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.. _sphx_glr_auto_examples_reliability_inverse_form_inverse_form_cantilever.py:


Inverse FORM - Cantilever Beam
-----------------------------------
The following example is Example 7.2 from Chapter 7 of :cite:`FORM_XDu`.

A cantilever beam is considered to fail if the displacement at the tip exceeds the threshold :math:`D_0`.
The performance function :math:`G(\textbf{U})` of this problem is given by

.. math:: G = D_0 - \frac{4L^3}{Ewt} \sqrt{ \left(\frac{P_x}{w^2}\right)^2 + \left(\frac{P_y}{t^2}\right)^2}

Where the external forces are modeled as random variables :math:`P_x \sim N(500, 100)` and :math:`P_y \sim N(1000,100)`.
The constants in the problem are length (:math:`L=100`), elastic modulus (:math:`E=30\times 10^6`), cross section width
(:math:`w=2`), cross section height (:math:`t=4`), and :math:`D_0=3`.

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First, we import the necessary modules.

.. GENERATED FROM PYTHON SOURCE LINES 21-29

.. code-block:: default


    import numpy as np
    from scipy import stats
    from UQpy.distributions import Normal
    from UQpy.reliability.taylor_series import InverseFORM
    from UQpy.run_model.RunModel import RunModel
    from UQpy.run_model.model_execution.PythonModel import PythonModel


.. GENERATED FROM PYTHON SOURCE LINES 30-33

Next, we initialize the :code:`RunModel` object.
The file defining the performance function file can be found on the UQpy GitHub.
It contains a function :code:`cantilever_beam` to compute the performance function :math:`G(\textbf{U})`.

.. GENERATED FROM PYTHON SOURCE LINES 36-40

.. code-block:: default


    model = PythonModel(model_script='local_pfn.py', model_object_name="cantilever_beam")
    runmodel_object = RunModel(model=model)


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Next, we define the external forces in the :math:`x` and :math:`y` direction as distributions that will be passed into
:code:`FORM`. Along with the distributions, :code:`FORM` takes in the previously defined :code:`runmodel_object`,
the specified probability of failure, and the tolerances. These tolerances are smaller than the defaults to ensure
convergence with the level of accuracy given in the problem.

.. GENERATED FROM PYTHON SOURCE LINES 48-57

.. code-block:: default


    p_fail = 0.04054
    distributions = [Normal(500, 100), Normal(1_000, 100)]
    inverse_form = InverseFORM(distributions=distributions,
                               runmodel_object=runmodel_object,
                               p_fail=p_fail,
                               tolerance_u=1e-5,
                               tolerance_gradient=1e-5)


.. GENERATED FROM PYTHON SOURCE LINES 58-63

With everything defined we are ready to run the inverse first-order reliability method and print the results.
The solution to this problem given by Du is :math:`\textbf{U}^*=(1.7367, 0.16376)` with a reliability index of
:math:`\beta_{HL}=||\textbf{U}^*||=1.7444` and probability of failure of
:math:`p_{fail} = \Phi(-\beta_{HL})=0.04054`. We expect this problem to converge in 4 iterations.
We confirm our design point matches this length, and therefore has a probability of failure specified by our input.

.. GENERATED FROM PYTHON SOURCE LINES 66-74

.. code-block:: default


    inverse_form.run()
    beta = np.linalg.norm(inverse_form.design_point_u)
    print('Design point in standard normal space (u^*):', inverse_form.design_point_u[0])
    print('Design point in original space:', inverse_form.design_point_x[0])
    print('Hasofer-Lind reliability index:', beta)
    print('Probability of failure at design point:', stats.norm.cdf(-beta))
    print('Number of iterations:', inverse_form.iteration_record[0])


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