37th AIAA/ASME/ASCE/AHS/ASC Structure, Structural Dynamics and Materials Conference

The paper presents a fuzzy formulation for design problems. Design variables, as well as problem parameters, are modeled as fuzzy numbers, characterized by membership functions. An optimization approach and a formulation based on functional requirements on performance parameters is presented. Approximation concepts for an efficient numerical implementation are introduced. Local and global approximations for response functions are used to estimate membership functions for system outputs. Example problems are presented to illustrate the ideas set forth. Introduction Traditional approaches to design assume that all the design data is known with a mathematical precision. That is, it is assumed that the constraints delimit a well defined set of feasible decisions and that the objective function is well defined and easy to formulate. In practice, however, most real systems are imprecise in nature, and their mathematical models can best be represented with some level of imprecision or approximation. The relationships and statements used for the description of the system are, in general, imprecise because of inherent fuzziness in the system (Refs. 1-3). In this context, fuzziness is a type imprecision which is represented by a range and a preference function to describe the possibility or desirability of using a particular value within the range. For problems involving fuzziness in the input data, the design criterion, as well as the performance requirements, constitute a class of alternatives whose boundaries are not well defined. From a mathematical viewpoint, these types of imprecisions have been formulated in the context of fuzzy mathematics (Refs. 4-6). The fuzzy design criterion and performance constraints are characterized by their membership functions. To evaluate these functions, non* Assistant Professor, Civil Engineering Department, Member ASME. **Assistant Professor, Mechanical and Aerospace Engineering Department, Member AIAA, ASME. Copyright © 1996 by H. Jensen and A.E. Sepulveda. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. linear implicit functions of the actual design variables and problem parameters, which are available only in an algorithmic or numerical sense (for example using a finite element method) need to be evaluated. In general, a fuzzy design problem can be formulated in a rather automated framework using optimization techniques or, especially for preliminary and conceptual design, as the evaluation and ranking of several design concepts by evaluating performance and degree of designers satisfaction. Both approaches will be discussed in later sections. In either case, numerically efficient algorithms are required to evaluate system responses and their membership functions. The main objective of this paper is to introduce approximation concepts for fuzzy design. In this context, an efficient numerical methodology is developed to define high quality approximations for response functions and design criterion. These approximations will be used in turn to construct accurate approximations for the corresponding membership functions. Local and global approximations are studied for design objective and constraints. The performance and accuracy of the approximations are studied by comparing the results with the ones obtained by direct simulation techniques. Design Problem Formulation Two approaches to design will be considered in this paper, namely, an optimization approach and a design procedure based on functional requirements on performance parameters. The input data for the problem involves design variables, which the designer can modify in order to increase the overall performance, and problem parameters, which determine certain system characteristics, and are fixed for design purposes. In the context of fuzzy design, both variables and parameters can be considered imprecise having a fuzzy representation. That is, each fuzzy input is assumed to belong to a certain fuzzy set and an associated membership function which characterizes the designers knowledge or preference to that particular input. First Approach: Optimization Problem For a general structural synthesis problem, the constraints are expressed in terms of bounds of displacements, stresses, frequencies, etc. All of these constraints