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jyvikoandClaude Opus 5.5 96c9992c92 oracle: 2D cases (Kirsch plane stress, Lamé plane strain and axisymmetric, rotating disc and cylinder, NAFEMS LE1 with quad8) and J2 uniaxial explicit 
The pending 2D LE1 case becomes a real one; the plasticity cases stay
pending with what they still need (static J2 for the collapse pressure,
the transcribed measurements for the Taylor bar).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
2026-10-13 21:42:19 -04:00

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# 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 ModelCraft 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.
**117 of 123 quantities pass** (6 known failure(s)) in 35 active cases: 26 theory, 9 published benchmark, 0 experiment; 9 pending cases wait for capabilities still being built.
## Theory: closed-form solutions, patch tests and mesh invariants
| Case | Quantity | Reference | ModelCraft | Error | Tolerance | Result | Fixture |
|---|---|---|---|---|---|---|---|
| [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 |
| [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 | 0.00000000402% | 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 | -0.00000000116% | 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 | 0.00000000174% | 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 | < 1e-12 | exact | pass | matches |
| [cdt-stepped-block](../tests/oracle/cdt-stepped-block/) | volume of the tetrahedra | 47000 | 47000 | < 1e-10% | 1e-12 rel. | pass | matches |
| [cdt-stepped-block](../tests/oracle/cdt-stepped-block/) | elements that fail the certified Jacobian check | 0 | 0 | < 1e-12 | exact | pass | matches |
| [cdt-stepped-block](../tests/oracle/cdt-stepped-block/) | elements the Jacobian check could not decide | 0 | 0 | < 1e-12 | exact | pass | matches |
| [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | sixth frequency (the last rigid-body mode) | 0 | 1.5486e-7 | 1.5e-7 | ± 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-10% | 1e-12 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 |
| [patch-hex20](../tests/oracle/patch-hex20/) | u_x at the corner (1, 1, 1) | -2.5000e-4 | -2.5000e-4 | < 1e-10% | 1e-9 rel. | pass | matches |
| [patch-hex20](../tests/oracle/patch-hex20/) | u_y at the corner (1, 1, 1) | 0.001 | 0.001 | < 1e-10% | 1e-9 rel. | pass | matches |
| [patch-hex20](../tests/oracle/patch-hex20/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-10% | 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-10% | 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [patch-hex20](../tests/oracle/patch-hex20/) | σ_xy at the inner node | 0 | 5.8265e-16 | < 1e-12 | ± 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-10% | 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-10% | 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-10% | 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-10% | 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [patch-hex20-reduced](../tests/oracle/patch-hex20-reduced/) | σ_xy at the inner node | 0 | -3.7386e-16 | < 1e-12 | ± 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-10% | 1e-9 rel. | pass | matches |
| [patch-hex8](../tests/oracle/patch-hex8/) | u_y at the corner (1, 1, 1) | 0.001 | 1.0000e-3 | < 1e-10% | 1e-9 rel. | pass | matches |
| [patch-hex8](../tests/oracle/patch-hex8/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-10% | 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-10% | 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [patch-hex8](../tests/oracle/patch-hex8/) | σ_xy at the inner node | 0 | -1.5987e-16 | < 1e-12 | ± 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-10% | 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-10% | 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-10% | 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-10% | 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [patch-hex8-incompatible](../tests/oracle/patch-hex8-incompatible/) | σ_xy at the inner node | 0 | -2.1491e-16 | < 1e-12 | ± 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [patch-tet10](../tests/oracle/patch-tet10/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-10% | 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-10% | 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [patch-tet10](../tests/oracle/patch-tet10/) | σ_xy at the inner node | 0 | 2.7299e-15 | < 1e-12 | ± 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [patch-tet4](../tests/oracle/patch-tet4/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-10% | 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-10% | 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [patch-tet4](../tests/oracle/patch-tet4/) | σ_xy at the inner node | 0 | 2.3336e-17 | < 1e-12 | ± 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-10% | 1e-9 rel. | pass | matches |
| [patch-wedge6](../tests/oracle/patch-wedge6/) | u_y at the corner (1, 1, 1) | 0.001 | 0.001 | < 1e-10% | 1e-9 rel. | pass | matches |
| [patch-wedge6](../tests/oracle/patch-wedge6/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-10% | 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-10% | 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [patch-wedge6](../tests/oracle/patch-wedge6/) | σ_xy at the inner node | 0 | -8.3791e-17 | < 1e-12 | ± 1.0000e-9 | 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 | ModelCraft | 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-10% | 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-10% | 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-10% | 1e-9 rel. | pass | matches |
| [mh-solid-patch](../tests/oracle/mh-solid-patch/) | τ_xy at inner node 7 | 400 | 400 | < 1e-10% | 1e-9 rel. | pass | matches |
| [mh-solid-patch](../tests/oracle/mh-solid-patch/) | τ_xz at inner node 7 | 400 | 400 | < 1e-10% | 1e-9 rel. | pass | matches |
| [mh-solid-patch](../tests/oracle/mh-solid-patch/) | τ_yz at inner node 7 | 400 | 400 | < 1e-10% | 1e-9 rel. | pass | matches |
| [mh-solid-patch](../tests/oracle/mh-solid-patch/) | u_x at inner node 7 | 0.0014565 | 0.0014565 | < 1e-10% | 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
- **[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%
- **[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
- **[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-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)
- **[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)
- **[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) |
| [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) |
| [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]) |
| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | theory | Elastic–perfectly plastic thick cylinder: collapse pressure | implicit (static) J2 plasticity: explicit J2 is in, a collapse load needs the static solver | collapse pressure for a = 10, b = 20, σ_y = 250 = 200.09 | Hill, The Mathematical Theory of Plasticity, Oxford 1950, ch. V §2: plane strain von Mises collapse pressure p = (2/√3)σ_y ln(b/a) |
| [scordelis-lo-roof-shell](../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) |
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