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.