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[[Image:PMOM32.png|thumb|600px450px|Planar hybrid and triangular meshes for rectangular patches.]][[Image:PMOM30.png|thumb|600px450px|Mesh of two rectangular patches at two different planes. The lower substrate layer has a higher permittivity.]]=== The Understanding the Planar MoM Mesh ===

The method of moments (MoM) discretizes all the finite-sized objects of a planar structure (excluding the background structure) into a set of elementary cells. The planar integral equations are then solved approximately on these elementary cells. As this method does not require a discretization of the entire computational domain, it is often computationally much more efficient than differential-based techniques like FEM or FDTD, which mesh the whole domain. The accuracy of the MoM numerical solution depends greatly on the quality and resolution of the generated mesh. The mesh density gives a measure of how electrically small these elementary cells are. Low mesh resolutions compromise the accuracy of the numerical solution. On the other hand, very high mesh densities may lead to numerical instability of the method of moments. As a rule of thumb, a mesh density of about 20-30 cells per effective wavelength usually yields acceptable satisfactory results. Yet, for structures with lots of fine geometrical details or for highly resonant structures, higher mesh densities may be required. Also, the particular simulation data that you seek in a project will also influence your choice of mesh resolution. For example, far field characteristics like radiation patterns are less sensitive to the mesh density than field distributions on a structure with a highly irregular shape and a rugged boundary.

It is well known that any planar geometry with any degree EM.Picasso generates two types of complexity can be reasonably discretized using mesh for a surface planar structure: a pure triangular meshand a hybrid triangular-rectangular. [[In both case, EM.Cube]]'s [[Planar Module]] provides a versatile triangular mesh generator for this purpose. This generates Picasso attempts to create a highly regular mesh, in which most of the triangular cells have almost equal areas. The hybrid mesh type tries to produce as many rectangular cells as possible especially in the case of objects with rectangular or linear boundaries. In connection or junction areas between adjacent objects or close to highly curved boundaries, the use of triangular cells is clearly inevitable. EM.Picasso's default mesh type is hybrid. The uniformity or regularity of mesh is an important factor in warranting a stable MoM 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.

Using the generated The mesh of density gives a planar structure, [[EM.Cube]] creates a set measure of vectorial basis functions the number of cells per effective wavelength that are passed to the input file placed in various regions of the Planar MoM simulation engine. This engine requires edge-based basis functionsyour planar structure. The common edges between adjacent higher the mesh density, the more cells are used to define edgecreated on the geometrical objects. Keep in mind that only the finite-based rooftop or RWG basis functionssized objects of your structure are discretized. These elementary basis functions indeed provide The free-space wavelength is defined as <math>\lambda_0 = \tfrac{2\pi f}{c}</math>, where f is the current flow center frequency of your project and warrant c is the continuity among speed of light in the mesh cellsfree space. Therefore, when two objects overlap or share a common edgeThe effective wavelength is defined as <math>\lambda_{eff} = \tfrac{\lambda_0}{\sqrt{\varepsilon_{eff}}}</math>, where e_eff is the connection between them must be translated into &quot;bridge&quot; basis functionseffective permittivity. By default, which carry the information about current flow to the simulation engine[[EM.Picasso]] generates a hybrid mesh with a mesh density of 20 cells per effective wavelength.

'''The most important rule effective permittivity is defined differently for different types of traces and embedded object connections in [[EMsets.Cube]]'s [[Planar Module]] This is to make sure that enough cells are placed in areas that only objects belonging to the same trace can be connected to one anothermight feature higher field concentration.''' For examplePEC and conductive sheet traces, if two objects reside on the same Z-plane effective permittivity is defined as the larger of the permittivity of the two substrate layers just above and geometrically have a common edge which you can clearly see in below the project workspace, but organizationally they belong to two different metal metallic trace. For slot traces, then the bridge basis functions will not be generated between them, and effective permittivity is defined as the mean (average) of the permittivity of the simulation engine will see them disconnected. If two objects belong to substrate layers just above and below the same metallic trace and have a common overlap area, [[EM.Cube]] first merges the two objects using the &quot;Boolean Union&quot; operation and converts them into a single For embedded object for sets, the purpose of meshing. The mesh of &quot;unioned&quot; areas effective permittivity is usually made up defined as the largest of triangular cells. If two objects reside on the same Z-plane permittivities of all the substrate layers 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 resultsembedded dielectric sets.

The most important rule of object connections in EM.Picasso 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.Picasso]] first merges the two objects using the &quot;Boolean Union&quot; operation and converts them into a single object for the purpose of meshing. The mesh of &quot;unioned&quot; areas is usually made up of triangular cells.  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 &quot;unions&quot; the two objects and generates a connected mesh. [[EM.Cube]]'s hybrid planar mesh generator, however, behave behaves differently when it comes to the connection between rectangular objects. The rule in this case is the following: