TET10 — 10-node quadratic tetrahedron
=====================================

The quadratic-tet kernel.  Four corner nodes + six mid-edge
nodes — 10 total, 30 DOFs per element.  Indispensable for
unstructured-mesh CAD geometries where a hex mesh isn't
buildable.

* **Spec:** ``ELEMENTS.TET10``

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

Lagrange in **volume coordinates**
:math:`(L_0, L_1, L_2, L_3)` with
:math:`L_0 + L_1 + L_2 + L_3 = 1`:

* **Corner nodes** (:math:`i = 0, 1, 2, 3`):

  .. math::

      N^{c}_{i} = L_i\, (2 L_i - 1).

* **Mid-edge nodes** (one per edge :math:`(i, j)`):

  .. math::

      N^{m}_{ij} = 4\, L_i\, L_j.

The basis interpolates a quadratic polynomial in
:math:`L_i` exactly across the tet, satisfies Kronecker-
delta at all 10 nodes, and is partition-of-unity to machine
precision (verified on every reference-element node by the
gallery example).

See
:ref:`sphx_glr_gallery_elements_solid187_example_tet10_shape_function_slices.py`
for contour plots on the :math:`\xi_3 = 0` face slice and a
verified Kronecker-delta basis-matrix check.

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

**Keast 4-point rule** (Keast 1986), used for **both stiffness
and mass**:

.. math::

    \boldsymbol{\xi}_q = (a, a, a, 1 - 3a),
    \qquad a = \tfrac{5 - \sqrt{5}}{20},
    \qquad w_q = \tfrac{1}{24},

with :math:`q = 1, \ldots, 4` and the four points related by
the symmetric permutations of the volume coordinates.  Exact
for polynomials of degree ≤ 2 — sufficient for the
quadratic-tet :math:`\mathbf{B}^{\!\top}\, \mathbf{C}\,
\mathbf{B}` integrand under a constant-Jacobian map.

See :doc:`../theory/quadrature` for Stroud-rule context.

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

* :ref:`sphx_glr_gallery_elements_solid187_example_solid187.py` — quadratic-tet uniaxial-strain test.
* :ref:`sphx_glr_gallery_elements_solid187_example_tet10_reference_geometry.py` — node layout + 4-point Keast Gauss locations.

Implementation: :mod:`femorph_solver.elements.tet10`.

References
----------

* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, Table 6.5-1 (TET10 volume-
  coordinate shape functions).
* Hughes, T. J. R. (2000) *The Finite Element Method —
  Linear Static and Dynamic Finite Element Analysis*, Dover,
  §3.8 (quadratic tetrahedron).
* Zienkiewicz, O. C. and Taylor, R. L. (2013) *The Finite
  Element Method: Its Basis and Fundamentals*, 7th ed.,
  §8.8.2 (TET10 basis on the unit tetrahedron).
* Keast, P. (1986) "Moderate-degree tetrahedral quadrature
  formulas," *Computer Methods in Applied Mechanics and
  Engineering* 55 (3), 339–348.
* Stroud, A. H. (1971) *Approximate Calculation of Multiple
  Integrals*, Prentice-Hall (foundational tetrahedral
  cubatures).