Verification gallery

Runnable examples that drive femorph-solver’s validation framework to reproduce canonical FEA problems against published reference values.

Each example instantiates one BenchmarkProblem subclass, runs validate() (or a multi-refinement ConvergenceStudy), prints the computed-vs-published comparison, and renders a figure of the solved model.

The reference values in this gallery are traceable to a textbook, paper, or handbook — not to any other solver’s output. Every example is also a regression-guarded test under tests/validation/, so a gallery failure surfaces as a CI failure before it can drift silently in the rendered docs.

Single-hex uniaxial

Hooke’s law + Poisson contraction on a unit hex. Machine-precision check (Hughes 2000 §2.7).

Single-hex uniaxial tension — Hooke’s law + Poisson check

Irons patch test

Constant-strain patch test — the canonical consistency check (Irons & Razzaque 1972).

Irons constant-strain patch test

Cantilever tip deflection

\(\\delta = P L^3 / (3 E I)\) — Euler-Bernoulli closed form (Timoshenko 1955 §5.4).

Cantilever tip deflection — Euler-Bernoulli closed form

Cantilever natural frequency

First bending frequency (Rao 2017 §8.5 Table 8.1).

Cantilever natural frequency — Rao 2017 Table 8.1

Irons constant-strain patch test

Irons constant-strain patch test

Single-hex uniaxial tension — Hooke’s law + Poisson check

Single-hex uniaxial tension — Hooke's law + Poisson check

Cantilever tip deflection — Euler-Bernoulli closed form

Cantilever tip deflection — Euler-Bernoulli closed form

NAFEMS LE1 — elliptic membrane (plane stress)

NAFEMS LE1 — elliptic membrane (plane stress)

Clamped square plate under uniform pressure (NAFEMS LE5)

Clamped square plate under uniform pressure (NAFEMS LE5)

Cantilever under uniformly distributed load — Euler–Bernoulli closed form

Cantilever under uniformly distributed load — Euler–Bernoulli closed form

Free-free axial rod — natural frequencies

Free-free axial rod — natural frequencies

Cantilever natural frequency — Rao 2017 Table 8.1

Cantilever natural frequency — Rao 2017 Table 8.1

Simply-supported beam — uniformly-distributed load

Simply-supported beam — uniformly-distributed load

Fixed-free axial rod — natural frequencies

Fixed-free axial rod — natural frequencies

Clamped-clamped beam — first three bending natural frequencies

Clamped-clamped beam — first three bending natural frequencies

Pinched ring — Castigliano diametrical deflection

Pinched ring — Castigliano diametrical deflection

Cantilever Saint-Venant torsion — rectangular cross-section

Cantilever Saint-Venant torsion — rectangular cross-section

Simply-supported plate — first transverse bending mode

Simply-supported plate — first transverse bending mode

Simply-supported beam — first three bending natural frequencies

Simply-supported beam — first three bending natural frequencies

Plate with a circular hole — Kirsch stress concentration

Plate with a circular hole — Kirsch stress concentration

Simply-supported plate under uniform pressure (Navier series)

Simply-supported plate under uniform pressure (Navier series)

Shear-locking demonstration — HEX8 integration variants

Shear-locking demonstration — HEX8 integration variants

Cantilever beam — first four bending modes

Cantilever beam — first four bending modes

Cantilever beam under a tip moment

Cantilever beam under a tip moment

Clamped-clamped beam under a central point load

Clamped-clamped beam under a central point load

Simply-supported beam under a central point load

Simply-supported beam under a central point load

Propped cantilever under uniformly-distributed load

Propped cantilever under uniformly-distributed load

Mesh-refinement convergence — cantilever Euler-Bernoulli

Mesh-refinement convergence — cantilever Euler-Bernoulli

Mesh-refinement convergence — Lamé thick cylinder

Mesh-refinement convergence — Lamé thick cylinder

L-shaped frame under tip load — Castigliano on a two-member portal

L-shaped frame under tip load — Castigliano on a two-member portal

Propped cantilever with central point load

Propped cantilever with central point load

Lamé thick-walled cylinder under internal pressure

Lamé thick-walled cylinder under internal pressure

Cantilever with off-tip point load — load-position shape function

Cantilever with off-tip point load — load-position shape function

Continuous beam over three supports — Clapeyron’s three-moment theorem

Continuous beam over three supports — Clapeyron's three-moment theorem

Cantilever under linearly-varying distributed load (triangular)

Cantilever under linearly-varying distributed load (triangular)

Simply-supported beam with off-center point load

Simply-supported beam with off-center point load

Cook’s membrane — mesh-distortion benchmark (1974)

Cook's membrane — mesh-distortion benchmark (1974)