TerreLenz — Validation: every case, the method and the limits

How to read the tables

What “error” means here.

Error is TerreLenz’s result minus the reference, as a percentage of the reference, at the mesh and element order the test uses. Signs are kept: “+” means TerreLenz reads high. Where a test checks several points or speeds, the table gives the worst of them.

Tolerance is what the automated test enforces, and it is usually looser than the error we measure. That way a mesher or library update that moves a result by a hair doesn’t break the build, but a real regression does. Where the tolerance is the only number we publish, the table says “within”.

References are of four kinds: closed-form solutions (textbook fields, inductances, forces and losses); FEMM 4.2 with mesh-converged models built from the same geometry; the TEAM problems (published measurements for TEAM 7, the published reference table for TEAM 30a); and internal cross-checks, where two independent routes to the same quantity must agree, such as a transient run to steady state against the AC solution, a 3D model against its axisymmetric twin, or Maxwell stress against virtual work.

Second-order elements as standard

First-order elements are not good enough for engineering results: near a magnet the error reaches 30 %. Every study therefore defaults to second order, with nodal elements in 2D and hierarchical edge elements in 3D.

Block magnet, 2 mm from the pole face: +0.09 %

Open boundaries, not a wall

A finite air box with a zero-potential wall is wrong in a known direction. TerreLenz maps the exterior to infinity with a shell transformation, in 2D and 3D, so the model can stop close to the part.

Block magnet in a 30 mm ball of air: within 0.3 % of exact. Same ball, plain wall: 6–13 % off

Bracketing the answer

Flux-tangential walls overestimate stored energy; field-normal walls underestimate it. The tests solve both and require the open-boundary result to land between them. If it ever doesn’t, something is wrong with the boundary treatment, not the mesh.

Block magnet: 1.296 J (FN) < 1.330 J (open) < 1.399 J (FT)

Forces from the Maxwell stress tensor

Integrated over a shell of air around each body, with a thin-film term for bodies that touch. Machine torque uses Arkkio’s method over the meshed air-gap ring. Virtual work, dW′/dx, is kept as an independent cross-check.

Plunger stroke force: Maxwell stress −20.09 N, virtual work −20.14 N

Saturation by Newton–Raphson

With the exact Jacobian, in 2D and 3D, and optional adaptive relaxation. Given a straight-line B-H curve, the Newton solver reproduces the linear solver to six digits. Curves are extended at μ₀ beyond the last point, so a solve never runs off the end of the table.

Steel sphere, 1.5 MA/m applied: +0.65 % at 3.2 T

Convergence you can watch

A Zienkiewicz–Zhu error estimator drives adaptive refinement in 2D and 3D. Check accuracy re-solves with a finer mesh, higher order, a tighter Newton tolerance, half the time step and more air, then reports how far each result moved. That spread is the error bar you put in the report.

Two-wire line, 4 adaptive passes: −6.2 % → +0.39 % → +0.024 % → −0.034 %

Results

The cases, by study type.

Geometry and settings are stated where they matter. Unless noted, every case uses second-order elements and the mesh the test specifies.

3D magnetostatics

Block magnet: 20 × 20 × 10 mm, B_r 1.3 T, μ_r 1, in a 180 × 180 × 170 mm air box. Sphere: radius 10 mm, Fröhlich-type steel with μ_i 1500 and J_s 1.95 T, checked against the exact relation ⅔·H + B(H)/3μ₀ = H₀.

