FORMResult

class FORMResult(*args)

Result of a FORM analysis.

Available constructors:
FORMResult(designPoint, limitStateVariable, isInFailureSpace)
Parameters:

designPoint : sequence of float

Design point in the standard space resulting from the optimization algorithm.

limitStateVariable : Event

Event of which the probability is calculated.

isInFailureSpace : bool

Indicates whether the origin of the standard space is in the failure space.

Notes

Structure created by the method run() of the FORM class and obtained thanks to its method getResult().

Methods

drawEventProbabilitySensitivity(*args) Draw the sensitivities of the FORM failure probability.
drawHasoferReliabilityIndexSensitivity(*args) Draw the sensitivity of the Hasofer Reliability Index.
drawImportanceFactors(*args) Draw the importance factors.
getClassName() Accessor to the object’s name.
getEventProbability() Accessor to the failure probability P_f.
getEventProbabilitySensitivity() Accessor to the sentivities of the FORM failure probability P_f.
getGeneralisedReliabilityIndex() Accessor to the Generalised Reliability Index.
getHasoferReliabilityIndex() Accessor to the Hasofer Reliability Index.
getHasoferReliabilityIndexSensitivity() Accessor to the sensitivities of the Hasofer Reliability Index.
getId() Accessor to the object’s id.
getImportanceFactors(*args) Accessor to the importance factors.
getIsStandardPointOriginInFailureSpace() Accessor to know if the standard point origin is in the failure space.
getLimitStateVariable() Accessor to the event of which the probability is calculated.
getMeanPointInStandardEventDomain() Accessor to the mean point in the standard event domain.
getName() Accessor to the object’s name.
getOptimizationResult() Accessor to the result of the optimization problem.
getPhysicalSpaceDesignPoint() Accessor to the design point in the physical space.
getShadowedId() Accessor to the object’s shadowed id.
getStandardSpaceDesignPoint() Accessor to the design point in the standard space.
getVisibility() Accessor to the object’s visibility state.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
setIsStandardPointOriginInFailureSpace(...) Accessor to specify if the standard point origin is in the failure space.
setMeanPointInStandardEventDomain(...) Accessor to the mean point in the standard event domain.
setName(name) Accessor to the object’s name.
setOptimizationResult(optimizationResult) Accessor to the result of the optimization problem.
setShadowedId(id) Accessor to the object’s shadowed id.
setStandardSpaceDesignPoint(...) Accessor to the design point in the standard space.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)
drawEventProbabilitySensitivity(*args)

Draw the sensitivities of the FORM failure probability.

Parameters:

width : float, optional

Value to calculate the shift position of the BarPlot. By default it is 1.0.

Returns:

graphCollection : list of two Graph containing a barplot

The first graph drawing the sensitivities of the FORM failure probability with regards to the parameters of the probabilistic input vector. The second graph drawing the sensitivities of the FORM failure probability with regards to the parameters of the dependence structure of the probabilistic input vector.

drawHasoferReliabilityIndexSensitivity(*args)

Draw the sensitivity of the Hasofer Reliability Index.

Parameters:

width : float, optional

Value to calculate the shift position of the BarPlot. By default it is 1.0.

Returns:

graphCollection : sequence of two Graph containing a barplot

The first graph drawing the sensitivity of the Hasofer Reliability Index to the parameters of the marginals of the probabilistic input vector. The second graph drawing the sensitivity of the Hasofer Reliability Index to the parameters of the dependence structure of the probabilistic input vector.

drawImportanceFactors(*args)

Draw the importance factors.

Parameters:

type : int, optional

Returns:

graph : Graph

Pie of the importance factors of the probabilistic variables.

getClassName()

Accessor to the object’s name.

Returns:

class_name : str

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

getEventProbability()

Accessor to the failure probability P_f.

Returns:

probability : positive float

The FORM failure probability P_f.

getEventProbabilitySensitivity()

Accessor to the sentivities of the FORM failure probability P_f.

Returns:

sentivities : Point

Sentivities of the FORM failure probability with regards to the parameters of the probabilistic input vector and to parameters of the dependence structure of the probabilistic input vector.

getGeneralisedReliabilityIndex()

Accessor to the Generalised Reliability Index.

Returns:

index : float

Generalised reliability index \beta_g from the FORM failure probability is equal to \pm the Hasofer reliability index \beta_{HL} according to the fact the standard space center fulfills the event or not.

getHasoferReliabilityIndex()

Accessor to the Hasofer Reliability Index.

Returns:

index : float

Hasofer Reliability Index which is the distance of the design point from the origin of the standard space \beta_{HL}=||\vect{u}^*||.

getHasoferReliabilityIndexSensitivity()

Accessor to the sensitivities of the Hasofer Reliability Index.

