.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery/elements/solid185/example_hex8_shape_function_surfaces.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_solid185_example_hex8_shape_function_surfaces.py>`
        to download the full example code.

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

.. _sphx_glr_gallery_elements_solid185_example_hex8_shape_function_surfaces.py:


HEX8 shape-function surfaces on the reference cube
==================================================

Renders all eight trilinear Lagrange shape functions
:math:`N_i(\xi, \eta, \zeta)` for the 8-node hexahedron on
the reference cube :math:`\hat\Omega = [-1, +1]^3`.  The
basis functions are

.. math::

    N_i(\xi, \eta, \zeta) = \tfrac{1}{8}
      (1 + \xi_i \xi)(1 + \eta_i \eta)(1 + \zeta_i \zeta),
    \qquad i = 1, \ldots, 8,

with :math:`(\xi_i, \eta_i, \zeta_i) \in \{-1, +1\}^3` the
nodal coordinates.  Each :math:`N_i` is a tensor product of
1-D linear "hat" functions, equal to 1 at corner :math:`i`
and 0 at every other corner — the Kronecker-delta
interpolation property.

The plot below shows :math:`N_i` evaluated on the
:math:`\zeta = -1` cube face (:math:`i = 1, \ldots, 4`) and
the :math:`\zeta = +1` face (:math:`i = 5, \ldots, 8`).

References
----------

* Hughes, T. J. R. (2000) *The Finite Element Method —
  Linear Static and Dynamic Finite Element Analysis*, Dover,
  §3.1 — trilinear shape functions.
* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, §6.2.

Implementation: :class:`femorph_solver.elements.hex8.Hex8`.

.. GENERATED FROM PYTHON SOURCE LINES 38-44

.. code-block:: Python


    from __future__ import annotations

    import matplotlib.pyplot as plt
    import numpy as np








.. GENERATED FROM PYTHON SOURCE LINES 45-46

Corner natural-coordinate map (matches VTK_HEXAHEDRON ordering)

.. GENERATED FROM PYTHON SOURCE LINES 46-67

.. code-block:: Python

    corners = np.array(
        [
            [-1, -1, -1],
            [+1, -1, -1],
            [+1, +1, -1],
            [-1, +1, -1],
            [-1, -1, +1],
            [+1, -1, +1],
            [+1, +1, +1],
            [-1, +1, +1],
        ],
        dtype=float,
    )


    def n_i(xi, eta, zeta, i):
        """Trilinear Lagrange shape function for corner i."""
        xi_i, eta_i, zeta_i = corners[i]
        return 0.125 * (1 + xi_i * xi) * (1 + eta_i * eta) * (1 + zeta_i * zeta)









.. GENERATED FROM PYTHON SOURCE LINES 68-69

Sample on a 41×41 grid on each ζ = ±1 face

.. GENERATED FROM PYTHON SOURCE LINES 69-74

.. code-block:: Python


    n = 41
    xi = eta = np.linspace(-1.0, 1.0, n)
    XI, ETA = np.meshgrid(xi, eta, indexing="ij")








.. GENERATED FROM PYTHON SOURCE LINES 75-76

4×2 surface plot — one column per ζ-face, 4 rows for the 4 corners

.. GENERATED FROM PYTHON SOURCE LINES 76-108

.. code-block:: Python


    fig, axes = plt.subplots(
        4,
        2,
        figsize=(10.0, 14.0),
        subplot_kw={"projection": "3d"},
        constrained_layout=True,
    )
    fig.suptitle("HEX8 shape functions $N_i(\\xi, \\eta, \\zeta)$ on cube faces", fontsize=13)

