OneCAD-Simulation/docs/oracle.md

Verification and validation oracle

Generated by cargo xtask oracle from tests/oracle/; the policy is verification.md. Every case is our own model, built with FieldCraft commands from the problem's published description, and compared with the reference value its expected.json cites, within the stated tolerance and for the stated reason. Accepted results are frozen in each case's fixture.json: a later change that moves one beyond its band (10⁻⁹ of its size) fails CI even while the reference still passes. No reference value was produced with a commercial product.

142 of 148 quantities pass (6 known failure(s)) in 40 active cases: 31 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.

Theory: closed-form solutions, patch tests and mesh invariants

Case	Quantity	Reference	FieldCraft	Error	Tolerance	Result	Fixture
bar-pressure-wave	displacement of the loaded end at t = 0.06	-6.0000e-4	-5.9980e-4	0.034%	1%	pass	matches
bar-pressure-wave	particle velocity at x = 5, 20 elements behind the front, at t = 0.06	-0.01	-0.0098552	1.45%	2%	pass	matches
bar-pressure-wave	energy balance error	0	0.001527	1.5e-3	± 0.01	pass	matches
bar-rigid-wall-impact	displacement of the struck end at t = 0.3 (in contact until 2L/c = 0.2, then leaving at v₀)	0.01	0.0099541	-0.459%	3%	pass	matches
bar-rigid-wall-impact	displacement of the far end at t = 0.3 (−v₀L/c, then +v₀ after the reflection)	0.01	0.0099488	-0.512%	3%	pass	matches
bar-rigid-wall-impact	kinetic energy after the rebound over the initial ½mv₀²	1	0.99463	-0.537%	2%	pass	matches
cantilever-frequencies	first bending frequency (modes 1 and 2 are the y and z pair)	0.012771	0.012751	-0.158%	1%	pass	matches
cantilever-frequencies	the other first bending frequency	0.012771	0.012751	-0.158%	1%	pass	matches
cantilever-frequencies	second bending frequency	0.080034	0.079167	-1.08%	2%	pass	matches
cantilever-hex20	tip deflection u_z at (10, 0.5, 0.5)	-4.024	-4.0235	0.0131%	0.5%	pass	matches
cantilever-hex20	total reaction z	1	1	< 1e-7%	1e-9 rel.	pass	matches
cantilever-hex8-incompatible	tip deflection u_z at (10, 0.5, 0.5)	-4.024	-4.0175	0.162%	0.5%	pass	matches
cantilever-hex8-incompatible	total reaction z	1	1	< 1e-7%	1e-9 rel.	pass	matches
cantilever-tet10	tip deflection u_z at (10, 0.5, 0.5)	-4.024	-4.0233	0.018%	0.5%	pass	matches
cantilever-tet10	total reaction z	1	1	< 1e-7%	1e-9 rel.	pass	matches
cdt-cylinder	volume of the tetrahedra	6267.5	6266.5	-0.0152%	0.25%	pass	matches
cdt-cylinder	elements that fail the certified Jacobian check	0	0	≈ 0	exact	pass	matches
cdt-stepped-block	volume of the tetrahedra	47000	47000	< 1e-7%	1e-12 rel.	pass	matches
cdt-stepped-block	elements that fail the certified Jacobian check	0	0	≈ 0	exact	pass	matches
cdt-stepped-block	elements the Jacobian check could not decide	0	0	≈ 0	exact	pass	matches
free-beam-frequencies	sixth frequency (the last rigid-body mode)	0	≈ 0	≈ 0	± 8.1264e-5	pass	matches
free-beam-frequencies	first free–free bending frequency (mode 7)	0.081264	0.080618	-0.795%	1.5%	pass	matches
free-beam-frequencies	total mass	20	20	< 1e-7%	1e-12 rel.	pass	matches
j2-uniaxial-explicit	axial Cauchy stress at a logarithmic strain of 0.2	643.56	643.57	0.000174%	1%	pass	matches
j2-uniaxial-explicit	equivalent plastic strain	0.19678	0.19678	-0.00000113%	1%	pass	matches
kirsch-plate	σ_xx at the top of the hole (0, 1, 0) over the remote stress	3	3.0121	0.402%	2%	pass	matches
kirsch-plate	σ_yy at the side of the hole (1, 0, 0) over the remote stress	-1	-1.011	-1.1%	3%	pass	matches
kirsch-plate-plane-stress	σ_xx at the top of the hole (0, 1) over the remote stress	3	3.0029	0.096%	2%	pass	matches
kirsch-plate-plane-stress	σ_yy at the side of the hole (1, 0) over the remote stress	-1	-1.0019	-0.187%	3%	pass	matches
lame-cylinder	radial displacement at the bore (10, 0, 0)	0.0090794	0.0090793	-0.000267%	0.1%	pass	matches
lame-cylinder	radial displacement outside (20, 0, 0)	0.0057778	0.0057778	-0.000301%	0.1%	pass	matches
lame-cylinder	hoop stress σ_θ = σ_yy at the bore (10, 0, 0)	166.67	166.36	-0.183%	1%	pass	matches
lame-cylinder-axisymmetric	radial displacement at the bore (r = 10)	0.0090794	0.0090793	-0.000416%	0.1%	pass	matches
lame-cylinder-axisymmetric	radial displacement outside (r = 20)	0.0057778	0.0057778	-0.000327%	0.1%	pass	matches
lame-cylinder-axisymmetric	hoop stress σ_θ at the bore (stress component 3 of an axisymmetric element)	166.67	165.98	-0.412%	1%	pass	matches
lame-cylinder-axisymmetric	axial stress σ_z = 2νA at the bore (held ends)	20	20.084	0.42%	1%	pass	matches
lame-cylinder-plane-strain	radial displacement at the bore (10, 0)	0.0090794	0.0090793	-0.000267%	0.1%	pass	matches
lame-cylinder-plane-strain	hoop stress σ_θ = σ_yy at the bore (10, 0)	166.67	166.36	-0.183%	1%	pass	matches
lame-cylinder-plane-strain	axial stress σ_zz = 2νA at the bore (plane strain)	20	20.083	0.415%	1%	pass	matches
nl-cylinder-inflation	radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15	5	5.004	0.0793%	0.5%	pass	matches
nl-cylinder-inflation	radial displacement of the outer wall (20, 0, 0)	2.9129	2.9173	0.152%	0.5%	pass	matches
nl-elastica-cantilever	tip deflection δ_v/L at P L²/(E I) = 1	0.30172	0.30226	0.179%	1%	pass	matches
nl-elastica-cantilever	tip shortening δ_h/L at P L²/(E I) = 1	0.056433	0.056633	0.354%	2%	pass	matches
nl-elastica-cantilever	tip deflection δ_v/L at P L²/(E I) = 2	0.49346	0.49455	0.221%	1%	pass	matches
