Damped Oscillator Reliability

import numpy as np
import matplotlib.pyplot as plt
import uqtestfuns as uqtf

The damped oscillator reliability problem is an eight-dimensional reliability analysis test function [DKDS91, Dub11, DKDS90, BDL11].

Note

The reliability analysis variant differs from the base model. The base model computes the mean-square relative displacement of the secondary spring without reference to the performance of the system.

Test function instance

To create a default instance of the test function:

my_testfun = uqtf.DampedOscillatorReliability()

Check if it has been correctly instantiated:

print(my_testfun)

Function ID      : DampedOscillatorReliability
Input Dimension  : 8 (fixed)
Output Dimension : 1
Parameterized    : True
Description      : Performance function from Der Kiureghian and De Stefano (1990)
Applications     : reliability

Description

The problem poses the reliability of a damped oscillator system as defined in [IK85] (see Damped Oscillator). The reliability of the system depends on the secondary spring as described by the following performance function [Dub11, DKDS90] [1]:

\[ g(\boldsymbol{x}; p) = F_s - p k_s \mathcal{M}(M_p, M_s, K_p, K_s, \zeta_p, \zeta_s, S_0), \]

\[ \mathcal{M}(M_p, M_s, K_p, K_s, \zeta_p, \zeta_s, S_0) = \left( \pi \frac{S_0}{4 \zeta_s \omega_s^3} \frac{\zeta_a \zeta_s}{\zeta_p \zeta_s (4 \zeta_a^2 + \theta^2) + \gamma \zeta_a^2} \frac{(\zeta_p \omega_p^3 + \zeta_s \omega_s^3) \omega_p}{4 \zeta_a \omega_a^4} \right)^{0.5} \]

\[\begin{split} \begin{aligned} \omega_p & = \left( \frac{k_p}{m_p}\right)^{0.5} & \omega_s & = \left(\frac{k_s}{m_s}\right)^{0.5} & \omega_a & = \frac{\omega_p + \omega_s}{2}\\ \gamma & = \frac{m_s}{m_p} & \zeta_a & = \frac{\zeta_p + \zeta_s}{2} & \theta & = \frac{\omega_p - \omega_s}{\omega_a} \\ \end{aligned} \end{split}\]

where \(\boldsymbol{x} = \{ M_p, M_s, K_p, K_s, \zeta_p, \zeta_s, S_0, F_s \}\) is the eight-dimensional vector of input variables probabilistically defined further below and \(p\) is a deterministic parameter of the function (i.e., the peak factor).

The failure state and the failure probability are defined as \(g(\boldsymbol{x}; p) \leq 0\) and \(\mathbb{P}[g(\boldsymbol{X}; p) \leq 0]\), respectively.

Probabilistic input

Based on [DKDS91], the probabilistic input model for the damped oscillator reliability consists of eight independent random variables with marginal distributions shown in the table below.

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print(my_testfun.prob_input)

Function ID     : DampedOscillatorReliability
Input ID        : DerKiureghian1990a
Input Dimension : 8
Description     : Input model #1 for the damped oscillator reliability from
                  Der Kiureghian and De Stefano (1990)
Marginals       :

 No.    Name    Distribution          Parameters                       Description
-----  ------  --------------  -------------------------  --------------------------------------
  1      Mp      lognormal      [0.40048994 0.09975135]                Primary mass
  2      Ms      lognormal     [-4.61014535  0.09975135]              Secondary mass
  3      Kp      lognormal     [-0.01961036  0.1980422 ]         Primary spring stiffness
  4      Ks      lognormal     [-4.62478054  0.1980422 ]        Secondary spring stiffness
  5    Zeta_p    lognormal     [-3.06994228  0.38525317]          Primary damping ratio
  6    Zeta_s    lognormal     [-4.02359478  0.47238073]         Secondary damping ratio
  7      S0      lognormal      [4.60019502 0.09975135]       White noise base acceleration
  8      Fs      lognormal      [2.70307504 0.09975135]   Force capacity of the secondary spring

Copulas         : Independence

In the literature, the force capacity of the secondary spring (i.e., \(F_s\)) have different probabilistic specifications as summarized in the table below.

\(F_S\)	Keyword	Source
\(\mathcal{N}_{\mathrm{log}}(15.0, 1.50)\)	DerKiureghian1990a (default)	[DKDS90]
\(\mathcal{N}_{\mathrm{log}}(21.5, 2.15)\)	DerKiureghian1990b	[DKDS90]
\(\mathcal{N}_{\mathrm{log}}(27.5, 2.75)\)	DerKiureghian1990c	[DKDS90]

Note that the parameters of the log-normal distribution given above correspond to the mean and standard deviation of the log-normal distribution (and not the mean and standard deviation of the underlying normal distribution).

With the higher mean of \(F_s\), the failure probability becomes smaller.

