Note

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Compute reference probability for RP55¶

The goal of this example is to present how we computed the reference probability for problem 55 of the BBRC. This calculation is based on the geometry of the failure domain. In order to compute its surface, we use the shapely module, which allows to compute the surface of bidimensional polygons.

import openturns as ot
import openturns.viewer as otv
import numpy as np
import otbenchmark as otb
from shapely.geometry import Polygon

problem = otb.ReliabilityProblem55()

event = problem.getEvent()
g = event.getFunction()

problem.getProbability()

0.5600144282863704

Compute the bounds of the domain¶

inputVector = event.getAntecedent()
distribution = inputVector.getDistribution()
X1 = distribution.getMarginal(0)
X2 = distribution.getMarginal(1)
alphaMin = 0.00001
alphaMax = 1 - alphaMin
lowerBound = ot.Point(
    [X1.computeQuantile(alphaMin)[0], X2.computeQuantile(alphaMin)[0]]
)
upperBound = ot.Point(
    [X1.computeQuantile(alphaMax)[0], X2.computeQuantile(alphaMax)[0]]
)

Print the iso-values of the distribution¶

sampleSize = 5000
sampleInput = inputVector.getSample(sampleSize)
sampleOutput = g(sampleInput)
drawEvent = otb.DrawEvent(event)

cloud = drawEvent.drawSampleCrossCut(sampleSize)
_ = otv.View(cloud)

Draw the limit state surface¶

bounds = ot.Interval(lowerBound, upperBound)
bounds

class=Interval name=Unnamed dimension=2 lower bound=class=Point name=Unnamed dimension=2 values=[-0.99998,-0.99998] upper bound=class=Point name=Unnamed dimension=2 values=[0.99998,0.99998] finite lower bound=[1,1] finite upper bound=[1,1]

domain = drawEvent.fillEventCrossCut(bounds)
_ = otv.View(domain)

From integration¶

interval = ot.Interval([-1.0, -1.0], [1.0, 1.0])
interval

class=Interval name=Unnamed dimension=2 lower bound=class=Point name=Unnamed dimension=2 values=[-1,-1] upper bound=class=Point name=Unnamed dimension=2 values=[1,1] finite lower bound=[1,1] finite upper bound=[1,1]

threshold = event.getThreshold()
threshold

0.0

mean = distribution.getMean()
mean

class=Point name=Unnamed dimension=2 values=[0,0]

def ComputeIntegrandPoint(x):
    """Evaluate the integrand on one single point."""
    y = g(x)
    if y[0] < threshold:
        indicator = 1.0
    else:
        indicator = 0.0
    pdf = distribution.computePDF(x)
    z = [indicator * pdf]
    return z


integrand = ot.PythonFunction(2, 1, ComputeIntegrandPoint)

integrand(mean)

class=Point name=Unnamed dimension=1 values=[0]

dimension = distribution.getDimension()
totalNumberOfPoints = 10 ** 4
numberOfPointsInX = int(totalNumberOfPoints ** (1.0 / dimension))
numberOfPointsInX

algoGL = ot.GaussLegendre([numberOfPointsInX] * dimension)
value = algoGL.integrate(integrand, interval)
value

class=Point name=Unnamed dimension=1 values=[0.558822]

def ComputeIntegrandSample(sample):
    """Evaluate the integrand on a sample."""
    indicatorBoolean = np.array(g(sample)) < threshold
    indicator = np.array(indicatorBoolean, float)
    pdf = np.array(distribution.computePDF(sample))
    z = indicator * pdf
    return z

integrandSample = ot.PythonFunction(2, 1, func_sample=ComputeIntegrandSample)

sample = distribution.getSample(5)
integrandSample(sample)

	y0
0	0.25
1	0.25
2	0.25
3	0.25
4	0

integrandSample(mean)

class=Point name=Unnamed dimension=1 values=[0]

dimension = distribution.getDimension()
totalNumberOfPoints = 10 ** 6
numberOfPointsInX = int(totalNumberOfPoints ** (1.0 / dimension))
numberOfPointsInX

discretization = [numberOfPointsInX] * dimension
algoGL = ot.GaussLegendre(discretization)
value = algoGL.integrate(integrandSample, interval)
value

class=Point name=Unnamed dimension=1 values=[0.559983]

