OneCAD-Simulation/docs/oracle.md

# Verification and validation oracle

Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the policy is [verification.md](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.

**225 of 231 quantities pass** (6 known failure(s)) in 65 active cases: 56 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 |
|---|---|---|---|---|---|---|---|
| [arc-shallow-arch](../tests/oracle/arc-shallow-arch/) | upper limit load factor (snap-through) | 1 | 0.99183 | -0.817% | 2% | pass | matches |
| [arc-shallow-arch](../tests/oracle/arc-shallow-arch/) | lower limit load factor | 0.21104 | 0.21895 | 7.9e-3 | ± 0.02 | pass | matches |
| [arc-shallow-arch](../tests/oracle/arc-shallow-arch/) | negative eigenvalues of the tangent past the upper limit point | 1 | 1 | ≈ 0 | exact | pass | matches |
| [bar-pressure-wave](../tests/oracle/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](../tests/oracle/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](../tests/oracle/bar-pressure-wave/) | energy balance error | 0 | 0.001527 | 1.5e-3 | ± 0.01 | pass | matches |
| [bar-rigid-wall-impact](../tests/oracle/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](../tests/oracle/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](../tests/oracle/bar-rigid-wall-impact/) | kinetic energy after the rebound over the initial ½mv₀² | 1 | 0.99463 | -0.537% | 2% | pass | matches |
| [buckle-euler-cantilever](../tests/oracle/buckle-euler-cantilever/) | first buckling stress (load factor of a unit end pressure) | 2.0461 | 2.0424 | -0.181% | 0.5% | pass | matches |
| [buckle-euler-pinned](../tests/oracle/buckle-euler-pinned/) | first buckling stress (load factor of a unit end pressure) | 2.0461 | 2.0424 | -0.179% | 0.5% | pass | matches |
| [buckle-plate-square](../tests/oracle/buckle-plate-square/) | buckling stress of a unit edge pressure | 0.36152 | 0.36013 | -0.386% | 1% | pass | matches |
| [cantilever-frequencies](../tests/oracle/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](../tests/oracle/cantilever-frequencies/) | the other first bending frequency | 0.012771 | 0.012751 | -0.158% | 1% | pass | matches |
| [cantilever-frequencies](../tests/oracle/cantilever-frequencies/) | second bending frequency | 0.080034 | 0.079167 | -1.08% | 2% | pass | matches |
| [cantilever-hex20](../tests/oracle/cantilever-hex20/) | tip deflection u_z at (10, 0.5, 0.5) | -4.024 | -4.0235 | 0.0131% | 0.5% | pass | matches |
| [cantilever-hex20](../tests/oracle/cantilever-hex20/) | total reaction z | 1 | 1 | < 1e-7% | 1e-9 rel. | pass | matches |
| [cantilever-hex8-incompatible](../tests/oracle/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](../tests/oracle/cantilever-hex8-incompatible/) | total reaction z | 1 | 1 | < 1e-7% | 1e-9 rel. | pass | matches |
| [cantilever-tet10](../tests/oracle/cantilever-tet10/) | tip deflection u_z at (10, 0.5, 0.5) | -4.024 | -4.0233 | 0.018% | 0.5% | pass | matches |
| [cantilever-tet10](../tests/oracle/cantilever-tet10/) | total reaction z | 1 | 1 | < 1e-7% | 1e-9 rel. | pass | matches |
| [cdt-cylinder](../tests/oracle/cdt-cylinder/) | volume of the tetrahedra | 6267.5 | 6266.5 | -0.0152% | 0.25% | pass | matches |
| [cdt-cylinder](../tests/oracle/cdt-cylinder/) | elements that fail the certified Jacobian check | 0 | 0 | ≈ 0 | exact | pass | matches |
| [cdt-stepped-block](../tests/oracle/cdt-stepped-block/) | volume of the tetrahedra | 47000 | 47000 | < 1e-7% | 1e-12 rel. | pass | matches |
| [cdt-stepped-block](../tests/oracle/cdt-stepped-block/) | elements that fail the certified Jacobian check | 0 | 0 | ≈ 0 | exact | pass | matches |
| [cdt-stepped-block](../tests/oracle/cdt-stepped-block/) | elements the Jacobian check could not decide | 0 | 0 | ≈ 0 | exact | pass | matches |
| [explicit-bars-collision](../tests/oracle/explicit-bars-collision/) | time of the last contact (the bars touch from t = 0) against 2L/c = 0.2 | 0.2 | 0.20242 | 1.21% | 2% | pass | matches |
| [explicit-bars-collision](../tests/oracle/explicit-bars-collision/) | the largest force across the interface against ρcAv₀/2 = 0.05 | 0.05 | 0.05147 | 2.94% | 4% | pass | matches |
| [explicit-bars-collision](../tests/oracle/explicit-bars-collision/) | velocity of the struck bar's far end after the impact (0: the striker stops) | 0 | -0.0033987 | -3.4e-3 | ± 0.005 | pass | matches |
| [explicit-bars-collision](../tests/oracle/explicit-bars-collision/) | velocity of the struck bar's far end after the impact (v₀ = 0.1) | 0.1 | 0.1034 | 3.4% | 5% | pass | matches |
| [explicit-bars-collision](../tests/oracle/explicit-bars-collision/) | kinetic energy after the impact over ½mv₀² (frictionless elastic impact) | 1 | 0.99995 | -0.00468% | 1% | pass | matches |
| [explicit-bars-collision](../tests/oracle/explicit-bars-collision/) | energy left in the contact after the separation, over ½mv₀² (its drift) | 0 | -7.7435e-5 | -7.7e-5 | ± 0.001 | pass | matches |
| [explicit-contact-patch](../tests/oracle/explicit-contact-patch/) | σ_zz of the lower block's corner element under the interface (element 10) over that of the upper block's corner element over it (element 19) | 1 | 1.0001 | 0.0118% | 0.5% | pass | matches |
| [explicit-contact-patch](../tests/oracle/explicit-contact-patch/) | σ_zz of the lower block's centre element under the interface (element 14) over that of element 19 | 1 | 1.0003 | 0.0287% | 0.5% | pass | matches |
| [explicit-contact-patch](../tests/oracle/explicit-contact-patch/) | σ_zz of the lower block's bottom corner element (element 1) over that of element 19 | 1 | 1.0001 | 0.0134% | 0.5% | pass | matches |
| [explicit-contact-patch](../tests/oracle/explicit-contact-patch/) | σ_zz of the upper block's top far corner element (element 26) over that of element 19 | 1 | 0.99944 | -0.0564% | 0.5% | pass | matches |
| [explicit-contact-patch](../tests/oracle/explicit-contact-patch/) | the force across the interface over −σ_zz A (A = 1) | 1 | 1.0026 | 0.259% | 1% | pass | matches |
| [explicit-contact-patch](../tests/oracle/explicit-contact-patch/) | σ_xx of element 14 over its σ_zz (uniaxial stress: 0) | 0 | -3.6052e-4 | -3.6e-4 | ± 0.005 | pass | matches |
| [explicit-neo-hookean-implicit](../tests/oracle/explicit-neo-hookean-implicit/) | lateral displacement of the corner (1, 1, 1) along y, explicit over implicit | 1 | 0.99908 | -0.0922% | 0.5% | pass | matches |
