15 © 2013 SRESA All rights reserved Advances in Interval Finite Element Modelling of Structures Rafi L. Muhanna 1 , M. V. Rama Rao 2 , and Robert L. Mullen 3 1 School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA 2 Vasavi College of Engineering, Hyderabad - 500 031 INDIA 3 Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC 29208, USA rafi.muhanna@gtsav.gatech.edu, dr.mvrr@gmail.com, rlm@sc.edu Abstract Finite element models of structures using interval numbers to model parameters uncertainty are presented. The difficulties with simple replacement of real numbers by intervals are reviewed. Methods to overcome these difficulties are presented. Recent advances in providing sharp solution bounds for linear static, linear dynamic, and non-linear static analysis of structures are described based on the authors’ work. Examples of successful interval finite element solutions to structural problems are given. Keywords: Finite Elements, Interval Arithmetic, Nonlinear Structural response Life Cycle Reliability and Safety Engineering Vol.2 Issue 3 (2013) 15-22 1. Introduction Accounting for uncertainty in modern structural analysis and design is unavoidable. Uncertainties can be classified in two general types: aleatory (stochastic or random) and epistemic (subjective) (Yager et al., 1994; Klir and Filger 1988; Oberkampf et al., 2001). Aleatory or irreducible uncertainty is related to inherent variability and is efficiently modeled using probability theory. When data is scarce or there is lack of information, the probability theory is not useful because the needed probability distributions cannot be accurately constructed. In this case, epistemic uncertainty, which describes subjectivity, ignorance or lack of information, can be used. Epistemic uncertainty is also called reducible uncertainty because it can be reduced with increased state of knowledge or collection of more data. Formal theories to handle epistemic uncertainty have been proposed including Fuzzy sets (Zadeh 1965, 1978), Dempster–Shafer evidence theory (Dempster 1967, Yager et al., 1994; Klir and Filger, 1988), possibility theory (Dubois and Prade, 1988), interval analysis (Moore, 1966), and imprecise probabilities (Walley, 1991). It is possible to represent uncertainty or imprecision using imprecise probabilities (Walley, 1991; Sarin, 1978; Weichselberger, 2000) which extend the traditional probability theory by allowing for intervals or sets of probabilities. In general, imprecise probabilities present computational challenges. By imposing some restrictions, Ferson and Donald (1998) have developed a formal Probability Bounds Analysis (PBA) that facilitates computation; Berleant and collaborators independently developed a similar approach (Berleant and Goodman-Strauss, 1998). Also, related methods were developed earlier for Dempster- Shafer representations of uncertainty (Yager, 1986). PBA can represent uncertainty or imprecision, and it has been shown to be useful in engineering design (Aughenbaugh, J. M., and Paredis, C. J. J., 2006). Intervals are the basis for analyzing uncertainty using fuzzy sets, possibility theory or imprecise probability theories. In this paper, we will focus on Interval Finite Element Methods (IFEM) itself and not their applications to various theories of uncertainty. In the next section, interval arithmetic will be summarized. The difficulty with naïve replacement of real numbers with interval values will be demonstrated. Approaches to effectively implement IFEM for linear problems are presented in sections 2 and 3. Finally, non-linear IFEM implementations are presented in section 4. 2. Short Review of Interval Arithmetic Detailed information about interval arithmetic can be found in a series of books and publications such as Hansen (1965); Moore (1966); Alefeld and Herzberger (1983); Neumaier (1990); Rump (1999); and Sun Microsystems (2002). In this paper, the notation follows the recommendation of (Kearfott et al, 2005). Accordingly, interval quantities (interval number, interval vector, interval matrix) are introduced in boldface. Real quantities are introduced in non-bold face.