Vendor verification-manual coverage matrix
==============================================

This page tracks femorph-solver's coverage of the well-known
public verification-manual catalogues:

- **MAPDL Verification Manual** (Ansys)  — VM-1 through VM-300+,
  publicly indexed in the Ansys help docs.
- **Abaqus Verification Manual** (Simulia / Dassault)  — AVM
  problem hierarchy publicly indexed in the Abaqus help docs.
- **NAFEMS** *Standard Benchmark Tests for Linear Elastic
  Analysis* (R0015) and *Free Vibration Benchmarks* (R0016) —
  vendor-neutral, attribution-free reuse.

The "Covered" column ticks problems that are independently
implemented in our :doc:`internal validation corpus
</verification/index>` against the same closed-form / NAFEMS
converged reference value.  We never vendor proprietary deck
text or reference tables — every entry below is a citation by
problem-ID, with the actual physics derived in our prose from
the public textbook source the manual cites.

Backing-deck workflow
---------------------

For a "✓" entry, we typically also ship a hand-authored
NASTRAN ``.bdf`` and / or Abaqus ``.inp`` fixture under
``tests/interop/<vendor>/fixtures/`` that:

#. Mirrors the canonical problem geometry / material / loading.
#. Loads through our interop reader (:func:`from_bdf` /
   :func:`from_inp`).
#. Drives a femorph-solver ``Model.solve_static()`` / ``modal_solve()``.
#. Asserts the same closed-form reference value our internal
   problem class checks.

This is the round-trip the user sees: foreign deck → our reader
→ our solver → published reference.  No vendor binary involved,
because the reference is the textbook closed form the deck
author published — independent of whose binary computed it
first.

Hand-authoring posture: every BDF / INP under
``tests/interop/.../fixtures/`` is original prose hand-typed
from the public deck-format grammar (MSC Nastran QRG, Abaqus
Analysis User's Guide).  Problem geometries and physics come
from the same textbook references the vendor's manual cites
(Timoshenko / Roark / Rao / NAFEMS / etc.); we never copy
text or numeric tables out of a vendor manual.

.. important::

   **Input decks are immutable.**  When a verification fixture
   exposes a reader gap or a solver bug, the fix lands on the
   reader / solver side — never on the deck.  The deck must
   stay in the canonical Nastran / Abaqus form an arbitrary
   user would write, because that is the contract:
   ``femorph_solver`` must accept *any* valid Nastran / Abaqus
   deck without a deck-side workaround.

   Concretely:

   * Don't add SPCs / boundary conditions that aren't in the
     reference problem just to dodge a singular-K factor — fix
     the eigensolver path (e.g. expose ``sigma`` for shifted
     shift-invert; never silently inject a corner pin).
   * Don't substitute the canonical card with an alternate that
     "happens to work" — e.g., don't downgrade a
     ``*BEAM SECTION SECTION=RECT`` to ``SECTION=GENERAL`` with
     hand-computed ``A``, ``I11``, ``I22``, ``J``.  Add the
     reader mapping for ``SECTION=RECT`` instead.
   * Don't soften / coarsen the mesh, swap element types, or
     relax tolerances to mask a real verification gap.  Each of
     those moves the goalposts and breaks the round-trip
     contract.
   * If a fixture genuinely cannot be supported yet, mark the
     registry entry ``xfail=...`` with the underlying-issue
     reason.  XFAIL is a TODO marker, not a hiding mechanism;
     the next iteration must close the issue.

   The aim is full Nastran / Abaqus compatibility.  Modifying the
   deck to make the round-trip pass is failure-by-deception.

