Fast probability integration: Difference between revisions

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'''Fast probability integration''' ('''FPI''') is a method of determining the probability of a class of events, particularly a failure event, that is faster to execute than [[Monte Carlo analysis]].<ref>Murthy ''et al.'', p. 128.</ref> It is used where large numbers of time-variant variables contribute to the reliability of a system. The method was proposed by Wen and Chen in 1987.<ref>Beck & Melchers, p. 2201.</ref>

For a simple failure analysis with one stress variable, there will be a time-variant failure barrier, <math>r(t)</math>, beyond which the system will fail. This simple case may have a deterministic solution, but for more complex systems, such as crack analysis of a large structure, there can be a very large number of variables, for instance, because of the large number of ways a crack can propagate. In many cases, it is infeasible to produce a deterministic solution even when the individual variables are all individually deterministic.<ref>Beck & Melchers, p. 2202.</ref> In this case, one defines a probabilistic failure barrier surface, <math> \mathbf R (t)</math>, over the [[vector space]] of the stress variables.<ref>Beck & Melchers, p. 2201.</ref>

If failure barrier crossings are assumed to comply with the [[Poisson counting process]], an expression for maximum probable failure can be developed for each stress variable. The overall probability of failure is obtained by averaging (that is, [[integral|integrating]]) over the entire variable vector space. FPI is a method of approximating this integral. The input to FPI is a time-variant expression, but the output is time-invariant, allowing it to be solved by [[first-order reliability method]] (FORM) or second-order reliability method (SORM).<ref>Beck & Melchers, p. 2201.</ref>

An FPI package is included as part of the core modules of the [[NASA]]-designed [[NESSUS Probabilistic Analysis Software|NESSUS]] software.<ref>Shah ''et al.'', p. 5.</ref> It was initially used to analyse risks and uncertainties concerning the [[Space Shuttle main engine]],<ref>Shah ''et al.'', p. 5.</ref> but is now used much more widely in a variety of industries.<ref>Riha ''et al.'', p. 3.</ref>

== References ==

{{reflist|23em}}

== Bibliography ==

* Beck, André T.; Melchers, Robert E., "Fatigue and fracture reliability analysis under random loading", pp. &nbsp;2201–2204 in, Bathe, K.J (ed), ''Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics'' June 17–20, 2003, Elsevier, 2003 {{ISBN|008052947X}}.

* ShahMurthy, A.R.;Pappu Shiao, ML.CN.; NagpalMital, V.Subodh K.; ChamisShah, C.CAshwin R., "Design tool developed for probabilistic modeling of ceramic matrix composite strength", [https://books.google.co.ukcom/books?id=oag3AQAAMAAJQ9eAxUlA4x0C&pg=PA127 ''Probabilisticpp. Evaluation127–128] ofin, Uncertainties''Research and& RisksTechnology in Aerospace Components1998''], NASA TechnicalLewis MemorandumResearch 105603Center, March 19921999.

* Riha, David S.; Thacker, Ben H.; Huyse, Luc J.; Enright, Mike P.; Waldhart, Chris J.; Francis, W. Loren; Nicolella, Dniel P.; Hudak, Stephen J.; Liang, Wuwei; Fitch, Simeon H.K., "Applications of reliability assessment for aerospace, automotive, bioengineering, and weapons systems", ch. 1 in, Nikolaidis, Efstratios; Ghiocel, Dan M.; Singhal, Suren, ''Engineering Design Reliability Applications: For the Aerospace, Automotive and Ship Industries'', CRC Press, 2007 {{ISBN|1420051334}}.

* Wen, Y.K.; Chen, H.C., [https://www.sciencedirect.com/science/article/abs/pii/0266892087900063 "On fast integration for time variant structural reliability"], ''Probabalistic Engineering Mechanics'', vol 2, iss. 3, pp. 156-162, September 1987.

* Shah, A.R.; Shiao, M.C.; Nagpal, V.K.; Chamis, C.C., [https://books.google.com/books?id=oag3AQAAMAAJ ''Probabilistic Evaluation of Uncertainties and Risks in Aerospace Components''], NASA Technical Memorandum 105603, March 1992.

* Wen, Y.K.; Chen, H.C., [https://www.sciencedirect.com/science/article/abs/pii/0266892087900063 "On fast integration for time variant structural reliability"], ''Probabalistic Engineering Mechanics'', vol. 2, iss. 3, pp. 156-162&nbsp;156–162, September 1987.

[[Category:Probabilistic models]]

[[Category:Reliability engineering]]