MultiFORM

class MultiFORM(*args)

FORM approximation with multiple design points.

This algorithm searches for several design points by restarting the nearest-point search. For each design point we add a bulge function that penalizes the limit-state G around it:

G_m(u) = G(u) + \sum_{i=1}^m B_i(u)

The bulge function can be chosed as:

B_i(u) = s_i (r_i^2 - \|u - u_i^{\ast}\|^2)^2

Once these points are found a FORM is computed with each point and the probability of interest is computed as a series system probability: the union of the approximated events:

P(E_{sys}) = P(\bigcup_{i=1}^N E_i) = 1 - \Phi_k (\vect{\beta}; \vect{\vect{\rho}})

Beware, as with the regular FORM method the search can fail, or the enumeration of design can be not incomplete.

More details can be found in [kiureghian1998].

Parameters
solverOptimizationAlgorithm

Optimization algorithm used to research the design point.

eventRandomVector

Failure system event, in disjunctive normal form (either a union of intersections, or a single intersection)

physicalStartingPointsequence of float

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

See also

FORM, MultiFORMResult

Examples

>>> import openturns as ot
>>> dim = 2
>>> f = ot.SymbolicFunction(['x0', 'x1'], ['5.0-x1-0.5*(x0-0.1)^2'])
>>> dist = ot.Normal(dim)
>>> X = ot.RandomVector(dist)
>>> Y = ot.CompositeRandomVector(f, X)
>>> event = ot.ThresholdEvent(Y, ot.Less(), 0.0)
>>> solver = ot.Cobyla()
>>> starting_pt = dist.getMean()
>>> algo = ot.MultiFORM(solver, event, starting_pt)
>>> algo.run()
>>> result = algo.getResult()
>>> n_design_pts = len(result.getFORMResultCollection())
>>> pf = result.getEventProbability()

Methods

getAnalyticalResult(self)

Accessor to the result.

getClassName(self)

Accessor to the object’s name.

getEvent(self)

Accessor to the event of which the probability is calculated.

getId(self)

Accessor to the object’s id.

getMaximumNumberOfDesignPoints(self)

Accessor to the maximum number of design points.

getName(self)

Accessor to the object’s name.

getNearestPointAlgorithm(self)

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

getPhysicalStartingPoint(self)

Accessor to the starting point of the optimization algorithm.

getResult(self)

Accessor to the result.

getShadowedId(self)

Accessor to the object’s shadowed id.

getVisibility(self)

Accessor to the object’s visibility state.

hasName(self)

Test if the object is named.

hasVisibleName(self)

Test if the object has a distinguishable name.

run(self)

Evaluate the failure probability.

setEvent(self, event)

Accessor to the event of which the probability is calculated.

setMaximumNumberOfDesignPoints(self, …)

Accessor to the maximum number of design points.

setName(self, name)

Accessor to the object’s name.

setNearestPointAlgorithm(self, solver)

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

setPhysicalStartingPoint(self, …)

Accessor to the starting point of the optimization algorithm.

setResult(self, formResult)

Accessor to the result of FORM.

setShadowedId(self, id)

Accessor to the object’s shadowed id.

setVisibility(self, visible)

Accessor to the object’s visibility state.
__init__(self, \*args)

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

getAnalyticalResult(self)

Accessor to the result.

Returns
resultAnalyticalResult

Result structure which contains the results of the optimisation problem.

getClassName(self)

Accessor to the object’s name.

Returns
class_namestr

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

getEvent(self)

Accessor to the event of which the probability is calculated.

Returns
eventRandomVector

Event of which the probability is calculated.

getId(self)

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getMaximumNumberOfDesignPoints(self)

Accessor to the maximum number of design points.

Returns
max_dpint

Maximum number of design points

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getNearestPointAlgorithm(self)

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

Returns
algorithmOptimizationAlgorithm

Optimization algorithm used to research the design point.

getPhysicalStartingPoint(self)

Accessor to the starting point of the optimization algorithm.

Returns
pointPoint

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

getResult(self)

Accessor to the result.

Returns
resultMultiFORMResult

Contains the global result as well as the sub FORM results.

getShadowedId(self)

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getVisibility(self)

Accessor to the object’s visibility state.

Returns
visiblebool

Visibility flag.

hasName(self)

Test if the object is named.

Returns
hasNamebool

True if the name is not empty.

hasVisibleName(self)

Test if the object has a distinguishable name.

Returns
hasVisibleNamebool

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

run(self)

Evaluate the failure probability.

Notes

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

setEvent(self, event)

Accessor to the event of which the probability is calculated.

Parameters
eventRandomVector

Event of which the probability is calculated.

setMaximumNumberOfDesignPoints(self, numberOfDesignPoints)

Accessor to the maximum number of design points.

Parameters
max_dpint

Maximum number of design points

setName(self, name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setNearestPointAlgorithm(self, solver)

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

Parameters
algorithmOptimizationAlgorithm

Optimization algorithm used to research the design point.

setPhysicalStartingPoint(self, 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(self, formResult)

Accessor to the result of FORM.

Parameters
resultFORMResult

Structure containing all the results of the FORM analysis.

setShadowedId(self, id)

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setVisibility(self, visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.