:orphan:

Elements
========

Per-element living documentation — one sub-gallery per registered
element.  Each script is a runnable companion to the corresponding
theory page under
:doc:`/user-guide/pre-processing/elements/index`, exercising the same
shape functions, quadrature rule, and formulation options on a tiny
self-contained problem.

These examples run on every docs build, so the theory pages stay
grounded in code that actually works.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. thumbnail-parent-div-close

.. raw:: html

    </div>

BEAM188 gallery
===============

Runnable BEAM188 (3D 2-node beam) examples.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The 2-node 3D beam element (Euler-Bernoulli slender-beam limit) maps to the natural-coordinate segment s \in [-1, +1]. Translations and torsion use linear shape functions; transverse displacement and slope use Hermite cubics so C^1 continuity holds across element boundaries.">

.. only:: html

  .. image:: /gallery/elements/beam188/images/thumb/sphx_glr_example_beam2_reference_geometry_thumb.png
    :alt:

  :doc:`/gallery/elements/beam188/example_beam2_reference_geometry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">BEAM2 reference geometry — Hermite cubic shape functions</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Slender steel cantilever modelled with a line of BEAM2 elements. Two validations:">

.. only:: html

  .. image:: /gallery/elements/beam188/images/thumb/sphx_glr_example_beam188_thumb.png
    :alt:

  :doc:`/gallery/elements/beam188/example_beam188`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">BEAM2 — cantilever tip deflection and first mode</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

COMBIN14 gallery
================

Runnable COMBIN14 (3D 2-node spring) examples.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The 2-node 3D longitudinal spring carries an axial-only stiffness k between two nodes connected by a unit direction vector \hat d = (l, m, n).  No mass.  Three translational DOFs per node, 6 DOFs per element.">

.. only:: html

  .. image:: /gallery/elements/combin14/images/thumb/sphx_glr_example_spring_reference_geometry_thumb.png
    :alt:

  :doc:`/gallery/elements/combin14/example_spring_reference_geometry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">SPRING — 2-node longitudinal spring stiffness</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Three co-linear nodes joined by two SPRING springs of stiffness k1 and k2. With the far end fixed and a tip load F the free-end displacement is F / k_eq where k_eq = k1 · k2 / (k1 + k2).">

.. only:: html

  .. image:: /gallery/elements/combin14/images/thumb/sphx_glr_example_combin14_thumb.png
    :alt:

  :doc:`/gallery/elements/combin14/example_combin14`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">SPRING — two springs in series</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

LINK180 gallery
===============

Gallery of runnable LINK180 (3D 2-node truss) examples.
See ``examples/link180/README.md`` in the repository for the element
theory write-up.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The 2-node 3D truss (axial-only spar) maps to the natural coordinate s \in [-1, +1].  Linear shape functions interpolate the three translational DOFs at each node:">

.. only:: html

  .. image:: /gallery/elements/link180/images/thumb/sphx_glr_example_truss2_reference_geometry_thumb.png
    :alt:

  :doc:`/gallery/elements/link180/example_truss2_reference_geometry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">TRUSS2 reference geometry — linear axial bar</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Single TRUSS2 spar fixed at node 1 and pulled along the x-axis at node 2. The tip displacement is compared to the closed-form rod solution u = P L / (E A) and the axial stress is plotted on the deformed mesh.">

.. only:: html

  .. image:: /gallery/elements/link180/images/thumb/sphx_glr_example_link180_thumb.png
    :alt:

  :doc:`/gallery/elements/link180/example_link180`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">TRUSS2 — axial truss under end load</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

MASS21 gallery
==============

Runnable MASS21 (1-node lumped mass) examples.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The 1-node concentrated-mass element contributes a diagonal 3 x 3 block to the global mass matrix:">

.. only:: html

  .. image:: /gallery/elements/mass21/images/thumb/sphx_glr_example_point_mass_reference_geometry_thumb.png
    :alt:

  :doc:`/gallery/elements/mass21/example_point_mass_reference_geometry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">POINT_MASS — single-node lumped mass</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Two nodes connected by a SPRING element. A POINT_MASS element sits on the free node. The first modal frequency is compared to the textbook SDOF result ω = √(k / m).">

.. only:: html

  .. image:: /gallery/elements/mass21/images/thumb/sphx_glr_example_mass21_thumb.png
    :alt:

  :doc:`/gallery/elements/mass21/example_mass21`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">POINT_MASS — single-DOF spring-mass oscillator</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