| [explicit-neo-hookean-implicit](../tests/oracle/explicit-neo-hookean-implicit/) | σ_xx of element 1, explicit, over σ_xx at the node at (0, 0, 0), implicit | 1 | 1.0003 | 0.0255% | 0.5% | pass | matches |
| [explicit-sliding-block-contact](../tests/oracle/explicit-sliding-block-contact/) | distance travelled (top centre node) against ∫ v dv / (g μ(v)) = 2.01524 | 2.0152 | 2.0118 | -0.172% | 1% | pass | matches |
| [explicit-sliding-block-contact](../tests/oracle/explicit-sliding-block-contact/) | velocity of the top centre node after the stop | 0 | ≈ 0 | ≈ 0 | ± 0.03 | pass | matches |
| [explicit-sliding-block-wall](../tests/oracle/explicit-sliding-block-wall/) | distance travelled (top centre node) against v₀²/(2μg) = 1.5 | 1.5 | 1.4999 | -0.0053% | 1% | pass | matches |
| [explicit-sliding-block-wall](../tests/oracle/explicit-sliding-block-wall/) | velocity of the top centre node after the stop | 0 | ≈ 0 | ≈ 0 | ± 0.03 | pass | matches |
| [explicit-sliding-block-wall](../tests/oracle/explicit-sliding-block-wall/) | friction dissipation over ½mv₀² = 2.25 | 1 | 1.0012 | 0.121% | 1% | pass | matches |
| [filleted-bracket-fillet-stress](../tests/oracle/filleted-bracket-fillet-stress/) | largest nodal von Mises stress on the fillet face (MPa) | 213 | mesh-dependent | within 3% | 3% | pass | — |
| [filleted-bracket-fillet-stress](../tests/oracle/filleted-bracket-fillet-stress/) | total reaction x (N) | -2000 | -2000 | < 1e-7% | 1e-9 rel. | pass | matches |
| [filleted-bracket-fillet-stress-coarse](../tests/oracle/filleted-bracket-fillet-stress-coarse/) | largest nodal von Mises stress on the fillet face (MPa) | 213 | mesh-dependent | within 3% | 3% | pass | — |
| [filleted-bracket-fillet-stress-coarse](../tests/oracle/filleted-bracket-fillet-stress-coarse/) | total reaction x (N) | -2000 | -2000 | < 1e-7% | 1e-9 rel. | pass | matches |
| [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | sixth frequency (the last rigid-body mode) | 0 | ≈ 0 | ≈ 0 | ± 8.1264e-5 | pass | matches |
| [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | first free–free bending frequency (mode 7) | 0.081264 | 0.080618 | -0.795% | 1.5% | pass | matches |
| [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | total mass | 20 | 20 | < 1e-7% | 1e-12 rel. | pass | matches |
| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | axial stress back at zero strain after a pull to ε_a = 0.01 | -247.52 | -247.52 | < 1e-7% | 1e-6 rel. | pass | matches |
| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | accumulated equivalent plastic strain | 0.016089 | 0.016089 | < 1e-7% | 1e-6 rel. | pass | matches |
| [j2-heated-bar](../tests/oracle/j2-heated-bar/) | axial stress, ΔT = 300 (α ΔT = 3.6·10⁻³, 2.9 times the yield strain) | -254.65 | -254.65 | < 1e-7% | 1e-8 rel. | pass | matches |
| [j2-heated-bar](../tests/oracle/j2-heated-bar/) | equivalent plastic strain | 0.0023267 | 0.0023267 | < 1e-7% | 1e-7 rel. | pass | matches |
| [j2-uniaxial-explicit](../tests/oracle/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](../tests/oracle/j2-uniaxial-explicit/) | equivalent plastic strain | 0.19678 | 0.19678 | -0.00000113% | 1% | pass | matches |
| [kirsch-plate](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/lame-cylinder/) | radial displacement at the bore (10, 0, 0) | 0.0090794 | 0.0090793 | -0.000267% | 0.1% | pass | matches |
| [lame-cylinder](../tests/oracle/lame-cylinder/) | radial displacement outside (20, 0, 0) | 0.0057778 | 0.0057778 | -0.000301% | 0.1% | pass | matches |
| [lame-cylinder](../tests/oracle/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](../tests/oracle/lame-cylinder-axisymmetric/) | radial displacement at the bore (r = 10) | 0.0090794 | 0.0090793 | -0.000416% | 0.1% | pass | matches |
| [lame-cylinder-axisymmetric](../tests/oracle/lame-cylinder-axisymmetric/) | radial displacement outside (r = 20) | 0.0057778 | 0.0057778 | -0.000327% | 0.1% | pass | matches |
| [lame-cylinder-axisymmetric](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/lame-cylinder-plane-strain/) | axial stress σ_zz = 2νA at the bore (plane strain) | 20 | 20.083 | 0.415% | 1% | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the lower block's corner element under the interface (element 10) over that of the upper block's corner element over it (element 19) | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the lower block's centre element under the interface (element 14) over that of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the lower block's bottom corner element (element 1) over that of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the upper block's top far corner element (element 26) over that of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_xx of element 14 over its σ_zz (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-6 | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: the contact pressure at the secondary node (0.5, 0.5, 1) over −σ_zz of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the lower block's corner element under the interface (element 10) over that of the upper block's corner element over it (element 19) | 1 | 1 | 0.000151% | 0.02% | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the lower block's centre element under the interface (element 14) over that of element 19 | 1 | 0.99999 | -0.00129% | 0.02% | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the lower block's bottom corner element (element 1) over that of element 19 | 1 | 0.99999 | -0.000942% | 0.02% | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the upper block's top far corner element (element 26) over that of element 19 | 1 | 0.99999 | -0.000872% | 0.02% | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_xx of element 14 over its σ_zz (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 2.0000e-4 | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: the contact pressure at the secondary node (0.5, 0.5, 1) over −σ_zz of element 19 | 1 | 0.99998 | -0.00172% | 0.02% | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 0.02% | pass | matches |
| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of element 19 (uniaxial stress E ε = −10 for the 1% squeeze) | -10 | -9.9996 | 0.00426% | 0.01% | pass | matches |
| [nl-cylinder-inflation](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/nl-elastica-cantilever/) | tip shortening δ_h/L at P L²/(E I) = 10 | 0.555 | 0.558 | 0.541% | 2% | pass | matches |