Exact computation¶

We solve the exact equation:

https://www.wolframalpha.com/input/?i=solve+min%280.2+%2B+0.6+*+%28x+-+y%29%5E4+-+%28x+-+y%29+%2F+sqrt%282%29%2C+0.2+%2B+0.6+*+%28x+-+y%29%5E4+%2B+%28x+-+y%29+%2F+sqrt%282%29%2C+%28x+-+y%29+%2B+5+%2F+sqrt%282%29+-+2.2%2C+%28y+-+x%29+%2B+5+%2F+sqrt%282%29+-+2.2%29+%3D+0

` solve (x - y) + 5 / sqrt(2) - 22/10 = 0 solve 2/10 + 6/10 * d^4 - d / sqrt(2) `

domain = drawEvent.fillEventCrossCut(bounds)
_ = otv.View(domain)

We first compute the exact equations defining the slanted lines at the limit state.

palette = [
    "dodgerblue3",
    "darkorange1",
    "forestgreen",
    "firebrick3",
    "mediumpurple3",
    "lightsalmon4",
    "orchid",
    "gray50",
    "goldenrod3",
    "turquoise3",
]

The goal is to define the failure domain.

root1 = 5.0 / np.sqrt(2.0) - 2.2
root1

1.3355339059327371

root2 = 0.2887406174573670970
root2

0.2887406174573671

root3 = 0.9370483776955247344172976
root3

0.9370483776955247

root2 = 0.2887406174573670970
f1full = ot.SymbolicFunction(["x", "root2"], ["x-root2"])
f1 = ot.ParametricFunction(f1full, [1], [root2])
graph1 = f1.draw(root2 - 1.0, 1.0, 100)
graph1.setColors([palette[0]])
graph1.setLegends(["f1"])

f2full = ot.SymbolicFunction(["x", "root3"], ["x-root3"])
f2 = ot.ParametricFunction(f2full, [1], [root3])
graph2 = f2.draw(root3 - 1.0, 1.0, 100)
graph2.setColors([palette[1]])
graph2.setLegends(["f2"])

f3full = ot.SymbolicFunction(["x", "root1"], ["x-root1"])
f3 = ot.ParametricFunction(f3full, [1], [root1])
graph3 = f3.draw(root1 - 1.0, 1.0, 100)
graph3.setColors([palette[2]])
graph3.setLegends(["f3"])

f4full = ot.SymbolicFunction(["x", "root2"], ["x+root2"])
f4 = ot.ParametricFunction(f4full, [1], [root2])
graph4 = f4.draw(-1.0, 1.0 - root2, 100)
graph4.setColors([palette[3]])
graph4.setLegends(["f4"])

f5full = ot.SymbolicFunction(["x", "root3"], ["x+root3"])
f5 = ot.ParametricFunction(f5full, [1], [root3])
graph5 = f5.draw(-1.0, 1.0 - root3, 100)
graph5.setColors([palette[4]])
graph5.setLegends(["f5"])

f6full = ot.SymbolicFunction(["x", "root1"], ["x+root1"])
f6 = ot.ParametricFunction(f6full, [1], [root1])
graph6 = f6.draw(-1.0, 1.0 - root1, 100)
graph6.setColors([palette[5]])
graph6.setLegends(["f6"])

domain.add(graph1)
domain.add(graph2)
domain.add(graph3)
domain.add(graph4)
domain.add(graph5)
domain.add(graph6)
_ = otv.View(domain)

Then we set the corners of the polygons in the failure domain.

