:orphan:

:orphan:

.. _verification-gallery:

Verification gallery
====================

Runnable examples that drive femorph-solver's
:doc:`validation framework </reference/validation>` to reproduce
canonical FEA problems against **published** reference values.

Each example instantiates one
:class:`~femorph_solver.validation.BenchmarkProblem` subclass,
runs ``validate()`` (or a multi-refinement
:class:`~femorph_solver.validation.ConvergenceStudy`), prints
the computed-vs-published comparison, and renders a figure of
the solved model.

**The reference values in this gallery are traceable to a
textbook, paper, or handbook** — not to any other solver's
output.  Every example is also a regression-guarded test under
``tests/validation/``, so a gallery failure surfaces as a CI
failure before it can drift silently in the rendered docs.

.. grid:: 1 2 2 2
    :gutter: 3

    .. grid-item-card:: Single-hex uniaxial
        :link: example_verify_single_hex_uniaxial
        :link-type: doc

        Hooke's law + Poisson contraction on a unit hex.
        Machine-precision check (Hughes 2000 §2.7).

    .. grid-item-card:: Irons patch test
        :link: example_verify_patch_test
        :link-type: doc

        Constant-strain patch test — the canonical
        consistency check (Irons & Razzaque 1972).

    .. grid-item-card:: Cantilever tip deflection
        :link: example_verify_cantilever_eb
        :link-type: doc

        :math:`\\delta = P L^3 / (3 E I)` — Euler-Bernoulli
        closed form (Timoshenko 1955 §5.4).

    .. grid-item-card:: Cantilever natural frequency
        :link: example_verify_cantilever_nf
        :link-type: doc

        First bending frequency (Rao 2017 §8.5 Table 8.1).


