.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery/elements/mass21/example_point_mass_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_mass21_example_point_mass_reference_geometry.py>`
        to download the full example code.

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

.. _sphx_glr_gallery_elements_mass21_example_point_mass_reference_geometry.py:


POINT_MASS — single-node lumped mass
=====================================

The 1-node concentrated-mass element contributes a diagonal
``3 x 3`` block to the global mass matrix:

.. math::

    M_e = \mathrm{diag}(m_x,\, m_y,\, m_z),

where :math:`(m_x, m_y, m_z)` are the per-axis masses (typically
all equal to a single isotropic mass :math:`m`).  Consistent
and lumped formulations coincide for a one-node element.

The rotary-inertia variant (``KEYOPT(3) != 2``) adds three
diagonal rotary-inertia entries; that path is signposted in
the kernel module header as a future addition.

References
----------
* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002)
  *Concepts and Applications of Finite Element Analysis*, 4th
  ed., Wiley, §11.3.
* Bathe, K.-J. (2014) *Finite Element Procedures*, 2nd ed.,
  Prentice Hall, §4.2.2.

Implementation: :class:`femorph_solver.elements.point_mass.PointMass`.

.. GENERATED FROM PYTHON SOURCE LINES 30-36

.. code-block:: Python


    from __future__ import annotations

    import numpy as np
    import pyvista as pv








.. GENERATED FROM PYTHON SOURCE LINES 37-39

Render the single-node element
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-59

.. code-block:: Python


    origin = np.array([[0.0, 0.0, 0.0]])

    plotter = pv.Plotter(off_screen=True, window_size=(480, 360))
    plotter.add_points(
        origin, render_points_as_spheres=True, point_size=40, color="#d62728", label="point-mass node"
    )
    # Three axis-aligned arrows showing the 3 mass DOFs (m_x, m_y, m_z).
    for vec, _label, col in [
        (np.array([1.0, 0.0, 0.0]), "m_x", "#1f77b4"),
        (np.array([0.0, 1.0, 0.0]), "m_y", "#2ca02c"),
        (np.array([0.0, 0.0, 1.0]), "m_z", "#ff7f0e"),
    ]:
        plotter.add_arrows(origin, vec[None, :], mag=0.8, color=col)
    plotter.add_axes(line_width=4, color="black")
    plotter.view_isometric()
    plotter.camera.zoom(0.9)
    plotter.add_legend(face=None, size=(0.22, 0.07), bcolor="white")
    plotter.show()




.. image-sg:: /gallery/elements/mass21/images/sphx_glr_example_point_mass_reference_geometry_001.png
   :alt: example point mass reference geometry
   :srcset: /gallery/elements/mass21/images/sphx_glr_example_point_mass_reference_geometry_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 60-66

Sanity — global mass contribution
---------------------------------

A point-mass at node ``n`` adds ``M_e`` to rows / columns
``(3n, 3n+1, 3n+2)`` of the global mass matrix.  Verify the
contribution is exactly diagonal.

.. GENERATED FROM PYTHON SOURCE LINES 66-73

.. code-block:: Python


    m_x, m_y, m_z = 7.5, 7.5, 7.5  # isotropic 7.5 kg point mass
    M_e = np.diag([m_x, m_y, m_z])
    print("point-mass contribution at one node:")
    print(f"  M_e (diag) = {np.diag(M_e)}")
    np.testing.assert_allclose(M_e - np.diag(np.diag(M_e)), 0.0, atol=1e-15)
    print("OK — M_e is purely diagonal.")




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

 .. code-block:: none

    point-mass contribution at one node:
      M_e (diag) = [7.5 7.5 7.5]
    OK — M_e is purely diagonal.





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

   **Total running time of the script:** (0 minutes 0.145 seconds)


.. _sphx_glr_download_gallery_elements_mass21_example_point_mass_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_point_mass_reference_geometry.ipynb <example_point_mass_reference_geometry.ipynb>`

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

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

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

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


.. only:: html

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

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