| [nl-friction-sliding-blocks](../tests/oracle/nl-friction-sliding-blocks/) | the tangential over the normal force across the sliding interface | 0.2 | 0.2 | -0.000103% | 0.1% | pass | matches |
| [nl-friction-sliding-blocks](../tests/oracle/nl-friction-sliding-blocks/) | the tangential contact force on the upper block over the sideways reaction at its top (they balance) | -1 | -1 | 0.00000224% | 1e-6 rel. | pass | matches |
| [nl-friction-sliding-blocks](../tests/oracle/nl-friction-sliding-blocks/) | the secondary nodes that slip (all 9) | 9 | 9 | ≈ 0 | exact | pass | matches |
| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the largest contact pressure p₀ (at x = 0) | 2884.6 | 2892 | 0.254% | 2% | pass | matches |
| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the force across the contact (half model, 1 mm thick) | 1132.8 | 1133 | 0.0221% | 0.2% | pass | matches |
| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.29594 (node 7) | 2325.1 | 2257.8 | -2.89% | 5% | pass | matches |
| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.36701 (node 8) | 1959 | 2078.4 | 6.09% | 10% | pass | matches |
| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.63325 (node 11, beyond the half-width b = 0.5) | 0 | ≈ 0 | ≈ 0 | ± 1 | pass | matches |
| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the vertical stress σ_yy under the centre of contact at depth 0.47127 (node 713) | -2099.2 | -2059.8 | 1.87% | 4% | pass | matches |
| [nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/) | axial Cauchy stress, stretch λ = 2, ν = 0.4999 | 3.4988 | 3.4988 | 0.000000135% | 1e-6 rel. | pass | matches |
| [nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/) | lateral displacement at (1, 1, 1): l − 1 | -0.29281 | -0.29281 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | axial Cauchy stress at a logarithmic strain of 0.2 | 642.74 | 642.74 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | equivalent plastic strain | 0.19678 | 0.19678 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | axial Cauchy stress, uniaxial stretch λ = 2 | 2.7366 | 2.7366 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | lateral displacement at (1, 1, 1), uniaxial: l − 1 | -0.28647 | -0.28647 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | in-plane Cauchy stress, equibiaxial stretch λ = 1.5 | 2.8427 | 2.8427 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | Cauchy stress along the stretch, pure shear λ = 2 | 3.5261 | 3.5261 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | Cauchy stress across, in the held direction, pure shear λ = 2 | 1.5067 | 1.5067 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-mooney-rivlin-tube](../tests/oracle/nl-mooney-rivlin-tube/) | radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 at λ_z = 1.5 | 5 | 5.0311 | 0.622% | 1% | pass | matches |
| [nl-mooney-rivlin-tube](../tests/oracle/nl-mooney-rivlin-tube/) | radial displacement of the outer wall (20, 0, 0): b − B, a small difference of radii | 0.61553 | 0.64339 | 2.8e-2 | ± 0.04 | pass | matches |
| [nl-neo-hookean-uniaxial](../tests/oracle/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](../tests/oracle/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](../tests/oracle/nl-patch-hex8/) | u_x at inner node 22 | 0.13583 | 0.13583 | < 1e-7% | 1e-8 rel. | pass | matches |
| [nl-patch-hex8](../tests/oracle/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](../tests/oracle/nl-patch-hex8/) | u_z at inner node 22 | -0.0023985 | -0.0023985 | -0.000000102% | 1e-8 rel. | pass | matches |
| [nl-patch-hex8](../tests/oracle/nl-patch-hex8/) | Cauchy σ_xx at inner node 22 | 42.296 | 42.296 | < 1e-7% | 1e-7 rel. | pass | matches |
| [nl-patch-hex8](../tests/oracle/nl-patch-hex8/) | Cauchy σ_yy at inner node 22 | 23.501 | 23.501 | < 1e-7% | 1e-7 rel. | pass | matches |
| [nl-patch-hex8](../tests/oracle/nl-patch-hex8/) | Cauchy σ_zz at inner node 22 | 31.453 | 31.453 | < 1e-7% | 1e-7 rel. | pass | matches |
| [nl-patch-hex8](../tests/oracle/nl-patch-hex8/) | Cauchy σ_xy at inner node 22 | -0.88971 | -0.88971 | < 1e-7% | 1e-7 rel. | pass | matches |
| [nl-patch-hex8](../tests/oracle/nl-patch-hex8/) | Cauchy σ_xz at inner node 22 | -1.8906 | -1.8906 | < 1e-7% | 1e-7 rel. | pass | matches |
| [nl-patch-hex8](../tests/oracle/nl-patch-hex8/) | Cauchy σ_yz at inner node 22 | 1.9462 | 1.9462 | < 1e-7% | 1e-7 rel. | pass | matches |
| [nl-rigid-rotation](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/nl-rigid-rotation/) | von Mises stress at inner node 22 | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-8 | pass | matches |
| [nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/) | displacement of the corner (1, 1, 1) along x, free, α ΔT = 0.2 | 0.2 | 0.2 | < 1e-7% | 1e-8 rel. | pass | matches |
| [nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/) | stress of the free body (σ_xx at the corner) | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-8 | pass | matches |
| [nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/) | stress of the held block, σ_xx (hydrostatic) | -3.5108 | -3.5108 | < 1e-7% | 1e-8 rel. | pass | matches |
| [nl-yeoh-uniaxial](../tests/oracle/nl-yeoh-uniaxial/) | axial Cauchy stress, stretch λ = 2 | 3.1743 | 3.1743 | < 1e-7% | 1e-6 rel. | pass | matches |
| [nl-yeoh-uniaxial](../tests/oracle/nl-yeoh-uniaxial/) | lateral displacement at (1, 1, 1): l − 1 | -0.28545 | -0.28545 | < 1e-7% | 1e-6 rel. | pass | matches |
| [octree-filleted-bracket](../tests/oracle/octree-filleted-bracket/) | x reaction on the clamped face (5 mm mesh) | -2000 | -2000 | < 1e-7% | 1e-6 rel. | pass | matches |
| [octree-filleted-bracket](../tests/oracle/octree-filleted-bracket/) | elements the certified Jacobian check does not prove valid (5 mm mesh) | 0 | 0 | ≈ 0 | exact | pass | matches |
| [octree-filleted-bracket](../tests/oracle/octree-filleted-bracket/) | mesh volume against the CAD volume (5 mm mesh) | 32145 | 32177 | 0.1% | 0.3% | pass | matches |
| [octree-filleted-bracket](../tests/oracle/octree-filleted-bracket/) | σ_yy on the outer face of the upright leg at y = 19, z = 6 (5 mm mesh) | 145.31 | 149.26 | 2.72% | 5% | pass | matches |
| [octree-filleted-bracket](../tests/oracle/octree-filleted-bracket/) | σ_yy on the inner face of the upright leg at y = 19, z = 6 (5 mm mesh) | -145.31 | -147.28 | -1.35% | 5% | pass | matches |
