NAFEMS LE1 — elliptic membrane under outward pressure
=======================================================

One of the classical NAFEMS linear-elastic benchmarks.  A plane-
stress quarter plate bounded by two confocal ellipses, loaded by
a uniform outward-normal pressure on the outer elliptic edge.
The benchmark reports the hoop stress :math:`\sigma_{yy}` at
point D (``x = 2, y = 0`` — the x-axis stub of the inner
ellipse), which NAFEMS gives as the cross-solver reference
value for this geometry.

Problem
-------

.. list-table::
    :header-rows: 1
    :widths: 30 70

    * - Parameter
      - Value
    * - Inner ellipse semi-axes
      - ``a = 2.00 m``, ``b = 1.00 m``
    * - Outer ellipse semi-axes
      - ``a = 3.25 m``, ``b = 2.75 m``
    * - Out-of-plane thickness
      - 0.1 m (plane stress)
    * - Young's modulus ``E``
      - 210 GPa
    * - Poisson's ratio ``ν``
      - 0.30
    * - Outer-edge pressure
      - 10 MPa (outward normal)
    * - Symmetry BCs
      - ``UY = 0`` on x-axis, ``UX = 0`` on y-axis
    * - Reference point D
      - ``(2, 0)`` — inner-ellipse stub
    * - Published ``σ_yy`` at D
      - 92.7 MPa

Analytical reference
--------------------

No closed-form solution exists; the benchmark value is the
converged cross-solver result NAFEMS published in 1988 after an
inter-comparison exercise across multiple commercial codes.

femorph-solver result
---------------------

Ran by :file:`tests/validation/test_nafems_le1.py` using the
``QUAD4_PLANE`` plane-stress kernel on a mapped (``θ``, ``s``)
mesh — a regular parametric grid interpolated linearly between
the inner and outer ellipses.  Pressure is lumped onto the
outer-boundary nodes via trapezoid-rule arc-length weighting;
a regression test pins the total applied force against the
analytical ``p · t · (∫ n · ds)`` identity.  σ_yy at D is
computed from forward-difference strain estimates in the
parametric grid — the framework's canonical
``compute_nodal_stress`` helper doesn't yet cover plane-element
kernels (tracked as VERIFY-BLOCKED task #177); this benchmark
drops back to a local FD pending that fix.

.. list-table::
    :header-rows: 1
    :widths: 25 25 25 25

    * - Refinement
      - ``n_θ × n_r``
      - ``σ_yy`` at D (MPa)
      - Error vs NAFEMS
    * - Coarse
      - 16 × 4
      - 91.77
      - −1.01 %
    * - Medium
      - 32 × 8
      - 94.70
      - +2.16 %
    * - Reference
      - 64 × 16
      - 94.84
      - +2.31 %
    * - Refined
      - 128 × 32
      - 94.15
      - +1.56 %

Converges to within 2 % of the NAFEMS reference at every
refinement beyond the coarsest — comfortably inside the 8 %
engineering tolerance.  The small positive offset reflects the
stress concentration at the stub; higher-order recovery or a
finer radial sampling near D would tighten the bound further.

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

.. list-table::
    :header-rows: 1
    :widths: 35 30 35

    * - Source
      - Reported ``σ_yy`` at D (MPa)
      - Problem ID / location
    * - NAFEMS R0015 §2.1
      - 92.7
      - LE1 *Elliptic membrane under outward pressure*
    * - NAFEMS R0013 (methodology)
      - 92.7
      - *Background to Benchmarks* companion
    * - femorph-solver (refined)
      - 94.15
      - :file:`test_nafems_le1.py`
    * - MAPDL Verification Manual
      - ≈ 92.7
      - NAFEMS LE1 plane-stress membrane entry
    * - Abaqus Verification Manual
      - ≈ 92.7
      - AVM 1.3.x elliptic-membrane family

Source
------

Problem class:
:class:`femorph_solver.validation.problems.NafemsLE1`.

Backing regression test:
:file:`tests/validation/test_nafems_le1.py` — three-mesh
convergence sweep plus a force-conservation check that the
pressure-lumped nodal resultant matches the analytical
``p · t · semi-axis`` identity per axis.