# Reference values Every benchmark's target, and where it comes from. A row is either **computed** (the example evaluates the closed form itself -- preferred) or **quoted** (a published constant, which needs a real citation and an independent check). | benchmark | quantity | reference | tolerance | source | |---|---|---|---|---| | `ex_analytic_compliance_gradient` | dC/dt (CSDL) / (-3 C/t) | 1 | 0.01 | Euler--Bernoulli pure-bending scaling: D = E t^3 / [12(1-nu^2)] | | `ex_cantilever_tip_load` | vertical displacement at the tip-edge midpoint | 0.000578843 | 0.02 | Computed Timoshenko beam formula in this file | | `ex_cantilever_tip_moment` | maximum relative error of tip deflection and y rotation | 0 | 1e-07 | Computed Euler-Bernoulli constant-curvature formulas in this file | | `ex_cantilever_uniform_pressure` | vertical displacement at the tip-edge midpoint | 0.00868333 | 0.02 | Computed Timoshenko beam formula in this file | | `ex_clamped_circular_plate` | centre transverse deflection | 17.0625 | 0.02 | Computed Kirchhoff--Love circular-plate solution, w=q a^4/(64 D) | | `ex_clamped_plate_uniform` | centre transverse deflection | 1.37592 | 0.01 | Timoshenko & Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (1959), Table 35, p. 202: quoted 0.00126 | | `ex_clt_abd` | maximum normalized A/B/D assembly discrepancy | 0 | 1e-10 | independent NumPy evaluation of the CLT thickness integrals in this file | | `ex_cooks_membrane` | vertical displacement at mid-right edge | 23.9 | 0.02 | Cook, 'Some elements for analysis of plane solid structures', Int. J. Numer. Methods Eng., 1974 | | `ex_crossply_bending` | midpoint transverse deflection | 0.000640869 | 0.06 | one-term Navier solution evaluated from the independently assembled D matrix | | `ex_curved_beam` | tip deflection in z | 0.5022 | 0.02 | MacNeal & Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1(1), 1985 | | `ex_elastic_energy_identity` | largest relative energy-identity residual for pressure and point loads | 0 | 1e-07 | Hermit outputs.py: elastic_energy and compliance conventions | | `ex_gradient_finite_difference` | 1 - best relative CSDL/FD gradient error | 1 | 0.002 | Second-order central-difference formula, evaluated by this script | | `ex_hypar_warped_quad` | global-z deflection at the centre of the free edge | 0.92118 | 0.02 | Computed in this file: triangle n=64 = 0.92117998; n=48 differs by 1.16e-4 relative on the same geometry. Triangles are planar and objective under the compared curvature formulations | | `ex_laminate_matches_isotropic` | maximum relative compliance, mass, or tip-displacement difference | 0 | 1e-08 | same-run hm.isotropic solution; identical isotropic CLT constitutive law | | `ex_mass_and_cg` | maximum absolute residual across uniform and graded exact mass/CG | 0 | 1e-10 | Closed-form area integrals computed in this file | | `ex_optimal_thickness_taper` | optimiser-recovered exponent p in t(x) = A ((L-x)/L)**p | 1 | 0.05 | Computed in this file: minimising int M^2/t^3 at fixed int t gives t proportional to sqrt(M), hence (L-x) under uniform load; cross-checked against a direct 1-D SLSQP solve to 2.4e-6 | | `ex_pinched_cylinder` | radial deflection under the point load | 1.8248e-05 | 0.02 | MacNeal & Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1(1), 1985 | | `ex_pinched_hemisphere` | radial deflection under the load | 0.0924 | 0.02 | MacNeal & Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1(1), 1985 | | `ex_pinched_hemisphere_quad` | radial deflection under the load | 0.0924 | 0.02 | MacNeal & Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1(1), 1985 | | `ex_scordelis_lo` | vertical deflection at free-edge midspan | 0.3024 | 0.02 | MacNeal & Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1(1), 1985 | | `ex_scordelis_lo_quad` | vertical deflection at free-edge midspan | 0.3024 | 0.02 | MacNeal & Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1(1), 1985 | | `ex_shape_derivative_direction_failure` | 1 - relative error between adjoint and FD | 1 | 0.005 | Tsai-Wu failure index with `fiber_direction` geometry sensitivity | | `ex_shape_derivative_fd` | 1 - best relative CSDL/FD shape-gradient error | 1 | 0.005 | UFL CoordinateDerivative; central differences evaluated by this script | | `ex_ss_plate_sinusoidal` | centre transverse deflection | 2.80261 | 0.02 | Computed Kirchhoff--Love single Navier mode | | `ex_ss_plate_uniform` | centre transverse deflection | 4.43609 | 0.02 | Computed Kirchhoff--Love odd Navier double series (801 terms/direction) | | `ex_thin_limit_shear` | FE / analytic Reissner--Mindlin centre-deflection ratio | 1 | 0.02 | Computed Reissner--Mindlin single mode: Kirchhoff bending + kappa G h shear | | `ex_tsai_wu_uniaxial` | maximum hand-recovered Tsai-Wu field or KS discrepancy | 0 | 1e-10 | Tsai--Wu coefficients and KS reduction independently evaluated from hermit/failure.py | | `ex_twisted_beam` | tip deflection in z | 0.005424 | 0.02 | MacNeal & Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1(1), 1985 | | `ex_warped_quad_consistency` | ratio of the warped-quad tip deflection to the triangle tip deflection | 1 | 0.02 | Measured on this fixture; the underlying twisted-beam reference is MacNeal & Harder, Finite Elements in Analysis and Design 1(1), 1985 | ## Checking a quoted constant Quoted values are the suite's weak point: a plausible-looking wrong number produces a benchmark that passes and verifies nothing. Each one gets an independent review pass that re-derives or re-sources it **without** seeing the original justification. Disagreements are resolved before the example lands, not by adjusting the tolerance. See `README.md`, *Reference-value integrity*.