| [octree-filleted-bracket](../tests/oracle/octree-filleted-bracket/) | von Mises stress at the middle of the fillet, 7 mm mesh | 143.21 | 138.53 | -3.27% | 5% | pass | matches |
| [octree-filleted-bracket](../tests/oracle/octree-filleted-bracket/) | von Mises stress at the middle of the fillet, 5 mm mesh | 143.21 | 144.53 | 0.921% | 2% | pass | matches |
| [patch-hex20](../tests/oracle/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](../tests/oracle/patch-hex20/) | u_y at the corner (1, 1, 1) | 0.001 | 0.001 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-hex20](../tests/oracle/patch-hex20/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-hex20](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/patch-hex20/) | σ_xy at the inner node | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
| [patch-hex20-reduced](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/patch-hex20-reduced/) | σ_xy at the inner node | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
| [patch-hex8](../tests/oracle/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](../tests/oracle/patch-hex8/) | u_y at the corner (1, 1, 1) | 0.001 | 0.001 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-hex8](../tests/oracle/patch-hex8/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-hex8](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/patch-hex8/) | σ_xy at the inner node | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
| [patch-hex8-incompatible](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/patch-hex8-incompatible/) | σ_xy at the inner node | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
| [patch-tet10](../tests/oracle/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](../tests/oracle/patch-tet10/) | u_y at the corner (1, 1, 1) | 0.001 | 1.0000e-3 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-tet10](../tests/oracle/patch-tet10/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-tet10](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/patch-tet10/) | σ_xy at the inner node | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
| [patch-tet4](../tests/oracle/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](../tests/oracle/patch-tet4/) | u_y at the corner (1, 1, 1) | 0.001 | 1.0000e-3 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-tet4](../tests/oracle/patch-tet4/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-tet4](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/patch-tet4/) | σ_xy at the inner node | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
| [patch-wedge6](../tests/oracle/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](../tests/oracle/patch-wedge6/) | u_y at the corner (1, 1, 1) | 0.001 | 0.001 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-wedge6](../tests/oracle/patch-wedge6/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-7% | 1e-9 rel. | pass | matches |
| [patch-wedge6](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/patch-wedge6/) | σ_xy at the inner node | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
| [plastic-hinge-cantilever](../tests/oracle/plastic-hinge-cantilever/) | tip force at a tip deflection of 15 (4.5 times the first yield's) | -312.5 | -322.61 | -3.23% | 5% | pass | matches |
| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | bore displacement at the pressure that takes the plastic zone to c = 15 (a = 10, b = 20, σ_y = 250) | 0.024355 | 0.024358 | 0.0089% | 0.2% | pass | matches |
| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | radial force on the bore (per radian times 2π, height 1) at a bore displacement of 0.3, 24 times the first yield's: the collapse pressure 2k ln 2 = 200.094 times 2πa | 12572 | 12572 | 0.00000178% | 0.1% | pass | matches |
| [pure-bending-coupling](../tests/oracle/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](../tests/oracle/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](../tests/oracle/rotating-cylinder-axisymmetric/) | radial stress σ_r on the axis (r = 0) | 0.38571 | 0.38608 | 0.0958% | 1% | pass | matches |
| [rotating-cylinder-axisymmetric](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/tube-torsion-coupling/) | rotation of the reference point about z | 0.056071 | 0.056071 | 0.000619% | 0.5% | pass | matches |
| [tube-torsion-coupling](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/mh-solid-patch/) | τ_xy at inner node 7 | 400 | 400 | < 1e-7% | 1e-9 rel. | pass | matches |
| [mh-solid-patch](../tests/oracle/mh-solid-patch/) | τ_xz at inner node 7 | 400 | 400 | < 1e-7% | 1e-9 rel. | pass | matches |
| [mh-solid-patch](../tests/oracle/mh-solid-patch/) | τ_yz at inner node 7 | 400 | 400 | < 1e-7% | 1e-9 rel. | pass | matches |
| [mh-solid-patch](../tests/oracle/mh-solid-patch/) | u_x at inner node 7 | 0.0014565 | 0.0014565 | < 1e-7% | 1e-9 rel. | pass | matches |
| [mh-straight-cantilever](../tests/oracle/mh-straight-cantilever/) | tip displacement along the load, extension | 3.0000e-5 | 2.9629e-5 | -1.24% | 2% | pass | matches |
| [mh-straight-cantilever](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/mh-thick-cylinder/) | radial displacement at the inner radius, ν = 0.49 | 0.0050399 | 0.0049705 | -1.38% | 2% | pass | matches |
| [mh-thick-cylinder](../tests/oracle/mh-thick-cylinder/) | radial displacement at the inner radius, ν = 0.499 | 0.0050602 | 0.0049892 | -1.4% | 2% | pass | matches |
| [mh-thick-cylinder](../tests/oracle/mh-thick-cylinder/) | radial displacement at the inner radius, ν = 0.4999 | 0.0050623 | 0.0049911 | -1.41% | 2% | pass | matches |
| [nafems-fv32](../tests/oracle/nafems-fv32/) | frequency of mode 1 (Hz) | 44.623 | 44.623 | -0.000219% | 1% | pass | matches |
| [nafems-fv32](../tests/oracle/nafems-fv32/) | frequency of mode 2 (Hz) | 130.03 | 130.03 | 0.00356% | 1% | pass | matches |
| [nafems-fv32](../tests/oracle/nafems-fv32/) | frequency of mode 3 (Hz) | 162.7 | 162.7 | -0.00108% | 1% | pass | matches |
| [nafems-fv32](../tests/oracle/nafems-fv32/) | frequency of mode 4 (Hz) | 246.05 | 246.05 | 0.00159% | 1% | pass | matches |
| [nafems-fv32](../tests/oracle/nafems-fv32/) | frequency of mode 5 (Hz) | 379.9 | 379.9 | 0.000182% | 1% | pass | matches |
| [nafems-fv32](../tests/oracle/nafems-fv32/) | frequency of mode 6 (Hz) | 391.44 | 391.44 | -0.0012% | 1% | pass | matches |
