Numerical Verification Guideanalytical & discrete referencesreproducible · pytest385 definitions

// numerical verification — not physical validation

Implementation claims,
made executable.

The engine executes 1331 scalar comparisons drawn from 385 problem definitions. References include analytical solutions, discrete governing equations, properties, and legacy literature targets. A PASS means only that the recorded quantity met its recorded threshold in this environment; it does not establish mesh independence or agreement with experiment. Run it yourself: python -m verification.suite.

1331

executed comparisons

385

case definitions

100%

passing (1331/1331)

Bars & axial members

112 comparisons112/112 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
BAR-0001	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.0001, P=1000	1.2500e-05	1.2500e-05	0.0e+00	✓
BAR-0002	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.0001, P=10000	1.2500e-04	1.2500e-04	0.0e+00	✓
BAR-0003	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.0001, P=50000	6.2500e-04	6.2500e-04	0.0e+00	✓
BAR-0004	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.0001, P=100000	0.00125	0.00125	0.0e+00	✓
BAR-0005	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.0005, P=1000	2.5000e-06	2.5000e-06	0.0e+00	✓
BAR-0006	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.0005, P=10000	2.5000e-05	2.5000e-05	0.0e+00	✓
BAR-0007	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.0005, P=50000	1.2500e-04	1.2500e-04	0.0e+00	✓
BAR-0008	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.0005, P=100000	2.5000e-04	2.5000e-04	0.0e+00	✓
BAR-0009	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.001, P=1000	1.2500e-06	1.2500e-06	0.0e+00	✓
BAR-0010	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.001, P=10000	1.2500e-05	1.2500e-05	0.0e+00	✓
BAR-0011	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.001, P=50000	6.2500e-05	6.2500e-05	0.0e+00	✓
BAR-0012	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.001, P=100000	1.2500e-04	1.2500e-04	0.0e+00	✓
BAR-0013	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.01, P=1000	1.2500e-07	1.2500e-07	0.0e+00	✓
BAR-0014	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.01, P=10000	1.2500e-06	1.2500e-06	0.0e+00	✓
BAR-0015	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.01, P=50000	6.2500e-06	6.2500e-06	0.0e+00	✓
BAR-0016	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.25, A=0.01, P=100000	1.2500e-05	1.2500e-05	0.0e+00	✓
BAR-0017	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.0001, P=1000	2.5000e-05	2.5000e-05	0.0e+00	✓
BAR-0018	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.0001, P=10000	2.5000e-04	2.5000e-04	0.0e+00	✓
BAR-0019	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.0001, P=50000	0.00125	0.00125	0.0e+00	✓
BAR-0020	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.0001, P=100000	0.0025	0.0025	0.0e+00	✓
BAR-0021	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.0005, P=1000	5.0000e-06	5.0000e-06	0.0e+00	✓
BAR-0022	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.0005, P=10000	5.0000e-05	5.0000e-05	0.0e+00	✓
BAR-0023	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.0005, P=50000	2.5000e-04	2.5000e-04	0.0e+00	✓
BAR-0024	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.0005, P=100000	5.0000e-04	5.0000e-04	0.0e+00	✓
BAR-0025	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.001, P=1000	2.5000e-06	2.5000e-06	0.0e+00	✓
BAR-0026	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.001, P=10000	2.5000e-05	2.5000e-05	0.0e+00	✓
BAR-0027	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.001, P=50000	1.2500e-04	1.2500e-04	0.0e+00	✓
BAR-0028	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.001, P=100000	2.5000e-04	2.5000e-04	0.0e+00	✓
BAR-0029	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.01, P=1000	2.5000e-07	2.5000e-07	0.0e+00	✓
BAR-0030	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.01, P=10000	2.5000e-06	2.5000e-06	0.0e+00	✓
BAR-0031	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.01, P=50000	1.2500e-05	1.2500e-05	0.0e+00	✓
BAR-0032	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=0.5, A=0.01, P=100000	2.5000e-05	2.5000e-05	0.0e+00	✓
BAR-0033	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.0001, P=1000	5.0000e-05	5.0000e-05	0.0e+00	✓
BAR-0034	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.0001, P=10000	5.0000e-04	5.0000e-04	0.0e+00	✓
BAR-0035	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.0001, P=50000	0.0025	0.0025	0.0e+00	✓
BAR-0036	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.0001, P=100000	0.005	0.005	0.0e+00	✓
BAR-0037	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.0005, P=1000	1.0000e-05	1.0000e-05	0.0e+00	✓
BAR-0038	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.0005, P=10000	1.0000e-04	1.0000e-04	0.0e+00	✓
BAR-0039	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.0005, P=50000	5.0000e-04	5.0000e-04	0.0e+00	✓
BAR-0040	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.0005, P=100000	0.001	0.001	0.0e+00	✓
BAR-0041	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.001, P=1000	5.0000e-06	5.0000e-06	0.0e+00	✓
BAR-0042	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.001, P=10000	5.0000e-05	5.0000e-05	0.0e+00	✓
BAR-0043	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.001, P=50000	2.5000e-04	2.5000e-04	0.0e+00	✓
BAR-0044	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.001, P=100000	5.0000e-04	5.0000e-04	0.0e+00	✓
BAR-0045	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.01, P=1000	5.0000e-07	5.0000e-07	0.0e+00	✓
BAR-0046	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.01, P=10000	5.0000e-06	5.0000e-06	0.0e+00	✓
BAR-0047	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.01, P=50000	2.5000e-05	2.5000e-05	0.0e+00	✓
BAR-0048	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=1.0, A=0.01, P=100000	5.0000e-05	5.0000e-05	0.0e+00	✓
BAR-0049	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.0001, P=1000	1.0000e-04	1.0000e-04	0.0e+00	✓
BAR-0050	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.0001, P=10000	0.001	0.001	0.0e+00	✓
BAR-0051	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.0001, P=50000	0.005	0.005	0.0e+00	✓
BAR-0052	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.0001, P=100000	0.01	0.01	0.0e+00	✓
BAR-0053	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.0005, P=1000	2.0000e-05	2.0000e-05	0.0e+00	✓
BAR-0054	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.0005, P=10000	2.0000e-04	2.0000e-04	0.0e+00	✓
BAR-0055	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.0005, P=50000	0.001	0.001	0.0e+00	✓
BAR-0056	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.0005, P=100000	0.002	0.002	0.0e+00	✓
BAR-0057	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.001, P=1000	1.0000e-05	1.0000e-05	0.0e+00	✓
BAR-0058	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.001, P=10000	1.0000e-04	1.0000e-04	0.0e+00	✓
BAR-0059	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.001, P=50000	5.0000e-04	5.0000e-04	0.0e+00	✓
BAR-0060	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.001, P=100000	0.001	0.001	0.0e+00	✓
BAR-0061	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.01, P=1000	1.0000e-06	1.0000e-06	0.0e+00	✓
BAR-0062	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.01, P=10000	1.0000e-05	1.0000e-05	0.0e+00	✓
BAR-0063	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.01, P=50000	5.0000e-05	5.0000e-05	0.0e+00	✓
BAR-0064	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=2.0, A=0.01, P=100000	1.0000e-04	1.0000e-04	0.0e+00	✓
BAR-0065	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.0001, P=1000	2.0000e-04	2.0000e-04	0.0e+00	✓
BAR-0066	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.0001, P=10000	0.002	0.002	0.0e+00	✓
BAR-0067	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.0001, P=50000	0.01	0.01	0.0e+00	✓
BAR-0068	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.0001, P=100000	0.02	0.02	0.0e+00	✓
BAR-0069	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.0005, P=1000	4.0000e-05	4.0000e-05	0.0e+00	✓
BAR-0070	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.0005, P=10000	4.0000e-04	4.0000e-04	0.0e+00	✓
BAR-0071	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.0005, P=50000	0.002	0.002	0.0e+00	✓
BAR-0072	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.0005, P=100000	0.004	0.004	0.0e+00	✓
BAR-0073	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.001, P=1000	2.0000e-05	2.0000e-05	0.0e+00	✓
BAR-0074	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.001, P=10000	2.0000e-04	2.0000e-04	0.0e+00	✓
BAR-0075	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.001, P=50000	0.001	0.001	0.0e+00	✓
BAR-0076	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.001, P=100000	0.002	0.002	0.0e+00	✓
BAR-0077	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.01, P=1000	2.0000e-06	2.0000e-06	0.0e+00	✓
BAR-0078	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.01, P=10000	2.0000e-05	2.0000e-05	0.0e+00	✓
BAR-0079	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.01, P=50000	1.0000e-04	1.0000e-04	0.0e+00	✓
BAR-0080	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=4.0, A=0.01, P=100000	2.0000e-04	2.0000e-04	0.0e+00	✓
BAR-0081	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.0001, P=1000	4.0000e-04	4.0000e-04	0.0e+00	✓
BAR-0082	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.0001, P=10000	0.004	0.004	0.0e+00	✓
BAR-0083	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.0001, P=50000	0.02	0.02	0.0e+00	✓
BAR-0084	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.0001, P=100000	0.04	0.04	0.0e+00	✓
BAR-0085	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.0005, P=1000	8.0000e-05	8.0000e-05	0.0e+00	✓
BAR-0086	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.0005, P=10000	8.0000e-04	8.0000e-04	0.0e+00	✓
BAR-0087	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.0005, P=50000	0.004	0.004	0.0e+00	✓
BAR-0088	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.0005, P=100000	0.008	0.008	0.0e+00	✓
BAR-0089	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.001, P=1000	4.0000e-05	4.0000e-05	0.0e+00	✓
BAR-0090	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.001, P=10000	4.0000e-04	4.0000e-04	0.0e+00	✓
BAR-0091	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.001, P=50000	0.002	0.002	0.0e+00	✓
BAR-0092	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.001, P=100000	0.004	0.004	0.0e+00	✓
BAR-0093	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.01, P=1000	4.0000e-06	4.0000e-06	0.0e+00	✓
BAR-0094	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.01, P=10000	4.0000e-05	4.0000e-05	0.0e+00	✓
BAR-0095	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.01, P=50000	2.0000e-04	2.0000e-04	0.0e+00	✓
BAR-0096	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=8.0, A=0.01, P=100000	4.0000e-04	4.0000e-04	0.0e+00	✓
BAR-0097	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.0001, P=1000	8.0000e-04	8.0000e-04	0.0e+00	✓
BAR-0098	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.0001, P=10000	0.008	0.008	0.0e+00	✓
BAR-0099	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.0001, P=50000	0.04	0.04	0.0e+00	✓
BAR-0100	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.0001, P=100000	0.08	0.08	0.0e+00	✓
BAR-0101	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.0005, P=1000	1.6000e-04	1.6000e-04	0.0e+00	✓
BAR-0102	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.0005, P=10000	0.0016	0.0016	0.0e+00	✓
BAR-0103	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.0005, P=50000	0.008	0.008	0.0e+00	✓
BAR-0104	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.0005, P=100000	0.016	0.016	0.0e+00	✓
BAR-0105	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.001, P=1000	8.0000e-05	8.0000e-05	0.0e+00	✓
BAR-0106	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.001, P=10000	8.0000e-04	8.0000e-04	0.0e+00	✓
BAR-0107	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.001, P=50000	0.004	0.004	0.0e+00	✓
BAR-0108	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.001, P=100000	0.008	0.008	0.0e+00	✓
BAR-0109	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.01, P=1000	8.0000e-06	8.0000e-06	0.0e+00	✓
BAR-0110	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.01, P=10000	8.0000e-05	8.0000e-05	0.0e+00	✓
BAR-0111	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.01, P=50000	4.0000e-04	4.0000e-04	0.0e+00	✓
BAR-0112	Prismatic bar under end load	u = PL / AE	Timoshenko, Strength of Materials I	L=16.0, A=0.01, P=100000	8.0000e-04	8.0000e-04	0.0e+00	✓

Beam bending

206 comparisons206/206 passmax err 3.6e-13

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
BEA-0113	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=0.5, I=8e-06, P=1000	2.4802e-05	2.4802e-05	1.3e-13	✓
BEA-0114	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=0.5, I=8e-06, P=1000	7.4405e-05	7.4405e-05	1.2e-13	✓
BEA-0115	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=0.5, I=8e-06, P=5000	1.2401e-04	1.2401e-04	1.3e-13	✓
BEA-0116	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=0.5, I=8e-06, P=5000	3.7202e-04	3.7202e-04	1.2e-13	✓
BEA-0117	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=0.5, I=8e-06, w=8000	3.7202e-05	3.7202e-05	2.8e-14	✓
BEA-0118	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=0.5, I=8e-06, M=4000	2.9762e-04	2.9762e-04	1.2e-13	✓
BEA-0119	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=0.5, I=4e-06, P=1000	4.9603e-05	4.9603e-05	1.3e-13	✓
BEA-0120	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=0.5, I=4e-06, P=1000	1.4881e-04	1.4881e-04	1.2e-13	✓
BEA-0121	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=0.5, I=4e-06, P=5000	2.4802e-04	2.4802e-04	1.3e-13	✓
BEA-0122	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=0.5, I=4e-06, P=5000	7.4405e-04	7.4405e-04	1.2e-13	✓
BEA-0123	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=0.5, I=4e-06, w=8000	7.4405e-05	7.4405e-05	2.8e-14	✓
BEA-0124	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=0.5, I=4e-06, M=4000	5.9524e-04	5.9524e-04	1.2e-13	✓
BEA-0125	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=0.5, I=2e-05, P=1000	9.9206e-06	9.9206e-06	6.8e-14	✓
BEA-0126	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=0.5, I=2e-05, P=1000	2.9762e-05	2.9762e-05	6.9e-14	✓
BEA-0127	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=0.5, I=2e-05, P=5000	4.9603e-05	4.9603e-05	6.8e-14	✓
BEA-0128	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=0.5, I=2e-05, P=5000	1.4881e-04	1.4881e-04	6.9e-14	✓
BEA-0129	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=0.5, I=2e-05, w=8000	1.4881e-05	1.4881e-05	1.5e-13	✓
BEA-0130	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=0.5, I=2e-05, M=4000	1.1905e-04	1.1905e-04	7.4e-14	✓
BEA-0131	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.0, I=8e-06, P=1000	1.9841e-04	1.9841e-04	1.3e-13	✓
BEA-0132	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.0, I=8e-06, P=1000	2.9762e-04	2.9762e-04	1.2e-13	✓
BEA-0133	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.0, I=8e-06, P=5000	9.9206e-04	9.9206e-04	1.3e-13	✓
BEA-0134	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.0, I=8e-06, P=5000	0.0014881	0.0014881	1.2e-13	✓
BEA-0135	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=1.0, I=8e-06, w=8000	5.9524e-04	5.9524e-04	2.8e-14	✓
BEA-0136	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=1.0, I=8e-06, M=4000	0.0011905	0.0011905	1.2e-13	✓
BEA-0137	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.0, I=4e-06, P=1000	3.9683e-04	3.9683e-04	1.3e-13	✓
BEA-0138	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.0, I=4e-06, P=1000	5.9524e-04	5.9524e-04	1.2e-13	✓
BEA-0139	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.0, I=4e-06, P=5000	0.0019841	0.0019841	1.3e-13	✓
BEA-0140	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.0, I=4e-06, P=5000	0.0029762	0.0029762	1.2e-13	✓
BEA-0141	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=1.0, I=4e-06, w=8000	0.0011905	0.0011905	2.8e-14	✓
BEA-0142	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=1.0, I=4e-06, M=4000	0.002381	0.002381	1.2e-13	✓
BEA-0143	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.0, I=2e-05, P=1000	7.9365e-05	7.9365e-05	6.8e-14	✓
BEA-0144	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.0, I=2e-05, P=1000	1.1905e-04	1.1905e-04	6.9e-14	✓
BEA-0145	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.0, I=2e-05, P=5000	3.9683e-04	3.9683e-04	6.8e-14	✓
BEA-0146	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.0, I=2e-05, P=5000	5.9524e-04	5.9524e-04	6.9e-14	✓
BEA-0147	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=1.0, I=2e-05, w=8000	2.3810e-04	2.3810e-04	1.5e-13	✓
BEA-0148	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=1.0, I=2e-05, M=4000	4.7619e-04	4.7619e-04	7.4e-14	✓
BEA-0149	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.5, I=8e-06, P=1000	6.6964e-04	6.6964e-04	1.6e-14	✓
BEA-0150	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.5, I=8e-06, P=1000	6.6964e-04	6.6964e-04	1.8e-14	✓
BEA-0151	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.5, I=8e-06, P=5000	0.0033482	0.0033482	1.6e-14	✓
BEA-0152	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.5, I=8e-06, P=5000	0.0033482	0.0033482	1.8e-14	✓
BEA-0153	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=1.5, I=8e-06, w=8000	0.0030134	0.0030134	1.1e-13	✓
BEA-0154	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=1.5, I=8e-06, M=4000	0.0026786	0.0026786	1.5e-14	✓
BEA-0155	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.5, I=4e-06, P=1000	0.0013393	0.0013393	1.6e-14	✓
BEA-0156	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.5, I=4e-06, P=1000	0.0013393	0.0013393	1.8e-14	✓
BEA-0157	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.5, I=4e-06, P=5000	0.0066964	0.0066964	1.6e-14	✓
BEA-0158	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.5, I=4e-06, P=5000	0.0066964	0.0066964	1.8e-14	✓
BEA-0159	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=1.5, I=4e-06, w=8000	0.0060268	0.0060268	1.1e-13	✓
BEA-0160	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=1.5, I=4e-06, M=4000	0.0053571	0.0053571	1.5e-14	✓
BEA-0161	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.5, I=2e-05, P=1000	2.6786e-04	2.6786e-04	6.6e-14	✓
BEA-0162	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.5, I=2e-05, P=1000	2.6786e-04	2.6786e-04	7.6e-14	✓
BEA-0163	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=1.5, I=2e-05, P=5000	0.0013393	0.0013393	6.6e-14	✓
BEA-0164	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=1.5, I=2e-05, P=5000	0.0013393	0.0013393	7.6e-14	✓
BEA-0165	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=1.5, I=2e-05, w=8000	0.0012054	0.0012054	3.6e-13	✓
BEA-0166	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=1.5, I=2e-05, M=4000	0.0010714	0.0010714	6.1e-14	✓
BEA-0167	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=2.0, I=8e-06, P=1000	0.0015873	0.0015873	2.0e-13	✓
BEA-0168	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=2.0, I=8e-06, P=1000	0.0011905	0.0011905	2.0e-13	✓
BEA-0169	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=2.0, I=8e-06, P=5000	0.0079365	0.0079365	2.0e-13	✓
BEA-0170	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=2.0, I=8e-06, P=5000	0.0059524	0.0059524	2.0e-13	✓
BEA-0171	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=2.0, I=8e-06, w=8000	0.0095238	0.0095238	1.7e-13	✓
BEA-0172	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=2.0, I=8e-06, M=4000	0.0047619	0.0047619	2.0e-13	✓
BEA-0173	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=2.0, I=4e-06, P=1000	0.0031746	0.0031746	2.0e-13	✓
BEA-0174	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=2.0, I=4e-06, P=1000	0.002381	0.002381	2.0e-13	✓
BEA-0175	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=2.0, I=4e-06, P=5000	0.015873	0.015873	2.0e-13	✓
BEA-0176	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=2.0, I=4e-06, P=5000	0.011905	0.011905	2.0e-13	✓
BEA-0177	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=2.0, I=4e-06, w=8000	0.019048	0.019048	1.7e-13	✓
BEA-0178	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=2.0, I=4e-06, M=4000	0.0095238	0.0095238	2.0e-13	✓
BEA-0179	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=2.0, I=2e-05, P=1000	6.3492e-04	6.3492e-04	8.1e-14	✓
BEA-0180	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=2.0, I=2e-05, P=1000	4.7619e-04	4.7619e-04	6.7e-14	✓
BEA-0181	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=2.0, I=2e-05, P=5000	0.0031746	0.0031746	8.1e-14	✓
BEA-0182	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=2.0, I=2e-05, P=5000	0.002381	0.002381	6.6e-14	✓
BEA-0183	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=2.0, I=2e-05, w=8000	0.0038095	0.0038095	8.8e-14	✓
BEA-0184	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=2.0, I=2e-05, M=4000	0.0019048	0.0019048	7.2e-14	✓
BEA-0185	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=3.0, I=8e-06, P=1000	0.0053571	0.0053571	3.1e-14	✓
BEA-0186	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=3.0, I=8e-06, P=1000	0.0026786	0.0026786	3.2e-14	✓
BEA-0187	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=3.0, I=8e-06, P=5000	0.026786	0.026786	3.2e-14	✓
BEA-0188	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=3.0, I=8e-06, P=5000	0.013393	0.013393	3.2e-14	✓
BEA-0189	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=3.0, I=8e-06, w=8000	0.048214	0.048214	2.0e-13	✓
BEA-0190	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=3.0, I=8e-06, M=4000	0.010714	0.010714	3.4e-14	✓
BEA-0191	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=3.0, I=4e-06, P=1000	0.010714	0.010714	3.1e-14	✓
BEA-0192	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=3.0, I=4e-06, P=1000	0.0053571	0.0053571	3.2e-14	✓
BEA-0193	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=3.0, I=4e-06, P=5000	0.053571	0.053571	3.2e-14	✓
BEA-0194	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=3.0, I=4e-06, P=5000	0.026786	0.026786	3.2e-14	✓
BEA-0195	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=3.0, I=4e-06, w=8000	0.096429	0.096429	2.0e-13	✓
BEA-0196	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=3.0, I=4e-06, M=4000	0.021429	0.021429	3.4e-14	✓
BEA-0197	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=3.0, I=2e-05, P=1000	0.0021429	0.0021429	2.0e-16	✓
BEA-0198	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=3.0, I=2e-05, P=1000	0.0010714	0.0010714	2.0e-16	✓
BEA-0199	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=3.0, I=2e-05, P=5000	0.010714	0.010714	4.9e-16	✓
BEA-0200	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=3.0, I=2e-05, P=5000	0.0053571	0.0053571	0.0e+00	✓
BEA-0201	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=3.0, I=2e-05, w=8000	0.019286	0.019286	4.9e-14	✓
BEA-0202	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=3.0, I=2e-05, M=4000	0.0042857	0.0042857	2.0e-16	✓
BEA-0203	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=4.0, I=8e-06, P=1000	0.012698	0.012698	3.5e-14	✓
BEA-0204	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=4.0, I=8e-06, P=1000	0.0047619	0.0047619	4.1e-14	✓
BEA-0205	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=4.0, I=8e-06, P=5000	0.063492	0.063492	3.5e-14	✓
BEA-0206	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=4.0, I=8e-06, P=5000	0.02381	0.02381	4.1e-14	✓
BEA-0207	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=4.0, I=8e-06, w=8000	0.15238	0.15238	8.6e-14	✓
BEA-0208	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=4.0, I=8e-06, M=4000	0.019048	0.019048	4.1e-14	✓
BEA-0209	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=4.0, I=4e-06, P=1000	0.025397	0.025397	3.5e-14	✓
BEA-0210	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=4.0, I=4e-06, P=1000	0.0095238	0.0095238	4.1e-14	✓
BEA-0211	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=4.0, I=4e-06, P=5000	0.12698	0.12698	3.5e-14	✓
BEA-0212	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=4.0, I=4e-06, P=5000	0.047619	0.047619	4.1e-14	✓
BEA-0213	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=4.0, I=4e-06, w=8000	0.30476	0.30476	8.6e-14	✓
BEA-0214	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=4.0, I=4e-06, M=4000	0.038095	0.038095	4.1e-14	✓
BEA-0215	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=4.0, I=2e-05, P=1000	0.0050794	0.0050794	7.7e-15	✓
BEA-0216	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=4.0, I=2e-05, P=1000	0.0019048	0.0019048	1.1e-14	✓
BEA-0217	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=4.0, I=2e-05, P=5000	0.025397	0.025397	7.8e-15	✓
BEA-0218	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=4.0, I=2e-05, P=5000	0.0095238	0.0095238	1.1e-14	✓
BEA-0219	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=4.0, I=2e-05, w=8000	0.060952	0.060952	2.3e-13	✓
BEA-0220	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=4.0, I=2e-05, M=4000	0.007619	0.007619	1.1e-14	✓
BEA-0221	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=5.0, I=8e-06, P=1000	0.024802	0.024802	3.9e-14	✓
BEA-0222	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=5.0, I=8e-06, P=1000	0.0074405	0.0074405	4.8e-14	✓
BEA-0223	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=5.0, I=8e-06, P=5000	0.12401	0.12401	3.9e-14	✓
BEA-0224	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=5.0, I=8e-06, P=5000	0.037202	0.037202	4.8e-14	✓
BEA-0225	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=5.0, I=8e-06, w=8000	0.37202	0.37202	2.3e-13	✓
BEA-0226	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=5.0, I=8e-06, M=4000	0.029762	0.029762	4.3e-14	✓
BEA-0227	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=5.0, I=4e-06, P=1000	0.049603	0.049603	3.9e-14	✓
BEA-0228	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=5.0, I=4e-06, P=1000	0.014881	0.014881	4.8e-14	✓
BEA-0229	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=5.0, I=4e-06, P=5000	0.24802	0.24802	3.9e-14	✓
BEA-0230	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=5.0, I=4e-06, P=5000	0.074405	0.074405	4.8e-14	✓
BEA-0231	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=5.0, I=4e-06, w=8000	0.74405	0.74405	2.3e-13	✓
BEA-0232	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=5.0, I=4e-06, M=4000	0.059524	0.059524	4.3e-14	✓
BEA-0233	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=5.0, I=2e-05, P=1000	0.0099206	0.0099206	1.2e-13	✓
BEA-0234	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=5.0, I=2e-05, P=1000	0.0029762	0.0029762	1.1e-13	✓
BEA-0235	Cantilever, tip point load — deflection	δ = PL³ / 3EI	Roark's Formulas, Table 8.1	L=5.0, I=2e-05, P=5000	0.049603	0.049603	1.2e-13	✓
BEA-0236	Cantilever, tip point load — end rotation	θ = PL² / 2EI	Roark's Formulas, Table 8.1	L=5.0, I=2e-05, P=5000	0.014881	0.014881	1.1e-13	✓
BEA-0237	Cantilever, uniform load — tip deflection	δ = wL⁴ / 8EI	Roark's Formulas, Table 8.1	L=5.0, I=2e-05, w=8000	0.14881	0.14881	1.1e-14	✓
BEA-0238	Cantilever, tip moment — deflection	δ = ML² / 2EI	Timoshenko, Strength of Materials I	L=5.0, I=2e-05, M=4000	0.011905	0.011905	1.0e-13	✓
BEA-0239	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=2.0, I=8e-06, P=10000	9.9206e-04	9.9206e-04	2.8e-14	✓
BEA-0240	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=2.0, I=8e-06, w=6000	7.4405e-04	7.4405e-04	2.6e-14	✓
BEA-0241	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=2.0, I=8e-06, P=12000	2.9762e-04	2.9762e-04	1.6e-15	✓
BEA-0242	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=2.0, I=8e-06, w=9000	2.2321e-04	2.2321e-04	8.5e-16	✓
BEA-0243	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=2.0, I=2e-05, P=10000	3.9683e-04	3.9683e-04	6.9e-14	✓
BEA-0244	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=2.0, I=2e-05, w=6000	2.9762e-04	2.9762e-04	6.6e-14	✓
BEA-0245	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=2.0, I=2e-05, P=12000	1.1905e-04	1.1905e-04	5.8e-15	✓
BEA-0246	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=2.0, I=2e-05, w=9000	8.9286e-05	8.9286e-05	5.9e-15	✓
BEA-0247	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=2.0, I=5e-05, P=10000	1.5873e-04	1.5873e-04	4.5e-14	✓
BEA-0248	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=2.0, I=5e-05, w=6000	1.1905e-04	1.1905e-04	4.1e-14	✓
BEA-0249	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=2.0, I=5e-05, P=12000	4.7619e-05	4.7619e-05	6.1e-15	✓
BEA-0250	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=2.0, I=5e-05, w=9000	3.5714e-05	3.5714e-05	6.1e-15	✓
BEA-0251	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=2.0, I=0.0001, P=10000	7.9365e-05	7.9365e-05	4.5e-14	✓
BEA-0252	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=2.0, I=0.0001, w=6000	5.9524e-05	5.9524e-05	4.1e-14	✓
BEA-0253	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=2.0, I=0.0001, P=12000	2.3810e-05	2.3810e-05	6.1e-15	✓
BEA-0254	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=2.0, I=0.0001, w=9000	1.7857e-05	1.7857e-05	6.1e-15	✓
BEA-0255	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=3.0, I=8e-06, P=10000	0.0033482	0.0033482	1.6e-14	✓
BEA-0256	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=3.0, I=8e-06, w=6000	0.0037667	0.0037667	1.6e-14	✓
BEA-0257	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=3.0, I=8e-06, P=12000	0.0010045	0.0010045	4.7e-15	✓
BEA-0258	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=3.0, I=8e-06, w=9000	0.00113	0.00113	5.2e-15	✓
BEA-0259	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=3.0, I=2e-05, P=10000	0.0013393	0.0013393	1.5e-15	✓
BEA-0260	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=3.0, I=2e-05, w=6000	0.0015067	0.0015067	1.4e-15	✓
BEA-0261	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=3.0, I=2e-05, P=12000	4.0179e-04	4.0179e-04	4.0e-15	✓
BEA-0262	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=3.0, I=2e-05, w=9000	4.5201e-04	4.5201e-04	4.4e-15	✓
BEA-0263	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=3.0, I=5e-05, P=10000	5.3571e-04	5.3571e-04	2.3e-14	✓
BEA-0264	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=3.0, I=5e-05, w=6000	6.0268e-04	6.0268e-04	2.3e-14	✓
BEA-0265	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=3.0, I=5e-05, P=12000	1.6071e-04	1.6071e-04	3.7e-15	✓
BEA-0266	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=3.0, I=5e-05, w=9000	1.8080e-04	1.8080e-04	4.0e-15	✓
BEA-0267	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=3.0, I=0.0001, P=10000	2.6786e-04	2.6786e-04	2.3e-14	✓
BEA-0268	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=3.0, I=0.0001, w=6000	3.0134e-04	3.0134e-04	2.3e-14	✓
BEA-0269	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=3.0, I=0.0001, P=12000	8.0357e-05	8.0357e-05	3.7e-15	✓
BEA-0270	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=3.0, I=0.0001, w=9000	9.0402e-05	9.0402e-05	4.0e-15	✓
BEA-0271	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=4.0, I=8e-06, P=10000	0.0079365	0.0079365	1.7e-14	✓
BEA-0272	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=4.0, I=8e-06, w=6000	0.011905	0.011905	1.7e-14	✓
BEA-0273	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=4.0, I=8e-06, P=12000	0.002381	0.002381	2.0e-15	✓
BEA-0274	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=4.0, I=8e-06, w=9000	0.0035714	0.0035714	1.5e-15	✓
BEA-0275	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=4.0, I=2e-05, P=10000	0.0031746	0.0031746	9.0e-15	✓
BEA-0276	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=4.0, I=2e-05, w=6000	0.0047619	0.0047619	1.0e-14	✓
BEA-0277	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=4.0, I=2e-05, P=12000	9.5238e-04	9.5238e-04	5.7e-16	✓
BEA-0278	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=4.0, I=2e-05, w=9000	0.0014286	0.0014286	9.1e-16	✓
BEA-0279	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=4.0, I=5e-05, P=10000	0.0012698	0.0012698	1.9e-14	✓
BEA-0280	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=4.0, I=5e-05, w=6000	0.0019048	0.0019048	1.9e-14	✓
BEA-0281	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=4.0, I=5e-05, P=12000	3.8095e-04	3.8095e-04	4.1e-15	✓
BEA-0282	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=4.0, I=5e-05, w=9000	5.7143e-04	5.7143e-04	4.6e-15	✓
BEA-0283	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=4.0, I=0.0001, P=10000	6.3492e-04	6.3492e-04	1.9e-14	✓
BEA-0284	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=4.0, I=0.0001, w=6000	9.5238e-04	9.5238e-04	1.9e-14	✓
BEA-0285	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=4.0, I=0.0001, P=12000	1.9048e-04	1.9048e-04	4.1e-15	✓
BEA-0286	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=4.0, I=0.0001, w=9000	2.8571e-04	2.8571e-04	4.6e-15	✓
BEA-0287	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=5.0, I=8e-06, P=10000	0.015501	0.015501	7.5e-15	✓
BEA-0288	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=5.0, I=8e-06, w=6000	0.029064	0.029064	7.5e-15	✓
BEA-0289	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=5.0, I=8e-06, P=12000	0.0046503	0.0046503	2.2e-15	✓
BEA-0290	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=5.0, I=8e-06, w=9000	0.0087193	0.0087193	2.6e-15	✓
BEA-0291	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=5.0, I=2e-05, P=10000	0.0062004	0.0062004	1.5e-14	✓
BEA-0292	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=5.0, I=2e-05, w=6000	0.011626	0.011626	1.5e-14	✓
BEA-0293	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=5.0, I=2e-05, P=12000	0.0018601	0.0018601	3.1e-15	✓
BEA-0294	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=5.0, I=2e-05, w=9000	0.0034877	0.0034877	3.4e-15	✓
BEA-0295	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=5.0, I=5e-05, P=10000	0.0024802	0.0024802	1.3e-14	✓
BEA-0296	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=5.0, I=5e-05, w=6000	0.0046503	0.0046503	1.4e-14	✓
BEA-0297	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=5.0, I=5e-05, P=12000	7.4405e-04	7.4405e-04	3.8e-15	✓
BEA-0298	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=5.0, I=5e-05, w=9000	0.0013951	0.0013951	3.9e-15	✓
BEA-0299	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=5.0, I=0.0001, P=10000	0.0012401	0.0012401	1.3e-14	✓
BEA-0300	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=5.0, I=0.0001, w=6000	0.0023251	0.0023251	1.4e-14	✓
BEA-0301	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=5.0, I=0.0001, P=12000	3.7202e-04	3.7202e-04	3.8e-15	✓
BEA-0302	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=5.0, I=0.0001, w=9000	6.9754e-04	6.9754e-04	3.9e-15	✓
BEA-0303	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=6.0, I=8e-06, P=10000	0.026786	0.026786	2.0e-14	✓
BEA-0304	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=6.0, I=8e-06, w=6000	0.060268	0.060268	2.0e-14	✓
BEA-0305	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=6.0, I=8e-06, P=12000	0.0080357	0.0080357	9.5e-15	✓
BEA-0306	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=6.0, I=8e-06, w=9000	0.01808	0.01808	1.0e-14	✓
BEA-0307	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=6.0, I=2e-05, P=10000	0.010714	0.010714	3.7e-15	✓
BEA-0308	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=6.0, I=2e-05, w=6000	0.024107	0.024107	4.0e-15	✓
BEA-0309	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=6.0, I=2e-05, P=12000	0.0032143	0.0032143	8.1e-16	✓
BEA-0310	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=6.0, I=2e-05, w=9000	0.0072321	0.0072321	1.3e-15	✓
BEA-0311	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=6.0, I=5e-05, P=10000	0.0042857	0.0042857	2.0e-16	✓
BEA-0312	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=6.0, I=5e-05, w=6000	0.0096429	0.0096429	1.8e-16	✓
BEA-0313	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=6.0, I=5e-05, P=12000	0.0012857	0.0012857	1.0e-15	✓
BEA-0314	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=6.0, I=5e-05, w=9000	0.0028929	0.0028929	9.0e-16	✓
BEA-0315	Simply-supported beam, central load	δ = PL³ / 48EI	Roark's Formulas, Table 8.1	L=6.0, I=0.0001, P=10000	0.0021429	0.0021429	2.0e-16	✓
BEA-0316	Simply-supported beam, uniform load	δ = 5wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=6.0, I=0.0001, w=6000	0.0048214	0.0048214	1.8e-16	✓
BEA-0317	Fixed-fixed beam, central load	δ = PL³ / 192EI	Roark's Formulas, Table 8.1	L=6.0, I=0.0001, P=12000	6.4286e-04	6.4286e-04	1.0e-15	✓
BEA-0318	Fixed-fixed beam, uniform load	δ = wL⁴ / 384EI	Roark's Formulas, Table 8.1	L=6.0, I=0.0001, w=9000	0.0014464	0.0014464	9.0e-16	✓

