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

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_dimension_reduction_pod_pod_diffusion.py>`
        to download the full example code or to run this example in your browser via Binder

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

.. _sphx_glr_auto_examples_dimension_reduction_pod_pod_diffusion.py:


POD on Diffusion Equation 2D
============================

.. GENERATED FROM PYTHON SOURCE LINES 8-29

In this example, the diffusion equation is solved and then methods from the POD class are used to decompose the output
solutions/dataset and extract its basis functions which can be used for the reconstruction of the solution.

2D Diffusion equation
---------------------
:math:`\frac{\partial U}{\partial t}=D\bigg(\frac{\partial^2 U}{\partial x^2}+\frac{\partial^2 U}{\partial y^2}\bigg)`

where :math:`D` is the diffusion coefficient. :math:`U` describes the behavior of the particles in Brownian motion,
resulting from their random movements and collisions.


Problem description:
~~~~~~~~~~~~~~~~~~~~

- A 2D metal plate is initially at temperature :math:`T_{cool}`.

- A disc of a specified size inside the plate is at temperature :math:`T_{hot}`.

- Suppose that the edges of the plate are held fixed at :math:`T_{cool}`.

- The diffusion equation is applied to follow the evolution of the temperature of the plate.

.. GENERATED FROM PYTHON SOURCE LINES 34-36

Import the necessary libraries. Here we import standard libraries such as numpy, matplotlib, and we also import the
POD class from UQpy.

.. GENERATED FROM PYTHON SOURCE LINES 39-46

.. code-block:: default


    import numpy as np
    import matplotlib.pyplot as plt
    from UQpy.dimension_reduction import DirectPOD, SnapshotPOD, HigherOrderSVD
    from DiffusionEquation import diffusion
    import time


.. GENERATED FROM PYTHON SOURCE LINES 47-59

The diffusion equation is solved by calling the 'diffusion' function and a dataset (list) is obtained. To run this function the following need to be specified:
- :math:`w, h` - Plate size, :math:`(mm)`

- :math:`dx, dy` - Intervals in :math:`x-, y-` directions, :math:`(mm)`

- :math:`D` - Thermal diffusivity, :math:`(mm^2/s)`

- :math:`T_{cool}, T_{hot}` - Plate and disc temperature

- :math:`r, cx, cy` - Initial conditions - ring of inner radius :math:`r`, width :math:`dr` centered at :math:`(cx,cy) (mm)`

- `nsteps` - Number of time steps

.. GENERATED FROM PYTHON SOURCE LINES 62-72

.. code-block:: default


    w = h = 5.
    dx, dy = 0.1, 0.1
    D = 5
    Tcool, Thot = 400, 700
    r, cx, cy = 1.5, 2.5, 2.5
    nsteps = 500

    Data = diffusion(w, h, dx, dy, D, Tcool, Thot, r, cx, cy, nsteps)


.. GENERATED FROM PYTHON SOURCE LINES 73-76

The Direct POD method is used to reconstruct the data for different values of spatial modes. Full reconstruction is
achieved when the number of modes chosen equals the number of dimensions. Since the dataset for this problem is large,
the Snapshot POD is recommended as the decomposition will be performed much faster.

.. GENERATED FROM PYTHON SOURCE LINES 79-98

.. code-block:: default


    start_time = time.time()

    n_modes = [1, 2, 3, 4, 40, 50]
    frame = 40

    Data_modes = []

    for n_mode in n_modes:
        pod = DirectPOD(solution_snapshots=Data, n_modes=n_mode)
        Data_reconstr = pod.reconstructed_solution
        Data_reduced = pod.reduced_solution
        Data_modes.append(Data_reconstr[:, :, frame])
        del Data_reconstr

    elapsed_time = time.time() - start_time
    time.strftime("%H:%M:%S", time.gmtime(elapsed_time))
    print('Elapsed time: ', elapsed_time)


.. GENERATED FROM PYTHON SOURCE LINES 99-100

Comparison of input and reduced solution.

.. GENERATED FROM PYTHON SOURCE LINES 103-126

.. code-block:: default


    # Plot input solution
    plt.figure()
    c = plt.imshow(Data[frame], cmap=plt.get_cmap('coolwarm'), vmin=Tcool, vmax=Thot)
    plt.colorbar(c)
    plt.title('Input solution', fontweight="bold", size=15)
    plt.show()

    # Plot reduced solution
    fig = plt.figure(figsize=(10, 7))
    fig.subplots_adjust(hspace=0.5, wspace=0.4)
    for i in range(len(n_modes)):
        ax = fig.add_subplot(2, 3, i + 1)
        im = ax.imshow(Data_modes[i], cmap=plt.get_cmap('coolwarm'), vmin=Tcool, vmax=Thot)
        ax.set_axis_off()
        ax.set_title('Mode {}'.format(n_modes[i]), size=15)

    fig.subplots_adjust(right=0.85)
    cbar_ax = fig.add_axes([0.9, 0.15, 0.03, 0.7])
    cbar_ax.set_xlabel('$T$ / K', labelpad=20)
    fig.colorbar(im, cax=cbar_ax)
    fig.suptitle('Reconstructed solution', fontweight="bold", size=15)
    plt.show()


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

   **Total running time of the script:** ( 0 minutes  0.000 seconds)


.. _sphx_glr_download_auto_examples_dimension_reduction_pod_pod_diffusion.py:

.. only:: html

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


    .. container:: binder-badge

      .. image:: images/binder_badge_logo.svg
        :target: https://mybinder.org/v2/gh/SURGroup/UQpy/master?urlpath=lab/tree/notebooks/auto_examples/dimension_reduction/pod/pod_diffusion.ipynb
        :alt: Launch binder
        :width: 150 px



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

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

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

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


.. only:: html

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

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