nl-elastica-cantilever	tip shortening δ_h/L at P L²/(E I) = 2	0.16064	0.16132	0.422%	2%	pass	matches
nl-elastica-cantilever	tip deflection δ_v/L at P L²/(E I) = 5	0.71379	0.71598	0.306%	1%	pass	matches
nl-elastica-cantilever	tip shortening δ_h/L at P L²/(E I) = 5	0.38763	0.38955	0.495%	2%	pass	matches
nl-elastica-cantilever	tip deflection δ_v/L at P L²/(E I) = 10	0.81061	0.81421	0.444%	1%	pass	matches
nl-elastica-cantilever	tip shortening δ_h/L at P L²/(E I) = 10	0.555	0.558	0.541%	2%	pass	matches
nl-neo-hookean-uniaxial	axial Cauchy stress at the corner (1, 1, 1), stretch λ = 2	8.0601	8.0601	< 1e-7%	1e-6 rel.	pass	matches
nl-neo-hookean-uniaxial	lateral displacement u_y at the corner (1, 1, 1), l − 1	-0.18686	-0.18686	< 1e-7%	1e-6 rel.	pass	matches
nl-patch-hex8	u_x at inner node 22	0.13583	0.13583	< 1e-7%	1e-8 rel.	pass	matches
nl-patch-hex8	u_y at inner node 22	3.4875e-4	3.4875e-4	-0.000000665%	1e-8 rel.	pass	matches
nl-patch-hex8	u_z at inner node 22	-0.0023985	-0.0023985	-0.000000102%	1e-8 rel.	pass	matches
nl-patch-hex8	Cauchy σ_xx at inner node 22	42.296	42.296	< 1e-7%	1e-7 rel.	pass	matches
nl-patch-hex8	Cauchy σ_yy at inner node 22	23.501	23.501	< 1e-7%	1e-7 rel.	pass	matches
nl-patch-hex8	Cauchy σ_zz at inner node 22	31.453	31.453	< 1e-7%	1e-7 rel.	pass	matches
nl-patch-hex8	Cauchy σ_xy at inner node 22	-0.88971	-0.88971	< 1e-7%	1e-7 rel.	pass	matches
nl-patch-hex8	Cauchy σ_xz at inner node 22	-1.8906	-1.8906	< 1e-7%	1e-7 rel.	pass	matches
nl-patch-hex8	Cauchy σ_yz at inner node 22	1.9462	1.9462	< 1e-7%	1e-7 rel.	pass	matches
nl-rigid-rotation	u_x at inner node 22: the rotation's	-0.011149	-0.011149	-1.2e-16	± 1.0000e-9	pass	matches
nl-rigid-rotation	u_y at inner node 22: the rotation's	0.078591	0.078591	-5.6e-17	± 1.0000e-9	pass	matches
nl-rigid-rotation	u_z at inner node 22: the rotation's	-0.067441	-0.067441	-1.2e-16	± 1.0000e-9	pass	matches
nl-rigid-rotation	von Mises stress at inner node 22	0	≈ 0	≈ 0	± 1.0000e-8	pass	matches
patch-hex20	u_x at the corner (1, 1, 1)	-2.5000e-4	-2.5000e-4	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20	u_y at the corner (1, 1, 1)	0.001	0.001	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20	u_z at the corner (1, 1, 1)	0.00225	0.00225	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20	σ_xx at the inner node (0.46, 0.55, 0.53)	1	1	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20	σ_yy at the inner node (0.46, 0.55, 0.53)	2	2	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20	σ_zz at the inner node (0.46, 0.55, 0.53)	3	3	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20	σ_xy at the inner node	0	≈ 0	≈ 0	± 1.0000e-9	pass	matches
patch-hex20-reduced	u_x at the corner (1, 1, 1)	-2.5000e-4	-2.5000e-4	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20-reduced	u_y at the corner (1, 1, 1)	0.001	1.0000e-3	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20-reduced	u_z at the corner (1, 1, 1)	0.00225	0.00225	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20-reduced	σ_xx at the inner node (0.46, 0.55, 0.53)	1	1	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20-reduced	σ_yy at the inner node (0.46, 0.55, 0.53)	2	2	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20-reduced	σ_zz at the inner node (0.46, 0.55, 0.53)	3	3	< 1e-7%	1e-9 rel.	pass	matches
patch-hex20-reduced	σ_xy at the inner node	0	≈ 0	≈ 0	± 1.0000e-9	pass	matches
patch-hex8	u_x at the corner (1, 1, 1)	-2.5000e-4	-2.5000e-4	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8	u_y at the corner (1, 1, 1)	0.001	0.001	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8	u_z at the corner (1, 1, 1)	0.00225	0.00225	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8	σ_xx at the inner node (0.46, 0.55, 0.53)	1	1	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8	σ_yy at the inner node (0.46, 0.55, 0.53)	2	2	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8	σ_zz at the inner node (0.46, 0.55, 0.53)	3	3	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8	σ_xy at the inner node	0	≈ 0	≈ 0	± 1.0000e-9	pass	matches
patch-hex8-incompatible	u_x at the corner (1, 1, 1)	-2.5000e-4	-2.5000e-4	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8-incompatible	u_y at the corner (1, 1, 1)	0.001	0.001	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8-incompatible	u_z at the corner (1, 1, 1)	0.00225	0.00225	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8-incompatible	σ_xx at the inner node (0.46, 0.55, 0.53)	1	1	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8-incompatible	σ_yy at the inner node (0.46, 0.55, 0.53)	2	2	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8-incompatible	σ_zz at the inner node (0.46, 0.55, 0.53)	3	3	< 1e-7%	1e-9 rel.	pass	matches
patch-hex8-incompatible	σ_xy at the inner node	0	≈ 0	≈ 0	± 1.0000e-9	pass	matches
patch-tet10	u_x at the corner (1, 1, 1)	-2.5000e-4	-2.5000e-4	< 1e-7%	1e-9 rel.	pass	matches
patch-tet10	u_y at the corner (1, 1, 1)	0.001	1.0000e-3	< 1e-7%	1e-9 rel.	pass	matches
patch-tet10	u_z at the corner (1, 1, 1)	0.00225	0.00225	< 1e-7%	1e-9 rel.	pass	matches
patch-tet10	σ_xx at the inner node (0.46, 0.55, 0.53)	1	1	< 1e-7%	1e-9 rel.	pass	matches
patch-tet10	σ_yy at the inner node (0.46, 0.55, 0.53)	2	2	< 1e-7%	1e-9 rel.	pass	matches
patch-tet10	σ_zz at the inner node (0.46, 0.55, 0.53)	3	3	< 1e-7%	1e-9 rel.	pass	matches
patch-tet10	σ_xy at the inner node	0	≈ 0	≈ 0	± 1.0000e-9	pass	matches
