.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery/elements/plane182/example_quad4_plane_reference_geometry.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_gallery_elements_plane182_example_quad4_plane_reference_geometry.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_gallery_elements_plane182_example_quad4_plane_reference_geometry.py:


QUAD4 plane reference geometry — bilinear shape functions + 2×2 Gauss
=====================================================================

The 4-node bilinear plane element maps to the natural
coordinates :math:`(\xi, \eta) \in [-1, 1]^2`.  Two
translational DOFs per node; 8 DOFs per element.

Shape functions (bilinear Lagrange on the reference square):

.. math::

    N_i(\xi, \eta) = \tfrac{1}{4}(1 + \xi_i\xi)(1 + \eta_i\eta),
    \qquad i = 1, \ldots, 4,

with :math:`(\xi_i, \eta_i) \in \{-1, +1\}^2` at the corresponding
corners.  Stiffness and consistent mass use 2×2 Gauss-Legendre
(4 points), exact for the bilinear :math:`B^T D B` and
:math:`N^T N` integrands.

Plane stress vs plane strain selected by the ``_PLANE_MODE``
material flag (Cook 2002 §3.5–§3.6).

References
----------
* Zienkiewicz, O. C. and Taylor, R. L. (2013) *The Finite Element
  Method*, 7th ed., §6 + App. G.
* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002)
  *Concepts and Applications of Finite Element Analysis*, 4th
  ed., Wiley, §5, §11.3.
* Hughes, T. J. R. (2000) *The Finite Element Method*, Dover, §3.1.

Implementation: :class:`femorph_solver.elements.quad4_plane.Quad4Plane`.

.. GENERATED FROM PYTHON SOURCE LINES 35-41

.. code-block:: Python


    from __future__ import annotations

    import numpy as np
    import pyvista as pv








.. GENERATED FROM PYTHON SOURCE LINES 42-44

Reference quad
--------------

.. GENERATED FROM PYTHON SOURCE LINES 44-50

.. code-block:: Python


    corners = np.array([[-1.0, -1.0, 0.0], [+1.0, -1.0, 0.0], [+1.0, +1.0, 0.0], [-1.0, +1.0, 0.0]])
    cells = np.hstack([[4], np.arange(4, dtype=np.int64)])
    cell_types = np.array([pv.CellType.QUAD], dtype=np.uint8)
    ref_quad = pv.UnstructuredGrid(cells, cell_types, corners)








.. GENERATED FROM PYTHON SOURCE LINES 51-52

2×2 Gauss-Legendre on the reference square

.. GENERATED FROM PYTHON SOURCE LINES 52-56

.. code-block:: Python


    g = 1.0 / np.sqrt(3.0)
    gauss = np.array([[a, b, 0.0] for a in (-g, +g) for b in (-g, +g)])








.. GENERATED FROM PYTHON SOURCE LINES 57-58

Render

.. GENERATED FROM PYTHON SOURCE LINES 58-74

.. code-block:: Python


    plotter = pv.Plotter(off_screen=True, window_size=(560, 480))
    plotter.add_mesh(
        ref_quad, style="wireframe", color="black", line_width=2.5, label="reference QUAD4"
    )
    plotter.add_points(
        corners, render_points_as_spheres=True, point_size=18, color="black", label="corner nodes (4)"
    )
    plotter.add_points(
        gauss, render_points_as_spheres=True, point_size=16, color="#d62728", label="2×2 Gauss"
    )
    plotter.view_xy()
    plotter.camera.zoom(1.4)
    plotter.add_legend(face=None, size=(0.22, 0.10), bcolor="white")
    plotter.show()




.. image-sg:: /gallery/elements/plane182/images/sphx_glr_example_quad4_plane_reference_geometry_001.png
   :alt: example quad4 plane reference geometry
   :srcset: /gallery/elements/plane182/images/sphx_glr_example_quad4_plane_reference_geometry_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 75-77

Verify partition of unity at the Gauss points
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 77-87

.. code-block:: Python


    xi_i = np.array([-1.0, +1.0, +1.0, -1.0])
    eta_i = np.array([-1.0, -1.0, +1.0, +1.0])
    sums = []
    for q in gauss:
        s = sum(0.25 * (1 + xi_i[i] * q[0]) * (1 + eta_i[i] * q[1]) for i in range(4))
        sums.append(s)
    sums = np.array(sums)
    np.testing.assert_allclose(sums, 1.0, atol=1e-14)
    print(f"4 Gauss points; Σ N_i(ξ_q, η_q) = {sums.round(15)} — partition of unity verified.")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    4 Gauss points; Σ N_i(ξ_q, η_q) = [1. 1. 1. 1.] — partition of unity verified.





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.168 seconds)


.. _sphx_glr_download_gallery_elements_plane182_example_quad4_plane_reference_geometry.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

      :download:`Download Jupyter notebook: example_quad4_plane_reference_geometry.ipynb <example_quad4_plane_reference_geometry.ipynb>`

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

      :download:`Download Python source code: example_quad4_plane_reference_geometry.py <example_quad4_plane_reference_geometry.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: example_quad4_plane_reference_geometry.zip <example_quad4_plane_reference_geometry.zip>`


.. only:: html

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

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