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

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

.. _sphx_glr_auto_examples_plot_inverse_form_cantilever_beam.py:


Inverse FORM with a cantilever beam
===================================

.. GENERATED FROM PYTHON SOURCE LINES 7-24

This example uses the cantilever beam model from the OpenTURNS use cases
to demonstrate the :class:`~otrobopt.InverseFORM` algorithm.

The beam tip displacement is given by:

.. math::

    d = \frac{F L^3}{3 E I}

where :math:`E` is Young's modulus, :math:`F` is the load,
:math:`L` is the length, and :math:`I` is the moment of inertia.

We treat the length :math:`L` as a parameter to calibrate. Given a
target reliability index :math:`\beta_t`, we seek the beam length
such that the Hasofer-Lind reliability index of the failure event
:math:`\{d > d_0\}` (displacement exceeding a threshold) equals
:math:`\beta_t`.

.. GENERATED FROM PYTHON SOURCE LINES 26-36

.. code-block:: Python

    import openturns as ot
    import otrobopt
    from openturns.usecases import cantilever_beam

    ot.RandomGenerator.SetSeed(0)

    # Load the cantilever beam use case
    cb = cantilever_beam.CantileverBeam()
    print(cb.model)





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

 .. code-block:: none

    [E,F,L,I]->[F*L^3/(3*E*I)]




.. GENERATED FROM PYTHON SOURCE LINES 37-39

Freeze the beam length L (input index 2) as a parameter with initial value 2.55 m.
The remaining random inputs are E, F, I.

.. GENERATED FROM PYTHON SOURCE LINES 39-48

.. code-block:: Python

    g = ot.ParametricFunction(cb.model, [2], [2.55])

    # Build the joint distribution of the random inputs (E, F, I)
    marginals = [cb.E, cb.F, cb.II]
    distribution = ot.JointDistribution(marginals)

    # Starting point: median values of the random inputs
    x0 = [dist.computeQuantile(0.5)[0] for dist in marginals]








.. GENERATED FROM PYTHON SOURCE LINES 49-50

Define the failure event: displacement > 0.15 m.

.. GENERATED FROM PYTHON SOURCE LINES 50-54

.. code-block:: Python

    vect = ot.RandomVector(distribution)
    output = ot.CompositeRandomVector(g, vect)
    event = ot.ThresholdEvent(output, ot.Greater(), 0.15)








.. GENERATED FROM PYTHON SOURCE LINES 55-57

Run the inverse FORM algorithm with a target reliability index
:math:`\beta_t = 3.0`.

.. GENERATED FROM PYTHON SOURCE LINES 57-66

.. code-block:: Python

    algo = otrobopt.InverseFORM(event, 'L', x0)
    algo.setTargetBeta(3.0)
    algo.run()

    result = algo.getResult()
    print('Calibrated L =', result.getParameter())
    print('Hasofer-Lind beta =', result.getHasoferReliabilityIndex())
    print('Convergence criteria =', result.getConvergenceCriteria())





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

 .. code-block:: none

    Calibrated L = [2.18001]
    Hasofer-Lind beta = 2.9999999999999996
    Convergence criteria = [4,0,1.83978e-05,4.44089e-16]




.. GENERATED FROM PYTHON SOURCE LINES 67-72

The calibrated length :math:`L^*` is the beam length that achieves
the target reliability index :math:`\beta_t = 3.0` for the given
failure threshold. If the initial length leads to a reliability
above the target, the algorithm increases it (making the beam more
flexible and thus less reliable), and vice versa.


.. _sphx_glr_download_auto_examples_plot_inverse_form_cantilever_beam.py:

.. only:: html

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

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

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

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

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

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

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