SORMResult

class SORMResult(*args)

Result of a SORM analysis.

Available constructors:

SORMResult(designPoint, limitStateVariable, isInFailureSpace)

Parameters
designPointsequence of float

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

limitStateVariableRandomVector

Event of which the probability is calculated.

isInFailureSpacebool

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

See also

Analytical, AnalyticalResult, FORM, FORMResult, StrongMaximumTest

Notes

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

Methods

drawHasoferReliabilityIndexSensitivity(*args)

Draw the sensitivity of the Hasofer Reliability Index.

drawImportanceFactors(*args)

Draw the importance factors.

getClassName()

Accessor to the object’s name.

getEventProbabilityBreitung()

Accessor to the failure probability P_{Breitung}.

getEventProbabilityHohenbichler()

Accessor to the failure probability P_{Hohenbichler}.

getEventProbabilityTvedt()

Accessor to the failure probability P_{Tvedt}.

getGeneralisedReliabilityIndexBreitung()

Accessor to the Generalised Reliability Index Breitung.

getGeneralisedReliabilityIndexHohenbichler()

Accessor to the Generalised Reliability Index Hohenbichler.

getGeneralisedReliabilityIndexTvedt()

Accessor to the Generalised Reliability Index Tvedt.

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.

getSortedCurvatures()

Accessor to the sorted curvatures.

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.

getEventProbabilityHohenBichler

getGeneralisedReliabilityIndexHohenBichler
__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

drawHasoferReliabilityIndexSensitivity(*args)

Draw the sensitivity of the Hasofer Reliability Index.

Parameters
widthfloat, optional

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

Returns
graphCollectionsequence 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
typeint, optional

See getImportanceFactors()

Returns
graphGraph

Pie of the importance factors of the probabilistic variables.

getClassName()

Accessor to the object’s name.

Returns
class_namestr

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

getEventProbabilityBreitung()

Accessor to the failure probability P_{Breitung}.

Returns
probabilitypositive float

The SORM failure probability P_{Breitung} according to the Breitung approximation.

getEventProbabilityHohenbichler()

Accessor to the failure probability P_{Hohenbichler}.

Returns
probabilitypositive float

The SORM failure probability P_{Hohenbichler} according to the Hohenbichler approximation.

getEventProbabilityTvedt()

Accessor to the failure probability P_{Tvedt}.

Returns
probabilitypositive float

The SORM failure probability P_{Tvedt} according to the Tvedt approximation.

getGeneralisedReliabilityIndexBreitung()

Accessor to the Generalised Reliability Index Breitung.

Returns
indexfloat

Generalised reliability index evaluated from the Breitung SORM failure probability.

\beta_{Breitung} = \left \{ \begin{array}{ll} \displaystyle -\Phi(P_{Breitung}) & \text{if the standard space origin is not in the failure space} \\ \displaystyle \Phi(P_{Breitung}) & \text{otherwise} \end{array} \right.

getGeneralisedReliabilityIndexHohenbichler()

Accessor to the Generalised Reliability Index Hohenbichler.

Returns
indexfloat

Generalised reliability index evaluated from the Hohenbichler SORM failure probability.

\beta_{Hohenbichler} = \left \{ \begin{array}{ll} \displaystyle -\Phi(P_{Hohenbichler}) & \text{if the standard space origin is not in the failure space} \\ \displaystyle \Phi(P_{Hohenbichler}) & \text{otherwise} \end{array} \right.

getGeneralisedReliabilityIndexTvedt()

Accessor to the Generalised Reliability Index Tvedt.

Returns
indexfloat

Generalised reliability index evaluated from the Tvedt SORM failure probability.

\beta_{Tvedt} = \left \{ \begin{array}{ll} \displaystyle -\Phi(P_{Tvedt}) & \text{if the standard space origin is not in the failure space} \\ \displaystyle \Phi(P_{Tvedt}) & \text{otherwise} \end{array} \right.

getHasoferReliabilityIndex()

Accessor to the Hasofer Reliability Index.

Returns
indexfloat

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.

Refer to Sensitivity Factors from FORM method.

Returns
sensitivityPointWithDescription

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
idint

Internal unique identifier.

getImportanceFactors(*args)

Accessor to the importance factors.

Refer Importance factors from FORM method.

Parameters
typeint, 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
factorsPointWithDescription

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
isInFailureSpacebool

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
limitStateVariableRandomVector

Event of which the probability is calculated.

getMeanPointInStandardEventDomain()

Accessor to the mean point in the standard event domain.

Returns
meanPointPoint

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
namestr

The name of the object.

getOptimizationResult()

Accessor to the result of the optimization problem.

Returns
resultOptimizationResult

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
designPointPoint

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

getShadowedId()

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getSortedCurvatures()

Accessor to the sorted curvatures.

Returns
curvaturesPoint

Curvatures of the standard limite state function at the standard design point (\kappa_i)_{1 \leq i \leq n-1} with n the dimension of the random vector \vect{X}.

getStandardSpaceDesignPoint()

Accessor to the design point in the standard space.

Returns
designPointPoint

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

getVisibility()

Accessor to the object’s visibility state.

Returns
visiblebool

Visibility flag.

hasName()

Test if the object is named.

Returns
hasNamebool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns
hasVisibleNamebool

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
isInFailureSpacebool

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
meanPointsequence 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
namestr

The name of the object.

setOptimizationResult(optimizationResult)

Accessor to the result of the optimization problem.

Parameters
resultOptimizationResult

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
idint

Internal unique identifier.

setStandardSpaceDesignPoint(standardSpaceDesignPoint)

Accessor to the design point in the standard space.

Parameters
designPointsequence of float

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

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.