Case	Checked against	Quantity	Error
Block magnet	Closed form	B_z, 2 mm from the pole face (0.3605 vs 0.3601 T)	+0.09 %
Block magnet	Closed form	B_z, 5 mm	+0.63 %
Block magnet	Closed form	B_z, 10 mm	+1.00 %
Block magnet	Closed form	B_z, 20 mm (air-box truncation starts to count)	+1.93 %
Block magnet, adaptive mesh from a 20 mm seed: 3 742 → 211 000 elements in 4 passes, 49 s	Closed form	B_z, 3 mm above the pole	0.14 %
Block magnet in a 30 mm ball of air, open boundary	Closed form	On-axis field	within 0.3 %
One eighth of the block magnet, three symmetry planes	Whole model	Stored energy × 8	0.015 %
Block magnet with field-normal / flux-tangential walls at ±25 and ±50 mm	Open-boundary model, 1.330 J	Stored energy brackets the open value	1.296 / 1.327 J below · 1.399 / 1.340 J above
Steel sphere, 10 kA/m (linear, μ_r 1000)	Exact solution	B at the center	+0.22 %
Steel sphere, 100 kA/m (nearly linear)	Exact solution	B at the center	+0.22 %
Steel sphere, 400 kA/m (at the knee)	Exact solution	B at the center (1.5069 vs 1.5036 T)	+0.22 %
Steel sphere, 1.5 MA/m (deep saturation)	Exact solution	B at the center (3.2051 vs 3.1844 T)	+0.65 %
Ring coil, second-order edge elements	Thick-coil formula	B on axis: center / 10 mm / 25 mm	+0.7 % / −0.4 % / −1.0 %
Ring coil, 3D model	Its own 2D axisymmetric model	Flux linkage (5.712 vs 5.701 mWb)	0.2 %
Two cuboid magnets	Magnetic charge model	F_z (−42.2 N) / F_x (−10.2 N) / torque	−0.08 % / +0.03 % / +1.2 %

Tolerances: block magnet 1 % to 5 mm, 1.5 % at 10 mm, 4 % at 20 mm; sphere 2 %; ring coil 1 % below 20 mm, 3 % at 25 mm. The ring coil’s +0.7 % at the center is discretization: the exact discrete solution on the same mesh sits at +0.14 %.

2D magnetostatics and saturation

C-core: a saturating core with a 100-turn coil and an armature across two 1 mm gaps, planar, 100 mm deep. The FEMM 4.2 models are mesh-converged and built from the same geometry description.

Case	Checked against	Quantity	Error
Two-wire line, planar	L′ = μ₀/π (ln(d/a) + ¼)	Inductance per length	−0.08 %
Two-wire line, planar	F′ = μ₀I²/2πd	Force per length	−0.4 %
Two-wire line, adaptive from 833 elements: 4 passes to 23 400 elements, 3 s	Analytic	Inductance per length, pass by pass	−6.2 → +0.39 → +0.024 → −0.034 %
Two coaxial rings, axisymmetric	Maxwell’s filament formula	Mutual inductance	−0.08 %
Two coaxial rings, axisymmetric	Analytic	Axial force	+0.06 %
Thick solenoid, axisymmetric	Analytic	Center field	+0.43 %
Permeable cylinder, μ_r 10, in an applied field	Analytic	Field inside	−0.20 %
Sphere in an axial applied field, axisymmetric	Analytic	Field inside	+0.08 %
Two-wire line / thin ring in a small air region, open boundary	Free-space values	Inductance	−0.03 % / +0.09 %
Same region with a plain zero-potential wall (for comparison)	Free-space values	Inductance	−8.7 % / −3.4 %
Planar magnet with field-normal / flux-tangential walls	Open-boundary model, 8.437 J	Stored energy brackets the open value	8.049 / 8.333 J below · 8.818 / 8.525 J above
C-core, 500 A-turns	FEMM 4.2	Gap B (0.29115 vs 0.29115 T)	0.00 %
C-core, 500 A-turns	FEMM 4.2	Armature force (−73.306 vs −73.307 N)	0.00 %
C-core, 500 A-turns	FEMM 4.2	Flux linkage (46.167 vs 46.166 mWb)	0.00 %
C-core, 3000 A-turns (saturated)	FEMM 4.2	Gap B (1.2128 vs 1.2137 T)	−0.07 %
C-core, 3000 A-turns (saturated)	FEMM 4.2	Armature force (−1266.96 vs −1267.46 N)	−0.04 %
C-core, 3000 A-turns (saturated)	FEMM 4.2	Flux linkage (194.18 vs 194.20 mWb)	−0.01 %
C-core, 5 A and 30 A	FEMM 4.2 block integrals	Energy and co-energy	within 0.5 %
Laminated ring, stacking factor 0.9	Analytic	Loss / flux density	0.03 % / 0.01 %
Magnet at 120 °C, library temperature coefficients	Scaled 20 °C solve	Field	exactly 0.88×

AC, eddy currents, transients and circuits

Where the reference is another TerreLenz study, the two routes are independent: a transient stepped to steady state against the time-harmonic solution, or a 3D model against its axisymmetric twin.

