.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/WeakForms/LinearElasticity1.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_examples_WeakForms_LinearElasticity1.py>`
        to download the full example code.

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

.. _sphx_glr_examples_WeakForms_LinearElasticity1.py:


LinearElasticity1
=================

A cantilever beam undergoing bending deformation.

Note that this simulation is also performed in `examples/Elastic/Elas1.py`.

.. GENERATED FROM PYTHON SOURCE LINES 14-113



.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /examples/WeakForms/images/sphx_glr_LinearElasticity1_001.png
         :alt: $uy$
         :srcset: /examples/WeakForms/images/sphx_glr_LinearElasticity1_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/WeakForms/images/sphx_glr_LinearElasticity1_002.png
         :alt: LinearElasticity1
         :srcset: /examples/WeakForms/images/sphx_glr_LinearElasticity1_002.png
         :class: sphx-glr-multi-img





.. code-block:: Python
   :lineno-start: 15


    import matplotlib.pyplot as plt
    import numpy as np

    from EasyFEA import Display, ElemType, Models, Simulations
    from EasyFEA.FEM import Field, BiLinearForm, FeArray, Sym_Grad, Trace
    from EasyFEA.Geoms import Domain

    if __name__ == "__main__":
        Display.Clear()

        # ----------------------------------------------
        # Configuration
        # ----------------------------------------------

        dim = 2

        # geom
        L = 120  # mm
        h = 13

        # load
        F = -800  # N

        # model
        E = 210000  # MPa
        v = 0.3

        elastic = Models.Elastic.Isotropic(dim, E, v, planeStress=True, thickness=h)
        lmbda = elastic.get_lambda()
        mu = elastic.get_mu()

        # ----------------------------------------------
        # Mesh
        # ----------------------------------------------

        contour = Domain((0, 0), (L, h), h / 3)

        if dim == 2:
            mesh = contour.Mesh_2D([], ElemType.QUAD9, isOrganised=True)
        else:
            mesh = contour.Mesh_Extrude(
                [], [0, 0, h], [3], ElemType.HEXA27, isOrganised=True
            )

        nodesX0 = mesh.Nodes_Conditions(lambda x, y, z: x == 0)
        nodesXL = mesh.Nodes_Conditions(lambda x, y, z: x == L)

        # ----------------------------------------------
        # Formulations
        # ----------------------------------------------

        field = Field(mesh.groupElem, dim)

        def S(u: Field) -> FeArray:
            Eps = Sym_Grad(u)
            return 2 * mu * Eps + lmbda * Trace(Eps) * np.eye(dim)

        @BiLinearForm
        def ComputeK(u: Field, v: Field):
            Sig = S(u)
            Eps = Sym_Grad(v)
            return Sig.ddot(Eps)

        weakForms = Models.WeakForms(field, ComputeK)

        # ----------------------------------------------
        # Simulations
        # ----------------------------------------------

        simu = Simulations.WeakForms(mesh, weakForms)

        simu.add_dirichlet(nodesX0, [0] * dim, simu.Get_unknowns())
        simu.add_surfLoad(nodesXL, [F / h**2], ["y"])

        simu.Solve()

        # ----------------------------------------------
        # Results
        # ----------------------------------------------

        Display.Plot_Result(simu, "uy", 10, plotMesh=True)

        def von_mises_stress(u):
            Stress = S(u)
            if dim == 2:
                xx = Stress[..., 0, 0]
                yy = Stress[..., 1, 1]
                xy = Stress[..., 0, 1]
                return np.sqrt(xx**2 + yy**2 - xx * yy + 3 * xy**2)
            else:
                Stress_d = Stress - 1 / 3 * Trace(Stress) * np.eye(3)
                return np.sqrt(3 / 2 * Stress_d.ddot(Stress_d))

        Svm_e = field.Evaluate_e(von_mises_stress, simu.u)

        Display.Plot_Result(simu, Svm_e, plotMesh=True, ncolors=11)

        plt.show()


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

   **Total running time of the script:** (0 minutes 0.206 seconds)


.. _sphx_glr_download_examples_WeakForms_LinearElasticity1.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: LinearElasticity1.ipynb <LinearElasticity1.ipynb>`

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

      :download:`Download Python source code: LinearElasticity1.py <LinearElasticity1.py>`

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

      :download:`Download zipped: LinearElasticity1.zip <LinearElasticity1.zip>`


.. only:: html

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

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