Time variant system reliability problem

The objective is to evaluate the outcrossing rate from a safe to a failure domain in a time variant reliability problem.

We consider the following limit state function, defined as the difference between a degrading resistance r(t) = R - bt and a time-varying load S(t):

..math:

begin{align*} g(t)= r(t) - S(t) = R - bt - S(t) quad forall t in [0,T] end{align*}

The failure domaine is defined by:

g(t) \leq 0

which means that the resistance at t is less thant the stress at t.

We propose the following probabilistic model:

  • R is the initial resistance, and R \sim \mathcal{N}(\mu_R, \sigma_R);

  • b is the deterioration rate of the resistance; it is deterministic;

  • S(\omega,t) is the time-varying stress, which is modeled by a stationary Gaussian process of mean value \mu_S, standard deviation \sigma_S and a squared exponential covariance model C(s,t).

The outcrossing rate from the safe to the failure domain at instant t is defined by:

\nu^+(t) = \lim_{\Delta t \rightarrow 0+} \dfrac{\mathbb{P}\{ g(t) \ge 0 \cap g(t+\Delta t) \leq 0\} }{\Delta t}

For each t, we note the random variable Z_t = R - bt - S_t where S_t = S(., t).

To evaluate \nu^+(t), we need to consider the bivariate random vector (Z_t, Z_{t+\Delta t}).

The event \{ g(t) \geq 0 \cap g(t+\Delta t) <0\} writes as the intersection of both events :

  • \mathcal{E}_t^1 = \{   Z_t \geq 0\} and

  • \mathcal{E}_t^2 = \{   Z_{t+\Delta t} \leq 0\}.

The objective is to evaluate:

\mathbb{P}\{\mathcal{E}_t^1 \cap \mathcal{E}_t^2\} \quad \forall t \in [0,T]

1. Define some useful functions

We define the bivariate random vector Y_t = (bt + S_t, b(t+\Delta t) + S_{t+\Delta t}). Here, Y_t is a bivariate Normal random vector:

  • whith mean [bt, b(t+\delta t)] and

  • whith covariance matrix \Sigma defined by:

..math::

begin{align*} Sigma = left( begin{array}{cc} C(t, t) & C(t, t+Delta t) \ C(t, t+Delta t) & C(t+Delta t, t+Delta t) end{array} right) end{align*}

This function buils Y_t =(Y_t^1, Y_t^2).

def buildNormal(b, t, mu_S, covariance, delta_t = 1e-5):
    sigma = CovarianceMatrix(2)
    sigma[0, 0] = covariance(t, t)[0,0]
    sigma[0, 1] = covariance(t, t+delta_t)[0,0]
    sigma[1, 1] = covariance(t+delta_t, t+delta_t)[0,0]
    return Normal([b*t + mu_S, b*(t+delta_t) + mu_S], sigma)

This function creates the trivariate random vector (R, Y_t^1, Y_t^2) where R is independant from (Y_t^1, Y_t^2). We need to create this random vector because both events \mathcal{E}_t^1 and \mathcal{E}_t^2 must be defined from the same random vector!

def buildCrossing(b, t, mu_S, covariance, R, delta_t = 1e-5):
    normal = buildNormal(b, t, mu_S, covariance, delta_t)
    #return BlockIndependentDistribution([R, normal]): only from the 1.16 version!
    marg = [R, normal.getMarginal(0), normal.getMarginal(1)]
    cop = ComposedCopula([IndependentCopula(1), normal.getCopula()])
    return ComposedDistribution(marg, cop)

This function evaluates the probability using the Monte Carlo sampling. It defines the intersection event \mathcal{E}_t^1 \cap \mathcal{E}_t^2.

def computeCrossingProbability_MonteCarlo(b, t, mu_S, covariance, R, delta_t, n_block, n_iter, CoV):
    full = buildCrossing(b, t, mu_S, covariance, R, delta_t)
    X = RandomVector(full)
    f1 = SymbolicFunction(["R", "X1", "X2"], ["X1 - R"])
    e1 = ThresholdEvent(CompositeRandomVector(f1, X), Less(), 0.0)
    f2 = SymbolicFunction(["R", "X1", "X2"], ["X2 - R"])
    e2 = ThresholdEvent(CompositeRandomVector(f2, X), GreaterOrEqual(), 0.0)
    event = IntersectionEvent([e1, e2])
    algo = ProbabilitySimulationAlgorithm(event, MonteCarloExperiment())
    algo.setBlockSize(n_block)
    algo.setMaximumOuterSampling(n_iter)
    algo.setMaximumCoefficientOfVariation(CoV)
    algo.run()
    return algo.getResult().getProbabilityEstimate() / delta_t

This function evaluates the probability using the Low Discrepancy sampling.

def computeCrossingProbability_QMC(b, t, mu_S, covariance, R, delta_t, n_block, n_iter, CoV):
    full = buildCrossing(b, t, mu_S, covariance, R, delta_t)
    X = RandomVector(full)
    f1 = SymbolicFunction(["R", "X1", "X2"], ["X1 - R"])
    e1 = ThresholdEvent(CompositeRandomVector(f1, X), Less(), 0.0)
    f2 = SymbolicFunction(["R", "X1", "X2"], ["X2 - R"])
    e2 = ThresholdEvent(CompositeRandomVector(f2, X), GreaterOrEqual(), 0.0)
    event = IntersectionEvent([e1, e2])
    algo = ProbabilitySimulationAlgorithm(event, LowDiscrepancyExperiment(SobolSequence(X.getDimension()), n_block, False))
    algo.setBlockSize(n_block)
    algo.setMaximumOuterSampling(n_iter)
    algo.setMaximumCoefficientOfVariation(CoV)
    algo.run()
    return algo.getResult().getProbabilityEstimate() / delta_t