patch-tet4	u_x at the corner (1, 1, 1)	-2.5000e-4	-2.5000e-4	< 1e-7%	1e-9 rel.	pass	matches
patch-tet4	u_y at the corner (1, 1, 1)	0.001	1.0000e-3	< 1e-7%	1e-9 rel.	pass	matches
patch-tet4	u_z at the corner (1, 1, 1)	0.00225	0.00225	< 1e-7%	1e-9 rel.	pass	matches
patch-tet4	σ_xx at the inner node (0.46, 0.55, 0.53)	1	1	< 1e-7%	1e-9 rel.	pass	matches
patch-tet4	σ_yy at the inner node (0.46, 0.55, 0.53)	2	2	< 1e-7%	1e-9 rel.	pass	matches
patch-tet4	σ_zz at the inner node (0.46, 0.55, 0.53)	3	3	< 1e-7%	1e-9 rel.	pass	matches
patch-tet4	σ_xy at the inner node	0	≈ 0	≈ 0	± 1.0000e-9	pass	matches
patch-wedge6	u_x at the corner (1, 1, 1)	-2.5000e-4	-2.5000e-4	< 1e-7%	1e-9 rel.	pass	matches
patch-wedge6	u_y at the corner (1, 1, 1)	0.001	0.001	< 1e-7%	1e-9 rel.	pass	matches
patch-wedge6	u_z at the corner (1, 1, 1)	0.00225	0.00225	< 1e-7%	1e-9 rel.	pass	matches
patch-wedge6	σ_xx at the inner node (0.46, 0.55, 0.53)	1	1	< 1e-7%	1e-9 rel.	pass	matches
patch-wedge6	σ_yy at the inner node (0.46, 0.55, 0.53)	2	2	< 1e-7%	1e-9 rel.	pass	matches
patch-wedge6	σ_zz at the inner node (0.46, 0.55, 0.53)	3	3	< 1e-7%	1e-9 rel.	pass	matches
patch-wedge6	σ_xy at the inner node	0	≈ 0	≈ 0	± 1.0000e-9	pass	matches
pure-bending-coupling	bending stress σ_xx at mid-span, top fibre (10, 0.5, 1)	15	15	-0.0000103%	0.5%	pass	matches
pure-bending-coupling	rotation of the reference point about y	0.3	0.30355	1.18%	1%	known failure: 1.2% high: the distributing coupling weights every node of the end face equally, corner and mid-side nodes of hex20 alike, which is not the consistent linear traction; the end face deforms locally and the fitted rotation includes it. Area-consistent weights would remove it	—
rotating-cylinder-axisymmetric	radial stress σ_r on the axis (r = 0)	0.38571	0.38608	0.0958%	1%	pass	matches
rotating-cylinder-axisymmetric	hoop stress σ_θ at the surface (r = 1)	0.12857	0.12873	0.123%	1%	pass	matches
rotating-disc-plane-stress	hoop stress σ_θ = σ_yy at the bore (20, 0)	65.312	65.188	-0.19%	1%	pass	matches
rotating-disc-plane-stress	radial stress σ_r = σ_xx at r = √(ab), its maximum	20.724	20.683	-0.2%	1%	pass	matches
rotating-disc-plane-stress	hoop stress σ_θ = σ_yy at the rim (100, 0)	16.328	16.352	0.149%	1%	pass	matches
tube-torsion-coupling	rotation of the reference point about z	0.056071	0.056071	0.000619%	0.5%	pass	matches
tube-torsion-coupling	surface shear stress τ_θz = σ_yz at (10, 0, 20)	1078.3	1081.4	0.288%	1%	pass	matches

Published benchmarks

Case	Quantity	Reference	FieldCraft	Error	Tolerance	Result	Fixture
mh-curved-cantilever	tip displacement along the in-plane load	0.08734	0.076833	-12%	2%	known failure: 0.880 of the theory, grade C: straight-sided hex8 on six elements over 90° (MacNeal–Harder's HEXA(8): 0.880, Table 9 p. 15)	—
mh-curved-cantilever	tip displacement along the out-of-plane load	0.5022	0.40095	-20.2%	2%	known failure: 0.798 of the theory, grade D: out-of-plane bending with torsion on one element across (MacNeal–Harder's HEXA(8): 0.849)	—
mh-solid-patch	σ_xx at inner node 7 (0.788, 0.693, 0.644)	2000	2000	< 1e-7%	1e-9 rel.	pass	matches
mh-solid-patch	σ_yy at inner node 7 (0.788, 0.693, 0.644)	2000	2000	< 1e-7%	1e-9 rel.	pass	matches
mh-solid-patch	σ_zz at inner node 7 (0.788, 0.693, 0.644)	2000	2000	< 1e-7%	1e-9 rel.	pass	matches
mh-solid-patch	τ_xy at inner node 7	400	400	< 1e-7%	1e-9 rel.	pass	matches
mh-solid-patch	τ_xz at inner node 7	400	400	< 1e-7%	1e-9 rel.	pass	matches
mh-solid-patch	τ_yz at inner node 7	400	400	< 1e-7%	1e-9 rel.	pass	matches
mh-solid-patch	u_x at inner node 7	0.0014565	0.0014565	< 1e-7%	1e-9 rel.	pass	matches
mh-straight-cantilever	tip displacement along the load, extension	3.0000e-5	2.9629e-5	-1.24%	2%	pass	matches
mh-straight-cantilever	tip displacement along the load, in-plane shear (width 0.2)	0.1081	0.10574	-2.18%	2%	known failure: 0.978 of the theory, grade B: one hex8 with incompatible modes through the depth of the prescribed mesh (MacNeal–Harder's HEXA(8): 0.981, Table 8(a) p. 15); refined meshes converge (cantilever-hex8-incompatible)	—
mh-straight-cantilever	tip displacement along the load, out-of-plane shear (depth 0.1)	0.4321	0.42037	-2.71%	2%	known failure: 0.973 of the theory, grade B, as for in-plane shear (MacNeal–Harder's HEXA(8): 0.981)	—
mh-straight-cantilever	tip rotation under a unit twisting moment	0.03208	0.028612	-10.8%	2%	known failure: 0.892 of the theory, grade C: one element across the section cannot represent the torsion warping (MacNeal–Harder's HEXA(8): 0.910, HEX20: 0.904)	—
mh-thick-cylinder	radial displacement at the inner radius, ν = 0.49	0.0050399	0.0049705	-1.38%	2%	pass	matches
mh-thick-cylinder	radial displacement at the inner radius, ν = 0.499	0.0050602	0.0049892	-1.4%	2%	pass	matches
mh-thick-cylinder	radial displacement at the inner radius, ν = 0.4999	0.0050623	0.0049911	-1.41%	2%	pass	matches
nafems-fv32	frequency of mode 1 (Hz)	44.623	44.623	-0.000219%	1%	pass	matches
nafems-fv32	frequency of mode 2 (Hz)	130.03	130.03	0.00356%	1%	pass	matches
nafems-fv32	frequency of mode 3 (Hz)	162.7	162.7	-0.00108%	1%	pass	matches
nafems-fv32	frequency of mode 4 (Hz)	246.05	246.05	0.00159%	1%	pass	matches
nafems-fv32	frequency of mode 5 (Hz)	379.9	379.9	0.000182%	1%	pass	matches
nafems-fv32	frequency of mode 6 (Hz)	391.44	391.44	-0.0012%	1%	pass	matches
nafems-le1	σ_yy at D (2, 0)	92.7	92.869	0.182%	1%	pass	matches
