Verification
============

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

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

#. **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.

.. toctree::
    :maxdepth: 1
    :caption: Vendor coverage

    vendor_matrix

.. toctree::
    :maxdepth: 1
    :caption: Structural benchmarks

    cantilever_beam_tip_deflection
    cantilever_tip_moment
    cantilever_torsion
    cantilever_udl
    ss_beam_central_load
    ss_beam_udl
    cc_beam_central_load
    propped_cantilever
    pinched_ring
    plate_with_hole
    cantilever_beam_natural_frequency
    cantilever_higher_modes
    ss_beam_modes
    cc_beam_modes
    axial_rod_nf
    free_free_rod_nf
    nafems_le1
    lame_cylinder
    conservation_and_orthogonality
    patch_test
    simply_supported_plate
    clamped_plate_static
    single_hex_uniaxial