FORM

class FORM(*args)

First Order Reliability Method (FORM).

Available constructors:
FORM(nearestPointAlgorithm, event, physicalStartingPoint)
Parameters:

nearestPointAlgorithm : OptimizationAlgorithm

Optimization algorithm used to research the design point.

event : Event

Failure event.

physicalStartingPoint : sequence of float

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

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 density 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.RandomVector(myFunction, vect)
>>> event = ot.Event(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()

Methods

getAnalyticalResult() Accessor to the result.
getClassName() Accessor to the object’s name.
getEvent() Accessor to the event of which the probability is calculated.
getId() Accessor to the object’s id.
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.
getShadowedId() Accessor to the object’s shadowed id.
getVisibility() Accessor to the object’s visibility state.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
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.
setShadowedId(id) Accessor to the object’s shadowed id.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)
getAnalyticalResult()

Accessor to the result.

Returns:

result : AnalyticalResult

Result structure which contains the results of the optimisation problem.

getClassName()

Accessor to the object’s name.

Returns:

class_name : str

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

getEvent()

Accessor to the event of which the probability is calculated.

Returns:

event : Event

Event of which the probability is calculated.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getNearestPointAlgorithm()

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

Returns:

algorithm : OptimizationAlgorithm

Optimization algorithm used to research the design point.

getPhysicalStartingPoint()

Accessor to the starting point of the optimization algorithm.

Returns:

point : Point

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

getResult()

Accessor to the result of FORM.

Returns:

result : FORMResult

Structure containing all the results of the FORM analysis.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

True if the name is not empty and not the default one.

run()

Evaluate the failure probability.

Notes

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

setEvent(event)

Accessor to the event of which the probability is calculated.

Parameters:

event : Event

Event of which the probability is calculated.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setNearestPointAlgorithm(solver)

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

Parameters:

algorithm : OptimizationAlgorithm

Optimization algorithm used to research the design point.

setPhysicalStartingPoint(physicalStartingPoint)

Accessor to the starting point of the optimization algorithm.

Parameters:

point : sequence of float

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

setResult(formResult)

Accessor to the result of FORM.

Parameters:

result : FORMResult

Structure containing all the results of the FORM analysis.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.