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<td>[[image:Cube-icon.png | link=Getting_Started_with_EM.Cube]] [[image:cad-ico.png | link=CubeCAD]] [[image:fdtd-ico.png | link=EM.Tempo]] [[image:prop-ico.png | link=EM.Terrano]] [[image:static-ico.png | link=EM.Ferma]] [[image:planar-ico.png | link=EM.Picasso]] [[image:metal-ico.png | link=EM.Libera]] [[image:po-ico.png | link=EM.Illumina]] </td>
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== An Overview of FDTD Modeling ==
In the Finite Difference Time Domain (FDTD) method, a discretized form of Maxwell’s equations is solved numerically and simultaneously in both the 3D space and time. During this process, the electric and magnetic fields are computed everywhere in the computational domain and as a function of time starting at t = 0. From knowledge of the primary fields in space and time, one can compute other secondary quantities including frequency domain characteristics like scattering parameters, input impedance, far-field radiation patterns, radar cross section, etc.
Since FDTD is a finite domain numerical technique, the computational domain of the problem must be truncated. At the boundaries of the computational domain, proper boundary conditions must be enforced. In a shielded structure, all objects are enclosed within a perfect electric (or magnetic) conductor box. In an open boundary problem like an antenna, some kind of absorbing boundary conditions such as a perfectly matched layer (PML) must be used to emulate the free space. The absorbing boundaries should act such that the incident fields and waves propagate through them without any back reflection. The FDTD simulation time depends directly on the size of the computational domain and on how close you can place the PML walls to the enclosed objects.
The FDTD computational domain must be discretized using an appropriate meshing scheme. [[EM.Tempo]] uses a non-uniform, adaptive, voxel-based Yee mesh with a mesh density that you can customize. A fixed-cell mesh generator is also available, where you can set constant cell dimensions along the three principal axes for the entire computational domain. The variable mesh density is specified in terms of the effective wavelength inside material media. As a result, the mesh resolution and average mesh cell size differ in regions that are filled with different types of material. [[EM.Tempo]]'s adaptive mesher generates more cells in the areas that are occupied by dielectric materials, fewer cells in the free-space regions and no cells inside PEC regions. [[EM.Tempo]]'s default adaptive mesh generator also refines the mesh around curved segments of lines, surface or solids to produce a far more accurate representation of your geometry.
The FDTD method provides a wideband simulation of your physical structure. In order to produce sufficient spectral information, an appropriate wideband temporal waveform is needed to excite the physical structure. The choice of the waveform, its bandwidth and time delay all affect the convergence behavior of the FDTD time marching loop. By default, [[EM.Tempo]] uses a modulated Gaussian waveform with optimal parameters. Another issue of concern is the numerical stability of the time marching scheme. You might expect to get better and more accurate results if you keep increasing the FDTD mesh resolution. However, in order to satisfy the Courant-Friedrichs-Levy (CFL) stability condition, the time step must be inversely proportional to the maximum grid cell size . A high resolution mesh requires a smaller time step. To let the fields in the computational domain fully evolve over time, a smaller time step will require a larger number of time steps to converge. [[EM.Tempo]] automatically chooses a time step that satisfies the CFL condition.
== Differential Form of Maxwell's Equations & the Yee Cell ==
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<td> [[Image:FDTD MAN10.png|thumb|left|550px480px|The boundary CPML cells placed outside the visible domain box.]] </td>
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:<math> E_y^{inc}(x,y,t) = -\cos\phi \; \exp \left(-\frac{(t-t_0)^2}{\tau^2} \right) \exp(j2\pi f_0 t) \exp(-jk_x x) \exp(-jk_y y)</math>
for TE<sub>z</sub> polarization. Here, f<sub>0</sub> is the center frequency of the modulated Gaussian pulse waveform, t<sub>0</sub> is the time delay, and τ is the Gaussian pulse width. The choices of the Gaussian waveform [[parameters]] are very critical in order to avoid possible resonances. For a fixed value of k<sub>l</sub>, the horizontal resonance occurs at:
:<math> f_{res} = \frac{k_l c}{2 \pi} </math>
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