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"""
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 :math:`r(t) = R - bt` and a time-varying load :math:`S(t)`:
#
# .. math::
# \begin{align*}
# g(t)= r(t) - S(t) = R - bt - S(t)
# \end{align*}
#
# We propose the following probabilistic model:
#
# - :math:`R` is the initial resistance, and :math:`R \sim \mathcal{N}(\mu_R, \sigma_R)`;
# - :math:`b` is the deterioration rate of the resistance; it is deterministic;
# - :math:`S(t)` is the time-varying stress, which is modeled by a stationary Gaussian process of mean value :math:`\mu_S`,
# standard deviation :math:`\sigma_S` and a squared exponential covariance model;
# - :math:`t` is the time, varying in :math:`[0,T]`.
#
# %%
# First, import the python modules:
# %%
import openturns as ot
import openturns.viewer as otv
import math as m
# %%
# 1. Create the Gaussian process :math:`(\omega, t) \rightarrow S(\omega,t)`
# --------------------------------------------------------------------------
# %%
# Create the mesh which is a regular grid on :math:`[0,T]`, with :math:`T=50`, by step =1:
# %%
b = 0.01
t0 = 0.0
step = 1
tfin = 50
n = round((tfin - t0) / step)
myMesh = ot.RegularGrid(t0, step, n)
# %%
# Create the squared exponential covariance model:
#
# .. math::
# C(s,t) = \sigma^2e^{-\frac{1}{2} \left( \dfrac{s-t}{l} \right)^2}
#
# where the scale parameter is :math:`l=\frac{10}{\sqrt{2}}` and the amplitude :math:`\sigma = 1`.
#
# %%
ll = 10 / m.sqrt(2)
myCovKernel = ot.SquaredExponential([ll])
print("cov model = ", myCovKernel)
# %%
# Create the Gaussian process :math:`S(t)`:
# %%
S_proc = ot.GaussianProcess(myCovKernel, myMesh)
# %%
# 2. Create the process :math:`(\omega, t) \rightarrow R(\omega)-bt`
# ------------------------------------------------------------------
# %%
# First, create the random variable :math:`R \sim \mathcal{N}(\mu_R, \sigma_R)`, with :math:`\mu_R = 5` and :math:`\sigma_R = 0.3`:
# %%
muR = 5
sigR = 0.3
R = ot.Normal(muR, sigR)
# %%
# The create the Dirac random variable :math:`B = b`:
# %%
B = ot.Dirac(b)
# %%
# Then create the process :math:`(\omega, t) \rightarrow R(\omega)-bt` using the :class:`~openturns.FunctionalBasisProcess` class
# and the functional basis :math:`\phi_1 : t \rightarrow 1` and :math:`\phi_2: -t \rightarrow t`:
#
# .. math::
# R(\omega)-bt = R(\omega)\phi_1(t) + B(\omega) \phi_2(t)
#
# with :math:`(R,B)` independent.
# %%
const_func = ot.SymbolicFunction(["t"], ["1"])
linear_func = ot.SymbolicFunction(["t"], ["-t"])
myBasis = ot.Basis([const_func, linear_func])
coef = ot.JointDistribution([R, B])
R_proc = ot.FunctionalBasisProcess(coef, myBasis, myMesh)
# %%
# 3. Create the process :math:`Z: (\omega, t) \rightarrow R(\omega)-bt + S(\omega, t)`
# ------------------------------------------------------------------------------------
# %%
# First, aggregate both processes into one process of dimension 2: :math:`(R_{proc}, S_{proc})`
# %%
myRS_proc = ot.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 :class:`~openturns.ValueFunction` class.
# We define the function :math:`g` on :math:`\mathbb{R}^2` by:
#
# .. math::
# g(x,y) = x-y
#
# in order to define the spatial field function :math:`g_{dyn}` that acts on fields, defined by:
#
# .. math::
# \forall t\in [0,T], g_{dyn}(X(\omega, t), Y(\omega, t)) = X(\omega, t) - Y(\omega, t)
#
# %%
g = ot.SymbolicFunction(["x1", "x2"], ["x1-x2"])
gDyn = ot.ValueFunction(g, myMesh)
# %%
# Now you have to create the final process :math:`Z` thanks to :math:`g_{dyn}`:
# %%
Z_proc = ot.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 $Z(\omega, t)$")
view = otv.View(graph)
# %%
# 5. Evaluate the probability that :math:`Z(\omega, t) \in \mathcal{D}`
# ---------------------------------------------------------------------
# %%
# We define the domain :math:`\mathcal{D} = [2,4]` and the event :math:`Z(\omega, t) \in \mathcal{D}`:
# %%
domain = ot.Interval([2], [4])
print("D = ", domain)
event = ot.ProcessEvent(Z_proc, domain)
# %%
# We use the Monte Carlo sampling to evaluate the probability:
# %%
MC_algo = ot.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, "]")
view.ShowAll()
|
