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1.4 The Finite Difference Time Domain Technique

by Jose Alves

The Finite Difference Time Domain (FDTD) technique is a prominent numerical method in computational electromagnetics, functioning as a time domain solver. It enables the prediction of a system's wideband response, accounts for nonlinearities, and allows for monitoring results during the computation itself.

The FDTD method involves discretizing the derivatives in Maxwell's curl equations using second-order central differences in the time domain. Originally proposed by K.S. Yee, the method employs a staggered grid system, with grid sizes denoted as (Δx, Δy, Δz), to evaluate the electric and magnetic fields separately. This algorithm calculates the solution at a specific time based on the solutions from previous time steps.

A key limitation of this explicit, marching-on-in-time algorithm is its requirement for the time step to be constrained by the smallest spatial mesh step to maintain numerical stability, as dictated by the Courant-Friedrichs-Lewy (CFL) criterion. Consequently, finer spatial discretization necessitates smaller time steps, extending the overall simulation duration, particularly for low-frequency problems that converge slowly.

FDTD allows for the application of various boundary conditions to truncate the domain and represent different scenarios effectively. It can be integrated with the Transmission Line Method to model cables, addressing Electromagnetic Compatibility issues, as demonstrated in the figure depicting antennas radiating under rocket fairings and coupling to a cable model.

One significant challenge with FDTD is its handling of curved structures, due to the grid mesh application leading to stair-casing effects on the materials in the model. However, this characteristic can be advantageous, as the grid mesh can accurately capture the structure while disregarding minor tolerance artifacts often present in CAD designs. FDTD is capable of simulating electrically large geometries with billions of cells, but it requires careful consideration of time steps when combining large structures with critical small geometries.

FDTD is readily implementable on cluster architectures and can achieve substantial speedups using Graphical Processing Units (GPUs), although memory bandwidth remains a primary bottleneck.

To address its inherent limitations, various solutions have been devised, including the use of conformal mesh and sub-gridding to mitigate stair-casing, sub-cell models to obviate the need for fine meshes, and permittivity scaling to accelerate the convergence of low-frequency transients.

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