SORM

class SORM(*args)

Second Order Reliability Method (SORM).

Refer to SORM.

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.

See also

Analytical, AnalyticalResult, FORM, StrongMaximumTest, SORMResult

Notes

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

The Second Order Reliability Method (SORM) consists in approximating the limit state surface in U-space at the design point P^* by a quadratic surface. SORM is usually more accurate than FORM e.g. in case when the event boundary is highly curved.

Let us denote by n the dimension of the random vector \vect{X} and (\kappa_i)_{1 \leq i \leq n-1} the n-1 main curvatures of the limit state function at the design point in the standard space.

Several approximations of the failure probability P_f are available in the library, and detailed here in the case where the origin of the standard space does not belong to the failure domain:

  • Breitung’s formula:

    P_{Breitung} = E(-\beta_{HL})\prod_{i=1}^{n-1} \frac{1}{\sqrt{1 + \beta_{HL}\kappa_i}}

    E the marginal cumulative distribution function of the spherical distributions in the standard space and \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.

  • Hohenbichler’s formula is an approximation of the previous equation:

    \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}

    where \Phi is the cumulative distribution function of the standard 1D normal distribution and \phi is the standard Gaussian probability density function.

  • Tvedt’s formula:

    \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} - {\cR}e \left( \prod_{j=1}^{N-1} \left( 1 + (i + \beta_{HL}) \kappa_j \right) ^{-1/2} \right)\right ] \end{array} \right.

    where {\cR}e(z) is the real part of the complex number z and i the complex number such that i^2 = -1.

The evaluation of the failure probability is stored in the data structure SORMResult 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.SORM(solver, event, [50.0, 1.0, 10.0, 5.0])
>>> algo.run()
>>> result = algo.getResult()

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 SORM.

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(sormResult)

Accessor to the result of SORM.
__init__(*args)
getAnalyticalResult()

Accessor to the result.

Returns:
resultAnalyticalResult

Result structure which contains the results of the optimisation problem.

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

The object class name (object.__class__.__name__).

getEvent()

Accessor to the event of which the probability is calculated.

Returns:
eventRandomVector

Event of which the probability is calculated.

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getNearestPointAlgorithm()

Accessor to the optimization algorithm used to find the design point.

Returns:
algorithmOptimizationAlgorithm

Optimization algorithm used to research the design point.

getPhysicalStartingPoint()

Accessor to the starting point of the optimization algorithm.

Returns:
pointPoint

Starting point of the optimization algorithm, declared in the physical space.

getResult()

Accessor to the result of SORM.

Returns:
resultSORMResult

Structure containing all the results of the SORM analysis.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

run()

Evaluate the failure probability.

Notes

Evaluate the failure probability and create a SORMResult, the structure result which is accessible with the method getResult().

setEvent(event)

Accessor to the event of which the probability is calculated.

Parameters:
eventRandomVector

Event of which the probability is calculated.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

setNearestPointAlgorithm(solver)

Accessor to the optimization algorithm used to find the design point.

Parameters:
algorithmOptimizationAlgorithm

Optimization algorithm used to research the design point.

setPhysicalStartingPoint(physicalStartingPoint)

Accessor to the starting point of the optimization algorithm.

Parameters:
pointsequence of float

Starting point of the optimization algorithm, declared in the physical space.

setResult(sormResult)

Accessor to the result of SORM.

Parameters:
resultSORMResult

Structure containing all the results of the SORM analysis.

Examples using the class

Estimate a buckling probability

Estimate a buckling probability

Use the FORM - SORM algorithms

Use the FORM - SORM algorithms

An illustrated example of a FORM probability estimate

An illustrated example of a FORM probability estimate

Using the FORM - SORM algorithms on a nonlinear function

Using the FORM - SORM algorithms on a nonlinear function