Conductor losses#

Magnelio offers two routes for wall losses on good conductors: a perturbative post-processing route (default) and a broadband time-domain surface-impedance boundary (opt-in).

Perturbative wall losses (post-processing)#

With lossless PEC walls in the field solve, the dissipated power is evaluated perturbatively from the tangential magnetic field,

\[ P_{\text{loss}}(f) = \tfrac12\, R_s(f) \sum_{\text{wall}} w\,|H_{\tan}|^2, \qquad R_s = \sqrt{\pi f \mu / \sigma}, \]

(monitors/wall_loss.py, postprocessing/wall_loss.py, DD-082). This is the classical power-loss perturbation method of microwave engineering [16, 38]. Two accuracy refinements are in-house (DD-087): exact conformal wall areas on curved conductors (removing the \(4/\pi\) staircase over-count) and a conformal tangential-H sampling rule (uncut-face booking with a normal-direction walk).

Surface roughness#

Roughness enters the perturbative chain as one real, frequency-dependent multiplier \(K(f)\) on the surface resistance, \(R_{s,\text{rough}} = K(f)\,R_{s,\text{smooth}}\) (materials/roughness.py, DD-088). Implemented models:

  • Hammerstad — the classical RMS-height curve fit of Hammerstad and Jensen [39].

  • Huray “snowball” — the physics-based sphere-cluster model of Huray et al. [40]; the loss-factor form implemented is Bracken’s eq. (5) [41].

  • Cannonball-Huray parameterisation — sphere radius and coverage from a single \(R_z\) datasheet number via close packing, after Simonovich [42].

A real \(K(f)\) is non-causal as a time-domain impedance (noted by Bracken [41]); this is admissible here because the perturbative chain evaluates power per frequency bin and never forms a time-domain impedance.

Broadband time-domain SIBC#

SIBC is currently an opt-in: wall_model="sibc". The surface-impedance boundary condition realises the Leontovich condition \(E_{\tan} = Z_s(\omega)\,(\hat n \times \mathbf H)\) [43, 44] directly in the leapfrog update (solver/sibc.py, DD-091):

  • \(Z_s(\omega)\) — smooth-metal \(\sqrt{j\omega\mu/\sigma}\) or the causally completed rough impedance — is fitted as a Foster/Stieltjes ladder \(c_0 + \sum_p c_p s/(s+b_p)\) with non-negative coefficients by NNLS on fixed log-spaced poles (materials/surface_impedance.py). Foster’s canonical positive- real ladder form is classical network synthesis [45]; NNLS is Lawson and Hanson [46]. Because every branch is elementarily passive, the time-domain recursion is dissipative by construction — stability is unconditional at the unchanged lossless CFL, independent of fit accuracy — an in-house result (internal derivation dossier investigations/sibc/DERIVATION.md, kept outside the public repository).

  • The causal reactance of a rough surface is completed from the real roughness excess \((K-1)R_s\) by a subtracted Kramers–Kronig quadrature [47, 48].

  • The per-branch states are advanced with the trapezoidal rule and folded into the H update like a magnetic surface conductivity.

Approximating a surface impedance by a low-order rational function and convolving it recursively in FDTD is an established technique: Maloney and Smith [49], Beggs, Luebbers, Yee and Kunz [50], and the first-order-section approach of Oh and Schutt-Ainé [51] are the closest published antecedents; Magnelio’s specific NNLS-Foster construction with its unconditional dissipation identity is in-house.

The conformal booking of SIBC faces (which faces carry the damping term and with which geometric weight \(G_f = A_f/l^2_{\text{dual}}\)) reuses the DD-087 conformal wall-area machinery.