Symbolic design optimization : a computer aided method to increase monotonicity through variable reformulation

Abstract: "Monotonicity analysis, developed by [Wilde 78] and [Papalambros 79], is an approach to simplifying and solving some nonlinear, constrained global optimization problems without iterative numerical calculations. Unfortunately, monotonicity analysis is limited to problems in which the objective function and constraints vary monotonically with the design variables. To alleviate this limitation it is sometimes possible to reformulate the design problem to increase the degree of monotonicity and thereby facilitate the complete or partial application of monotonicity analysis procedures.These useful reformulations are accomplished by a transformation to alternative design parameters, such as a critical ratio, a nondimensional parameter, or a simple difference; e.g. the ratio of surface area to volume for heat transfer loss, the Reynold's number in fluid mechanics, or the velocity difference across a fluid coupling. We have developed a method by which the alternative parameters are chosen for physical significance and for the ability to reduce the number of nonmonotonic variables in a system of constraints. Rules have been developed for the creation of physically significant new parameters from the algebraic combination of the original parameters.The rules are based on engineering principles and rely on knowledge about what a parameter physically represents rather than other qualitiessuch as dimensions. A computer based system, called EUDOXUS, has been developed to automate this procedure. The method and its implementation have demonstrated successful results for highly nonlinear, nonmonotonic, and coupled parameterized designs in various mechanical engineering domains.