Elastic stability

24 comparisons24/24 passmax err 3.9e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ELA-0319	Euler column — pinned-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=1.0, L=3.0, I=8e-06	1.8424e+06	1.8423e+06	1.3e-05	✓
ELA-0320	Euler column — pinned-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=1.0, L=3.0, I=3e-05	6.9088e+06	6.9087e+06	1.3e-05	✓
ELA-0321	Euler column — pinned-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=1.0, L=5.0, I=8e-06	6.6325e+05	6.6324e+05	1.3e-05	✓
ELA-0322	Euler column — pinned-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=1.0, L=5.0, I=3e-05	2.4872e+06	2.4871e+06	1.3e-05	✓
ELA-0323	Euler column — pinned-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=1.0, L=8.0, I=8e-06	2.5908e+05	2.5908e+05	1.3e-05	✓
ELA-0324	Euler column — pinned-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=1.0, L=8.0, I=3e-05	9.7155e+05	9.7154e+05	1.3e-05	✓
ELA-0325	Euler column — fixed-free	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=2.0, L=3.0, I=8e-06	4.6058e+05	4.6058e+05	8.4e-07	✓
ELA-0326	Euler column — fixed-free	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=2.0, L=3.0, I=3e-05	1.7272e+06	1.7272e+06	8.4e-07	✓
ELA-0327	Euler column — fixed-free	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=2.0, L=5.0, I=8e-06	1.6581e+05	1.6581e+05	8.4e-07	✓
ELA-0328	Euler column — fixed-free	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=2.0, L=5.0, I=3e-05	6.2179e+05	6.2179e+05	8.4e-07	✓
ELA-0329	Euler column — fixed-free	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=2.0, L=8.0, I=8e-06	64769	64769	8.4e-07	✓
ELA-0330	Euler column — fixed-free	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=2.0, L=8.0, I=3e-05	2.4289e+05	2.4288e+05	8.4e-07	✓
ELA-0331	Euler column — fixed-fixed	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.5, L=3.0, I=8e-06	7.3709e+06	7.3693e+06	2.1e-04	✓
ELA-0332	Euler column — fixed-fixed	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.5, L=3.0, I=3e-05	2.7641e+07	2.7635e+07	2.1e-04	✓
ELA-0333	Euler column — fixed-fixed	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.5, L=5.0, I=8e-06	2.6535e+06	2.6529e+06	2.1e-04	✓
ELA-0334	Euler column — fixed-fixed	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.5, L=5.0, I=3e-05	9.9507e+06	9.9486e+06	2.1e-04	✓
ELA-0335	Euler column — fixed-fixed	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.5, L=8.0, I=8e-06	1.0365e+06	1.0363e+06	2.1e-04	✓
ELA-0336	Euler column — fixed-fixed	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.5, L=8.0, I=3e-05	3.8870e+06	3.8862e+06	2.1e-04	✓
ELA-0337	Euler column — fixed-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.699, L=3.0, I=8e-06	3.7691e+06	3.7706e+06	3.9e-04	✓
ELA-0338	Euler column — fixed-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.699, L=3.0, I=3e-05	1.4134e+07	1.4140e+07	3.9e-04	✓
ELA-0339	Euler column — fixed-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.699, L=5.0, I=8e-06	1.3569e+06	1.3574e+06	3.9e-04	✓
ELA-0340	Euler column — fixed-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.699, L=5.0, I=3e-05	5.0883e+06	5.0903e+06	3.9e-04	✓
ELA-0341	Euler column — fixed-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.699, L=8.0, I=8e-06	5.3004e+05	5.3024e+05	3.9e-04	✓
ELA-0342	Euler column — fixed-pinned	P_cr = π²EI / (KL)²	Timoshenko & Gere, Theory of Elastic Stability	K=0.699, L=8.0, I=3e-05	1.9876e+06	1.9884e+06	3.9e-04	✓

Dynamics — natural frequency

54 comparisons54/54 passmax err 3.0e-05

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
DYN-0343	cantilever beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=8e-06, mode=1	81.864	81.864	9.9e-08	✓
DYN-0344	cantilever beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=8e-06, mode=2	513.03	513.03	1.1e-06	✓
DYN-0345	cantilever beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=8e-06, mode=3	1436.5	1436.5	8.0e-06	✓
DYN-0346	cantilever beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=2e-05, mode=1	91.526	91.526	9.9e-08	✓
DYN-0347	cantilever beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=2e-05, mode=2	573.59	573.59	1.1e-06	✓
DYN-0348	cantilever beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=2e-05, mode=3	1606.1	1606.1	8.0e-06	✓
DYN-0349	cantilever beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=8e-06, mode=1	20.466	20.466	9.9e-08	✓
DYN-0350	cantilever beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=8e-06, mode=2	128.26	128.26	1.1e-06	✓
DYN-0351	cantilever beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=8e-06, mode=3	359.13	359.12	8.0e-06	✓
DYN-0352	cantilever beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=2e-05, mode=1	22.882	22.882	9.9e-08	✓
DYN-0353	cantilever beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=2e-05, mode=2	143.4	143.4	1.1e-06	✓
DYN-0354	cantilever beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=2e-05, mode=3	401.52	401.51	8.0e-06	✓
DYN-0355	cantilever beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=8e-06, mode=1	9.096	9.096	1.0e-07	✓
DYN-0356	cantilever beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=8e-06, mode=2	57.003	57.003	1.1e-06	✓
DYN-0357	cantilever beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=8e-06, mode=3	159.61	159.61	8.0e-06	✓
DYN-0358	cantilever beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=2e-05, mode=1	10.17	10.17	9.9e-08	✓
DYN-0359	cantilever beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=2e-05, mode=2	63.732	63.732	1.1e-06	✓
DYN-0360	cantilever beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=2e-05, mode=3	178.45	178.45	8.0e-06	✓
DYN-0361	fixed-fixed beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=8e-06, mode=1	520.92	520.92	9.4e-07	✓
DYN-0362	fixed-fixed beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=8e-06, mode=2	1435.9	1435.9	7.8e-06	✓
DYN-0363	fixed-fixed beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=8e-06, mode=3	2815.1	2815	3.0e-05	✓
DYN-0364	fixed-fixed beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=2e-05, mode=1	582.41	582.4	9.4e-07	✓
DYN-0365	fixed-fixed beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=2e-05, mode=2	1605.4	1605.4	7.8e-06	✓
DYN-0366	fixed-fixed beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=2e-05, mode=3	3147.4	3147.3	3.0e-05	✓
DYN-0367	fixed-fixed beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=8e-06, mode=1	130.23	130.23	9.4e-07	✓
DYN-0368	fixed-fixed beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=8e-06, mode=2	358.99	358.98	7.8e-06	✓
DYN-0369	fixed-fixed beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=8e-06, mode=3	703.77	703.75	3.0e-05	✓
DYN-0370	fixed-fixed beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=2e-05, mode=1	145.6	145.6	9.4e-07	✓
DYN-0371	fixed-fixed beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=2e-05, mode=2	401.36	401.36	7.8e-06	✓
DYN-0372	fixed-fixed beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=2e-05, mode=3	786.84	786.82	3.0e-05	✓
DYN-0373	fixed-fixed beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=8e-06, mode=1	57.88	57.88	9.4e-07	✓
DYN-0374	fixed-fixed beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=8e-06, mode=2	159.55	159.55	7.8e-06	✓
DYN-0375	fixed-fixed beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=8e-06, mode=3	312.79	312.78	3.0e-05	✓
DYN-0376	fixed-fixed beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=2e-05, mode=1	64.712	64.712	9.4e-07	✓
DYN-0377	fixed-fixed beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=2e-05, mode=2	178.38	178.38	7.8e-06	✓
DYN-0378	fixed-fixed beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=2e-05, mode=3	349.71	349.7	3.0e-05	✓
DYN-0379	simply-supported beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=8e-06, mode=1	229.79	229.79	2.0e-07	✓
DYN-0380	simply-supported beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=8e-06, mode=2	919.18	919.18	3.3e-06	✓
DYN-0381	simply-supported beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=8e-06, mode=3	2068.2	2068.2	1.6e-05	✓
DYN-0382	simply-supported beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=2e-05, mode=1	256.92	256.92	2.0e-07	✓
DYN-0383	simply-supported beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=2e-05, mode=2	1027.7	1027.7	3.3e-06	✓
DYN-0384	simply-supported beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=1.0, I=2e-05, mode=3	2312.3	2312.3	1.6e-05	✓
DYN-0385	simply-supported beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=8e-06, mode=1	57.449	57.449	2.0e-07	✓
DYN-0386	simply-supported beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=8e-06, mode=2	229.8	229.79	3.3e-06	✓
DYN-0387	simply-supported beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=8e-06, mode=3	517.05	517.04	1.6e-05	✓
DYN-0388	simply-supported beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=2e-05, mode=1	64.23	64.23	2.0e-07	✓
DYN-0389	simply-supported beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=2e-05, mode=2	256.92	256.92	3.3e-06	✓
DYN-0390	simply-supported beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=2.0, I=2e-05, mode=3	578.08	578.07	1.6e-05	✓
DYN-0391	simply-supported beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=8e-06, mode=1	25.533	25.533	2.0e-07	✓
DYN-0392	simply-supported beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=8e-06, mode=2	102.13	102.13	3.3e-06	✓
DYN-0393	simply-supported beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=8e-06, mode=3	229.8	229.79	1.6e-05	✓
DYN-0394	simply-supported beam, mode 1	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=2e-05, mode=1	28.546	28.546	2.0e-07	✓
DYN-0395	simply-supported beam, mode 2	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=2e-05, mode=2	114.19	114.19	3.3e-06	✓
DYN-0396	simply-supported beam, mode 3	fn = (bn L)^2 /2pi . sqrt(EI/rho A L^4)	Blevins, Formulas for Natural Frequency and Mode Shape	L=3.0, I=2e-05, mode=3	256.92	256.92	1.6e-05	✓

Prestressed modal (stress stiffening)

12 comparisons12/12 passmax err 1.0e-06

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
PRE-0397	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=2.0, N/Ncr=-0.5	40.622	40.622	1.0e-06	✓
PRE-0398	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=2.0, N/Ncr=+0.5	70.36	70.36	3.4e-07	✓
PRE-0399	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=2.0, N/Ncr=+1	81.245	81.245	5.2e-07	✓
PRE-0400	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=2.0, N/Ncr=+2	99.504	99.504	6.9e-07	✓
PRE-0401	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=3.0, N/Ncr=-0.5	18.054	18.054	1.0e-06	✓
PRE-0402	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=3.0, N/Ncr=+0.5	31.271	31.271	3.4e-07	✓
PRE-0403	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=3.0, N/Ncr=+1	36.109	36.109	5.2e-07	✓
PRE-0404	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=3.0, N/Ncr=+2	44.224	44.224	6.9e-07	✓
PRE-0405	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=5.0, N/Ncr=-0.5	6.4996	6.4996	1.0e-06	✓
PRE-0406	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=5.0, N/Ncr=+0.5	11.258	11.258	3.4e-07	✓
PRE-0407	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=5.0, N/Ncr=+1	12.999	12.999	5.2e-07	✓
PRE-0408	Pinned-pinned beam bending frequency under axial preload	f1(N) = f1(0) sqrt(1 + N/Ncr), Ncr = pi^2 EI/L^2	Timoshenko & Gere, Theory of Elastic Stability	L=5.0, N/Ncr=+2	15.921	15.921	6.9e-07	✓

Dynamics — damped modal

12 comparisons12/12 passmax err 2.2e-11

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
DYN-0409	Rayleigh proportional modal damping ratio	zeta_i = alpha/(2 w_i) + beta w_i/2	Clough & Penzien, Dynamics of Structures	alpha=5, beta=1e-05, mode=0	0.047894	0.047894	2.4e-13	✓
DYN-0410	Damped natural frequency f_d = f_n sqrt(1-zeta^2)	f_d = f_n sqrt(1 - zeta^2)	linear structural dynamics	alpha=5, beta=1e-05, mode=0	8.3439	8.3439	2.2e-11	✓
DYN-0411	Rayleigh proportional modal damping ratio	zeta_i = alpha/(2 w_i) + beta w_i/2	Clough & Penzien, Dynamics of Structures	alpha=5, beta=1e-05, mode=1	0.0092449	0.0092449	4.3e-13	✓
DYN-0412	Damped natural frequency f_d = f_n sqrt(1-zeta^2)	f_d = f_n sqrt(1 - zeta^2)	linear structural dynamics	alpha=5, beta=1e-05, mode=1	52.35	52.35	9.2e-13	✓
DYN-0413	Rayleigh proportional modal damping ratio	zeta_i = alpha/(2 w_i) + beta w_i/2	Clough & Penzien, Dynamics of Structures	alpha=5, beta=1e-05, mode=2	0.0073199	0.0073199	1.9e-13	✓
DYN-0414	Damped natural frequency f_d = f_n sqrt(1-zeta^2)	f_d = f_n sqrt(1 - zeta^2)	linear structural dynamics	alpha=5, beta=1e-05, mode=2	146.62	146.62	1.9e-13	✓
DYN-0415	Rayleigh proportional modal damping ratio	zeta_i = alpha/(2 w_i) + beta w_i/2	Clough & Penzien, Dynamics of Structures	alpha=2, beta=5e-06, mode=0	0.019184	0.019184	1.3e-13	✓
DYN-0416	Damped natural frequency f_d = f_n sqrt(1-zeta^2)	f_d = f_n sqrt(1 - zeta^2)	linear structural dynamics	alpha=2, beta=5e-06, mode=0	8.352	8.352	1.8e-11	✓
DYN-0417	Rayleigh proportional modal damping ratio	zeta_i = alpha/(2 w_i) + beta w_i/2	Clough & Penzien, Dynamics of Structures	alpha=2, beta=5e-06, mode=1	0.0038624	0.0038624	2.7e-13	✓
DYN-0418	Damped natural frequency f_d = f_n sqrt(1-zeta^2)	f_d = f_n sqrt(1 - zeta^2)	linear structural dynamics	alpha=2, beta=5e-06, mode=1	52.352	52.352	1.7e-12	✓
DYN-0419	Rayleigh proportional modal damping ratio	zeta_i = alpha/(2 w_i) + beta w_i/2	Clough & Penzien, Dynamics of Structures	alpha=2, beta=5e-06, mode=2	0.0033886	0.0033886	2.0e-13	✓
DYN-0420	Damped natural frequency f_d = f_n sqrt(1-zeta^2)	f_d = f_n sqrt(1 - zeta^2)	linear structural dynamics	alpha=2, beta=5e-06, mode=2	146.62	146.62	2.3e-13	✓

Dynamics — nonlinear continuum

1 comparisons1/1 passmax err 1.1e-06

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
DYN-0421	Small-amplitude finite-strain continuum transient equals the linear transient	max\|u_nl\| = max\|u_lin\| (Neo-Hookean linearises at F=I)	consistency vs the verified linear Newmark transient	quad4 strip, tip step load	4.5948e-06	4.5948e-06	1.1e-06	✓

Fluids — Stokes flow

2 comparisons2/2 passmax err 1.1e-02

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
FLU-0422	Poiseuille channel velocity converges to (G/2mu)y(H-y)	u_x(y) = (G/2mu) y (H-y)	White, Viscous Fluid Flow (Stokes limit)	24x24 mesh, relative error	0.0041573	0	4.2e-03	✓
FLU-0423	Poiseuille velocity error shows ~second-order convergence	err(h) / err(h/2) ~ 4 (O(h^2))	mesh refinement study	err(12)/err(24)	3.9567	4	1.1e-02	✓

Fluids — Navier-Stokes

2 comparisons2/2 passmax err 4.9e-02

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
FLU-0424	Kovasznay flow velocity converges to the exact N-S solution	u = 1 - e^{lam x} cos(2 pi y), v = (lam/2pi) e^{lam x} sin(2 pi y)	Kovasznay (1948); exact steady Navier-Stokes	Re=40, 16x16 mesh, relative error	0.049369	0	4.9e-02	✓
FLU-0425	Kovasznay velocity error drops under mesh refinement	err(h)/err(h/1.6) > 1	mesh refinement study	err(10)/err(16)	2.0503	2	2.5e-02	✓

Fluids — Transient Navier-Stokes

3 comparisons3/3 passmax err 6.3e-02

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
FLU-0426	Taylor-Green vortex decays at the exact analytic rate	peak \|u\|(T) = exp(-2 nu T) (unit initial peak)	Taylor-Green (1937); exact unsteady Navier-Stokes	12x12 mesh, dt=0.02, 5 steps	0.9802	0.9802	0.0e+00	✓
FLU-0427	Taylor-Green interior velocity matches the exact field	u = -cos x sin y e^{-2 nu t}, v = sin x cos y e^{-2 nu t}	Taylor-Green exact solution	12x12 relative error	0.063003	0	6.3e-02	✓
FLU-0428	Taylor-Green error drops under mesh refinement	err(8x8)/err(12x12) > 1	mesh refinement study	coarse/fine error ratio	1.2714	1.25	1.7e-02	✓

Multiphysics — Piezoelectric

6 comparisons6/6 passmax err 1.3e-15

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MUL-0429	Converse effect: applied voltage produces the exact tip strain	u_L = -(e/c) V (free bar, uniform field)	linear piezoelectricity, exact 1D closed form	V=60 V	-1.2857e-08	-1.2857e-08	0.0e+00	✓
MUL-0430	Converse effect: applied voltage produces the exact tip strain	u_L = -(e/c) V (free bar, uniform field)	linear piezoelectricity, exact 1D closed form	V=150 V	-3.2143e-08	-3.2143e-08	0.0e+00	✓
MUL-0431	Direct effect: imposed strain generates the exact open-circuit voltage	phi_L = (e/kappa) delta (open circuit, D=0)	linear piezoelectricity, exact 1D closed form	delta=1e-07 m	100	100	1.3e-15	✓
MUL-0432	Direct effect: imposed strain generates the exact open-circuit voltage	phi_L = (e/kappa) delta (open circuit, D=0)	linear piezoelectricity, exact 1D closed form	delta=3e-07 m	300	300	9.5e-16	✓
MUL-0433	Short-circuit compliance recovers the bare elastic modulus c	u_L = F L / (A c)	linear piezoelectricity, exact 1D closed form	tip force, phi=0 everywhere	7.1429e-07	7.1429e-07	0.0e+00	✓
MUL-0434	Open-circuit bar is stiffened to c_D = c + e^2/kappa	u_L = F L / (A (c + e^2/kappa))	linear piezoelectricity, exact 1D closed form	tip force, open circuit	5.8824e-07	5.8824e-07	1.8e-16	✓

Multiphysics — Vibro-acoustics

4 comparisons4/4 passmax err 2.0e-05

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MUL-0435	Coupled fundamental frequency matches the exact characteristic equation	k - m w^2 + A rho c w cot(wL/c) = 0	exact piston-on-fluid-column coupled eigenproblem	mode 1, 300 elements	32.078	32.078	2.8e-09	✓
MUL-0436	Coupled second frequency matches the exact characteristic equation	k - m w^2 + A rho c w cot(wL/c) = 0	exact piston-on-fluid-column coupled eigenproblem	mode 2, 300 elements	171.61	171.61	4.6e-06	✓
MUL-0437	Short-cavity coupled mode recovers the trapped-gas-spring limit	w^2 = (k + rho c^2 A / L) / m	exact incompressible/low-frequency limit	L=0.02 m	43.671	43.672	2.0e-05	✓
MUL-0438	Vanishing fluid density decouples to the in-vacuo piston frequency	f = sqrt(k/m)/(2 pi)	exact decoupled limit	rho -> 0	31.831	31.831	6.5e-09	✓

Multiphysics — Magnetostatics

5 comparisons5/5 passmax err 1.6e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MUL-0439	Vector potential at the slab centre matches the exact field	A_z(d/2) = mu J d^2 / 8	infinite current slab, exact solution	mu=mu0, J=1e6, d=0.1 m	0.0015708	0.0015708	3.2e-14	✓
MUL-0440	Vector potential at the quarter point matches the exact field	A_z(x) = (mu J / 2) x (d - x)	infinite current slab, exact solution	x = d/4	0.0011781	0.0011781	2.3e-14	✓
MUL-0441	Slab flux density is purely transverse (B_x = 0)	B_x = dA_z/dy = 0	current slab symmetry	max \|B_x\| over elements	5.5511e-16	0	5.6e-16	✓
MUL-0442	Magnetic energy converges to the exact slab energy	W = h mu J^2 d^3 / 24	integral \|B\|^2/2mu, exact	80x4 mesh, relative energy error	1.5625e-04	0	1.6e-04	✓
MUL-0443	Magnetic energy error drops at second order under refinement	err(h)/err(h/2) = 4	O(h^2) convergence study	err(40)/err(80)	4	4	4.7e-10	✓

