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ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering | Volume 2 | Issue 4 | Research Paper
Research Papers

Semi-Analytic Probability Density Function for System Uncertainty

[+] Author and Article Information
Ahmad Bani Younes

Assistant ProfessorMem. ASME Department of Aerospace Engineering, Khalifa University, P. O. Box 127788, Abu Dhabi, UAE e-mail: ahmad.younes@kustar.ac.ae

James Turner

Visiting ProfessorMem. ASME Department of Aerospace Engineering, Khalifa University, P. O. Box 127788, Abu Dhabi, UAE e-mail: james.turner@kustar.ac.ae

Manuscript received January 28, 2016; final manuscript received June 13, 2016; published online August 19, 2016. Assoc. Editor: Athanasios Pantelous.

ASME J. Risk Uncertainty Part B 2(4), 041007 (Aug 19, 2016) (7 pages) Paper No: RISK-16-1041; doi: 10.1115/1.4033886 History: Received January 28, 2016; Accepted June 13, 2016
Article
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In general, the behavior of science and engineering is predicted based on nonlinear math models. Imprecise knowledge of the model parameters alters the system response from the assumed nominal model data. One proposes an algorithm for generating insights into the range of variability that can be expected due to model uncertainty. An automatic differentiation tool builds the exact partial derivative models required to develop a state transition tensor series (STTS)-based solution for nonlinearly mapping initial uncertainty models into instantaneous uncertainty models. The fully nonlinear statistical system properties are recovered via series approximations. The governing nonlinear probability distribution function is approximated by developing an inverse mapping algorithm for the forward series model. Numerical examples are presented, which demonstrate the effectiveness of the proposed methodology.
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Figures

Grahic Jump Location
Fig. 1

Performance impact of the STTS over MC

+Performance impact of the STTS over MC
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Performance impact of the STTS over MC
Grahic Jump Location
Fig. 2

Block diagram shows the partial derivative execution of a predefined model

+Block diagram shows the partial derivative execution of a predefined model
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Block diagram shows the partial derivative execution of a predefined model
Grahic Jump Location
Fig. 3

Schematic diagram demonstrates the application of Liouvilles theorem to compute truncation error

+Schematic diagram demonstrates the application of Liouvilles theorem to compute truncation error
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Schematic diagram demonstrates the application of Liouvilles theorem to compute truncation error
Grahic Jump Location
Fig. 4

Phase portrait and uncertainty propagation about the nominal trajectory using fourth-order state transition tensor. (a) Phase portrait of the dynamic motion (for ϵ=1) and the uncertainty propagation about the nominal trajectory using fourth-order state transition tensor. (b) STTS solution validated by MC and exact integration along the nominal trajectory (for ϵ=1) using fourth-order state transition tensor. (c) Fourth-order STTS 3σ solution predicts a good match with the exact propagation.

+Phase portrait and uncertainty propagation about the nominal trajectory using fourth-order state transition tensor. (a) Phase portrait of the dynamic motion (for ϵ=1) and the uncertainty propagation about the nominal trajectory using fourth-order state transition tensor. (b) STTS solution validated by MC and exact integration along the nominal trajectory (for ϵ=1) using fourth-order state transition tensor. (c) Fourth-order STTS 3σ solution predicts a good match with the exact propagation.
View Large  |  Save Figure  |  Download Slide (.ppt)
Phase portrait and uncertainty propagation about the nominal trajectory using fourth-order state transition tensor. (a) Phase portrait of the dynamic motion (for ϵ=1) and the uncertainty propagation about the nominal trajectory using fourth-order state transition tensor. (b) STTS solution validated by MC and exact integration along the nominal trajectory (for ϵ=1) using fourth-order state transition tensor. (c) Fourth-order STTS 3σ solution predicts a good match with the exact propagation.
Grahic Jump Location
Fig. 5

Propagation of PDF over time for ϵ={0,0.2,0.4}: (a) ϵ=0, (b) ϵ=0.2, and (c) ϵ=0.4

+Propagation of PDF over time for ϵ={0,0.2,0.4}: (a) ϵ=0, (b) ϵ=0.2, and (c) ϵ=0.4
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Propagation of PDF over time for ϵ={0,0.2,0.4}: (a) ϵ=0, (b) ϵ=0.2, and (c) ϵ=0.4
Grahic Jump Location
Fig. 6

Propagation of PDF over time for ϵ={0.6,0.8,1.0}: (a) ϵ=0.6, (b) ϵ=0.8, and (c) ϵ=1.0

+Propagation of PDF over time for ϵ={0.6,0.8,1.0}: (a) ϵ=0.6, (b) ϵ=0.8, and (c) ϵ=1.0
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Propagation of PDF over time for ϵ={0.6,0.8,1.0}: (a) ϵ=0.6, (b) ϵ=0.8, and (c) ϵ=1.0

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