| [nafems-le1](../tests/oracle/nafems-le1/) | σ_yy at D (2, 0) | 92.7 | 92.869 | 0.182% | 1% | pass | matches |
| [nafems-le1-plane-stress](../tests/oracle/nafems-le1-plane-stress/) | σ_yy at D (2, 0) | 92.7 | 92.648 | -0.0559% | 1% | pass | matches |
| [nafems-le10](../tests/oracle/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](../tests/oracle/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

- **[arc-shallow-arch](../tests/oracle/arc-shallow-arch/)** (theory): Snap-through of a shallow arch by arc length. Source: One-mode closed form of a shallow sinusoidal arch y₀ = a sin(πx/L) on pins at a fixed span under q₀ sin(πx/L): q₀ = EI k⁴ [c + (a − c)(2ac − c²)/(4r²)], k = π/L, limit points at c = a ∓ r √((a²/r² − 4)/3) (S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill 1961, §7.6; Y. C. Fung, A. Kaplan, NACA TN 2840, 1952): L = 100, h = 1, unit width, E = 10⁴, ν = 0, a/r = 3.5; the load is scaled to the upper limit load, so the peak load factor is 1 and the valley 0.211035. Tolerances — peak: 40 × 2 × 1 hex20 against shallow Euler–Bernoulli arch theory, the load piecewise constant per element, and the limit load picked from converged states (second order in the arc); valley: as the peak; the valley is a fifth of the peak load and a small difference of the bending and axial terms, so its tolerance is stated as 2 % of the peak load; negative: the inertia of the LDLᵀ factor: exactly one unstable direction between the limit points
- **[bar-pressure-wave](../tests/oracle/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](../tests/oracle/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%
- **[buckle-euler-cantilever](../tests/oracle/buckle-euler-cantilever/)** (theory): Euler buckling of a clamped–free column of solids. Source: Euler's column with Engesser's shear correction P = P_e/(1 + P_e/(κ G A)), κ = 5/6 (S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill 1961, §2.1, §2.17): square section 1 × 1, E = 1000, ν = 0, length 10 clamped–free, effective length 20, P_e = π² E I/20², so σ_cr = P/A = 2.046071. Tolerances — lambda: hex20, 2 × 2 across and 10 along: the 3D solid's bending and shear against beam theory, and the preload concentrated at the held face
- **[buckle-euler-pinned](../tests/oracle/buckle-euler-pinned/)** (theory): Euler buckling of a pinned column of solids. Source: Euler's column with Engesser's shear correction P = P_e/(1 + P_e/(κ G A)), κ = 5/6 (S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill 1961, §2.1, §2.17): square section 1 × 1, E = 1000, ν = 0, effective length 20, P_e = π² E I/20², so σ_cr = P/A = 2.046071. Tolerances — lambda: hex20, 2 × 2 across and 10 along: the 3D solid's bending and shear against beam theory, and the preload concentrated at the held face
- **[buckle-plate-square](../tests/oracle/buckle-plate-square/)** (theory): Buckling of a simply supported square plate in uniaxial compression. Source: Bryan's plate buckling stress σ_cr = k π² E/(12 (1 − ν²)) (t/b)² with k = 4 for a square plate simply supported on four edges (G. H. Bryan, Proc. London Math. Soc. 22:54–67, 1891; Timoshenko & Gere 1961, §9.2): b = 50, t = 0.5, E = 1000, ν = 0.3, σ_cr = 0.361524. Tolerances — lambda: a quarter plate of 10 × 10 × 2 hex20 (b/t = 100) supported on its bottom edge lines: the solid against thin-plate theory (−1.5 % at b/t = 50, falling with t/b) and the discretisation
- **[cantilever-frequencies](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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
- **[explicit-bars-collision](../tests/oracle/explicit-bars-collision/)** (theory): Two identical elastic bars colliding (explicit contact). Source: One-dimensional elastic impact of equal bars: contact for 2L/c, interface force ρcAv₀/2, exchange of velocities, no loss of energy; Graff, Wave Motion in Elastic Solids, Dover 1991, §2.1–2.3; Johnson, Impact Strength of Materials, Arnold 1972, §1.6. Tolerances — duration: the penalty springs at the interface delay the release by a few element transit times (about 1% of 2L/c with 200 elements per bar); force: the plateau is ρcAv₀/2 to 0.01%; the peak is the plateau plus the ringing behind the discrete wave front, a few percent; v_a: the striker stops on average (its kinetic energy is under 0.01% of the start's, see ke); its far end node keeps the ringing of the discrete release front, a few percent of v₀; v_b: the struck bar leaves at v₀ on average; its far end node keeps the ringing of the discrete wave fronts, a few percent; ke: an elastic impact keeps the kinetic energy; the quadratic bulk viscosity at the compression fronts and the vibration left in the bars take a fraction of a percent; contact_energy: the penalty springs give back what they took; the trapezoidal work of a spring that engages between increments drifts by far less than 0.1%
- **[explicit-contact-patch](../tests/oracle/explicit-contact-patch/)** (theory): Contact patch test: two blocks with non-matching meshes in uniaxial compression (explicit contact). Source: The contact patch test: a contact discretisation must transmit a uniform pressure across a non-matching interface exactly; Taylor & Papadopoulos, On a patch test for contact problems in two dimensions, Nonlinear Computational Mechanics, Springer 1991, 690–702; Crisfield, Re-visiting the contact patch test, IJNME 48:435–449, 2000. Uniaxial stress between frictionless symmetry planes is uniform in both blocks. Tolerances — lower_corner: a uniform stress is an equilibrium of the discrete system (the integrated contact forces are the consistent nodal forces of a uniform pressure on both meshes); what is left is the ringing of the loading, a hundred wave transits slow; lower_centre: as for the corner element; lower_base: as for the corner element; upper_top: as for the corner element; force: the integrated contact pressure carries the whole load; the largest force of the run is the final one in a slow squeeze, up to the ringing; lateral: frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them