This function evaluates the probability using the FORM algorithm for event systems..

def computeCrossingProbability_FORM(b, t, mu_S, covariance, R, delta_t):
    full = buildCrossing(b, t, mu_S, covariance, R, delta_t)
    X = RandomVector(full)
    f1 = SymbolicFunction(["R", "X1", "X2"], ["X1 - R"])
    e1 = ThresholdEvent(CompositeRandomVector(f1, X), Less(), 0.0)
    f2 = SymbolicFunction(["R", "X1", "X2"], ["X2 - R"])
    e2 = ThresholdEvent(CompositeRandomVector(f2, X), GreaterOrEqual(), 0.0)
    event = IntersectionEvent([e1, e2])
    algo = SystemFORM(SQP(), event, X.getMean())
    algo.run()
    return algo.getResult().getEventProbability() / delta_t

2. Evaluate the probability

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

First, fix some parameters: (\mu_R, \sigma_R, \mu_S, \sigma_S, \Delta t, T, b) and the covariance model wich is the Squared Exponential model. Be careful to the parameter \Delta t which is of great importance: if it is too small, the simulation methods have problems to converge because the correlation rate is too high between the instants t and t+\Delta t. We advice to take \Delta t \simeq 10^{-1}.

mu_S = 3.0
sigma_S = 0.5
l = 10

b = 0.01

mu_R = 5.0
sigma_R = 0.3
R = Normal(mu_R, sigma_R)

covariance = SquaredExponential([l/sqrt(2)], [sigma_S])

t0 = 0.0
t1 = 50.0
N = 26

# Get all the time steps t
times = RegularGrid(t0, (t1 - t0) / (N - 1.0), N).getVertices()

delta_t = 1e-1

Use all the methods previously described:

  • Monte Carlo: values in values_MC

  • Low discrepancy suites: values in values_QMC

  • FORM: values in values_FORM

values_MC = list()
values_QMC = list()
values_FORM = list()

for tick in times:
    values_MC.append(computeCrossingProbability_MonteCarlo(b, tick[0], mu_S, covariance, R, delta_t, 2**12, 2**3, 1e-2))
    values_QMC.append(computeCrossingProbability_QMC(b, tick[0], mu_S, covariance, R, delta_t, 2**12, 2**3, 1e-2))
    values_FORM.append(computeCrossingProbability_FORM(b, tick[0], mu_S, covariance, R, delta_t))
print('Values MC = ', values_MC)
print('Values QMC = ', values_QMC)
print('Values FORM = ', values_FORM)

Out:

Values MC =  [0.0, 0.0, 0.0, 0.00030517578125, 0.00030517578125, 0.0, 0.00030517578125, 0.00030517578125, 0.00030517578125, 0.0, 0.00091552734375, 0.0, 0.0006103515625, 0.0, 0.0, 0.0, 0.0, 0.0006103515625, 0.00091552734375, 0.0, 0.00030517578125, 0.0006103515625, 0.00091552734375, 0.00030517578125, 0.00091552734375, 0.00030517578125]
Values QMC =  [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0006103515625, 0.0, 0.0006103515625, 0.0, 0.0, 0.0, 0.00030517578125, 0.00030517578125, 0.0, 0.00030517578125, 0.0, 0.00091552734375, 0.0006103515625, 0.0, 0.00030517578125, 0.0, 0.0006103515625, 0.001220703125, 0.00091552734375, 0.0006103515625]
Values FORM =  [6.407247221452685e-05, 7.202731340860951e-05, 8.087457491593016e-05, 9.070179169300293e-05, 0.0001016035263802752, 0.00011368175169091608, 0.00012704623305297141, 0.00014181490835112135, 0.00015811426182631293, 0.00017607968850349097, 0.0001958558454373012, 0.00021759698560569734, 0.0002414672698574692, 0.00026764105252706364, 0.0002963031350828803, 0.0003276489830651007, 0.00036188490016252284, 0.00039922815388919713, 0.0004399070467586194, 0.00048416092659680056, 0.0005322401297909951, 0.0005844058510196042, 0.0006409299329987239, 0.0007020945699345352, 0.0007681919182910387, 0.0008395236089949951]

Draw the graphs!

g = Graph()
g.setAxes(True)
g.setGrid(True)
c = Curve(times, [[p] for p in values_MC])
g.add(c)
c = Curve(times, [[p] for p in values_QMC])
g.add(c)
c = Curve(times, [[p] for p in values_FORM])
g.add(c)
g.setLegends(["MC", "QMC", "FORM"])
g.setColors(["red", "blue", 'black'])
g.setLegendPosition("topleft")
g.setXTitle("t")
g.setYTitle("Outcrossing rate")
view = View(g)
view.ShowAll()
plot proba system event

Total running time of the script: ( 0 minutes 8.016 seconds)

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