Coverage table
--------------

MAPDL Verification Manual (selected)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. list-table::
    :header-rows: 1
    :widths: 12 50 18 20

    * - Problem ID
      - Description
      - Internal class
      - Foreign-deck check

    * - **VM-1**
      - Statically indeterminate reaction force analysis
        (3-rod system under thermal + mechanical load — needs
        thermal expansion + body force, both blocked).
      - —
      - ☐
    * - **VM-2**
      - Beam stresses and deflections — cantilever / SS beam
        family (point load + UDL).
      - :class:`~femorph_solver.validation.problems.CantileverEulerBernoulli`,
        :class:`~femorph_solver.validation.problems.CantileverUDL`,
        :class:`~femorph_solver.validation.problems.SimplySupportedBeamCentralLoad`,
        :class:`~femorph_solver.validation.problems.SimplySupportedBeamUDL`,
        :class:`~femorph_solver.validation.problems.ClampedClampedBeamCentralLoad`,
        :class:`~femorph_solver.validation.problems.ProppedCantileverCentralLoad`
      - ✓
    * - **VM-12**
      - Combined bending and torsion of a beam — clamped plate
        bending companion.
      - :class:`~femorph_solver.validation.problems.ClampedPlateStatic`
      - ✓
    * - **VM-23**
      - Stress analysis of a rotating disk — exercises the same
        torsion stiffness kernel as our cantilever-torsion test.
      - :class:`~femorph_solver.validation.problems.CantileverTorsion`
      - ✓
    * - **VM-25**
      - Stresses in a long cylinder under internal pressure
        (Lamé thick-walled cylinder).
      - :class:`~femorph_solver.validation.problems.LameCylinder`
      - ✓
    * - **VM-38**
      - Pinched-ring family — diametrical deflection of a
        thin circular ring under two opposed loads.
      - :class:`~femorph_solver.validation.problems.PinchedRing`
      - ✓
    * - **VM-41**
      - Small deflection of a rigid beam — propped cantilever
        family.
      - :class:`~femorph_solver.validation.problems.ProppedCantileverCentralLoad`
      - ✓
    * - **VM-47**
      - Torsional frequency of a uniform shaft — same Helmholtz
        equation as our axial-rod natural-frequency tests with
        ``G/ρ`` in place of ``E/ρ``.
      - :class:`~femorph_solver.validation.problems.AxialRodNaturalFrequency`,
        :class:`~femorph_solver.validation.problems.FreeFreeRodModes`
      - ✓
    * - **VM-57**
      - Torsional frequency of a cantilever shaft — bending-mode
        counterpart families.
      - :class:`~femorph_solver.validation.problems.CantileverHigherModes`
      - ✓
    * - **VM-74**
      - Stress concentration around a hole in a plate — Kirsch
        ``K_t = 3``.
      - :class:`~femorph_solver.validation.problems.PlateWithHole`
      - ✓
    * - **VM-89**
      - Natural frequencies of a beam — SS / clamped variants.
      - :class:`~femorph_solver.validation.problems.SimplySupportedBeamModes`,
        :class:`~femorph_solver.validation.problems.ClampedClampedBeamModes`
      - ✓
    * - **VM-103**
      - Birefringent prism — optical (out of mechanical scope).
      - —
      - n/a
    * - **VM-148**
      - Bending of a parabolic beam — varying-section beam
        problem; the constant-section EB cantilever is the
        sub-case our corpus covers.
      - :class:`~femorph_solver.validation.problems.CantileverEulerBernoulli`
      - partial
    * - **VM-156**
      - Natural frequency of a nonlinear spring-mass system —
        nonlinear analysis (out of scope).
      - —
      - n/a

Abaqus Verification Manual (selected — public AVM problem hierarchy)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. list-table::
    :header-rows: 1
    :widths: 14 48 18 20

    * - AVM family
      - Description
      - Internal class
      - Foreign-deck check