nafems-le1-plane-stress	σ_yy at D (2, 0)	92.7	92.648	-0.0559%	1%	pass	matches
nafems-le10	σ_yy at D (2, 0, 0.3) on the loaded face	-5.38	-5.3339	0.857%	2%	pass	matches
scordelis-lo-roof-solid	vertical displacement at the midpoint of the free edge, mid-surface	-0.3024	-0.3014	0.33%	2%	pass	matches

Sources and tolerances

• bar-pressure-wave (theory): Elastic wave in a bar from a pressure step (explicit). Source: d'Alembert's solution of the one-dimensional wave equation: c = √(E/ρ), particle velocity v = p/(ρc); Graff, Wave Motion in Elastic Solids, Dover 1991, §2.1. Tolerances — u_end: the end moves at the particle velocity p/(ρc) from the start; the discrete wave's ringing behind the front changes the end's displacement by well under 1%; v_mid: central differences ring behind a step front (dispersion); bulk viscosity damps it within a few elements, leaving about 1% twenty elements behind; balance: external work = kinetic + internal + dissipated energy; the solver warns above 1%
• bar-rigid-wall-impact (theory): Elastic bar striking a rigid wall (explicit). Source: One-dimensional elastic impact on a rigid wall: contact time 2L/c, rebound at the impact speed, contact stress ρcv₀; Graff, Wave Motion in Elastic Solids, Dover 1991, §2.3; Johnson, Impact Strength of Materials, Arnold 1972, §1.6. Tolerances — u_front: the contact ends at 2L/c in the elastic theory; the discrete release front is spread over a few elements, a few tenths of a percent of the time, and the rebound speed is v₀ to about 1%; u_back: as for the struck end; ke: an elastic impact keeps the kinetic energy; quadratic bulk viscosity in the compression front and the vibration left in the bar take about 1%
• cantilever-frequencies (theory): Natural frequencies of a cantilever (Euler–Bernoulli). Source: Euler–Bernoulli beam: f_n = (β_n L)²/(2π L²)·√(EI/ρA), β₁L = 1.8751, β₂L = 4.6941; Timoshenko, Young and Weaver, Vibration Problems in Engineering, 4th ed., Wiley 1974, §5.3. Tolerances — f1: Euler–Bernoulli neglects shear and rotary inertia, which lower f1 by about 0.2% at L/h = 20; f1_pair: as f1: a square section bends alike in y and z; f2: shear and rotary inertia lower the second mode by about 1% at L/h = 20 (Timoshenko)
• cantilever-hex20 (theory): Cantilever under a tip load, Timoshenko deflection (hex20). Source: Timoshenko and Gere, Mechanics of Materials (beam with shear deformation: δ = PL³/3EI + PL/κGA, κ = 5/6 for a rectangle); Cowper, J. Appl. Mech. 33 (1966) 335–340 for the shear coefficient. Tolerances — tip_uz: beam theory vs 3D elasticity: the shear coefficient and the fully clamped end change the tip deflection by a few tenths of a percent at L/h = 10; reaction: equilibrium holds to rounding
• cantilever-hex8-incompatible (theory): Cantilever under a tip load, Timoshenko deflection (hex8-incompatible). Source: Timoshenko and Gere, Mechanics of Materials (beam with shear deformation: δ = PL³/3EI + PL/κGA, κ = 5/6 for a rectangle); Cowper, J. Appl. Mech. 33 (1966) 335–340 for the shear coefficient. Tolerances — tip_uz: beam theory vs 3D elasticity (a few tenths of a percent at L/h = 10); incompatible modes represent the bending exactly in the constant-moment part; reaction: equilibrium holds to rounding
• cantilever-tet10 (theory): Cantilever under a tip load, Timoshenko deflection (tet10). Source: Timoshenko and Gere, Mechanics of Materials (beam with shear deformation: δ = PL³/3EI + PL/κGA, κ = 5/6 for a rectangle); Cowper, J. Appl. Mech. 33 (1966) 335–340 for the shear coefficient. Tolerances — tip_uz: beam theory vs 3D elasticity (a few tenths of a percent at L/h = 10); quadratic tetrahedra are converged on this mesh; reaction: equilibrium holds to rounding
• cdt-cylinder (theory): Constrained Delaunay mesh of a cylinder: volume within the chordal bound, valid elements. Source: mesh invariants: an inscribed polygon of sagitta s loses at most 2s/R of a circle's area (elementary geometry); constrained Delaunay tetrahedralization after Diazzi et al., ACM TOG 42(6) 2023. Tolerances — volume: the tessellation inscribes the cylinder within the chordal tolerance s = 0.05: the polygon section loses between 0 and 2s/R = 0.5% of the area; the reference is the middle of that band; invalid: every element of a valid mesh has positive volume
• cdt-stepped-block (theory): Constrained Delaunay mesh of a stepped block: exact volume, valid elements. Source: mesh invariants: a tetrahedralization of a polyhedron fills its volume exactly (Si, TetGen paper ACM TOMS 41(2) 2015 for the definition of a CDT; Diazzi et al., ACM TOG 42(6) 2023); the volume is 60·40·10 + 40·40·10 + 20·40·10 − 10·10·10. Tolerances — volume: a planar boundary is tessellated exactly and the constrained Delaunay mesh fills it exactly: only rounding; invalid: every element of a valid mesh has positive volume; undetermined: a linear tetrahedron's sign is exact
• free-beam-frequencies (theory): Natural frequencies of a free beam: rigid-body modes and the first bending mode. Source: Euler–Bernoulli free–free beam, β₁L = 4.7300; Timoshenko, Young and Weaver, Vibration Problems in Engineering, 4th ed., Wiley 1974, §5.3. Tolerances — rigid_6: a free body has six zero frequencies; numerical rigid-body frequencies below 10⁻³ of the first elastic one (eigenvalues below 10⁻⁶ of it) are rounding of the shifted solve; f7: Euler–Bernoulli neglects shear and rotary inertia, about 0.8% at L/h = 20 for the free–free mode; mass: the mass matrix integrates the density exactly