Materials — finite-strain plasticity

18 comparisons18/18 passmax err 3.0e-09

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MAT-0444	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=1.05, σy=2.5e+08, H=2e+09	3.4414e+08	3.4414e+08	1.7e-16	✓
MAT-0445	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=1.2, σy=2.5e+08, H=2e+09	6.0856e+08	6.0856e+08	2.0e-16	✓
MAT-0446	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=1.5, σy=2.5e+08, H=2e+09	1.0504e+09	1.0504e+09	4.9e-15	✓
MAT-0447	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=2.0, σy=2.5e+08, H=2e+09	1.6201e+09	1.6201e+09	2.9e-16	✓
MAT-0448	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=1.05, σy=3e+08, H=0	3.0000e+08	3.0000e+08	0.0e+00	✓
MAT-0449	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=1.2, σy=3e+08, H=0	3.0000e+08	3.0000e+08	0.0e+00	✓
MAT-0450	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=1.5, σy=3e+08, H=0	3.0000e+08	3.0000e+08	0.0e+00	✓
MAT-0451	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=2.0, σy=3e+08, H=0	3.0000e+08	3.0000e+08	3.6e-14	✓
MAT-0452	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=1.05, σy=2e+08, H=5e+09	4.3312e+08	4.3312e+08	4.1e-16	✓
MAT-0453	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=1.2, σy=2e+08, H=5e+09	1.0845e+09	1.0845e+09	2.2e-16	✓
MAT-0454	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=1.5, σy=2e+08, H=5e+09	2.1730e+09	2.1730e+09	2.6e-15	✓
MAT-0455	Large-stretch J2 bar — true stress vs logarithmic strain	σ(ε) = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes, Computational Inelasticity (finite-strain J2)	λ=2.0, σy=2e+08, H=5e+09	3.5763e+09	3.5763e+09	0.0e+00	✓
MAT-0456	Continuum uniaxial finite-strain J2 — log-strain material point	axial Kirchhoff σ = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes; Miehe, logarithmic strain-space plasticity	λ=1.1, σy=2.5e+08, H=2e+09	4.3626e+08	4.3626e+08	2.9e-11	✓
MAT-0457	Continuum uniaxial finite-strain J2 — log-strain material point	axial Kirchhoff σ = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes; Miehe, logarithmic strain-space plasticity	λ=1.2, σy=2.5e+08, H=2e+09	6.0856e+08	6.0856e+08	7.7e-10	✓
MAT-0458	Continuum uniaxial finite-strain J2 — log-strain material point	axial Kirchhoff σ = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes; Miehe, logarithmic strain-space plasticity	λ=1.3, σy=2.5e+08, H=2e+09	7.6706e+08	7.6706e+08	5.6e-10	✓
MAT-0459	Continuum uniaxial finite-strain J2 — log-strain material point	axial Kirchhoff σ = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes; Miehe, logarithmic strain-space plasticity	λ=1.1, σy=3e+08, H=0	3.0000e+08	3.0000e+08	2.3e-09	✓
MAT-0460	Continuum uniaxial finite-strain J2 — log-strain material point	axial Kirchhoff σ = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes; Miehe, logarithmic strain-space plasticity	λ=1.2, σy=3e+08, H=0	3.0000e+08	3.0000e+08	3.0e-09	✓
MAT-0461	Continuum uniaxial finite-strain J2 — log-strain material point	axial Kirchhoff σ = (σy + Hε)/(1 + H/E), ε = ln λ	Simo & Hughes; Miehe, logarithmic strain-space plasticity	λ=1.3, σy=3e+08, H=0	3.0000e+08	3.0000e+08	4.8e-11	✓

Materials — hyperelasticity

36 comparisons36/36 passmax err 4.1e-09

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MAT-0462	Mooney-Rivlin 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = C10(I1-3)+C01(I2-3)+vol	Bonet & Wood, Nonlinear Continuum Mechanics for FEA	C10=350000, C01=150000, E=(0.12, -0.05, 0.03)	3.5721e+05	3.5721e+05	3.2e-11	✓
MAT-0463	Mooney-Rivlin 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = C10(I1-3)+C01(I2-3)+vol	Bonet & Wood, Nonlinear Continuum Mechanics for FEA	C10=350000, C01=150000, E=(0.2, 0.08, -0.04)	9.5539e+05	9.5539e+05	1.1e-10	✓
MAT-0464	Mooney-Rivlin 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = C10(I1-3)+C01(I2-3)+vol	Bonet & Wood, Nonlinear Continuum Mechanics for FEA	C10=350000, C01=150000, E=(-0.06, 0.15, 0.05)	1.4337e+05	1.4337e+05	3.2e-10	✓
MAT-0465	Mooney-Rivlin 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = C10(I1-3)+C01(I2-3)+vol	Bonet & Wood, Nonlinear Continuum Mechanics for FEA	C10=400000, C01=50000, E=(0.12, -0.05, 0.03)	3.1934e+05	3.1934e+05	5.2e-13	✓
MAT-0466	Mooney-Rivlin 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = C10(I1-3)+C01(I2-3)+vol	Bonet & Wood, Nonlinear Continuum Mechanics for FEA	C10=400000, C01=50000, E=(0.2, 0.08, -0.04)	8.3789e+05	8.3789e+05	1.0e-10	✓
MAT-0467	Mooney-Rivlin 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = C10(I1-3)+C01(I2-3)+vol	Bonet & Wood, Nonlinear Continuum Mechanics for FEA	C10=400000, C01=50000, E=(-0.06, 0.15, 0.05)	1.2502e+05	1.2502e+05	2.6e-10	✓
MAT-0468	Mooney-Rivlin (C01=0, C10=μ/2) reduces to Neo-Hookean	S_MR = S_NH	consistency / limit test	E=(0.12, -0.05, 0.03)	2.0296e+05	2.0296e+05	0.0e+00	✓
MAT-0469	Mooney-Rivlin (C01=0, C10=μ/2) reduces to Neo-Hookean	S_MR = S_NH	consistency / limit test	E=(0.2, 0.08, -0.04)	6.3516e+05	6.3516e+05	0.0e+00	✓
MAT-0470	Mooney-Rivlin (C01=0, C10=μ/2) reduces to Neo-Hookean	S_MR = S_NH	consistency / limit test	E=(-0.06, 0.15, 0.05)	1.8599e+05	1.8599e+05	0.0e+00	✓
MAT-0471	Ogden 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = Σ(μ_p/α_p)(Σλ^α_p−3)+vol	Ogden (1972); Bonet & Wood, Nonlinear Continuum Mechanics for FEA	mu=(130000.0, -40000.0), alpha=(1.8, -2.1), E=(0.12, -0.05, 0.03)	54154	54154	2.5e-10	✓
MAT-0472	Ogden 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = Σ(μ_p/α_p)(Σλ^α_p−3)+vol	Ogden (1972); Bonet & Wood, Nonlinear Continuum Mechanics for FEA	mu=(130000.0, -40000.0), alpha=(1.8, -2.1), E=(0.2, 0.08, -0.04)	1.4103e+05	1.4103e+05	1.2e-10	✓
MAT-0473	Ogden 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = Σ(μ_p/α_p)(Σλ^α_p−3)+vol	Ogden (1972); Bonet & Wood, Nonlinear Continuum Mechanics for FEA	mu=(130000.0, -40000.0), alpha=(1.8, -2.1), E=(-0.06, 0.15, 0.05)	20370	20370	5.6e-10	✓
MAT-0474	Ogden 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = Σ(μ_p/α_p)(Σλ^α_p−3)+vol	Ogden (1972); Bonet & Wood, Nonlinear Continuum Mechanics for FEA	mu=(200000.0, 60000.0), alpha=(2.6, 1.1), E=(0.12, -0.05, 0.03)	83312	83312	1.0e-10	✓
MAT-0475	Ogden 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = Σ(μ_p/α_p)(Σλ^α_p−3)+vol	Ogden (1972); Bonet & Wood, Nonlinear Continuum Mechanics for FEA	mu=(200000.0, 60000.0), alpha=(2.6, 1.1), E=(0.2, 0.08, -0.04)	1.8671e+05	1.8671e+05	2.8e-11	✓
MAT-0476	Ogden 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = Σ(μ_p/α_p)(Σλ^α_p−3)+vol	Ogden (1972); Bonet & Wood, Nonlinear Continuum Mechanics for FEA	mu=(200000.0, 60000.0), alpha=(2.6, 1.1), E=(-0.06, 0.15, 0.05)	3399.6	3399.6	4.1e-09	✓
MAT-0477	Ogden single term (μ, α=2) reduces to Neo-Hookean	S_Ogden = S_NH	consistency / limit test	E=(0.12, -0.05, 0.03)	99555	99555	7.3e-16	✓
MAT-0478	Ogden single term (μ, α=2) reduces to Neo-Hookean	S_Ogden = S_NH	consistency / limit test	E=(0.2, 0.08, -0.04)	2.0942e+05	2.0942e+05	2.8e-16	✓
MAT-0479	Ogden single term (μ, α=2) reduces to Neo-Hookean	S_Ogden = S_NH	consistency / limit test	E=(-0.06, 0.15, 0.05)	-9928	-9928	5.1e-15	✓
MAT-0480	Transversely isotropic 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = W_NH + (k1/2k2)(exp(k2(I4-1)^2)-1)	Holzapfel, Gasser & Ogden (2000); Bonet & Wood	k1=300000, k2=8, deg=30, E=(0.15, -0.04, 0.05)	3.1045e+05	3.1045e+05	3.0e-11	✓
MAT-0481	Transversely isotropic 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = W_NH + (k1/2k2)(exp(k2(I4-1)^2)-1)	Holzapfel, Gasser & Ogden (2000); Bonet & Wood	k1=300000, k2=8, deg=30, E=(0.2, 0.08, -0.04)	4.9915e+05	4.9915e+05	4.3e-11	✓
MAT-0482	Transversely isotropic 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = W_NH + (k1/2k2)(exp(k2(I4-1)^2)-1)	Holzapfel, Gasser & Ogden (2000); Bonet & Wood	k1=300000, k2=8, deg=30, E=(-0.06, 0.15, 0.05)	2889.4	2889.4	3.6e-09	✓
MAT-0483	Transversely isotropic 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = W_NH + (k1/2k2)(exp(k2(I4-1)^2)-1)	Holzapfel, Gasser & Ogden (2000); Bonet & Wood	k1=500000, k2=2, deg=60, E=(0.15, -0.04, 0.05)	1.4215e+05	1.4215e+05	3.3e-11	✓
MAT-0484	Transversely isotropic 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = W_NH + (k1/2k2)(exp(k2(I4-1)^2)-1)	Holzapfel, Gasser & Ogden (2000); Bonet & Wood	k1=500000, k2=2, deg=60, E=(0.2, 0.08, -0.04)	2.5906e+05	2.5906e+05	4.4e-11	✓
MAT-0485	Transversely isotropic 2nd-PK stress equals strain-energy gradient	S11 = ∂W/∂E11, W = W_NH + (k1/2k2)(exp(k2(I4-1)^2)-1)	Holzapfel, Gasser & Ogden (2000); Bonet & Wood	k1=500000, k2=2, deg=60, E=(-0.06, 0.15, 0.05)	56813	56813	6.1e-11	✓
MAT-0486	Transversely isotropic (k1=0) reduces to Neo-Hookean	S_aniso = S_NH	consistency / limit test	E=(0.15, -0.04, 0.05)	1.2747e+05	1.2747e+05	0.0e+00	✓
MAT-0487	Transversely isotropic (k1=0) reduces to Neo-Hookean	S_aniso = S_NH	consistency / limit test	E=(0.2, 0.08, -0.04)	2.0942e+05	2.0942e+05	0.0e+00	✓
MAT-0488	Transversely isotropic (k1=0) reduces to Neo-Hookean	S_aniso = S_NH	consistency / limit test	E=(-0.06, 0.15, 0.05)	-9928	-9928	0.0e+00	✓
MAT-0489	GOH dispersion/two-family fibre 2nd-PK stress equals dW/dE	S11 = ∂W/∂E11, E=kappaI1+(1-3kappa)I4-1 per family	Gasser, Ogden & Holzapfel (2006)	kappa=0.15, deg=(40.0, -40.0), E=(0.12, 0.05, 0.03)	2.7819e+05	2.7819e+05	1.2e-10	✓
MAT-0490	GOH dispersion/two-family fibre 2nd-PK stress equals dW/dE	S11 = ∂W/∂E11, E=kappaI1+(1-3kappa)I4-1 per family	Gasser, Ogden & Holzapfel (2006)	kappa=0.15, deg=(40.0, -40.0), E=(0.2, -0.03, -0.04)	3.1604e+05	3.1604e+05	3.0e-12	✓
MAT-0491	GOH dispersion/two-family fibre 2nd-PK stress equals dW/dE	S11 = ∂W/∂E11, E=kappaI1+(1-3kappa)I4-1 per family	Gasser, Ogden & Holzapfel (2006)	kappa=0.15, deg=(40.0, -40.0), E=(0.08, 0.14, 0.02)	3.1861e+05	3.1861e+05	7.8e-11	✓
MAT-0492	GOH dispersion/two-family fibre 2nd-PK stress equals dW/dE	S11 = ∂W/∂E11, E=kappaI1+(1-3kappa)I4-1 per family	Gasser, Ogden & Holzapfel (2006)	kappa=0.05, deg=(25.0, -25.0), E=(0.12, 0.05, 0.03)	4.5256e+05	4.5256e+05	1.5e-10	✓
MAT-0493	GOH dispersion/two-family fibre 2nd-PK stress equals dW/dE	S11 = ∂W/∂E11, E=kappaI1+(1-3kappa)I4-1 per family	Gasser, Ogden & Holzapfel (2006)	kappa=0.05, deg=(25.0, -25.0), E=(0.2, -0.03, -0.04)	7.3897e+05	7.3897e+05	1.7e-11	✓
MAT-0494	GOH dispersion/two-family fibre 2nd-PK stress equals dW/dE	S11 = ∂W/∂E11, E=kappaI1+(1-3kappa)I4-1 per family	Gasser, Ogden & Holzapfel (2006)	kappa=0.05, deg=(25.0, -25.0), E=(0.08, 0.14, 0.02)	4.0624e+05	4.0624e+05	6.4e-11	✓
MAT-0495	GOH (kappa=0, one family, tension) reduces to single-fibre model	S_GOH = S_single_fibre	consistency / limit test	E=(0.2, 0.02, 0.01)	1.0230e+06	1.0230e+06	0.0e+00	✓
MAT-0496	GOH (kappa=0, one family, tension) reduces to single-fibre model	S_GOH = S_single_fibre	consistency / limit test	E=(0.15, 0.05, 0.0)	5.7996e+05	5.7996e+05	0.0e+00	✓
MAT-0497	GOH (kappa=0, one family, tension) reduces to single-fibre model	S_GOH = S_single_fibre	consistency / limit test	E=(0.25, -0.02, 0.03)	1.9908e+06	1.9908e+06	0.0e+00	✓

Materials — viscoplasticity

7 comparisons7/7 passmax err 7.3e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MAT-0498	Perzyna 1D return equals N=1 overstress closed form	dp = (dt/eta)f_tr / (1 + (dt/eta)(E+H))	Perzyna (1966); Simo & Hughes, Computational Inelasticity	eta=1000	0.0085366	0.0085366	2.6e-14	✓
MAT-0499	Perzyna 1D return equals N=1 overstress closed form	dp = (dt/eta)f_tr / (1 + (dt/eta)(E+H))	Perzyna (1966); Simo & Hughes, Computational Inelasticity	eta=1e+09	0.0084951	0.0084951	1.9e-13	✓
MAT-0500	Perzyna 1D return equals N=1 overstress closed form	dp = (dt/eta)f_tr / (1 + (dt/eta)(E+H))	Perzyna (1966); Simo & Hughes, Computational Inelasticity	eta=1e+14	1.7464e-05	1.7464e-05	5.5e-11	✓
MAT-0501	Perzyna eta->0 recovers rate-independent plasticity (1D)	dp -> f_tr/(E+H)	consistency / limit test	eta=1e-6	0.0085366	0.0085366	4.5e-13	✓
MAT-0502	Perzyna eta->0 recovers rate-independent radial return (3D)	dp -> (sigma_e_tr - sy)/(3G+H)	consistency / limit test	eta=1e-8	0.0054649	0.0054649	9.1e-13	✓
MAT-0503	Perzyna held-strain bar follows evp(t)=evp_inf(1-e^{-t/tau})	evp(tau) = evp_inf(1-1/e), tau=eta/(E+H)	linear-ODE closed form; backward Euler	t=tau	0.0010784	0.0010792	7.3e-04	✓
MAT-0504	Perzyna held-strain bar relaxes to rate-independent plasticity	sigma(t->inf) = sy + H*evp_inf	steady-state / limit test	t=12tau	2.5854e+08	2.5854e+08	8.4e-06	✓

Materials — 2D creep

6 comparisons6/6 passmax err 3.3e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MAT-0505	Plane-strain J2 creep e_c,xx matches the 3D radial return	in-plane creep increment = 3D reference (e_zz=0, traceless creep)	consistency vs verified 3D J2 secondary creep	eps=(0.002, -0.001, 0.0015)	1.2697e-04	1.2697e-04	2.1e-16	✓
MAT-0506	Plane-strain J2 creep gamma_c,xy matches the 3D radial return	in-plane shear creep increment = 3D reference	consistency vs verified 3D J2 secondary creep	eps=(0.002, -0.001, 0.0015)	6.5132e-05	6.5132e-05	2.1e-16	✓
MAT-0507	Plane-strain J2 creep e_c,xx matches the 3D radial return	in-plane creep increment = 3D reference (e_zz=0, traceless creep)	consistency vs verified 3D J2 secondary creep	eps=(-0.001, 0.003, -0.002)	-1.1651e-04	-1.1651e-04	2.3e-16	✓
MAT-0508	Plane-strain J2 creep gamma_c,xy matches the 3D radial return	in-plane shear creep increment = 3D reference	consistency vs verified 3D J2 secondary creep	eps=(-0.001, 0.003, -0.002)	-1.1104e-04	-1.1104e-04	0.0e+00	✓
MAT-0509	Plane-strain J2 creep e_c,xx matches the 3D radial return	in-plane creep increment = 3D reference (e_zz=0, traceless creep)	consistency vs verified 3D J2 secondary creep	eps=(0.0025, 0.001, 0.0005)	4.0842e-05	4.0842e-05	3.3e-16	✓
MAT-0510	Plane-strain J2 creep gamma_c,xy matches the 3D radial return	in-plane shear creep increment = 3D reference	consistency vs verified 3D J2 secondary creep	eps=(0.0025, 0.001, 0.0005)	-6.5258e-05	-6.5258e-05	0.0e+00	✓

Multiphysics — temperature-dependent stiffness

4 comparisons4/4 passmax err 1.9e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MUL-0511	Heated bar under load: u = PL/(A E(T))	E(T)=E(1+c(T-Tref)); u=PL/(A E(T))	closed form; one-way thermo-mechanical with E(T)	c=-0.002, T=100	0.00625	0.00625	0.0e+00	✓
MUL-0512	Heated bar under load: u = PL/(A E(T))	E(T)=E(1+c(T-Tref)); u=PL/(A E(T))	closed form; one-way thermo-mechanical with E(T)	c=0.0015, T=60	0.0045872	0.0045872	1.9e-16	✓
MUL-0513	Restrained heated bar: sigma = -E(T) alpha dT	sigma = -E(T) alpha (T - Tref)	closed form; one-way thermo-mechanical with E(T)	c=-0.002, T=100	-1.9200e+08	-1.9200e+08	0.0e+00	✓
MUL-0514	Restrained heated bar: sigma = -E(T) alpha dT	sigma = -E(T) alpha (T - Tref)	closed form; one-way thermo-mechanical with E(T)	c=0.001, T=50	-2.1000e+08	-2.1000e+08	0.0e+00	✓

Multiphysics — hygroscopic swelling

7 comparisons7/7 passmax err 1.8e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MUL-0515	Restrained bar under moisture uptake: sigma = -E beta dC	sigma = -E * moisture_expansion * (C - C_ref)	closed form; one-way hygro-mechanical	beta=0.003, dC=0.8	-1.6800e+08	-1.6800e+08	0.0e+00	✓
MUL-0516	Restrained bar under moisture uptake: sigma = -E beta dC	sigma = -E * moisture_expansion * (C - C_ref)	closed form; one-way hygro-mechanical	beta=0.0005, dC=1	-3.5000e+07	-3.5000e+07	0.0e+00	✓
MUL-0517	Free bar swells to u = beta dC L	u(L) = moisture_expansion * dC * L	closed form; unconstrained swelling	beta=0.003, dC=0.8	0.0024	0.0024	1.8e-16	✓
MUL-0518	Unconstrained swelling is stress-free	sigma = 0 for a free bar	eigenstrain relieves stress	free bar	0	0	0.0e+00	✓
MUL-0519	Swelling stress is linear in the concentration change	sigma(2 dC)/sigma(dC) = 2	linearity of the eigenstrain	dC doubled	2	2	0.0e+00	✓
MUL-0520	Zero moisture-expansion coefficient produces no stress	sigma = 0 for beta = 0	no-swelling limit	beta=0	0	0	0.0e+00	✓
MUL-0521	Fickian gradient: element dC is the mean surface concentration	dC = (C0+CL)/2 - C_ref on a linear profile	steady Fickian field	C0=0.2, CL=1.0	0.6	0.6	0.0e+00	✓

Reduction — static condensation (superelement)

8 comparisons8/8 passmax err 1.8e-14

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
RED-0522	Truss chain: condensed solution vs full	max \|u_condensed - u_full\| / scale (static condensation is exact)	exact Guyan condensation	retain {0,5}	8.6736e-17	0	8.7e-17	✓
RED-0523	Truss chain: reduced stiffness symmetry	Schur complement K_c is symmetric	exact Guyan condensation	retain {0,5}	0	0	0.0e+00	✓
RED-0524	Truss chain: strain energy match	reduced-solve energy equals the full energy	exact Guyan condensation	retain {0,5}	0	0	0.0e+00	✓
RED-0525	Interior DOFs are condensed out of the reduced system	reduced + condensed = free DOFs	superelement DOF count	6-node chain	5	5	0.0e+00	✓
RED-0526	Cantilever beam: condensed solution vs full	max \|u_condensed - u_full\| / scale (static condensation is exact)	exact Guyan condensation	retain tip	6.1474e-16	0	6.1e-16	✓
RED-0527	Cantilever beam: reduced stiffness symmetry	Schur complement K_c is symmetric	exact Guyan condensation	retain tip	0	0	0.0e+00	✓
RED-0528	Cantilever beam: strain energy match	reduced-solve energy equals the full energy	exact Guyan condensation	retain tip	1.8325e-14	0	1.8e-14	✓
RED-0529	Reduced superelement stiffness is positive definite	min eig(K_c) > 0	well-posed condensed operator	cantilever tip	1	1	0.0e+00	✓

Infrastructure — matrix-free operator

22 comparisons22/22 passmax err 1.1e-14

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
INF-0530	Element-by-element matvec equals assembled K_ff (truss2d)	K_mf v == K_ff v (no assembly)	assembled reference operator	truss2d	1.1535e-16	0	1.2e-16	✓
INF-0531	Operator diagonal equals assembled diagonal (truss2d)	diag(K_mf) == diag(K_ff)	assembled diagonal	truss2d	0	0	0.0e+00	✓
INF-0532	Matrix-free CG equals the direct solve (truss2d)	u_mfcg == u_direct	direct factorization reference	truss2d	2.2768e-16	0	2.3e-16	✓
INF-0533	Cache and recompute modes agree (truss2d)	u(cache) == u(recompute)	mode consistency	truss2d	0	0	0.0e+00	✓
INF-0534	K is never assembled by the matrix-free solve (truss2d)	assembled_K_never_formed	assembly-free guarantee	truss2d	1	1	0.0e+00	✓
INF-0535	Element-by-element matvec equals assembled K_ff (quad4)	K_mf v == K_ff v (no assembly)	assembled reference operator	quad4	2.4695e-16	0	2.5e-16	✓
INF-0536	Operator diagonal equals assembled diagonal (quad4)	diag(K_mf) == diag(K_ff)	assembled diagonal	quad4	0	0	0.0e+00	✓
INF-0537	Matrix-free CG equals the direct solve (quad4)	u_mfcg == u_direct	direct factorization reference	quad4	1.0907e-14	0	1.1e-14	✓
INF-0538	Cache and recompute modes agree (quad4)	u(cache) == u(recompute)	mode consistency	quad4	7.6819e-15	0	7.7e-15	✓
INF-0539	K is never assembled by the matrix-free solve (quad4)	assembled_K_never_formed	assembly-free guarantee	quad4	1	1	0.0e+00	✓
INF-0540	Element-by-element matvec equals assembled K_ff (hex8)	K_mf v == K_ff v (no assembly)	assembled reference operator	hex8	2.5778e-16	0	2.6e-16	✓
INF-0541	Operator diagonal equals assembled diagonal (hex8)	diag(K_mf) == diag(K_ff)	assembled diagonal	hex8	0	0	0.0e+00	✓
INF-0542	Matrix-free CG equals the direct solve (hex8)	u_mfcg == u_direct	direct factorization reference	hex8	2.9088e-16	0	2.9e-16	✓
INF-0543	Cache and recompute modes agree (hex8)	u(cache) == u(recompute)	mode consistency	hex8	1.4544e-16	0	1.5e-16	✓
INF-0544	K is never assembled by the matrix-free solve (hex8)	assembled_K_never_formed	assembly-free guarantee	hex8	1	1	0.0e+00	✓
INF-0545	Iterative routing of a reduced system equals the direct solve	solve_reduced(cg) == solve_reduced(direct)	nonlinear/dynamic inner-solve routing	quad4 SPD	4.6559e-15	0	4.7e-15	✓
INF-0546	Threaded element loop matches the serial apply	K_mf(4 threads) v == K_mf(1) v to roundoff	parallel reduction consistency	hex8, 4 threads	0	0	0.0e+00	✓
INF-0547	Matrix-free CG is independent of the worker count	u(4 threads) == u(1 thread)	deterministic parallel solve	hex8, 4 threads	0	0	0.0e+00	✓
INF-0548	Recompute operator memory is far below assembled K	mem(recompute) < mem(assembled K)	O(n) memory footprint	quad4 grid	1	1	0.0e+00	✓
INF-0549	Interrupted solve leaves a checkpoint that is not yet converged	capped run status == incomplete	checkpoint durability	maxiter cap	1	1	0.0e+00	✓
INF-0550	Checkpoint on disk records the paused iterate and progress	checkpoint iterate persisted	durable checkpoint file	on disk	1	1	0.0e+00	✓
INF-0551	Warm-start resume from a checkpoint equals the direct solve	u(resumed) == u_direct	exact checkpoint/resume	resume	4.9134e-15	0	4.9e-15	✓

