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Simulation Library

Finite element mesh used in numerical field simulation. — Finite-element meshes discretize complex geometry so field equations can be solved numerically. Image source: Wikimedia Commons, Bruno Sudret, CC BY 4.0.

Simulation principleUse the simplest model that preserves the physics needed for the question. A model is useful when its assumptions are explicit, its limiting cases are checked, and at least one observable can be compared with experiment.

Downloadable Python starting points

The repository includes small, readable scripts that produce numerical output and plots:

Model hierarchyThese scripts are intentionally compact first models. Geometry, loss, collisions, multilevel structure, uncertainty and measured instrument transfer functions can be added when required by the physical problem.

Model-selection map

Problem	First model	When a more complete model is needed
Plane-wave propagation	analytic phasor	layered/inhomogeneous medium
Transmission line	telegrapher equations	full-wave launch/discontinuity
Antenna array	array factor	mutual coupling / installed platform
Waveguide	modal solution	bends, irises, dielectric loading
Complex 3-D RF structure	FEM / FDTD / MoM	hybrid/asymptotic for huge electrical size
Charged particles	Lorentz ODE	collisions / space charge / PIC
Atomic two-level system	optical Bloch equations	hyperfine/Zeeman multilevel density matrix
Strong periodic atomic drive	dressed-state / Floquet	full dissipative Floquet master equation

1. Plane wave

A minimal complex representation is

import numpy as np
z = np.linspace(0, 2, 1000)
f = 1e9
c = 299_792_458
beta = 2*np.pi*f/c
E = np.exp(-1j*beta*z)

Checks include linear phase advance, constant $

$ in a lossless medium, and wavelength $2\pi/\beta$.

2. Transmission line and reflection

For a lossless line,

\[Z_{in}=Z_0\frac{Z_L+jZ_0\tan\beta l}{Z_0+jZ_L\tan\beta l}.\]

Plots of $Z_{in}(l)$ and $\Gamma(l)$ on a Smith chart show the effect of electrical length directly.

3. Antenna array factor

For a linear array,

\[AF(\theta)=\sum_{n=0}^{N-1}w_ne^{jn(kd\sin\theta+\beta_s)}.\]

Useful parameters include element count, spacing, steering phase, amplitude taper, phase/amplitude errors, grating lobes, and scan loss after multiplication by an element pattern.

A full-wave model becomes important when mutual coupling and finite element patterns matter.

4. Radar waveform

An FMCW model contains

transmit chirp → delayed/Doppler-shifted echo → mixer → beat signal → FFT → range/velocity estimate.

Range resolution can be checked against $c/(2B)$ and Doppler against $2v/\lambda$.

5. Lorentz-force particle tracking

Integrate

\[m\dot{\mathbf v}=q(\mathbf E+\mathbf v\times\mathbf B),\qquad \dot{\mathbf r}=\mathbf v.\]

A Boris pusher is useful for long charged-particle trajectories in magnetic fields because it preserves gyromotion better than naive Euler stepping.

6. Monte Carlo transport + Shockley–Ramo

For particle $j$,

\[i_j=q_j\mathbf v_j\cdot\mathbf E_w.\]

A complete transport calculation can include sampled creation positions and velocities, actual $\mathbf E$ and $\mathbf B$ fields, stochastic collisions, wall/electrode interactions, weighting-field interpolation, particle summation, and the measured $Z_T(\omega)$ of the readout electronics.

This separates transport physics from signal induction.

7. Optical Bloch equations

For a two-level atom,

\[\dot\rho=-\frac{i}{\hbar}[H,\rho]+\mathcal L(\rho).\]

Useful consistency checks include the zero-drive limit, weak-drive Lorentzian response, saturation/power broadening, expected detuning symmetry, $\mathrm{Tr}\rho=1$, and physical populations.

8. Three-level EIT

For a ladder system $

g\rangle\rightarrow

e\rangle\rightarrow

r\rangle$, calculate steady-state $\rho_{ge}$ and use

\[\chi\propto\rho_{ge}/\Omega_p.\]

Realistic models may include Doppler averaging, transit time, laser linewidth, spatial Rabi variation, Zeeman structure and RF coupling.

9. Floquet / Shirley model

For a periodic Hamiltonian $H(t+T)=H(t)$, expand in photon replicas and diagonalize the enlarged Floquet Hamiltonian.

Useful outputs include quasienergies, bare-state overlap, target-state shift, avoided-crossing gaps, replica index $q$, pathway amplitudes, and eigenvector continuity by overlap between neighboring field points.

10. Magnetostatics and Halbach arrays

Solve

\[\nabla\cdot\mathbf B=0,\qquad \mathbf B=\mu_0(\mathbf H+\mathbf M).\]

For atomic sensing, useful post-processing includes both $

$ and the distribution of Zeeman shifts across the illuminated vapor volume.

11. FEM, FDTD and MoM

FEM: bounded complex geometries, waveguides, cavities and dielectric structures.
FDTD: broadband time-domain propagation and transients.
MoM: open-region conducting surfaces, wires, radiation and scattering.

See Comparison Tables and Computational Methods.

12. Reproducibility checklist

State units and coordinate system. State sign/phasor convention. Check at least one analytic limiting case. Perform time-step / mesh / basis convergence. Save parameter values with results. Compare an actual observable, not only hidden state variables. Distinguish fitted parameters from independently measured inputs.

Advanced simulation standards — verification before visualization

A numerical field picture is not a validated simulation. For every computed observable record the governing equations, boundary conditions, mesh or basis truncation, discretization order, solver tolerance, conserved quantities and computational cost.

Mesh-convergence method: calculate observable (Q(h)) at several element sizes (h), compare the asymptotic behavior against discretization order, and quantify discretization uncertainty separately from uncertainty in material properties. For an open-region EM problem, vary both the near-field mesh and absorbing-boundary/PML distance; a converged mesh can still produce an incorrect result if the boundary reflects energy.

Validation hierarchy: (1) compare against a closed-form limit, (2) compare an independent numerical formulation, (3) compare with calibrated physical measurements, and (4) perform sensitivity and uncertainty analysis. Agreement obtained only after fitting multiple unconstrained parameters is not independent validation.

Research extension: reconstruct scattered far fields from equivalent surface currents and compare them with a direct radiation calculation; investigate errors introduced by finite surface truncation.

Further reading: MIT Advanced Electromagnetism; A. Taflove and S. C. Hagness, Computational Electrodynamics.

Reproducible advanced laboratory models

Magnetic cleanliness versus boom length: field from a dipole and background versus spacecraft sensor location.
Coherent SDR channel timing: distinguish constant phase, channel delay and frequency drift.
Direction finding with an antenna array: Bartlett AoA estimator and additive noise.
Infrared radiometric photon budget: Planck radiance, passband integration, quantum efficiency and SNR.

The Advanced Engineering Learning Map connects these simulations to their governing equations and measurement methods.