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RP24 reliability problem

set_id problem_id
1 2

challenge set_1

Overview

Category Value
Type DACC
Number of random variables 2
Failure probability, $P_\mathrm{f}$ DACC
Reliability index, $\beta=-\Phi^{-1}(P_\mathrm{f})$ DACC
Number of performance functions 1
Continuity DACC
Reference [Dai2016]

Performance function

DACC

Random variables

The parametrization of distributions follows that of in sec:distributions.

Variable Description Distribution $\theta_1$ $\theta_2$ Mean Std
$X_1$ NA Normal 10.0 3.0 10.0 3.0
$X_2$ NA Normal 10.0 3.0 10.0 3.0

The random variables are mutually independent.

Visualization

DACC

Implementation

DACC

[RP24] https://rprepo.readthedocs.io/en/latest/reliability_problems.html#rp24

[Dai2016] https://rprepo.readthedocs.io/en/latest/references.html#dai2016

from sampling import linesampling as ls
from sampling import dists as dists
from reliability.tnochallenge import problem
RP24 = problem('RP24')
RP24.info
{'description': 'Non-linear performance of implicit type with two normal random variables.',
 'name': 'RP24',
 'kind': 'Challenge set 1',
 'latexFormula': 'NA',
 'latexPF': 'NA',
 'targetBeta': 'NA',
 'Nv': 2,
 'Ng': 1,
 'Distributions': ['normal'],
 'Inputs': {'X[1]': {'dist': 'normal',
   't1': 10.0,
   't2': 3.0,
   'mean': 10.0,
   'std': 3.0},
  'X[2]': {'dist': 'normal', 't1': 10.0, 't2': 3.0, 'mean': 10.0, 'std': 3.0}},
 'index': 3,
 'reference': 'https://rprepo.readthedocs.io/en/latest/references.html#dai2016'}

C24 = RP24.inputs() # Copula function for the input distributions
C24.marginals() # notice the Gumbel-max distribution has rounded values in the table above
(X1 ~ normal(t1=10, t2=3), X2 ~ normal(t1=10, t2=3))

alpha = ls.initialiseAlpha(RP24,C24,gradient=True)
print(alpha)
[0.7071064061528444, -0.7071071562200517]

LS = ls.LineSampling(lines=20,alpha=alpha,linegrid=[2,3,4,5,6])
cl,cr,data = LS.doLineZero(C24,RP24,additional=3)
print('Euclidean norm of design point: [%g, %g]'%(cl,cr))
Euclidean norm of design point: [2.375, 2.75]

RP24.evaluations()
11

pF, b, dp, LSdata, LSdata2, cvar, PFLine, lines = LS.failureProbability(C24,RP24,additional=4,seed=7)
print('failure probability:   [%.2e, %.2e]'%(pF[0],pF[1]))
print('reliability index:     [%g, %g]'%(b[0],b[1]))
print('coeff. of variation:   %g'%cvar)
print('total number of runs:  %i'%RP24.evaluations())
print('total number of lines: %i'%lines)
Line 0 is entirely in the safe domain. This line will be discarded
Line 3 is entirely in the safe domain. This line will be discarded
Line 7 is entirely in the safe domain. This line will be discarded
Line 8 is entirely in the safe domain. This line will be discarded
Line 14 is entirely in the safe domain. This line will be discarded
Line 15 is entirely in the safe domain. This line will be discarded
Line 18 is entirely in the safe domain. This line will be discarded
Line 19 is entirely in the safe domain. This line will be discarded
failure probability:   [2.98e-03, 5.04e-03]
reliability index:     [2.57331, 2.75048]
coeff. of variation:   0.21963
total number of runs:  159
total number of lines: 12

dp
(2.565801785382751, array([ 0.74205755, -0.67033618]))

# there might be a better important direction, but the analysis on line zero needs more definition
cl,cr,data = LS.doLineZero(C24,RP24,additional=10)
print('Euclidean norm of design point: [%g, %g]'%(cl,cr))
Euclidean norm of design point: [2.49805, 2.50098]

LS.plot([LSdata,LSdata2], space=['X'])

png

LS.plotLines2(LSdata2)

png

# Given that the upper bound of the norm of the design point (dp) is smaller than any other norm found on the limit state surface, we can't establish a new direction.
# We therefore run the analysis with more lines to decrease the CoV and possibly look for a better direction.
LS = ls.LineSampling(lines=100,alpha=alpha,linegrid=[0,1,2,3,4,5])
pF, b, dp, LSdata, LSdata2, cvar, PFLine, lines = LS.failureProbability(C24,RP24,additional=20)
print('failure probability:   [%.2e, %.2e]'%(pF[0],pF[1]))
print('reliability index:     [%g, %g]'%(b[0],b[1]))
print('coeff. of variation:   %g'%cvar)
print('total number of runs:  %i'%RP24.evaluations())
print('total number of lines: %i'%lines)
Line 3 is entirely in the safe domain. This line will be discarded
Line 4 is entirely in the safe domain. This line will be discarded
Line 5 is entirely in the safe domain. This line will be discarded
Line 9 is entirely in the safe domain. This line will be discarded
Line 10 is entirely in the safe domain. This line will be discarded
Line 16 is entirely in the safe domain. This line will be discarded
Line 19 is entirely in the safe domain. This line will be discarded
Line 22 is entirely in the safe domain. This line will be discarded
Line 26 is entirely in the safe domain. This line will be discarded
Line 29 is entirely in the safe domain. This line will be discarded
Line 33 is entirely in the safe domain. This line will be discarded
Line 45 is entirely in the safe domain. This line will be discarded
Line 46 is entirely in the safe domain. This line will be discarded
Line 47 is entirely in the safe domain. This line will be discarded
Line 49 is entirely in the safe domain. This line will be discarded
Line 54 is entirely in the safe domain. This line will be discarded
Line 55 is entirely in the safe domain. This line will be discarded
Line 60 is entirely in the safe domain. This line will be discarded
Line 63 is entirely in the safe domain. This line will be discarded
Line 66 is entirely in the safe domain. This line will be discarded
Line 72 is entirely in the safe domain. This line will be discarded
Line 79 is entirely in the safe domain. This line will be discarded
Line 97 is entirely in the safe domain. This line will be discarded
Line 98 is entirely in the safe domain. This line will be discarded
failure probability:   [3.79e-03, 3.79e-03]
reliability index:     [2.67044, 2.67045]
coeff. of variation:   0.0761469
total number of runs:  2294
total number of lines: 76

dp
(2.500036049024229, array([ 0.70903726, -0.70517102]))

LS.plot([LSdata,LSdata2], space='X')

png

LS.plotLines(LSdata)

png

N = 10_000_000
x,z,u = C24.sample(N=N) # shape of x: (6, N), void produces N=100 samples  
g24 = RP24.g()(*x)
i=0
for gi in g24:
    if gi>1:
        i+=1
print('%e'%(i/N))
1.059000e-04

sn = dists.normal(0,1,'z')
beta = -sn.ppf(i/N)
print('%g'%beta)
3.70451