On the Number of Exits Across the Boundary of a Region by a Vector Stochastic Process

Belyaev, Yu. K.

doi:10.1137/1113036

Cited by 106 publications

(28 citation statements)

References 4 publications

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“…Several analytical results regarding the outcrossing rate of a random process through a threshold are available in the literature: † Rice's formula [18,19] for scalar differentiable processes, which has been generalized by Belayev [20] for vector differentiable processes. † Specialization to differentiable Gaussian processes [21] † Case of scalar and vector rectangular wave renewal processes [22].…”

Section: Analytical Formulations and Asymptotic Integrationmentioning

confidence: 99%

The PHI2 method: a way to compute time-variant reliability

Andrieu-Renaud¹,

Sudret²,

Lemaire³

2004

Reliability Engineering & System Safety

366

124

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Time-variant reliability problems appear in the engineering practice when (a) the material properties of the structure deteriorate in time or (b) random loading modelled as random processes is involved. This paper presents a method called PHI2 which is based on the outcrossing approach and allows to solve such problems using classical time-invariant reliability tools such as FORM/SORM methods. The PHI2 method is first presented. Then it is benchmarked with the well-established 'asymptotic methods' [Stochast. Process. Appl. 13 (1988) The PHI2 method appears more accurate in all cases. As an application example, the method is finally applied on a case representing a mechanical system (a beam) placed in an environment that can have exceptional configuration. q

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Section: Analytical Formulations and Asymptotic Integrationmentioning

confidence: 99%

The PHI2 method: a way to compute time-variant reliability

Andrieu-Renaud¹,

Sudret²,

Lemaire³

2004

Reliability Engineering & System Safety

366

124

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“…For a vector process, the mean out-crossing rate may be calculated by considering out-crossings perpendicular to the boundary as discussed by Belyaev [3]. Consider a hypercubic safe region described by Equation (2).…”

Section: 22mentioning

confidence: 99%

“…References [1,2]), while the vector process counterpart has received much less attention. Theoretical discussions have been made [3], but results have been presented only for the cases of independent processes and very small dimensions [4,5], presumably because the hardware speed required for the calculations was not available at the time that the interest was focused on this topic.…”

Section: Introductionmentioning

confidence: 99%

Analytical approximation for stationary reliability of certain and uncertain linear dynamic systems with higher-dimensional output

Taflanidis

Beck

2006

Earthquake Engng Struct. Dyn.

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SUMMARYAn analytical approximation for the calculation of the stationary reliability of linear dynamic systems with higher-dimensional output under Gaussian excitation is presented. For systems with certain parameters theoretical and computational issues are discussed for two topics: (1) the correlation of failure events at di erent parts of the failure boundary and (2) the approximation of the conditional out-crossing rate across the failure boundary by the unconditional one. The correlation in the ÿrst topic is approximated by a multivariate integral, which is evaluated numerically by an e cient algorithm. For the second topic some existing semi-empirical approximations are discussed and a new one is introduced. The extension to systems with uncertain parameters requires the calculation of a multi-dimensional reliability integral over the space of the uncertain parameters. An existing asymptotic approximation is used for this task and an e cient scheme for numerical calculation of the ÿrst-and second-order derivatives of the integrand is presented. Stochastic simulation using an importance sampling approach is also considered as an alternative method, especially for cases where the dimension of the uncertain parameters is moderately large. Comparisons between the proposed approximations and Monte Carlo simulation for some examples related to earthquake excitation are made. It is suggested that the proposed analytical approximations are appropriate for problems that require a large number of consistent error estimates of the probability of failure, as occurs in reliability-based design optimization. Numerical problems regarding computational e ciency may arise when the dimension of both the output and the uncertain parameters is large.

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“…This concludes an integration of local out-crossing rates over boundary of a fixed safe domain [2], an integration of (conditional) out-crossing rates over time and an integration of (conditional) failure probabilities over random resistance parameter [3].A large amount of literature on time-variant reliability is based on so called asymptotic approaches, see [4] for a comprehensive review. However the mathematics of these approaches as well as their implementation reveal complex and can hardly be found in commercial software COMREL (RCP Consult, 1998) being a remarkable exception.…”

Section: Introductionmentioning

confidence: 99%

A crossing rate method to reliability analysis of antenna structure under fluctuating wind

Wang

Zhan

et al. 2012

Proceedings of the IEEE 2012 Prognostics and System Health Management Conference (PHM-2012 Beijing)

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In order to investigate the complete problem of antenna design under wind load in a statistical way, a time variant reliability procedure is proposed in this paper, which differs from conventional reliability approaches for antenna design. It considers the influence of the Gaussian behavior of wind-induced pressures of antenna in its lifetime on the reliability index of antenna. It is shown that the reliability of under wind load significantly reduces when time is considered. A time independent reliability approach which does not consider degradation of may greatly over estimate the reliability index for antenna under fluctuating wind load.The design recommendations are also given in this paper. The procedures proposed in this paper represent a suitable way to evaluate the safety of antenna under wind load.

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On the Number of Exits Across the Boundary of a Region by a Vector Stochastic Process

Cited by 106 publications

References 4 publications

The PHI2 method: a way to compute time-variant reliability

The PHI2 method: a way to compute time-variant reliability

Analytical approximation for stationary reliability of certain and uncertain linear dynamic systems with higher-dimensional output

A crossing rate method to reliability analysis of antenna structure under fluctuating wind

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