Infrastructure — distributed matrix-free operator

17 comparisons17/17 passmax err 3.0e-12

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
INF-0552	Distributed matvec equals the assembled K_ff	K_dist v == K_ff v (no assembly)	assembled reference operator	quad4, 2 processes	1.8914e-16	0	1.9e-16	✓
INF-0553	Distributed matvec is independent of the process count	K_dist(3) v == K_dist(2) v to roundoff	partition reduction consistency	quad4, 2 vs 3 processes	1.8914e-16	0	1.9e-16	✓
INF-0554	Distributed operator diagonal equals the assembled diagonal	diag(K_dist) == diag(K_ff)	assembled diagonal	quad4	1.1275e-16	0	1.1e-16	✓
INF-0555	Distributed CG equals the direct solve	u_distcg == u_direct	direct factorization reference	quad4, 2 processes	1.0492e-14	0	1.0e-14	✓
INF-0556	Distributed CG is independent of the process count	u(3 processes) == u(2 processes)	deterministic distributed solve	quad4, 2 vs 3 processes	4.1939e-15	0	4.2e-15	✓
INF-0557	K is never assembled by the distributed solve	assembled_K_never_formed	assembly-free guarantee	quad4	1	1	0.0e+00	✓
INF-0558	GPU-capable matvec equals the assembled K_ff	K_gpu v == K_ff v (device or host)	assembled reference operator	quad4	1.8914e-16	0	1.9e-16	✓
INF-0559	GPU-capable CG equals the direct solve	u_gpucg == u_direct	direct factorization reference	quad4	1.0907e-14	0	1.1e-14	✓
INF-0560	Halo-exchange matvec equals the assembled K_ff	K_halo v == K_ff v (boundary-only exchange)	assembled reference operator	quad4, 2 processes	1.8829e-16	0	1.9e-16	✓
INF-0561	Halo exchange moves less data than the full-vector exchange	sum(touched_p) < n_free * processes	communication-volume property	quad4, 4 processes	1	1	0.0e+00	✓
INF-0562	Halo-exchange CG equals the direct solve	u_halocg == u_direct	direct factorization reference	quad4, 2 processes	4.3459e-13	0	4.3e-13	✓
INF-0563	Halo-exchange CG is independent of the process count	u(4 processes) == u(2 processes)	deterministic distributed solve	quad4, 2 vs 4 processes	5.0557e-14	0	5.1e-14	✓
INF-0564	Additive Schwarz CG equals the direct solve	u_schwarz == u_direct	direct factorization reference	clamped plate, 2 blocks	2.9643e-12	0	3.0e-12	✓
INF-0565	Single-block additive Schwarz converges in a few iterations	iterations(1 block) small	exact local solve property	clamped plate, 1 block	1	1	0.0e+00	✓
INF-0566	Two-block additive Schwarz beats diagonal Jacobi	iterations(schwarz-2) < iterations(jacobi)	preconditioner-quality property	clamped plate	1	1	0.0e+00	✓
INF-0567	Node-block Jacobi CG equals the direct solve	u_blockjacobi == u_direct	direct factorization reference	clamped plate	1.8640e-12	0	1.9e-12	✓
INF-0568	Node-block Jacobi preconditioner is symmetric	x^T M y == y^T M x	symmetric-positive-definite property	clamped plate	0	0	0.0e+00	✓

Reduction — p-refinement

5 comparisons5/5 passmax err 3.2e-01

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
RED-0569	Promotion inserts one shared node per unique edge	n_nodes(p2) = n_nodes(p1) + n_unique_edges	conforming promotion	quad4 grid	29	29	0.0e+00	✓
RED-0570	All promotable elements become quadratic	quad4 -> quad8	element order raised	quad4 grid	1	1	0.0e+00	✓
RED-0571	p-refined mesh reproduces a linear field exactly (conforming)	linear patch test on the quadratic mesh	conforming p2 mesh	quad8 grid	1.0842e-19	0	1.1e-19	✓
RED-0572	Quadratic solve is far sharper than linear at the same mesh	err(p2) << err(p1)	p-refinement accuracy	12x4 cantilever	1	1	0.0e+00	✓
RED-0573	Quadratic elements converge at a higher rate than linear	rate(p2) > rate(p1)	higher-order convergence	rate ratio	0.68229	1	3.2e-01	✓

Multiphysics — temperature-dependent properties

4 comparisons4/4 passmax err 1.1e-12

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MUL-0574	Nonlinear conductivity k(T) matches the Kirchhoff closed form	integral k dT linear in x (1-D bar)	closed form; Picard fixed point	ck=0.002	52.268	52.268	1.8e-13	✓
MUL-0575	Nonlinear conductivity k(T) matches the Kirchhoff closed form	integral k dT linear in x (1-D bar)	closed form; Picard fixed point	ck=0.005	54.951	54.951	1.1e-12	✓
MUL-0576	Temperature-dependent yield sy(T) plastic response	sigma = sy(T) + H p, sy(T)=sy(1+cy(T-Tref))	closed form	cy=-0.0008, T=220	2.2167e+08	2.2167e+08	1.3e-16	✓
MUL-0577	Temperature-dependent yield sy(T) plastic response	sigma = sy(T) + H p, sy(T)=sy(1+cy(T-Tref))	closed form	cy=-0.0005, T=300	2.2414e+08	2.2414e+08	1.3e-16	✓

Materials — damage-plasticity

7 comparisons7/7 passmax err 7.8e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MAT-0578	Ductile damage-plasticity nominal stress (monotonic stretch)	sigma = (1-D(p))(sy + H p), p=(E eps-sy)/(E+H)	Lemaitre strain-equivalence; closed form	eps=0.001	2.0000e+08	2.0000e+08	0.0e+00	✓
MAT-0579	Ductile damage-plasticity nominal stress (monotonic stretch)	sigma = (1-D(p))(sy + H p), p=(E eps-sy)/(E+H)	Lemaitre strain-equivalence; closed form	eps=0.005	2.3896e+08	2.3896e+08	1.2e-16	✓
MAT-0580	Ductile damage-plasticity nominal stress (monotonic stretch)	sigma = (1-D(p))(sy + H p), p=(E eps-sy)/(E+H)	Lemaitre strain-equivalence; closed form	eps=0.01	1.8454e+08	1.8454e+08	0.0e+00	✓
MAT-0581	Ductile damage-plasticity nominal stress (monotonic stretch)	sigma = (1-D(p))(sy + H p), p=(E eps-sy)/(E+H)	Lemaitre strain-equivalence; closed form	eps=0.03	3.3498e+07	3.3498e+07	7.8e-16	✓
MAT-0582	Damage-plasticity with Dc=0 reduces to J2 plasticity	sigma(Dc=0) = sigma_plasticity	consistency / limit test	eps=1e-2	2.7586e+08	2.7586e+08	0.0e+00	✓
MAT-0583	Continuum (hex8) ductile damage stress = (1-D(p)) * J2 stress	sigma_vm = (1 - D(p)) sigma_vm,J2	Lemaitre strain-equivalence; consistency vs verified finite-J2	lambda=1.08	3.9960e+07	3.9960e+07	7.5e-16	✓
MAT-0584	Continuum (hex8) ductile damage stress = (1-D(p)) * J2 stress	sigma_vm = (1 - D(p)) sigma_vm,J2	Lemaitre strain-equivalence; consistency vs verified finite-J2	lambda=1.15	5.2373e+07	5.2373e+07	2.8e-16	✓

Acoustics — cavity modes

6 comparisons6/6 passmax err 2.3e-05

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ACO-0585	Rigid-rigid duct acoustic mode f_n = n c/(2L)	f_n = n c/(2L)	1-D Helmholtz duct closed form	n=1, L=1, c=343	171.5	171.5	2.6e-06	✓
ACO-0586	Rigid-rigid duct acoustic mode f_n = n c/(2L)	f_n = n c/(2L)	1-D Helmholtz duct closed form	n=2, L=1, c=343	343	343	1.0e-05	✓
ACO-0587	Rigid-rigid duct acoustic mode f_n = n c/(2L)	f_n = n c/(2L)	1-D Helmholtz duct closed form	n=3, L=1, c=343	514.51	514.5	2.3e-05	✓
ACO-0588	Open-open duct acoustic mode f_n = n c/(2L)	f_n = n c/(2L)	1-D Helmholtz duct closed form	n=1, L=1, c=343	171.5	171.5	2.6e-06	✓
ACO-0589	Open-open duct acoustic mode f_n = n c/(2L)	f_n = n c/(2L)	1-D Helmholtz duct closed form	n=2, L=1, c=343	343	343	1.0e-05	✓
ACO-0590	Open-open duct acoustic mode f_n = n c/(2L)	f_n = n c/(2L)	1-D Helmholtz duct closed form	n=3, L=1, c=343	514.51	514.5	2.3e-05	✓

Acoustics — driven response

2 comparisons2/2 passmax err 7.0e-06

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ACO-0591	Driven duct pressure p(x)=p0 cos(k(L-x))/cos(kL)	1-D Helmholtz forced response closed form	Kinsler & Frey, Fundamentals of Acoustics	f=100 Hz	-2.36	-2.36	3.1e-06	✓
ACO-0592	Driven duct pressure p(x)=p0 cos(k(L-x))/cos(kL)	1-D Helmholtz forced response closed form	Kinsler & Frey, Fundamentals of Acoustics	f=150 Hz	-0.21187	-0.21187	7.0e-06	✓

Meshing — unstructured triangulation

4 comparisons4/4 passmax err 5.9e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MES-0593	Delaunay triangles tile the rectangle (area conservation)	sum(tri areas) = W*H	computational geometry	W=2, H=1	2	2	2.2e-16	✓
MES-0594	Heat solve on the auto-mesh reproduces T = x/W	linear field exact on CST triangulation	manufactured solution	W=2, H=1	4.4409e-16	0	4.4e-16	✓
MES-0595	Delaunay triangles tile the rectangle (area conservation)	sum(tri areas) = W*H	computational geometry	W=1, H=1.5	1.5	1.5	5.9e-16	✓
MES-0596	Heat solve on the auto-mesh reproduces T = x/W	linear field exact on CST triangulation	manufactured solution	W=1, H=1.5	4.9960e-16	0	5.0e-16	✓

Fatigue — stress life

6 comparisons6/6 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
FAT-0597	Basquin S-N life inverts sigma_a = sigma_f (2 Nf)^b	Nf = 0.5 (sigma_a/sigma_f)^(1/b)	Basquin (1910); ASTM E739	sa=3e+08	3.0000e+08	3.0000e+08	0.0e+00	✓
FAT-0598	Basquin S-N life inverts sigma_a = sigma_f (2 Nf)^b	Nf = 0.5 (sigma_a/sigma_f)^(1/b)	Basquin (1910); ASTM E739	sa=4.5e+08	4.5000e+08	4.5000e+08	0.0e+00	✓
FAT-0599	Basquin S-N life inverts sigma_a = sigma_f (2 Nf)^b	Nf = 0.5 (sigma_a/sigma_f)^(1/b)	Basquin (1910); ASTM E739	sa=6e+08	6.0000e+08	6.0000e+08	0.0e+00	✓
FAT-0600	Miner's-rule cumulative damage sums block damages	D = sum n_i / Nf_i	Palmgren-Miner	two blocks	2.1225	2.1225	0.0e+00	✓
FAT-0601	Goodman mean-stress correction sar = sa/(1 - sm/su)	sar = sa/(1 - sm/su)	Goodman diagram	sa=200MPa, sm=200MPa, su=1GPa	2.5000e+08	2.5000e+08	0.0e+00	✓
FAT-0602	Rainflow (ASTM E1049) interior closed-loop range	inner 1<->-1 loop -> range 2	ASTM E1049 four-point method	history [0,3,-1,1,-3,0]	2	2	0.0e+00	✓

Fatigue — strain life

5 comparisons5/5 passmax err 4.3e-15

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
FAT-0603	Basquin-Coffin-Manson life inverts the strain amplitude	eps_a = (sf/E)(2N)^b + ef(2N)^c	Coffin (1954); Manson (1953)	N=100	100	100	2.4e-15	✓
FAT-0604	Basquin-Coffin-Manson life inverts the strain amplitude	eps_a = (sf/E)(2N)^b + ef(2N)^c	Coffin (1954); Manson (1953)	N=10000	10000	10000	1.5e-15	✓
FAT-0605	Basquin-Coffin-Manson life inverts the strain amplitude	eps_a = (sf/E)(2N)^b + ef(2N)^c	Coffin (1954); Manson (1953)	N=1e+06	1.0000e+06	1.0000e+06	0.0e+00	✓
FAT-0606	Elastic-only strain-life reduces to Basquin	Nf = 0.5(eps_a E/sf)^(1/b)	consistency / limit test	ef=0	145.87	145.87	4.3e-15	✓
FAT-0607	Plastic-only strain-life reduces to Coffin-Manson	Nf = 0.5(eps_a/ef)^(1/c)	consistency / limit test	sf=0	459.79	459.79	2.8e-15	✓

Fatigue — crack growth

3 comparisons3/3 passmax err 1.9e-08

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
FAT-0608	Paris-law life matches the constant-Y closed form	N = (af^(1-m/2)-a0^(1-m/2))/(C(dsigma Y sqrt(pi))^m(1-m/2))	Paris & Erdogan (1963)	m=3	7.7663e+06	7.7663e+06	9.2e-09	✓
FAT-0609	Paris-law life matches the constant-Y closed form	N = (af^(1-m/2)-a0^(1-m/2))/(C(dsigma Y sqrt(pi))^m(1-m/2))	Paris & Erdogan (1963)	m=4	0.91189	0.91189	1.9e-08	✓
FAT-0610	Critical crack size reaches the fracture toughness	sigma_max Y sqrt(pi a_c) = KIC	linear elastic fracture mechanics	smax=200MPa, KIC=30MPa√m	3.0000e+07	3.0000e+07	0.0e+00	✓

Fatigue — from FE stress field

3 comparisons3/3 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
FAT-0611	Per-element stress-life from the FE equivalent stress	Nf_e = 0.5 (sigma_e/sigma_f)^(1/b)	Basquin S-N applied to FE stresses	A=0.0002	1.7434e+09	1.7434e+09	0.0e+00	✓
FAT-0612	Per-element stress-life from the FE equivalent stress	Nf_e = 0.5 (sigma_e/sigma_f)^(1/b)	Basquin S-N applied to FE stresses	A=0.0001	1.7025e+06	1.7025e+06	0.0e+00	✓
FAT-0613	Critical element is the shortest-life location	min over elements	consistency / decision output	two-bar	1.7025e+06	1.7025e+06	0.0e+00	✓

Contact — node-to-segment

4 comparisons4/4 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
CON-0614	Non-matching contact force distribution	reaction split N1:N2 = (1-xi):xi	Wriggers, Computational Contact Mechanics	xi=0.25	3	3	0.0e+00	✓
CON-0615	Non-matching contact force distribution	reaction split N1:N2 = (1-xi):xi	Wriggers, Computational Contact Mechanics	xi=0.5	1	1	0.0e+00	✓
CON-0616	Non-matching contact force distribution	reaction split N1:N2 = (1-xi):xi	Wriggers, Computational Contact Mechanics	xi=0.75	0.33333	0.33333	0.0e+00	✓
CON-0617	Node-to-segment reduces to node-to-node at a vertex	identical displacements when xi = 0	patch/consistency test	xi=0	-2.7500e-04	-2.7500e-04	0.0e+00	✓

Thermal stress

36 comparisons36/36 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
THE-0618	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=20.0, L=0.5	-4.8000e+07	-4.8000e+07	0.0e+00	✓
THE-0619	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=20.0, L=1.0	-4.8000e+07	-4.8000e+07	0.0e+00	✓
THE-0620	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=20.0, L=2.0	-4.8000e+07	-4.8000e+07	0.0e+00	✓
THE-0621	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=20.0, L=4.0	-4.8000e+07	-4.8000e+07	0.0e+00	✓
THE-0622	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=50.0, L=0.5	-1.2000e+08	-1.2000e+08	0.0e+00	✓
THE-0623	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=50.0, L=1.0	-1.2000e+08	-1.2000e+08	0.0e+00	✓
THE-0624	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=50.0, L=2.0	-1.2000e+08	-1.2000e+08	0.0e+00	✓
THE-0625	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=50.0, L=4.0	-1.2000e+08	-1.2000e+08	0.0e+00	✓
THE-0626	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=100.0, L=0.5	-2.4000e+08	-2.4000e+08	0.0e+00	✓
THE-0627	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=100.0, L=1.0	-2.4000e+08	-2.4000e+08	0.0e+00	✓
THE-0628	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=100.0, L=2.0	-2.4000e+08	-2.4000e+08	0.0e+00	✓
THE-0629	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.2e-05, ΔT=100.0, L=4.0	-2.4000e+08	-2.4000e+08	0.0e+00	✓
THE-0630	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=20.0, L=0.5	-6.8000e+07	-6.8000e+07	0.0e+00	✓
THE-0631	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=20.0, L=1.0	-6.8000e+07	-6.8000e+07	0.0e+00	✓
THE-0632	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=20.0, L=2.0	-6.8000e+07	-6.8000e+07	0.0e+00	✓
THE-0633	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=20.0, L=4.0	-6.8000e+07	-6.8000e+07	0.0e+00	✓
THE-0634	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=50.0, L=0.5	-1.7000e+08	-1.7000e+08	0.0e+00	✓
THE-0635	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=50.0, L=1.0	-1.7000e+08	-1.7000e+08	0.0e+00	✓
THE-0636	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=50.0, L=2.0	-1.7000e+08	-1.7000e+08	0.0e+00	✓
THE-0637	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=50.0, L=4.0	-1.7000e+08	-1.7000e+08	0.0e+00	✓
THE-0638	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=100.0, L=0.5	-3.4000e+08	-3.4000e+08	0.0e+00	✓
THE-0639	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=100.0, L=1.0	-3.4000e+08	-3.4000e+08	0.0e+00	✓
THE-0640	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=100.0, L=2.0	-3.4000e+08	-3.4000e+08	0.0e+00	✓
THE-0641	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=1.7e-05, ΔT=100.0, L=4.0	-3.4000e+08	-3.4000e+08	0.0e+00	✓
THE-0642	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=20.0, L=0.5	-9.2000e+07	-9.2000e+07	0.0e+00	✓
THE-0643	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=20.0, L=1.0	-9.2000e+07	-9.2000e+07	0.0e+00	✓
THE-0644	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=20.0, L=2.0	-9.2000e+07	-9.2000e+07	0.0e+00	✓
THE-0645	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=20.0, L=4.0	-9.2000e+07	-9.2000e+07	0.0e+00	✓
THE-0646	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=50.0, L=0.5	-2.3000e+08	-2.3000e+08	0.0e+00	✓
THE-0647	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=50.0, L=1.0	-2.3000e+08	-2.3000e+08	0.0e+00	✓
THE-0648	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=50.0, L=2.0	-2.3000e+08	-2.3000e+08	0.0e+00	✓
THE-0649	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=50.0, L=4.0	-2.3000e+08	-2.3000e+08	0.0e+00	✓
THE-0650	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=100.0, L=0.5	-4.6000e+08	-4.6000e+08	0.0e+00	✓
THE-0651	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=100.0, L=1.0	-4.6000e+08	-4.6000e+08	0.0e+00	✓
THE-0652	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=100.0, L=2.0	-4.6000e+08	-4.6000e+08	0.0e+00	✓
THE-0653	Fully-restrained bar, temperature rise	σ = −EαΔT	Timoshenko, Strength of Materials II	α=2.3e-05, ΔT=100.0, L=4.0	-4.6000e+08	-4.6000e+08	0.0e+00	✓

Torsion

27 comparisons27/27 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
TOR-0654	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=0.5, J=1e-06, T=100.0	6.5000e-04	6.5000e-04	0.0e+00	✓
TOR-0655	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=0.5, J=1e-06, T=500.0	0.00325	0.00325	0.0e+00	✓
TOR-0656	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=0.5, J=1e-06, T=2000.0	0.013	0.013	0.0e+00	✓
TOR-0657	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=0.5, J=5e-06, T=100.0	1.3000e-04	1.3000e-04	0.0e+00	✓
TOR-0658	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=0.5, J=5e-06, T=500.0	6.5000e-04	6.5000e-04	0.0e+00	✓
TOR-0659	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=0.5, J=5e-06, T=2000.0	0.0026	0.0026	0.0e+00	✓
TOR-0660	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=0.5, J=2e-05, T=100.0	3.2500e-05	3.2500e-05	0.0e+00	✓
TOR-0661	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=0.5, J=2e-05, T=500.0	1.6250e-04	1.6250e-04	0.0e+00	✓
TOR-0662	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=0.5, J=2e-05, T=2000.0	6.5000e-04	6.5000e-04	0.0e+00	✓
TOR-0663	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=1.0, J=1e-06, T=100.0	0.0013	0.0013	0.0e+00	✓
TOR-0664	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=1.0, J=1e-06, T=500.0	0.0065	0.0065	0.0e+00	✓
TOR-0665	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=1.0, J=1e-06, T=2000.0	0.026	0.026	0.0e+00	✓
TOR-0666	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=1.0, J=5e-06, T=100.0	2.6000e-04	2.6000e-04	0.0e+00	✓
TOR-0667	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=1.0, J=5e-06, T=500.0	0.0013	0.0013	0.0e+00	✓
TOR-0668	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=1.0, J=5e-06, T=2000.0	0.0052	0.0052	0.0e+00	✓
TOR-0669	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=1.0, J=2e-05, T=100.0	6.5000e-05	6.5000e-05	0.0e+00	✓
TOR-0670	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=1.0, J=2e-05, T=500.0	3.2500e-04	3.2500e-04	0.0e+00	✓
TOR-0671	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=1.0, J=2e-05, T=2000.0	0.0013	0.0013	0.0e+00	✓
TOR-0672	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=2.0, J=1e-06, T=100.0	0.0026	0.0026	0.0e+00	✓
TOR-0673	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=2.0, J=1e-06, T=500.0	0.013	0.013	0.0e+00	✓
TOR-0674	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=2.0, J=1e-06, T=2000.0	0.052	0.052	0.0e+00	✓
TOR-0675	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=2.0, J=5e-06, T=100.0	5.2000e-04	5.2000e-04	0.0e+00	✓
TOR-0676	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=2.0, J=5e-06, T=500.0	0.0026	0.0026	0.0e+00	✓
TOR-0677	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=2.0, J=5e-06, T=2000.0	0.0104	0.0104	0.0e+00	✓
TOR-0678	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=2.0, J=2e-05, T=100.0	1.3000e-04	1.3000e-04	0.0e+00	✓
TOR-0679	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=2.0, J=2e-05, T=500.0	6.5000e-04	6.5000e-04	0.0e+00	✓
TOR-0680	Circular shaft, end torque — twist	φ = TL / GJ	Timoshenko, Strength of Materials II	L=2.0, J=2e-05, T=2000.0	0.0026	0.0026	0.0e+00	✓

Continuum patch tests

5 comparisons5/5 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
CON-0681	Uniform-strain patch — quad4 (plane_stress)	σₓₓ = C·εₓₓ (constant-strain patch)	MacNeal & Harder (1985); NAFEMS	element=quad4, ε=0.001	2.1333e+08	2.1333e+08	0.0e+00	✓
CON-0682	Uniform-strain patch — cst (plane_stress)	σₓₓ = C·εₓₓ (constant-strain patch)	MacNeal & Harder (1985); NAFEMS	element=cst, ε=0.001	2.1333e+08	2.1333e+08	0.0e+00	✓
CON-0683	Uniform-strain patch — quad8 (plane_stress)	σₓₓ = C·εₓₓ (constant-strain patch)	MacNeal & Harder (1985); NAFEMS	element=quad8, ε=0.001	2.1333e+08	2.1333e+08	0.0e+00	✓
CON-0684	Uniform-strain patch — tri6 (plane_stress)	σₓₓ = C·εₓₓ (constant-strain patch)	MacNeal & Harder (1985); NAFEMS	element=tri6, ε=0.001	2.1333e+08	2.1333e+08	0.0e+00	✓
CON-0685	Uniform-strain patch — quad4 (plane_strain)	σₓₓ = C·εₓₓ (constant-strain patch)	MacNeal & Harder (1985); NAFEMS	element=quad4, ε=0.001	2.4000e+08	2.4000e+08	0.0e+00	✓

3D solids

3 comparisons3/3 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
3D-0686	Uniform-strain patch — hex8	σₓₓ = C·εₓₓ	MacNeal & Harder (1985)	ε=0.001	2.6923e+08	2.6923e+08	0.0e+00	✓
3D-0687	Uniform-strain patch - tet4	sigma_xx = C.eps_xx	MacNeal & Harder (1985)	tet #1, eps=0.001	2.6923e+08	2.6923e+08	0.0e+00	✓
3D-0688	Uniform-strain patch - tet4	sigma_xx = C.eps_xx	MacNeal & Harder (1985)	tet #2, eps=0.001	2.6923e+08	2.6923e+08	0.0e+00	✓

3D frames

10 comparisons10/10 passmax err 2.8e-13

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
3D-0689	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=1.0, Iz=8e-06, P=5000	9.9206e-04	9.9206e-04	1.4e-13	✓
3D-0690	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=1.0, Iz=3e-05, P=5000	2.6455e-04	2.6455e-04	2.8e-13	✓
3D-0691	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=2.0, Iz=8e-06, P=5000	0.0079365	0.0079365	1.6e-13	✓
3D-0692	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=2.0, Iz=3e-05, P=5000	0.0021164	0.0021164	1.3e-13	✓
3D-0693	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=3.0, Iz=8e-06, P=5000	0.026786	0.026786	3.2e-14	✓
3D-0694	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=3.0, Iz=3e-05, P=5000	0.0071429	0.0071429	4.1e-14	✓
3D-0695	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=4.0, Iz=8e-06, P=5000	0.063492	0.063492	9.2e-14	✓
3D-0696	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=4.0, Iz=3e-05, P=5000	0.016931	0.016931	1.7e-14	✓
3D-0697	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=5.0, Iz=8e-06, P=5000	0.12401	0.12401	3.9e-14	✓
3D-0698	Space-frame cantilever, transverse tip load	delta = PL^3 / 3E Iz	Timoshenko, Strength of Materials I	L=5.0, Iz=3e-05, P=5000	0.033069	0.033069	1.7e-13	✓