Parameters

The performance function contains a single parameter whose default value is shown below [DKDS91].

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print(my_testfun.parameters)

Function ID  : DampedOscillatorReliability
Parameter ID : DerKiureghian1990
Description  : Parameter set for the damped oscillator reliability
               problem from Der Kiureghian and De Stefano (1990)

 No.    Keyword      Value      Type     Description
-----  ---------  -----------  ------  ---------------
  1       pf      3.00000e+00  float   Peak factor [-]

Reference results

This section provides several reference results of typical UQ analyses involving the test function.

Sample histogram

Shown below is the histogram of the output based on \(10^6\) random points:

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xx_test = my_testfun.prob_input.get_sample(1000000)
yy_test = my_testfun(xx_test)
idx_pos = yy_test > 0
idx_neg = yy_test <= 0

hist_pos = plt.hist(yy_test, bins="auto", color="#0571b0")
plt.hist(yy_test[idx_neg], bins=hist_pos[1], color="#ca0020")
plt.axvline(0, linewidth=1.0, color="#ca0020")

plt.grid()
plt.ylabel("Counts [-]")
plt.xlabel("$\mathcal{M}(\mathbf{X})$")
plt.gcf().set_dpi(150);

Failure probability

Some reference values for the failure probability \(P_f\) and from the literature are summarized in the table below (\(\mu_{F_s}\) is the log-normal distribution mean of \(F_s\)).

\(\mu_{F_s}\)	Method	\(N\)	\(\hat{P}_f\)	\(\mathrm{CoV}[\hat{P}_f]\)	Source
\(15\)	FORM	\(1'179\)	\(2.19 \times 10^{-2}\)	—	[DKDS90, BDL11]
	SS	\(3 \times 10^5\)	\(4.63 \times 10^{-3}\)	\(< 3\%\)	[Dub11]
	Meta-IS[2]	\(464 + 600\)	\(4.80 \times 10^{-3}\)	\(< 5\%\)	[Dub11]
	SVM + SS	\(1'719\)	\(4.78 \times 10^{-3}\)	\(< 4\%\)	[BDL11]
\(21.5\)	FORM	\(2'520\)	\(3.50 \times 10^{-4}\)	—	[DKDS90, BDL11]
	SS	\(5 \times 10^5\)	\(4.75 \times 10^{-5}\)	\(< 4\%\)	[Dub11]
	Meta-IS	\(336 + 400\)	\(4.46 \times 10^{-5}\)	\(< 5\%\)	[Dub11]
	SVM + SS	\(2'865\)	\(4.42 \times 10^{-5}\)	\(< 7\%\)	[BDL11]
\(27.5\)	FORM	\(2'727\)	\(3.91 \times 10^{-6}\)	—	[DKDS90, BDL11]
	SS	\(7 \times 10^5\)	\(3.47 \times 10^{-7}\)	\(< 5\%\)	[Dub11]
	Meta-IS	\(480 + 200\)	\(3.76 \times 10^{-7}\)	\(< 5\%\)	[Dub11]
	SVM + SS	\(4'011\)	\(3.66 \times 10^{-7}\)	\(< 10\%\)	[BDL11]

Note that in the table above the total number of model evaluations for metamodel-based Importance Sampling (Meta-IS) is the sum of training runs and the correction runs [Dub11].

References

[IK85]

Takeru Igusa and Armen Der Kiureghian. Dynamic characterization of two-degree-of-freedom equipment-structure systems. Journal of Engineering Mechanics, 111(1):1–19, 1985. doi:10.1061/(asce)0733-9399(1985)111:1(1).

[DKDS91] (1,2,3)

Armen Der Kiureghian and Mario De Stefano. Efficient algorithm for second-order reliability analysis. Journal of Engineering Mechanics, 117(12):2904–2923, 1991. doi:10.1061/(asce)0733-9399(1991)117:12(2904).

[Dub11] (1,2,3,4,5,6,7,8,9,10)

Vincent Dubourg. Adaptive surrogate models for reliability analysis and reliability-based design optimization. phdthesis, Université Blaise Pascal - Clermont II, Clermont-Ferrand, France, 2011.

[DKDS90] (1,2,3,4,5,6,7,8)

Armen Der Kiureghian and Mario De Stefano. An efficient algorithm for second-order reliability analysis. techreport UCB/SEMM-90/20, Department of Civil and Environmental Engineering, University of California, Berkeley, 1990.

[BDL11] (1,2,3,4,5,6,7)

J.-M. Bourinet, F. Deheeger, and M. Lemaire. Assessing small failure probabilities by combined subset simulation and support vector machines. Structural Safety, 33(6):343–353, 2011. doi:10.1016/j.strusafe.2011.06.001.

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[1]

see, for instance, Eqs. (5.5) and (5.7), pp. 184-185 in [Dub11].

[2]

Metamodel-based Importance Sampling