PLANE182 gallery
================

Runnable PLANE182 (2D 4-node quad) examples.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The 4-node bilinear plane element maps to the natural coordinates (\xi, \eta) \in [-1, 1]^2.  Two translational DOFs per node; 8 DOFs per element.">

.. only:: html

  .. image:: /gallery/elements/plane182/images/thumb/sphx_glr_example_quad4_plane_reference_geometry_thumb.png
    :alt:

  :doc:`/gallery/elements/plane182/example_quad4_plane_reference_geometry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">QUAD4 plane reference geometry — bilinear shape functions + 2×2 Gauss</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Renders the four bilinear (4-node) shape functions N_i(\xi, \eta) on the reference quadrilateral \hat\Omega = [-1, +1]^2:">

.. only:: html

  .. image:: /gallery/elements/plane182/images/thumb/sphx_glr_example_quad4_shape_functions_thumb.png
    :alt:

  :doc:`/gallery/elements/plane182/example_quad4_shape_functions`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">QUAD4 bilinear shape functions on the reference square</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A square piece of plate is fixed on its left edge and pulled on its right edge. We check the tip displacement against Hooke&#x27;s law u_x = σ L / E and the Poisson contraction u_y = −ν u_x in plane stress, then render the deformed mesh.">

.. only:: html

  .. image:: /gallery/elements/plane182/images/thumb/sphx_glr_example_plane182_thumb.png
    :alt:

  :doc:`/gallery/elements/plane182/example_plane182`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">QUAD4_PLANE — plane-stress uniaxial tension</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

SHELL181 gallery
================

Runnable SHELL181 (4-node shell) examples.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The 4-node flat Mindlin–Reissner shell maps to the natural coordinates (\xi, \eta) \in [-1, 1]^2 and carries six DOFs per node — three translations (u_x, u_y, u_z) plus three rotations (\theta_x, \theta_y, \theta_z) — for 24 DOFs per element.">

.. only:: html

  .. image:: /gallery/elements/shell181/images/thumb/sphx_glr_example_quad4_shell_reference_geometry_thumb.png
    :alt:

  :doc:`/gallery/elements/shell181/example_quad4_shell_reference_geometry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">QUAD4 shell reference geometry — Mindlin–Reissner shape functions</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Single QUAD4_SHELL element (a 1 m × 1 m flat plate of thickness t) in pure in-plane tension. Out-of-plane DOFs are suppressed so the problem reduces to plane-stress membrane behaviour: u_x = σ L / E and the Poisson contraction u_y = −ν u_x.">

.. only:: html

  .. image:: /gallery/elements/shell181/images/thumb/sphx_glr_example_shell181_thumb.png
    :alt:

  :doc:`/gallery/elements/shell181/example_shell181`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">QUAD4_SHELL — uniaxial stretch of a flat square plate</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

SOLID185 gallery
================

Runnable SOLID185 (8-node hex) examples.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Renders the 8-node trilinear hexahedron&#x27;s reference element \hat\Omega = [-1, 1]^3 with three layers of annotation:">

.. only:: html

  .. image:: /gallery/elements/solid185/images/thumb/sphx_glr_example_hex8_reference_geometry_thumb.png
    :alt:

  :doc:`/gallery/elements/solid185/example_hex8_reference_geometry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">HEX8 reference geometry — nodes, shape functions, Gauss points</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Renders all eight trilinear Lagrange shape functions N_i(\xi, \eta, \zeta) for the 8-node hexahedron on the reference cube \hat\Omega = [-1, +1]^3.  The basis functions are">

.. only:: html

  .. image:: /gallery/elements/solid185/images/thumb/sphx_glr_example_hex8_shape_function_surfaces_thumb.png
    :alt:

  :doc:`/gallery/elements/solid185/example_hex8_shape_function_surfaces`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">HEX8 shape-function surfaces on the reference cube</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Single HEX8 element (a 1 m unit cube) with one face clamped in the pulling direction and the opposite face loaded. The displacement field is uniform — u_x = σ L / E on the loaded face — which gives a clean check on the full 24 × 24 stiffness plus the boundary/load handling.">

.. only:: html

  .. image:: /gallery/elements/solid185/images/thumb/sphx_glr_example_solid185_thumb.png
    :alt:

  :doc:`/gallery/elements/solid185/example_solid185`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">HEX8 — uniaxial tension of a unit cube</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

