10 th World Congress on Structural and Multidisciplinary Optimization May 19 - 24, 2013, Orlando, Florida, USA Approximate Fuzzy Structural Analysis Applying Taylor Series and Intervening Variables M.A. Valdebenito 1 , H.A. Jensen 1 , M. Beer 2 and C.A. P´ erez 1 1 Universidad Tecnica Federico Santa Maria, Dept. de Obras Civiles, Av. Espa˜ na 1680, Valparaiso, Chile ({marcos.valdebenito,hector.jensen}@usm.cl) 2 Institute of Risk and Uncertainty, School of Engineering, University of Liverpool, Liverpool, UK (M.Beer@liverpool.ac.uk) 1. Abstract Fuzzy structural analysis is a well developed tool for uncertainty quantification in computational mechan- ics. However, its application may be numerically demanding, as it involves the solution of optimization problems in order to determine extrema of the structural response. This contribution explores the ap- plication of Taylor series and intervening variables for performing fuzzy structural analysis efficiently. Results presented indicate the application of an intervening variable of the reciprocal type may offer significant improvement in terms of accuracy with respect to traditional approximations. 2. Keywords: fuzzy structural analysis, Taylor series, intervening variable, reciprocal variable. 3. Introduction The importance of explicitly considering the effects of uncertainties in structural analysis has been widely acknowledged by the engineering community [8, 13]. A number of approaches for uncertainty quantifica- tion have been developed within the framework of classical probabilities as well as Bayesian techniques, see e.g. [5, 39], etc. Recently, the so-called non traditional approaches for uncertainty quantification have gained considerable attention as well, e.g. [3, 7, 14, 25]. In particular, approaches based on interval analysis and fuzzy analysis have been the subject of active research [29, 30]. It should be noted that irre- spective of the approach used to quantify the effects of uncertainty, the application of such procedures is usually much more involved from a numerical viewpoint than performing deterministic analyses. This is due to the fact the structural performance is not quantified by means of a unique, deterministic quantity but by a set of possible outcomes. In interval analysis (see e.g. [19, 37]), uncertainty in the value of one or more parameters of a model is quantified in terms of bounds. Then, the objective of interval analysis is determining the bounds of a structural response of interest given bounds associated with the unknown input parameters. Fuzzy structural analysis can be interpreted as a sequence of interval analyses as pointed out in [29]. That is, uncertain input parameters are assigned a membership which varies between 0 and 1. For each different value of the membership function, the input variables of a model can be characterized by means of in- tervals. Hence, the structural response can be characterized as an interval as well. In this manner, it is possible to determine the membership for the response function. The latter procedure has been termed in the literature as α-level optimization (see e.g. [29, 31]). A major challenge for the practical implementation of fuzzy structural analysis using the α-level optimiza- tion is the associated numerical costs. For each α-level analyzed (i.e. value of the membership function), it is necessary to determine the minimum and maximum of the structural response given that the un- certain structural parameters lie on a certain interval. Clearly, this is an optimization problem that can be extremely challenging due to issues such as non linearities of the functions involved and the inherent difficulties associated with the determination of global optima [1, 21]. Hence, a number of approaches have been devised in order to overcome this issue, see e.g. [6, 11, 15, 24]. Among these approaches, Taylor series expansion has received considerable attention [10, 17, 26, 37, 27]. This is due to the fact numerical efforts associated with the construction of a Taylor series expansion are limited to a single structural analysis (calculation of stiffness matrix inverse) plus additional assembly and multiplication of structural matrices for calculating sensitivities [20]. Nonetheless, Taylor series may not be always appropriate as they may fail in capturing nonlinearities of the functions being approximated. In view of this issue, this contribution explores the application of Taylor series considering intervening variables. The latter vari- ables have been applied customarily in the field of structural optimization (see e.g. [21, 35, 38]) and have also been considered within the framework of structural reliability and classical probabilities [16, 40, 42]. 1