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The canonical FE consistency test: a 2×2×2 hex patch with an interior node displaced from its regular position, subjected to boundary displacements u = \varepsilon_0\, x consistent with a uniform axial strain field, must recover \varepsilon = \varepsilon_0 everywhere to machine precision.">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_patch_test_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_patch_test`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Irons constant-strain patch test</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The smallest possible verification: a unit cube of isotropic linear elastic material under a uniform axial traction must reproduce 1-D Hooke&#x27;s law \sigma_{xx} = E\, \varepsilon_{xx} and Poisson contraction \varepsilon_{yy} = -\nu\, \varepsilon_{xx} to machine precision.">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_single_hex_uniaxial_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_single_hex_uniaxial`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Single-hex uniaxial tension — Hooke's law + Poisson check</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Slender clamped cantilever under a transverse tip load has tip deflection">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_cantilever_eb_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_cantilever_eb`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Cantilever tip deflection — Euler-Bernoulli closed form</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Canonical plane-stress benchmark from the NAFEMS Standard Benchmark Tests for Linear Elastic Analysis, R0015 (1990), problem LE1.  A quarter elliptic-membrane plate bounded by two confocal ellipses">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_nafems_le1_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_nafems_le1`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">NAFEMS LE1 — elliptic membrane (plane stress)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical fully-clamped thin-plate verification: a square plate of side a and thickness h is clamped on all four edges (every DOF pinned, full thickness) and loaded by a uniform transverse pressure q.  The maximum centre deflection has no elementary closed form but is published from the converged double-Fourier solution (NAFEMS R0015 §2.5 LE5; Timoshenko &amp; Woinowsky-Krieger 1959 §32 Table 12) as">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_clamped_plate_static_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_clamped_plate_static`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Clamped square plate under uniform pressure (NAFEMS LE5)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="For a clamped-free prismatic cantilever of length L, flexural rigidity EI, and uniform transverse line load w (force per unit length), the tip deflection is">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_cantilever_udl_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_cantilever_udl`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Cantilever under uniformly distributed load — Euler–Bernoulli closed form</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="First transverse bending frequency of a clamped-free prismatic beam:">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_cantilever_nf_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_cantilever_nf`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Cantilever natural frequency — Rao 2017 Table 8.1</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The longitudinal natural frequencies of a uniform rod fixed at one end and free at the other are (Rao 2017 §8.5; Timoshenko 1974 §5.1)">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_axial_rod_nf_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_axial_rod_nf`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Fixed-free axial rod — natural frequencies</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical Bernoulli-beam eigenvalue problem.  A prismatic beam clamped at both ends has bending natural frequencies (Rao 2017 §8.5 Table 8.1; Timoshenko 1974 §5.3)">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_cc_beam_modes_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_cc_beam_modes`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Clamped-clamped beam — first three bending natural frequencies</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical thin-circular-ring elasticity benchmark: a ring of mean radius R and rectangular cross-section b \times t (with b \ll R and t \ll R) is loaded by two equal opposing point loads P along a diameter.  Castigliano&#x27;s theorem applied to the curved-beam strain energy gives the diametrical deflection along the loading axis (Timoshenko &amp; Young 1968 §79; Roark Table 9 case 1):">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_pinched_ring_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_pinched_ring`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Pinched ring — Castigliano diametrical deflection</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical Saint-Venant elasticity verification: a clamped-free straight beam carrying an end torque T at its free tip develops a uniform twist rate.  The total tip rotation is (Roark &amp; Young 1989 Table 20 case 4; Timoshenko &amp; Goodier 1970 §90):">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_cantilever_torsion_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_cantilever_torsion`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Cantilever Saint-Venant torsion — rectangular cross-section</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical thin-plate eigenvalue problem.  A simply-supported square plate of side a and thickness h has Kirchhoff natural frequencies (Timoshenko &amp; Woinowsky-Krieger 1959 §63 eq. 151)">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_ss_plate_modes_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_ss_plate_modes`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Simply-supported plate — first transverse bending mode</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical 2D elasticity benchmark: a thin rectangular plate with a small circular hole, loaded by uniform remote tension \sigma_{\infty} along the x-axis.  Kirsch&#x27;s 1898 closed form for the infinite plate gives a stress field that, on the hole edge r = a, peaks at the points perpendicular to the loading direction (\theta = \pm\pi/2):">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_plate_with_hole_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_plate_with_hole`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Plate with a circular hole — Kirsch stress concentration</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical thin-plate verification: a simply-supported square plate of side a and thickness h carries a uniform transverse pressure q.  The Navier double-sine expansion (Timoshenko &amp; Woinowsky-Krieger 1959 §30 eqs. 115–117) gives the centre deflection">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_ss_plate_static_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_ss_plate_static`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Simply-supported plate under uniform pressure (Navier series)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The textbook trap of low-order solid elements: a fully- integrated 8-node trilinear hex (HEX8 with full 2 × 2 × 2 Gauss, no B-bar / enhanced strain) cannot reproduce linear-bending kinematics, because the constant-strain Gauss point at the element centre overstates the bending stiffness (Cook §6.6, §6.13).  The result is shear locking: even on a slender cantilever where the Euler-Bernoulli analytical answer is exact, the FE displacement is too small — sometimes by orders of magnitude — and the missing strain energy goes into spurious transverse shear at the Gauss point.">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_shear_locking_demo_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_shear_locking_demo`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Shear-locking demonstration — HEX8 integration variants</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Pure-bending end-loaded prismatic beam, the simplest test of moment-driven Bernoulli kinematics.  A clamped-free cantilever of length L carries a moment M_0 about the out-of-plane axis applied at its tip; equilibrium and elementary integration give a closed form (Roark &amp; Young 1989, Table 8 case 4; Timoshenko 1955 §32):">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_cantilever_tip_moment_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_cantilever_tip_moment`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Cantilever beam under a tip moment</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical Bernoulli-beam eigenvalue problem.  A clamped-free prismatic cantilever has bending natural frequencies (Rao 2017 §8.5 Table 8.1; Timoshenko 1974 §5.3)">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_cantilever_higher_modes_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_cantilever_higher_modes`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Cantilever beam — first four bending modes</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Statically-indeterminate first-order beam with both ends fully clamped, carrying a single transverse point load P at midspan.  Symmetry plus compatibility at the clamps gives the elementary closed form (Roark &amp; Young 1989, Table 8 case 5; Timoshenko 1955 §40):">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_cc_beam_central_load_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_cc_beam_central_load`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Clamped-clamped beam under a central point load</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical statically-determinate first-order beam: simple supports (pin + roller) at x = 0 and x = L, with a single transverse point load P at midspan. Symmetry plus equilibrium gives the elementary closed form">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_ss_beam_central_load_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_ss_beam_central_load`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Simply-supported beam under a central point load</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Statically-indeterminate first-order beam: clamped at x = 0, simply supported at x = L, with a uniform downward distributed load q over the full span.  The closed-form Euler-Bernoulli solution (Roark &amp; Young 1989, Table 8 case 7; Timoshenko 1955 §40) gives">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_propped_cantilever_udl_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_propped_cantilever_udl`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Propped cantilever under uniformly-distributed load</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The textbook demonstration of FE convergence: for a fixed problem with a known closed form, refine the mesh and watch the discretisation error decrease at the asymptotic rate set by the element&#x27;s polynomial order.  For the standard 8-node hex with enhanced-strain (Simo–Rifai) under a tip point load on a slender cantilever, the expected convergence rate in the energy / displacement norm is">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_convergence_cantilever_eb_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_convergence_cantilever_eb`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mesh-refinement convergence — cantilever Euler-Bernoulli</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Companion to the sphx_glr_gallery_verification_example_verify_convergence_cantilever_eb.py beam study, this example shows the same O(h^{2}) displacement-norm convergence rate on a 3D axisymmetric elasticity problem: the Lamé thick cylinder under internal pressure.  Different physics (purely radial deformation, no bending), different element geometry (annular sector instead of beam-aspect prism), but the same Galerkin error theorem holds — Strang &amp; Fix (2008) §3.7 promises \|u^{h} - u\|_{L^{2}} = O(h^{p+1}) with p = 1 for trilinear hex.">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_convergence_lame_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_convergence_lame`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mesh-refinement convergence — Lamé thick cylinder</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical pressure-vessel benchmark: a long cylinder with inner radius a, outer radius b, subjected to internal pressure p_i and held at zero axial strain (plane strain).  The closed-form Lamé (1852) solution gives purely radial displacement u_r(r) and the radial / hoop stresses">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_lame_cylinder_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_lame_cylinder`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Lamé thick-walled cylinder under internal pressure</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Classical statically-indeterminate beam benchmark: a prismatic straight beam of total length 2L rests on three simple supports — two at the ends and one at the centre — and carries a uniformly distributed load q over both spans.  Because there is one redundant reaction, the moment at the central support follows from the three-moment theorem of Clapeyron (1857):">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_continuous_beam_3supports_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_continuous_beam_3supports`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Continuous beam over three supports — Clapeyron's three-moment theorem</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Robert D. Cook&#x27;s 1974 International Journal for Numerical Methods in Engineering paper introduced what became the universal stress- test for low-order solid elements under mesh distortion: a tapered plate clamped on its short edge and loaded by a uniform transverse shear on its slanted edge.  The non-rectangular geometry forces every interior element to be a parallelogram (or worse) — the canonical setting where a fully-integrated 8-node hex (plain 2 × 2 × 2 Gauss) shear-locks badly even though the material is moderately compressible (\nu = 1/3) and no volumetric pathology is present.">

