Validation

Each case below is an automated test (native C++, unit or end-to-end) that compares MagFEM with an analytical solution. They run on every code change.

Case	Reference	Error
Strip with uniform current (A and energy)	\(A = \mu_0 J x(L-x)/2\)	< 0.2 %
Magnet filling the domain	\(B = B_r\)	~1e-14
Infinite axisymmetric solenoid	\(B_z = \mu_0 J (R_2 - R_1)\)	< 0.01 %
Periodic strip	\(A = \mu_0 J y(2H-y)/2\)	< 0.2 %
Circuit in the strip	\(\tfrac{1}{2} L I^2 =\) energy; analytical λ	< 1 %
Quadratic interpolation	exact quadratic function	≪ linear
Mixed boundary (planar)	linear \(A(x)\)	~1e-14
Mixed boundary (axisymmetric)	\(A = C_1 r/2 + C_2/r\)	3e-6
Uniform axial magnet (axisymmetric)	\(B = B_r\) (\(r > R/4\))	1.8 % (drops with refinement)
Nonlinear	exact \(H(B)\) in the saturated range	drops with refinement
Transient diffusion	analytical series	6e-4
Coupled circuit (RL) and transformer	discrete RL; \(V_2 = (N_2/N_1) V_1\)	~1e-15
AC: skin effect in a plate	\(A = A_0 \cosh(k(L-x))/\cosh(kL)\)	1.3e-4
AC: negative J	\(\hat{A}(-J) = -\hat{A}(J)\)	exact
Force on a conductor in a uniform field	\(F = I \times B\)	0.3 % (surface) / 1.4 % (line)
Iron losses in a uniform field	\(k_h f B^2 V\)	exact
Laminated sheets: losses with factor \(f\) and \(k_e = \pi^2\sigma d^2/6\)	steel volume \(f V\) with \(B/f\)	exact
Laminated sheets: equivalent material	\(\nu = 1/(\mu_0(f\mu_r + 1 - f))\); \(B = fB_{steel} + (1-f)\mu_0H\)	exact
DC R with AWG wire (axisymmetric)	\(N\,2\pi r/(\sigma A_{wire})\)	< 0.2 %
Skin, round wire	\(1 + (a/\delta)^4/48\) and \(a/(2\delta) + 1/4\)	< 1e-5
Proximity, round wire in a uniform field	\(N\,\ell\,\pi\sigma\omega^2\hat B^2 a^4/8\) (low frequency)	< 0.1 %
Thermal: slab with generation and convection	\(T = T_{amb} + qL/2h + q x(L-x)/2k\)	< 0.1 K
Thermal: axisymmetric cylinder	\(T = T_{amb} + qR/2h + q(R^2-r^2)/4k\)	< 0.1 K
Thermal: fan (air balance)	\(T_{out} = T_{in} + P/(\rho c_p Q)\)	< 1e-3
Thermal with σ(T) (DC block)	\(T = T_{amb} + P_{20}(1+\alpha(T-20))/(hS)\)	< 0.05 K
Proximity, rectangular wire (Dowell)	\(\sigma\omega^2\hat B^2 t^3/24\) per width	< 0.1 %
Eddy-current losses	\(\propto f^2\) at low frequency	ratio 3.97 (≈ 4)

The tests live in core/tests (native core), web/src/**/*.test.ts (unit) and web/e2e (end-to-end). To run them all: ./scripts/check.

Comparison with FEMM

MagFEM uses the same mesh generator (Tangle) and the same boundary and material conventions as FEMM, and imports .fem files: it is easy to solve the same model in both and compare energy, flux linkage and force.