# EM.Picasso's simulation engine is based on a 2.5-D MoM formulation. Only vertical volume currents and no circumferential components are allowed on embedded objects. The 2.5-D assumption holds very well in two cases: (a) when embedded objects are very thin with a very small cross section (with lateral dimensions less than 2-5% of the material wavelength) or (b) when embedded objects are very short and sandwiched between two closely spaced PEC traces or grounds from the top and bottom.
== Discretizing the Planar Structure == [[Image:PMOM31.png|thumb|400px|The Planar Mesh Settings dialog.]]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. Both the quality and resolution of the generated mesh greatly affect the accuracy of the MoM numerical solution. The mesh density gives a measure of the number of cells per effective wavelength that are placed in various regions of your planar structure. The higher the mesh density, the more cells are created on the finite-sized geometrical objects. As a rule of thumb, a mesh density of about 20-30 cells per effective wavelength usually yields satisfactory results. But for structures with lots of fine geometrical details or for highly resonant structures, higher mesh densities may be required. The particular output data that you seek in a simulation 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 structures with a highly irregular shapes and boundaries. EM.Picasso provides two types of mesh for a planar structure: a pure triangular surface mesh and a hybrid triangular-rectangular surface mesh. In both case, EM.Picasso attempts to create a highly regular mesh, in which most of the cells have almost equal areas. For planar structures with regular, mostly rectangular shapes, the hybrid mesh generator usually leads to faster computation times.  [[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_Triangular_Surface_Mesh_Generator | 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_Hybrid_Planar_Mesh_Generator | Hybrid Planar Mesh Generator]]'''. [[Image:Info_icon.png|40px]] Click here to learn more about '''[[Mesh_Generation_Schemes_in_EM.Cube#General_Rules_of_Planar_Hybrid_Mesh_Generator| General Rules of Planar Hybrid Mesh Generator]]'''. [[Image:Info_icon.png|40px]] Click here to learn more about '''[[Mesh_Generation_Schemes_in_EM.Cube#Refining_the_Planar_Mesh_Locally| Refining the Planar Mesh Locally]]'''. <table><tr><td> [[Image:PMOM48F.png|thumb|350px|Geometry of a multilayer slot-coupled patch array.]] </td><td> [[Image:PMOM48G.png|thumb|370px|Hybrid planar mesh of the slot-coupled patch array.]] </td></tr></table><table><tr><td> [[Image:PMOM48H.png|thumb|350px|Details of the hybrid planar mesh of the slot-coupled patch array around discontinuities.]] </td></tr></table> == Excitation Sources ==
Your planar structure must be excited by some sort of signal source that induces electric surface currents on metal parts, magnetic surface currents on slot traces, and conduction or polarization volume currents on vertical vias and embedded objects. The excitation source you choose depends on the observables you seek in your project. EM.Picasso provides the following source types for exciting planar structures (click on each type to learn more about it):
[[Image:Info_icon.png|40px]] Click here to learn more about '''[[Common_Excitation_Source_Types_in_EM.Cube#Modeling_Coupled_Ports | Modeling Coupled Ports]]'''.
== Running Planar MoM Simulations == === EM.Picasso's Simulation Modes === EM.Picasso offers five Planar MoM simulation modes (click on each type to learn more about it): * '''[[#Running a Single-Frequency Planar MoM Analysis | Single-Frequency Analysis]]'''* '''[[Parametric_Modeling,_Sweep_%26_Optimization#Running_Frequency_Sweep_Simulations_in_EM.Cube | Frequency Sweep]]''' including uniform and adaptive frequency sweeps * '''[[Parametric_Modeling,_Sweep_%26_Optimization#Running_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]'''* '''[[Parametric_Modeling,_Sweep_%26_Optimization#Optimization | Optimization]]'''* '''[[Building_Reusable_Models#Running_an_HDMR_Sweep | HDMR Sweep]]''' You can set the simulation mode from EM.Picasso's "Simulation Run Dialog". A single-frequency analysis is a single-run simulation. All the other simulation modes in the above list are considered multi-run simulations. If you run a simulation without having defined any observables, no data will be generated at the end of the simulation. In multi-run simulation modes, certain [[parameters]] are varied and a collection of simulation data files are generated. At the end of a sweep simulation, you can graph the simulation results in EM.Grid or you can animate the 3D simulation data from the navigation tree. === Running a Single-Frequency Planar MoM Analysis === [[Image:PMOM80.png|thumb|400px|EM.Picasso's Simulation Run dialog.]]A single-frequency analysis is the simplest type of EM.Picasso simulation and involves the following steps: * Set the units of your project and the frequency of operation. Note that the default project unit is '''millimeter'''. * Define you background structure and its layer properties and trace types. * Construct your planar structure using [[CubeCAD]]'s drawing tools to create all the finite-sized metal and slot trace objects and possibly embedded metal or dielectric objects that are interspersed among the substrate layers.* Define an excitation source and observables for your project.* Examine the planar mesh, verify its integrity and change the mesh density if necessary.* Run the Planar MoM simulation engine.