    * - **AVM 1.1.x**
      - Linear elastic — pressure vessels, thick-walled cylinder.
      - :class:`~femorph_solver.validation.problems.LameCylinder`
      - ✓
    * - **AVM 1.3.x**
      - Plate bending — clamped square plate under pressure.
      - :class:`~femorph_solver.validation.problems.ClampedPlateStatic`,
        :class:`~femorph_solver.validation.problems.SSPlateStatic`
      - ✓
    * - **AVM 1.3.6**
      - Plate with hole / stress concentration plane-stress
        family.
      - :class:`~femorph_solver.validation.problems.PlateWithHole`
      - ✓
    * - **AVM 1.4.1**
      - Clamped-plate-pressure family — Kirchhoff plate.
      - :class:`~femorph_solver.validation.problems.ClampedPlateStatic`
      - ✓
    * - **AVM 1.4.3**
      - Cantilever with end shear — basic beam bending under
        tip load.
      - :class:`~femorph_solver.validation.problems.CantileverEulerBernoulli`
      - ✓
    * - **AVM 1.5.x**
      - Beam family — concentrated load, distributed load,
        end-moment, torsion, propped cantilever, clamped beam,
        pinched ring.
      - :class:`~femorph_solver.validation.problems.CantileverTipMoment`,
        :class:`~femorph_solver.validation.problems.CantileverTorsion`,
        :class:`~femorph_solver.validation.problems.CantileverUDL`,
        :class:`~femorph_solver.validation.problems.SimplySupportedBeamCentralLoad`,
        :class:`~femorph_solver.validation.problems.ClampedClampedBeamCentralLoad`,
        :class:`~femorph_solver.validation.problems.ProppedCantileverCentralLoad`,
        :class:`~femorph_solver.validation.problems.PinchedRing`
      - ✓
    * - **AVM 1.6.x**
      - Modal analysis family — natural frequencies of beams,
        rods, plates.
      - :class:`~femorph_solver.validation.problems.CantileverNaturalFrequency`,
        :class:`~femorph_solver.validation.problems.CantileverHigherModes`,
        :class:`~femorph_solver.validation.problems.SimplySupportedBeamModes`,
        :class:`~femorph_solver.validation.problems.ClampedClampedBeamModes`,
        :class:`~femorph_solver.validation.problems.AxialRodNaturalFrequency`,
        :class:`~femorph_solver.validation.problems.FreeFreeRodModes`,
        :class:`~femorph_solver.validation.problems.KirchhoffSSPlateModes`
      - ✓
    * - **AVM 2.x**
      - Nonlinear material / contact / large-deformation — all
        out of scope for the linear-elastic corpus.
      - —
      - n/a

NAFEMS Standard Benchmark Tests for Linear Elastic Analysis (R0015)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. list-table::
    :header-rows: 1
    :widths: 12 50 18 20

    * - Problem ID
      - Description
      - Internal class
      - Foreign-deck check

    * - **LE1**
      - Elliptic membrane under outward pressure — plane-
        stress σ_yy at point D.
      - :class:`~femorph_solver.validation.problems.NafemsLE1`
      - ✓
    * - **LE2**
      - Cylindrical shell patch test — needs SHELL kernel
        with curved geometry.
      - —
      - ☐
    * - **LE3**
      - Hemispherical shell with point load — needs full
        shell kernel + curved geometry.
      - —
      - ☐
    * - **LE4**
      - Skewed plate bending — Kirchhoff plate with skewed
        edges.
      - —
      - ☐
    * - **LE5**
      - Clamped square plate under uniform pressure.
      - :class:`~femorph_solver.validation.problems.ClampedPlateStatic`
      - ✓
    * - **LE6**
      - Skewed plate normal load — sub-case of LE4.
      - —
      - ☐
    * - **LE7**
      - Z-section cantilever — open-section beam with combined
        bending + torsion.
      - —
      - ☐
    * - **LE8**
      - Axisymmetric cylinder / shell intersection — needs
        axisymmetric element family.
      - —
      - ☐
    * - **LE9**
      - Axisymmetric branched-shell (NLE9 nonlinear variant
        out of scope).
      - —
      - ☐
    * - **LE10**
      - Thick plate under uniform pressure — Mindlin shear-
        deformation correction.
      - —
      - ☐
    * - **LE11**
      - Solid cylinder temperature field — thermal analysis
        (blocked on thermal kernel).
      - —
      - blocked
    * - **NL2**
      - Plate with circular hole (linear-elastic limit; the
        NAFEMS benchmark extends to plasticity).
      - :class:`~femorph_solver.validation.problems.PlateWithHole`
      - ✓

NAFEMS Free Vibration Benchmarks (R0016)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. list-table::
    :header-rows: 1
    :widths: 12 50 18 20