SOLID186 gallery
================

Runnable SOLID186 (20-node serendipity hex) examples.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Single HEX20 brick (a unit cube, 20 nodes including the mid-edge nodes of the serendipity family) loaded in uniaxial tension.  The consistent-load vector for a serendipity 8-node face has corner weight -1/12 and mid-edge weight +1/3; applying these produces a uniform σxx field, so εxx = σ/E and εyy = εzz = −ν·εxx exactly at every node.">

.. only:: html

  .. image:: /gallery/elements/solid186/images/thumb/sphx_glr_example_solid186_thumb.png
    :alt:

  :doc:`/gallery/elements/solid186/example_solid186`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">HEX20 — uniaxial tension on a 20-node hex</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Renders the 20-node serendipity hexahedron&#x27;s reference element \hat\Omega = [-1, 1]^3:">

.. only:: html

  .. image:: /gallery/elements/solid186/images/thumb/sphx_glr_example_hex20_reference_geometry_thumb.png
    :alt:

  :doc:`/gallery/elements/solid186/example_hex20_reference_geometry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">HEX20 reference geometry — corner + mid-edge nodes, Gauss points</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Renders the eight HEX20 serendipity shape functions that do not vanish on the \zeta = -1 face of the reference cube \hat\Omega = [-1, +1]^3: four corner functions N^c_i whose nodes lie on that face, plus four mid-edge functions N^m_j whose mid-edge nodes lie on the same face.  The other twelve basis functions (top-face corners, top-face mid-edges, and the four vertical-edge mid-edges with \zeta_j = 0) all vanish at \zeta = -1 and are omitted from the figure.">

.. only:: html

  .. image:: /gallery/elements/solid186/images/thumb/sphx_glr_example_hex20_shape_function_slices_thumb.png
    :alt:

  :doc:`/gallery/elements/solid186/example_hex20_shape_function_slices`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">HEX20 shape-function slices on the bottom face</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

SOLID187 gallery
================

Runnable SOLID187 (10-node quadratic tetrahedron) examples.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Single TET10 (10-node quadratic tet) driven by a fully-prescribed displacement field that matches a uniform strain state. Under all-Dirichlet BCs the solve has a unique solution equal to the prescribed field, so the recovered strain tensor matches the analytical one to machine precision. Useful as a self-contained smoke test that TET10 + femorph_solver.Model.eel are in lockstep.">

.. only:: html

  .. image:: /gallery/elements/solid187/images/thumb/sphx_glr_example_solid187_thumb.png
    :alt:

  :doc:`/gallery/elements/solid187/example_solid187`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">TET10 — prescribed uniform strain on a 10-node tet</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Renders the ten quadratic-tetrahedron shape functions N_i^c(L) (4 corners) and N_{ij}^m(L) (6 mid-edges) sliced through the \xi_3 = 0 face of the unit tetrahedron.  In volume coordinates L_i (where L_0 = 1 - L_1 - L_2 - L_3) the basis is">

.. only:: html

  .. image:: /gallery/elements/solid187/images/thumb/sphx_glr_example_tet10_shape_function_slices_thumb.png
    :alt:

  :doc:`/gallery/elements/solid187/example_tet10_shape_function_slices`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">TET10 shape-function slices on the unit tetrahedron</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Renders the 10-node quadratic tetrahedron&#x27;s reference element \{(\xi_1, \xi_2, \xi_3) : \xi_i \ge 0, \sum_i \xi_i \le 1\}:">

.. only:: html

  .. image:: /gallery/elements/solid187/images/thumb/sphx_glr_example_tet10_reference_geometry_thumb.png
    :alt:

  :doc:`/gallery/elements/solid187/example_tet10_reference_geometry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">TET10 reference geometry — corner + mid-edge nodes, Keast points</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:
   :includehidden:


   /gallery/elements/beam188/index.rst
   /gallery/elements/combin14/index.rst
   /gallery/elements/link180/index.rst
   /gallery/elements/mass21/index.rst
   /gallery/elements/plane182/index.rst
   /gallery/elements/shell181/index.rst
   /gallery/elements/solid185/index.rst
   /gallery/elements/solid186/index.rst
   /gallery/elements/solid187/index.rst


.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-gallery

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download all examples in Python source code: elements_python.zip </gallery/elements/elements_python.zip>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download all examples in Jupyter notebooks: elements_jupyter.zip </gallery/elements/elements_jupyter.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_