Validation
Every number below comes from a case's results.json, produced by running the case - not by hand.
Nabla is not compared against itself. Every case states its reference source - a closed form, a published benchmark, another code on the same model, or an internal invariance the other tiers can't see - and is judged against it with a tolerance fixed before the run.
Cases
V01 — Coaxial cable, long solenoid and current loop
A round conductor, a thick solenoid and a current loop, judged against closed-form magnetostatics in both the planar and axisymmetric formulations.
V02 — C-core electromagnet with an airgap
A wound C-core across an airgap, nonlinear and saturated, judged against FEMM on the identical model and against Ampere's law on a contour in the window air.
V08 — Steady and transient heat conduction
Six Fourier-conduction problems with closed-form solutions, each solved at both element orders on the uncoupled thermal module.
V09 — P1 vs P2 convergence study
Three exact problems, meshed at five densities spanning 16x in element count, checking that the solver's error falls at the theoretical order as the mesh refines.
V10 — Nabla against FEMM on identical models
Four physics and six models built, solved and read in both Nabla and FEMM 4.2 inside one run - the comparison a prospective user makes anyway, done here with the models published.
V11 — TEAM Problem 30 - induction motor with an analytic solution
The one rotating-machine benchmark with a closed-form answer: a three-phase exposed-winding induction motor, judged against TEAM Problem 30's own published solution.
V12 — Skin effect: round conductor, slab and slot conductor
Three one-dimensional eddy-current problems with closed-form solutions, spanning a wide frequency sweep on each geometry.
V13 — Magnetized cylinder and far-field truncation study
A uniformly magnetized cylinder solved with the outer boundary at increasing radii and with the far-field (balloon) boundary condition, against the closed-form exterior field.
V14 — Sector vs full model; sliding band vs conforming mesh
One machine as sixteen models: periodic sector against full ring, and sliding band against a conforming mesh, checking that two constructions Nabla calls exact really are.
V15 — Force methods cross-check on a linear actuator
A C-core pulling on a plunger across two airgaps, with force computed three ways - the eggshell region integral, a Maxwell-stress contour, and virtual work - and checked for mutual agreement.
V16 — Transformer: magnetizing and short-circuit impedance
A two-winding transformer solved open-circuit and short-circuit, with its magnetizing and leakage impedance against closed-form winding theory and against FEMM.
V17 — Circular loop self and mutual inductance, axisymmetric
One circular turn asked for its own inductance and for its mutual coupling to eight coaxial neighbours, axisymmetric, against the elliptic-integral closed form.
V18 — Iron loss post-processing vs Steinmetz by hand
The iron-loss post-processor against the Steinmetz model its own coefficients are defined for. This case shipped FAIL, found three defects, and now passes.
V19 — Backward Euler vs Crank-Nicolson order study
Backward Euler against Crank-Nicolson on transients whose exact answer is known, measuring the observed order of each scheme rather than asserting it.
V20 — Solver-option invariance matrix
Half a dozen solver switches that are choices of method, not of answer - linear solver, time scheme, element order, band type, axial slices, field-circuit coupling - each moved to see whether the answer moves with it.
V22 — Linear machine and the linear sliding band
A flat linear machine's force pair at eight mover positions, by three constructions and two codes, with the mover carried by the linear sliding band.
V23 — Loss to thermal handoff on a machine
The loss-to-temperature chain end to end: a machine's Joule and iron losses become heat sources in a nine-layer stack, judged against the layered-cylinder closed form and against conservation.