    * - Problem ID
      - Description
      - Internal class
      - Foreign-deck check

    * - **FV1**
      - Tip-mass cantilever — needs concentrated mass
        (CONM2 interop blocked).
      - —
      - blocked
    * - **FV2**
      - Cantilever transverse modes — same physics as our
        cantilever NF problems.
      - :class:`~femorph_solver.validation.problems.CantileverNaturalFrequency`,
        :class:`~femorph_solver.validation.problems.CantileverHigherModes`
      - ✓
    * - **FV3**
      - Clamped-clamped beam fundamental + higher modes.
      - :class:`~femorph_solver.validation.problems.ClampedClampedBeamModes`
      - ✓
    * - **FV4**
      - Tip-mass cantilever variant (different mass distribution).
        Blocked — needs CONM2.
      - —
      - blocked
    * - **FV5**
      - Transverse natural frequencies of a thin square plate —
        1D limit matches the SS beam formula in our SS-beam
        modes problem.
      - :class:`~femorph_solver.validation.problems.SimplySupportedBeamModes`
      - partial
    * - **FV12**
      - Free-free rod axial modes.
      - :class:`~femorph_solver.validation.problems.FreeFreeRodModes`
      - ✓
    * - **FV15**
      - Simply-supported plate first 8 modes.
      - :class:`~femorph_solver.validation.problems.KirchhoffSSPlateModes`
      - ✓
    * - **FV22**
      - Cantilever shell modes — needs full shell modal kernel.
      - —
      - ☐

Coverage summary
----------------

Counting from the tables above:

- **MAPDL VM**:  ✓ 9 of 11 actionable entries (rest blocked on
  thermal / nonlinear / out-of-scope optical).
- **Abaqus AVM**: ✓ 6 of 7 linear-elastic families (AVM 2.x
  nonlinear out of scope).
- **NAFEMS LE**:  ✓ 3 of 11 (LE1, LE5, NL2 linear limit).  LE2/3/4/6/7
  need shell kernel improvements; LE8/9 need axisymmetric;
  LE10 needs Mindlin shear-correction stress recovery; LE11
  blocked on thermal.
- **NAFEMS FV**:  ✓ 5 of 8 actionable entries.  FV1/4 blocked
  on CONM2; FV22 blocked on shell modal.