Case	Checked against	Quantity	Error
Round copper wire, 5 kHz (δ 0.93 mm, radius 5 mm), adaptive from 240 to 6 300 elements	Bessel-function solution	AC resistance	+0.04 %
Round copper wire, 50 / 300 / 1000 Hz	Bessel-function solution	AC resistance	within 1 %
Conducting cylinder as a shield, a/δ = 2	B₀ / \|J₀(ka)\|	Field inside	within 0.16 %
Same cylinder under a transient sine	Its AC solution	Loss	within 1.8 %
Conducting sphere in a uniform AC field, 3D	Closed form	Eddy-current loss	within 3 %
Aluminum disk above a ring coil, 200 Hz, 3D	Its axisymmetric AC model	Eddy loss / coil inductance	1.2 % / 0.2 %
TEAM 7 asymmetric conductor with a hole, 50 and 200 Hz, 675 000 unknowns	Published measurements	B_z 15 mm above the plate	within 4 % of the 78 G peak at all but two points per line; worst 5.4 G
TEAM 7	Published measurements	J_y on the face toward the coil	within 15 % of the extreme, except at the hole’s edge
Plate over a coil, Crank–Nicolson transient to steady state	Its AC solution	Current / loss / force	0.02 % / 0.04 % / 0.06 %
Saturating C-core, nonlinear transient at DC	Its magnetostatic solution	All quantities	identical
Stranded coil, voltage step	V/R (1 − e^−tR/L)	Current rise	passes
Same step with adaptive BDF2 time steps	Exact	Current, in 29 steps	0.26 %
Ring around a wire	Analytic	Iron loss (Bertotti)	0.007 %
Jiles–Atherton ring, 50 Hz	Loop area × f	Hysteresis loss (0.5446 W, equal to electrical input to six digits)	within 0.4 %
Jiles–Atherton ring	Independent integration of the model	Remanence (0.627 vs 0.624 T)	+0.5 %
Saturated C-core, AC study with the effective B-H curve	Fundamental of the transient	Flux linkage	+5.9 % (see Limits)
Induction heating of a round coil, 2D	Closed form	Temperature	within 0.5 %
Capacitor discharge into a coil (netlist)	RLC solution	Current / overshoot	0.03 % / 0.003 %
Freewheeling diode	Analytic	Decay	within 0.5 %
Crowbar decay	Analytic	Current	within 0.001 %
Star and delta netlists	Nodal analysis	Currents	6 digits

TEAM 7: at 200 Hz and ωt = 90° the field is 3.5–4 G high along the line; a phase offset in the measurement is suspected but not confirmed, so the test keeps the tighter tolerance and the discrepancy is reported as is.

Motion, machines and magnets

TEAM 30a: the three-phase induction motor, 60 Hz, 1 m deep, torque checked at 0, 200, 400 and 1200 rad/s against the published table. Machine torque is Arkkio’s method over the meshed air-gap ring.

Case	Checked against	Quantity	Error
TEAM 30a, 0 / 200 / 400 / 1200 rad/s	Published reference table	Torque	within 0.02 %
TEAM 30a	Independent FE reference	Rotor losses	≤ 4 %
TEAM 30a, slip model in AC	Motion solution	Torque at s = 0.47 / s = −2.18	+0.7 % / −4.9 %
TEAM 30a, locked rotor, 1 ms time step	AC solution	Torque: Crank–Nicolson (implicit Euler for comparison)	−0.9 % (−9 %)
4-pole rotor, 12-slot stator	FEMM 4.2	Cogging torque	within 6 % of amplitude
12-slot, 4-pole sector sweep	Whole machine	Cogging torque	within 1.6 % of amplitude
One pole of a 4-pole rotor (periodic sector)	Whole machine	Fields / energy × 4	0.3 % / 0.001 %
Anti-periodic sector, AC and transient	Whole machine	Bar loss / flux linkage × 4	0.005 % / 1 %
Translational periodic array	Whole model	Field at every probe	within 1 %
Rotating magnet	ω · dλ/dθ from a static sweep	Back-EMF	passes
Magnetic pendulum, load-driven	2π √(J/k)	Period	−0.05 %
Rotor on a torsion spring, load-driven	Analytic	Period	0.01 %
Load-driven sector model	Whole machine	Rotor angle	within 0.15°
Magnetic gear, two rotors	Exact ratio 2.5	Torque ratio; sum of the three torques	2.44–2.55; 0.4 %
Magnetic gear, free outer rotor	Gear ratio −120	Lock-in speed	−119.997 rpm
Plunger, blocked	V/R (1 − e^−tR/L)	Current	within 0.2 %
Plunger, free	Energy balance	Electrical in vs mechanical + stored	within 0.3 %
Plunger stroke force	Virtual work dW′/dy (−20.14 N)	Maxwell stress force (−20.09 N)	0.2 %
Eddy-current brake	Low-speed formula (−0.26054 N)	Drag (−0.26033 N)	0.08 %
45° of rotation at 1000 rpm, adaptive time steps	Fixed fine steps	Flux linkage	within 0.4 % of peak
Irreversible demagnetization with recoil	Demagnetization law, per element	B at every step	within 0.2 mT
Cylindrical magnet	Analytic	Mean demagnetization margin	within 0.3 mT
Magnetizing fixture, half-field curve	Exact	Magnetized fraction	50.001 %