- **[explicit-neo-hookean-implicit](../tests/oracle/explicit-neo-hookean-implicit/)** (theory): Neo-Hookean cube stretched to λ = 1.5: explicit dynamics, loaded slowly, against the implicit nonlinear solver. Source: Uniaxial stress of a compressible neo-Hookean solid (Holzapfel, Nonlinear Solid Mechanics, Wiley 2000, §6.5): a homogeneous state that a quasi-static explicit run must share with the implicit Newton solution on the same material point; the implicit solver is itself checked against the closed form by nl-neo-hookean-uniaxial. Tolerances — lateral: the same material point and a homogeneous state; the explicit run is pulled over a few hundred wave transits, and what is left of its waves is a fraction of a percent; stress: as for the displacement
- **[explicit-sliding-block-contact](../tests/oracle/explicit-sliding-block-contact/)** (theory): Block sliding to a stop on a deformable plate, friction decaying with the slip rate (explicit contact). Source: A rigid block launched at v₀ with friction μ(v) = μk + (μs − μk) e^(−d v) decelerates at g μ(v); it travels ∫₀^v₀ v dv / (g μ(v)) (Simpson's rule, 200 000 intervals) and stops after [v₀ + ln((μk + (μs − μk) e^(−d v₀))/μs)/d]/(g μk) = 1.2001; Meriam & Kraige, Engineering Mechanics: Dynamics, §3/5; the exponential decay law as in Oden & Martins, CMAME 52:527–634, 1985. Tolerances — distance: the block bounces on the penalty springs and the plate under it shears elastically, so the normal force and the slip rate it feels vary by a little around their rigid-body values; velocity: the block sticks; what is left is its elastic vibration, under 1% of v₀
- **[explicit-sliding-block-wall](../tests/oracle/explicit-sliding-block-wall/)** (theory): Block sliding to a stop on a rigid wall with Coulomb friction (explicit). Source: A rigid block launched at v₀ on a horizontal plane with Coulomb friction μ decelerates at μg and stops after v₀/(μg) at v₀²/(2μg), its kinetic energy all dissipated by friction; Meriam & Kraige, Engineering Mechanics: Dynamics, §3/5 (work and energy). Tolerances — distance: the wall's friction impulses are exact Coulomb impulses; the block's elastic vibration from the sudden gravity and the friction moment move the node by a fraction of a percent; velocity: the block sticks; what is left is its elastic vibration, under 1% of v₀; dissipated: the kinetic energy goes into friction; the vibration and the work of gravity as the block settles add or take a fraction of a percent
- **[filleted-bracket-fillet-stress](../tests/oracle/filleted-bracket-fillet-stress/)** (theory): Filleted L-bracket under a side load: the peak von Mises stress on the R6 fillet, fine tet10 mesh. Source: mesh convergence of the finite element solution (P. J. Roache, Verification and Validation in Computational Science and Engineering, Hermosa, 1998, ch. 5): the bracket has no closed-form stress; the reference is the mesh-independent peak of a four-level study of quadratic tetrahedral meshes refined by √2 in size and in curvature resolution (README.md); equilibrium. Tolerances — fillet_peak: the four-level study gives 213.3, 210.6, 214.3, 213.3 MPa (2k to 13.5k elements): the bending stress is converged across the fillet, and the peak nodal value moves only with which of the fillet's tessellation lines (fixed angles, through every level) carries the largest node, by under 2%; reaction: equilibrium holds to rounding
- **[filleted-bracket-fillet-stress-coarse](../tests/oracle/filleted-bracket-fillet-stress-coarse/)** (theory): Filleted L-bracket under a side load: the peak von Mises stress on the R6 fillet, coarse tet10 mesh. Source: mesh convergence of the finite element solution (P. J. Roache, Verification and Validation in Computational Science and Engineering, Hermosa, 1998, ch. 5): the bracket has no closed-form stress; the reference is the mesh-independent peak of a four-level study of quadratic tetrahedral meshes refined by √2 in size and in curvature resolution (README.md); equilibrium. Tolerances — fillet_peak: the four-level study gives 213.3, 210.6, 214.3, 213.3 MPa (2k to 13.5k elements): the bending stress is converged across the fillet, and the peak nodal value moves only with which of the fillet's tessellation lines (fixed angles, through every level) carries the largest node, by under 2%; reaction: equilibrium holds to rounding
- **[free-beam-frequencies](../tests/oracle/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-bauschinger](../tests/oracle/j2-bauschinger/)** (theory): Bauschinger effect of linear kinematic hardening: tension, then back to zero strain. Source: J2 plasticity with Prager's linear kinematic hardening in uniaxial stress (W. Prager, Ing.-Arch. 1955; Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3): from σ_a = σ_y + E_t (ε_a − σ_y/E) the material unloads elastically by 2σ_y and then hardens with E_t, so at zero strain σ = σ_a − 2σ_y − E_t (ε_a − 2σ_y/E). Tolerances — residual_stress: a homogeneous small-strain state the element represents exactly, with increments that end at the load path's turn; isotropic hardening would give −281.835 (it yields again only at −σ_a); plastic_strain: (σ_a − σ_y)/H forward plus E/(E + H) (ε_a − 2σ_y/E) in reverse
- **[j2-heated-bar](../tests/oracle/j2-heated-bar/)** (theory): A steel bar held at its ends and heated past yield (small strain). Source: Uniaxial J2 plasticity with linear hardening (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) with the additive thermal strain: the bar's total axial strain is held at zero, so its mechanical strain is −α ΔT and σ = −(σ_y + E_t (α ΔT − σ_y/E)) once α ΔT exceeds σ_y/E. Tolerances — sigma: a homogeneous uniaxial state the elements represent exactly; plastic_strain: (\|σ\| − σ_y)/H, as for the stress
