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"""
Use the FORM - SORM algorithms
==============================
"""
# %%
# In this example we estimate a failure probability with the `FORM` algorithm on the :ref:`cantilever beam <use-case-cantilever-beam>` example.
# More precisely, we show how to use the associated results:
#
# - the design point in both physical and standard space,
# - the probability estimation according to the FORM approximation, and the following SORM ones: Tvedt, Hohenbichler and Breitung,
# - the Hasofer reliability index and the generalized ones evaluated from the Breitung, Tvedt and Hohenbichler approximations,
# - the importance factors defined as the normalized director factors of the design point in the :math:`U`-space
# - the sensitivity factors of the Hasofer reliability index and the FORM probability,
# - the coordinates of the mean point in the standard event space.
#
# See :ref:`FORM <form_approximation>` and :ref:`SORM <sorm_approximation>` for theoretical details.
# %%
# Model definition
# ----------------
# %%
from openturns.usecases import cantilever_beam
import openturns as ot
import openturns.viewer as otv
# %%
# We load the model from the usecases module :
cb = cantilever_beam.CantileverBeam()
# %%
# We use the input parameters distribution from the data class :
distribution = cb.distribution
distribution.setDescription(["E", "F", "L", "I"])
# %%
# We define the model
model = cb.model
# %%
# Create the event whose probability we want to estimate.
# %%
vect = ot.RandomVector(distribution)
G = ot.CompositeRandomVector(model, vect)
event = ot.ThresholdEvent(G, ot.Greater(), 0.3)
event.setName("deviation")
# %%
# FORM Analysis
# -------------
# %%
# Define a solver, here we use a :class:`~openturns.MultiStart` optimization based on :class:`~openturns.Cobyla`
startingSample = distribution.getSample(10)
optimAlgo = ot.MultiStart(ot.Cobyla(), startingSample)
optimAlgo.setMaximumCallsNumber(1000)
optimAlgo.setMaximumAbsoluteError(1.0e-4)
optimAlgo.setMaximumRelativeError(1.0e-4)
optimAlgo.setMaximumResidualError(1.0e-4)
optimAlgo.setMaximumConstraintError(1.0e-4)
# %%
# Run FORM
algo = ot.FORM(optimAlgo, event)
algo.run()
result = algo.getResult()
# %%
# Analysis of the results
# -----------------------
# %%
# Probability
result.getEventProbability()
# %%
# Hasofer reliability index
result.getHasoferReliabilityIndex()
# %%
# Design point in the standard U* space.
# %%
print(result.getStandardSpaceDesignPoint())
# %%
# Design point in the physical X space.
# %%
print(result.getPhysicalSpaceDesignPoint())
# %%
# Importance factors
graph = result.drawImportanceFactors()
view = otv.View(graph)
# %%
marginalSensitivity, otherSensitivity = result.drawHasoferReliabilityIndexSensitivity()
marginalSensitivity.setLegends(["E", "F", "L", "I"])
marginalSensitivity.setLegendPosition("bottom")
view = otv.View(marginalSensitivity)
# %%
marginalSensitivity, otherSensitivity = result.drawEventProbabilitySensitivity()
marginalSensitivity.setLegends(["E", "F", "L", "I"])
marginalSensitivity.setLegendPosition("bottom")
view = otv.View(marginalSensitivity)
# %%
# Error history
optimResult = result.getOptimizationResult()
graphErrors = optimResult.drawErrorHistory()
graphErrors.setLegendPosition("bottom")
graphErrors.setYMargin(0.0)
view = otv.View(graphErrors)
# %%
# Get additional results with SORM
algo = ot.SORM(optimAlgo, event)
algo.run()
sorm_result = algo.getResult()
# %%
# Reliability index with Breitung approximation
sorm_result.getGeneralisedReliabilityIndexBreitung()
# %%
# ... with Hohenbichler approximation
sorm_result.getGeneralisedReliabilityIndexHohenbichler()
# %%
# .. with Tvedt approximation
sorm_result.getGeneralisedReliabilityIndexTvedt()
# %%
# SORM probability of the event with Breitung approximation
sorm_result.getEventProbabilityBreitung()
# %%
# ... with Hohenbichler approximation
sorm_result.getEventProbabilityHohenbichler()
# %%
# ... with Tvedt approximation
sorm_result.getEventProbabilityTvedt()
# %%
# FORM analysis with finite difference gradient
# ---------------------------------------------
# %%
# When the considered function has no analytical expression, the gradient may not be known.
# In this case, a constant step finite difference gradient definition may be used.
# %%
def cantilever_beam_python(X):
E, F, L, II = X
return [F * L**3 / (3 * E * II)]
cbPythonFunction = ot.PythonFunction(4, 1, func=cantilever_beam_python)
epsilon = [1e-8] * 4 # Here, a constant step of 1e-8 is used for every dimension
gradStep = ot.ConstantStep(epsilon)
cbPythonFunction.setGradient(
ot.CenteredFiniteDifferenceGradient(gradStep, cbPythonFunction.getEvaluation())
)
G = ot.CompositeRandomVector(cbPythonFunction, vect)
event = ot.ThresholdEvent(G, ot.Greater(), 0.3)
event.setName("deviation")
# %%
# However, given the different nature of the model variables, a blended (variable)
# finite difference step may be preferable:
# The step depends on the location in the input space
gradStep = ot.BlendedStep(epsilon)
cbPythonFunction.setGradient(
ot.CenteredFiniteDifferenceGradient(gradStep, cbPythonFunction.getEvaluation())
)
G = ot.CompositeRandomVector(cbPythonFunction, vect)
event = ot.ThresholdEvent(G, ot.Greater(), 0.3)
event.setName("deviation")
# %%
# We can then run the FORM analysis in the same way as before:
algo = ot.FORM(optimAlgo, event)
algo.run()
result = algo.getResult()
# %%
otv.View.ShowAll()
|
