.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery/elements/link180/example_truss2_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_link180_example_truss2_reference_geometry.py>`
        to download the full example code.

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

.. _sphx_glr_gallery_elements_link180_example_truss2_reference_geometry.py:


TRUSS2 reference geometry — linear axial bar
=============================================

The 2-node 3D truss (axial-only spar) maps to the natural
coordinate :math:`s \in [-1, +1]`.  Linear shape functions
interpolate the three translational DOFs at each node:

.. math::

    N_1(s) = \tfrac{1 - s}{2}, \qquad
    N_2(s) = \tfrac{1 + s}{2}.

Axial strain :math:`\varepsilon_{xx} = (u_2 - u_1) / L` is
constant across the element; the resulting 6×6 stiffness in
the global frame is

.. math::

    K_e = \frac{E A}{L} \begin{bmatrix} +T & -T \\ -T & +T \end{bmatrix},
    \qquad T_{ij} = d_i d_j

with :math:`(d_1, d_2, d_3)` the unit direction cosines along
the I→J axis.

References
----------
* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002)
  *Concepts and Applications of Finite Element Analysis*, 4th
  ed., Wiley, §3.2.
* Bathe, K.-J. (2014) *Finite Element Procedures*, 2nd ed.,
  Prentice Hall, §5.3.1.
* Przemieniecki, J. S. (1968) *Theory of Matrix Structural
  Analysis*, McGraw-Hill, §5.1.

Implementation: :class:`femorph_solver.elements.truss2.Truss2`.

.. GENERATED FROM PYTHON SOURCE LINES 38-45

.. code-block:: Python


    from __future__ import annotations

    import matplotlib.pyplot as plt
    import numpy as np
    import pyvista as pv








.. GENERATED FROM PYTHON SOURCE LINES 46-47

Linear shape functions on [-1, +1]

.. GENERATED FROM PYTHON SOURCE LINES 47-66

.. code-block:: Python


    s = np.linspace(-1.0, 1.0, 200)
    N1 = 0.5 * (1.0 - s)
    N2 = 0.5 * (1.0 + s)

    fig, ax = plt.subplots(1, 1, figsize=(6.4, 3.6))
    ax.plot(s, N1, label="$N_1(s) = (1 - s) / 2$", color="#1f77b4", lw=2)
    ax.plot(s, N2, label="$N_2(s) = (1 + s) / 2$", color="#ff7f0e", lw=2)
    ax.scatter([-1.0, 1.0], [1.0, 1.0], color="black", zorder=5)
    ax.set_xlabel(r"$s$")
    ax.set_ylabel("$N_i(s)$")
    ax.set_title("TRUSS2 linear shape functions")
    ax.legend(loc="lower center", ncol=2, fontsize=10)
    ax.grid(True, ls=":", alpha=0.5)
    ax.set_xlim(-1.0, 1.0)
    ax.set_ylim(-0.05, 1.1)
    fig.tight_layout()
    fig.show()




.. image-sg:: /gallery/elements/link180/images/sphx_glr_example_truss2_reference_geometry_001.png
   :alt: TRUSS2 linear shape functions
   :srcset: /gallery/elements/link180/images/sphx_glr_example_truss2_reference_geometry_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 67-68

Reference element with the 2-point Gauss-Legendre rule

.. GENERATED FROM PYTHON SOURCE LINES 68-95

.. code-block:: Python


    cells = np.array([2, 0, 1])
    cell_types = np.array([pv.CellType.LINE], dtype=np.uint8)
    truss = pv.UnstructuredGrid(cells, cell_types, np.array([[-1.0, 0.0, 0.0], [+1.0, 0.0, 0.0]]))
    g = 1.0 / np.sqrt(3.0)
    gauss = np.array([[-g, 0.0, 0.0], [+g, 0.0, 0.0]])

    plotter = pv.Plotter(off_screen=True, window_size=(640, 200))
    plotter.add_mesh(truss, color="black", line_width=4)
    plotter.add_points(
        np.array([[-1.0, 0.0, 0.0], [+1.0, 0.0, 0.0]]),
        render_points_as_spheres=True,
        point_size=18,
        color="black",
        label="end nodes",
    )
    plotter.add_points(
        gauss,
        render_points_as_spheres=True,
        point_size=14,
        color="#d62728",
        label="2-pt Gauss-Legendre",
    )
    plotter.view_xy()
    plotter.camera.zoom(1.6)
    plotter.show()




.. image-sg:: /gallery/elements/link180/images/sphx_glr_example_truss2_reference_geometry_002.png
   :alt: example truss2 reference geometry
   :srcset: /gallery/elements/link180/images/sphx_glr_example_truss2_reference_geometry_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 96-97

Sanity — partition of unity at every Gauss point

.. GENERATED FROM PYTHON SOURCE LINES 97-101

.. code-block:: Python


    sums = N1[np.searchsorted(s, [-g, +g])] + N2[np.searchsorted(s, [-g, +g])]
    np.testing.assert_allclose(sums, 1.0, atol=1e-12)
    print("OK — N_1(s) + N_2(s) = 1 at every Gauss point.")




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

 .. code-block:: none

    OK — N_1(s) + N_2(s) = 1 at every Gauss point.





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

   **Total running time of the script:** (0 minutes 0.258 seconds)


.. _sphx_glr_download_gallery_elements_link180_example_truss2_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_truss2_reference_geometry.ipynb <example_truss2_reference_geometry.ipynb>`

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

      :download:`Download Python source code: example_truss2_reference_geometry.py <example_truss2_reference_geometry.py>`

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

      :download:`Download zipped: example_truss2_reference_geometry.zip <example_truss2_reference_geometry.zip>`


.. only:: html

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

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