== Discretizing Planar Structures ==
[[Image:PMOM32.png|thumb|450px|Planar hybrid and triangular meshes for rectangular patches.]]
[[Image:PMOM30.png|thumb|450px|Mesh of two rectangular patches at two different planes. The lower substrate layer has a higher permittivity.]]
[[Image:PMOM31.png|thumb|400px|The Planar Mesh Settings dialog.]]
=== Understanding the Planar MoM Mesh ===
[[Image:Info_icon.png|40px]] Click here to learn more about '''[[Mesh_Generation_Schemes_in_EM.Cube#Working_with_Mesh_Generator | Working with Mesh Generator ]]'''.
[[Image:Info_icon.png|40px]] Click here to learn more about EM.Picasso's '''[[Mesh_Generation_Schemes_in_EM.Cube#The_Hybrid_Planar_Mesh_Generator The_Triangular_Surface_Mesh_Generator | Hybrid Planar Triangular Surface Mesh Generator ]]'''.
[[Image:Info_icon.png|40px]] Click here to learn more about EM.Picasso's '''[[Mesh_Generation_Schemes_in_EM.Cube#The_Triangular_Surface_Mesh_Generator The_Hybrid_Planar_Mesh_Generator | Triangular Surface Hybrid Planar Mesh Generator ]]'''.
[[Image:PMOM44Info_icon.png|thumb|400px|Deleting or curing defective triangular cells: Case 1.40px]]Click here to learn more about '''[[Image:PMOM42Mesh_Generation_Schemes_in_EM.pngCube#General_Rules_of_Planar_Hybrid_Mesh_Generator|thumb|400px|Deleting or curing defective triangular cells: Case 2.General Rules of Planar Hybrid Mesh Generator]][[Image:PMOM45'''.png|thumb|300px|Locking the mesh density of an object group from its property dialog.]]=== Generating, Viewing & Customizing a Planar Mesh ===
You can generate and view a planar mesh by clicking the '''Show Mesh''' [[FileImage:mesh_toolInfo_icon.png|40px]] button of the Click here to learn more about '''Simulate Toolbar''' or by selecting '''Menu > Simulate > Discretization > Show Mesh''' or using the keyboard shortcut '''Ctrl+M'''. When the mesh of the planar structure is displayed in [[EMMesh_Generation_Schemes_in_EM.Cube]]âs project workspace, its "Mesh View" mode is enabled. In this mode you can perform view operations like rotate view, pan or zoom, but you cannot create new objects or edit existing ones. To exit the mesh view mode, press the keyboard's '''Esc Key''' or click the '''Show Mesh''' [[File:mesh_tool.png]] button once again. Once a mesh is generated, it stays in the memory until the structure is changed or the mesh density or other settings are modified. Every time you view mesh, the one in the memory is displayed. You can force [[EM.Picasso]] to create a new mesh from the ground up by selecting '''Menu > Simulate > Discretization > Regenerate Mesh''' or by right clicking on the '''Planar Mesh''' item in the '''Discretization''' section of the navigation tree and selecting '''Regenerate''' from the contextual menu. You can change the settings of the planar mesh including the mesh type and density from the Planar Mesh Settings Dialog. You can also change these settings while in the mesh view mode, and you can update the changes to view the new mesh. To open the mesh settings dialog, either click the '''Mesh Settings''' [[File:mesh_settings.png]] button of the '''Simulate Toolbar''' or select '''Menu > Simulate > Discretization > Mesh Settings...''', or by right click on the '''Planar Mesh''' item in the '''Discretization''' section of the Navigation Tree and select '''Mesh Settings...''' from the contextual menu, or use the keyboard shortcut '''Ctrl+G'''. You can change the mesh algorithm from the dropdown list labeled '''Mesh Type''', which offers two options: '''Hybrid''' and '''Triangular'''. You can also enter a different value for '''Mesh Density''' in cells per effective wavelength (λ<sub>eff</sub>). For each value of mesh density, the dialog also shows the average "Cell Edge Length" in the free space. To get an idea of the size of mesh cells on the traces and embedded object sets, divide this edge length by the square root of the effective permittivity a particular trace or set. Click the '''Apply''' button to make the changes effective. === A Note on the Junction Mesh === The integrity of the planar mesh and its continuity in the junction areas where adjacent objects are connected directly affects the simulation results. 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. If two objects belong to the same trace (residing on the same Z-plane) and have a common overlap area, EM.Picasso first merges the two objects using the "Boolean Union" operation and converts them into a single object for the purpose of meshing. EM.Picasso's hybrid planar mesh generator has some additional rules: * If two connected rectangular objects have the same side dimensions along the common linear edge with perfect alignment, a rectangular junction mesh is produced.* If two connected rectangular objects have different side dimensions along the common linear edge or have edge offset, a set of triangular cells is generated along the edge of the object with the large side.