Returns:

sensitivity : PointWithDescription

Sequence containing the sensitivities of the Hasofer Reliability Index to the parameters of the probabilistic input vector (marginals and dependence structure) with a description for each component.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getImportanceFactors(*args)

Accessor to the importance factors.

Parameters:

type : int, optional

  • When ot.AnalyticalResult.ELLIPTICAL, the importance factors are evaluated as the square of the co-factors of the design point in the elliptical space of the iso-probabilistic transformation (Y-space).
  • When ot.AnalyticalResult.CLASSICAL they are evaluated as the square of the co-factors of the design point in the U-space.
  • When ot.AnalyticalResult.PHYSICAL, the importance factors are evaluated as the square of the physical sensitivities.

By default type = ot.AnalyticalResult.ELLIPTICAL.

Returns:

factors : PointWithDescription

Sequence containing the importance factors with a description for each component.

Notes

  • If the importance factors are evaluated as the square of the co-factors of the design point in the U-space :

    \alpha_i^2 = \frac{(u_i^*)^2}
                  {\beta_{HL}^2}

  • If the importance factors are evaluated as the square of the co-factors of the design point in the Y-space :

    \alpha_i^2 = \frac{(y_i^*)^2}
                  {\|\vect{y}^*\|^2}

    where

    Y^* = \left(
      \begin{array}{c}
          E^{-1}\circ F_1(X_1^*) \\
          E^{-1}\circ F_2(X_2^*) \\
          \vdots \\
          E^{-1}\circ F_n(X_n^*)
      \end{array}
      \right)

    with \vect{X}^* is the design point in the physical space and E the univariate standard CDF of the elliptical space. In the case where the input distribution of \vect{X} has an elliptical copula C_E, then E has the same type as C_E. In the case where the input distribution of \vect{X} has a copula C which is not elliptical, then E=\Phi where \Phi is the CDF of the standard normal.

  • If the importance factors are evaluated as the square of the physical sensitivities :

    \alpha_i^2 = \displaystyle \frac{s_i^2}{{\|s\|}^2}

    where

    s_i = \displaystyle \frac{\partial \beta}{\partial x_i} (x^*)
    = \sum_{j=1}^n \frac{\partial \beta}{\partial u_i} \frac{\partial u_j}{\partial x_i} (x^*)

getIsStandardPointOriginInFailureSpace()

Accessor to know if the standard point origin is in the failure space.

Returns:

isInFailureSpace : bool

Indicates whether the origin of the standard space is in the failure space.

getLimitStateVariable()

Accessor to the event of which the probability is calculated.

Returns:

limitStateVariable : Event

Event of which the probability is calculated.

getMeanPointInStandardEventDomain()

Accessor to the mean point in the standard event domain.

Returns:

meanPoint : Point

Mean point of the standard space distribution restricted to the event domain: \displaystyle \frac{1}{E_1(-\beta)}\int_{\beta}^{\infty} u_1 p_1(u_1)\di{u_1} where E_1 is the spheric univariate distribution of the standard space and \beta the reliability index.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getOptimizationResult()

Accessor to the result of the optimization problem.

Returns:

result : OptimizationResult

Contains the design point in the standard space and information concerning the convergence of the optimization algorithm.

getPhysicalSpaceDesignPoint()

Accessor to the design point in the physical space.

Returns:

designPoint : Point

Design point in the physical space resulting from the optimization algorithm.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getStandardSpaceDesignPoint()

Accessor to the design point in the standard space.

Returns:

designPoint : Point

Design point in the standard space resulting from the optimization algorithm.

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.

setIsStandardPointOriginInFailureSpace(isStandardPointOriginInFailureSpace)

Accessor to specify if the standard point origin is in the failure space.

Parameters:

isInFailureSpace : bool

Indicates whether the origin of the standard space is in the failure space.

setMeanPointInStandardEventDomain(meanPointInStandardEventDomain)

Accessor to the mean point in the standard event domain.

Parameters:

meanPoint : sequence of float

Mean point of the standard space distribution restricted to the event domain: \displaystyle \frac{1}{E_1(-\beta)}\int_{\beta}^{\infty} u_1 p_1(u_1)\di{u_1} where E_1 is the spheric univariate distribution of the standard space and \beta the reliability index.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setOptimizationResult(optimizationResult)

Accessor to the result of the optimization problem.

Parameters:

result : OptimizationResult

Contains the design point in the standard space and information concerning the convergence of the optimization algorithm.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setStandardSpaceDesignPoint(standardSpaceDesignPoint)

Accessor to the design point in the standard space.

Parameters:

designPoint : sequence of float

Design point in the standard space resulting from the optimization algorithm.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.