Irons patch test — constant-strain completeness
================================================

The Irons patch test is the *ur*-completeness check for any
finite-element formulation: a patch of elements under a
displacement field consistent with a uniform strain tensor must
produce that exact strain, to machine precision, on every
element.

If a formulation fails the patch test it can't converge — not
"converges slowly", not "inaccurate on coarse meshes" but
**cannot reach the correct answer at any refinement**.  The
patch test is the shibboleth separating elements that work from
elements that don't.

Problem
-------

Take a unit cube subdivided into a 2 × 2 × 2 patch of HEX8
elements.  Apply a uniform affine displacement field on every
boundary node:

.. math::

    \mathbf{u}(\mathbf{x}) = \mathbf{F}\,\mathbf{x}

where :math:`\mathbf{F}` is a constant deformation gradient.  Per
the kinematic relation :math:`\varepsilon = \tfrac{1}{2}(\mathbf{F} +
\mathbf{F}^T) - \mathbf{I}`, the interior strain field must be
that constant tensor on every cell.

Any node-to-node variation in the recovered strain — even at the
10⁻⁸ level — indicates the element formulation is not
constant-strain-complete.

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

This is not a convergence test; it is an identity test.  The
expected outcome is exact:

.. math::

    \varepsilon_{ij}(\mathbf{x}) = \tfrac{1}{2}(F_{ij} + F_{ji}) - \delta_{ij},
    \quad \forall \mathbf{x} \in \Omega

on every cell, in every component.

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

Ran by :file:`tests/analytical/test_patch_test.py` on a 2 × 2 × 2
HEX8 patch with a non-trivial constant strain tensor applied
through the affine boundary condition.  Per-cell strain matches
the imposed :math:`\mathbf{F}`-derived tensor to machine
precision — the regression test asserts ≤ 10⁻¹⁰ absolute error
on every strain component of every cell.

This confirms the SOLID185 kernel's B̄ formulation (Hughes §4.4)
satisfies the Irons completeness criterion.

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

.. list-table::
    :header-rows: 1
    :widths: 45 25 30

    * - Source
      - Expected
      - Problem ID / location
    * - Irons–Razzaque 1972 (original)
      - Identity (exact)
      - *The Mathematical Foundations of the FEM*, pp. 557–587
    * - Taylor–Beresford–Wilson 1976
      - Identity (exact)
      - IJNME 10 (1976), pp. 1211–1219
    * - Hughes *The Finite Element Method*
      - Identity (exact)
      - Dover (2000), §4.4.3
    * - NAFEMS *Background to Benchmarks*
      - Identity (exact)
      - BtB-4.2 (patch-test completeness)
    * - femorph-solver
      - ≤ 10⁻¹⁰ abs
      - ``test_patch_test.py``
    * - Abaqus Verification Manual
      - Identity (exact)
      - AVM 1.4.4 (C3D8 patch test)
    * - MAPDL Verification Manual
      - Identity (exact)
      - VM-82 (patch test)

All public sources agree this is an identity test — there is no
tolerance band to argue about.  A formulation that doesn't pass
at machine precision is broken.

femorph-solver's HEX8 kernel clears the patch test; the ≤ 10⁻¹⁰
residual is pure floating-point arithmetic noise on the
Gauss-integration path.

Source
------

Backing regression test:
:file:`tests/analytical/test_patch_test.py`
— landed with Agent 2's TA-9b analytical suite (#150).