* Rectangular objects that contain gap source or lumped elements, always have a rectangular mesh around the gap area. 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.Picassoâs planar mesh generator automatically handles situations of this kind and generates all the required connection meshes.  <table><tr><td> [[File:PMOM36.png#Refining_the_Planar_Mesh_Locally|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|375px]] [[File:PMOM40.png|375px]] </td></tr><tr><td> Meshes of short and long vertical PEC vias connecting two horizontal metallic strips. </td></tr></table> === Refining the Planar Mesh Locally === It is very important to apply the right mesh density to capture all the geometrical details of your planar structure. This is especially true for "field discontinuity" regions such as junction areas between objects of different side dimensions, where larger current concentrations are usually observed at sharp corners, or at the connection areas between metallic traces and PEC vias, as well as the areas around gap sources and lumped elements, as these create voltage or current discontinuities. For large planar structures, using a higher mesh density may not always be a practical option since it will quickly lead to a very large MoM matrix and thus growing the size of the numerical problem. EM.Picasso provides several ways of controlling the mesh of a planar structure locally.  The Planar Mesh Settings dialog gives a few options for customizing your planar mesh around geometrical and field discontinuities. You can check the check box labeled "'''Refine Mesh at Junctions'''", which increases the mesh resolution at the connection area between rectangular objects. Or you can check the check box labeled "'''Refine Mesh at Gap Locations'''", which may prove particularly useful when gap sources or lumped elements are placed on a short transmission line connected from both ends. Or you can check the check box labeled "'''Refine Mesh at Vias'''", which increases the mesh resolution on the cross section of embedded object sets and at the connection regions of the metallic objects connected to them. [[EM.Picasso]] typically doubles the mesh resolution locally at the discontinuity areas when the respective boxes are checked. You should always visually inspect EM.Picasso's default generated mesh to see if the current mesh settings have produced an acceptable mesh. You may often need to change the mesh density or other [[parameters]] and regenerate the mesh. Sometimes EM.Picasso's default mesh may contain very narrow triangular cells due to very small angles between two edges. In some rare cases, extremely small triangular cells may be generated, whose area is a small fraction of the average mesh cell. These cases typically happen at the junctions and other discontinuity regions or at the boundary of highly irregular geometries with extremely fine details. In such cases, increasing or decreasing the mesh density by one or few cells per effective wavelength often resolves that problem and eliminates those defective cells. Nonetheless, EM.Picasso's planar mesh generator offers an option to identify the defective triangular cells and either delete them or cure them. By curing we mean removing a narrow triangular cell and merging its two closely spaced nodes to fill the crack left behind. EM.Picasso by default deletes or cures all the triangular cells that have angles less than 10º. Sometimes removing defective cells may inadvertently cause worse problems in the mesh. You may choose to disable this feature and uncheck the box labeled "'''Remove Defective Triangular Cells'''" in the Planar Mesh Settings dialog. You can also change the value of the minimum allowable cell angle. Another way of local mesh control is to lock the mesh density of certain traces or object sets. The mesh density that you specify in the Planar Mesh Settings dialog is a global parameter and applies to all the traces and embedded object sets in your project. However, you can lock the mesh of individual PEC, PMC and conductive sheet traces or embedded objects sets. In that case, the locked mesh density takes precedence over the global density. Note that locking mesh of object groups, in principle, is different than refining the mesh at discontinuities. In the latter case, the mesh of connection areas is affected. However, objects belonging to different traces cannot be connected to one another. Therefore, locking mesh can be useful primarily for isolated object groups that may require a higher (or lower) mesh resolution. You can lock the local mesh density by accessing the property dialog of a specific trace or embedded object set and checking the box labeled '''Lock Mesh'''. This will enable the '''Mesh Density''' box, where you can accept the default global value or set any desired new value.
== Excitation Sources ==