Finite Element Structural Analysis using Imprecise Probabilities Based on P-Box Representation Hao Zhang 1 , R. L. Mullen 2 , and R. L. Muhanna 3 1 University of Sydney, Sydney 2006, Australia, haozhang@usyd.edu.au 2 University of South Carolina, Columbia, SC 29208, USA, rlm@cec.sc.edu 3 Georgia Institute of Technology, Atlanta 31407, USA, rafi.muhanna@gtsav.gatech.edu Abstract: Imprecise probability identifies a number of various mathematical frameworks for making decisions when precise probabilities (or PDF) are not known. Imprecise probabilities are normally associated with epistemic sources of uncertainty where the available knowledge is insufficient to construct precise probabilities. While there is no “unified theory of imprecise probabilities”, most frameworks describe the uncertainty in terms of bounded possibilities or unknown probabilities between a specified lower and upper bounds. In this paper, the authors’ previous developed interval finite element methods are extended to compute p- box structures of a finite element solution where loading parameters are described by p-box structures. Both discrete p-box structures and interval p-box Monte Carlo algorithms are presented along with example problems that illustrate the capabilities of the new methods. The computational efficiency of the p-box finite element methods is also presented. Keywords: Uncertainty; Imprecise Probability; P-Box; Interval Finite Elements. 1. Introduction Engineering analysis and design require users to make several assumptions, typically at different levels. One of these levels is the choice of the underlying mathematical model of the system. Another level is the description of the model parameters. Assumptions are usually made to facilitate processing the analysis and design, which result in that the nature of engineering computation is conditioned by a priori assumptions. In a deterministic analysis, the geometry, loads, and material properties are assumed to have specific values. Conversely, if these assumptions do not hold, a new model with a new set of assumptions has to be used to reflect the real variations in geometry, load and material properties. The best candidate methods for handling this new situation are probabilistic methods. However, they assume complete information about the variability of all parameters and their respective Probability Density Functions (PDFs) are available. The available information, however, might range from scarce or limited to comprehensive. When there is limited or insufficient data, designers fall back to deterministic analysis, which is an irrational approach that does not account for parameters variability. On the other hand when more data is available but insufficient to completely justify a particular PDF, designers would assume a probability distribution that might not represent the real behavior of the system. More advanced Bayesian methods can treat the incomplete information in describing PDFs by updating parameters of a probability distribution. With a Bayesian approach, subjective judgments are required to estimate the random variables (i.e., form of the PDF), and thus the approach remains a subjective representation of uncertainty. In such a case, we find ourselves in 4th International Workshop on Reliable Engineering Computing (REC 2010) Edited by Michael Beer, Rafi L. Muhanna and Robert L. Mullen Copyright © 2010 Professional Activities Centre, National University of Singapore. ISBN: 978-981-08-5118-7. Published by Research Publishing Services. doi:10.3850/978-981-08-5118-7 013 211