Fuzzy finite element analysis of heat conduction problems with uncertain parameters Bart M. Nicolaï a,⇑ , Jose A. Egea b , Nico Scheerlinck c , Julio R. Banga d , Ashim K. Datta e a Flanders Centre of Postharvest Technology/BIOSYST-MeBioS, Catholic University of Leuven, Willem de Croylaan 42, 3001 Leuven, Belgium b Department of Applied Mathematics and Statistics, Technical University of Cartagena (UPCT), Paseo Alfonso XIII 52, 30203 Cartagena, Spain c Scientific Computing Group, Computer Science Department, Catholic University of Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium d (Bio)Process Engineering Group, IIM-CSIC, C/Eduardo Cabello 6, 36208 Vigo, Spain e Biological and Environmental Engineering, Cornell University 208 Riley-Robb Hall Ithaca, NY 14853-5701, USA article info Article history: Received 27 July 2010 Received in revised form 18 September 2010 Accepted 25 September 2010 Available online 1 October 2010 Keywords: Fuzzy Interval Finite element Heat conduction Uncertainty Numerical solution abstract In this article we have used four different global optimisation algorithms for interval finite element anal- ysis of (non)linear heat conduction problems: (i) sequential quadratic programming (SQP), (ii) a scatter search method (SSm), (iii) the vertex algorithm, and (iv) the response surface method (RSM). Their per- formance was compared based on a thermal sterilisation problem and a food freezing problem. The ver- tex method proved to be by far the fastest method but is only effective if the solution is a monotonic function of the uncertain parameters. The RSM was also fast albeit much less than the vertex method. Both SQP and SSm were considerably slower than the former methods; SQP did not converge to the real solution in the food freezing test problem. The interval finite element method was used as a building block for a fuzzy finite element analysis based on the a-cuts method. The RSM fuzzy finite element method was identified as the fastest algorithm among all the tested methods. It was shown that uncer- tain parameters may cause large uncertainties in the process variables. The algorithms can be used to obtain more realistic modelling of food processes that often have significant uncertainty in the model parameters. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction A wide variety of heat and mass transfer models have been developed to describe the most important food processes, includ- ing sterilisation, freezing and thawing, microwave processing, dry- ing, etc. (Sablani et al., 2006; Datta, 2008). Such models have great prospects for food process design and optimisation. Because of their complexity they are typically solved using the finite element or some other numerical methods. Most models to date are deter- ministic in the sense that they are based on the assumption that the product and process parameters are known in advance. In real- ity, for many food processes this is not the case: the parameters are either variable or uncertain, or a combination of both. Variability means that parameters such as the thermal conductivity may vary between objects or even within an object; variability can be de- scribed by probabilistic measures. Uncertainty has been defined as ‘a potential deficiency in any phase or activity of the modelling process that is due to lack of knowledge’ (Moens and Vandepitte, 2005). Uncertainty is essentially caused by incomplete information resulting from vagueness, nonspecificity (the availability of differ- ent models for the same phenomenon), or even dissonance (con- flicting evidence of the modelled process). There is overlap of both definitions: for example, if the probabilistic description of a variable parameter is not available this can also be considered as an uncertainty. In contrast to such an uncertain variability, a cer- tain variability is completely specified in a probabilistic sense. Uncertainty can in principle be reduced through appropriate experiments; variability is irreducible as it is an inherent property of the object or process. Obviously, if a model parameter is vari- able, so is the model prediction. Likewise, uncertain parameters re- sult in uncertain model predictions. In food process design both variable parameters and uncertain- ties are often taken into account through safety factors. These are often implicitly introduced by basing the design calculations on a ‘worst case’ (fail-safe) process, in which the values leading to the most conservative process are assigned to the model parameters. Alternatively, process design could be based on statistical consid- erations, taking into account all uncertainties associated with the process. Such an analysis relies on methods to calculate the vari- ability or uncertainty of the model predictions. Several probabilis- tic methods have been introduced to calculate the propagation of variability through the heat and mass transfer model. The Monte 0260-8774/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2010.09.017 ⇑ Corresponding author. Tel.: +32 16 322375; fax: +32 16 322955. E-mail address: bart.nicolai@biw.kuleuven.be (B.M. Nicolaï). Journal of Food Engineering 103 (2011) 38–46 Contents lists available at ScienceDirect Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng