Skip to content

reliability_TNO_RP22.md

Latest commit

 

History

History
207 lines (122 loc) · 4.49 KB

reliability_TNO_RP22.md

File metadata and controls

207 lines (122 loc) · 4.49 KB

RP22 reliability problem

set_id problem_id
-1 1

Tutorial set_1

Overview

Category Value
Type Symbolic
Number of random variables 5
Failure probability, $P_\mathrm{f}$ 4.16 1e-3
Reliability index, $\beta=-\Phi^{-1}(P_\mathrm{f})$ 2.64
Number of performance functions 1
Reference [Grooteman2011]

Performance function

$$g({\bf X}) = 2.5 - (X_1 + X_2)/ \sqrt{2} + 0.1\ (X_1 - X_2)^2$$

Random variables

The parametrization of distributions follows that of in.

Variable Description Distribution $\theta_1$ $\theta_2$ Mean Std
$X_1$ NA Normal 0.0 1.0 0.0 1.0
$X_2$ NA Normal 0.0 1.0 0.0 1.0

The random variables are mutually independent.

[Grooteman2011] https://rprepo.readthedocs.io/en/latest/references.html#grooteman2011

from sampling import linesampling as ls
from sampling import dists as dists
from reliability.tnochallenge import problem
RP22 = problem('RP22')
RP22.info
{'description': 'Tutorial set problem. Quadratic function with mixed term, convex.',
 'name': 'RP22',
 'kind': 'Tutorial set 1',
 'latexFormula': 'NA',
 'latexPF': 'NA',
 'targetBeta': 'NA',
 'Nv': 2,
 'Ng': 1,
 'Distributions': ['normal'],
 'Inputs': {'X[1]': {'dist': 'normal',
   't1': 0.0,
   't2': 1.0,
   'mean': 0.0,
   'std': 1.0},
  'X[2]': {'dist': 'normal', 't1': 0.0, 't2': 1.0, 'mean': 0.0, 'std': 1.0}},
 'index': 3,
 'reference': 'https://rprepo.readthedocs.io/en/latest/references.html#dai2016'}

C22 = RP22.inputs() # Pi-copula (independent) function for the input distributions
C22.marginals() # notice the Gumbel-max distribution has rounded values in the table above
(X1 ~ normal(t1=0, t2=1), X2 ~ normal(t1=0, t2=1))

alpha = ls.initialiseAlpha(RP22,C22,gradient=True)
print(alpha)
[0.7071067811865476, 0.7071067811865476]

LS = ls.LineSampling(lines=20,alpha=alpha,linegrid=[1,2,3,4])
cl,cr,data = LS.doLineZero(C22,RP22,additional=3)
print('Euclidean norm of design point: [%g, %g]'%(cl,cr))
Euclidean norm of design point: [2.25, 2.625]

RP22.evaluations()
10

pF, b, dp, LSdata, LSdata2, cvar, PFLine, lines = LS.failureProbability(C22,RP22,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'%RP22.evaluations())
print('total number of lines: %i'%lines)
failure probability:   [3.68e-03, 6.09e-03]
reliability index:     [2.50678, 2.67968]
coeff. of variation:   0.0866671
total number of runs:  170
total number of lines: 20

dp
(2.562598274716007, array([0.70088701, 0.71327231]))

# There are no new candidate design points. So the analysis can stop

Plotting the results

LS.plotLines3([LSdata,LSdata2])

png

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

png

LS = ls.LineSampling(lines=100,alpha=alpha,linegrid=[1,2,3,4,5,6])
pF, b, dp, LSdata, LSdata2, cvar, PFLine, lines = LS.failureProbability(C22,RP22,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'%RP22.evaluations())
print('total number of lines: %i'%lines)
failure probability:   [3.46e-03, 5.74e-03]
reliability index:     [2.52744, 2.70049]
coeff. of variation:   0.0512113
total number of runs:  1170
total number of lines: 100

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

png