Probability density evolution analysis of engineering structures via cubature points

Authors:
Jun Xu;Jianbing Chen;Jie Li
Affiliations:
State Key Laboratory of Disaster Reduction in Civil Engineering, School of Civil Engineering, Tongji University, Shanghai, People's Republic of China 200092;State Key Laboratory of Disaster Reduction in Civil Engineering, School of Civil Engineering, Tongji University, Shanghai, People's Republic of China 200092;State Key Laboratory of Disaster Reduction in Civil Engineering, School of Civil Engineering, Tongji University, Shanghai, People's Republic of China 200092
Venue:
Computational Mechanics
Year:
2012

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Abstract

The probability density evolution method (PDEM) is a new approach for stochastic dynamics whereby the dynamic response and reliability evaluation of multi-degree-of-freedom nonlinear systems could be carried out. The apparent similarity and subtle distinction between the ordinary cubature and PDEM are explored with the aid of the concept of the rank of an integral. It is demonstrated that the ordinary cubature are rank-1 integrals, whereas an rank-驴 integral is involved in PDEM. This interprets the puzzling phenomenon that some cubature formulae doing well in ordinary high-dimensional integration may fail in PDEM. A criterion that the stability index does not exceed unity is then put forward. This distinguishes the cubature formulae by their applicability to higher-rank integrals and the adaptability to PDEM. Several kinds of cubature formulae are discussed and tested based on the criterion. The analysis is verified by numerical examples, demonstrating that some strategies, e.g. the quasi-symmetric point method, are preferred in different scenarios. Problems to be further studied are pointed out.