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

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.