* Visualize the output simulation data. To run a planar MoM analysis of your project structure, open the Run Simulation Dialog by clicking the '''Run''' [[File:run_icon.png]] button on the '''Simulate Toolbar''' or select '''Menu > Simulate > Run''' or use the keyboard shortcut {{key|Ctrl+R}}. The '''Single-Frequency Analysis''' option of the '''Simulation Mode''' dropdown list is selected by default. Once you click the {{key|Run}} button, the simulation starts. A new window called the "Output Window" opens up that reports the different stages of simulation and the percentage of the tasks completed at any time. After the simulation is successfully completed, a message pops up and reports the end of simulation. In certain cases like calculating scattering [[parameters]] of a circuit or reflection / transmission characteristics of a periodic surface, some results are also reported in the output window.  === Setting Numerical Parameters === A planar MoM simulation involves a number of numerical [[parameters]] that take preset default values unless you change them. You can access these [[parameters]] and change their values by clicking the '''Settings''' button next to the '''Select Engine''' dropdown list in the [[Planar Module]]'s Simulation Run dialog. In most cases, you do not need to open this dialog and you can leave all the default numerical parameter values intact. However, it is useful to familiarize yourself Working with these [[parameters]], as they may affect the accuracy of your numerical results. The Planar MoM Engine Settings Dialog is organized in a number of sections. Here we describe some of the numerical [[parameters]]. The "'''Matrix Fill'''" section of the dialog deals with the operations involving the dyadic Green's functions. You can set a value for the '''Convergence Rate for Integration''', which is 1E-5 by default. This is used for the convergence test of all the infinite integrals in the calculation of the Hankel transform of spectral-domain dyadic Green's functions. When the substrate is lossy, the surface wave poles are captured in the complex integration plane using contour deformation. You can change the maximum number of iterations involved in this deformed contour integration, whose default value is 20. When the substrate is very thin with respect to the wavelength, the dyadic Green's functions exhibit numerical instability. Additional singularity extraction measures are taken to avoid numerical instability but at the expense of increased computation time. By default, a thin substrate layer is defined to a have a thickness less than 0.01λ<sub>eff</sub>, where λ<sub>eff</sub> is the effective wavelength. You can modify the definition of "Thin Substrate" by entering a value for '''Thin Substrate Threshold''' different than the default 0.01. The parameter '''Max Coupling Range''' determines the distance threshold in wavelength between the observation and source points after which the Green's interactions are neglected. This distance by default is set to 1,000 wavelengths. For electrically small structures, the phase variation across the structure may be negligible. In such cases, a fast quasi-static analysis can be carried out. You can set this threshold in wavelengths in the box labeled '''Max Dimensions for Quasi-Static Analysis'''. In the "Spectral Domain Integration" section of the dialog, you can set a value to '''Max Spectral Radius in k0''', which has a default value of 30. This means that the infinite spectral-domain integrals in the spectral variable k<sub>ρ</sub> are pre-calculated and tabulated up to a limit of 30k<sub>0</sub>, where k<sub>0</sub> is the free space propagation constant. These integrals may converge much faster based on the specified Convergence Rate for Integration described earlier. However, in certain cases involving highly oscillatory integrands, much larger integration limits like 100k<sub>0</sub> might be needed to warrant adequate convergence. For spectral-domain integration along the real k<sub>ρ</sub> axis, the interval [0, Nk<sub>0</sub>] is subdivided into a large number of sub-intervals, within each an 8-point Gauss-Legendre quadrature is applied. The next parameter, '''No. Radial Integration Divisions per k<sub>0</sub>''', determines how small these intervals should be. By default, 2 divisions are used for the interval [0, k<sub>0</sub>]. In other words, the length of each integration sub-interval is k<sub>0</sub>/2. You can increase the resolution of integration by increasing this value above 2. Finally, instead of 2D Cartesian integration in the spectral domain, a polar integration is performed. You can set the '''No. of Angular Integration Points''', which has a default value of 100. EM.Picasso provides a large selection of linear system solvers including both direct and iterative methods. EM.Picasso, by default, provides a "'''Automatic'''" solver option that picks the best method based on the settings and size of the numerical problem. For linear systems with a size less than N = 3,000, the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. 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 "'''Output MoM Matrix and Vectors'''" 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. <table><tr><td> [[Image:PMOM79.png|thumb|700px|EM.Picasso's Planar MoM Engine Settings dialog.]] </td></tr></table> == Working with EM.Picasso Simulation Data ==
Depending on the source type and the types of observables defined in a project, a number of output data are generated at the end of a planar MoM simulation. Some of these data are 2D by nature and some are 3D. The output simulation data generated by EM.Picasso can be categorized into the following groups (click on each type to learn more about it):
<td> [[File:PMOM124.png|thumb|300px|EM.Picasso's Radar Cross Section dialog]] </td>
<td> [[Image:PMOM125.png|thumb|420px|An example of the 3D mono-static radar cross section plot of a patch antenna.]] </td>
</tr>
</table>
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== Discretizing a Planar Structure in EM.Picasso ==
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[[Image:PMOM31.png|thumb|400px|The Planar Mesh Settings dialog.]]
