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FORM

In FORM :cite:`FORM_XDu`, the performance function is linearized according to

G(\textbf{U})  \approx  G(\textbf{U}^\star) + \nabla G(\textbf{U}^\star)(\textbf{U}-\textbf{U}^\star)^\intercal

where \textbf{U}^\star is the expansion point, G(\textbf{U}) is the performance function evaluated in the standard normal space and \nabla G(\textbf{U}^\star) is the gradient of G(\textbf{U}) evaluated at \textbf{U}^\star. The probability failure can be calculated by

P_{f, \text{form}} = \Phi(-\beta_{HL})

where \Phi(\cdot) is the standard normal cumulative distribution function and \beta_{HL}=||\textbf{U}^*|| is the norm of the design point known as the Hasofer-Lind reliability index calculated with the iterative Hasofer-Lind-Rackwitz-Fiessler (HLRF) algorithm. The convergence criteria used for HLRF algorithm are:

\text{tolerance}_{\textbf{U}}:\ ||\textbf{U}^{k} - \textbf{U}^{k-1}||_2 \leq 10^{-3}

\text{tolerance}_\beta:\ ||\beta_{HL}^{k} - \beta_{HL}^{k-1}||_2 \leq 10^{-3}

\text{tolerance}_{\nabla G(\textbf{U})}:\ ||\nabla G(\textbf{U}^{k})- \nabla G(\textbf{U}^{k-1})||_2 \leq 10^{-3}

The :class:`.FORM` class is imported using the following command:

>>> from UQpy.reliability.taylor_series.FORM import FORM

Methods

.. autoclass:: UQpy.reliability.taylor_series.FORM
    :members: run

Attributes

.. autoattribute:: UQpy.reliability.taylor_series.FORM.alpha


.. autoattribute:: UQpy.reliability.taylor_series.FORM.alpha_record


.. autoattribute:: UQpy.reliability.taylor_series.FORM.beta


.. autoattribute:: UQpy.reliability.taylor_series.FORM.beta_record


.. autoattribute:: UQpy.reliability.taylor_series.FORM.design_point_u


.. autoattribute:: UQpy.reliability.taylor_series.FORM.design_point_x


.. autoattribute:: UQpy.reliability.taylor_series.FORM.error_record


.. autoattribute:: UQpy.reliability.taylor_series.FORM.failure_probability


.. autoattribute:: UQpy.reliability.taylor_series.FORM.iterations


.. autoattribute:: UQpy.reliability.taylor_series.FORM.state_function_record


.. autoattribute:: UQpy.reliability.taylor_series.FORM.state_function_gradient_record


.. autoattribute:: UQpy.reliability.taylor_series.FORM.u_record


.. autoattribute:: UQpy.reliability.taylor_series.FORM.x_record

Examples

.. toctree::

   FORM Examples <../auto_examples/reliability/form/index>