FORM¶

class FORM(*args)¶

First Order Reliability Method (FORM).

Refer to FORM.

Parameters:

nearestPointAlgorithmOptimizationAlgorithm: Optimization algorithm used to search the design point.
eventRandomVector: Failure event.
physicalStartingPointsequence of float: Starting point of the optimization algorithm, declared in the physical space.

Methods

`getAnalyticalResult`()	Accessor to the result.
`getClassName`()	Accessor to the object's name.
`getEvent`()	Accessor to the event of which the probability is calculated.
`getName`()	Accessor to the object's name.
`getNearestPointAlgorithm`()	Accessor to the optimization algorithm used to find the design point.
`getPhysicalStartingPoint`()	Accessor to the starting point of the optimization algorithm.
`getResult`()	Accessor to the result of FORM.
`hasName`()	Test if the object is named.
`run`()	Evaluate the failure probability.
`setEvent`(event)	Accessor to the event of which the probability is calculated.
`setName`(name)	Accessor to the object's name.
`setNearestPointAlgorithm`(solver)	Accessor to the optimization algorithm used to find the design point.
`setPhysicalStartingPoint`(physicalStartingPoint)	Accessor to the starting point of the optimization algorithm.
`setResult`(formResult)	Accessor to the result of FORM.

See also

Analytical, AnalyticalResult, SORM, StrongMaximumTest, FORMResult

Notes

See Analytical for the description of the first steps of the FORM analysis.

The First Order Reliability Method (FORM) consists in linearizing the limit state function $G(\vect{U}\,,\,\vect{d})$ at the design point, denoted $P^*$ , which is the point on the limit state surface $G(\vect{U}\,,\,\vect{d})=0$ that is closest to the origin of the standard space.

Then, the probability $P_f$ where the limit state surface has been approximated by a linear surface (hyperplane) can be obtained exactly, thanks to the rotation invariance of the standard distribution $f_{\vect{U}}$ :

$P_f = \left\{ \begin{array}{ll} \displaystyle E(-\beta_{HL}) & \text{if the origin of the }\vect{u}\text{-space lies in the domain }\cD_f \\ \displaystyle E(+\beta_{HL}) & \text{otherwise} \end{array} \right.$

where $\beta_{HL}$ is the Hasofer-Lind reliability index, defined as the distance of the design point $\vect{u}^*$ to the origin of the standard space and $E$ the marginal cumulative distribution function of the spherical distributions in the standard space.

The evaluation of the failure probability is stored in the data structure FORMResult recoverable with the getResult() method.

Examples

>>> import openturns as ot
>>> myFunction = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['-F*L^3/(3*E*I)'])
>>> myDistribution = ot.Normal([50.0, 1.0, 10.0, 5.0], [1.0]*4, ot.IdentityMatrix(4))
>>> vect = ot.RandomVector(myDistribution)
>>> output = ot.CompositeRandomVector(myFunction, vect)
>>> event = ot.ThresholdEvent(output, ot.Less(), -3.0)
>>> # We create an OptimizationAlgorithm algorithm
>>> solver = ot.AbdoRackwitz()
>>> algo = ot.FORM(solver, event, [50.0, 1.0, 10.0, 5.0])
>>> algo.run()
>>> result = algo.getResult()