.. _sorm_approximation:
SORM
----
The Second Order Reliability Method is used in the same context as the
First Order Reliability: refer to for further details. The objective of
SORM is to evaluate the probability content of the event
:math:`\cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\}`
:
.. math::
:label: PfX4
P_f = \Prob{g(\vect{X}\,,\,\vect{d})\leq 0} = \int_{\cD_f} \pdf\, d\vect{x}
| The principle is the same as for FORM. After having mapped the
physical space into the standard through an isoprobabilistic
transformation :eq:`PfX4` becomes:
.. math::
:label: PfU2
P_f = \Prob{G(\vect{U}\,,\,\vect{d})\leq 0} = \int_{\Rset^n} \boldsymbol{1}_{G(\vect{u}\,,\,\vect{d}) \leq 0}\,f_{\vect{U}}(\vect{u})\,d\vect{u}
where :math:`f_{\vect{U}}` is the density function of the
distribution in the standard space : that distribution is spherical
(invariant by rotation by definition). That property implies that
:math:`f_{\vect{U}}` is a function of :math:`||\vect{U}||^2` only.
Furthermore, we suppose that outside the sphere which tangents the
limit state surface in the standard space, :math:`f_{\vect{U}}` is
decreasing.
| The difference with FORM comes from the approximation of the limit
state surface at the design point :math:`\vect{P}^*` in the standard
space: SORM approximates it by a quadratic surface that has the same
main curvatures at the design point.
| Let us denote by :math:`n` the dimension of the random vector
:math:`\vect{X}` and :math:`(\kappa_i)_{1 \leq i \leq n-1}` the
:math:`n-1` main curvatures of the limit state function at the design
point in the standard space.
| Several approximations are available,
detailed here in the case where the origin of the standard
space does not belong to the failure domain :
- Breitung’s formula is an asymptotic results: the
usual formula used in the normal standard space, has been generalized
in [lebrun2009b]_ to standard spaces where the
distribution is spherical, with :math:`E` the marginal cumulative
density function of the spherical distributions in the standard space:
.. math::
:label: PfSORM_B
P_{Breitung}^{generalized} \stackrel{\beta\rightarrow\infty}{=} E(-\beta)\prod_{i=1}^{n-1}\frac{1}{\sqrt{1+\beta\kappa_i}}
where :math:`\Phi` is the cumulative distribution function of the
standard 1D normal distribution.
- Hohenbichler’s formula is an approximation of :eq:`PfSORM_B`:
.. math::
:label: PfSORM_HB
\displaystyle P_{Hohenbichler} = \Phi(-\beta_{HL}) \prod_{i=1}^{n-1} \left( 1+\frac{\phi(-\beta_{HL})}{\Phi(-\beta_{HL})}\kappa_i \right) ^{-1/2}
**This formula is valid only in the normal standard space and if**
:math:`\boldsymbol{\forall i, 1+\frac{\phi(-\beta_{HL})}{\Phi(-\beta_{HL})}\kappa_i > 0}`.
- | Tvedt’s formula (Tvedt, 1988):
.. math::
:label: PfSORM_T
\left\{
\begin{array}{lcl}
\displaystyle P_{Tvedt} & = & A_1 + A_2 + A_3 \\
\displaystyle A_1 & = & \displaystyle \Phi(-\beta_{HL}) \prod_{i=1}^{N-1} \left( 1+\beta_{HL} \kappa_i \right) ^{-1/2}\\
\displaystyle A_2 & = & \displaystyle\left[ \beta_{HL} \Phi(-\beta_{HL}) - \phi(\beta_{HL})\right ] \left[ \prod_{j=1}^{N-1} \left( 1+\beta_{HL} \kappa_i \right) ^{-1/2} - \prod_{j=1}^{N-1} \left( 1+(1 + \beta_{HL}) \kappa_i \right) ^{-1/2} \right ] \\
\displaystyle A_3 & = & \displaystyle(1 + \beta_{HL}) \left[ \beta_{HL} \Phi(-\beta_{HL}) - \phi(\beta_{HL})\right ] \left[ \prod_{j=1}^{N-1} \left( 1+\beta_{HL} \kappa_i \right) ^{-1/2} \right.\\
& & \displaystyle\left. - {\cR}e \left( \prod_{j=1}^{N-1} \left( 1+(i + \beta_{HL}) \kappa_j \right) ^{-1/2} \right)\right ]
\end{array}
\right.
where :math:`{\cR}e(z)` is the real part of the complex number
:math:`z` and :math:`i` the complex number such that
:math:`i^2 = -1` and :math:`\Phi` the cumulative distribution
function of the standard 1D normal distribution.
**This formula is valid only in the normal standard space and if**
:math:`\boldsymbol{\forall i, 1+\beta \kappa_i > 0}` and
:math:`\boldsymbol{\forall i, 1+(1 + \beta) \kappa_i> 0}`.
.. topic:: API:
- See :class:`~openturns.SORM`
.. topic:: Examples:
- See :doc:`/examples/reliability_sensitivity/estimate_probability_form`
.. topic:: References:
- Breitung K. a, "Asymptotic approximation for probability integral," Probability Engineering Mechanics, 1989, Vol 4, No. 4.
- Breitung K. b, 1984, "Asymptotic Approximation for multinormal Integrals," Journal of Engineering Mechanics, ASCE, 110(3), 357-366.
- Hohenbichler M., Rackwitz R., 1988, "Improvement of second order reliability estimates by importance sampling," Journal of Engineering Mechanics, ASCE,114(12), pp 2195-2199.
- [lebrun2009b]_
- [lebrun2009c]_
- Tvedt L. 1988, "Second order reliability by an exact integral," proc. of the IFIP Working Conf. Reliability and Optimization of Structural Systems, Thoft-Christensen (Ed), pp377-384.
- Zhao Y. G., Ono T., 1999, "New approximations for SORM : part 1", Journal of Engineering Mechanics, ASCE,125(1), pp 79-85.
- Zhao Y. G., Ono T., 1999, "New approximations for SORM : part 2", Journal of Engineering Mechanics, ASCE,125(1), pp 86-93.
- Adhikari S., 2004, "Reliability analysis using parabolic failure surface approximation", Journal of Engineering Mechanics, ASCE,130(12), pp 1407-1427.