.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/ex10_poisson-equation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_examples_ex10_poisson-equation.py>`
        to download the full example code.

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

.. _sphx_glr_examples_ex10_poisson-equation.py:


Poisson Equation
----------------
The `Poisson equation <https://en.wikipedia.org/wiki/Poisson%27s_equation>`_ with fixed
boundaries on the bottom, top, left and right end-edges and a unit load, as given
in Eq. :eq:`poisson` and Eq. :eq:`poisson-boundaries`, is solved on a rectangle.

.. math::
   :label: poisson

   \text{div}(\boldsymbol{\nabla} u) + f = 0 \quad \text{in} \quad \Omega

.. math::
   :label: poisson-boundaries

   u &= 0 \quad \text{on} \quad \Gamma_u

   f &= 1 \quad \text{in} \quad \Omega

The Poisson equation is transformed into integral form representation by the
`divergence (Gauss's) theorem <https://en.wikipedia.org/wiki/Divergence_theorem>`_, see
Eq. :eq:`poisson-integral-form`.

.. math::
   :label: poisson-integral-form

   \int_\Omega \boldsymbol{\nabla} (\delta u) \cdot \boldsymbol{\nabla} (\Delta u)
       \ d\Omega = \int_\Omega  \delta u \cdot f \ d\Omega

.. GENERATED FROM PYTHON SOURCE LINES 31-55







.. tab-set::



   .. tab-item:: Static Scene



            
     .. image-sg:: /examples/images/sphx_glr_ex10_poisson-equation_001.png
        :alt: ex10 poisson equation
        :srcset: /examples/images/sphx_glr_ex10_poisson-equation_001.png
        :class: sphx-glr-single-img
     


   .. tab-item:: Interactive Scene



       .. offlineviewer:: /home/docs/checkouts/readthedocs.org/user_builds/felupe-ja/checkouts/latest/felupe/docs/examples/images/sphx_glr_ex10_poisson-equation_001.vtksz






.. tab-set::



   .. tab-item:: Static Scene



            
     .. image-sg:: /examples/images/sphx_glr_ex10_poisson-equation_002.png
        :alt: ex10 poisson equation
        :srcset: /examples/images/sphx_glr_ex10_poisson-equation_002.png
        :class: sphx-glr-single-img
     


   .. tab-item:: Interactive Scene



       .. offlineviewer:: /home/docs/checkouts/readthedocs.org/user_builds/felupe-ja/checkouts/latest/felupe/docs/examples/images/sphx_glr_ex10_poisson-equation_002.vtksz






.. code-block:: Python


    import felupe as fem

    mesh = fem.Rectangle(n=2**5).triangulate()
    region = fem.RegionTriangle(mesh)
    u = fem.Field(region, dim=1)
    field = fem.FieldContainer([u])

    boundaries = fem.BoundaryDict(
        bottom=fem.Boundary(u, fy=0),
        top=fem.Boundary(u, fy=1),
        left=fem.Boundary(u, fx=0),
        right=fem.Boundary(u, fx=1),
    )
    boundaries.plot(show_lines=False).show()

    solid = fem.SolidBody(umat=fem.Laplace(), field=field)
    load = fem.SolidBodyForce(field=field, values=1.0)

    step = fem.Step([solid, load], boundaries=boundaries)
    job = fem.Job([step]).evaluate()

    view = mesh.view(point_data={"Field": u.values})
    view.plot("Field").show()


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

   **Total running time of the script:** (0 minutes 0.493 seconds)


.. _sphx_glr_download_examples_ex10_poisson-equation.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: ex10_poisson-equation.ipynb <ex10_poisson-equation.ipynb>`

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

      :download:`Download Python source code: ex10_poisson-equation.py <ex10_poisson-equation.py>`

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

      :download:`Download zipped: ex10_poisson-equation.zip <ex10_poisson-equation.zip>`


.. only:: html

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

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