Element kernels
===============

femorph-solver ships a structural element library.  Each kernel
is independently implemented from the open literature; every
non-trivial algorithm (shape function, integration rule,
formulation variant) carries a ``References`` block in its
source module.  The pre-processing pages under
:doc:`/user-guide/pre-processing/elements/index` are the
user-facing how-to; this section is the **engineering
reference** — topology, shape functions, integration rules, and
cites for every element.

.. toctree::
    :maxdepth: 1
    :caption: Solid kernels

    hex8
    hex20
    tet10
    wedge15_pyr13

.. toctree::
    :maxdepth: 1
    :caption: 2D & shell kernels

    quad4_plane
    quad4_shell

.. toctree::
    :maxdepth: 1
    :caption: Line / point kernels

    beam2
    truss2
    spring
    point_mass

Reference-element topology
--------------------------

Each kernel works on a canonical parametric reference
element.  These figures show the local node numbering used by
femorph-solver internally; downstream assembly maps the
parametric coordinates :math:`(\xi, \eta, \zeta)` to physical
space through the element's :term:`Jacobian`.

.. grid:: 1 2 2 2
    :gutter: 3

    .. grid-item::

        .. figure:: /_static/figures/element_topology_hex8.svg
            :alt: HEX8 reference element with 8 corner nodes
                numbered 1-8.
            :width: 90%

            HEX8 — 8 corner nodes on the bi-unit cube.

    .. grid-item::

        .. figure:: /_static/figures/element_topology_tet10.svg
            :alt: TET10 reference element with 4 corners and 6
                mid-edge nodes.
            :width: 90%

            TET10 — 4 corners + 6 mid-edges in the unit
            tetrahedron.

    .. grid-item::

        .. figure:: /_static/figures/element_topology_quad4.svg
            :alt: QUAD4 reference element with 4 corner nodes.
            :width: 90%

            QUAD4 — 4 corners on the bi-unit square.

    .. grid-item::

        .. figure:: /_static/figures/element_topology_beam2.svg
            :alt: BEAM2 reference element with 2 endpoints,
                6 DOFs each.
            :width: 90%

            BEAM2 — 2 endpoints, 6 DOFs each (3 translations + 3 rotations).

HEX20 inherits HEX8's corner numbering and adds 12 mid-edge
nodes 9-20 in the standard MAPDL SOLID186 order
(bottom-face mid-edges, top-face mid-edges, vertical
mid-edges).  See :doc:`hex20` for the full per-element page.


Quick reference table
---------------------

.. list-table::
    :header-rows: 1
    :widths: 18 14 32 32

    * - Kernel
      - DOFs / node
      - Stiffness rule
      - Mass rule
    * - :doc:`hex8`
      - 3
      - 2 × 2 × 2 Gauss + B-bar / EAS
      - 2 × 2 × 2 Gauss
    * - :doc:`hex20`
      - 3
      - 2 × 2 × 2 reduced (Bedrosian)
      - 14-pt Irons
    * - :doc:`tet10`
      - 3
      - 4-pt Keast
      - 4-pt Keast
    * - :doc:`wedge15_pyr13`
      - 3
      - Dunavant × Gauss / Duffy
      - Dunavant × Gauss / Duffy
    * - :doc:`quad4_plane`
      - 2
      - 2 × 2 Gauss
      - 2 × 2 Gauss
    * - :doc:`quad4_shell`
      - 6
      - 2 × 2 + 1 × 1 SR shear
      - 2 × 2 Gauss
    * - :doc:`beam2`
      - 6
      - Closed-form Hermite cubic
      - Closed-form (consistent / lumped)
    * - :doc:`truss2`
      - 3
      - Closed-form linear
      - Closed-form (consistent / lumped)
    * - :doc:`spring`
      - 3
      - Closed-form 1×1
      - —
    * - :doc:`point_mass`
      - 3
      - —
      - Diagonal lumped 3×3

.. seealso::

    :doc:`/user-guide/pre-processing/elements/index` — per-element
    integration flags, material-property slots, and runnable demos.