.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery/elements/combin14/example_combin14.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_combin14_example_combin14.py>`
        to download the full example code.

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

.. _sphx_glr_gallery_elements_combin14_example_combin14.py:


.. _ref_combin14_example:

SPRING — two springs in series
================================

Three co-linear nodes joined by two SPRING springs of stiffness
``k1`` and ``k2``. With the far end fixed and a tip load ``F`` the
free-end displacement is ``F / k_eq`` where
``k_eq = k1 · k2 / (k1 + k2)``.

.. GENERATED FROM PYTHON SOURCE LINES 12-22

.. code-block:: Python


    from __future__ import annotations

    import numpy as np
    import pyvista as pv
    from vtkmodules.util.vtkConstants import VTK_LINE

    import femorph_solver
    from femorph_solver import ELEMENTS








.. GENERATED FROM PYTHON SOURCE LINES 23-27

Problem data
------------
Two springs. ``k1`` at half the stiffness of ``k2``; their series
combination sits in between.

.. GENERATED FROM PYTHON SOURCE LINES 27-33

.. code-block:: Python

    k1 = 1000.0  # N/m
    k2 = 2000.0  # N/m
    F = 50.0  # N tip load

    k_eq = k1 * k2 / (k1 + k2)








.. GENERATED FROM PYTHON SOURCE LINES 34-39

Build the model
---------------
Three nodes along x. SPRING only carries an axial spring force; the
transverse DOFs have zero stiffness and the solver's zero-pivot guard
pins them automatically.

.. GENERATED FROM PYTHON SOURCE LINES 39-62

.. code-block:: Python

    points = np.array(
        [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [2.0, 0.0, 0.0]],
        dtype=np.float64,
    )
    cells = np.array([2, 0, 1, 2, 1, 2], dtype=np.int64)
    cell_types = np.array([VTK_LINE, VTK_LINE], dtype=np.uint8)
    grid = pv.UnstructuredGrid(cells, cell_types, points)
    # Two real-constant ids: cell 0 uses real-id 1 (k1), cell 1 uses real-id 2 (k2).
    grid.cell_data["ansys_real_constant"] = np.array([1, 2], dtype=np.int32)

    m = femorph_solver.Model.from_grid(grid)
    m.assign(ELEMENTS.SPRING, real=(k1,), real_id=1)
    m.assign(ELEMENTS.SPRING, real=(k2,), real_id=2)

    m.fix(nodes=[1], dof="ALL")  # clamp node 1
    # Restrain transverse translations on the free nodes so the DOFs the
    # solver actually keeps are just UX at nodes 2 and 3.
    for nn in (2, 3):
        m.fix(nodes=[nn], dof="UY")
        m.fix(nodes=[nn], dof="UZ")

    m.apply_force(3, fx=F)








.. GENERATED FROM PYTHON SOURCE LINES 63-69

Static solve + analytical comparison
------------------------------------
With both springs in equilibrium the tip displacement must equal
``F / k_eq``. Because the springs are in series the force is the
same in each and we can check the intermediate displacement too
(``u2 = F / k1``).

.. GENERATED FROM PYTHON SOURCE LINES 69-81

.. code-block:: Python

    res = m.solve()
    dof = m.dof_map()

    u2 = res.displacement[np.where((dof[:, 0] == 2) & (dof[:, 1] == 0))[0][0]]
    u3 = res.displacement[np.where((dof[:, 0] == 3) & (dof[:, 1] == 0))[0][0]]

    print(f"u at node 2 = {u2:.6e} m (expected {F / k1:.6e})")
    print(f"u at node 3 = {u3:.6e} m (expected {F / k_eq:.6e})")

    assert np.isclose(u2, F / k1, rtol=1e-12)
    assert np.isclose(u3, F / k_eq, rtol=1e-12)





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

 .. code-block:: none

    u at node 2 = 5.000000e-02 m (expected 5.000000e-02)
    u at node 3 = 7.500000e-02 m (expected 7.500000e-02)




.. GENERATED FROM PYTHON SOURCE LINES 82-87

Visualise the deformation
-------------------------
Build a point_data displacement vector and warp the mesh. Because the
real deflection is ~7.5 cm on a 2 m baseline a modest exaggeration
makes the difference between the two springs visible.

.. GENERATED FROM PYTHON SOURCE LINES 87-108

.. code-block:: Python

    grid = m.grid.copy()
    displacement = np.zeros((grid.n_points, 3), dtype=np.float64)
    for i, nn in enumerate(grid.point_data["ansys_node_num"]):
        rows = np.where(dof[:, 0] == int(nn))[0]
        for r in rows:
            displacement[i, int(dof[r, 1])] = res.displacement[r]
    grid.point_data["displacement"] = displacement

    warped = grid.warp_by_vector("displacement", factor=1.0)
    plotter = pv.Plotter(off_screen=True)
    plotter.add_mesh(grid, style="wireframe", color="gray", line_width=3)
    plotter.add_mesh(
        warped,
        scalars=np.linalg.norm(displacement, axis=1),
        line_width=6,
        render_points_as_spheres=True,
        point_size=14,
        scalar_bar_args={"title": "|u| [m]"},
    )
    plotter.add_axes()
    plotter.show()



.. image-sg:: /gallery/elements/combin14/images/sphx_glr_example_combin14_001.png
   :alt: example combin14
   :srcset: /gallery/elements/combin14/images/sphx_glr_example_combin14_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.169 seconds)


.. _sphx_glr_download_gallery_elements_combin14_example_combin14.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: example_combin14.ipynb <example_combin14.ipynb>`

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

      :download:`Download Python source code: example_combin14.py <example_combin14.py>`

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

      :download:`Download zipped: example_combin14.zip <example_combin14.zip>`


.. only:: html

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

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