Analytical

class Analytical(*args)

Base class to evaluate the probability of failure of a system.

Available constructors:
Analytical(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

Used in reliability analysis, Analytical is a base class for the approximation methods FORM and SORM enabling to evaluate the failure probability of a system. A failure event is defined as follows : \cD_f = \{\vect{X} \in \Rset^n \, | \, g(\vect{X},\vect{d}) \le 0\} where \vect{X} denotes a random input vector representing the sources of uncertainties, \vect{d} is a determinist vector representing the fixed variables. g(\vect{X},\vect{d}) is the limit state function of the model separating the failure domain from the safe domain. Considering f_\vect{X}(\vect{x}) the joint probability density function of the random variables \vect{X}, the probability of failure of the event \cD_f is :

P_f = \int_{g(\vect{X},\vect{d})\le 0}f_\vect{X}(\vect{x})\di{\vect{x}}

The analytical methods use an isoprobabilistic transformation to move from the physical space to the standard normal space (U-space) where distributions are spherical (invariant by rotation by definition), with zero mean, unit variance and unit correlation matrix. The usual isoprobabilistic transformations are the Generalized Nataf transformation and the Rosenblatt one.

In that new U-space, the event has the new expression defined from the transformed limit state function of the model G : \cD_f = \{\vect{U} \in \Rset^n \, | \, G(\vect{U}\,,\,\vect{d}) \le 0\} and its boundary : \{\vect{U} \in \Rset^n \, | \,G(\vect{U}\,,\,\vect{d}) = 0\}. Then, the event probability P_f rewrites :

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})\di{\vect{u}}

where f_{\vect{U}} is the density function of the distribution in the standard space.

The analytical methods rely on the assumption that most of the contribution to P_f comes from points located in the vicinity of a particular point P^*, the design point, defined in the U-space as the point located on the limit state surface verifying the event of maximum likelihood. Given the probabilistic characteristics of the U-space, P^* has a geometrical interpretation: it is the point located on the event boundary and at minimal distance from the origin of the U-space. Thus, considering \vect{u}^* its coordinates in the U-space, the design point is the result of the constrained optimization problem :

\vect{u}^* = argmin \{||\vect{u}|| \, | \, G(\vect{u}) = 0 \}

Then the limit state surface is approximated in the standard space by a linear surface (FORM) or by a quadratic surface (SORM) at the design point in order to evaluate the failure probability. For more information on this evaluation, see the documentation associated with these two methods.

The result of the optimization problem is recoverable thanks to the method getAnalyticalResult().

The unicity and the strongness of the design point can be checked thanks to the Strong Maximum Test.

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)
>>> myEvent = ot.Event(output, ot.Less(), -3.0)
>>> # We create an OptimizationAlgorithm algorithm
>>> myOptim = ot.AbdoRackwitz()
>>> myAlgo = ot.Analytical(myOptim, myEvent, [50.0, 1.0, 10.0, 5.0])

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.
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() Perform the research of the design point.
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.
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 : NumericalPoint

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

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()

Perform the research of the design point.

Notes

Performs the research of the design point and creates a AnalyticalResult, the structure result which is accessible with the method getAnalyticalResult().

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.

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.