    face_zeta = (-1.0, +1.0)
    node_indices_per_face = ((0, 1, 2, 3), (4, 5, 6, 7))
    for col, (zeta, idxs) in enumerate(zip(face_zeta, node_indices_per_face, strict=True)):
        for row, idx in enumerate(idxs):
            ax = axes[row, col]
            N = n_i(XI, ETA, zeta, idx)
            ax.plot_surface(XI, ETA, N, cmap="plasma", linewidth=0, antialiased=True)
            x_i, y_i, _ = corners[idx]
            ax.set_title(
                f"$N_{{{idx + 1}}}$ at corner $({int(x_i):+d}, {int(y_i):+d}, {int(zeta):+d})$",
                fontsize=10,
            )
            ax.set_xlabel(r"$\xi$")
            ax.set_ylabel(r"$\eta$")
            ax.set_zlabel(r"$N$")
            ax.set_xticks([-1, 0, 1])
            ax.set_yticks([-1, 0, 1])
            ax.set_zticks([0, 0.5, 1])
            ax.view_init(elev=22, azim=-55)

    fig.show()




.. image-sg:: /gallery/elements/solid185/images/sphx_glr_example_hex8_shape_function_surfaces_001.png
   :alt: HEX8 shape functions $N_i(\xi, \eta, \zeta)$ on cube faces, $N_{1}$ at corner $(-1, -1, -1)$, $N_{5}$ at corner $(-1, -1, +1)$, $N_{2}$ at corner $(+1, -1, -1)$, $N_{6}$ at corner $(+1, -1, +1)$, $N_{3}$ at corner $(+1, +1, -1)$, $N_{7}$ at corner $(+1, +1, +1)$, $N_{4}$ at corner $(-1, +1, -1)$, $N_{8}$ at corner $(-1, +1, +1)$
   :srcset: /gallery/elements/solid185/images/sphx_glr_example_hex8_shape_function_surfaces_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 109-114

Verify partition of unity at every interior sample point

:math:`\sum_{i=1}^8 N_i(\xi, \eta, \zeta) = 1` for any
:math:`(\xi, \eta, \zeta) \in \hat\Omega`.  Sample on a
21×21×21 lattice and assert.

.. GENERATED FROM PYTHON SOURCE LINES 114-124

.. code-block:: Python


    n3 = 21
    xi3 = eta3 = zeta3 = np.linspace(-1.0, 1.0, n3)
    XI3, ETA3, ZETA3 = np.meshgrid(xi3, eta3, zeta3, indexing="ij")
    total = sum(n_i(XI3, ETA3, ZETA3, idx) for idx in range(8))
    max_dev = float(np.max(np.abs(total - 1.0)))
    assert max_dev < 1.0e-14, f"partition-of-unity violated: max |Σ N_i - 1| = {max_dev:.3e}"
    print(f"max |Σ N_i - 1| over a 21^3 grid = {max_dev:.3e}")
    print("OK — eight trilinear HEX8 shape functions form a partition of unity.")





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

 .. code-block:: none

    max |Σ N_i - 1| over a 21^3 grid = 4.441e-16
    OK — eight trilinear HEX8 shape functions form a partition of unity.




.. GENERATED FROM PYTHON SOURCE LINES 125-131

Verify Kronecker-delta interpolation at every corner

:math:`N_i(\boldsymbol{\xi}_j) = \delta_{ij}`: each shape
function equals 1 at its own corner and 0 at all other
corners — required so the nodal-displacement DOFs are
literally the displacement values.

.. GENERATED FROM PYTHON SOURCE LINES 131-138

.. code-block:: Python


    print()
    for i in range(8):
        row = [n_i(*corners[j], i) for j in range(8)]
        expected = [1.0 if j == i else 0.0 for j in range(8)]
        np.testing.assert_allclose(row, expected, atol=1.0e-15)
    print("OK — Kronecker-delta interpolation verified at every corner.")




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

 .. code-block:: none


    OK — Kronecker-delta interpolation verified at every corner.





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

   **Total running time of the script:** (0 minutes 1.943 seconds)


.. _sphx_glr_download_gallery_elements_solid185_example_hex8_shape_function_surfaces.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: example_hex8_shape_function_surfaces.ipynb <example_hex8_shape_function_surfaces.ipynb>`

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

      :download:`Download Python source code: example_hex8_shape_function_surfaces.py <example_hex8_shape_function_surfaces.py>`

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

      :download:`Download zipped: example_hex8_shape_function_surfaces.zip <example_hex8_shape_function_surfaces.zip>`


.. only:: html

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

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