How the cases are set up
Each case builds its part, meshes it, refines until the answer settles and runs its checks, and the number is then compared with a known answer.
Textbook formulas
The cantilever cases use a 200 x 20 x 10 mm steel bar clamped at one end with 100 N at the
other. The tip deflection is compared with Timoshenko beam theory, which counts both bending and
shear, and the stress on the top surface halfway along with M c / I. The clamped end makes the
3D model slightly stiffer than beam theory, by under 1%.
The plate with a hole is 400 x 200 x 4 mm with a 20 mm hole, pulled at 10 MPa and held by rollers on three faces. The peak at the hole is compared with Kirsch’s solution with Peterson’s finite width correction, a stress concentration of 3.035 on the gross section.
The thick walled pressure vessel is a quarter of a tube with radii of 50 and 100 mm under 100 MPa of internal pressure, in plane strain with symmetry rollers on all the cut faces. Its von Mises stress at the bore is compared with Lame’s solution.
The torsion case is a 20 mm steel shaft, 100 mm long, clamped at one end with 50 N m applied
as a distributed moment at the other. The surface von Mises stress is compared with
sqrt(3) x 16 T / (pi d^3).
The self weight case is a 1 m steel cantilever, 20 x 20 mm, under 1 g, and its tip deflection is
compared with q L^4 / (8 E I).
Industry benchmarks
NAFEMS LE1 is the elliptic membrane, a quarter of an elliptical ring with 10 MPa of outward pressure on its outer edge and symmetry on both axes. sigma_yy at point D, at (2000, 0), is compared with the target of 92.7 MPa.
NAFEMS LE10 is the thick plate, a 600 mm thick elliptical ring under 1 MPa on its top face, with the outer edge held in x and y and the outer mid plane line held in z, which Wobbl models as a 2 mm band. sigma_yy at point D on the top surface is compared with the target of -5.38 MPa.
Real parts from the bake-off
These are the parts from the bake-off. A sand cast motor mount carries a belt pull off the part through four bolts. A molded ABS enclosure shell with bosses, a snap tongue and a USB opening is pressed near a corner. A PA12 drone arm with Warren truss windows takes thrust and torque from a rigid motor. A PLA L bracket, once with a sharp 0.5 mm inside corner and once with a 6 mm fillet, is screwed to a wall it can lift off.
The references are gmsh and CalculiX models built by hand and refined over four to six meshes
of up to 400,000 nodes. Their tolerances follow how settled each reference was, so the sharp
bracket, whose reference was still moving 5% on its last mesh, gets 8%. The parts ship with
Wobbl in src/wobbl/validation/parts/.
Heat
The slab is a 100 mm aluminum bar held at 100 C on one end and cooled by 50 W/(m2 K) air at
20 C on the other, insulated elsewhere. The temperature of the cooled face is compared with the
series resistance L/k + 1/h.
The lumped case is a 20 mm aluminum cube at 100 C cooling in 10 W/(m2 K) air for ten minutes. Its mean temperature is compared with lumped capacitance, which is exact to well under 1% here because the Biot number is 2e-4.
The sudden face case is a 100 mm aluminum bar at 20 C whose end is held at 120 C from the start,
insulated elsewhere. The temperature rise 10 mm in after 2 s is compared with the erfc solution
for a semi infinite solid, 100 erfc(x / (2 sqrt(alpha t))), which is exact here because the
heat has not reached the far end yet. Like every held face that starts away from the part’s
temperature, the end ramps to 120 C over a few milliseconds instead of jumping at once.
Circuits
A red LED on 5 V through 150 ohm is compared with the diode equation solved by Newton iteration
with the same model parameters. An RC network’s voltage at one time constant is compared with
1 - e^-1. A 12 V to 5 V buck converter’s output ripple is compared with the ideal triangular
ripple current through the output capacitor and its ESR.
Motion
The motion cases go through the same integrator, events and checks as a flight of your own, with
gravity held at its ground value and either no air or air of one density, so each one has an
exact answer. A ball dropped 300 m without air is compared with sqrt(2 h / g) for its fall
time. The granite sphere from the falling rock example, 9 cm across
with a drag coefficient of 0.6, drops 300 m through air of one density, and its impact speed is
compared with the quadratic drag solution vt tanh(g t / vt), with the fall time from
h = vt^2 / g ln cosh(g t / vt). A ball thrown at 30 m/s and 40 degrees without air is compared
with v^2 sin(2 theta) / g for its range.
The rocket climbs straight up in a vacuum off a 1 m rail, with 1000 N of thrust for 4 s that
burns 4 kg of propellant and takes it from 11 kg to 7 kg. Its speed at burnout is compared with
ve ln(m0 / m1) - g tb, and its height at burnout with
ve tb - ve (m1 / mdot) ln(m0 / m1) - g tb^2 / 2. A 5 kg payload let go at 500 m under a 2 m
parachute with a drag coefficient of 1.5 is compared with its terminal speed
sqrt(2 m g / (rho Cd A)) as it reaches the ground. Each of the six lands within a millionth of
its answer.