data = [
    [-1.0, 1.0],
    [1.0, f1([1.0])[0]],
    [1.0, f2([1.0])[0]],
    [1.0, f3([1.0])[0]],
    [-1.0, f4([-1.0])[0]],
    [-1.0, f5([-1.0])[0]],
    [-1.0, f6([-1.0])[0]],
    [root1 - 1.0, -1.0],
    [root2 - 1.0, -1.0],
    [root3 - 1.0, -1.0],
    [1.0 - root1, 1.0],
    [1.0 - root2, 1.0],
    [1.0 - root3, 1.0],
    [1.0, -1.0],
]
sample = ot.Sample(data)
sample

	v0	v1
0	-1	1
1	1	0.7112594
2	1	0.06295162
3	1	-0.3355339
4	-1	-0.7112594
5	-1	-0.06295162
6	-1	0.3355339
7	0.3355339	-1
8	-0.7112594	-1
9	-0.06295162	-1
10	-0.3355339	1
11	0.7112594	1
12	0.06295162	1
13	1	-1

cloud = ot.Cloud(sample)
domain.add(cloud)
extendedBounds = ot.Interval([-1.5, -1.5], [2.0, 2.0])
domain.setBoundingBox(extendedBounds)
domain

class=Graph name=Domain where g(x) < 0.0 implementation=class=GraphImplementation name=Domain where g(x) < 0.0 title=Domain where g(x) < 0.0 xTitle=x1 yTitle=x2 axes=ON grid=ON legendposition=topright legendFontSize=1 drawables=[class=Drawable name=Unnamed implementation=class=PolygonArray name=Unnamed derived from class=DrawableImplementation name=Unnamed legend=In data=class=Sample name=Unnamed implementation=class=SampleImplementation name=Unnamed size=11352 dimension=2 data=[[-0.679986,-0.99998],[-0.639987,-0.99998],[-0.679986,-0.959981],...,[0.719986,0.99998],[0.679986,0.99998],[0.679986,0.99998]] color=blue isColorExplicitlySet=true fillStyle=solid lineStyle=solid pointStyle=plus lineWidth=1 vertices number=4 palette=[lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1,lightsalmon1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color=lightsalmon4 isColorExplicitlySet=true fillStyle=solid lineStyle=solid pointStyle=none lineWidth=1,class=Drawable name=Unnamed implementation=class=Cloud name=Unnamed derived from class=DrawableImplementation name=Unnamed legend= data=class=Sample name=Unnamed implementation=class=SampleImplementation name=Unnamed size=14 dimension=2 data=[[-1,1],[1,0.711259],[1,0.0629516],[1,-0.335534],[-1,-0.711259],[-1,-0.0629516],[-1,0.335534],[0.335534,-1],[-0.711259,-1],[-0.0629516,-1],[-0.335534,1],[0.711259,1],[0.0629516,1],[1,-1]] color=#bcbd22 isColorExplicitlySet=false fillStyle=solid lineStyle=solid pointStyle=plus lineWidth=1]

In order to identify these points, we add the legends as text in the plot.

texts = ["p" + str(i) for i in range(sample.getSize())]
texts

['p0', 'p1', 'p2', 'p3', 'p4', 'p5', 'p6', 'p7', 'p8', 'p9', 'p10', 'p11', 'p12', 'p13']

text = ot.Text(sample, texts)

domain.add(text)
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The probability is the ratio between the failure domain and the domain itself. The area of the domain is $2^2$ . The area of each polygon is computed based on the shapely module.

corners = np.array([sample[i] for i in [0, 6, 10]])
corners

array([[-1.        ,  1.        ],
       [-1.        ,  0.33553391],
       [-0.33553391,  1.        ]])

polygon = Polygon(corners)
area1 = polygon.area
area1

0.2207575950825023

corners = np.array([sample[i] for i in [4, 11, 12, 5]])
corners
polygon = Polygon(corners)
area2 = polygon.area
area2

0.8992712614902384

corners = np.array([sample[i] for i in [8, 9, 2, 1]])
corners
polygon = Polygon(corners)
area3 = polygon.area
area3

0.8992712614902386

corners = np.array([sample[i] for i in [7, 13, 3]])
corners
polygon = Polygon(corners)
area4 = polygon.area
area4

0.2207575950825023

pf = (area1 + area2 + area3 + area4) / 2.0 ** 2
pf

0.5600144282863704

print("pf=%.17e" % (pf))

pf=5.60014428286370380e-01

otv.View.ShowAll()

Total running time of the script: (0 minutes 3.814 seconds)

otbenchmark

Module otbenchmark

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