Orthotropic & composites

120 comparisons120/120 passmax err 2.1e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ORT-0699	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=0 deg	2.1817e+08	2.1817e+08	0.0e+00	✓
ORT-0700	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=0 deg	3.4763e+06	3.4763e+06	0.0e+00	✓
ORT-0701	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=0 deg	1.8181e+08	1.8181e+08	0.0e+00	✓
ORT-0702	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=0 deg	2.8969e+06	2.8969e+06	0.0e+00	✓
ORT-0703	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=0 deg	5.7360e+06	5.7360e+06	1.6e-16	✓
ORT-0704	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=15 deg	1.9256e+08	1.9256e+08	0.0e+00	✓
ORT-0705	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=15 deg	1.5303e+07	1.5303e+07	0.0e+00	✓
ORT-0706	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=15 deg	4.6203e+07	4.6203e+07	0.0e+00	✓
ORT-0707	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=15 deg	1.9127e+08	1.9127e+08	0.0e+00	✓
ORT-0708	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=15 deg	1.6243e+07	1.6243e+07	0.0e+00	✓
ORT-0709	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=15 deg	5.2123e+07	5.2123e+07	0.0e+00	✓
ORT-0710	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=30 deg	1.3126e+08	1.3126e+08	0.0e+00	✓
ORT-0711	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=30 deg	3.8955e+07	3.8955e+07	0.0e+00	✓
ORT-0712	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=30 deg	6.5032e+07	6.5032e+07	0.0e+00	✓
ORT-0713	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=30 deg	1.5273e+08	1.5273e+08	0.0e+00	✓
ORT-0714	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=30 deg	4.8505e+07	4.8505e+07	1.5e-16	✓
ORT-0715	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=30 deg	8.3582e+07	8.3582e+07	0.0e+00	✓
ORT-0716	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=45 deg	6.7989e+07	6.7989e+07	0.0e+00	✓
ORT-0717	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=45 deg	5.0781e+07	5.0781e+07	0.0e+00	✓
ORT-0718	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=45 deg	5.1439e+07	5.1439e+07	0.0e+00	✓
ORT-0719	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=45 deg	9.0951e+07	9.0951e+07	0.0e+00	✓
ORT-0720	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=45 deg	7.6611e+07	7.6611e+07	1.9e-16	✓
ORT-0721	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=45 deg	8.0139e+07	8.0139e+07	1.9e-16	✓
ORT-0722	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=60 deg	2.8376e+07	2.8376e+07	0.0e+00	✓
ORT-0723	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=60 deg	3.8955e+07	3.8955e+07	0.0e+00	✓
ORT-0724	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=60 deg	2.4064e+07	2.4064e+07	0.0e+00	✓
ORT-0725	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=60 deg	3.9690e+07	3.9690e+07	0.0e+00	✓
ORT-0726	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=60 deg	7.5817e+07	7.5817e+07	0.0e+00	✓
ORT-0727	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=60 deg	4.9442e+07	4.9442e+07	0.0e+00	✓
ORT-0728	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=75 deg	1.4372e+07	1.4372e+07	0.0e+00	✓
ORT-0729	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=75 deg	1.5303e+07	1.5303e+07	0.0e+00	✓
ORT-0730	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=75 deg	5.2361e+06	5.2361e+06	0.0e+00	✓
ORT-0731	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=75 deg	1.5468e+07	1.5468e+07	0.0e+00	✓
ORT-0732	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=75 deg	4.3554e+07	4.3554e+07	1.7e-16	✓
ORT-0733	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=75 deg	1.7984e+07	1.7984e+07	0.0e+00	✓
ORT-0734	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=90 deg	1.2415e+07	1.2415e+07	0.0e+00	✓
ORT-0735	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=90 deg	3.4763e+06	3.4763e+06	0.0e+00	✓
ORT-0736	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=90 deg	1.0346e+07	1.0346e+07	0.0e+00	✓
ORT-0737	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=90 deg	2.8969e+06	2.8969e+06	0.0e+00	✓
ORT-0738	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	T300/5208 graphite-epoxy, theta=90 deg	5.7360e+06	5.7360e+06	1.6e-16	✓
ORT-0739	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=0 deg	4.7001e+07	4.7001e+07	0.0e+00	✓
ORT-0740	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=0 deg	2.6182e+06	2.6182e+06	0.0e+00	✓
ORT-0741	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=0 deg	3.9167e+07	3.9167e+07	0.0e+00	✓
ORT-0742	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=0 deg	2.1818e+06	2.1818e+06	0.0e+00	✓
ORT-0743	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=0 deg	3.3120e+06	3.3120e+06	1.4e-16	✓
ORT-0744	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=15 deg	4.2529e+07	4.2529e+07	0.0e+00	✓
ORT-0745	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=15 deg	4.6158e+06	4.6158e+06	0.0e+00	✓
ORT-0746	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=15 deg	8.0764e+06	8.0764e+06	0.0e+00	✓
ORT-0747	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=15 deg	4.0825e+07	4.0825e+07	0.0e+00	✓
ORT-0748	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=15 deg	4.6174e+06	4.6174e+06	0.0e+00	✓
ORT-0749	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=15 deg	1.1374e+07	1.1374e+07	0.0e+00	✓
ORT-0750	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=30 deg	3.1775e+07	3.1775e+07	0.0e+00	✓
ORT-0751	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=30 deg	8.6111e+06	8.6111e+06	0.0e+00	✓
ORT-0752	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=30 deg	1.1456e+07	1.1456e+07	0.0e+00	✓
ORT-0753	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=30 deg	3.4116e+07	3.4116e+07	0.0e+00	✓
ORT-0754	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=30 deg	1.0200e+07	1.0200e+07	0.0e+00	✓
ORT-0755	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=30 deg	1.6854e+07	1.6854e+07	0.0e+00	✓
ORT-0756	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=45 deg	2.0545e+07	2.0545e+07	0.0e+00	✓
ORT-0757	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=45 deg	1.0609e+07	1.0609e+07	0.0e+00	✓
ORT-0758	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=45 deg	9.2327e+06	9.2327e+06	0.0e+00	✓
ORT-0759	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=45 deg	2.3276e+07	2.3276e+07	0.0e+00	✓
ORT-0760	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=45 deg	1.4996e+07	1.4996e+07	0.0e+00	✓
ORT-0761	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=45 deg	1.6333e+07	1.6333e+07	1.1e-16	✓
ORT-0762	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=60 deg	1.3310e+07	1.3310e+07	0.0e+00	✓
ORT-0763	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=60 deg	8.6111e+06	8.6111e+06	0.0e+00	✓
ORT-0764	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=60 deg	4.5358e+06	4.5358e+06	0.0e+00	✓
ORT-0765	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=60 deg	1.4115e+07	1.4115e+07	0.0e+00	✓
ORT-0766	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=60 deg	1.4813e+07	1.4813e+07	0.0e+00	✓
ORT-0767	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=60 deg	1.1087e+07	1.1087e+07	0.0e+00	✓
ORT-0768	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=75 deg	1.0546e+07	1.0546e+07	0.0e+00	✓
ORT-0769	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=75 deg	4.6158e+06	4.6158e+06	0.0e+00	✓
ORT-0770	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=75 deg	1.1563e+06	1.1563e+06	0.0e+00	✓
ORT-0771	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=75 deg	9.5593e+06	9.5593e+06	0.0e+00	✓
ORT-0772	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=75 deg	9.2307e+06	9.2307e+06	0.0e+00	✓
ORT-0773	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=75 deg	5.6074e+06	5.6074e+06	1.7e-16	✓
ORT-0774	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=90 deg	1.0070e+07	1.0070e+07	0.0e+00	✓
ORT-0775	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=90 deg	2.6182e+06	2.6182e+06	0.0e+00	✓
ORT-0776	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=90 deg	8.3915e+06	8.3915e+06	0.0e+00	✓
ORT-0777	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=90 deg	2.1818e+06	2.1818e+06	0.0e+00	✓
ORT-0778	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Scotchply glass-epoxy, theta=90 deg	3.3120e+06	3.3120e+06	1.4e-16	✓
ORT-0779	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=0 deg	2.4598e+08	2.4598e+08	0.0e+00	✓
ORT-0780	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=0 deg	5.1306e+06	5.1306e+06	0.0e+00	✓
ORT-0781	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=0 deg	2.0498e+08	2.0498e+08	0.0e+00	✓
ORT-0782	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=0 deg	4.2755e+06	4.2755e+06	0.0e+00	✓
ORT-0783	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=0 deg	4.4720e+06	4.4720e+06	0.0e+00	✓
ORT-0784	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=15 deg	2.1655e+08	2.1655e+08	0.0e+00	✓
ORT-0785	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=15 deg	1.9580e+07	1.9580e+07	0.0e+00	✓
ORT-0786	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=15 deg	5.2987e+07	5.2987e+07	0.0e+00	✓
ORT-0787	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=15 deg	2.1578e+08	2.1578e+08	0.0e+00	✓
ORT-0788	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=15 deg	1.8271e+07	1.8271e+07	0.0e+00	✓
ORT-0789	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=15 deg	5.8261e+07	5.8261e+07	0.0e+00	✓
ORT-0790	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=30 deg	1.4671e+08	1.4671e+08	0.0e+00	✓
ORT-0791	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=30 deg	4.8479e+07	4.8479e+07	0.0e+00	✓
ORT-0792	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=30 deg	7.3454e+07	7.3454e+07	0.0e+00	✓
ORT-0793	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=30 deg	1.7123e+08	1.7123e+08	1.7e-16	✓
ORT-0794	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=30 deg	5.5999e+07	5.5999e+07	1.3e-16	✓
ORT-0795	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=30 deg	9.4583e+07	9.4583e+07	0.0e+00	✓
ORT-0796	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=45 deg	7.6345e+07	7.6345e+07	0.0e+00	✓
ORT-0797	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=45 deg	6.2929e+07	6.2929e+07	0.0e+00	✓
ORT-0798	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=45 deg	5.5918e+07	5.5918e+07	0.0e+00	✓
ORT-0799	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=45 deg	1.0090e+08	1.0090e+08	1.5e-16	✓
ORT-0800	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=45 deg	8.9720e+07	8.9720e+07	1.7e-16	✓
ORT-0801	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=45 deg	8.9603e+07	8.9603e+07	1.7e-16	✓
ORT-0802	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=60 deg	3.4876e+07	3.4876e+07	0.0e+00	✓
ORT-0803	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=60 deg	4.8479e+07	4.8479e+07	0.0e+00	✓
ORT-0804	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=60 deg	2.3399e+07	2.3399e+07	0.0e+00	✓
ORT-0805	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=60 deg	4.4663e+07	4.4663e+07	0.0e+00	✓
ORT-0806	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=60 deg	8.9369e+07	8.9369e+07	1.7e-16	✓
ORT-0807	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=60 deg	5.2871e+07	5.2871e+07	1.4e-16	✓
ORT-0808	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=75 deg	2.2841e+07	2.2841e+07	0.0e+00	✓
ORT-0809	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=75 deg	1.9580e+07	1.9580e+07	0.0e+00	✓
ORT-0810	Lamina stress sigma_xy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=75 deg	2.9317e+06	2.9317e+06	0.0e+00	✓
ORT-0811	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=75 deg	2.0988e+07	2.0988e+07	0.0e+00	✓
ORT-0812	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=75 deg	5.1641e+07	5.1641e+07	1.4e-16	✓
ORT-0813	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=75 deg	1.6548e+07	1.6548e+07	1.1e-16	✓
ORT-0814	Lamina stress sigma_xx (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=90 deg	2.2307e+07	2.2307e+07	0.0e+00	✓
ORT-0815	Lamina stress sigma_yy (uniaxial-x)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=90 deg	5.1306e+06	5.1306e+06	0.0e+00	✓
ORT-0816	Lamina stress sigma_xx (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=90 deg	1.8589e+07	1.8589e+07	0.0e+00	✓
ORT-0817	Lamina stress sigma_yy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=90 deg	4.2755e+06	4.2755e+06	0.0e+00	✓
ORT-0818	Lamina stress sigma_xy (shear-coupled)	sigma = Qbar(theta) . epsilon (transformed lamina stiffness)	Jones, Mechanics of Composite Materials	Boron-epoxy, theta=90 deg	4.4720e+06	4.4720e+06	2.1e-16	✓

Composite laminates (CLT)

22 comparisons22/22 passmax err 2.2e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
COM-0819	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=7e+10, H=0.002	51.282	51.282	0.0e+00	✓
COM-0820	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=7e+10, H=0.004	410.26	410.26	0.0e+00	✓
COM-0821	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=7e+10, H=0.006	1384.6	1384.6	1.6e-16	✓
COM-0822	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=7e+10, H=0.004	410.26	410.26	0.0e+00	✓
COM-0823	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=7e+10, H=0.008	3282.1	3282.1	0.0e+00	✓
COM-0824	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=7e+10, H=0.012	11077	11077	1.6e-16	✓
COM-0825	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=1.4e+11, H=0.002	102.56	102.56	0.0e+00	✓
COM-0826	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=1.4e+11, H=0.004	820.51	820.51	0.0e+00	✓
COM-0827	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=1.4e+11, H=0.006	2769.2	2769.2	1.6e-16	✓
COM-0828	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=1.4e+11, H=0.004	820.51	820.51	0.0e+00	✓
COM-0829	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=1.4e+11, H=0.008	6564.1	6564.1	0.0e+00	✓
COM-0830	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=1.4e+11, H=0.012	22154	22154	1.6e-16	✓
COM-0831	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=2.1e+11, H=0.002	153.85	153.85	0.0e+00	✓
COM-0832	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=2.1e+11, H=0.004	1230.8	1230.8	0.0e+00	✓
COM-0833	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=2.1e+11, H=0.006	4153.8	4153.8	2.2e-16	✓
COM-0834	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=2.1e+11, H=0.004	1230.8	1230.8	0.0e+00	✓
COM-0835	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=2.1e+11, H=0.008	9846.2	9846.2	0.0e+00	✓
COM-0836	Isotropic laminate bending stiffness D11	D11 = E H^3 / 12(1-nu^2)	Jones, Mechanics of Composite Materials	E=2.1e+11, H=0.012	33231	33231	2.2e-16	✓
COM-0837	Unidirectional laminate, effective Ex	Ex = E1 (0-deg lamina)	Jones, Mechanics of Composite Materials	E1=1.4e+11	1.4000e+11	1.4000e+11	0.0e+00	✓
COM-0838	Unidirectional laminate, effective Ey	Ey = E2 (0-deg lamina)	Jones, Mechanics of Composite Materials	E2=1e+10	1.0000e+10	1.0000e+10	0.0e+00	✓
COM-0839	Unidirectional laminate, effective Ex	Ex = E1 (0-deg lamina)	Jones, Mechanics of Composite Materials	E1=1.81e+11	1.8100e+11	1.8100e+11	1.7e-16	✓
COM-0840	Unidirectional laminate, effective Ey	Ey = E2 (0-deg lamina)	Jones, Mechanics of Composite Materials	E2=1.03e+10	1.0300e+10	1.0300e+10	0.0e+00	✓

Micromechanics

14 comparisons14/14 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MIC-0841	Longitudinal modulus, rule of mixtures	E1 = Vf Ef + Vm Em	Chamis (1989); Jones	Vf=0.0	3.4000e+09	3.4000e+09	0.0e+00	✓
MIC-0842	Transverse modulus within Voigt/Reuss bounds	Reuss <= E2 <= Voigt	Halpin & Tsai (1969)	Vf=0.0	1	1	0.0e+00	✓
MIC-0843	Longitudinal modulus, rule of mixtures	E1 = Vf Ef + Vm Em	Chamis (1989); Jones	Vf=0.2	4.8720e+10	4.8720e+10	0.0e+00	✓
MIC-0844	Transverse modulus within Voigt/Reuss bounds	Reuss <= E2 <= Voigt	Halpin & Tsai (1969)	Vf=0.2	1	1	0.0e+00	✓
MIC-0845	Longitudinal modulus, rule of mixtures	E1 = Vf Ef + Vm Em	Chamis (1989); Jones	Vf=0.4	9.4040e+10	9.4040e+10	0.0e+00	✓
MIC-0846	Transverse modulus within Voigt/Reuss bounds	Reuss <= E2 <= Voigt	Halpin & Tsai (1969)	Vf=0.4	1	1	0.0e+00	✓
MIC-0847	Longitudinal modulus, rule of mixtures	E1 = Vf Ef + Vm Em	Chamis (1989); Jones	Vf=0.5	1.1670e+11	1.1670e+11	0.0e+00	✓
MIC-0848	Transverse modulus within Voigt/Reuss bounds	Reuss <= E2 <= Voigt	Halpin & Tsai (1969)	Vf=0.5	1	1	0.0e+00	✓
MIC-0849	Longitudinal modulus, rule of mixtures	E1 = Vf Ef + Vm Em	Chamis (1989); Jones	Vf=0.6	1.3936e+11	1.3936e+11	0.0e+00	✓
MIC-0850	Transverse modulus within Voigt/Reuss bounds	Reuss <= E2 <= Voigt	Halpin & Tsai (1969)	Vf=0.6	1	1	0.0e+00	✓
MIC-0851	Longitudinal modulus, rule of mixtures	E1 = Vf Ef + Vm Em	Chamis (1989); Jones	Vf=0.7	1.6202e+11	1.6202e+11	0.0e+00	✓
MIC-0852	Transverse modulus within Voigt/Reuss bounds	Reuss <= E2 <= Voigt	Halpin & Tsai (1969)	Vf=0.7	1	1	0.0e+00	✓
MIC-0853	Longitudinal modulus, rule of mixtures	E1 = Vf Ef + Vm Em	Chamis (1989); Jones	Vf=1.0	2.3000e+11	2.3000e+11	0.0e+00	✓
MIC-0854	Transverse modulus within Voigt/Reuss bounds	Reuss <= E2 <= Voigt	Halpin & Tsai (1969)	Vf=1.0	1	1	0.0e+00	✓

Progressive failure (Hashin)

8 comparisons8/8 passmax err 7.1e-08

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
PRO-0855	First-ply-failure: 90-deg ply matrix tension	sigma_22 = Yt at first-ply-failure	Hashin (1980)	Yt=4e+07	4.0000e+07	4.0000e+07	0.0e+00	✓
PRO-0856	Last-ply-failure load (fibre tension)	N_lpf = Xt * t(0-deg plies)	Hashin (1980); force balance	Xt=1.5e9	3.7500e+05	3.7500e+05	5.6e-08	✓
PRO-0857	First-ply-failure: 90-deg ply matrix tension	sigma_22 = Yt at first-ply-failure	Hashin (1980)	Yt=6e+07	6.0000e+07	6.0000e+07	3.7e-16	✓
PRO-0858	Last-ply-failure load (fibre tension)	N_lpf = Xt * t(0-deg plies)	Hashin (1980); force balance	Xt=1.5e9	3.7500e+05	3.7500e+05	5.6e-08	✓
PRO-0859	First-ply-failure: 90-deg ply matrix tension	sigma_22 = Yt at first-ply-failure	Hashin (1980)	Yt=4e+07	4.0000e+07	4.0000e+07	0.0e+00	✓
PRO-0860	Last-ply-failure load (fibre tension)	N_lpf = Xt * t(0-deg plies)	Hashin (1980); force balance	Xt=1.5e9	3.7500e+05	3.7500e+05	7.1e-08	✓
PRO-0861	First-ply-failure: 90-deg ply matrix tension	sigma_22 = Yt at first-ply-failure	Hashin (1980)	Yt=6e+07	6.0000e+07	6.0000e+07	0.0e+00	✓
PRO-0862	Last-ply-failure load (fibre tension)	N_lpf = Xt * t(0-deg plies)	Hashin (1980); force balance	Xt=1.5e9	3.7500e+05	3.7500e+05	7.1e-08	✓

Oxidation (reaction-diffusion)

8 comparisons8/8 passmax err 1.8e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
OXI-0863	Reactant profile at sealed face	c(L) = c0 / cosh(beta L), beta=sqrt(k/D)	Thiele (1939)	beta*L=1.0	0.64805	0.64805	8.8e-06	✓
OXI-0864	Reactant profile at mid-depth	c(x) = c0 cosh(beta(L-x))/cosh(beta L)	Thiele (1939)	beta*L=1.0	0.73076	0.73076	6.1e-06	✓
OXI-0865	Reactant profile at sealed face	c(L) = c0 / cosh(beta L), beta=sqrt(k/D)	Thiele (1939)	beta*L=1.5	0.42508	0.4251	3.5e-05	✓
OXI-0866	Reactant profile at mid-depth	c(x) = c0 cosh(beta(L-x))/cosh(beta L)	Thiele (1939)	beta*L=1.5	0.55035	0.55036	2.3e-05	✓
OXI-0867	Reactant profile at sealed face	c(L) = c0 / cosh(beta L), beta=sqrt(k/D)	Thiele (1939)	beta*L=2.0	0.26578	0.2658	8.9e-05	✓
OXI-0868	Reactant profile at mid-depth	c(x) = c0 cosh(beta(L-x))/cosh(beta L)	Thiele (1939)	beta*L=2.0	0.41013	0.41015	5.4e-05	✓
OXI-0869	Reactant profile at sealed face	c(L) = c0 / cosh(beta L), beta=sqrt(k/D)	Thiele (1939)	beta*L=2.5	0.16304	0.16307	1.8e-04	✓
OXI-0870	Reactant profile at mid-depth	c(x) = c0 cosh(beta(L-x))/cosh(beta L)	Thiele (1939)	beta*L=2.5	0.30792	0.30795	1.0e-04	✓

User material (UMAT hook)

5 comparisons5/5 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
USE-0871	Registered isotropic law reproduces built-in	sigma(UMAT) = sigma(built-in)	hook self-consistency	isotropic	1	1	0.0e+00	✓
USE-0872	Damage-degraded stiffness law	sigma = (1-d) . sigma_base	ply-discount / CDM	d=0.0	2.1333e+08	2.1333e+08	0.0e+00	✓
USE-0873	Damage-degraded stiffness law	sigma = (1-d) . sigma_base	ply-discount / CDM	d=0.3	1.4933e+08	1.4933e+08	0.0e+00	✓
USE-0874	Damage-degraded stiffness law	sigma = (1-d) . sigma_base	ply-discount / CDM	d=0.6	8.5333e+07	8.5333e+07	0.0e+00	✓
USE-0875	Damage-degraded stiffness law	sigma = (1-d) . sigma_base	ply-discount / CDM	d=0.9	2.1333e+07	2.1333e+07	0.0e+00	✓

Composite literature benchmarks

20 comparisons20/20 passmax err 4.5e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
COM-0876	Quasi-isotropic laminate modulus	Ex = (U1^2 - U4^2)/U1	Tsai & Pagano (1968), laminate invariants	T300/BSL914C (WWFE)	5.4136e+10	5.4136e+10	1.4e-16	✓
COM-0877	Quasi-isotropic: Ex = Ey (in-plane isotropy)	Ex = Ey	Tsai & Pagano (1968)	T300/BSL914C (WWFE)	5.4136e+10	5.4136e+10	4.2e-16	✓
COM-0878	Quasi-isotropic shear modulus Gxy = U5	Gxy = U5 = (U1-U4)/2	Tsai & Pagano (1968)	T300/BSL914C (WWFE)	2.0717e+10	2.0717e+10	1.8e-16	✓
COM-0879	Quasi-isotropic rotation invariance	Ex(phi) = Ex(0) for a quasi-isotropic laminate	Tsai & Pagano (1968)	T300/BSL914C (WWFE), phi=17.0	5.4136e+10	5.4136e+10	0.0e+00	✓
COM-0880	Quasi-isotropic rotation invariance	Ex(phi) = Ex(0) for a quasi-isotropic laminate	Tsai & Pagano (1968)	T300/BSL914C (WWFE), phi=31.0	5.4136e+10	5.4136e+10	0.0e+00	✓
COM-0881	Quasi-isotropic laminate modulus	Ex = (U1^2 - U4^2)/U1	Tsai & Pagano (1968), laminate invariants	E-glass/LY556 (WWFE)	2.5312e+10	2.5312e+10	3.0e-16	✓
COM-0882	Quasi-isotropic: Ex = Ey (in-plane isotropy)	Ex = Ey	Tsai & Pagano (1968)	E-glass/LY556 (WWFE)	2.5312e+10	2.5312e+10	4.5e-16	✓
COM-0883	Quasi-isotropic shear modulus Gxy = U5	Gxy = U5 = (U1-U4)/2	Tsai & Pagano (1968)	E-glass/LY556 (WWFE)	9.7004e+09	9.7004e+09	0.0e+00	✓
COM-0884	Quasi-isotropic rotation invariance	Ex(phi) = Ex(0) for a quasi-isotropic laminate	Tsai & Pagano (1968)	E-glass/LY556 (WWFE), phi=17.0	2.5312e+10	2.5312e+10	1.5e-16	✓
COM-0885	Quasi-isotropic rotation invariance	Ex(phi) = Ex(0) for a quasi-isotropic laminate	Tsai & Pagano (1968)	E-glass/LY556 (WWFE), phi=31.0	2.5312e+10	2.5312e+10	0.0e+00	✓
COM-0886	Quasi-isotropic laminate modulus	Ex = (U1^2 - U4^2)/U1	Tsai & Pagano (1968), laminate invariants	AS4/3501-6 (WWFE)	5.1061e+10	5.1061e+10	3.0e-16	✓
COM-0887	Quasi-isotropic: Ex = Ey (in-plane isotropy)	Ex = Ey	Tsai & Pagano (1968)	AS4/3501-6 (WWFE)	5.1061e+10	5.1061e+10	4.5e-16	✓
COM-0888	Quasi-isotropic shear modulus Gxy = U5	Gxy = U5 = (U1-U4)/2	Tsai & Pagano (1968)	AS4/3501-6 (WWFE)	1.9768e+10	1.9768e+10	0.0e+00	✓
COM-0889	Quasi-isotropic rotation invariance	Ex(phi) = Ex(0) for a quasi-isotropic laminate	Tsai & Pagano (1968)	AS4/3501-6 (WWFE), phi=17.0	5.1061e+10	5.1061e+10	1.5e-16	✓
COM-0890	Quasi-isotropic rotation invariance	Ex(phi) = Ex(0) for a quasi-isotropic laminate	Tsai & Pagano (1968)	AS4/3501-6 (WWFE), phi=31.0	5.1061e+10	5.1061e+10	1.5e-16	✓
COM-0891	Tsai-Wu axis strength: fibre tension	criterion reduces to the uniaxial strength	Tsai & Wu (1971)	fibre tension	1.5000e+09	1.5000e+09	1.6e-16	✓
COM-0892	Tsai-Wu axis strength: fibre compression	criterion reduces to the uniaxial strength	Tsai & Wu (1971)	fibre compression	9.0000e+08	9.0000e+08	1.3e-16	✓
COM-0893	Tsai-Wu axis strength: transverse tension	criterion reduces to the uniaxial strength	Tsai & Wu (1971)	transverse tension	2.7000e+07	2.7000e+07	4.1e-16	✓
COM-0894	Tsai-Wu axis strength: transverse compression	criterion reduces to the uniaxial strength	Tsai & Wu (1971)	transverse compression	2.0000e+08	2.0000e+08	0.0e+00	✓
COM-0895	Tsai-Wu axis strength: in-plane shear	criterion reduces to the uniaxial strength	Tsai & Wu (1971)	in-plane shear	8.0000e+07	8.0000e+07	0.0e+00	✓

Self-verification (error estimator)

3 comparisons3/3 passmax err 3.9e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
SEL-0896	ZZ error ~ 0 on a constant-strain patch	eta -> 0 for an exactly-representable field	Zienkiewicz & Zhu (1987)	6x4 patch test	3.9389e-16	0	3.9e-16	✓
SEL-0897	Under-resolved mesh flagged (no false pass)	coarse-mesh guard	Zienkiewicz & Zhu (1987)	4x1 bending	1	1	0.0e+00	✓
SEL-0898	Error estimate decreases under refinement	eta(fine) < eta(coarse)	Zienkiewicz & Zhu (1987)	8x2 -> 16x4 bending	1	1	0.0e+00	✓

Adaptive refinement

2 comparisons2/2 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ADA-0899	Error estimate decreases as the mesh auto-refines	eta(round n+1) < eta(round n)	Zienkiewicz & Zhu (1987), h-adaptivity	4x2 -> refined twice	1	1	0.0e+00	✓
ADA-0900	Refinement quadruples the element count each round	uniform h-refinement: 1 quad -> 4	conforming h-refinement	round 0 -> round 1	4	4	0.0e+00	✓

Targeted refinement

3 comparisons3/3 passmax err 2.8e-12

ID	Problem	Reference	Source	Parameters	Computed	Rel. err	✓
TAR-0901	Hanging-node constraint passes the linear patch test	u_hang = 1/2(u_a + u_b) -> linear field exact	FE consistency (patch test)	refined quad adjacent to a coarse quad	2.7778e-12	2.8e-12	✓
TAR-0902	Stress is continuous across the coarse/fine T-junction	max(sigma_xx) - min(sigma_xx) = 0	FE consistency (patch test)	5 elements, 1 hanging node	2.1198e-12	2.1e-12	✓
TAR-0903	All-marked targeted refinement equals uniform refinement	mark every element -> conforming, 0 hanging nodes	refinement-operator consistency	4x2 mesh	0	0.0e+00	✓

3D error estimate

3 comparisons3/3 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
3D-0904	Constant-stress hex8 block estimates ~0 discretization error	recovered stress = FE stress -> eta = 0	Zienkiewicz & Zhu (1987), 3D	2x2x2 uniform stretch	0	0	0.0e+00	✓
3D-0905	hex8 error estimate decreases under refinement	eta(fine) < eta(coarse)	Zienkiewicz & Zhu (1987), 3D	sheared block 2^3 -> 4^3	1	1	0.0e+00	✓
3D-0906	Mindlin-plate bending-moment estimate decreases under refinement	eta(fine) < eta(coarse)	Zienkiewicz & Zhu (1987), plate bending	clamped plate, central load, 4x4 -> 8x8	1	1	0.0e+00	✓

Result assessment

4 comparisons4/4 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
RES-0907	Peak stress is reported in the root-adjacent element	first element centroid x = L/(2 nx) for centroidal stress output	beam theory locates the peak at the support; element output is centroidal	16x4 cantilever, end shear	0.25	0.25	0.0e+00	✓
RES-0908	Peak deflection is reported at the free tip (x = L)	max \|u\| at the loaded free end	beam theory (deflection peaks at the tip)	16x4 cantilever, end shear	8	8	0.0e+00	✓
RES-0909	Safety factor equals allowable stress / peak stress	SF = sigma_allow / sigma_max	definition of the factor of safety	allowable = 2x peak	2	2	0.0e+00	✓
RES-0910	Verdict is FAIL when peak stress exceeds the allowable	SF < 1 -> FAIL	definition of the factor of safety	allowable = 0.5x peak	1	1	0.0e+00	✓

Higher-order elements

4 comparisons4/4 passmax err 4.6e-03

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
HIG-0911	tet10 passes the uniform-strain patch test	recovered strain = exact	FE consistency (patch test)	10-node quadratic tet	6.5052e-19	0	6.5e-19	✓
HIG-0912	hex20 passes the uniform-strain patch test	recovered strain = exact	FE consistency (patch test)	20-node serendipity hex	4.3368e-19	0	4.3e-19	✓
HIG-0913	MITC4 plate is rank-sufficient (3 zero-energy modes)	zero modes = 3 (rigid body)	Dvorkin & Bathe (1984)	single element eigenvalues	3	3	0.0e+00	✓
HIG-0914	MITC4 thin plate does not shear-lock (Kirchhoff limit)	w -> 0.00406 q a^4/D as t/a -> 0	Timoshenko, Theory of Plates	t/a = 1e-3, simply supported	2.2066e-04	2.2168e-04	4.6e-03	✓