Quick design against finite elements

Quick design is an analytic (magnetic-equivalent-circuit) estimate of the template machine, about 0.1 ms per design. It is a sizing tool: this is how far it sits from the full FE model of the default 12-slot, 10-pole machine at 10 A, using the tabulated M270-35A steel.

Quantity	Quick design	Finite elements	Difference
Magnet flux linkage ψ_m	67.7 mWb	67.2 mWb	+0.7 %
Phase EMF at 1500 rpm, peak	53.2 V	52.8 V	+0.8 %
Torque at 10 A, q axis	5.08 N·m	5.01 N·m	+1.4 %
L_d (2D part)	1.19 mH	1.20 mH	−0.8 %
L_q (2D part)	1.28 mH	1.18 mH	+8 %
Open-circuit iron loss	7.4 W	8.2 W	−10 %

Buried-magnet rotors (flat, V and spoke) are rougher: flux linkage within 15 %, best torque within 20 %, L_d and L_q within 35 % of FE, because the estimate has no leakage-bridge saturation model. The induction-machine estimate reads 5–8 % above FE on torque at every slip and within 3 % on current. That is why the button next to the estimate builds the FE model.

Where the error comes from

Four things move a finite-element result. TerreLenz shows you each.

The air region

A finite box moves far-field results by about 1.6 % depending on whether the wall is field-normal or flux-tangential, and the effect grows with distance: +1.93 % on the block magnet at 20 mm. Open boundaries remove it; Check accuracy adds more air and shows what changed.

The mesh

On a typical wire-force model, a mesh refinement moves the answer by 0.4 % and the air-region choice by 7.4 %, so the tests always vary both. Adaptive refinement shows the convergence pass by pass rather than asking you to trust one mesh.

The element order

First-order elements reach 30 % error near a magnet, which is why second order is the default and not an option to remember. Orders above two are on the roadmap.

The time step

On TEAM 30a with the rotor locked and a 1 ms step, implicit Euler reads the torque 9 % low; Crank–Nicolson reads it 0.9 % low. Crank–Nicolson is the default, adaptive BDF2 is available, and every transient starts from a converged static solve.

Limits

What TerreLenz doesn’t do yet.

You should know these before you buy, not after. The tester build reports the same list under Help.

Motion is 2D. Rotation, periodic sectors, netlists and the controller in the loop are 2D today. 3D covers magnetostatics, AC and transients with coils on current or voltage sources (series R, L and C, no switches), eddy currents and saturation, at fixed time steps. 3D motion is next on the roadmap.
3D AC is linear. A saturating body in a 3D eddy-current study is refused with a message, not silently approximated.
2D AC treats saturation with an effective B-H curve, fundamental only. On the saturated C-core that reads +5.9 % on flux linkage, so for waveforms distorted by saturation use a transient study.
Element order is 1 or 2. Higher orders are planned.
The built-in soft materials are generic approximations, not datasheet data. Enter your own B-H curves and loss coefficients for design work. The magnet catalog (129 grades) is fitted to the standard grade tables and measured production lots.
Iron loss is Bertotti per harmonic. Minor loops and DC bias are not modeled, and stranded (Litz) coils have no proximity-loss model. Magnet eddy loss comes from transient FE, not from quick design.
2D heating is steady state, air gaps don’t conduct, planar models have no end windings, and the machine thermal network is not yet calibrated against heat runs.
Per-body torque on a rotor at the default mesh reads 3–6 % below Arkkio’s method, so machine models use Arkkio. Cogging against FEMM is within 6 % of amplitude; the sector sweep is within 1.6 %.

