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

.. only:: html

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

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

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

.. _sphx_glr_gallery_verification_example_verify_cantilever_nf.py:


Cantilever natural frequency — Rao 2017 Table 8.1
==================================================

First transverse bending frequency of a clamped-free prismatic
beam:

.. math::

    f_{1} = \frac{(\beta_{1} L)^{2}}{2 \pi L^{2}}
           \sqrt{\frac{E I}{\rho A}}, \qquad \beta_{1} L = 1.8751.

Reference: Rao, S. S. *Mechanical Vibrations*, 6th ed.,
Pearson, 2017, §8.5, Table 8.1 first entry.

.. GENERATED FROM PYTHON SOURCE LINES 16-25

.. code-block:: Python


    from __future__ import annotations

    import numpy as np
    import pyvista as pv

    from femorph_solver.validation import ConvergenceStudy
    from femorph_solver.validation.problems import CantileverNaturalFrequency








.. GENERATED FROM PYTHON SOURCE LINES 26-28

Build + solve + convergence study
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-57

.. code-block:: Python


    problem = CantileverNaturalFrequency()
    study = ConvergenceStudy(
        problem=problem,
        refinements=[
            {"nx": 20, "ny": 3, "nz": 3},
            {"nx": 40, "ny": 3, "nz": 3},
            {"nx": 80, "ny": 3, "nz": 3},
        ],
    )
    records = study.run()
    rec = records[0]
    pub = rec.results[0].published

    print(f"{problem.name}: {problem.description}")
    print(f"\n  source : {pub.source}")
    print(f"  formula: {pub.formula}")
    print(f"  published: {pub.value:.6e} {pub.unit}\n")
    print(f"  {'mesh':<22s} {'n_dof':>8s} {'f_1 (Hz)':>14s} {'rel err':>10s}  pass")
    for r in rec.results:
        mesh_str = "  ".join(f"{k}={v}" for k, v in r.mesh_params.items())
        pass_str = "✓" if r.passed else "✗"
        print(
            f"  {mesh_str:<22s} {r.n_dof:>8d} {r.computed:+14.4f} "
            f"{r.rel_error * 100:+9.2f}%   {pass_str}"
        )
    if rec.convergence_rate is not None:
        print(f"\n  fitted rate: |err| ∝ n_dof^(-{rec.convergence_rate:.2f})")





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

 .. code-block:: none

    cantilever_nf: First transverse bending frequency of a clamped-free prismatic beam — Rao 2017 §8.5 Table 8.1.

      source : Rao 2017 §8.5 Table 8.1
      formula: f1 = (beta1 L)^2 / (2 pi L^2) sqrt(E I / (rho A))
      published: 4.076904e+01 Hz

      mesh                      n_dof       f_1 (Hz)    rel err  pass
      nx=20  ny=3  nz=3          1008       +41.0046     +0.58%   ✓
      nx=40  ny=3  nz=3          1968       +40.8698     +0.25%   ✓
      nx=80  ny=3  nz=3          3888       +40.8137     +0.11%   ✓

      fitted rate: |err| ∝ n_dof^(-1.20)




.. GENERATED FROM PYTHON SOURCE LINES 58-60

Render the first bending mode shape
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 60-94

.. code-block:: Python


    m = problem.build_model(nx=40, ny=3, nz=3)
    modal = m.modal_solve(n_modes=10)

    shapes = np.asarray(modal.mode_shapes).reshape(-1, 3, modal.mode_shapes.shape[-1])
    # Re-run the mode filter from the problem's extract to pick the
    # actual first-bending mode (the square cross-section makes the
    # UY/UZ bending pair degenerate).
    selected = None
    for k in range(shapes.shape[-1]):
        ux_rms = float(np.sqrt((shapes[:, 0, k] ** 2).mean()))
        uy_rms = float(np.sqrt((shapes[:, 1, k] ** 2).mean()))
        uz_rms = float(np.sqrt((shapes[:, 2, k] ** 2).mean()))
        if max(uy_rms, uz_rms) > 2.0 * ux_rms:
            selected = k
            break
    if selected is None:
        selected = 0

    phi = shapes[:, :, selected]
    warped = m.grid.copy()
    # Normalise the mode shape so the tip amplitude is ~10% of L,
    # just for a visible figure.
    scale = 0.1 * problem.L / max(np.linalg.norm(phi, axis=1).max(), 1.0e-30)
    warped.points = m.grid.points + scale * phi
    warped["|phi|"] = np.linalg.norm(phi, axis=1)

    plotter = pv.Plotter(off_screen=True)
    plotter.add_mesh(m.grid, style="wireframe", color="#888888", opacity=0.3)
    plotter.add_mesh(warped, scalars="|phi|", cmap="plasma", show_edges=False)
    plotter.view_isometric()
    plotter.camera.zoom(1.1)
    plotter.show()




.. image-sg:: /gallery/verification/images/sphx_glr_example_verify_cantilever_nf_001.png
   :alt: example verify cantilever nf
   :srcset: /gallery/verification/images/sphx_glr_example_verify_cantilever_nf_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 95-96

Acceptance

.. GENERATED FROM PYTHON SOURCE LINES 96-102

.. code-block:: Python


    finest = rec.results[-1]
    assert finest.passed, (
        f"finest-mesh f1 failed: {finest.rel_error:.2%} "
        f"above tolerance {finest.published.tolerance:.0%}"
    )








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

   **Total running time of the script:** (0 minutes 1.821 seconds)


.. _sphx_glr_download_gallery_verification_example_verify_cantilever_nf.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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