:orphan:

Analyses
========

Runnable examples organised by the *kind of analysis* rather than
which element happens to be used.  Each sub-gallery walks through the
full loop — build the model, solve, interpret the result — for one
analysis family.

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

    .. grid-item-card:: Static
        :link: static/index
        :link-type: doc

        :math:`K u = F`.  Cantilever deflections, plane-stress /
        plane-strain problems, single-element patch tests.

    .. grid-item-card:: Modal
        :link: modal/index
        :link-type: doc

        :math:`K \varphi = \omega^2 M \varphi`.  Plate modes, SDOF
        oscillators, beam cantilever frequencies.

    .. grid-item-card:: Transient
        :link: transient/index
        :link-type: doc

        :math:`M \ddot u + C \dot u + K u = F(t)` via Newmark-β.

    .. grid-item-card:: Cyclic
        :link: cyclic/index
        :link-type: doc

        Cyclic-symmetry modal on a base sector with phase constraints.


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


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Cyclic-symmetry modal
=====================

Modal analysis of rotationally periodic structures — compute the full
rotor's spectrum from a single base sector plus phase constraints.


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


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    <div class="sphx-glr-thumbcontainer" tooltip="A rotor with N identical sectors spanning 360° has a spectrum that factors by harmonic index k \in \{0, 1, \dots, N/2\}. Analysing one base sector plus the constraint">

.. only:: html

  .. image:: /gallery/analyses/cyclic/images/thumb/sphx_glr_example_rotor_sector_thumb.png
    :alt:

  :doc:`/gallery/analyses/cyclic/example_rotor_sector`

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      <div class="sphx-glr-thumbnail-title">Cyclic-symmetry modal on a rotor sector</div>
    </div>


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    </div>

Modal analysis
==============

Free-vibration modes — :math:`K \varphi = \omega^2 M \varphi`.  Plate
modes, SDOF oscillators, beam cantilever frequencies.


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    <div class="sphx-glr-thumbcontainer" tooltip="End-to-end modal-analysis example: build a 1 m × 1 m × 10 mm steel plate as a 2-through-thickness, 40 × 40 in-plane hex mesh in pyvista, wrap it in femorph_solver.Model, clamp the x = 0 edge, and extract the first 10 modes with femorph_solver.Model.modal_solve.">

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  .. image:: /gallery/analyses/modal/images/thumb/sphx_glr_example_plate_modes_thumb.png
    :alt:

  :doc:`/gallery/analyses/modal/example_plate_modes`

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      <div class="sphx-glr-thumbnail-title">SOLID185 — cantilever-plate modal (2 × 40 × 40 hex mesh)</div>
    </div>


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    </div>

Static analysis
===============

:math:`K u = F` with Dirichlet BCs.  The workhorse analysis — one
factorisation, one back-solve.


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


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    <div class="sphx-glr-thumbcontainer" tooltip="Static analysis of a cantilever plate under a distributed tip load. This example walks through the full femorph_solver.Model.solve → StaticResult → femorph_solver.io.static_result_to_grid → pyvista rendering loop and checks static equilibrium via StaticResult.reaction.">

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  .. image:: /gallery/analyses/static/images/thumb/sphx_glr_example_cantilever_plate_thumb.png
    :alt:

  :doc:`/gallery/analyses/static/example_cantilever_plate`

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      <div class="sphx-glr-thumbnail-title">SOLID185 — cantilever plate static analysis</div>
    </div>


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Transient analysis
==================

Time-domain simulations of the equation of motion
:math:`M \ddot u + C \dot u + K u = F(t)` using Newmark-β.


.. raw:: html

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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="A single-DOF mass-spring-damper subjected to a step load shows off the Newmark-β transient integrator in solve_transient.  We integrate for long enough to watch the underdamped oscillation settle toward the new static equilibrium, then compare the result to the textbook second-order response.">

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  .. image:: /gallery/analyses/transient/images/thumb/sphx_glr_example_sdof_transient_thumb.png
    :alt:

  :doc:`/gallery/analyses/transient/example_sdof_transient`

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      <div class="sphx-glr-thumbnail-title">SDOF transient — step-load response</div>
    </div>


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.. toctree::
   :hidden:
   :includehidden:


   /gallery/analyses/cyclic/index.rst
   /gallery/analyses/modal/index.rst
   /gallery/analyses/static/index.rst
   /gallery/analyses/transient/index.rst


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

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

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

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

      :download:`Download all examples in Jupyter notebooks: analyses_jupyter.zip </gallery/analyses/analyses_jupyter.zip>`


.. only:: html

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

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