NLOscillatorSensitivity

class NLOscillatorSensitivity

Class to define a Oscillator sensitivity benchmark problem.

Methods

getFirstOrderIndices()

Returns the first order Sobol' sensitivity indices.

getFunction()

Returns the function.

getInputDistribution()

Returns the input distribution.

getName()

Returns the name of the problem.

getTotalOrderIndices()

Returns the total order Sobol' sensitivity indices.
__init__()

Create a nonlinear oscillator sensitivity problem.

The function is defined by the equation:

g(x) = fs - 3*ks*\sqrt{\pi*S0/(4.*xis*omegas^3)* xi_a*xis/(xip*xis*(4.*xi_a^2+theta^2)+gamma*xi_a^2)* (xip*omegap^3+xis*omegas^3)*omegap/(4.*xi_a*omegaa^4)}

where * omegap = np.sqrt(kp/mp) * omegas = np.sqrt(ks/ms) * omegaa = 0.5*(omegap+omegas) * gamma = ms/mp * xi_a = 0.5*(xip+xis) * theta = 1./omegaa*(omegap-omegas)

The input random variables are independent.

The aim is to assess reliability of a two-degree-of-freedom primary-secondary system under a white noise base acceleration.

The basic variables characterizing the physical behavior are * the masses mp and ms * spring stiffnesses kp and ks * natural frequencies ωp and ωs * damping ratios ξp and ξs

where the subscripts p and s respectively refer to the primary and secondary oscillators.

The variables in the model are: * Fs : the force capacity of the secondary spring, * S0 is the intensity of the white noise, * ωp=(kp/mp)1/2, * ωs=(ks/ms)1/2, * ωa=(ωp+ωs)/2 the average frequency ratio, * γ=ms/mp the mass ratio, * ξa=(ξp+ξs)/2 the average damping ratio and * r=(ωp−ωs)/ωa a tuning parameter.

Parameters:
None.

Notes

The dimension of this problem cannot be changed.

The model was first introduced in (Iooss, 2015).

The two interesting characteristics of this application test-case are its set of non-normal basic random variables and the fact that it suffers from a highly nonlinear limit-state surface (which prevents from using any FORM-based approach). Moreover, following, it seems relevant to consider the mean of the force capacity Fs as the most influent distribution parameter on the failure probability.

The analysis is the following.

  • The most influential parameter is Fs with first order indice equal to 0.4. It has no interaction with other variables.

  • The second most influential parameter is kp, with first order indice equal to 0.18. It has small interactions with other parameters, since its total order indice is equal to 0.23.

  • The variable S0 is insignificant.

References

  • A. Der Kiureghian, M. De Stefano, Efficient algorithm for second-order reliability analysis, J. Eng. Mech. 117 (12) (1991) 2904–2923.

  • J.-M. Bourinet, F. Deheeger, M. Lemaire, Assessing small failure probabilities by combined subset simulation and Support Vector Machines, Struct. Saf. 33 (6) (2011) 343–353.

  • J.-M. Bourinet, Rare-event probability estimation with adaptive support vector regression surrogates, Reliab. Eng. Syst. Saf. 150 (2016) 210–221.

  • V. Dubourg, Adaptive surrogate models for reliability analysis and reliability-based design optimization, PhD thesis, Université Blaise Pascal – Clermont II, 2011.

  • Vincent Chabridon, Mathieu Balesdent, Jean-Marc Bourinet, Jérôme Morio, Nicolas Gayton. Evaluation of failure probability under parameter epistemic uncertainty: application to aerospace system reliability assessment. Aerospace Science and Technology, Elsevier, 2017, 69, pp.526-537.

  • Analyse de sensibilité fiabiliste avec prise en compte d’incertitudes sur le modèle probabiliste, Thèse présentée par Vincent Chabridon, 2019.

Examples

>>> import otbenchmark as otb
>>> problem = otb.FloodingSensitivity()
getFirstOrderIndices()

Returns the first order Sobol’ sensitivity indices.

Parameters:
None.
Returns:
firstOrderIndices: ot.Point

The first order sensitivity indices.

getFunction()

Returns the function.

Parameters:
None.
Returns:
function: ot.Function

The function.

getInputDistribution()

Returns the input distribution.

Parameters:
None.
Returns:
distribution: ot.Distribution

The distribution.

getName()

Returns the name of the problem.

Parameters:
None.
Returns:
name: str

The name.

getTotalOrderIndices()

Returns the total order Sobol’ sensitivity indices.

Parameters:
None.
Returns:
totalOrderIndices: ot.Point

The total order sensitivity indices.