Advanced analyses

5 comparisons5/5 passmax err 9.4e-06

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ADV-0915	Topology optimization hits the volume budget	final volume fraction = target	SIMP (Bendsoe & Sigmund)	cantilever, 30x15	0.4	0.4	9.4e-06	✓
ADV-0916	Topology optimization increases stiffness	compliance(final) < compliance(initial)	SIMP compliance minimization	cantilever, 30x15	1	1	0.0e+00	✓
ADV-0917	XFEM recovers the handbook stress-intensity factor	K_I within a few % of SENT handbook	Tada, Paris & Irwin handbook	a/W = 0.4, 40x41	1	1	0.0e+00	✓
ADV-0918	Phase-field fracture shows the peak-then-drop signature	final reaction < peak reaction (crack severs)	Miehe et al. (2010)	SENT, 32x32	1	1	0.0e+00	✓
ADV-0919	Moving heat source forms a melt pool at the Rosenthal scale	peak temperature > 1500 C, melt pool present	Rosenthal (1946)	80x40 plate, laser sweep	1	1	0.0e+00	✓

Heat transfer

36 comparisons36/36 passmax err 1.3e-15

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
HEA-0920	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=0.5, T0=0.0, TL=100.0	50	50	0.0e+00	✓
HEA-0921	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=0.5, T0=20.0, TL=80.0	50	50	2.8e-16	✓
HEA-0922	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=0.5, T0=-40.0, TL=120.0	40	40	3.6e-16	✓
HEA-0923	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=1.0, T0=0.0, TL=100.0	50	50	0.0e+00	✓
HEA-0924	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=1.0, T0=20.0, TL=80.0	50	50	2.8e-16	✓
HEA-0925	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=1.0, T0=-40.0, TL=120.0	40	40	3.6e-16	✓
HEA-0926	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=2.0, T0=0.0, TL=100.0	50	50	0.0e+00	✓
HEA-0927	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=2.0, T0=20.0, TL=80.0	50	50	2.8e-16	✓
HEA-0928	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=2.0, T0=-40.0, TL=120.0	40	40	3.6e-16	✓
HEA-0929	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=4.0, T0=0.0, TL=100.0	50	50	0.0e+00	✓
HEA-0930	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=4.0, T0=20.0, TL=80.0	50	50	2.8e-16	✓
HEA-0931	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=10.0, L=4.0, T0=-40.0, TL=120.0	40	40	3.6e-16	✓
HEA-0932	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=0.5, T0=0.0, TL=100.0	50	50	1.3e-15	✓
HEA-0933	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=0.5, T0=20.0, TL=80.0	50	50	1.3e-15	✓
HEA-0934	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=0.5, T0=-40.0, TL=120.0	40	40	1.2e-15	✓
HEA-0935	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=1.0, T0=0.0, TL=100.0	50	50	1.3e-15	✓
HEA-0936	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=1.0, T0=20.0, TL=80.0	50	50	1.3e-15	✓
HEA-0937	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=1.0, T0=-40.0, TL=120.0	40	40	1.2e-15	✓
HEA-0938	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=2.0, T0=0.0, TL=100.0	50	50	1.3e-15	✓
HEA-0939	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=2.0, T0=20.0, TL=80.0	50	50	1.3e-15	✓
HEA-0940	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=2.0, T0=-40.0, TL=120.0	40	40	1.2e-15	✓
HEA-0941	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=4.0, T0=0.0, TL=100.0	50	50	1.3e-15	✓
HEA-0942	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=4.0, T0=20.0, TL=80.0	50	50	1.3e-15	✓
HEA-0943	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=45.0, L=4.0, T0=-40.0, TL=120.0	40	40	1.2e-15	✓
HEA-0944	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=0.5, T0=0.0, TL=100.0	50	50	1.4e-16	✓
HEA-0945	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=0.5, T0=20.0, TL=80.0	50	50	2.8e-16	✓
HEA-0946	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=0.5, T0=-40.0, TL=120.0	40	40	0.0e+00	✓
HEA-0947	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=1.0, T0=0.0, TL=100.0	50	50	1.4e-16	✓
HEA-0948	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=1.0, T0=20.0, TL=80.0	50	50	2.8e-16	✓
HEA-0949	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=1.0, T0=-40.0, TL=120.0	40	40	0.0e+00	✓
HEA-0950	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=2.0, T0=0.0, TL=100.0	50	50	1.4e-16	✓
HEA-0951	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=2.0, T0=20.0, TL=80.0	50	50	2.8e-16	✓
HEA-0952	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=2.0, T0=-40.0, TL=120.0	40	40	0.0e+00	✓
HEA-0953	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=4.0, T0=0.0, TL=100.0	50	50	1.4e-16	✓
HEA-0954	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=4.0, T0=20.0, TL=80.0	50	50	2.8e-16	✓
HEA-0955	1D bar, prescribed end temperatures	T(x) linear → T_mid = (T₀+T_L)/2	Incropera, Heat & Mass Transfer	k=200.0, L=4.0, T0=-40.0, TL=120.0	40	40	0.0e+00	✓

Nonlinear — hyperelastic

4 comparisons4/4 passmax err 1.0e-11

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
NON-0956	Neo-Hookean uniaxial (plane strain)	σₓₓ from W=(μ/2)(I₁−3)−μ lnJ+(λ/2)ln²J	Bonet & Wood, Nonlinear Continuum Mechanics	λ₁=1.2	2.0551e+05	2.0551e+05	9.9e-16	✓
NON-0957	Neo-Hookean uniaxial (plane strain)	σₓₓ from W=(μ/2)(I₁−3)−μ lnJ+(λ/2)ln²J	Bonet & Wood, Nonlinear Continuum Mechanics	λ₁=1.4	3.9511e+05	3.9511e+05	3.1e-14	✓
NON-0958	Neo-Hookean uniaxial (plane strain)	σₓₓ from W=(μ/2)(I₁−3)−μ lnJ+(λ/2)ln²J	Bonet & Wood, Nonlinear Continuum Mechanics	λ₁=1.6	5.8049e+05	5.8049e+05	6.3e-13	✓
NON-0959	Neo-Hookean uniaxial (plane strain)	σₓₓ from W=(μ/2)(I₁−3)−μ lnJ+(λ/2)ln²J	Bonet & Wood, Nonlinear Continuum Mechanics	λ₁=1.8	7.6806e+05	7.6806e+05	1.0e-11	✓

Contact & friction

40 comparisons40/40 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
CON-0960	Block on incline — Coulomb stick/slip	holds ⇔ tan θ ≤ μ	Johnson, Contact Mechanics	θ=10.0°, μ=0.5	1	1	0.0e+00	✓
CON-0961	Block on incline — Coulomb stick/slip	holds ⇔ tan θ ≤ μ	Johnson, Contact Mechanics	θ=20.0°, μ=0.5	1	1	0.0e+00	✓
CON-0962	Block on incline — Coulomb stick/slip	holds ⇔ tan θ ≤ μ	Johnson, Contact Mechanics	θ=26.0°, μ=0.5	1	1	0.0e+00	✓
CON-0963	Block on incline — Coulomb stick/slip	holds ⇔ tan θ ≤ μ	Johnson, Contact Mechanics	θ=35.0°, μ=0.5	0	0	0.0e+00	✓
CON-0970	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=0, P·f1/g=0.4	1228.1	1228.1	0.0e+00	✓
CON-0971	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=0, P·f1/g=1.0	3070.2	3070.2	0.0e+00	✓
CON-0972	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=0, P·f1/g=3.0	9210.5	9210.5	0.0e+00	✓
CON-0973	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=0, P·f1/g=8.0	24561	24561	0.0e+00	✓
CON-0974	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=1e-05, P·f1/g=0.4	0	0	0.0e+00	✓
CON-0975	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=1e-05, P·f1/g=1.0	0	0	0.0e+00	✓
CON-0976	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=1e-05, P·f1/g=3.0	2456.1	2456.1	0.0e+00	✓
CON-0977	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=1e-05, P·f1/g=8.0	8596.5	8596.5	0.0e+00	✓
CON-0978	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=5e-05, P·f1/g=0.4	0	0	0.0e+00	✓
CON-0979	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=5e-05, P·f1/g=1.0	0	0	0.0e+00	✓
CON-0980	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=5e-05, P·f1/g=3.0	12281	12281	0.0e+00	✓
CON-0981	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.0, L2=2.0, g=5e-05, P·f1/g=8.0	42982	42982	0.0e+00	✓
CON-0982	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=0, P·f1/g=0.4	445.86	445.86	0.0e+00	✓
CON-0983	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=0, P·f1/g=1.0	1114.6	1114.6	0.0e+00	✓
CON-0984	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=0, P·f1/g=3.0	3343.9	3343.9	0.0e+00	✓
CON-0985	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=0, P·f1/g=8.0	8917.2	8917.2	0.0e+00	✓
CON-0986	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=1e-05, P·f1/g=0.4	0	0	0.0e+00	✓
CON-0987	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=1e-05, P·f1/g=1.0	0	0	0.0e+00	✓
CON-0988	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=1e-05, P·f1/g=3.0	1783.4	1783.4	0.0e+00	✓
CON-0989	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=1e-05, P·f1/g=8.0	6242	6242	0.0e+00	✓
CON-0990	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=5e-05, P·f1/g=0.4	0	0	0.0e+00	✓
CON-0991	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=5e-05, P·f1/g=1.0	0	0	0.0e+00	✓
CON-0992	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=5e-05, P·f1/g=3.0	8917.2	8917.2	0.0e+00	✓
CON-0993	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=1.0, L2=3.0, g=5e-05, P·f1/g=8.0	31210	31210	0.0e+00	✓
CON-0994	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=0, P·f1/g=0.4	4117.6	4117.6	0.0e+00	✓
CON-0995	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=0, P·f1/g=1.0	10294	10294	0.0e+00	✓
CON-0996	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=0, P·f1/g=3.0	30882	30882	0.0e+00	✓
CON-0997	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=0, P·f1/g=8.0	82353	82353	0.0e+00	✓
CON-0998	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=1e-05, P·f1/g=0.4	0	0	0.0e+00	✓
CON-0999	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=1e-05, P·f1/g=1.0	0	0	0.0e+00	✓
CON-1000	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=1e-05, P·f1/g=3.0	6588.2	6588.2	0.0e+00	✓
CON-1001	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=1e-05, P·f1/g=8.0	23059	23059	0.0e+00	✓
CON-1002	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=5e-05, P·f1/g=0.4	0	0	0.0e+00	✓
CON-1003	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=5e-05, P·f1/g=1.0	0	0	0.0e+00	✓
CON-1004	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=5e-05, P·f1/g=3.0	32941	32941	0.0e+00	✓
CON-1005	Two-bar gap — deformable-to-deformable transmitted force	Fc = (P·L1/E1A1 − g)/(L1/E1A1 + L2/E2A2)	Wriggers, Computational Contact Mechanics	L1=2.5, L2=0.5, g=5e-05, P·f1/g=8.0	1.1529e+05	1.1529e+05	0.0e+00	✓

Dynamics — harmonic

5 comparisons5/5 passmax err 1.3e-16

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
DYN-0964	SDOF forced response amplitude	\|U\| = F/√((k−ω²m)²+(ωc)²)	Den Hartog, Mechanical Vibrations	ω/ωₙ=0.5	6.3491e-06	6.3491e-06	0.0e+00	✓
DYN-0965	SDOF forced response amplitude	\|U\| = F/√((k−ω²m)²+(ωc)²)	Den Hartog, Mechanical Vibrations	ω/ωₙ=0.8	1.3225e-05	1.3225e-05	1.3e-16	✓
DYN-0966	SDOF forced response amplitude	\|U\| = F/√((k−ω²m)²+(ωc)²)	Den Hartog, Mechanical Vibrations	ω/ωₙ=1.0	5.0038e-04	5.0038e-04	0.0e+00	✓
DYN-0967	SDOF forced response amplitude	\|U\| = F/√((k−ω²m)²+(ωc)²)	Den Hartog, Mechanical Vibrations	ω/ωₙ=1.2	1.0819e-05	1.0819e-05	0.0e+00	✓
DYN-0968	SDOF forced response amplitude	\|U\| = F/√((k−ω²m)²+(ωc)²)	Den Hartog, Mechanical Vibrations	ω/ωₙ=1.5	3.8093e-06	3.8093e-06	0.0e+00	✓

Shells

1 comparisons1/1 passmax err 3.5e-03

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
SHE-0969	Scordelis-Lo roof — free-edge deflection	reference = 0.3024	MacNeal & Harder (1985), standard shell benchmark	12×12 quarter mesh	0.30134	0.3024	3.5e-03	✓

Dynamics — nonlinear transient

8 comparisons8/8 passmax err 1.7e-08

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
DYN-1006	Small-amplitude limit — SDOF step response peak	u_peak = 2F/k (Newmark, consistent mass)	Chopra, Dynamics of Structures	E=2.1e+11, L=1.0, rho=7850.0	9.5238e-08	9.5238e-08	1.7e-08	✓
DYN-1007	Undamped energy conservation	\|KE+U−W\| / E_peak < 2%	Newmark (1959), average acceleration	E=2.1e+11, L=1.0, rho=7850.0	1	1	0.0e+00	✓
DYN-1008	Small-amplitude limit — SDOF step response peak	u_peak = 2F/k (Newmark, consistent mass)	Chopra, Dynamics of Structures	E=7e+10, L=2.0, rho=2700.0	5.7143e-07	5.7143e-07	1.7e-08	✓
DYN-1009	Undamped energy conservation	\|KE+U−W\| / E_peak < 2%	Newmark (1959), average acceleration	E=7e+10, L=2.0, rho=2700.0	1	1	0.0e+00	✓
DYN-1010	Small-amplitude limit — SDOF step response peak	u_peak = 2F/k (Newmark, consistent mass)	Chopra, Dynamics of Structures	E=1e+11, L=0.5, rho=4000.0	1.0000e-07	1.0000e-07	1.7e-08	✓
DYN-1011	Undamped energy conservation	\|KE+U−W\| / E_peak < 2%	Newmark (1959), average acceleration	E=1e+11, L=0.5, rho=4000.0	1	1	0.0e+00	✓
DYN-1012	Dynamic snap-through — von Mises truss	step load 1.5x static limit ⇒ apex crosses mirror (u_y < −2h)	Bathe, Finite Element Procedures	h/half-span=0.1	1	1	0.0e+00	✓
DYN-1013	Dynamic snap-through — von Mises truss	step load 1.5x static limit ⇒ apex crosses mirror (u_y < −2h)	Bathe, Finite Element Procedures	h/half-span=0.16	1	1	0.0e+00	✓

Composites — laminated shells

5 comparisons5/5 passmax err 1.9e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
COM-1014	Isotropic-ply stack equals isotropic plate (plate4)	identical D ⇒ identical deflection	Classical lamination theory	E=7e+10, t=0.004	-0.10008	-0.10008	1.4e-12	✓
COM-1015	Isotropic-ply stack equals isotropic plate (mitc4)	identical D ⇒ identical deflection	Classical lamination theory	E=7e+10, t=0.004	-0.098719	-0.098719	6.1e-12	✓
COM-1016	SSSS specially orthotropic plate — Navier series	w_max = 16qa⁴/π⁶ ΣΣ …/(mn·Dmn) with CLT D	Reddy, Mechanics of Laminated Composite Plates	stack=[0, 90, 90, 0]	-8.788	-8.7897	1.9e-04	✓
COM-1017	SSSS specially orthotropic plate — Navier series	w_max = 16qa⁴/π⁶ ΣΣ …/(mn·Dmn) with CLT D	Reddy, Mechanics of Laminated Composite Plates	stack=[0, 0, 90, 90, 90, 90, 0, 0]	-1.0985	-1.0987	1.6e-04	✓
COM-1018	Quasi-isotropic coupon — effective Ex	Ex = (σx/εx) from FE equals CLT effective Ex	Tsai & Pagano invariants	[0/±45/90]s	5.4068e+10	5.4068e+10	9.9e-16	✓

Materials — viscoelastic & creep

11 comparisons11/11 passmax err 3.5e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MAT-1019	SLS stress relaxation	σ(t) = ε₀(E_inf + E₁e^{−t/τ})	Simo & Hughes, Computational Inelasticity	E0=1e+10, E1=4e+09, τ=2.0	6.0271e+05	6.0272e+05	2.3e-05	✓
MAT-1020	SLS stress relaxation	σ(t) = ε₀(E_inf + E₁e^{−t/τ})	Simo & Hughes, Computational Inelasticity	E0=1e+10, E1=7e+09, τ=0.5	3.0474e+05	3.0476e+05	7.8e-05	✓
MAT-1021	SLS stress relaxation	σ(t) = ε₀(E_inf + E₁e^{−t/τ})	Simo & Hughes, Computational Inelasticity	E0=7e+10, E1=2e+10, τ=10.0	5.0135e+06	5.0136e+06	1.4e-05	✓
MAT-1022	Rubbery long-term modulus	σ(∞) = E_inf·ε₀	Ferry, Viscoelastic Properties of Polymers	E_inf=6e+09	6.0000e+05	6.0000e+05	1.4e-09	✓
MAT-1023	Norton power-law creep — constant-stress bar	ε(t) = σ/E + A·σⁿ·t	Norton (1929); Kraus, Creep Analysis	n=3.0, σ=5e+07	0.005	0.005	9.0e-12	✓
MAT-1024	Norton power-law creep — constant-stress bar	ε(t) = σ/E + A·σⁿ·t	Norton (1929); Kraus, Creep Analysis	n=4.0, σ=4e+07	0.004	0.004	2.2e-16	✓
MAT-1025	Norton power-law creep — constant-stress bar	ε(t) = σ/E + A·σⁿ·t	Norton (1929); Kraus, Creep Analysis	n=5.0, σ=3e+07	0.003	0.003	0.0e+00	✓
MAT-1026	Continuum viscoelastic relaxation (quad4)	σ(t) = ε(E_inf + E₁e^{−t/τ})	Simo & Hughes, Computational Inelasticity	E0=1e+10, E1=6e+09, τ=1.0	4.3002e+05	4.3017e+05	3.5e-04	✓
MAT-1027	Continuum viscoelastic relaxation (hex8)	σ(t) = ε(E_inf + E₁e^{−t/τ})	Simo & Hughes, Computational Inelasticity	E0=1e+10, E1=6e+09, τ=1.0	4.3002e+05	4.3017e+05	3.5e-04	✓
MAT-1028	Continuum J2 creep — uniaxial (hex8)	ε_c(t) = A·σⁿ·t	Norton (1929); Simo & Hughes, Computational Inelasticity	n=3.0, σ=1e+08	1.0000e-03	0.001	7.2e-08	✓
MAT-1029	Continuum J2 creep — uniaxial (hex8)	ε_c(t) = A·σⁿ·t	Norton (1929); Simo & Hughes, Computational Inelasticity	n=5.0, σ=8e+07	3.2768e-06	3.2768e-06	2.2e-07	✓

Dynamics — explicit

72 comparisons72/72 passmax err 1.2e-13

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
DYN-1030	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=7e+10, rho=2700, L=0.5	1.3887e-04	1.3887e-04	0.0e+00	✓
DYN-1031	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=7e+10, rho=2700, L=0.5	-5.0000e-04	-5.0000e-04	1.9e-14	✓
DYN-1032	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=7e+10, rho=2700, L=1.0	2.7775e-04	2.7775e-04	0.0e+00	✓
DYN-1033	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=7e+10, rho=2700, L=1.0	-5.0000e-04	-5.0000e-04	1.9e-14	✓
DYN-1034	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=7e+10, rho=2700, L=2.0	5.5549e-04	5.5549e-04	0.0e+00	✓
DYN-1035	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=7e+10, rho=2700, L=2.0	-5.0000e-04	-5.0000e-04	1.9e-14	✓
DYN-1036	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=7e+10, rho=7850, L=0.5	2.3679e-04	2.3679e-04	0.0e+00	✓
DYN-1037	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=7e+10, rho=7850, L=0.5	-5.0000e-04	-5.0000e-04	2.4e-14	✓
DYN-1038	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=7e+10, rho=7850, L=1.0	4.7359e-04	4.7359e-04	2.3e-16	✓
DYN-1039	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=7e+10, rho=7850, L=1.0	-5.0000e-04	-5.0000e-04	6.3e-14	✓
DYN-1040	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=7e+10, rho=7850, L=2.0	9.4718e-04	9.4718e-04	0.0e+00	✓
DYN-1041	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=7e+10, rho=7850, L=2.0	-5.0000e-04	-5.0000e-04	2.4e-14	✓
DYN-1042	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=7e+10, rho=4500, L=0.5	1.7928e-04	1.7928e-04	1.5e-16	✓
DYN-1043	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=7e+10, rho=4500, L=0.5	-5.0000e-04	-5.0000e-04	7.1e-14	✓
DYN-1044	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=7e+10, rho=4500, L=1.0	3.5857e-04	3.5857e-04	0.0e+00	✓
DYN-1045	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=7e+10, rho=4500, L=1.0	-5.0000e-04	-5.0000e-04	4.2e-14	✓
DYN-1046	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=7e+10, rho=4500, L=2.0	7.1714e-04	7.1714e-04	1.5e-16	✓
DYN-1047	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=7e+10, rho=4500, L=2.0	-5.0000e-04	-5.0000e-04	7.1e-14	✓
DYN-1048	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=2e+11, rho=2700, L=0.5	8.2158e-05	8.2158e-05	1.6e-16	✓
DYN-1049	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=2e+11, rho=2700, L=0.5	-5.0000e-04	-5.0000e-04	6.6e-14	✓
DYN-1050	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=2e+11, rho=2700, L=1.0	1.6432e-04	1.6432e-04	0.0e+00	✓
DYN-1051	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=2e+11, rho=2700, L=1.0	-5.0000e-04	-5.0000e-04	8.6e-14	✓
DYN-1052	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=2e+11, rho=2700, L=2.0	3.2863e-04	3.2863e-04	1.6e-16	✓
DYN-1053	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=2e+11, rho=2700, L=2.0	-5.0000e-04	-5.0000e-04	6.6e-14	✓
DYN-1054	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=2e+11, rho=7850, L=0.5	1.4009e-04	1.4009e-04	0.0e+00	✓
DYN-1055	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=2e+11, rho=7850, L=0.5	-5.0000e-04	-5.0000e-04	8.4e-14	✓
DYN-1056	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=2e+11, rho=7850, L=1.0	2.8018e-04	2.8018e-04	1.9e-16	✓
DYN-1057	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=2e+11, rho=7850, L=1.0	-5.0000e-04	-5.0000e-04	4.7e-14	✓
DYN-1058	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=2e+11, rho=7850, L=2.0	5.6036e-04	5.6036e-04	0.0e+00	✓
DYN-1059	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=2e+11, rho=7850, L=2.0	-5.0000e-04	-5.0000e-04	8.4e-14	✓
DYN-1060	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=2e+11, rho=4500, L=0.5	1.0607e-04	1.0607e-04	1.3e-16	✓
DYN-1061	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=2e+11, rho=4500, L=0.5	-5.0000e-04	-5.0000e-04	1.9e-14	✓
DYN-1062	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=2e+11, rho=4500, L=1.0	2.1213e-04	2.1213e-04	1.3e-16	✓
DYN-1063	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=2e+11, rho=4500, L=1.0	-5.0000e-04	-5.0000e-04	1.9e-14	✓
DYN-1064	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=2e+11, rho=4500, L=2.0	4.2426e-04	4.2426e-04	1.3e-16	✓
DYN-1065	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=2e+11, rho=4500, L=2.0	-5.0000e-04	-5.0000e-04	1.9e-14	✓
DYN-1066	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=1.13e+11, rho=2700, L=0.5	1.0930e-04	1.0930e-04	0.0e+00	✓
DYN-1067	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=1.13e+11, rho=2700, L=0.5	-5.0000e-04	-5.0000e-04	1.2e-13	✓
DYN-1068	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=1.13e+11, rho=2700, L=1.0	2.1860e-04	2.1860e-04	0.0e+00	✓
DYN-1069	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=1.13e+11, rho=2700, L=1.0	-5.0000e-04	-5.0000e-04	1.2e-13	✓
DYN-1070	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=1.13e+11, rho=2700, L=2.0	4.3721e-04	4.3721e-04	0.0e+00	✓
DYN-1071	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=1.13e+11, rho=2700, L=2.0	-5.0000e-04	-5.0000e-04	1.2e-13	✓
DYN-1072	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=1.13e+11, rho=7850, L=0.5	1.8637e-04	1.8637e-04	0.0e+00	✓
DYN-1073	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=1.13e+11, rho=7850, L=0.5	-5.0000e-04	-5.0000e-04	1.9e-14	✓
DYN-1074	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=1.13e+11, rho=7850, L=1.0	3.7274e-04	3.7274e-04	0.0e+00	✓
DYN-1075	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=1.13e+11, rho=7850, L=1.0	-5.0000e-04	-5.0000e-04	1.9e-14	✓
DYN-1076	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=1.13e+11, rho=7850, L=2.0	7.4549e-04	7.4549e-04	0.0e+00	✓
DYN-1077	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=1.13e+11, rho=7850, L=2.0	-5.0000e-04	-5.0000e-04	1.9e-14	✓
DYN-1078	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=1.13e+11, rho=4500, L=0.5	1.4111e-04	1.4111e-04	1.9e-16	✓
DYN-1079	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=1.13e+11, rho=4500, L=0.5	-5.0000e-04	-5.0000e-04	1.4e-14	✓
DYN-1080	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=1.13e+11, rho=4500, L=1.0	2.8222e-04	2.8222e-04	1.9e-16	✓
DYN-1081	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=1.13e+11, rho=4500, L=1.0	-5.0000e-04	-5.0000e-04	1.4e-14	✓
DYN-1082	Fixed-free bar critical step equals the wave-speed limit	dt_crit = L*sqrt(2)/c, c = sqrt(E/rho) (lumped 2-node bar)	central-difference stability, discrete	E=1.13e+11, rho=4500, L=2.0	5.6443e-04	5.6443e-04	1.9e-16	✓
DYN-1083	Free vibration matches the central-difference dispersion relation	u_N = u0cos(Omegat), cos(Omegadt) = 1 - (omegadt)^2/2	central-difference integrator identity, discrete	E=1.13e+11, rho=4500, L=2.0	-5.0000e-04	-5.0000e-04	1.4e-14	✓
DYN-1084	Free-free bar conserves linear momentum	p_x = M*v0 (rigid translation, zero internal force)	conservation property, discrete	rho=2700, L=0.5	1.35	1.35	0.0e+00	✓
DYN-1085	Free-free rigid translation conserves kinetic energy	KE_final = 0.5Mv0^2	conservation property, discrete	rho=2700, L=0.5	6.75	6.75	0.0e+00	✓
DYN-1086	Free-free bar conserves linear momentum	p_x = M*v0 (rigid translation, zero internal force)	conservation property, discrete	rho=2700, L=1.0	2.7	2.7	0.0e+00	✓
DYN-1087	Free-free rigid translation conserves kinetic energy	KE_final = 0.5Mv0^2	conservation property, discrete	rho=2700, L=1.0	13.5	13.5	0.0e+00	✓
DYN-1088	Free-free bar conserves linear momentum	p_x = M*v0 (rigid translation, zero internal force)	conservation property, discrete	rho=2700, L=2.0	5.4	5.4	0.0e+00	✓
DYN-1089	Free-free rigid translation conserves kinetic energy	KE_final = 0.5Mv0^2	conservation property, discrete	rho=2700, L=2.0	27	27	0.0e+00	✓
DYN-1090	Free-free bar conserves linear momentum	p_x = M*v0 (rigid translation, zero internal force)	conservation property, discrete	rho=7850, L=0.5	3.925	3.925	1.1e-16	✓
DYN-1091	Free-free rigid translation conserves kinetic energy	KE_final = 0.5Mv0^2	conservation property, discrete	rho=7850, L=0.5	19.625	19.625	0.0e+00	✓
DYN-1092	Free-free bar conserves linear momentum	p_x = M*v0 (rigid translation, zero internal force)	conservation property, discrete	rho=7850, L=1.0	7.85	7.85	1.1e-16	✓
DYN-1093	Free-free rigid translation conserves kinetic energy	KE_final = 0.5Mv0^2	conservation property, discrete	rho=7850, L=1.0	39.25	39.25	0.0e+00	✓
DYN-1094	Free-free bar conserves linear momentum	p_x = M*v0 (rigid translation, zero internal force)	conservation property, discrete	rho=7850, L=2.0	15.7	15.7	1.1e-16	✓
DYN-1095	Free-free rigid translation conserves kinetic energy	KE_final = 0.5Mv0^2	conservation property, discrete	rho=7850, L=2.0	78.5	78.5	0.0e+00	✓
DYN-1096	Free-free bar conserves linear momentum	p_x = M*v0 (rigid translation, zero internal force)	conservation property, discrete	rho=4500, L=0.5	2.25	2.25	0.0e+00	✓
DYN-1097	Free-free rigid translation conserves kinetic energy	KE_final = 0.5Mv0^2	conservation property, discrete	rho=4500, L=0.5	11.25	11.25	0.0e+00	✓
DYN-1098	Free-free bar conserves linear momentum	p_x = M*v0 (rigid translation, zero internal force)	conservation property, discrete	rho=4500, L=1.0	4.5	4.5	0.0e+00	✓
DYN-1099	Free-free rigid translation conserves kinetic energy	KE_final = 0.5Mv0^2	conservation property, discrete	rho=4500, L=1.0	22.5	22.5	0.0e+00	✓
DYN-1100	Free-free bar conserves linear momentum	p_x = M*v0 (rigid translation, zero internal force)	conservation property, discrete	rho=4500, L=2.0	9	9	0.0e+00	✓
DYN-1101	Free-free rigid translation conserves kinetic energy	KE_final = 0.5Mv0^2	conservation property, discrete	rho=4500, L=2.0	45	45	0.0e+00	✓

