubly infinite periodic Green's functions are said to be expressed in terms of &quot;Floquet Modes&quot;. The exact formulation involves an infinite set of these periodic Floquet modes. During the MoM matrix fill process for a periodic structure, a finite number of Floquet modes are calculated. By default, [[EM.Cube]]'s planar MoM engine considers M_x = M_y = 25. This implies a total of 51 modes along the X direction and a total of 51 modes along the Y direction, or a grand total of 51² = 2,601 Floquet modes. You can increase the number of Floquet modes for your project from the Planar MoM Engine Settings Dialog. In the section titled &quot;Periodic Simulation&quot;, you can change the values of '''Number of Floquet Modes''' in the two boxes designated X and Y.

After the MoM impedance matrix '''[Z]''' (not to be confused with the impedance parameters) and excitation vector '''[V]''' have been computed through the matrix fill process, the planar MoM simulation engine is ready to solve the system of linear equations:

where '''[I]''' is the solution vector, which contains the unknown amplitudes of all the basis functions that represent the unknown electric and magnetic currents of finite extents in your planar structure. In the above equation, N is the dimension of the linear system and equal to the total number of basis functions in the planar mesh. [[EM.Cube]]'s linear solvers compute the solution vector'''[I]''' of the above system. You can instruct [[EM.Cube]] to write the MoM matrix and excitation and solution vectors into output data files for your examination. To do so, check the box labeled &quot;'''Output MoM Matrix and Vectors'''&quot; in the Matrix Fill section of the Planar MoM Engine Settings dialog. These are written into three files called mom.dat1, exc.dat1 and soln.dat1, respectively.

# LU Decomposition Method

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