• j2-uniaxial-explicit (theory): Uniaxial tension of a J2 bilinear-hardening cube to large strain (explicit). Source: von Mises plasticity with linear isotropic hardening under uniaxial stress (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3); logarithmic strain for a hypoelastic–plastic law under coaxial stretching. Tolerances — sigma: uniaxial J2 with linear hardening: σ = σ_y + E_t (ε − σ_y/E) in logarithmic strain; the stretch takes 10⁴ wave transits, so inertia and the ringing of the start (ρcv ≈ 0.1% of σ) stay well under 1%; plastic_strain: ε_p = ε − σ/E, as for the stress
• kirsch-plate (theory): Plate with a circular hole in tension (Kirsch). Source: Kirsch, Zeitschrift des VDI 42 (1898) 797–807; Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §35. Tolerances — kt: the reference is the infinite plate; the finite plate (W = 60a, Heywood's width correction about +0.1%, and the loaded ends 30a away) raises the peak by a few tenths of a percent, and the stress extrapolated to the hole surface adds up to about 1%; side: infinite-plate value −S; finite width and extrapolation as for Kt, on a smaller value
• kirsch-plate-plane-stress (theory): Plate with a circular hole in tension, 2D plane stress (Kirsch). Source: Kirsch, Zeitschrift des VDI 42 (1898) 797–807; Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §35. Tolerances — kt: the reference is the infinite plate; the finite plate (W = 60a, loaded ends 30a away) raises the peak by a few tenths of a percent, and the stress extrapolated to the hole surface adds up to about 1%; side: infinite-plate value −S; finite width and extrapolation as for Kt, on a smaller value
• lame-cylinder (theory): Thick-walled cylinder under internal pressure, plane strain (Lamé). Source: Lamé's solution, Timoshenko and Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill 1970, §28 (thick-walled cylinder). Tolerances — ur_a: quadratic elements on 8 radial divisions resolve u_r = A r + B/r to well under 0.1%; ur_b: as for the bore; hoop_a: stress extrapolated from the integration points to the bore surface and averaged over two elements: about 0.5% at the steepest gradient
• lame-cylinder-axisymmetric (theory): Thick-walled cylinder under internal pressure, axisymmetric (Lamé). Source: Lamé's solution, Timoshenko and Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill 1970, §28. Tolerances — ur_a: quadratic elements on 8 radial divisions resolve u_r = A r + B/r to well under 0.1%; ur_b: as at the bore; hoop_a: stress extrapolated to the bore surface: about 0.5%; axial: as for the hoop stress
• lame-cylinder-plane-strain (theory): Thick-walled cylinder under internal pressure, 2D plane strain (Lamé). Source: Lamé's solution, Timoshenko and Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill 1970, §28. Tolerances — ur_a: quadratic elements on 8 radial divisions resolve u_r = A r + B/r to well under 0.1%; hoop_a: stress extrapolated to the bore surface and averaged over two elements: about 0.5%; szz: σ_zz = ν(σ_r + σ_θ) follows the in-plane stresses, extrapolated as they are
• mh-curved-cantilever (benchmark): MacNeal–Harder curved cantilever, hex8 with incompatible modes. Source: MacNeal and Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1 (1985) 3–20: Fig. 5 (p. 7) geometry, material and mesh; Table 3 (p. 10) theoretical tip displacements. Tolerances — in_plane: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14); out_of_plane: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14)
• mh-solid-patch (benchmark): MacNeal–Harder patch test for solids (hex8 with incompatible modes). Source: MacNeal and Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1 (1985) 3–20: Fig. 3 (p. 6) geometry and inner node locations, Table 2(c) (p. 10) boundary displacements and theoretical solution. Tolerances — sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; txy: a patch test is passed exactly: the tolerance only admits rounding; txz: a patch test is passed exactly: the tolerance only admits rounding; tyz: a patch test is passed exactly: the tolerance only admits rounding; u7: a patch test is passed exactly: the tolerance only admits rounding
• mh-straight-cantilever (benchmark): MacNeal–Harder straight cantilever, regular hex8 mesh with incompatible modes. Source: MacNeal and Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1 (1985) 3–20: Fig. 4(a) (p. 7) geometry, material and mesh; Table 3 (p. 10) theoretical tip displacements. Tolerances — extension: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14); in_plane: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14); out_of_plane: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14); twist: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14); the moment reaches the end through a distributing coupling
• mh-thick-cylinder (benchmark): MacNeal–Harder thick-walled cylinder, nearly incompressible (hex8 with incompatible modes). Source: MacNeal and Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1 (1985) 3–20: Fig. 10 (p. 9) geometry, material and radial mesh; Table 5(c) (p. 11) radial displacements at the inner radius. Tolerances — ur_1: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14); ur_2: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14); ur_3: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14)
