Electromagnetic waves

Overview

The electrostatic and magnetostatic chapters solved for scalar potentials — the electric potential $\varphi$ and the magnetic scalar potential $\psi$ — one value per node. Full-wave electromagnetics needs a vector field, because Maxwell's equations relate the curl of one field to another. This chapter covers the vector-field solvers: the electric field $E$ and the magnetic vector potential $A$.

The extra machinery is the Nédélec edge element — a finite element whose degrees of freedom live on mesh edges (their tangential component), the natural home for fields whose curl matters. The same solve(Physics(), …) marker pattern still applies; only the field type changes.

Cavity modes

A perfectly-conducting (PEC) enclosure traps electromagnetic standing waves. Their frequencies are the eigenvalues of the curl–curl problem

\[\nabla \times \nabla \times E = \omega^2 E ,\]

the vector analogue of the acoustic cavity problem. Each $\omega$ is a resonant frequency and each $E$ a cavity mode.

ElectromagneticWave <: AbstractPhysics

A worked example: a resonant cavity

m = tetrahedralize_box(1.0, 1.0, 1.0, 4, 4, 4)
mat = Material(youngs_modulus=200e9, poisson_ratio=0.3,
               relative_permittivity=1.0, relative_permeability=1.0)

ω, modes = solve(ElectromagneticWave(), m, mat; nev=3)
round.(ω; digits=4)

3-element Vector{Float64}:
 4.3545
 4.4658
 4.4658

The first three resonant frequencies of a unit PEC cavity are $4.35$, $4.47$, and $4.47$ (the last two are the two degenerate transverse modes).

Magnetostatics with the vector potential

In three dimensions it is convenient to solve magnetostatics for the magnetic vector potential $A$, with $B = \nabla \times A$. Ampère's law becomes the vector Poisson problem

\[\nabla \times (\nu\, \nabla \times A) = J ,\]

where $\nu = 1/\mu$ is the magnetic reluctivity and $J$ the current density. The result is the potential $A$ on the edge elements and the reconstructed flux density $B$.

MagnetostaticsVectorPotential

r = solve(MagnetostaticsVectorPotential(), m, mat)
keys(r)

(:A, :B)

Time-domain Maxwell

To follow how a field evolves — a pulse bouncing around a cavity — Maxwell's equations are marched in time with the finite-element time-domain (FETD) method and the Newmark scheme:

TransientMaxwell <: AbstractPhysics

r = solve(TransientMaxwell(), m, mat; nsteps=20, dt=1e-3)
length(r.times)

21

Magneto-elastic and thermo-electro-mechanical coupling

Two markers couple the electromagnetic field to the solid. Magnetostriction couples the displacement $u$ to the magnetic scalar potential $\psi$ (a magnetized material strains); pyroelectricity couples displacement, potential, and temperature together — the full $u$–$\varphi$–$T$ block:

Magnetostriction <: AbstractPhysics

r = solve(Magnetostriction(), m, mat)
keys(r)

(:u, :ψ)

Pyroelectricity

r = solve(Pyroelectricity(), m, pzt)
keys(r)

(:u, :φ, :T)

Next steps

The edge elements and the curl–curl operators assemble onto the same sparse linear algebra as every other physics — Sparse linear algebra — and the electromagnetic resonance, like the structural one, is an eigenproblem, covered alongside its siblings in Eigenproblems.