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. Both the quality and resolution of the generated mesh greatly affect the accuracy of the MoM numerical solution. The mesh density gives a measure of the number of cells per effective wavelength that are placed in various regions of your planar structure. The higher the mesh density, the more cells are created on the finite-sized geometrical objects. As a rule of thumb, a mesh density of about 20-30 cells per effective wavelength usually yields satisfactory results. But for structures with lots of fine geometrical details or for highly resonant structures, higher mesh densities may be required. The particular output data that you seek in a simulation 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 structures with a highly irregular shapes and boundaries.
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EM.Picasso provides two types of mesh for a planar structure: a pure triangular surface mesh and a hybrid triangular-rectangular surface mesh. In both case, EM.Picasso attempts to create a highly regular mesh, in which most of the cells have almost equal areas. For planar structures with regular, mostly rectangular shapes, the hybrid mesh generator usually leads to faster computation times.
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[[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 ]]'''.
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[[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 | Triangular Surface Mesh Generator]]'''.
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[[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 | Hybrid Planar Mesh Generator]]'''.
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[[Image:Info_icon.png|40px]] Click here to learn more about '''[[Mesh_Generation_Schemes_in_EM.Cube#General_Rules_of_Planar_Hybrid_Mesh_Generator| General Rules of Planar Hybrid Mesh Generator]]'''.
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[[Image:Info_icon.png|40px]] Click here to learn more about '''[[Mesh_Generation_Schemes_in_EM.Cube#Refining_the_Planar_Mesh_Locally| Refining the Planar Mesh Locally]]'''.
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<table>
<tr>
<td> [[Image:PMOM48F.png|thumb|350px|Geometry of a multilayer slot-coupled patch array.]] </td>
<td> [[Image:PMOM48G.png|thumb|370px|Hybrid planar mesh of the slot-coupled patch array.]] </td>
</tr>
</table>
<table>
<tr>
<td> [[Image:PMOM48H.png|thumb|350px|Details of the hybrid planar mesh of the slot-coupled patch array around discontinuities.]] </td>
</tr>
</table>
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== Running Planar MoM Simulations in EM.Picasso ==
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=== EM.Picasso's Simulation Modes ===
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EM.Picasso offers five Planar MoM simulation modes (click on each type to learn more about it):
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* '''[[#Running a Single-Frequency Planar MoM Analysis | Single-Frequency Analysis]]'''
* '''[[Parametric_Modeling,_Sweep_%26_Optimization#Running_Frequency_Sweep_Simulations_in_EM.Cube | Frequency Sweep]]''' including uniform and adaptive frequency sweeps
* '''[[Parametric_Modeling,_Sweep_%26_Optimization#Running_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]'''
* '''[[Parametric_Modeling,_Sweep_%26_Optimization#Optimization | Optimization]]'''
* '''[[Building_Reusable_Models#Running_an_HDMR_Sweep | HDMR Sweep]]'''
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You can set the simulation mode from EM.Picasso's "Simulation Run Dialog". A single-frequency analysis is a single-run simulation. All the other simulation modes in the above list are considered multi-run simulations. If you run a simulation without having defined any observables, no data will be generated at the end of the simulation. In multi-run simulation modes, certain [[parameters]] are varied and a collection of simulation data files are generated. At the end of a sweep simulation, you can graph the simulation results in EM.Grid or you can animate the 3D simulation data from the navigation tree.
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=== Running a Single-Frequency Planar MoM Analysis ===
[[Image:PMOM80.png|thumb|400px|EM.Picasso's Simulation Run dialog.]]
A single-frequency analysis is the simplest type of EM.Picasso simulation and involves the following steps:
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* Set the units of your project and the frequency of operation. Note that the default project unit is '''millimeter'''.
* Define you background structure and its layer properties and trace types.
* Construct your planar structure using [[CubeCAD]]'s drawing tools to create all the finite-sized metal and slot trace objects and possibly embedded metal or dielectric objects that are interspersed among the substrate layers.