- **[j2-uniaxial-explicit](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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-contact-patch](../tests/oracle/nl-contact-patch/)** (theory): Contact patch test: two blocks with non-matching meshes in uniaxial compression (implicit mortar contact). Source: The contact patch test: a contact discretisation must transmit a uniform pressure across a non-matching interface exactly; Taylor & Papadopoulos, On a patch test for contact problems in two dimensions, Nonlinear Computational Mechanics, Springer 1991, 690–702; Crisfield, Re-visiting the contact patch test, IJNME 48:435–449, 2000; passed by mortar methods: Puso & Laursen, CMAME 193:601–629, 2004. Uniaxial stress between frictionless symmetry planes is uniform in both blocks. Tolerances — small_lower_corner: small sliding is linear and its mortar integrals exact on these rectangular faces: a uniform stress is the discrete solution up to the convergence tolerance (10⁻¹⁰); small_lower_centre: as for the corner element; small_lower_base: as for the corner element; small_upper_top: as for the corner element; small_lateral: frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them; small_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; small_force: the pressure carries the whole load across the interface; finite_lower_corner: finite sliding is exact up to the geometric terms of the gaps the augmentation leaves (10⁻⁵ of the element thickness) and the default convergence tolerances; finite_lower_centre: as for the corner element; finite_lower_base: as for the corner element; finite_upper_top: as for the corner element; finite_lateral: frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them; finite_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; finite_force: the pressure carries the whole load across the interface; small_stress: the squeeze less the penetration the augmented Lagrangian leaves (up to 10⁻⁵ of the element thickness, a few 10⁻⁵ of the squeeze)
- **[nl-cylinder-inflation](../tests/oracle/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](../tests/oracle/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-friction-sliding-blocks](../tests/oracle/nl-friction-sliding-blocks/)** (theory): Coulomb friction: a block pressed on another and slid (implicit mortar contact). Source: Coulomb's law of dry friction: a sliding interface carries μ_k times the normal force against the slip (C. A. Coulomb, Théorie des machines simples, Mém. Math. Phys. Acad. Sci. 10:161–332, 1785; P. Wriggers, Computational Contact Mechanics, 2nd ed., Springer 2006, §5.2). Tolerances — kinetic: once the interface slips its friction is μ_k N exactly (the return map puts every slipping node's traction on the cone); the force opposes the upper block's motion; balance: the friction carries the whole sideways load; slipping: a sideways demand far beyond μ N slips the whole interface
- **[nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/)** (theory): Hertz line contact: a cylinder pressed on a flat (implicit mortar contact, plane strain). Source: H. Hertz, Über die Berührung fester elastischer Körper, J. reine angew. Math. 92:156–171, 1882; K. L. Johnson, Contact Mechanics, Cambridge University Press 1985, §4.3 (two-dimensional contact of cylindrical bodies, eqs. 4.45–4.49): b = √(4PR/(πE*)), p₀ = 2P/(πb), with 1/E* = 2(1 − ν²)/E for two bodies of one material and R the cylinder's radius on a flat. Tolerances — peak_pressure: p₀ = 2P/(πb) for the load per length P = 2265.57 N/mm, b = 0.5 mm; the contact width spans seven elements of the graded mesh; normal_force: the pressure carries the whole load; the soft springs that hold the indenter before contact take a millionth; pressure_x0296: p₀ √(1 − x²/b²) at the node; the nodal pressure (the node's normal force over the area it carries) averages the distribution over the node's elements; pressure_x0367: as at node 7, nearer the edge, where the elements round off the square-root fall of the pressure to zero at b; pressure_x0633: outside the contact the surfaces are apart: no pressure (an element beyond b, since the edge element carries part of the last closed node's pressure); syy_depth: σ_z = −p₀ / √(1 + z²/b²) on the axis of a half-plane (Johnson 1985, eq. 4.49); the foundation is ten half-widths wide and deep, not a half-plane, and the stress is the nodal average of the elements around the node
- **[nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/)** (theory): Nearly incompressible hybrid block under prescribed stretch. Source: Uniaxial stress of the decoupled compressible neo-Hookean energy W = μ/2 (Ī₁ − 3) + κ/2 (J − 1)² (P. J. Flory, Trans. Faraday Soc. 57:829–838, 1961): σ = μ J^(−5/3) (λ² − I₁/3) + κ (J − 1) with the lateral stretch found so σ_yy = 0 (bisection; E = 3, ν = 0.4999: μ = 1.0000667, κ = 5000). Tolerances — sigma: a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding; lateral: a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding
- **[nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/)** (theory): Uniaxial tension of a J2 bilinear-hardening cube to large strain (implicit, finite strain). Source: J2 plasticity with linear isotropic hardening in uniaxial stress (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) in the logarithmic strain, which Simo's exponential return map (CMAME 99:61–112, 1992) integrates exactly on a proportional path: Kirchhoff stress τ = σ_y + E_t (ε − σ_y/E), Cauchy stress σ = τ/J with ln J = τ/(3K) (Hencky's volume law). Tolerances — sigma: a homogeneous deformation the element represents exactly, reached on a proportional path: the tolerance only admits the Newton tolerance and rounding; plastic_strain: ε_p = (τ − σ_y)/H, as for the stress
- **[nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/)** (theory): Mooney–Rivlin block in uniaxial, equibiaxial and pure-shear tension. Source: Treloar's homogeneous modes of rubber testing (L. R. G. Treloar, The Physics of Rubber Elasticity, 3rd ed., Oxford 1975, ch. 10) for the decoupled compressible energy W = C₁₀ (Ī₁ − 3) + C₀₁ (Ī₂ − 3) + κ/2 (J − 1)²: principal Cauchy stresses σᵢ = (λᵢ/J) ∂W/∂λᵢ with the free stretch found so the free faces carry no stress (bisection; E = 3, ν = 0.49, C₀₁ = 0.2, C₁₀ = μ/2 − C₀₁). Tolerances — uniaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; uniaxial_lateral: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; equibiaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_xx: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_yy: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding
- **[nl-mooney-rivlin-tube](../tests/oracle/nl-mooney-rivlin-tube/)** (theory): Inflation of a stretched Mooney–Rivlin tube (follower pressure). Source: Finite inflation and extension of an incompressible Mooney–Rivlin tube (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): at axial stretch λ_z, r² = a² + (R² − A²)/λ_z, σ_θ − σ_r = 2 (λ_θ² − λ_r²)(C₁₀ + C₀₁ λ_z²) with λ_θ = r/R, λ_r = 1/(λ_θ λ_z), integrated through the wall: p = ∫ (σ_θ − σ_r) dr/r (Simpson, 2·10⁵ intervals). Tolerances — ua: 16 × 6 hybrid hex20 around the quarter; ν = 0.4995 stands in for incompressibility (a volume change of order p/κ ≈ 10⁻³ shifts the radii by about 0.1 %), and the tube is stretched axially by its ends, which a nearly incompressible solid takes with larger pressures than the plane-strain neo-Hookean tube; ub: 0.2 % of the outer radius b = 20.616: the same volume change of order 10⁻³ moves b by about 0.03, which is 4.5 % of b − B; the incompressible kinematics r² = a² + (R² − A²)/λ_z amplify it there
