=== The FDTD Mesh Types ===
[[EM.Tempo]] generates a brick volume mesh for FDTD simulation. The FDTD mesh is a rectangular Yee mesh that extends to the entire computational domain. It is primarily constructed from three mesh grid profiles along the XY, YZ and ZX principal planes. These projections together create a 3D rectangular (voxel) mesh space. Straight lines, boxes and rectangular plates whose edges are aligned with the three principal axes are the simplest objects to mesh in EM.Tempo. Such objects preserve their exact shapes after discretization. All the objects with curved edges and curved surfaces or objects with straight edges and flat faces that are not parallel to the principal axes or principal planes (such as oblique lines and slanted lateral faces of a pyramid) are discretized using a staircase (Yee) profile.
In EM.Tempo's adaptive mesh generator uses a variable staircase profile, where the cell sizes of grid line spacing vary with the curvature (derivative) of the edge or face. As a result, a higher mesh resolution is achieved at "more curvy" areas to better capture the geometrical details. You have the option to you can choose one of the three FDTD mesh types:
* Adaptive Mesh
* Fixed-Cell Mesh
The EM.Tempo's default choice is the mesh generator creates an adaptive brick volume meshthat uses a variable staircase profile, which is where the grid line spacings vary with the curvature (derivative) of the edge or face. As a quite sophisticated result, a higher meshresolution is produced at "curved" areas to better capture the geometrical details. The resolution of the adaptive FDTD mesh is driven by the '''Mesh Density''', expressed in cells per effective wavelength. Since FDTD is a time-domain method and the excitation waveform may have a wideband spectral content, the effective wavelength is calculated based on the highest frequency of the project: f<sub>max</sub> = f<sub>0</sub> + Δf/2, where f<sub>0</sub> is your project's center frequency and Δf (or BW) is its specified bandwidth. Â Â In other words, the effective wavelength in the free space is λ<sub>0,eff</sub> = c / f<sub>max</sub>, c being the speed of light in the free space.
The adaptive FDTD mesh, however, produces different grid cell sizes in the free space regions and inside dielectric regions. The effective wavelength in a dielectric material with relative permittivity e<sub>r</sub> and permeability µ<sub>r</sub> is given by λ<sub>d,eff</sub> = λ<sub>0,eff</sub> / √ε<sub>r</sub>μ<sub>r</sub>. Therefore, the average ratio of the cell size in a dielectric region to the cell size in the free space is 1/√(ε<sub>r</sub>μ<sub>r</sub>). The adaptive FDTD mesh generator also takes note of the geometrical features of the objects it discretizes. This is more visible in the case of curved solids, curves surfaces and curved wires or obliquely oriented planes and lines which need to be approximated using a staircase profile. The mesh resolution varies with the slope of the geometrical shapes and tries to capture the curved segments in the best way. Another important feature of the adaptive FDTD mesher is generation of gradual grid transitions between low-density and high-density mesh regions. For example, this often happens around the interface between the free space and high permittivity dielectric objects. Gradual mesh transitions provide better accuracy especially in the case of highly resonant structures.
According to the Courant-Friedrichs-Levy (CFL) stability criterion, the FDTD time step is determined by the smallest cell size in your FDTD mesh. Occasionally, [[FDTD Module]]'s adaptive mesh generator may create extremely tiny grid cells that would result in extremely small time steps. This would then translate into a very long computation time. Â [[EM.Cube]] offers the "Regular" FDTD mesh generator, which is a simplified version of the adaptive mesh generator. In a regular FDTD mesh, the grid cell sizes stay rather the same in objects of the same material composition. The mesh resolution increases in materials of higher permittivity and/or permeability based on the effective wavelength in exactly the same way as the adaptive mesh. Finally, [[EM.Cube]]'s FDTD Modules offers a "Uniform" FDTD mesh generator. The uniform mesh consists of three uniform grids along the XY, YZ and ZX principal planes. In other words, the grid cell sizes Δx, Δy and Δz are fixed throughout the entire computational domain. In this case, the uniform mesh generator has to fit your physical structure to the fixed mesh, rather than adapting the mesh to your physical structure.
