Verification

femorph-solver’s correctness story rests on three independent pillars — each benchmark page in this section sits on at least one of them:

1. Analytical reference — a closed-form solution derived from classical mechanics (Timoshenko, Cook, Hughes, Bathe, …). The unambiguous ground truth where one exists.

2. Vendor-neutral benchmark corpus — published for the express purpose of cross-solver comparison:

   • NAFEMS Benchmark Tests for Linear Elastic Analysis (NAFEMS BENCHMARK-LA series) — the industry standard for linear-structural verification. Vendor-neutral, attribution-free reuse.

   • NAFEMS Background to Benchmarks — methodology and derivation companion.

   • ASME boiler-code benchmark suite — free with attribution.

3. Cross-reference citations to proprietary verification manuals — short factual citations only (one line, one problem ID + one numeric comparison value per source). This follows the academic-publishing convention used by every serious FEA validation paper:

   • Abaqus Verification Manual (Simulia) — referenced by problem ID (e.g., AVM 1.4.3). We never vendor AVM decks, text, or expected-result tables.

   • Abaqus Benchmarks Manual — same posture.

   • MAPDL Verification Manual (Ansys) — referenced by problem ID (e.g., VM-1). Same posture.

   • NX Nastran Verification Manual (Siemens) — same.

Each benchmark page follows the same four-section template:

• Problem — our prose, physics derived from textbook.

• Analytical reference (where applicable) — closed-form + cited textbook derivation.

• femorph-solver result — computed value, committed test.

• Cross-references — one-line citations to NAFEMS / AVM / MAPDL-VM / etc. for readers who want to consult multiple independent verification sources.

Commercial-ship posture

Everything under doc/source/verification/ is original content or legally-reusable benchmark spec. We never vendor proprietary verification manuals’ decks or their tables of expected results. See the fair-use citation pattern below.

Fair-use citation pattern

A typical cross-reference row in a benchmark page reads:

| Source            | Reference value | Problem ID |
|-------------------|-----------------|------------|
| Closed form       | 0.001 000 m     | (Timoshenko §5.4) |
| NAFEMS LE1        | 0.000 998 m     | NAFEMS BENCHMARK-LE1 |
| femorph-solver    | 0.001 012 m     | this test |
| Abaqus VM         | 0.000 999 m     | AVM 1.4.3 |
| MAPDL VM          | 0.000 998 m     | VM-1 |

Short, factual, purposeful — a single number for comparison plus the problem ID. This is the same table you’ll find in published FEA-vendor cross-comparison papers going back decades.

Vendor coverage

• Vendor verification-manual coverage matrix

Structural benchmarks

• Cantilever beam — tip deflection under end load
• Cantilever beam — tip deflection + rotation under pure tip moment
• Cantilever under tip torque — Saint-Venant torsion
• Cantilever beam — tip deflection under uniform distributed load
• Simply-supported beam — central point load
• Simply-supported beam — uniformly-distributed load
• Clamped-clamped beam — central point load
• Propped cantilever — central point load
• Pinched ring — diametrical deflection
• Plate with a circular hole under uniaxial tension — Kirsch
• Cantilever beam — first bending natural frequency
• Cantilever beam — higher bending natural frequencies
• Simply-supported beam — first three bending natural frequencies
• Clamped-clamped beam — first three bending natural frequencies
• Axial rod — fundamental natural frequency (fixed-free)
• Free-free axial rod — natural frequencies
• NAFEMS LE1 — elliptic membrane under outward pressure
• Thick-walled cylinder under internal pressure (Lamé)
• Conservation + orthogonality — four-in-one sanity check
• Irons patch test — constant-strain completeness
• Simply-supported plate — Kirchhoff bending frequencies
• NAFEMS LE5 — clamped square plate under uniform pressure
• Single hex — uniaxial tension (Hooke’s law)