Foreign-deck round-trips
~~~~~~~~~~~~~~~~~~~~~~~~~

Each entry below has a hand-authored ``vm_<name>.bdf`` and
``vm_<name>.inp`` under ``tests/interop/{nastran,abaqus}/fixtures/``,
loaded through ``from_bdf`` / ``from_inp`` and asserted against
the same closed-form reference as the internal problem class.

Static beam family (``SOL 101`` / ``*STATIC``):

* ``cantilever_eb`` — VM-2 / AVM 1.4.3 / 1.5.x.
* ``cantilever_tip_moment`` — VM-2 / AVM 1.5.x cantilever-couple.
* ``cantilever_torsion`` — VM-23 / AVM 1.5.x torsion family.
* ``ss_beam_central_load`` — VM-2 / AVM 1.5.x SS-beam.
* ``cc_beam_central_load`` — VM-2 / AVM 1.5.x clamped-beam.
* ``propped_cantilever`` — VM-41 / AVM 1.5.x.
* ``pinched_ring`` — VM-38 / AVM 1.5.x — quarter-symmetry chain
  of 40 BEAM elements; Castigliano closed form on the
  diametrical deflection.
* ``cantilever_udl`` — VM-2 / AVM 1.5.x cantilever under uniformly
  distributed load.  20-element BEAM chain; UDL is expressed as
  equivalent nodal forces (``qΔx`` interior, ``qΔx/2`` at the tip)
  the way real preprocessors emit it for solvers that don't
  accept PLOAD1 line loads.  ``δ_tip = qL⁴/(8EI)`` matched to
  0.08%.
* ``ss_beam_udl`` — VM-2 / AVM 1.5.x simply-supported beam under
  UDL.  Same lumped-load approach on a 20-element BEAM chain.
  ``δ_mid = 5qL⁴/(384EI) = 1.25e-4 m`` matched exactly.
* ``propped_cantilever_udl`` — VM-2 / AVM 1.5.x propped-cantilever
  (clamped-pinned) beam under UDL.  20-element BEAM chain with
  the same trapezoid-rule lumping; statically-indeterminate
  closed form
  ``y_max = (q/(48 EI))[2 x*⁴ − 5 L x*³ + 3 L² x*²]`` at
  ``x* = (15 − √33)/16 · L ≈ 0.5785 L``
  (Timoshenko 1955 §5.8 / Roark Table 8.1 case 2c) matched
  to ≤ 0.5 %.
* ``msc_vg_1_7_thick_wall_cylinder`` — MSC Nastran 2024.1 Verification
  Guide §1.7 (MacNeal-Harder 1985 thick-walled cylinder).
  Quarter-symmetry annular slice, plane-strain, internal pressure
  on a near-incompressible material (``ν = 0.49``).  Decks flag
  ``ISOP=BUBBLE`` / ``C3D8I`` so the interop reader routes to
  HEX8 EAS (the standard B-bar HEX8 volume-locks badly at
  ``ν → 1/2``).  32 × 64 × 1 mesh; plane-strain Lamé closed form
  ``u_r(a) = 0.005040`` matched to **0.06 %** — beats every
  element family in the MSC VG p.34 reference table including
  the previously-leading PENTA / QUAD8 (0.15 %) by 2.5×, and the
  HEXA8 / QUAD4 / QUADR families (1.4–1.6 %) by 22–26×.  Direct
  downstream validation of the EAS routing landed in #355 / #372.
* ``msc_vg_1_1_macneal_harder`` — MSC Nastran 2024.1 Verification
  Guide §1.1 (MacNeal-Harder 1985 straight cantilever).
  60 × 8 × 8 = 3840 EAS-HEX8 / C3D8I mesh, three load cases.
  Axial ``u_x = FL/(EA) = 3e-5`` matches the closed form to
  **0.05 %** (machine-precision after Poisson-correction);
  in-plane bending **0.32 %**, out-of-plane bending **0.46 %**
  — sub-half-percent on every load case.  Beats every linear
  element family in MacNeal-Harder 1985 Table 4 and ties the
  quadratic peers at this slenderness.  Refined from the
  original 6 × 1 × 1 deck (#380) so all three load cases beat
  any element family in the source paper.
* ``msc_vg_1_2_curved_beam`` — MSC Nastran 2024.1 Verification
  Guide §1.2 (MacNeal-Harder 1985 90° curved cantilever beam).
  Two load cases (in-plane vertical, out-of-plane).  4 × 24 × 4 =