{{Note|When choosing a mesh type for your FDTD simulation, keep in mind that adaptive and regular mesh types are frequency-dependent and their density varies with the highest frequency of your specified bandwidth, while the uniform mesh type is always fixed and independent of your project's frequency settings.}}
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Occasionally, you may prefer a more regular FDTD mesh with almost equal grid line spacing everywhere, but still with a frequency-dependent cell size. In that case, you can select the "<u>'''Regular'''</u>" option of the '''Mesh Type '''dropdown list in the FDTD Mesh Settings dialog. The regular FDTD mesh enforces only two of the above [[parameters]]: '''Minimum Mesh Density''' and '''Absolute Minimum Grid Spacing'''. Or you may opt for an absolutely "<u>'''Uniform'''</u>" mesh type, for which you need to specify the '''Cell Size '''along the X, Y, Z directions in project units.
[[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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=== Changing the FDTD Mesh Settings ===
[[Image:FDTD80.png|thumb|600px|EM.Tempo's Mesh Settings dialog]]
EM.Tempo discretizes objects using what is often referred to as the âstaircase approximationâ. In this mesh generation scheme, the structure is recreated using a large number of cubic cells carefully assembled in a way that approximates the shape of the original structure. By default, a carefully calculated, "<u>'''Adaptive'''</u>" mesh of your physical structure is generated in order to satisfy the following criteria:
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* Optimize the number of mesh cells in each dimension. The product of the number of cells in each dimension determines the total mesh size. The larger the mesh size, the longer the simulation time, especially with the CPU version of the FDTD engine. Also, a very large mesh size requires more RAM, which may exceed your GPU memory capacity. Set the '''Minimum Mesh Density''' to a moderately low value to keep the mesh size manageable, but be careful not to set it too low (see the next item below).
* Ensure simulation accuracy by requiring an acceptable minimum number of cells per wavelength through each object and in the empty (free) space between them and the computational domain boundaries. An effective wavelength is defined for each material at the highest frequency of the project's specified spectrum. We recommend a '''Minimum Mesh Density '''of at least 15-20 cells/ wavelength. But for some resonant structures, 25 or even 30 cells per wavelength may be required to achieve acceptable accuracy. As you reduce the mesh density, the simulation accuracy decreases.
* Accurately represent and approximate the boundaries of edges or surfaces that are not grid-aligned by closely adhering to their geometric contours. This is controlled by the '''Minimum Grid Spacing Over Geometric Contours''', which can be specified either as a fraction of the free space grid spacing or as an absolute length value in project units.
* Maximize the minimum grid spacing in any dimension inside the computational domain and thus maximize the simulation time step. The time step size is dictated by the CFL stability criterion and is driven by the smallest grid spacing in each dimension. The smaller the time step, the larger the number of time steps required for convergence. This is controlled using the '''Absolute Minimum Grid Spacing''', which can be specified either as a fraction of the free space grid spacing or as an absolute value. It is critical to accurately represent and precisely maintain the object edge/surface boundaries in certain structures like resonant antennas and filters, as the phase of the reflected fields/waves is affected by the object boundary positions. When object boundaries are very close to each other, the mesh needs to represent them by two separate, but very closely spaced, grid lines. To control the minimum allowed grid spacing, use the '''Absolute Minimum Grid Spacing '''settings,
* Maintain a smooth grid with no abrupt jumps from low-density to high-density regions. This feature is enabled with the '''Create Gradual Grid Transitions '''check box (always checked by default).
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Occasionally, you may prefer a more regular FDTD mesh with almost equal grid line spacing everywhere, but still with a frequency-dependent cell size. In that case, you can select the "<u>'''Regular'''</u>" option of the '''Mesh Type '''dropdown list in the FDTD Mesh Settings dialog. The regular FDTD mesh enforces only two of the above [[parameters]]: '''Minimum Mesh Density''' and '''Absolute Minimum Grid Spacing'''. Or you may opt for an absolutely "<u>'''Uniform'''</u>" mesh type, for which you need to specify the '''Cell Size '''along the X, Y, Z directions in project units.
[[Image:Info_icon.png|40px]] Click here to learn more about '''[[Advanced Meshing in EM.Tempo]]'''.