• nafems-fv32 (benchmark): NAFEMS FV32 cantilevered tapered membrane, free vibration (hex20 plane stress layer). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test FV32 (frequencies 44.623, 130.03, 162.70, 246.05, 379.90, 391.44 Hz); as reproduced in the Abaqus 2017 Benchmarks Guide, 'FV32: Cantilevered tapered membrane'. Tolerances — f1: targets published to five figures for 2D plane stress; a thin solid layer with a free face differs from plane stress by O((t/L)²) and the 16 × 8 quadratic mesh is converged to about 0.1%; f2: targets published to five figures for 2D plane stress; a thin solid layer with a free face differs from plane stress by O((t/L)²) and the 16 × 8 quadratic mesh is converged to about 0.1%; f3: targets published to five figures for 2D plane stress; a thin solid layer with a free face differs from plane stress by O((t/L)²) and the 16 × 8 quadratic mesh is converged to about 0.1%; f4: targets published to five figures for 2D plane stress; a thin solid layer with a free face differs from plane stress by O((t/L)²) and the 16 × 8 quadratic mesh is converged to about 0.1%; f5: targets published to five figures for 2D plane stress; a thin solid layer with a free face differs from plane stress by O((t/L)²) and the 16 × 8 quadratic mesh is converged to about 0.1%; f6: targets published to five figures for 2D plane stress; a thin solid layer with a free face differs from plane stress by O((t/L)²) and the 16 × 8 quadratic mesh is converged to about 0.1%
• nafems-le1 (benchmark): NAFEMS LE1 elliptic membrane (hex20 plane stress layer). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE1 (target σ_yy = 92.7 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE1: Plane stress elements—elliptic membrane'. Tolerances — syy_D: the target is published to three figures; plane stress modelled by a thin solid layer and the stress extrapolated to the edge add a few tenths of a percent
• nafems-le1-plane-stress (benchmark): NAFEMS LE1 elliptic membrane with 2D plane stress elements (quad8). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE1 (target σ_yy = 92.7 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE1: Plane stress elements—elliptic membrane'. Tolerances — syy_D: the target is published to three figures; the stress extrapolated to the edge on a converged mesh adds a few tenths of a percent
• nafems-le10 (benchmark): NAFEMS LE10 thick plate under pressure (hex20). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE10 (target σ_yy = −5.38 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE10: Thick plate under pressure'. Tolerances — syy_D: the target is published to three figures; at D the stress is extrapolated to a corner where the hole edge meets the loaded face, and published converged solutions scatter by 1–2% (Abaqus Benchmarks Guide fine meshes: −5.51 to −5.89)
• nl-cylinder-inflation (theory): Inflation of a thick neo-Hookean cylinder by a follower pressure (plane strain). Source: Finite inflation of an incompressible neo-Hookean thick-walled tube in plane strain: r² = R² + a² − A², σ_θ − σ_r = μ (λ² − λ⁻²) with λ = r/R, integrated through the wall: p = (μ/2)[(ln λ_a² − λ_a⁻²) − (ln λ_b² − λ_b⁻²)] (R. S. Rivlin, Large elastic deformations of isotropic materials VI, Phil. Trans. R. Soc. A 242:173–195, 1949; R. W. Ogden, Non-Linear Elastic Deformations, Dover 1997, §5.3.2). Tolerances — ua: the closed form is incompressible and our material nearly so (κ/μ = 1000, a volume change of order p/κ ≈ 4·10⁻⁴); hybrid hex20, 6 × 16, follow the wall's strain gradient well inside; ub: as at the bore
• nl-elastica-cantilever (theory): Large-deflection cantilever under a dead tip load against the elastica (solids). Source: The elastica of a cantilever under a vertical end load: K. E. Bisshopp, D. C. Drucker, Large deflection of cantilever beams, Quarterly of Applied Mathematics 3(3):272–275, 1945 (elliptic-integral solution); values computed by integrating θ″ = −(P/EI) cos θ by shooting (our own RK4, 20000 steps), agreeing with the table of K. Mattiasson, Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals, IJNME 17(1):145–153, 1981, to all six digits. Tolerances — v1: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u1: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v2: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u2: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v5: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u5: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v10: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u10: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's
• nl-neo-hookean-uniaxial (theory): Uniaxial stretch of a neo-Hookean block to twice its length. Source: Closed form of the compressible neo-Hookean model W = μ/2 (Ī₁ − 3) + κ/2 (J − 1)² in uniaxial stress (G. A. Holzapfel, Nonlinear Solid Mechanics, Wiley 2000, §6.5, eq. 6.148 with the volumetric term of §6.4): σ_yy = μ J^(−5/3)(l² − Ī/3) + κ(J − 1) = 0 fixes the lateral stretch l, then σ_xx = μ J^(−5/3)(λ² − l²). Tolerances — sigma: a homogeneous deformation every element represents exactly: the tolerance only admits rounding and the Newton tolerance; lateral: a homogeneous deformation every element represents exactly: the tolerance only admits rounding and the Newton tolerance
