ElectromagneticsEngineering methods

Electromagnetics (FDTD)

Solving Maxwell's equations on a grid — the electric and magnetic fields leapfrogging through space and time. A self-contained FDTD solver reproduces the canonical wave phenomena: radiation, refraction, diffraction, scattering and cavity resonance. The field-solver sibling to CFD, in electromagnetics.

DjiniousLabModelling & simulation

Concentric electromagnetic wavefronts radiating outward from a point source on a grid

wave speed (cells/step, magic step): 1.000wave speed (cells/step, magic step)
cavity resonance vs analytic mode: 0.5%cavity resonance vs analytic mode
Snell refraction index vs √εr: +2.5%Snell refraction index vs √εr
reflection cut by the absorbing boundary: 14 dBreflection cut by the absorbing boundary
design notebooks: 10design notebooks
requirements PASS: 5 / 5requirements PASS

Watch a wave obey Maxwell's equations, cell by cell.

Electromagnetic fields fill space and evolve by Maxwell's equations — and you can solve them directly on a grid, letting the electric and magnetic fields leapfrog through time in step. That's the Finite-Difference Time-Domain method, the workhorse of computational electromagnetics (HFSS, CST, COMSOL-RF). This program builds an FDTD solver from scratch and uses it to reproduce, one notebook at a time, the wave phenomena every physics course teaches: a wave radiating, refracting, diffracting, scattering, and resonating. It's the field-solver sibling to the CFD flagship.

Two curl equations, leapfrogging on a staggered grid.

Maxwell's curl equations say a changing magnetic field makes an electric one and vice versa. FDTD discretizes them on the Yee grid — electric and magnetic field components interleaved in space and updated alternately in time — so the simulation literally marches the wave forward. The one rule you can't break is the Courant stability limit: step too far per tick and the field explodes. Get it right and a point source radiates clean circular wavefronts at exactly the speed of light.

Concentric electromagnetic wavefronts radiating from a point source across the grid — The hero: a point source radiating concentric electromagnetic wavefronts across the Yee grid — the electric field marching outward at the speed of light, exactly as Maxwell's equations demand. The whole program is built on this update; everything else is what you put in the wave's way.

Put glass in the path and the wave slows and bends.

Drop a region of higher permittivity — glass, water, any dielectric — into the grid and the wave slows down inside it. Part of it reflects at the interface, part transmits with a compressed wavelength, and the ratio of wavelengths is exactly the refractive index √εr — Snell's law, emerging from the field update with no law of refraction coded in. The solver reproduces it to a couple percent, dispersion error and all.

A wave entering a higher-permittivity region, partly reflecting and transmitting with a shorter wavelength — Refraction from first principles: a wave crossing into a dielectric (right half) partly reflects and partly transmits with a compressed wavelength; the wavelength ratio is the refractive index. Snell's law isn't programmed in — it falls out of solving Maxwell's equations on the grid.

A plane wave passing through a narrow slit and spreading into circular Huygens wavefronts — Diffraction: a plane wave hitting a conducting screen with a narrow slit doesn't cast a sharp shadow — it spreads into circular wavefronts beyond the aperture, every point of the slit acting as a new source. Huygens' principle, watched directly.

A conductor casts a shadow and scatters the rest.

Send a plane wave at a conducting cylinder and two things happen at once: a standing-wave pattern builds up in front as the incident and reflected waves interfere, and a quiet shadow forms behind. Scattering and shadowing are how radar sees, how an antenna's surroundings distort it, and how a stealth shape is judged — and they come straight out of forcing the field to zero on the conductor.

A plane wave scattering off a conducting cylinder, with interference in front and a shadow behind — Scattering: a plane wave striking a conducting cylinder builds an interference pattern in front and leaves a shadow behind. The same physics that lets radar detect a target and forces antenna designers to account for everything near the aperture — solved on the grid.

Close the box and only certain notes survive.

Trap the field in a conducting box and it can't radiate away — instead it rings at the cavity's resonant frequencies, the electromagnetic equivalent of an organ pipe's notes. Drive a broadband pulse and Fourier-transform the response, and the spectrum peaks land on the analytic cavity modes to within half a percent; drive it at a resonance and the corresponding standing-wave mode pattern fills the box. Resonators, filters and waveguides all live on this physics.

The standing-wave mode pattern of a resonant cavity driven at its fundamental frequency — Cavity resonance: driven at its fundamental, a conducting box rings in the TM₁₁ standing-wave mode — a single central antinode. The measured resonant frequencies match the analytic cavity modes to 0.5%. The basis of every microwave resonator, filter and waveguide.

Every number re-derived at sign-off.

The V&V notebook rebuilds the solver from scratch and re-derives each requirement against theory, printing a PASS/FAIL board.

Result

Wave speed (magic step): 1.000
Courant stability: bounded
Wavelength vs c/f: +3.7%
Snell index vs √εr: 2.05 vs 2.00
Cavity resonance: 0.5%

Requirement

Wave speed (magic step): R-01 = 1.000
Courant stability: R-02
Wavelength vs c/f: R-03
Snell index vs √εr: R-04
Cavity resonance: R-05

A real field solver, at teaching fidelity.

The deliverable is ten notebooks and the dossier — a discretized Maxwell field, not an acausal lumped network, so there's no custom block or canvas. The solver is 2D (transverse-magnetic), in normalized units, at modest grid resolution — so there's measurable numerical dispersion, the cylinder and screens use staircase conducting boundaries, the absorbing boundary is a tapered lossy rim rather than a full split-field PML, and the media are non-dispersive and linear. What the program proves is the capability the portfolio lacked alongside CFD — solving Maxwell's equations as a field on a grid — with a self-contained solver (no external EM dependency) that reproduces radiation, refraction, diffraction, scattering and resonance, every number re-runnable and checked against theory. It pairs naturally with the SAR radar flagship.