Use the Directional Sampling Algorithm

In this example we estimate a failure probability with the directional simulation algorithm provided by the DirectionalSampling class.

Introduction

The directional simulation algorithm operates in the standard space based on:

1. a root strategy to evaluate the nearest failure point along each direction and take the contribution of each direction to the failure event probability into account. The available strategies are: - RiskyAndFast - MediumSafe - SafeAndSlow

2. a sampling strategy to choose directions in the standard space. The available strategies are: - RandomDirection - OrthogonalDirection

Let us consider the analytical example of the cantilever beam described here.

from __future__ import print_function
from openturns.usecases import cantilever_beam as cantilever_beam
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)

We load the model from the usecases module :

cb = cantilever_beam.CantileverBeam()

We load the joint probability distribution of the input parameters :

distribution = cb.distribution

We load the model giving the displacement at the end of the beam :

model = cb.model

We create the event whose probability we want to estimate.

vect = ot.RandomVector(distribution)
G = ot.CompositeRandomVector(model, vect)
event = ot.ThresholdEvent(G, ot.Greater(), 0.30)

Root finding algorithm.

solver = ot.Brent()
rootStrategy = ot.MediumSafe(solver)

Direction sampling algorithm.

samplingStrategy = ot.OrthogonalDirection()

Create a simulation algorithm.

algo = ot.DirectionalSampling(event, rootStrategy, samplingStrategy)
algo.setMaximumCoefficientOfVariation(0.1)
algo.setMaximumOuterSampling(40000)
algo.setConvergenceStrategy(ot.Full())
algo.run()

Retrieve results.

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

Out:

Pf= 4.6231314140367655e-07

We can observe the convergence history with the drawProbabilityConvergence method.

graph = algo.drawProbabilityConvergence()
graph.setLogScale(ot.GraphImplementation.LOGX)
view = viewer.View(graph)