Acoustics — damped response

13 comparisons13/13 passmax err 7.0e-06

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ACO-1102	Damped driven duct matches the complex Helmholtz closed form	\|p_FE - p0 cos(k~(L-x))/cos(k~L)\| / \|p_exact\|, k~=ksqrt(1-ieta)	Kinsler & Frey, Fundamentals of Acoustics (hysteretic loss)	f=100 Hz, eta=0.02	3.1186e-06	0	3.1e-06	✓
ACO-1103	Loss factor produces a complex (out-of-phase) pressure field	max\|Im p\| / max\|Re p\| > 0 for eta>0 (1 if true)	hysteretic damping property, discrete	f=100 Hz, eta=0.02	1	1	0.0e+00	✓
ACO-1104	Damped driven duct matches the complex Helmholtz closed form	\|p_FE - p0 cos(k~(L-x))/cos(k~L)\| / \|p_exact\|, k~=ksqrt(1-ieta)	Kinsler & Frey, Fundamentals of Acoustics (hysteretic loss)	f=100 Hz, eta=0.05	3.0783e-06	0	3.1e-06	✓
ACO-1105	Loss factor produces a complex (out-of-phase) pressure field	max\|Im p\| / max\|Re p\| > 0 for eta>0 (1 if true)	hysteretic damping property, discrete	f=100 Hz, eta=0.05	1	1	0.0e+00	✓
ACO-1106	Damped driven duct matches the complex Helmholtz closed form	\|p_FE - p0 cos(k~(L-x))/cos(k~L)\| / \|p_exact\|, k~=ksqrt(1-ieta)	Kinsler & Frey, Fundamentals of Acoustics (hysteretic loss)	f=100 Hz, eta=0.1	2.9479e-06	0	2.9e-06	✓
ACO-1107	Loss factor produces a complex (out-of-phase) pressure field	max\|Im p\| / max\|Re p\| > 0 for eta>0 (1 if true)	hysteretic damping property, discrete	f=100 Hz, eta=0.1	1	1	0.0e+00	✓
ACO-1108	Damped driven duct matches the complex Helmholtz closed form	\|p_FE - p0 cos(k~(L-x))/cos(k~L)\| / \|p_exact\|, k~=ksqrt(1-ieta)	Kinsler & Frey, Fundamentals of Acoustics (hysteretic loss)	f=150 Hz, eta=0.02	6.9977e-06	0	7.0e-06	✓
ACO-1109	Loss factor produces a complex (out-of-phase) pressure field	max\|Im p\| / max\|Re p\| > 0 for eta>0 (1 if true)	hysteretic damping property, discrete	f=150 Hz, eta=0.02	1	1	0.0e+00	✓
ACO-1110	Damped driven duct matches the complex Helmholtz closed form	\|p_FE - p0 cos(k~(L-x))/cos(k~L)\| / \|p_exact\|, k~=ksqrt(1-ieta)	Kinsler & Frey, Fundamentals of Acoustics (hysteretic loss)	f=150 Hz, eta=0.05	6.9064e-06	0	6.9e-06	✓
ACO-1111	Loss factor produces a complex (out-of-phase) pressure field	max\|Im p\| / max\|Re p\| > 0 for eta>0 (1 if true)	hysteretic damping property, discrete	f=150 Hz, eta=0.05	1	1	0.0e+00	✓
ACO-1112	Damped driven duct matches the complex Helmholtz closed form	\|p_FE - p0 cos(k~(L-x))/cos(k~L)\| / \|p_exact\|, k~=ksqrt(1-ieta)	Kinsler & Frey, Fundamentals of Acoustics (hysteretic loss)	f=150 Hz, eta=0.1	6.6023e-06	0	6.6e-06	✓
ACO-1113	Loss factor produces a complex (out-of-phase) pressure field	max\|Im p\| / max\|Re p\| > 0 for eta>0 (1 if true)	hysteretic damping property, discrete	f=150 Hz, eta=0.1	1	1	0.0e+00	✓
ACO-1114	Zero loss factor recovers the undamped closed form	eta=0 -> p(x)=p0 cos(k(L-x))/cos(kL)	Kinsler & Frey, Fundamentals of Acoustics	f=120 Hz, eta=0	-0.77405	-0.77405	2.9e-06	✓

Multiphysics — thermo-plastic

22 comparisons22/22 passmax err 1.6e-14

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MUL-1115	Statically determinate series bar carries uniform axial stress	sigma = P/A at every element (equilibrium)	thermo-elasto-plastic closed form	Tl=0, Tr=200	3.8873e-16	0	3.9e-16	✓
MUL-1116	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=0	0	0	0.0e+00	✓
MUL-1117	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=1	0	0	0.0e+00	✓
MUL-1118	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=2	0	0	0.0e+00	✓
MUL-1119	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=3	0	0	0.0e+00	✓
MUL-1120	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=4	5.0000e-04	5.0000e-04	1.6e-14	✓
MUL-1121	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=5	0.0015	0.0015	8.7e-16	✓
MUL-1122	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=6	0.0025	0.0025	3.5e-15	✓
MUL-1123	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=7	0.0035	0.0035	2.8e-15	✓
MUL-1124	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=8	0.0045	0.0045	1.7e-15	✓
MUL-1125	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=0, Tr=200, elem=9	0.0055	0.0055	0.0e+00	✓
MUL-1126	Statically determinate series bar carries uniform axial stress	sigma = P/A at every element (equilibrium)	thermo-elasto-plastic closed form	Tl=50, Tr=250	2.4835e-16	0	2.5e-16	✓
MUL-1127	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=0	1.0000e-03	0.001	3.9e-15	✓
MUL-1128	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=1	0.002	0.002	1.7e-15	✓
MUL-1129	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=2	0.003	0.003	8.7e-16	✓
MUL-1130	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=3	0.004	0.004	8.7e-16	✓
MUL-1131	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=4	0.005	0.005	1.0e-15	✓
MUL-1132	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=5	0.006	0.006	1.6e-15	✓
MUL-1133	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=6	0.007	0.007	1.7e-15	✓
MUL-1134	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=7	0.008	0.008	8.7e-16	✓
MUL-1135	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=8	0.009	0.009	1.9e-16	✓
MUL-1136	Per-element plastic strain from the coupled yield field	eps_p,e = max(0, (sigma - sy0(1+c(T_e-Tref)))/H)	thermo-elasto-plastic closed form	Tl=50, Tr=250, elem=9	0.01	0.01	1.7e-16	✓

Multiphysics — Nonlinear magnetostatics

8 comparisons8/8 passmax err 2.5e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MUL-1137	Saturable-slab potential at the centre matches the Ampere + B-H closed form	A_z(d/2) = -int_0^{d/2} B(J(s-d/2)) ds, nu0 B(1+c B^2)=H	nonlinear current slab, semi-analytic reference	mu0=0.001, c=0.02, J=100000	0.10741	0.1074	5.1e-05	✓
MUL-1138	Saturable-slab potential at the quarter point matches the closed form	A_z(d/4) from the same Ampere + B-H reference	nonlinear current slab, semi-analytic reference	mu0=0.001, c=0.02, J=100000	0.077747	0.077744	3.5e-05	✓
MUL-1139	Nonlinear Picard iteration reaches a flux-density fixed point	final \|B\|^2 relative change < 1e-8 (1 if converged)	fixed-point convergence property	mu0=0.001, c=0.02	1	1	0.0e+00	✓
MUL-1140	Saturable-slab potential at the centre matches the Ampere + B-H closed form	A_z(d/2) = -int_0^{d/2} B(J(s-d/2)) ds, nu0 B(1+c B^2)=H	nonlinear current slab, semi-analytic reference	mu0=0.002, c=0.05, J=80000	0.13059	0.13057	1.0e-04	✓
MUL-1141	Saturable-slab potential at the quarter point matches the closed form	A_z(d/4) from the same Ampere + B-H reference	nonlinear current slab, semi-analytic reference	mu0=0.002, c=0.05, J=80000	0.089863	0.08986	3.7e-05	✓
MUL-1142	Nonlinear Picard iteration reaches a flux-density fixed point	final \|B\|^2 relative change < 1e-8 (1 if converged)	fixed-point convergence property	mu0=0.002, c=0.05	1	1	0.0e+00	✓
MUL-1143	Nonlinear potential converges to the closed form at second order	err(h)/err(h/2) ~ 4	O(h^2) convergence study	err(40)/err(80)	4.001	4	2.5e-04	✓
MUL-1144	Zero reluctivity coefficient recovers the exact linear potential	c=0 -> A_z(d/2) = mu0 J d^2 / 8	linear current slab, exact solution	c=0	0.125	0.125	4.0e-14	✓

Multiphysics — Piezoelectric 2D

4 comparisons4/4 passmax err 1.4e-12

ID	Problem	Reference	Source	Parameters	Computed	Rel. err	✓
MUL-1145	Coupled patch test reproduces a uniform state (displacement)	linear u on boundary -> exact interior u	piezoelectric patch test	distorted 2x2 mesh	1.5247e-20	1.5e-20	✓
MUL-1146	Coupled patch test reproduces a uniform state (potential)	linear phi on boundary -> exact interior phi	piezoelectric patch test	distorted 2x2 mesh	1.3507e-12	1.4e-12	✓
MUL-1147	Uniaxial strip converse effect matches the verified 1-D bar	2-D strip (uy=0) == 1-D piezo bar, node-for-node	reduction to the exact 1-D piezoelectric solution	V=200, 10 elements	5.8175e-16	5.8e-16	✓
MUL-1148	Uniaxial strip direct effect matches the verified 1-D bar	2-D strip open-circuit voltage == 1-D piezo bar	reduction to the exact 1-D piezoelectric solution	delta=1e-4, 10 elements	1.2878e-14	1.3e-14	✓

Multiphysics — Piezoelectric 3D

4 comparisons4/4 passmax err 2.1e-14

ID	Problem	Reference	Source	Parameters	Computed	Rel. err	✓
MUL-1149	Coupled patch test reproduces a uniform state (displacement)	linear u on boundary -> exact interior u	3-D piezoelectric patch test	distorted 2x2x2 brick	1.0164e-20	1.0e-20	✓
MUL-1150	Coupled patch test reproduces a uniform state (potential)	linear phi on boundary -> exact interior phi	3-D piezoelectric patch test	distorted 2x2x2 brick	2.1316e-14	2.1e-14	✓
MUL-1151	Uniaxial column converse effect matches the verified 1-D bar	3-D column (ux=uy=0) == 1-D piezo bar, node-for-node	reduction to the exact 1-D piezoelectric solution	V=150, 8 elements	6.4930e-15	6.5e-15	✓
MUL-1152	Uniaxial column direct effect matches the verified 1-D bar	3-D column open-circuit voltage == 1-D piezo bar	reduction to the exact 1-D piezoelectric solution	delta=1e-4, 8 elements	7.0941e-16	7.1e-16	✓

Multiphysics — Eddy currents

8 comparisons8/8 passmax err 7.0e-02

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
MUL-1153	Skin-effect slab potential at the centre matches the closed form	A(x)=A0 sinh(k(L-x))/sinh(kL), k=sqrt(jomegamu*sigma)	time-harmonic conducting slab, exact solution	f=2000 Hz	3.3741e-04	0	3.4e-04	✓
MUL-1154	Skin-effect slab potential at the quarter point matches the closed form	same skin-effect closed form	time-harmonic conducting slab, exact solution	f=2000 Hz	1.5402e-04	0	1.5e-04	✓
MUL-1155	A nonzero frequency produces an out-of-phase (complex) field	max\|Im A\| / max\|Re A\| > 0 for omega>0 (1 if true)	eddy-current phase lag property	f=2000 Hz	1	1	0.0e+00	✓
MUL-1156	Skin-effect slab potential at the centre matches the closed form	A(x)=A0 sinh(k(L-x))/sinh(kL), k=sqrt(jomegamu*sigma)	time-harmonic conducting slab, exact solution	f=5000 Hz	0.0013226	0	1.3e-03	✓
MUL-1157	Skin-effect slab potential at the quarter point matches the closed form	same skin-effect closed form	time-harmonic conducting slab, exact solution	f=5000 Hz	6.1489e-04	0	6.1e-04	✓
MUL-1158	A nonzero frequency produces an out-of-phase (complex) field	max\|Im A\| / max\|Re A\| > 0 for omega>0 (1 if true)	eddy-current phase lag property	f=5000 Hz	1	1	0.0e+00	✓
MUL-1159	Eddy-current potential converges to the closed form at second order	err(h)/err(h/2) ~ 4	O(h^2) convergence study	err(60)/err(120)	3.7181	4	7.0e-02	✓
MUL-1160	Low-frequency limit recovers the linear magnetostatic profile	omega->0 -> A(x)=A0 (L-x)/L	quasistatic (Laplace) limit	f->0, A(L/2)	0.5	0.5	3.6e-14	✓

Fracture — cohesive zone

12 comparisons12/12 passmax err 1.5e-15

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
FRA-1161	Peak cohesive traction equals K*delta0	T_max = K*delta0	bilinear traction-separation, exact	K=1e+12, d0=1e-05, df=0.0001	1.0000e+07	1.0000e+07	0.0e+00	✓
FRA-1162	Dissipated energy equals the fracture energy Gc	G_c = 0.5Kdelta0*deltaf (area under T-delta)	bilinear cohesive law, exact	K=1e+12, d0=1e-05, df=0.0001	500	500	1.5e-15	✓
FRA-1163	Softening branch follows the linear-softening law	T = T_max (deltaf-delta)/(deltaf-delta0)	bilinear cohesive law, exact	K=1e+12, d0=1e-05, df=0.0001	5.0000e+06	5.0000e+06	7.5e-16	✓
FRA-1164	Damaged unloading returns to the origin along the secant	T = (1-D(kappa)) K delta on unload	irreversible bilinear damage, exact	K=1e+12, d0=1e-05, df=0.0001	2.5000e+06	2.5000e+06	0.0e+00	✓
FRA-1165	Peak cohesive traction equals K*delta0	T_max = K*delta0	bilinear traction-separation, exact	K=5e+11, d0=2e-05, df=8e-05	1.0000e+07	1.0000e+07	0.0e+00	✓
FRA-1166	Dissipated energy equals the fracture energy Gc	G_c = 0.5Kdelta0*deltaf (area under T-delta)	bilinear cohesive law, exact	K=5e+11, d0=2e-05, df=8e-05	400	400	1.3e-15	✓
FRA-1167	Softening branch follows the linear-softening law	T = T_max (deltaf-delta)/(deltaf-delta0)	bilinear cohesive law, exact	K=5e+11, d0=2e-05, df=8e-05	5.0000e+06	5.0000e+06	1.9e-16	✓
FRA-1168	Damaged unloading returns to the origin along the secant	T = (1-D(kappa)) K delta on unload	irreversible bilinear damage, exact	K=5e+11, d0=2e-05, df=8e-05	2.5000e+06	2.5000e+06	0.0e+00	✓
FRA-1169	Peak cohesive traction equals K*delta0	T_max = K*delta0	bilinear traction-separation, exact	K=2e+12, d0=5e-06, df=5e-05	1.0000e+07	1.0000e+07	0.0e+00	✓
FRA-1170	Dissipated energy equals the fracture energy Gc	G_c = 0.5Kdelta0*deltaf (area under T-delta)	bilinear cohesive law, exact	K=2e+12, d0=5e-06, df=5e-05	250	250	1.5e-15	✓
FRA-1171	Softening branch follows the linear-softening law	T = T_max (deltaf-delta)/(deltaf-delta0)	bilinear cohesive law, exact	K=2e+12, d0=5e-06, df=5e-05	5.0000e+06	5.0000e+06	7.5e-16	✓
FRA-1172	Damaged unloading returns to the origin along the secant	T = (1-D(kappa)) K delta on unload	irreversible bilinear damage, exact	K=2e+12, d0=5e-06, df=5e-05	2.5000e+06	2.5000e+06	0.0e+00	✓

Fracture — mixed-mode cohesive

12 comparisons12/12 passmax err 1.7e-07

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
FRA-1173	Proportional dissipation equals the BK mixed-mode fracture energy	Gc(beta) = GIc + (GIIc-GIc) (GII/(GI+GII))^eta	Benzeggagh-Kenane criterion, exact (proportional loading)	beta=0	500	500	1.3e-15	✓
FRA-1174	Mixed-mode onset separation matches the Camanho quadratic form	delta_m0 = dI0 dII0 sqrt((1+b^2)/(dII0^2 + b^2 dI0^2))	Camanho-Davila mixed-mode onset, exact	beta=0	1.0000e-05	1.0000e-05	0.0e+00	✓
FRA-1175	Proportional dissipation equals the BK mixed-mode fracture energy	Gc(beta) = GIc + (GIIc-GIc) (GII/(GI+GII))^eta	Benzeggagh-Kenane criterion, exact (proportional loading)	beta=0.5	548.47	548.47	1.2e-07	✓
FRA-1176	Mixed-mode onset separation matches the Camanho quadratic form	delta_m0 = dI0 dII0 sqrt((1+b^2)/(dII0^2 + b^2 dI0^2))	Camanho-Davila mixed-mode onset, exact	beta=0.5	1.0607e-05	1.0607e-05	0.0e+00	✓
FRA-1177	Proportional dissipation equals the BK mixed-mode fracture energy	Gc(beta) = GIc + (GIIc-GIc) (GII/(GI+GII))^eta	Benzeggagh-Kenane criterion, exact (proportional loading)	beta=1	683.01	683.01	1.7e-07	✓
FRA-1178	Mixed-mode onset separation matches the Camanho quadratic form	delta_m0 = dI0 dII0 sqrt((1+b^2)/(dII0^2 + b^2 dI0^2))	Camanho-Davila mixed-mode onset, exact	beta=1	1.1767e-05	1.1767e-05	0.0e+00	✓
FRA-1179	Proportional dissipation equals the BK mixed-mode fracture energy	Gc(beta) = GIc + (GIIc-GIc) (GII/(GI+GII))^eta	Benzeggagh-Kenane criterion, exact (proportional loading)	beta=2	861.78	861.78	1.3e-07	✓
FRA-1180	Mixed-mode onset separation matches the Camanho quadratic form	delta_m0 = dI0 dII0 sqrt((1+b^2)/(dII0^2 + b^2 dI0^2))	Camanho-Davila mixed-mode onset, exact	beta=2	1.3416e-05	1.3416e-05	0.0e+00	✓
FRA-1181	Proportional dissipation equals the BK mixed-mode fracture energy	Gc(beta) = GIc + (GIIc-GIc) (GII/(GI+GII))^eta	Benzeggagh-Kenane criterion, exact (proportional loading)	beta=1e+06	1000	1000	1.6e-15	✓
FRA-1182	Mixed-mode onset separation matches the Camanho quadratic form	delta_m0 = dI0 dII0 sqrt((1+b^2)/(dII0^2 + b^2 dI0^2))	Camanho-Davila mixed-mode onset, exact	beta=1e+06	1.5000e-05	1.5000e-05	0.0e+00	✓
FRA-1183	Pure mode I recovers G_Ic	beta=0 -> Gc=GIc	mixed-mode reduction, exact	beta=0	500	500	0.0e+00	✓
FRA-1184	Pure mode II recovers G_IIc	beta->inf -> Gc=GIIc	mixed-mode reduction, exact	beta->inf	1000	1000	7.2e-13	✓

Electromagnetics — full-wave

6 comparisons6/6 passmax err 5.0e-03

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ELE-1185	Lossless dielectric standing wave matches the Helmholtz closed form	u(x)=sin(k(L-x))/sin(kL), k=omega sqrt(mu eps), c=1/sqrt(mu eps)	time-harmonic slab, exact solution	f=6e+07 Hz, kL=1.26	1.2880e-06	0	1.3e-06	✓
ELE-1186	Lossless dielectric standing wave matches the Helmholtz closed form	u(x)=sin(k(L-x))/sin(kL), k=omega sqrt(mu eps), c=1/sqrt(mu eps)	time-harmonic slab, exact solution	f=1.2e+08 Hz, kL=2.51	4.0165e-05	0	4.0e-05	✓
ELE-1187	Full-wave potential converges to the closed form at second order	err(h)/err(h/2) ~ 4	O(h^2) convergence study	err(80)/err(160)	4.0198	4	5.0e-03	✓
ELE-1188	Good-conductor limit recovers the skin-effect field	sigma dominant -> g=(1+j)/delta diffusion	time-harmonic slab, exact solution	sigma=1e6, f=5 kHz	0.0013226	0	1.3e-03	✓
ELE-1189	General lossy medium matches the complex-propagation closed form	g=sqrt(jomegamusigma - omega^2mueps), sigma=omegaeps (loss tangent 1)	time-harmonic slab, exact solution	f=200 MHz, conduction=displacement	5.8121e-05	0	5.8e-05	✓
ELE-1190	Low-frequency limit recovers the linear profile	omega->0 -> u(x)=u0 (L-x)/L	quasistatic (Laplace) limit	f->0, u(L/2)	0.5	0.5	1.1e-14	✓

Plasticity — kinematic hardening

8 comparisons8/8 passmax err 4.1e-05

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
PLA-1191	Hardening slope in plastic strain equals H + C	d(sigma)/d(eps_p) = H + C	combined linear hardening, exact	H=0, C=5e+09	5.0000e+09	5.0000e+09	7.8e-15	✓
PLA-1192	Bauschinger reverse-yield elastic span equals 2(sy + H*alpha)	sigma_peak - sigma_reverse-yield = 2(sy + H*alpha)	translated yield surface, exact	H=0, C=5e+09	5.0002e+08	5.0000e+08	4.1e-05	✓
PLA-1193	Hardening slope in plastic strain equals H + C	d(sigma)/d(eps_p) = H + C	combined linear hardening, exact	H=2e+09, C=3e+09	5.0000e+09	5.0000e+09	2.9e-15	✓
PLA-1194	Bauschinger reverse-yield elastic span equals 2(sy + H*alpha)	sigma_peak - sigma_reverse-yield = 2(sy + H*alpha)	translated yield surface, exact	H=2e+09, C=3e+09	5.1464e+08	5.1463e+08	1.1e-05	✓
PLA-1195	Hardening slope in plastic strain equals H + C	d(sigma)/d(eps_p) = H + C	combined linear hardening, exact	H=4e+09, C=0	4.0000e+09	4.0000e+09	3.5e-15	✓
PLA-1196	Bauschinger reverse-yield elastic span equals 2(sy + H*alpha)	sigma_peak - sigma_reverse-yield = 2(sy + H*alpha)	translated yield surface, exact	H=4e+09, C=0	5.2942e+08	5.2941e+08	1.7e-05	✓
PLA-1197	Pure-kinematic Bauschinger span is 2*sy independent of prior strain	sigma_peak - sigma_reverse = 2*sy (H=0)	linear kinematic, exact	eps1=0.003	5.0000e+08	5.0000e+08	9.8e-06	✓
PLA-1198	Pure-kinematic Bauschinger span is 2*sy independent of prior strain	sigma_peak - sigma_reverse = 2*sy (H=0)	linear kinematic, exact	eps1=0.006	5.0000e+08	5.0000e+08	9.8e-06	✓

Plasticity — Chaboche kinematic

10 comparisons10/10 passmax err 2.1e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
PLA-1199	Back-stress saturates at C/gamma	X_sat = C/gamma (monotonic Armstrong-Frederick)	nonlinear kinematic hardening, exact saturation	C=6e+09, gamma=300	2.0000e+07	2.0000e+07	2.3e-08	✓
PLA-1200	Saturated stress equals sy + C/gamma	sigma_sat = sy + C/gamma	nonlinear kinematic hardening, exact	C=6e+09, gamma=300	2.7000e+08	2.7000e+08	1.7e-09	✓
PLA-1201	Back-stress follows X(p)=(C/gamma)(1-e^{-gamma p})	Armstrong-Frederick integrated ODE	exponential closed form	C=6e+09, gamma=300	1.9181e+07	1.9182e+07	5.1e-05	✓
PLA-1202	Onset tangent equals the linear-kinematic tangent	Et(onset) = E(H+C)/(E+H+C)	consistent tangent at first yield	C=6e+09, gamma=300	5.8251e+09	5.8252e+09	2.9e-05	✓
PLA-1203	Back-stress saturates at C/gamma	X_sat = C/gamma (monotonic Armstrong-Frederick)	nonlinear kinematic hardening, exact saturation	C=1e+10, gamma=500	2.0000e+07	2.0000e+07	2.0e-13	✓
PLA-1204	Saturated stress equals sy + C/gamma	sigma_sat = sy + C/gamma	nonlinear kinematic hardening, exact	C=1e+10, gamma=500	2.7000e+08	2.7000e+08	1.4e-14	✓
PLA-1205	Back-stress follows X(p)=(C/gamma)(1-e^{-gamma p})	Armstrong-Frederick integrated ODE	exponential closed form	C=1e+10, gamma=500	1.9902e+07	1.9903e+07	1.6e-05	✓
PLA-1206	Onset tangent equals the linear-kinematic tangent	Et(onset) = E(H+C)/(E+H+C)	consistent tangent at first yield	C=1e+10, gamma=500	9.5234e+09	9.5238e+09	4.7e-05	✓
PLA-1207	Back-stress integration converges at first order	err(dp)/err(dp/2) ~ 2 (backward Euler)	convergence study	err(1000)/err(2000)	2.0004	2	2.1e-04	✓
PLA-1208	Zero recovery recovers linear-kinematic hardening X = C*p	gamma->0 -> X = C*eps_p	linear-kinematic limit	gamma=1e-6	1.3420e+07	1.3420e+07	1.3e-09	✓

Plasticity — Chaboche multi-back-stress

5 comparisons5/5 passmax err 2.5e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
PLA-1209	Superposed back-stress X(p)=sum_i (C_i/g_i)(1-e^{-g_i p})	sum of Armstrong-Frederick terms	superposition closed form	3 terms, one linear	1.0715e+08	1.0715e+08	3.2e-06	✓
PLA-1210	Linear (gamma=0) term is exactly C*p — the ratcheting term	X_i = C_i eps_p for gamma_i=0	unbounded linear term	C_3=1.5e9, gamma_3=0	2.7321e+07	2.7321e+07	0.0e+00	✓
PLA-1211	Bounded terms saturate at sum_i C_i/gamma_i	X_sat = sum C_i/gamma_i (gamma_i>0)	exact multi-term saturation	two AF terms	8.0000e+07	8.0000e+07	3.1e-11	✓
PLA-1212	Onset tangent uses the summed modulus sum_i C_i	Et(onset) = E(H+sum C_i)/(E+H+sum C_i)	consistent tangent at yield	3 terms	6.3698e+10	6.3714e+10	2.5e-04	✓
PLA-1213	Single term (N=1) reduces to scalar Armstrong-Frederick	chaboche(N=1) == _return_map_af	exact reduction	N=1	1.8509e+07	1.8509e+07	0.0e+00	✓