Engineers’ FAQ

The questions we’d ask.

Which formulations does it solve?

2D planar and axisymmetric magnetostatics use the vector potential A_z (r·A_φ in axisymmetry) with reluctivity ν(|B|²). 2D AC and transient studies use the a-v formulation with stranded and massive conductors, coupled to circuits. 3D magnetostatics without coils uses the total scalar potential; 3D with coils, and 3D AC, use an ungauged vector potential on second-order edge elements, with each coil’s current density taken from a conduction solve and made exactly divergence-free.

How are force and torque computed?

Force per body is the Maxwell stress tensor integrated over a shell of air around the body, with a thin-film term for bodies in contact. Torque on a rotor uses Arkkio’s method over the meshed air-gap ring, which is far less mesh-sensitive than a surface integral. Virtual work is available as a cross-check; the two agree to 0.2 % on the plunger benchmark.

How are flux linkage and inductance defined?

Flux linkage is the integral of A over the coil section, times turns and depth (or 2π). The inductance matrix is reported two ways: apparent, from each circuit alone at its test current, and incremental, from ±1 % steps about the operating point. For permanent-magnet devices the incremental value is (λ − λ₀)/i. AC studies give the coil impedance R + jωL directly.

What linear solvers are used, and on how many cores?

2D and small 3D systems (below about 20 000 unknowns) use a sparse direct solver. Larger 3D systems use conjugate gradients with domain-decomposition (BDDC) or algebraic multigrid preconditioning, threaded across up to 12 physical cores. Sweeps run several solves in parallel, each in its own process, so a failed variant is reported rather than stopping the sweep.

For scale: TEAM 7 at 675 000 second-order edge unknowns solves in about 22 s per frequency on a desktop PC.

How is motion handled?

Rotation uses a moving band in the air gap, at a set speed, from an angle table, or load-driven: a semi-implicit Euler step on the rotor’s inertia, friction and load with Arkkio torque evaluated each step. Up to two rotors can turn, which is how magnetic gears are modeled. Translation is a static position sweep (re-meshed at each position), a quasi-static plunger model on λ(x, i) and F(x, i) tables, or a full 2D transient with eddy currents, re-meshed every step. Skew is modeled by slices.

How does time stepping work?

A theta scheme with implicit Euler or Crank–Nicolson (the default for new projects), or adaptive BDF2 with error control. Every transient starts from a converged static solve, so there is no artificial start-up transient. Nonlinear steps use Newton–Raphson with the exact Jacobian.

How is temperature handled?

Each material carries temperature coefficients or tables: B_r and H_cJ for magnets (library values −0.12 / −0.6 %/°C for NdFeB, −0.2 / +0.3 for ferrite, −0.035 / −0.25 for SmCo), resistivity for conductors, and saturation polarization for steels, with per-temperature demagnetization curves. A 2D heating study feeds its temperatures back into all of them.

Are results reproducible?

Yes. Meshing is deterministic, and a test checks that two runs of the same model produce byte-identical meshes. Every solve is written as plain files you can archive and re-run, and a project file is a zip of JSON you can read without TerreLenz.

Why a rainbow colormap, and can I change it?

By default, magnitudes use turbo, a rainbow map whose lightness is smoothed so it doesn’t show the false bands of the classic “jet”, and signed values use cool-warm. Rainbow maps separate more levels by eye, but their lightness doesn’t rise evenly, so they can overstate some features and they don’t survive a grayscale print.

When that matters, switch to a perceptual map in the Lenz · Results panel under Colors: Viridis or Inferno, whose lightness rises evenly, or Cividis, which is also safe for the most common color blindness. Grayscale is there too. Your choice is remembered and used for the report’s field plots, and the report’s method notes name the map.

Can I run these cases myself?

Ask us for any benchmark on this page as a project file and we’ll send it with the tester build. Or bring your own reference case: the fastest way to judge a solver is on a problem whose answer you already know.

Every number, and how we got it.