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

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

.. _sphx_glr_gallery_verification_example_verify_cantilever_eb.py:


Cantilever tip deflection — Euler-Bernoulli closed form
=======================================================

Slender clamped cantilever under a transverse tip load has
tip deflection

.. math::

    \delta = \frac{P L^{3}}{3 E I}, \qquad I = \frac{w h^{3}}{12}.

Reference: Timoshenko, S. P. *Strength of Materials, Part I*,
3rd ed., Van Nostrand, 1955, §5.4.

This gallery example drives a multi-refinement
:class:`~femorph_solver.validation.ConvergenceStudy` and
renders the deformed shape at the finest mesh.

.. GENERATED FROM PYTHON SOURCE LINES 19-28

.. 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 CantileverEulerBernoulli








.. GENERATED FROM PYTHON SOURCE LINES 29-35

Problem + convergence ladder
----------------------------

Three in-plane refinements (slenderness L/h = 20); three
transverse layers keeps the shear-strain field resolved at
the HEX8 EAS formulation used under the hood.

.. GENERATED FROM PYTHON SOURCE LINES 35-47

.. code-block:: Python


    problem = CantileverEulerBernoulli()
    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()








.. GENERATED FROM PYTHON SOURCE LINES 48-49

Print the comparison table

.. GENERATED FROM PYTHON SOURCE LINES 49-67

.. code-block:: Python


    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':<20s} {'n_dof':>8s} {'computed':>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:<20s} {r.n_dof:>8d} {r.computed:+14.6e} "
            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_eb: Slender clamped cantilever under transverse tip load — Euler-Bernoulli closed-form (Timoshenko 1955 §5.4).

      source : Timoshenko 1955 §5.4
      formula: delta = P L^3 / (3 E I)
      published: 3.200000e-03 m

      mesh                    n_dof       computed    rel err  pass
      nx=20  ny=3  nz=3        1008  +3.168413e-03     -0.99%   ✓
      nx=40  ny=3  nz=3        1968  +3.184714e-03     -0.48%   ✓
      nx=80  ny=3  nz=3        3888  +3.191451e-03     -0.27%   ✓

      fitted rate: |err| ∝ n_dof^(-0.85)




.. GENERATED FROM PYTHON SOURCE LINES 68-70

Render the deformed shape at the finest mesh
--------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 70-86

.. code-block:: Python


    m = problem.build_model(nx=80, ny=3, nz=3)
    res = m.solve()
    u = np.asarray(res.displacement).reshape(-1, 3)

    warped = m.grid.copy()
    warped.points = m.grid.points + 1.0 * u
    warped["|u|"] = np.linalg.norm(u, 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="|u|", cmap="viridis", show_edges=False)
    plotter.view_isometric()
    plotter.camera.zoom(1.1)
    plotter.show()




.. image-sg:: /gallery/verification/images/sphx_glr_example_verify_cantilever_eb_001.png
   :alt: example verify cantilever eb
   :srcset: /gallery/verification/images/sphx_glr_example_verify_cantilever_eb_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 87-89

Acceptance
----------

.. GENERATED FROM PYTHON SOURCE LINES 89-95

.. code-block:: Python


    finest = rec.results[-1]
    assert finest.passed, (
        f"finest-mesh tip_deflection 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 1.747 seconds)


.. _sphx_glr_download_gallery_verification_example_verify_cantilever_eb.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_eb.ipynb <example_verify_cantilever_eb.ipynb>`

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

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

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

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


.. only:: html

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

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