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 Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...

. 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. For a simple failure analysis with one stress variable, there will be a time-variant failure barrier,

r(t)

, 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. In this case, one defines a probabilistic failure barrier surface,

\mathbf R (t)

, over the

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

of the stress variables. 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, 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 The first-order reliability method, (FORM), is a semi- probabilistic reliability analysis method devised to evaluate the reliability of a system. The accuracy of the method can be improved by averaging over many samples, which is known as Line Samp ...

(FORM) or second-order reliability method (SORM). An FPI package is included as part of the core modules of the

NASA The National Aeronautics and Space Administration (NASA ) is an independent agency of the US federal government responsible for the civil space program, aeronautics research, and space research. NASA was established in 1958, succeeding t ...

-designed NESSUS software. It was initially used to analyse risks and uncertainties concerning the

Space Shuttle main engine The Aerojet Rocketdyne RS-25, also known as the Space Shuttle Main Engine (SSME), is a liquid-fuel cryogenic rocket engine that was used on NASA's Space Shuttle and is currently used on the Space Launch System (SLS). Designed and manufacture ...

, but is now used much more widely in a variety of industries.Riha ''et al.'', p. 3.

References

Bibliography

* Beck, André T.; Melchers, Robert E., "Fatigue and fracture reliability analysis under random loading", pp. 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 . * Murthy, Pappu L.N.; Mital, Subodh K.; Shah, Ashwin R., "Design tool developed for probabilistic modeling of ceramic matrix composite strength"
pp. 127–128
in, ''Research & Technology 1998'', NASA Lewis Research Center, 1999. * 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. * Shah, A.R.; Shiao, M.C.; Nagpal, V.K.; Chamis, C.C.
''Probabilistic Evaluation of Uncertainties and Risks in Aerospace Components''
NASA Technical Memorandum 105603, March 1992. * Wen, Y.K.; Chen, H.C.
"On fast integration for time variant structural reliability"
''Probabalistic Engineering Mechanics'', vol. 2, iss. 3, pp. 156–162, September 1987. Probabilistic models Reliability engineering