* Define an excitation source and observables for your project.
* Examine the planar mesh, verify its integrity and change the mesh density if necessary.
* Run the Planar MoM simulation engine.
* Visualize the output simulation data.
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To run a planar MoM analysis of your project structure, open the Run Simulation Dialog by clicking the '''Run''' [[File:run_icon.png]] button on the '''Simulate Toolbar''' or select '''Menu > Simulate > Run''' or use the keyboard shortcut {{key|Ctrl+R}}. The '''Single-Frequency Analysis''' option of the '''Simulation Mode''' dropdown list is selected by default. Once you click the {{key|Run}} button, the simulation starts. A new window called the "Output Window" opens up that reports the different stages of simulation and the percentage of the tasks completed at any time. After the simulation is successfully completed, a message pops up and reports the end of simulation. In certain cases like calculating scattering [[parameters]] of a circuit or reflection / transmission characteristics of a periodic surface, some results are also reported in the output window.
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=== Setting Numerical Parameters ===
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A planar MoM simulation involves a number of numerical [[parameters]] that take preset default values unless you change them. You can access these [[parameters]] and change their values by clicking the '''Settings''' button next to the '''Select Engine''' dropdown list in the [[Planar Module]]'s Simulation Run dialog. In most cases, you do not need to open this dialog and you can leave all the default numerical parameter values intact. However, it is useful to familiarize yourself with these [[parameters]], as they may affect the accuracy of your numerical results.
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The Planar MoM Engine Settings Dialog is organized in a number of sections. Here we describe some of the numerical [[parameters]]. The "'''Matrix Fill'''" section of the dialog deals with the operations involving the dyadic Green's functions. You can set a value for the '''Convergence Rate for Integration''', which is 1E-5 by default. This is used for the convergence test of all the infinite integrals in the calculation of the Hankel transform of spectral-domain dyadic Green's functions. When the substrate is lossy, the surface wave poles are captured in the complex integration plane using contour deformation. You can change the maximum number of iterations involved in this deformed contour integration, whose default value is 20. When the substrate is very thin with respect to the wavelength, the dyadic Green's functions exhibit numerical instability. Additional singularity extraction measures are taken to avoid numerical instability but at the expense of increased computation time. By default, a thin substrate layer is defined to a have a thickness less than 0.01λ<sub>eff</sub>, where λ<sub>eff</sub> is the effective wavelength. You can modify the definition of "Thin Substrate" by entering a value for '''Thin Substrate Threshold''' different than the default 0.01. The parameter '''Max Coupling Range''' determines the distance threshold in wavelength between the observation and source points after which the Green's interactions are neglected. This distance by default is set to 1,000 wavelengths. For electrically small structures, the phase variation across the structure may be negligible. In such cases, a fast quasi-static analysis can be carried out. You can set this threshold in wavelengths in the box labeled '''Max Dimensions for Quasi-Static Analysis'''.
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In the "Spectral Domain Integration" section of the dialog, you can set a value to '''Max Spectral Radius in k0''', which has a default value of 30. This means that the infinite spectral-domain integrals in the spectral variable k<sub>ρ</sub> are pre-calculated and tabulated up to a limit of 30k<sub>0</sub>, where k<sub>0</sub> is the free space propagation constant. These integrals may converge much faster based on the specified Convergence Rate for Integration described earlier. However, in certain cases involving highly oscillatory integrands, much larger integration limits like 100k<sub>0</sub> might be needed to warrant adequate convergence. For spectral-domain integration along the real k<sub>ρ</sub> axis, the interval [0, Nk<sub>0</sub>] is subdivided into a large number of sub-intervals, within each an 8-point Gauss-Legendre quadrature is applied. The next parameter, '''No. Radial Integration Divisions per k<sub>0</sub>''', determines how small these intervals should be. By default, 2 divisions are used for the interval [0, k<sub>0</sub>]. In other words, the length of each integration sub-interval is k<sub>0</sub>/2. You can increase the resolution of integration by increasing this value above 2. Finally, instead of 2D Cartesian integration in the spectral domain, a polar integration is performed. You can set the '''No. of Angular Integration Points''', which has a default value of 100.
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EM.Picasso provides a large selection of linear system solvers including both direct and iterative methods. EM.Picasso, by default, provides a "'''Automatic'''" solver option that picks the best method based on the settings and size of the numerical problem. For linear systems with a size less than N = 3,000, the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. 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 "'''Output MoM Matrix and Vectors'''" 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.
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<table>
<tr>
<td> [[Image:PMOM79.png|thumb|700px|EM.Picasso's Planar MoM Engine Settings dialog.]] </td>
</tr>
</table>