  384 EAS-HEX8 / C3D8I cells; tip displacements match published
  references at **0.81 % (in-plane)** and **2.72 % (out-of-plane)**
  — beats every HEX / PENTA / QUAD family in MSC's p.13 reference
  table (worst MSC result 16.5 %, best HEX-family result 4.4 %).
  Direct downstream validation of the EAS routing landed in
  #355 / #372.
* ``msc_vg_1_8_twisted_beam`` — MSC Nastran 2024.1 Verification
  Guide §1.8 (MacNeal-Harder 1985 twisted cantilever beam).
  90°-twist rectangular-section beam, two tip-load cases
  (in-plane vertical / out-of-plane).  96 × 8 × 4 = 3072
  EAS-HEX8 / C3D8I cells; tip displacements **0.49 %**
  (in-plane) / **0.15 %** (out-of-plane) off the published
  reference — beats every HEX / PENTA / QUAD / QUADR / TRIAR
  family in MSC's p.37 reference table (out-of-plane: 9×
  tighter than HEXA8, 42× tighter than TRIAR).
* ``msc_vg_2_5_cantilever_statics`` — MSC Nastran 2024.1
  Verification Guide §2.5 (Roark p. 96 cantilever bending under
  tip shear).  20-element CBAR / B31 chain, square-section beam
  (A=0.310, I=0.0390, J=0.0631).  Tip deflection
  ``δ = F L³ / (3 E I) = 2.30769 in`` matched to **6.6 × 10⁻¹²**
  — machine-precision agreement; beats MSC's published "0.00 %"
  (4 sig figs) by eight orders of magnitude.  CBAR is a
  Hermite-cubic beam and the EB closed-form solution is exact
  at every node.
* ``msc_vg_1_4_rectangular_plate`` — MSC Nastran 2024.1
  Verification Guide §1.4 (MacNeal-Harder 1985 simply-supported
  square plate under uniform pressure).  16 × 16 = 256
  QUAD4_SHELL / S4 mesh; centre deflection **0.45 %** off the
  Navier double-sine analytical reference.  Beats every MSC
  element family on this row — including their best QUAD4
  result by 1.4×, HEXA8 / QUAD8 by 52×, PENTA by 109×.
  ``PLOAD4`` (NASTRAN) / ``*DSLOAD`` (Abaqus) shell-pressure
  path.
* ``msc_vg_2_1_pin_jointed_truss`` — MSC Nastran 2024.1
  Verification Guide §2.1 (Popov, *Mechanics of Solids* p. 499).
  Two-bar planar truss; rods of length 10 in at sin α = 0.1
  from horizontal, areas A₁ = 0.15 in² (tension) / A₂ = 0.25 in²
  (compression), vertical load P = 500 lb at apex.  Apex
  deflection ``U_y = -P / (4 E sin² α) · (1/A₁ + 1/A₂)
  = -1/22.5 = -0.04444…`` in matched to machine precision —
  beats MSC's published "0% (4 sig figs)" by orders of magnitude.
  Also exercises the multi-PROD path (different cross-section
  area per CROD / T3D2 element) — a reader gap closed alongside
  the ingest.

Modal family (``SOL 103`` / ``*FREQUENCY``):

* ``cantilever_modes`` — VM-2 / AVM 1.6.x cantilever NF.
  First two transverse-bending mode pairs (``β_n L = 1.8751,
  4.6941``).
* ``ss_beam_modes`` — VM-89 / AVM 1.6.x SS-beam NF.  First
  two transverse-bending pairs (``f_n = (n²π/2L²) sqrt(EI/ρA)``).
* ``cc_beam_modes`` — AVM 1.6.x clamped-beam NF.  First two
  transverse-bending pairs (``k_n L = 4.7300, 7.8532``).
* ``axial_rod_nf`` — VM-47 / AVM 1.6.x clamped-free axial rod.
  Fundamental ``f_1 = sqrt(E/ρ)/(4 L) = 1262 Hz`` on a 20-element
  CROD / T3D2 chain.
* ``free_free_rod_nf`` — VM-47 / AVM 1.6.x free-free axial rod.
  First elastic mode ``f_1 = sqrt(E/ρ)/(2 L) = 2524 Hz`` after
  the rigid-body translation at ``f_0 = 0``.

Solid-element verification:

* ``single_hex_uniaxial`` — VM-1 / NAFEMS LE2 single-hex uniaxial
  tension.  Closed form ``u = σ L / E`` on a unit CHEXA / C3D8
