HEX20 — 20-node serendipity hexahedron
======================================

The quadratic-hex serendipity kernel.  Eight corner nodes plus
twelve mid-edge nodes — 20 total, 60 DOFs per element.  The
serendipity basis omits the body / face / mid-element nodes of
the full Lagrange family while preserving exact reproduction of
quadratic polynomials (Ergatoudis, Irons & Zienkiewicz 1968).

Dispatches collapsed-corner input to :doc:`wedge15_pyr13`
automatically.

* **Spec:** ``ELEMENTS.HEX20``

Shape functions
---------------

Two families on the reference cube
:math:`\hat\Omega = [-1, +1]^{3}`:

**Corner nodes** :math:`(\xi_i, \eta_i, \zeta_i) \in
\{-1, +1\}^{3}`:

.. math::

    N^{c}_{i}(\xi, \eta, \zeta) =
    \tfrac{1}{8}
    (1 + \xi_i \xi)(1 + \eta_i \eta)(1 + \zeta_i \zeta)
    (\xi_i \xi + \eta_i \eta + \zeta_i \zeta - 2).

**Mid-edge nodes** with one of :math:`\xi_j, \eta_j, \zeta_j`
zero (the axis along which the edge runs).  For
:math:`\xi_j = 0` mid-edges:

.. math::

    N^{m}_{j}(\xi, \eta, \zeta) =
    \tfrac{1}{4} (1 - \xi^{2})(1 + \eta_j \eta)(1 + \zeta_j \zeta),

with analogous expressions for :math:`\eta_j = 0` and
:math:`\zeta_j = 0` mid-edges (the missing-variable axis carries
the :math:`(1 - \cdot^{2})` factor).

The basis satisfies Kronecker-delta interpolation at all 20
nodes and partition-of-unity on :math:`\hat\Omega`.  See
:ref:`sphx_glr_gallery_elements_solid186_example_hex20_shape_function_slices.py`
for slice plots of the eight functions that don't vanish on the
:math:`\zeta = -1` face.

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

* **Stiffness:** 2 × 2 × 2 reduced Gauss-Legendre (8 points)
  by default — Bedrosian (1992) and the standard practitioner
  choice for serendipity hex.  Bending-mode response stays
  exact; spurious hourglass modes are absent because the 8
  points still couple all 60 stiffness rows.

* **Mass:** 14-point **Irons rule** (Irons 1971).  Integrates
  polynomials of degree ≤ 5 exactly on the cube — a non-tensor-
  product cubature constructed by hand for the cube's symmetry
  group with the minimum point count for this degree.  Preferred
  over the 27-point :math:`3 \times 3 \times 3` rule because it
  preserves the consistent-mass sparsity pattern without
  overshooting the integrand polynomial degree.

The full 27-point :math:`3 \times 3 \times 3` rule is also
available via ``ELEMENTS.HEX20(integration="full")``; useful
only for cross-checking the reduced rule on a sensitive bending
mode.

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

* :ref:`sphx_glr_gallery_elements_solid186_example_solid186.py` — single-hex uniaxial via mid-edge consistent loads.
* :ref:`sphx_glr_gallery_elements_solid186_example_hex20_reference_geometry.py` — corner + mid-edge node layout, stiffness and mass Gauss points.

Implementation: :mod:`femorph_solver.elements.hex20`;
collapsed-corner forms in
:mod:`femorph_solver.elements._wedge15_pyr13`.

References
----------

* Ergatoudis, J. G., Irons, B. M. and Zienkiewicz, O. C. (1968)
  "Curved isoparametric quadrilateral elements for finite
  element analysis," *International Journal of Solids and
  Structures* 4 (1), 31–42.
* Hughes, T. J. R. (2000) *The Finite Element Method —
  Linear Static and Dynamic Finite Element Analysis*, Dover,
  §3.7 (serendipity family).
* Zienkiewicz, O. C. and Taylor, R. L. (2013) *The Finite
  Element Method: Its Basis and Fundamentals*, 7th ed.,
  Butterworth-Heinemann, §8.7.3.
* Irons, B. M. (1971) "Quadrature rules for brick based
  finite elements," *International Journal for Numerical
  Methods in Engineering* 3 (2), 293–294 (14-point degree-5
  rule for the consistent mass integral).
* Bedrosian, G. (1992) "Shape functions and integration
  formulas for three-dimensional finite element analysis,"
  *International Journal for Numerical Methods in
  Engineering* 35 (1), 95–108 (reduced 2 × 2 × 2 stiffness).
* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, §6.5 (quadratic hex).