== Discretizing Planar Structures ==
[[Image:PMOM32.png|thumb|600px|Planar hybrid and triangular meshes for rectangular patches.]]
=== The Planar MoM Mesh ===
It is well known that any planar geometry with any degree of complexity can be reasonably discretized using a surface triangular mesh. [[EM.Cube]]'s [[Planar Module]] provides a versatile triangular mesh generator for this purpose. This generates a regular mesh, in which most of the triangular cells have almost equal areas. The uniformity or regularity of mesh is an important factor in warranting a stable numerical solution. A highly incongruous mesh may even produce completely erroneous results. [[EM.Cube]]'s [[Planar Module]] also offers another mesh generator that creates a "Hubrid" planar mesh combining triangular and rectangular cells. Although triangular cells are more versatile than rectangular cells in adapting to arbitrary geometries, many practical planar structures contain a large number of rectangular parts like patch antennas, microstrip lines and components, etc.
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[[File:PMOM32.png|800px]]
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Figure 1: Planar hybrid and triangular meshes for rectangular patches.
Using the generated mesh of a planar structure, [[EM.Cube]] creates a set of vectorial basis functions that are passed to the input file of the Planar MoM simulation engine. This engine requires edge-based basis functions. The common edges between adjacent cells are used to define edge-based rooftop or RWG basis functions. These elementary basis functions indeed provide the current flow and warrant the continuity among the mesh cells. Therefore, when two objects overlap or share a common edge, the connection between them must be translated into "bridge" basis functions, which carry the information about current flow to the simulation engine.
'''The most important rule of object connections in [[EM.Cube]]'s [[Planar Module]] is that only objects belonging to the same trace can be connected to one another.''' For example, if two objects reside on the same Z-plane and geometrically have a common edge which you can clearly see in the project workspace, but organizationally they belong to two different metal traces, then the bridge basis functions will not be generated between them, and the simulation engine will see them disconnected. If two objects belong to the same trace and have a common overlap area, [[EM.Cube]] first merges the two objects using the "Boolean Union" operation and converts them into a single object for the purpose of meshing. The mesh of "unioned" areas is usually made up of triangular cells. If two objects reside on the same Z-plane and geometrically overlap with each other but organizationally belong to two different trace groups, incongruous, overlapped cells will be generated that will either blow up the linear system or produce completely wrong simulation results.
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[[File:PMOM36.png|250px]] [[File:PMOM38.png|250px]] [[File:PMOM37.png|250px]]
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Figure 1: Two overlapping planar objects and their triangular and hybrid planar meshes.
When two planar objects belonging to the same trace are connected via a common edge, it is critical to generate a consistent mesh at the connection area and properly transition and merge the meshes of the individual objects. [[EM.Cube]]'s triangular planar mesh generator simply "unions" the two objects and generates a connected mesh. [[EM.Cube]]'s hybrid planar mesh generator, however, behave differently when it comes to the connection between rectangular objects. The rule in this case is the following:
* Rectangular objects that contain gap source or lumped elements, always have a rectangular mesh around the gap area.
[[File:PMOM33.png|250px]] [[File:PMOM35.png|250px]] [[File:PMOM34.png|250px]] Figure 2: Edge-connected rectangular planar objects and their triangular and hybrid planar meshes. [[EM.Cube]]'s [[Planar Module]] models embedded objects as vertical volume currents. The vectorial basis functions in this case are Z-directed prisms as opposed to rooftop basis functions. If an embedded object like an interconnect via is located under or above a metallic trace or connected from both top and bottom, it is critical to create mesh continuity between the embedded object and its connected metallic traces. In other words, the generated mesh must ensure current continuity between the vertical volume currents and horizontal surface currents. [[EM.Cube]]âs Picassoâs planar mesh generator automatically handles situations of this kind and generates all the required connection meshes. Keep in mind that [[EM.Cube]]âs Planar MoM engine uses a 2.5-D approximation, whereby only vertical volume currents are assumed inside embedded objects. When the height of an embedded object is small (as should typically be under the 2.5-D assumption), one prismatic cell is placed across the object along the Z-axis. Long PEC vias with a very small radius do also satisfy the 2.5-D assumption. In this case, the long via objects are discretized further along the Z direction and generate multiple stacked cells. Several prismatic cells along the Z-axis may increase the simulation time drastically. This is due to the fact that the host layer is effectively subdivided into a number of sub-layers and the stacked cells are treated as stacked vias embedded inside these sub-layers. As a result, the simulation engine needs to compute all the dyadic Greenâs functions accounting for the interactions between all such sub-layers. [[File:PMOM39.png|400px]] [[File:PMOM40.png|400px]]
<table><tr><td> [[File:PMOM36.png|250px]] [[File:PMOM38.png|250px]] [[File:PMOM37.png|250px]] </td></tr><tr><td> Two overlapping planar objects and their triangular and hybrid planar meshes. </td></tr><tr><td> [[File:PMOM33.png|250px]] [[File:PMOM35.png|250px]] [[File:PMOM34.png|250px]] </td></tr><tr><td> Edge-connected rectangular planar objects and their triangular and hybrid planar meshes. </td></tr><tr><td> [[File:PMOM39.png|400px]] [[File:PMOM40.png|400px]] </td></tr><tr><td> Mesh of a vertical PEC via connecting two horizontal metallic strips. The shorter via has one prismatic cell along the Z direction, while the longer via is discretized into several stacked cells.</td></tr></table>
=== Generating A Planar Mesh ===