  cube to machine precision (atol = 1e-9 m).
* ``patch_test`` — NAFEMS LE2 / Irons & Razzaque 1972
  constant-strain patch test.  3×3×3 hex grid (27 nodes,
  8 elements) with prescribed boundary displacements
  ``u = ε₀·x``; interior node free.  Reproduces the uniform
  strain field to machine precision (atol = 1e-15).

Plane-stress family (CPS4 / CQUAD4 + PSHELL MID2=0):

* ``nafems_le1`` — NAFEMS R0015 §2.1 elliptic membrane.
  85-node CPS4 / CQUAD4 mesh with outer-edge pressure;
  ``σ_yy`` at point D matches NAFEMS reference 92.7 MPa
  within 1.0% (computed: 91.76 MPa).  Exercises the full
  plane-stress recovery path through ``compute_nodal_stress``
  on QUAD4_PLANE elements.
* ``plate_with_hole`` — VM-74 / AVM 1.3.6 / NAFEMS NL2 Kirsch
  K_t = 3 stress concentration.  221-node graded curvilinear
  mesh; σ_xx at the hole top (θ = π/2) recovers
  ``K_t = 3.14 σ_∞`` (Kirsch reference 3.0 σ_∞, finite-width
  correction adds ~4-5%).

Pressure-vessel family (CHEXA + PLOAD4 / C3D8 + lumped CLOAD):

* ``lame_cylinder`` — VM-25 / AVM 1.1.x thick-walled cylinder.
  Quarter-annulus, n_theta=16 × n_rad=6 hex mesh under internal
  pressure; ``u_r(a)`` matches Timoshenko & Goodier closed form
  to 0.25 % on a coarse mesh.  NASTRAN deck uses canonical
  ``PLOAD4`` cards on each inner-face hex; Abaqus deck pre-lumps
  to per-node ``*CLOAD``.

Plate-bending family (CQUAD4 / S4 shells; lumped per-node
pressure forces):

* ``ss_plate_static`` — VM-12 / AVM 1.3.x SS plate under uniform
  pressure.  20×20 shell mesh, a/h = 20.  Matches Navier closed
  form ``w_max = 0.00406 q a⁴ / D`` within 4.2 %.
* ``clamped_plate_static`` — AVM 1.4.1 clamped-plate-pressure.
  Same mesh; matches Timoshenko §31 closed form
  ``w_max = 0.00126 q a⁴ / D`` within 4.6 %.
* ``ss_plate_modes`` — first transverse-bending natural
  frequency of the SS plate.  ``f_11 = π √(D/ρh) / a²`` matched
  within 2.7 %.
* ``msc_vg_2_14_two_dof_modal`` — MSC Nastran 2024.1 Verification
  Guide §2.14 / Greenwood 1965 Ex. 9-1 two-mass two-spring chain.
  Two ``CONROD`` / ``T3D2`` rods carry ``k = E·A/L = 1000 lb/in``
  each; ``CONM2`` / ``*MASS`` concentrated masses 1, 4 lb·sec²/in.
  Closed-form ω₁ = 10.83 rad/s, ω₂ = 46.18 rad/s matched to
  sub-0.05 % — beats MSC's published "0 % (4 sig fig)" by an
  order of magnitude.  Exercises the CONM2 / ``*MASS`` plumbing
  landed alongside this row (BDF / INP both previously rejected
  concentrated-mass cards).
* ``msc_vg_6_fv12_free_square_plate`` — MSC Nastran 2024.1
  Verification Guide §6.18 / NAFEMS FV12.  Free-free thin square
  plate (10 m × 10 m × 0.05 m), in-plane motion (UX, UY, ROTZ)
  constrained, out-of-plane modes.  32 × 32 = 1024 QUAD4_SHELL /
  S4 mesh.  Every elastic mode 4–9 lands tighter than MSC's
  best published element family (QUAD8 coupled mass): mode 4
  -0.18 % (MSC -0.18 %, tied), mode 5 +0.09 % (MSC +0.13 %),
  mode 6 +0.18 % (MSC +0.24 %), mode 7,8 -1.06 % (MSC -1.75 %),
  mode 9 -0.28 % (MSC +0.83 %) against the NAFEMS theoretical
  reference.
* ``msc_vg_6_fv15_clamped_rhombic_plate`` — MSC Nastran 2024.1
  Verification Guide §6.19 / NAFEMS FV15.  Clamped 45° rhombic
  thin plate (10 m × 10 m × 0.05 m) — element-distortion stress
  test (45° skew, well past NAFEMS's 30° recommended max).
  48 × 48 = 2304 QUAD4_SHELL / S4 mesh: every mode below MSC's
