WEDGE15 / PYR13 — degenerate-corner serendipity hex
===================================================

The 15-node quadratic wedge and the 13-node apex-singular
pyramid are **collapsed-corner forms** of :doc:`hex20` —
HEX20 with one or two pairs of corner nodes coincident.
Both ship in :mod:`femorph_solver.elements._wedge15_pyr13`
and are auto-dispatched from the HEX20 entry when the input
deck supplies a degenerate connectivity.

* **Specs:** ``ELEMENTS.WEDGE15``, ``ELEMENTS.PYR13``

WEDGE15
-------

15 nodes — 6 corners + 9 mid-edges.  45 DOFs per element.
Reference shape is a triangle :math:`\times` line:

.. math::

    \hat\Omega_{\mathrm{wedge}} =
    \{ (L_0, L_1, L_2, \zeta) :
       L_0 + L_1 + L_2 = 1,
       \zeta \in [-1, +1] \}.

**Shape functions.** Quadratic serendipity on the triangle (in
volume coordinates :math:`L_i`) tensored with a quadratic
shape function in :math:`\zeta`.  Bedrosian (1992) gives the
explicit basis; ZT §8.8.3 walks the construction.

**Integration.**

* **Stiffness:** 9-point composite — 3-point Dunavant
  triangle (degree-2 exact) :math:`\times` 3-point
  Gauss-Legendre line.
* **Mass:** 21-point — 7-point Dunavant triangle (degree-5
  exact) :math:`\times` 3-point Gauss-Legendre line.  Higher
  point count tracks the higher polynomial degree of
  :math:`\mathbf{N}^{\!\top}\, \mathbf{N}` vs.
  :math:`\mathbf{B}^{\!\top}\, \mathbf{C}\, \mathbf{B}`.

PYR13
-----

13 nodes — 5 corners (4 base + 1 apex) + 8 mid-edges.  39
DOFs per element.  The pyramid is the most awkward of the
standard solid shapes because the apex collapse makes the
shape functions **rational** (a removable
:math:`(1 - \zeta)^{-1}` factor) — Bedrosian's 1992 explicit
basis carries the rational form directly.

**Shape functions.** Apex-singular polynomial / rational basis
(Bedrosian 1992).  Alternatives: Zgainski-Coulomb-Marechal
(IEEE Mag. 1996) and Wachspress (*A Rational Finite Element
Basis*, Academic Press, 1975).

**Integration.** 2 × 2 × 2 **Duffy collapsed-hex** rule
(Duffy 1982).  The Duffy transformation maps the pyramid to a
collapsed cube where the rational integrand becomes finite at
every Gauss point.  8 points, exact for the degree-3 polynomial
limit of the rational basis.  An opt-in 27-point ``"consistent"``
rule is also available via ``ELEMENTS.PYR13(pyramid_rule="consistent")``;
it is mathematically more accurate but ~13 % apart in Frobenius
norm on a single element.

References
----------

* 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 (WEDGE15 + PYR13 shape
  functions).
* Zienkiewicz, O. C. and Taylor, R. L. (2013) *The Finite
  Element Method: Its Basis and Fundamentals*, 7th ed.,
  §8.8.3 (wedge serendipity).
* Dunavant, D. A. (1985) "High degree efficient symmetrical
  Gaussian quadrature rules for the triangle," *International
  Journal for Numerical Methods in Engineering* 21 (6),
  1129–1148.
* Duffy, M. G. (1982) "Quadrature over a pyramid or cube of
  integrands with a singularity at a vertex," *SIAM Journal on
  Numerical Analysis* 19 (6), 1260–1262.
* Zgainski, F.-X., Coulomb, J.-L. and Marechal, Y. (1996)
  "A new family of finite elements: the pyramidal elements,"
  *IEEE Transactions on Magnetics* 32 (3), 1393–1396
  (alternative pyramid basis).
* Wachspress, E. L. (1975) *A Rational Finite Element Basis*,
  Academic Press (rational-basis foundation).

Implementation:
:mod:`femorph_solver.elements._wedge15_pyr13`.