• nl-patch-hex8 (theory): Nonlinear patch test, hex8: a homogeneous finite deformation of a distorted mesh. Source: The patch test (B. M. Irons, A. Razzaque, in A. K. Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, 557–587) at finite strain: a boundary moved by x = F X makes F the solution everywhere (J. Bonet, R. D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd ed., Cambridge 2008, §8); the neo-Hookean Cauchy stress σ = μ J^(−5/3) dev b + κ (J − 1) I of that F. Tolerances — ux: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; uy: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; uz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; sxx: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; syy: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; szz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; sxy: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; sxz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; syz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance
• nl-rigid-rotation (theory): A quarter turn strains and stresses nothing (objectivity). Source: Objectivity of the total Lagrangian formulation and of the hyperelastic stress: a rigid motion x = R X + c gives C = I and σ = 0 whatever the rotation (Bonet & Wood 2008, §3.4 and §5.2; T. Belytschko, W. K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, 2nd ed., Wiley 2014, §5.5). Tolerances — ux: the inner nodes follow the rigid rotation exactly; uy: the inner nodes follow the rigid rotation exactly; uz: the inner nodes follow the rigid rotation exactly; von_mises: zero up to rounding of stresses of order E = 1000: a 90° turn leaves no strain
• patch-hex20 (theory): Constant-stress patch test, hex20. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
• patch-hex20-reduced (theory): Constant-stress patch test, hex20-reduced. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
• patch-hex8 (theory): Constant-stress patch test, hex8. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
• patch-hex8-incompatible (theory): Constant-stress patch test, hex8-incompatible. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
• patch-tet10 (theory): Constant-stress patch test, tet10. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
• patch-tet4 (theory): Constant-stress patch test, tet4. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
• patch-wedge6 (theory): Constant-stress patch test, wedge6. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
• pure-bending-coupling (theory): Pure bending through a distributing coupling. Source: Euler–Bernoulli pure bending, σ = M_y z/I, θ = ML/(EI) (exact in elasticity for ν = 0); Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §102. Tolerances — sigma_top: with ν = 0 pure bending is a quadratic displacement field that hex20 represents exactly; the coupling's nodal force pattern differs from the linear traction only within a depth or two of the end (Saint-Venant), five depths from the probe; rotation: the reference point follows the best-fit rigid motion of the end section, which the end zone's local deformation disturbs by well under 1%
• rotating-cylinder-axisymmetric (theory): Long rotating solid cylinder, axisymmetric. Source: Timoshenko and Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill 1970, §32 (long rotating cylinder, ε_z = 0). Tolerances — radial_axis: stress extrapolated to the axis, where σ_r = σ_θ: under 1% with sixteen quadratic elements across; hoop_rim: stress extrapolated to the surface: under 1%
• rotating-disc-plane-stress (theory): Rotating disc with a central hole, plane stress. Source: Timoshenko and Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill 1970, §32 (rotating disc of uniform thickness with a hole). Tolerances — hoop_bore: stress extrapolated to the bore surface, where the a²b²/r² term is steepest: 16 radial quadratic elements graded towards the bore leave about 0.2% (8 uniform: −2.4%; 32 graded: −0.05%); radial_max: the nearest node to √(ab) is within half an element of it, where σ_r is stationary (an error of order (h/r)², under 0.5%), and nodal averaging adds a few tenths of a percent; hoop_rim: as at the bore
• scordelis-lo-roof-solid (benchmark): Scordelis–Lo roof with one layer of hex20 reduced-integration solids. Source: MacNeal and Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1 (1985) 3–20: Fig. 8 (p. 8) geometry, material and load; Table 5(a) (p. 11) reference displacement 0.3024; Belytschko et al., Computer Methods in Applied Mechanics and Engineering 51 (1985) 221–258 (shell obstacle course). Tolerances — uz: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14); MacNeal and Harder normalise by 0.3024 (the deep-shell value quoted in their [5] is 0.3086), and a solid model of R/t = 100 differs from shell theory by O(t/R)
• tube-torsion-coupling (theory): Torsion of a tube through a kinematically coupled reference point. Source: Saint-Venant torsion of a circular tube, φ = TL/(GJ), τ = Tr/J with J = π(b⁴ − a⁴)/2; Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §104. Tolerances — twist: Saint-Venant torsion of a circular tube is exact in elasticity; the rigid end coupling only stiffens a short zone near the tip; quadratic elements with 16 divisions around are within 0.1%; shear: stress extrapolated to the outer surface and averaged where two blocks meet: about 0.3% with four radial quadratic elements (two gave 1.1%)

Pending cases

Reference values already cited; each case gets its recipe when the capability it needs lands.