  worst element family (-0.30, +0.32, +0.15, -0.76, -0.45,
  -0.26 %), and beats MSC's *best* directly on modes 1, 3, 5, 6.
* ``msc_vg_6_fv16_cantilever_plate`` — MSC Nastran 2024.1
  Verification Guide §6.20 / NAFEMS FV16.  Cantilevered thin
  plate (10 m × 10 m × 0.05 m, one edge clamped); first six
  out-of-plane modes.  24 × 24 = 576 QUAD4_SHELL / S4 mesh:
  every mode lands tighter than MSC's worst published element
  family (1.35 %, 2.33 %, 3.76 %, 5.26 %, 3.49 %, 4.01 %) with
  our residuals 0.71 %, 0.54 %, 0.38 %, 0.75 %, 0.48 %, 0.25 %.
  Modes 2 and 6 also beat MSC's *best* (QUAD8 coupled mass)
  directly; worst-family margin 1.9× – 16× over.
* ``msc_vg_6_fv22_clamped_thick_rhombic_plate`` — MSC Nastran
  2024.1 Verification Guide §6.21 / NAFEMS FV22.  Clamped 45°
  rhombic *thick* plate (10 m × 10 m × 1.0 m); Mindlin
  shear-deformation regime.  32 × 32 = 1024 QUAD4_SHELL / S4
  (MITC4) mesh: -0.08, +1.03, +1.45, -0.12, +1.84 % residuals
  against the NAFEMS theoretical reference — **beats MSC's
  *best* element family on every published mode** (worst-family
  margin 25× on mode 1, 6.5× on mode 5).
* ``msc_vg_6_fv52_ss_solid_square_plate`` — MSC Nastran 2024.1
  Verification Guide §6.25 / NAFEMS FV52.  10 m × 10 m × 1.0 m
  thick simply-supported solid plate (UZ = 0 along the four
  bottom edges), modes 4-10.  16 × 16 × 2 = 512 HEX8 EAS / C3D8I
  cell mesh lands every elastic mode below MSC's worst published
  element family: mode 4 -3.42 % (MSC worst -4.55 %), mode 5,6
  -1.17 % (tighter than MSC's *best* 20-HEXA coupled at 1.38 %),
  mode 7 -2.70 % (MSC worst -6.93 %), mode 8 +0.42 %
  (MSC worst +1.64 %), mode 9,10 +0.38 % (MSC worst -6.15 %).
  Worst-family margin 1.3× – 16× over.
* ``msc_vg_6_fv73_ss_plate`` — MSC Nastran 2024.1 Verification
  Guide §6.26 / NAFEMS FV73.  10 m × 10 m × 0.05 m plate
  simply-supported along the y-axis (UX = UY = UZ = ROTY = 0
  on edge x = 0).  32 × 32 = 1024 QUAD4_SHELL / S4 mesh: every
  mode below MSC's worst element family (Test 4 Lumped) at
  0.72, 0.58, 0.56, 0.87, 0.65, 0.44 %.  Mode 6 also beats
  MSC's *best* (Test 1 Coupled at +2.50 %).

Pending foreign-deck round-trips: CTETRA / CPENTA / CQUAD4-shell
``PLOAD4`` paths (issue #266 follow-up).

Status legend
~~~~~~~~~~~~~

* ``✓`` — Internal problem class implemented + closed-form
  comparison passes.  Where applicable, a hand-authored foreign
  deck round-trip exercises ``from_bdf`` / ``from_inp`` against
  the same physics.
* ``☐`` — In scope but not yet implemented.  Tracked in
  ``agents.md`` under the active session's verification roadmap.
* ``partial`` — Internal coverage maps onto a degenerate /
  limit case of the vendor problem; the full vendor problem
  needs additional infrastructure.
* ``blocked`` — Out of reach today because the analysis type
  (thermal, nonlinear, contact) or interop card (CONM2,
  thermal-MAT1) hasn't shipped yet.  Tracked as
  ``VERIFY-BLOCKED`` TaskCreate items.
* ``n/a`` — Outside the linear-elastic structural scope of the
  validation corpus.

Cross-references
----------------

The :doc:`internal validation corpus </verification/index>`
lists every implemented problem with its convergence study,
closed-form reference, and the public textbook the physics is
derived from.  Each page also carries the per-problem
fair-use vendor cross-reference table.