QUAD4_SHELL — 4-node Mindlin-Reissner flat shell
================================================

The 4-node thin-shell kernel.  Four corner nodes × 6 DOFs
each = 24 DOFs per element.  First-order shear-deformation
theory (Mindlin 1951; Reissner 1945) — independent rotational
DOFs decoupled from the displacement gradient, allowing the
element to capture transverse shear (unlike the rotation-from-
displacement Kirchhoff family).

* **Spec:** ``ELEMENTS.QUAD4_SHELL``

Kinematics
----------

The Mindlin-Reissner displacement field at a through-thickness
position :math:`\zeta \in [-t/2, +t/2]`:

.. math::

    \mathbf{u}(\xi, \eta, \zeta) =
    \mathbf{u}_{\mathrm{m}}(\xi, \eta) +
    \zeta\, \begin{bmatrix} \theta_y \\ -\theta_x \\ 0 \end{bmatrix}.

Membrane displacements :math:`\mathbf{u}_{\mathrm{m}}` and
section rotations :math:`(\theta_x, \theta_y)` are
**independent** Lagrange interpolants on the four corners — the
defining feature that admits transverse shear strain
:math:`\boldsymbol{\gamma}` as a primary variable.

Integration
-----------

Three integration tracks fold into the 24 × 24 stiffness:

* **Membrane + bending:** 2 × 2 Gauss-Legendre on the mid-
  surface (4 points).
* **Transverse shear:** 1 × 1 Gauss (centre point) — Malkus &
  Hughes' 1978 *selective-reduced integration*.  Suppresses
  shear locking on thin shells (Reissner's :math:`\xi_3 = 0`
  shear strain converges to zero in the Kirchhoff limit; the
  full 2 × 2 rule penalises the discrete approximation, the
  1 × 1 rule does not).
* **Drilling-DOF stabilisation:** :math:`\alpha\, G\, t\, A`
  per-node penalty (Allman 1984; Hughes-Brezzi 1989) keeps the
  local 24 × 24 stiffness non-singular when :math:`\theta_z` is
  free.  The default :math:`\alpha = 10^{-3}` matches the
  practitioner choice in Cook §17.3.

Real constants
--------------

* ``REAL[0]`` — :math:`t`, through-thickness shell thickness.

Verification cross-references
-----------------------------

* :ref:`sphx_glr_gallery_elements_shell181_example_shell181.py` — pure-membrane test (UZ + ROTX/Y/Z constrained).
* :ref:`sphx_glr_gallery_elements_shell181_example_quad4_shell_reference_geometry.py` — reference-frame plot.

The MITC4 variant of Bathe & Dvorkin (1985) — locking-free even
on warped meshes — is on the roadmap.

Implementation: :mod:`femorph_solver.elements.quad4_shell`.

References
----------

* Mindlin, R. D. (1951) "Influence of rotary inertia and
  shear on flexural motions of isotropic, elastic plates,"
  *J. Appl. Mech.* 18, 31–38.
* Reissner, E. (1945) "The effect of transverse shear
  deformation on the bending of elastic plates," *J. Appl.
  Mech.* 12, A69–A77.
* Malkus, D. S. and Hughes, T. J. R. (1978) "Mixed finite
  element methods — reduced and selective integration
  techniques: a unification of concepts," *CMAME* 15 (1),
  63–81 (selective-reduced shear).
* Allman, D. J. (1984) "A compatible triangular element
  including vertex rotations for plane elasticity analysis,"
  *Computers & Structures* 19 (1–2), 1–8 (drilling DOF).
* Hughes, T. J. R. and Brezzi, F. (1989) "On drilling degrees
  of freedom," *CMAME* 72 (1), 105–121.
* Bathe, K.-J. and Dvorkin, E. N. (1985) "A four-node plate
  bending element based on Mindlin / Reissner plate theory
  and a mixed interpolation," *IJNME* 21 (2), 367–383
  (MITC4 — roadmap).
* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, §17 (shells).