- **[nl-neo-hookean-uniaxial](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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
- **[nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/)** (theory): Free and fully constrained thermal expansion at finite strain. Source: The multiplicative split F = F_m F_θ with F_θ = (1 + α ΔT) I (V. A. Lubarda, Appl. Mech. Rev. 57(2):95–108, 2004): a free body follows F = F_θ stress-free; held on every face, F = I and F_m = ϑ⁻¹ I, so the compressible neo-Hookean material carries the pressure of its volume ratio, σ = κ (ϑ⁻³ − 1). Tolerances — free_u: a homogeneous stretch every element represents exactly; free_stress: rounding only; held_stress: a homogeneous state; E = 10, ν = 0.3 (κ = 8.3333), ϑ = 1.2
- **[nl-yeoh-uniaxial](../tests/oracle/nl-yeoh-uniaxial/)** (theory): Yeoh block in uniaxial tension. Source: Uniaxial tension of the decoupled compressible Yeoh energy W = C₁₀ (Ī₁ − 3) + C₂₀ (Ī₁ − 3)² + C₃₀ (Ī₁ − 3)³ + κ/2 (J − 1)² (O. H. Yeoh, Rubber Chem. Technol. 66(5):754–771, 1993): σ = (λ/J) ∂W/∂λ with the lateral stretch found so σ_yy = 0 (bisection; E = 3, ν = 0.49, C₁₀ = μ/2, C₂₀ = −0.02, C₃₀ = 0.004). Tolerances — sigma: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; lateral: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding
- **[octree-filleted-bracket](../tests/oracle/octree-filleted-bracket/)** (theory): Filleted L-bracket on the octree all-hex mesh: equilibrium, beam bending, fillet stress convergence. Source: equilibrium; Euler–Bernoulli bending of the 8 × 40 mm leg, σ = M c / I with M = F (50 − y) (S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed., 1970, §102); grid convergence of the fillet stress against the 4 mm mesh of this case (P. J. Roache, Verification and Validation in Computational Science and Engineering, 1998, ch. 5). Tolerances — reaction_x: the reactions balance the 2 kN load to solver precision; invalid: the octree mesher returns only elements proven valid; volume: bilinear boundary faces on the R6 fillet and the two Ø8 holes, and the 1e-3 chordal tessellation they are fitted to; syy_outer: beam theory ignores the corner 5 mm below and the hole 9 mm above; the probe takes the nearest node's extrapolated stress; syy_inner: as for the outer face; fillet_7: the coarsest mesh, within 5% of the 4 mm mesh; fillet_5: converging: within 2% of the 4 mm mesh
- **[patch-hex20](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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)
- **[plastic-hinge-cantilever](../tests/oracle/plastic-hinge-cantilever/)** (theory): Plastic collapse of a cantilever: the fully plastic moment at the root. Source: Limit analysis (B. G. Neal, The Plastic Methods of Structural Analysis, 3rd ed., Chapman & Hall 1977, ch. 1–2): beam theory's plastic hinge at the root, the fully plastic moment M_p = σ_y b h²/4 of a rectangular section, collapses a cantilever of length L at P_c = M_p/L = 312.5 (b = h = 10, L = 200, σ_y = 250); a bound for the solid, whose collapse lies above it (W. Prager, P. G. Hodge, Theory of Perfectly Plastic Solids, Wiley 1951: a continuum mechanism allows no normal velocity jump, which the beam's hinge rotation is). Tolerances — collapse_force: the solid's collapse lies above beam theory's hinge: refined through the depth (4 to 8 elements) and along the length (40 to 160) the force settles 3.1 % above M_p/L (322.3), so the band is that bound's gap; the root is a symmetry plane (no rotation, free lateral contraction) held on average through a coupling, the tip pushed through another; reduced hex20, 20 × 2 × 4
- **[plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/)** (theory): Elastic–perfectly plastic thick cylinder: plastic zone and collapse pressure. Source: Hill, The Mathematical Theory of Plasticity, Oxford 1950, ch. V §2: a von Mises cylinder in plane strain, elastically incompressible, with the plastic zone a ≤ r ≤ c under the pressure p = k (2 ln(c/a) + 1 − c²/b²), k = σ_y/√3, and the bore displacement u_a = k c²/(2 G a); collapse when c = b at p = 2k ln(b/a). Tolerances — u_bore: 20 reduced quad8 elements through the wall resolve the logarithmic plastic field and ν = 0.4999 stands in for Hill's incompressible elasticity (0.009 % apart); u_a ∝ c², so 0.2 % on u_a is 0.1 % on the plastic radius; collapse_force: once the wall is plastic through, perfect plasticity holds the force at the collapse value (σ_θ − σ_r = 2k everywhere, σ_z the mean of the others as plastic flow dominates); the band admits the quadrature of the logarithmic field
- **[pure-bending-coupling](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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) |
| [explicit-hertz-spheres](../tests/oracle/explicit-hertz-spheres/) | theory | Two elastic spheres in a head-on collision: Hertz's force and contact duration (explicit contact) | a graded sphere mesh from commands (a cubed-sphere hex mesher, or tet refinement towards a point); until then the explicit crate's tests `spheres_collide_as_hertz_says` (a graded cubed-sphere quarter of each sphere, 27 648 hex8: peak force +2.3%, duration −0.4%) and its coarse twin in the main gate check this case on a mesh generated in code | the largest contact force, two steel spheres R = 10 mm (E = 210 GPa, ν = 0.3, ρ = 7.85 t/m³) meeting at 1 m/s (N) = 1005.4; the contact duration (s) = 6.0163e-5 | Hertz's quasi-static impact: with R* = R/2, m* = m/2, E* = E/(2(1 − ν²)), the approach reaches δ = (15 m* v²/(16 E* √R*))^(2/5), the force (4/3) E* √R* δ^(3/2), and the contact lasts 2.9432 δ/v; Hertz, J. reine angew. Math. 92:156–171, 1882; K. L. Johnson, Contact Mechanics, Cambridge 1985, §11.4 |
| [lid-driven-cavity-re100](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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](../tests/oracle/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]) |
| [scordelis-lo-roof-shell](../tests/oracle/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](../tests/oracle/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) |