Fuzzy finite element approach for the analysis of imprecisely defined systems

Many engineering systems are too complex to be defined in precise mathematical terms. They often contain information and features that are vague, imprecise, qualitative, linguistic, or incomplete. The traditional deterministic and probabilistic techniques are not adequate to analyze such systems. This paper aims at developing a fuzzy finite element approach for the analysis of imprecisely defined systems. The development of the methodology starts from the basic concepts of fuzzy numbers of fuzzy arithmetic and implements suitably defined fuzzy calculus concepts such as differentiation and integration for the derivation, manipulation, and solution of the finite element equations. Simple stress analysis problems involving vaguely defined geometry, material properties, external loads, and boundary conditions are solved to establish and to illustrate the new procedure. The approach developed is applicable to systems that are described in linguistic terms as well as those that are described by incomplete information. If complete data are known, the method handles the information similar to that of a probabilistic approach. The present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures.