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The Application of the Finite-Difference Time-Domain (FDTD) Method

Key Takeaways

  • Learn about the benefits of the finite-difference time-domain method.

  • Gain a greater understanding of the application of the finite-difference time-domain.

  • Learn more about the formulations associated with the finite-difference time-domain.

Colorful physical formulas over black background

Electrodynamics, quantum physics, electrostatics, thermodynamics, and Maxwell's equations.

In the fields of science and electronics alike, there are always problems or equations that require solving. Throughout the advancement of these and similar areas of study or focus, there are methods in use for this precise purpose. In smartphone marketing, the saying is, "There's an App for that."

However, in the areas of electronics and science, there is always a method for that. In this case, the technique or method in question is the finite-difference time-domain (FDTD). We utilize this method to model computational electrodynamics or find approximate solutions to the associated system of differential equations.

Finite-Difference Time-Domain (FDTD)

Kane S. Yee first introduced the numerical analysis technique we call the finite-difference time-domain method in 1966. To this day, many still refer to this method as Yee's method. It did not officially receive the FDTD method designation until 1980. 

The FDTD method is a discrete approximation of James Clerk Maxwell's equations that numerically and simultaneously solve in both time and 3-dimensional space. Throughout this process, the magnetic and electric fields are calculated everywhere within the computational domain and as a function of time beginning at t = 0.

Note: James Clerk Maxwell's equations symbolize one of the most refined and aphoristic ways to state the fundamentals of magnetism and electricity. These equations include:

  • Gauss' law for electricity
  • Gauss' law for magnetism
  • Faraday's law of induction
  • Ampere's law

Maxwell's equations are the basis for FDTD and describe the effect and behavior of electromagnetism. In summary, it is the technique for the simulation of computational electromagnetism. Those in the field consider it the easiest and most effective way to model the effects of electromagnetism on a specific object or material.

The FDTD Approach

Utilizing the FDTD method will divide both time and space into distinct segments. It provides the segmentation of space into box-shaped cells that are small in comparison to the overall wavelength. The precise location of the electric fields is on the edges of the cube. However, the position of the magnetic fields is on the faces of the box-shaped cells.

In summary, there are three steps in the FDTD computation:

  • The discretization of the whole area into small cells (defined as Δx, Δy, and Δz) in the Yee cell.

  • Defining the electrical properties (conductivity and permittivity) of the model in a mesh grid.

  • Substitution of partial derivatives by differential quotient in James Clerk Maxwell's equations.

Note: In the above summary, the upper-case Delta (Δ) represents a change.

If you obtain the parameters of the primary fields in time and space, you can also calculate other secondary measures. These quantities include frequency-domain characteristics such as impedance (input), radar cross-section, scattering parameters, and far-field radiation patterns, to name a few.

Finite-Difference Time-Domain Applications and Formulations

The FDTD method is a finite domain numerical method; therefore, we must truncate the computational domain of the problem. We must also enforce the proper boundary conditions at the boundaries of the computational domain. For example, in a structure (shielded), we enclose all objects within a perfect magnetic or electric conductor box.

With open boundary problems, such as an antenna, we utilize an absorbing boundary condition like a PML (perfectly matched layer) to mimic the free space. In this case, the absorbing boundaries allow the incident waves and fields to flow through them without back reflection. In terms of FDTD simulation time, it directly correlates to the following parameters:

  • The size of the computational domain.

  • The proximity of the PML walls to the enclosed objects.

We must approximate the computational domain of the FDTD by utilizing an appropriate meshing scheme, but the method itself provides a wideband simulation of a physical structure. To produce the necessary spectral information, we require a suitable wideband temporal waveform to excite the physical structure. Your choice of a waveform, time delay, and bandwidth will affect the merging behavior of the FDTD loop (time marching).

Using the Finite-Difference Time-Domain Method

One particular issue of concern is the numerical stability of the FDTD method’s time marching scheme. Therefore, to satisfy the Courant-Friedrichs-Lewy (CFL) stability condition, your time step needs to be inversely proportional to the maximum grid cell size.

Note: The Courant-Friedrichs-Lewy condition is a necessary condition for convergence while numerically solving certain PDEs. This emerges in the numerical analysis of distinct time integration schemes when we utilize these for a numerical solution.

Functionally, a high-resolution mesh requires a smaller time step. Operationally, a smaller time step requires a more significant number of time steps to converge if letting the fields in the computational domain fully evolve.

Using the Finite-Difference Time-Domain Method, Continued

We must establish a computational domain when implementing an FDTD solution of James Clerk Maxwell's equations. We define a computational domain as the physical region in which we perform the simulation. In this case, we determine the H and E fields at each point in space within the computational domain. We specify the material of each cell within this computational domain as well. The material in question is generally either metal, dielectric, or free-space (air).

You can utilize any material, provided you specify its conductivity, permittivity, and permeability. Keep in mind that you cannot directly substitute the dielectric constant of dispersive materials in tabular form into the FDTD scheme. Therefore, you approximate by utilizing multiple Debye, Drude, Lorentz, modified Lorentz, QCRF (quadratic complex rational function), or CCPR (complex-conjugate pole-residue) models.

Upon establishing both the grid materials and the computational domain, we then specify a source. The source can be either an applied electric field, impinging plane wave, or current on a wire. With regard to an impinging plane wave, we can use the FDTD method to simulate light scattering from planar periodic structures, the photonic band structure of infinite periodic structures, and arbitrarily shaped objects.

Since we directly determine the E and H fields, the simulation's output is usually the H or E field at a series of points or a point within the computational domain. During this process, the simulation will evolve the H and E fields forward in time. Generally, it does the processing on the H and E fields returned by the simulation. Data processing can occur during the ongoing simulation. Lastly, during the time the FDTD method calculates the electromagnetic fields inside a compact spatial region, you can obtain radiated and/or scattered far-fields via near-to-far-field transformations.

The finite-difference time-domain is a numerical analysis method utilized for modeling computational electrodynamics. Though it is still a time-domain technique, FDTD solutions are capable of covering a wide range of frequencies within a single simulation run.

A graphic representation of magnetic flux and faraday’s law of induction

Magnetic Flux and Faraday's Law of Induction.

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