.. only:: html

  .. image:: /gallery/verification/images/thumb/sphx_glr_example_verify_cooks_membrane_thumb.png
    :alt:

  :doc:`/gallery/verification/example_verify_cooks_membrane`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Cook's membrane — mesh-distortion benchmark (1974)</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /gallery/verification/example_verify_patch_test
   /gallery/verification/example_verify_single_hex_uniaxial
   /gallery/verification/example_verify_cantilever_eb
   /gallery/verification/example_verify_nafems_le1
   /gallery/verification/example_verify_clamped_plate_static
   /gallery/verification/example_verify_cantilever_udl
   /gallery/verification/example_verify_cantilever_nf
   /gallery/verification/example_verify_axial_rod_nf
   /gallery/verification/example_verify_cc_beam_modes
   /gallery/verification/example_verify_pinched_ring
   /gallery/verification/example_verify_cantilever_torsion
   /gallery/verification/example_verify_ss_plate_modes
   /gallery/verification/example_verify_plate_with_hole
   /gallery/verification/example_verify_ss_plate_static
   /gallery/verification/example_verify_shear_locking_demo
   /gallery/verification/example_verify_cantilever_tip_moment
   /gallery/verification/example_verify_cantilever_higher_modes
   /gallery/verification/example_verify_cc_beam_central_load
   /gallery/verification/example_verify_ss_beam_central_load
   /gallery/verification/example_verify_propped_cantilever_udl
   /gallery/verification/example_verify_convergence_cantilever_eb
   /gallery/verification/example_verify_convergence_lame
   /gallery/verification/example_verify_lame_cylinder
   /gallery/verification/example_verify_continuous_beam_3supports
   /gallery/verification/example_verify_cooks_membrane


.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-gallery

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

      :download:`Download all examples in Python source code: verification_python.zip </gallery/verification/verification_python.zip>`

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

      :download:`Download all examples in Jupyter notebooks: verification_jupyter.zip </gallery/verification/verification_jupyter.zip>`


.. only:: html

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

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