Case	Kind	Problem	Needs	Reference values	Source
cylinder-channel-2d1	benchmark	Flow around a cylinder, steady 2D-1 (Re = 20)	lattice Boltzmann CFD (M12)	drag coefficient (published interval 5.57–5.59) = 5.58; lift coefficient (interval 0.0104–0.0110) = 0.0107	Schäfer and Turek, 'Benchmark computations of laminar flow around a cylinder', Notes on Numerical Fluid Mechanics 52 (1996) 547–566, Table 4 (reference intervals)
lid-driven-cavity-re100	benchmark	Lid-driven cavity, Re = 100	lattice Boltzmann CFD (M12)	u/U at y = 0.4531 on the vertical centreline = -0.2109	Ghia, Ghia and Shin, Journal of Computational Physics 48 (1982) 387–411, Table I (u along the vertical centreline)
mh-twisted-beam	benchmark	MacNeal–Harder twisted beam	shell elements (M8) or mapped blocks with helical edges	tip displacement along the in-plane load = 0.005424; tip displacement along the out-of-plane load = 0.001754	MacNeal and Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1 (1985) 3–20, Fig. 6 (p. 8), Table 3 (p. 10)
nafems-le11	benchmark	NAFEMS LE11 solid cylinder/taper/sphere under a temperature field	thermal loads (M8 heat transfer)	σ_zz at A = -105	NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE11 (target σ_zz = −105 MPa at A)
nl-cook-membrane	benchmark	Cook's membrane, nearly incompressible neo-Hookean, large load (plane strain)	the published finite-strain reference value, transcribed from the paper with its load and material constants	vertical displacement of the top corner C (48, 60) = to be transcribed	R. D. Cook, Improved two-dimensional finite element, J. Struct. Div. ASCE 100(ST9):1851–1863, 1974 (the tapered panel (0, 0), (48, 44), (48, 60), (0, 44), clamped on x = 0, shear on x = 48); finite-strain versions with a neo-Hookean material are published by J. C. Simo, F. Armero, IJNME 33:1413–1449, 1992; the values to compare with are to be transcribed from the paper
pinched-cylinder	benchmark	Pinched cylinder with rigid diaphragms	shell elements (M8)	radial displacement under the load = -1.8248e-5	Belytschko, Stolarski, Liu, Carpenter and Ong, 'Stress projection for membrane and shear locking in shell finite elements', Computer Methods in Applied Mechanics and Engineering 51 (1985) 221–258 (the shell obstacle course); reference 1.8248 × 10⁻⁵ under the load
pinched-hemisphere	benchmark	Hemispherical shell with an 18° hole under alternating point loads	shell elements (M8), or mapped blocks that follow doubly curved faces	displacement under the load along the load = 0.094	MacNeal and Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1 (1985) 3–20, Fig. 9 (p. 9), Table 5(b) (p. 11): 0.0940 (lower bound 0.0924 from their [7])
plastic-thick-cylinder	theory	Elastic–perfectly plastic thick cylinder: collapse pressure	implicit (static) J2 plasticity: explicit J2 is in, a collapse load needs the static solver	collapse pressure for a = 10, b = 20, σ_y = 250 = 200.09	Hill, The Mathematical Theory of Plasticity, Oxford 1950, ch. V §2: plane strain von Mises collapse pressure p = (2/√3)σ_y ln(b/a)
scordelis-lo-roof-shell	benchmark	Scordelis–Lo roof with shell elements	shell elements (MITC4, M8)	vertical displacement at the midpoint of the free edge = -0.3024	MacNeal and Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1 (1985) 3–20, Fig. 8 (p. 8), Table 5(a) (p. 11); Belytschko, Stolarski, Liu, Carpenter and Ong, 'Stress projection for membrane and shear locking in shell finite elements', Computer Methods in Applied Mechanics and Engineering 51 (1985) 221–258 (the shell obstacle course)
taylor-bar	experiment	Taylor bar impact (copper)	the measured final shapes, to be transcribed from the paper (explicit Johnson–Cook plasticity is in)	final over initial length = to be transcribed	Wilkins and Guinan, 'Impact of cylinders on a rigid boundary', Journal of Applied Physics 44 (1973) 1200–1206 (final length and mushroom diameter vs impact speed)