.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/sensitivity/sobol/mechanical_oscillator_ODE.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_sensitivity_sobol_mechanical_oscillator_ODE.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_sensitivity_sobol_mechanical_oscillator_ODE.py:


Mechanical oscillator model (multioutput)
==============================================

In this example, we consider the mechanical oscillator is governed by the following 
second-order ODE as demonstrated in [1]_:

.. math::
    m \ddot{x} + c \dot{x} + k x = 0

.. math::
    x(0) = \ell, \dot{x}(0) = 0.

The parameteres of the oscillator are modeled as follows:

.. math::
    m \sim \mathcal{U}(10, 12), c \sim \mathcal{U}(0.4, 0.8), k \sim \mathcal{U}(70, 90), \ell \sim \mathcal{U}(-1, -0.25).

Here, we compute the Sobol indices for each point in time and are called 
pointwise-in-time Sobol indices. These indices describe the sensitivity of the model 
parameters at each point in time.

.. [1] Gamboa, F., Janon, A., Klein, T., & Lagnoux, A. (2014). Sensitivity analysis for multidimensional and functional outputs. Electronic Journal of Statistics, 8(1), 575-603.

.. GENERATED FROM PYTHON SOURCE LINES 29-38

.. code-block:: default

    import numpy as np
    import matplotlib.pyplot as plt

    from UQpy.run_model.RunModel import RunModel
    from UQpy.run_model.model_execution.PythonModel import PythonModel
    from UQpy.distributions import Uniform
    from UQpy.distributions.collection.JointIndependent import JointIndependent
    from UQpy.sensitivity.SobolSensitivity import SobolSensitivity


.. GENERATED FROM PYTHON SOURCE LINES 39-40

**Define the model and input distributions**

.. GENERATED FROM PYTHON SOURCE LINES 40-58

.. code-block:: default


    # Create Model object
    model = PythonModel(
        model_script="local_mechanical_oscillator_ODE.py",
        model_object_name="mech_oscillator",
        var_names=[r"$m$", "$c$", "$k$", "$\ell$"],
        delete_files=True,
    )

    runmodel_obj = RunModel(model=model)

    # Define distribution object
    M = Uniform(10, (12 - 10))
    C = Uniform(0.4, (0.8 - 0.4))
    K = Uniform(70, (90 - 70))
    L = Uniform(-1, (-0.25 - -1))
    dist_object = JointIndependent([M, C, K, L])


.. GENERATED FROM PYTHON SOURCE LINES 59-60

**Compute Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 62-66

.. code-block:: default

    SA = SobolSensitivity(runmodel_obj, dist_object)

    SA.run(n_samples=500)


.. GENERATED FROM PYTHON SOURCE LINES 67-68

**Plot the Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 68-104

.. code-block:: default


    t_0 = 0
    t_f = 40
    dt = 0.05
    n_t = int((t_f - t_0) / dt)
    T = np.linspace(t_0, t_f, n_t)

    fig, ax = plt.subplots(1, 2, figsize=(16, 8))

    ax[0].plot(T, SA.total_order_indices[0, :], "r", label=r"$m$")
    ax[0].plot(T, SA.total_order_indices[1, :], "g", label=r"$c$")
    ax[0].plot(T, SA.total_order_indices[2, :], label=r"$k$", color="royalblue")
    ax[0].plot(T, SA.total_order_indices[3, :], label=r"$\ell$", color="aquamarine")

    ax[0].set_title("Total order Sobol indices", fontsize=16)
    ax[0].set_xlabel("time (s)", fontsize=16)
    ax[0].set_ylabel(r"$S_{T_i}$", fontsize=16)
    ax[0].set_xbound(0, t_f)
    ax[0].set_ybound(-0.2, 1.2)
    ax[0].legend()

    ax[1].plot(T, SA.first_order_indices[0, :], "r", label=r"$m$")
    ax[1].plot(T, SA.first_order_indices[1, :], "g", label=r"$c$")
    ax[1].plot(T, SA.first_order_indices[2, :], label=r"$k$", color="royalblue")
    ax[1].plot(T, SA.first_order_indices[3, :], label=r"$\ell$", color="aquamarine")

    ax[1].set_title("First order Sobol indices", fontsize=16)
    ax[1].set_xlabel("time (s)", fontsize=16)
    ax[1].set_ylabel(r"$S_i$", fontsize=16)
    ax[1].set_xbound(0, t_f)
    ax[1].set_ybound(-0.2, 1.2)
    ax[1].legend(fontsize=12)

    fig.suptitle("Pointwise-in-time Sobol indices", fontsize=20)

    plt.show()


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

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


.. _sphx_glr_download_auto_examples_sensitivity_sobol_mechanical_oscillator_ODE.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/sensitivity/sobol/mechanical_oscillator_ODE.ipynb
        :alt: Launch binder
        :width: 150 px



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

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

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

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


.. only:: html

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

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