# Create a process from random vectors and processes¶

The objective is to create a process defined from a random vector and a process.

We consider the following limit state function, defined as the difference between a degrading resistance and a time-varying load :

We propose the following probabilistic model: - is the initial resistance, and ; - is the deterioration rate of the resistance; it is deterministic; - is the time-varying stress, which is modeled by a stationary Gaussian process of mean value , standard deviation and a squared exponential covariance model; - is the time, varying in .

First, import the python modules:

from  openturns import *
from openturns.viewer import View
from math import *


## 1. Create the gaussian process ¶

Create the mesh which is a regular grid on , with , by step =1:

b = 0.01
t0 = 0.0
step = 1
tfin = 50
n = round((tfin-t0)/step)
myMesh = RegularGrid(t0, step, n)


Create the squared exeponential covariance model:

where the scale parameter is and the amplitude .

l = 10/sqrt(2)
myCovKernel = SquaredExponential([l])
print('cov model = ', myCovKernel)


Out:

cov model =  SquaredExponential(scale=[7.07107], amplitude=[1])


Create the gaussian process :

S_proc = GaussianProcess(myCovKernel, myMesh)


## 2. Create the process ¶

First, create the random variable , with and :

muR = 5
sigR = 0.3
R = Normal(muR, sigR)


The create the Dirac random variable :

B = Dirac(b)


Then create the process using the class and the functional basis and :

with independent.

const_func = SymbolicFunction(['t'], ['1'])
linear_func = SymbolicFunction(['t'], ['-t'])
myBasis = Basis([const_func, linear_func])

coef = ComposedDistribution([R, B])

R_proc = FunctionalBasisProcess(coef, myBasis, myMesh)


## 3. Create the process ¶

First, aggregate both processes into one process of dimension 2:

myRS_proc = AggregatedProcess([R_proc, S_proc])


Then create the spatial field function that acts only on the values of the process, keeping the mesh unchanged, using the ValueFunction class. We define the function on by:

in order to define the spatial field function that acts on fields, defined by:

g = SymbolicFunction(['x1', 'x2'], ['x1-x2'])
gDyn = ValueFunction(g, myMesh)


Now you have to create the final process thanks to :

Z_proc = CompositeProcess(gDyn, myRS_proc)


## 4. Draw some realizations of the process¶

N=10
sampleZ_proc = Z_proc.getSample(N)
graph = sampleZ_proc.drawMarginal(0)
graph.setTitle(r'Some realizations of $Z(\omega, t)$')

Show(graph)


## 5. Evaluate the probability that ¶

We define the domaine and the event :

domain = Interval([2], [4])
print('D = ', domain)
event = ProcessEvent(Z_proc, domain)


Out:

D =  [2, 4]


We use the Monte Carlo sampling to evaluate the probability:

MC_algo = ProbabilitySimulationAlgorithm(event)
MC_algo.setMaximumOuterSampling(1000000)
MC_algo.setBlockSize(100)
MC_algo.setMaximumCoefficientOfVariation(0.01)
MC_algo.run()

result = MC_algo.getResult()

proba = result.getProbabilityEstimate()
print('Probability = ', proba)
variance =  result.getVarianceEstimate()
print('Variance Estimate = ', variance)
IC90_low = proba- result.getConfidenceLength(0.90)/2
IC90_upp = proba + result.getConfidenceLength(0.90)/2
print('IC (90%) = [', IC90_low, ', ', IC90_upp, ']')


Out:

Probability =  0.7551515151515152
Variance Estimate =  5.659164649247292e-05
IC (90%) = [ 0.7427777057653888 ,  0.7675253245376417 ]


Total running time of the script: ( 0 minutes 0.129 seconds)

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