# Probability estimation: importance simulation¶

In this basic example we are going to estimate a failure probability.

Let’s consider the following analytical example of a cantilever beam, of Young’s modulus E, length L, section modulus I.

One end is built in a wall and we apply a concentrated bending load F at the other end of the beam, resulting in a deviation: It is considered that failure occurs when the beam deviation is too important: Four independent random variables are considered:

• E (the Young’s modulus) [Pa]

• L (the length) # [m]

• I (the section) # [m^4]

Stochastic model (simplified model, no units):

• E ~ Beta(0.93, 3.2, 2.8e7, 4.8e7)

• F ~ LogNormal(30000, 9000, 15000)

• L ~ Uniform(250, 260)

• I ~ Beta(2.5, 4.0, 3.1e2, 4.5e2)

:

from __future__ import print_function
import openturns as ot

:

# Create the marginal distributions of the parameters
dist_E = ot.Beta(0.93, 2.27, 2.8e7, 4.8e7)
dist_F = ot.LogNormalMuSigma(30000, 9000, 15000).getDistribution()
dist_L = ot.Uniform(250, 260)
dist_I = ot.Beta(2.5, 1.5, 3.1e2, 4.5e2)
marginals = [dist_E, dist_F, dist_L, dist_I]

:

# Create the Copula
RS = ot.CorrelationMatrix(4)
RS[2, 3] = -0.2
# Evaluate the correlation matrix of the Normal copula from RS
R = ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(RS)
# Create the Normal copula parametrized by R
copula = ot.NormalCopula(R)

:

# Create the joint probability distribution
distribution = ot.ComposedDistribution(marginals, copula)

:

# create the model
model = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['F*L^3/(3*E*I)'])

:

# create the event we want to estimate the probability
vect = ot.RandomVector(distribution)
G = ot.CompositeRandomVector(model, vect)
event = ot.ThresholdEvent(G, ot.Greater(), 30.0)

:

# Define a solver
optimAlgo = ot.Cobyla()
optimAlgo.setMaximumEvaluationNumber(1000)
optimAlgo.setMaximumAbsoluteError(1.0e-10)
optimAlgo.setMaximumRelativeError(1.0e-10)
optimAlgo.setMaximumResidualError(1.0e-10)
optimAlgo.setMaximumConstraintError(1.0e-10)

:

# Run FORM
algo = ot.FORM(optimAlgo, event, distribution.getMean())
algo.run()
result = algo.getResult()
standardSpaceDesignPoint = result.getStandardSpaceDesignPoint()

:

# define the importance distribution
myImportance = ot.Normal(standardSpaceDesignPoint, [1.0] * 4, ot.CorrelationMatrix(4))

:

# Create a simulation algorithm
experiment = ot.ImportanceSamplingExperiment(myImportance)
algo = ot.ProbabilitySimulationAlgorithm(ot.StandardEvent(event), experiment)
algo.setMaximumCoefficientOfVariation(0.1)
algo.setMaximumOuterSampling(40000)
algo.setConvergenceStrategy(ot.Full())
algo.run()

:

# retrieve results
result = algo.getResult()
probability = result.getProbabilityEstimate()
print('Pf=', probability)

Pf= 0.005064811687882659