Plasticity — Hill anisotropic yield

13 comparisons13/13 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
PLA-1214	Uniaxial x-direction yield equals sx0	seq(sx0,0,0)=sigma0 (rolling direction)	Hill48 calibration, exact	sx0=2.5e+08, R0=1.8	2.5000e+08	2.5000e+08	0.0e+00	✓
PLA-1215	Uniaxial y-direction yield equals sy0	seq(0,sy0,0)=sigma0 (transverse)	Hill48 calibration, exact	sy0=3e+08, R0=1.8	3.0000e+08	3.0000e+08	0.0e+00	✓
PLA-1216	Pure-shear yield equals tau0	seq(0,0,tau0)=sigma0	Hill48 calibration, exact	tau0=1.6e+08	1.6000e+08	1.6000e+08	0.0e+00	✓
PLA-1217	Equibiaxial yield equals sigma0/sqrt(F+G)	seq(sb,sb,0)=sigma0 -> sb=sigma0/sqrt(F+G)	Hill48 biaxial closed form	sx0=2.5e+08, sy0=3e+08	3.9104e+08	3.9104e+08	0.0e+00	✓
PLA-1218	Directional material-point test yields at the Hill directional stress	uniaxial-y onset stress = sy0	Hill48 material point, exact	sy0=3e+08	3.0000e+08	3.0000e+08	0.0e+00	✓
PLA-1219	Uniaxial x-direction yield equals sx0	seq(sx0,0,0)=sigma0 (rolling direction)	Hill48 calibration, exact	sx0=3e+08, R0=0.7	3.0000e+08	3.0000e+08	0.0e+00	✓
PLA-1220	Uniaxial y-direction yield equals sy0	seq(0,sy0,0)=sigma0 (transverse)	Hill48 calibration, exact	sy0=2.5e+08, R0=0.7	2.5000e+08	2.5000e+08	0.0e+00	✓
PLA-1221	Pure-shear yield equals tau0	seq(0,0,tau0)=sigma0	Hill48 calibration, exact	tau0=1.75e+08	1.7500e+08	1.7500e+08	0.0e+00	✓
PLA-1222	Equibiaxial yield equals sigma0/sqrt(F+G)	seq(sb,sb,0)=sigma0 -> sb=sigma0/sqrt(F+G)	Hill48 biaxial closed form	sx0=3e+08, sy0=2.5e+08	2.3596e+08	2.3596e+08	0.0e+00	✓
PLA-1223	Directional material-point test yields at the Hill directional stress	uniaxial-y onset stress = sy0	Hill48 material point, exact	sy0=2.5e+08	2.5000e+08	2.5000e+08	0.0e+00	✓
PLA-1224	Isotropic Hill48 reduces to plane-stress von Mises	F=G=H=1/2, N=3/2 -> seq = sqrt(sx^2 - sx sy + sy^2 + 3 sxy^2)	Hill48 -> von Mises reduction, exact	sigma=[200.0, 100.0, 50.0] MPa	1.9365e+08	1.9365e+08	0.0e+00	✓
PLA-1225	Isotropic Hill48 reduces to plane-stress von Mises	F=G=H=1/2, N=3/2 -> seq = sqrt(sx^2 - sx sy + sy^2 + 3 sxy^2)	Hill48 -> von Mises reduction, exact	sigma=[150.0, -80.0, 60.0] MPa	2.2738e+08	2.2738e+08	0.0e+00	✓
PLA-1226	Isotropic Hill48 reduces to plane-stress von Mises	F=G=H=1/2, N=3/2 -> seq = sqrt(sx^2 - sx sy + sy^2 + 3 sxy^2)	Hill48 -> von Mises reduction, exact	sigma=[0.0, 220.0, -90.0] MPa	2.6963e+08	2.6963e+08	0.0e+00	✓

Poroelasticity — Terzaghi consolidation

11 comparisons11/11 passmax err 1.5e-02

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
POR-1227	Consolidation coefficient c_v = kappa D_c / alpha^2	c_v = kappa/(1/M + alpha^2/D_c)	Biot 1-D reduction	alpha=1, 1/M=0	0.01	0.01	0.0e+00	✓
POR-1228	Final settlement equals the drained oedometer value q0 L/D_c	s_inf = q0 L / D_c	oedometer compliance	q0=1e4, L=1, D_c=1e7	0.001	0.001	0.0e+00	✓
POR-1229	Settlement converges to s_inf at large time factor	s(Tv~3) -> q0 L/D_c	long-time consolidation	Tv~3	0.99948	0.99951	2.3e-05	✓
POR-1230	Degree of consolidation U(Tv~0.1) matches Terzaghi	U = 1 - sum (2/M^2) exp(-M^2 Tv)	Terzaghi settlement series	Tv=0.100	0.35459	0.35682	6.2e-03	✓
POR-1231	Degree of consolidation U(Tv~0.2) matches Terzaghi	U = 1 - sum (2/M^2) exp(-M^2 Tv)	Terzaghi settlement series	Tv=0.200	0.50232	0.50409	3.5e-03	✓
POR-1232	Degree of consolidation U(Tv~0.5) matches Terzaghi	U = 1 - sum (2/M^2) exp(-M^2 Tv)	Terzaghi settlement series	Tv=0.500	0.76217	0.76395	2.3e-03	✓
POR-1233	Degree of consolidation U(Tv~0.848) matches Terzaghi	U = 1 - sum (2/M^2) exp(-M^2 Tv)	Terzaghi settlement series	Tv=0.850	0.8992	0.90047	1.4e-03	✓
POR-1234	Half-consolidation time factor Tv(U=0.5)=0.197	Tv_50 = 0.197 (Terzaghi)	characteristic time factor	U=0.5	0.2	0.197	1.5e-02	✓
POR-1235	Pore-pressure profile p(z)/p0 matches the Terzaghi sine series	p/p0 = sum (2/M) sin(M Z) exp(-M^2 Tv)	Terzaghi pressure series	Tv=0.200	0.0026352	0	2.6e-03	✓
POR-1236	Undrained initial excess pore pressure equals the applied load	p(0+) = q0 (alpha=1, 1/M=0, Skempton B=1)	undrained limit	alpha=1, 1/M=0	10000	10000	0.0e+00	✓
POR-1237	Undrained pressure with fluid+solid compressibility	p0 = alpha q0 / (D_c(1/M + alpha^2/D_c))	compressible undrained	1/M=5e-08	6666.7	6666.7	0.0e+00	✓

Elasticity — rotating disk

12 comparisons12/12 passmax err 2.3e-03

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ELA-1238	Solid disk center stress equals (3+nu)/8 rho omega^2 R^2	sigma(0) = (3+nu)/8 q R^2	rotating-disk closed form (plane stress)	solid, plane stress	2.8958e+08	2.8958e+08	0.0e+00	✓
ELA-1239	Free rim radial stress vanishes	sigma_r(R) = 0	traction-free rim	solid rim	0	0	0.0e+00	✓
ELA-1240	Free rim hoop stress equals (1-nu)/4 rho omega^2 R^2	sigma_theta(R) = (1-nu)/4 q R^2	rotating-disk closed form	solid rim	1.2285e+08	1.2285e+08	0.0e+00	✓
ELA-1241	Solid plane-stress: radial stress vs closed form	max \|sigma_r - exact\| / stress scale over the radius	FE vs exact profile	solid, plane stress	0.0021858	0	2.2e-03	✓
ELA-1242	Solid plane-stress: hoop stress vs closed form	max \|sigma_theta - exact\| / stress scale over the radius	FE vs exact profile	solid, plane stress	6.5812e-04	0	6.6e-04	✓
ELA-1243	Solid plane-stress: peak hoop stress vs closed form	peak sigma_theta vs exact	FE vs exact profile	solid, plane stress	1.2570e-05	0	1.3e-05	✓
ELA-1244	Long cylinder center stress uses nu*=nu/(1-nu)	sigma(0) = (3+nu*)/8 q R^2	rotating cylinder (plane strain)	solid, plane strain	3.0086e+08	3.0086e+08	0.0e+00	✓
ELA-1245	Annulus bore hoop stress matches closed form (peak)	sigma_theta(a) exact	rotating annulus, bore	annulus a/b=0.25	5.8683e+08	5.8683e+08	0.0e+00	✓
ELA-1246	Annulus plane-stress: radial stress vs closed form	max \|sigma_r - exact\| / stress scale over the radius	FE vs exact profile	annulus, plane stress	0.0022832	0	2.3e-03	✓
ELA-1247	Annulus plane-stress: hoop stress vs closed form	max \|sigma_theta - exact\| / stress scale over the radius	FE vs exact profile	annulus, plane stress	6.9067e-04	0	6.9e-04	✓
ELA-1248	Annulus plane-stress: peak hoop stress vs closed form	peak sigma_theta vs exact	FE vs exact profile	annulus, plane stress	2.3399e-04	0	2.3e-04	✓
ELA-1249	Centrifugal stress scales with omega^2	sigma(2 omega)/sigma(omega) = 4	quadratic speed scaling	omega doubled	4	4	0.0e+00	✓

Elasticity — thick cylinder (Lame)

10 comparisons10/10 passmax err 2.4e-03

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
ELA-1250	Bore radial stress equals -p_i (internal pressure)	sigma_r(a) = -p_i	Lame thick-cylinder solution	p_i=1e8, p_o=0	-1.0000e+08	-1.0000e+08	0.0e+00	✓
ELA-1251	Bore hoop stress equals p_i (a^2+b^2)/(b^2-a^2)	sigma_theta(a) = p_i (a^2+b^2)/(b^2-a^2)	Lame closed form	p_i=1e8, p_o=0	1.6667e+08	1.6667e+08	0.0e+00	✓
ELA-1252	Outer rim is traction-free under internal pressure	sigma_r(b) = 0	free outer surface	p_i=1e8, p_o=0	2.2352e-17	0	2.2e-17	✓
ELA-1253	Outer hoop stress equals 2 p_i a^2/(b^2-a^2)	sigma_theta(b) = 2 p_i a^2/(b^2-a^2)	Lame closed form	p_i=1e8, p_o=0	6.6667e+07	6.6667e+07	0.0e+00	✓
ELA-1254	Internal pressure: radial stress vs lame	max \|sigma_r - Lame\| / stress scale	FE vs exact profile	p_i=1e8, p_o=0	0.0016695	0	1.7e-03	✓
ELA-1255	Internal pressure: hoop stress vs lame	max \|sigma_theta - Lame\| / stress scale	FE vs exact profile	p_i=1e8, p_o=0	7.1457e-04	0	7.1e-04	✓
ELA-1256	Internal pressure: stress sum invariant	sigma_r + sigma_theta = 2A constant through the wall	FE vs exact profile	p_i=1e8, p_o=0	0.0023841	0	2.4e-03	✓
ELA-1257	External pressure: rim radial stress equals -p_o	sigma_r(b) = -p_o	Lame external-pressure case	p_i=0, p_o=6e7	-6.0000e+07	-6.0000e+07	0.0e+00	✓
ELA-1258	External pressure: bore hoop stress equals -2 p_o b^2/(b^2-a^2)	sigma_theta(a) = -2 p_o b^2/(b^2-a^2)	Lame closed form	p_i=0, p_o=6e7	-1.6000e+08	-1.6000e+08	0.0e+00	✓
ELA-1259	In-plane Lame field is independent of E/nu and plane assumption	sigma_theta(a) same as plane strain	Lame invariance	plane stress, E=70e9	1.6667e+08	1.6667e+08	0.0e+00	✓

Thermoelasticity — thermal cylinder

12 comparisons12/12 passmax err 4.6e-03

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
THE-1260	Inner rim is traction-free (sigma_r=0)	sigma_r(a) = 0	free inner surface	plane stress	0	0	0.0e+00	✓
THE-1261	Outer rim is traction-free (sigma_r=0)	sigma_r(b) = 0	free outer surface	plane stress	-2.9026e-17	0	2.9e-17	✓
THE-1262	Bore hoop stress matches the thermal-stress closed form	sigma_theta(a) exact	Timoshenko thermal cylinder	plane stress	-1.4688e+08	-1.4688e+08	0.0e+00	✓
THE-1263	Outer hoop stress matches the thermal-stress closed form	sigma_theta(b) exact	Timoshenko thermal cylinder	plane stress	9.3123e+07	9.3123e+07	0.0e+00	✓
THE-1264	Plane stress: radial thermal stress vs closed form	max \|sigma_r - exact\| / stress scale	FE vs exact profile	plane stress	0.0037458	0	3.7e-03	✓
THE-1265	Plane stress: hoop thermal stress vs closed form	max \|sigma_theta - exact\| / stress scale	FE vs exact profile	plane stress	0.0011271	0	1.1e-03	✓
THE-1266	Plane stress: free surface radial stress	sigma_r = 0 on both free rims	FE vs exact profile	plane stress	0.0030296	0	3.0e-03	✓
THE-1267	Plane-strain hoop stress equals plane-stress / (1-nu)	sigma_theta_strain(a) = sigma_theta_stress(a)/(1-nu)	plane-stress -> plane-strain scaling	plane strain	-2.0982e+08	-2.0982e+08	0.0e+00	✓
THE-1268	Plane strain: radial thermal stress vs closed form	max \|sigma_r - exact\| / stress scale	FE vs exact profile	plane strain	0.0045895	0	4.6e-03	✓
THE-1269	Plane strain: hoop thermal stress vs closed form	max \|sigma_theta - exact\| / stress scale	FE vs exact profile	plane strain	0.0019702	0	2.0e-03	✓
THE-1270	Plane strain: free surface radial stress	sigma_r = 0 on both free rims	FE vs exact profile	plane strain	0.0038682	0	3.9e-03	✓
THE-1271	Uniform temperature produces no thermal stress	sigma_theta = 0 for T_inner = T_outer	free thermal expansion	uniform T	2.3221e-16	0	2.3e-16	✓

Plates — circular plate bending

8 comparisons8/8 passmax err 1.6e-04

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
PLA-1272	Clamped central deflection w(0)=qR^4/(64D)	w(0) = q R^4 / (64 D)	Kirchhoff thin-plate closed form	clamped, uniform load	0.0010664	0.0010664	1.0e-08	✓
PLA-1273	Clamped central moment M(0)=qR^2(1+nu)/16	M_r(0)=M_theta(0)=q R^2 (1+nu)/16	Kirchhoff closed form	clamped center	812.63	812.5	1.6e-04	✓
PLA-1274	Clamped edge radial moment M_r(R)=-qR^2/8	M_r(R) = -q R^2 / 8	Kirchhoff closed form	clamped edge	-1249.9	-1250	6.9e-05	✓
PLA-1275	Simply-supported central deflection w(0)=qR^4(5+nu)/(64D(1+nu))	w(0) = q R^4 (5+nu)/(64 D (1+nu))	Kirchhoff closed form	simply supported	0.0043477	0.0043477	2.7e-09	✓
PLA-1276	Simply-supported central moment M(0)=qR^2(3+nu)/16	M(0) = q R^2 (3+nu)/16	Kirchhoff closed form	simply supported center	2062.6	2062.5	6.4e-05	✓
PLA-1277	Simply-supported edge radial moment vanishes	M_r(R) = 0	free-moment edge	simply supported edge	8.6515e-06	0	8.7e-06	✓
PLA-1278	Central deflection scales as R^4	w(2R)/w(R) = 16	R^4 scaling of plate bending	R doubled	16	16	0.0e+00	✓
PLA-1279	Central deflection scales as 1/t^3	w(t/2)/w(t) = 8	t^-3 scaling of flexural rigidity	thickness halved	8	8	0.0e+00	✓

Plates — rectangular plate (Navier)

6 comparisons6/6 passmax err 3.1e-03

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
PLA-1280	Central deflection matches the exact single-term Navier solution	w = p0/(D pi^4 (1/a^2+1/b^2)^2)	MITC4 vs exact Navier	a/b=1	1.3981e-04	1.4013e-04	2.3e-03	✓
PLA-1281	Deflection shape is sin(pi x/a) sin(pi y/b)	w(x,y)/w_center vs sin(pi x/a) sin(pi y/b)	MITC4 vs exact mode shape	a/b=1	1.0356e-04	0	1.0e-04	✓
PLA-1282	Central deflection matches the exact single-term Navier solution	w = p0/(D pi^4 (1/a^2+1/b^2)^2)	MITC4 vs exact Navier	a/b=1.5	2.6782e-04	2.6865e-04	3.1e-03	✓
PLA-1283	Deflection shape is sin(pi x/a) sin(pi y/b)	w(x,y)/w_center vs sin(pi x/a) sin(pi y/b)	MITC4 vs exact mode shape	a/b=1.5	6.6157e-05	0	6.6e-05	✓
PLA-1284	MITC4 mesh stays within tolerance of the exact solution	\|w_fe - w_navier\|/w_navier < 1e-2	mesh accuracy	nel=12	0.0022681	0	2.3e-03	✓
PLA-1285	Central deflection scales linearly with the load amplitude	w(2 p0) / w(p0) = 2	linearity in load	p0 doubled	2	2	0.0e+00	✓

Dynamics — beam free vibration

11 comparisons11/11 passmax err 1.1e-05

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
DYN-1286	Fundamental frequency (simply) matches Euler-Bernoulli	f1 = (beta_1 L/L)^2 sqrt(EI/rho A)/(2 pi)	Euler-Bernoulli closed form	simply	58.632	58.632	2.7e-08	✓
DYN-1287	Third overtone frequency (simply) matches Euler-Bernoulli	f4 vs closed form	Euler-Bernoulli closed form	simply	938.12	938.11	6.7e-06	✓
DYN-1288	Fundamental frequency (cantilever) matches Euler-Bernoulli	f1 = (beta_1 L/L)^2 sqrt(EI/rho A)/(2 pi)	Euler-Bernoulli closed form	cantilever	20.887	20.887	5.3e-09	✓
DYN-1289	Third overtone frequency (cantilever) matches Euler-Bernoulli	f4 vs closed form	Euler-Bernoulli closed form	cantilever	718.24	718.24	3.9e-06	✓
DYN-1290	Fundamental frequency (fixed) matches Euler-Bernoulli	f1 = (beta_1 L/L)^2 sqrt(EI/rho A)/(2 pi)	Euler-Bernoulli closed form	fixed	132.91	132.91	1.4e-07	✓
DYN-1291	Third overtone frequency (fixed) matches Euler-Bernoulli	f4 vs closed form	Euler-Bernoulli closed form	fixed	1187.3	1187.3	1.1e-05	✓
DYN-1292	Simply-supported overtone ratio f2/f1 = 2^2	f_n/f_1 = n^2 (SS beam)	harmonic overtone series	n=2	4	4	1.9e-07	✓
DYN-1293	Simply-supported overtone ratio f3/f1 = 3^2	f_n/f_1 = n^2 (SS beam)	harmonic overtone series	n=3	9	9	1.0e-06	✓
DYN-1294	Simply-supported overtone ratio f4/f1 = 4^2	f_n/f_1 = n^2 (SS beam)	harmonic overtone series	n=4	16	16	3.2e-06	✓
DYN-1295	Fundamental frequency scales as sqrt(E)	f(4E)/f(E) = 2	sqrt(EI) frequency scaling	E x4	2	2	0.0e+00	✓
DYN-1296	Fundamental frequency scales as 1/L^2	f(2L)/f(L) = 1/4	1/L^2 frequency scaling	L x2	0.25	0.25	4.1e-10	✓

Dynamics — rod axial vibration

11 comparisons11/11 passmax err 1.8e-03

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
DYN-1297	Fundamental longitudinal frequency (fixed-free)	f1 vs 1-D bar closed form	bar wave equation	fixed-free	646.54	646.52	2.9e-05	✓
DYN-1298	Fourth longitudinal frequency (fixed-free)	f4 vs 1-D bar closed form	bar wave equation	fixed-free	4532	4525.7	1.4e-03	✓
DYN-1299	Fundamental longitudinal frequency (fixed-fixed)	f1 vs 1-D bar closed form	bar wave equation	fixed-fixed	1293.2	1293	1.1e-04	✓
DYN-1300	Fourth longitudinal frequency (fixed-fixed)	f4 vs 1-D bar closed form	bar wave equation	fixed-fixed	5181.7	5172.2	1.8e-03	✓
DYN-1301	Fundamental longitudinal frequency (free-free)	f1 vs 1-D bar closed form	bar wave equation	free-free	1293.2	1293	1.1e-04	✓
DYN-1302	Fourth longitudinal frequency (free-free)	f4 vs 1-D bar closed form	bar wave equation	free-free	5181.7	5172.2	1.8e-03	✓
DYN-1303	Fixed-free overtone ratio f2/f1 = 3	f_n/f_1 = (2n-1) (fixed-free bar)	odd-harmonic series	n=2	3.0004	3	1.3e-04	✓
DYN-1304	Fixed-free overtone ratio f3/f1 = 5	f_n/f_1 = (2n-1) (fixed-free bar)	odd-harmonic series	n=3	5.0019	5	3.9e-04	✓
DYN-1305	Fixed-free overtone ratio f4/f1 = 7	f_n/f_1 = (2n-1) (fixed-free bar)	odd-harmonic series	n=4	7.0054	7	7.7e-04	✓
DYN-1306	Fixed-free fundamental equals c/(4L)	f1 = c/(4L), c=sqrt(E/rho)	quarter-wave resonance	fixed-free	646.54	646.52	2.9e-05	✓
DYN-1307	Fundamental scales with wave speed sqrt(E)	f(4E)/f(E) = 2	sqrt(E/rho) wave-speed scaling	E x4	2	2	0.0e+00	✓

Beams — Timoshenko shear deflection

7 comparisons7/7 passmax err 4.6e-02

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
BEA-1308	cantilever deflection matches Timoshenko (stubby)	delta = bending + shear closed form	Timoshenko beam theory	cantilever, L/h=4	2.5570e-05	2.5570e-05	6.2e-15	✓
BEA-1309	cantilever deflection matches Timoshenko (slender)	delta = bending + shear closed form	Timoshenko beam theory	cantilever, L/h=40	0.024393	0.024393	3.6e-12	✓
BEA-1310	simply deflection matches Timoshenko (stubby)	delta = bending + shear closed form	Timoshenko beam theory	simply, L/h=4	1.8210e-06	1.8210e-06	9.3e-15	✓
BEA-1311	simply deflection matches Timoshenko (slender)	delta = bending + shear closed form	Timoshenko beam theory	simply, L/h=40	0.0015268	0.0015268	2.2e-13	✓
BEA-1312	Slender cantilever recovers Euler-Bernoulli PL^3/(3EI)	delta -> P L^3/(3 E I) as h/L -> 0	Euler-Bernoulli limit	L/h=40	0.024393	0.024381	4.9e-04	✓
BEA-1313	Cantilever shear deflection equals P L/(ks G A)	delta_shear = P L/(ks G A)	transverse-shear compliance	cantilever	1.1886e-06	1.1886e-06	0.0e+00	✓
BEA-1314	Shear share is larger for a short/deep beam than a slender one	shear_fraction(L/h=4) > 10x shear_fraction(L/h=40)	shear scales as (h/L)^2	ratio	95.398	100	4.6e-02	✓

Per-solve numerical verification

6 comparisons6/6 passmax err 2.6e-10

ID	Problem	Reference	Source	Parameters	Computed	Rel. err	✓
PER-1315	Modal eigenpair residual	normalized governing-equation residual = 0	discrete Galerkin/Newmark/eigenvalue equations	maximum_eigenpair_residual	0	0.0e+00	✓
PER-1316	Transient dynamic equilibrium	normalized governing-equation residual = 0	discrete Galerkin/Newmark/eigenvalue equations	maximum_dynamic_equilibrium_residual	5.9286e-14	5.9e-14	✓
PER-1317	Harmonic dynamic equilibrium	normalized governing-equation residual = 0	discrete Galerkin/Newmark/eigenvalue equations	maximum_dynamic_equilibrium_residual	5.5511e-17	5.6e-17	✓
PER-1318	Geometrically nonlinear equilibrium	normalized governing-equation residual = 0	discrete Galerkin/Newmark/eigenvalue equations	final_nonlinear_equilibrium_residual	2.6318e-10	2.6e-10	✓
PER-1319	Unilateral contact complementarity	normalized governing-equation residual = 0	discrete Galerkin/Newmark/eigenvalue equations	maximum_contact_constraint_violation	2.3218e-24	2.3e-24	✓
PER-1320	Steady field equilibrium	normalized Kphi-f residual = 0	discrete Galerkin equation	field_equilibrium_residual	0	0.0e+00	✓

Time discretization error estimation

5 comparisons5/5 passmax err 7.7e-02

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
TIM-1321	Newmark Richardson estimate vs exact SDOF error	u(t)=F/k(1-cos(omega t)); coarse error from dt/dt/2	Newmark (1959); Richardson extrapolation	steps/period=10	0.12345	0.12437	7.4e-03	✓
TIM-1322	Newmark Richardson estimate vs exact SDOF error	u(t)=F/k(1-cos(omega t)); coarse error from dt/dt/2	Newmark (1959); Richardson extrapolation	steps/period=20	0.031758	0.031797	1.2e-03	✓
TIM-1323	Newmark estimator recovers second-order convergence	eta(dt)/eta(dt/2) = 2^2	Newmark method order	dt halved	3.8873	4	2.8e-02	✓
TIM-1324	Backward Euler estimator recovers order 1	eta(dt)/eta(dt/2) = 2^1	theta-method order p=1	dt halved	2.1538	2	7.7e-02	✓
TIM-1325	Crank-Nicolson estimator recovers order 2	eta(dt)/eta(dt/2) = 2^2	theta-method order p=2	dt halved	3.9782	4	5.4e-03	✓

Contact spatial convergence

6 comparisons6/6 passmax err 0.0e+00

ID	Problem	Reference	Source	Parameters	Computed	Reference	Rel. err	✓
CON-1326	Refined interface force matches two-bar compatibility	Fc=(P L/EA-g)/(2 L/EA)	closed-form two-bar compatibility	load_factor=2	5000	5000	0.0e+00	✓
CON-1327	Refined interface force matches two-bar compatibility	Fc=(P L/EA-g)/(2 L/EA)	closed-form two-bar compatibility	load_factor=3	10000	10000	0.0e+00	✓
CON-1328	Refined interface force matches two-bar compatibility	Fc=(P L/EA-g)/(2 L/EA)	closed-form two-bar compatibility	load_factor=5	20000	20000	0.0e+00	✓
CON-1329	Contact pressure is force divided by declared tributary area	p=\|Fc\|/A_contact	discrete contact pressure definition	area=0.02	1.0000e+06	1.0000e+06	0.0e+00	✓
CON-1330	Two-grid estimator detects an active-set transition	one changed declared contact	discrete two-grid active-set property	coarse open; refined closed	1	1	0.0e+00	✓
CON-1331	Active-set transition prevents a false PASS	verdict=REVIEW	discrete evidence-contract property	coarse open; refined closed	1	1	0.0e+00	✓

Implementation claims,made executable.

Bars & axial members

Beam bending

Elastic stability

Dynamics — natural frequency

Prestressed modal (stress stiffening)

Dynamics — damped modal

Dynamics — nonlinear continuum

Fluids — Stokes flow

Fluids — Navier-Stokes

Fluids — Transient Navier-Stokes

Multiphysics — Piezoelectric

Multiphysics — Vibro-acoustics

Multiphysics — Magnetostatics

Materials — finite-strain plasticity

Materials — hyperelasticity

Materials — viscoplasticity

Materials — 2D creep

Multiphysics — temperature-dependent stiffness

Multiphysics — hygroscopic swelling

Reduction — static condensation (superelement)

Infrastructure — matrix-free operator

Infrastructure — distributed matrix-free operator

Reduction — p-refinement

Multiphysics — temperature-dependent properties

Materials — damage-plasticity

Acoustics — cavity modes

Acoustics — driven response

Meshing — unstructured triangulation

Fatigue — stress life

Fatigue — strain life

Fatigue — crack growth

Fatigue — from FE stress field

Contact — node-to-segment

Thermal stress

Torsion

Continuum patch tests

3D solids

3D frames

Orthotropic & composites

Composite laminates (CLT)

Micromechanics

Progressive failure (Hashin)

Oxidation (reaction-diffusion)

User material (UMAT hook)

Composite literature benchmarks

Self-verification (error estimator)

Adaptive refinement

Targeted refinement

3D error estimate

Result assessment

Higher-order elements

Advanced analyses

Heat transfer

Nonlinear — hyperelastic

Contact & friction

Dynamics — harmonic

Shells

Dynamics — nonlinear transient

Composites — laminated shells

Materials — viscoelastic & creep

Dynamics — explicit

Acoustics — damped response

Multiphysics — thermo-plastic

Multiphysics — Nonlinear magnetostatics

Multiphysics — Piezoelectric 2D

Multiphysics — Piezoelectric 3D

Multiphysics — Eddy currents

Fracture — cohesive zone

Fracture — mixed-mode cohesive

Electromagnetics — full-wave

Plasticity — kinematic hardening

Plasticity — Chaboche kinematic

Plasticity — Chaboche multi-back-stress

Plasticity — Hill anisotropic yield

Poroelasticity — Terzaghi consolidation

Elasticity — rotating disk

Elasticity — thick cylinder (Lame)

Thermoelasticity — thermal cylinder

Plates — circular plate bending

Implementation claims,
made executable.