Achille Messac

Pareto Frontier Based Concept Selection Under Uncertainty, with Visualization

In a recent publication, we presented a new multiobjective decision-making tool for use in conceptual engineering design. In the present paper, we provide important developments that support the next phase in the evolution of the tool. These developments, together with those of our previous work, provide a concept selection approach that capitalizes on the benefits of computational optimization. Specifically, the new approach uses the efficiency and effectiveness of optimization to rapidly compare numerous designs, and characterize the tradeoff properties within the multiobjective design space. As such, the new approach differs significantly from traditional (non-optimization based) concept selection approaches where, comparatively speaking, significant time is often spent evaluating only a few points in the design space. Under the new approach, design concepts are evaluated using a so-calleds-Pareto frontier; this frontier originates from the Pareto frontiers of various concepts, and is the Pareto frontier for thesetof design concepts. An important characteristic of the s-Pareto frontier is that it provides a foundation for analyzing tradeoffs between design objectives and the tradeoffs between design concepts. The new developments presented in this paper include; (i) the notion ofminimally representingthe s-Pareto frontier, (ii) the quantification of concept goodness using s-Pareto frontiers, (iii) the development of an interactive design space exploration approach that can be used to visualizen-dimensional s-Pareto frontiers, and (iv) s-Pareto frontier-based approaches for considering uncertainty in concept selection. Simple structural examples are presented that illustrate representative applications of the proposed method.

Concept Selection Using s-Pareto Frontiers

We introduce the notion of s-Pareto optimality and show how it can be used to improve concept selection in engineering design. Specie c design alternatives are classie ed as s-Pareto optimal when there are no other alternatives from the same or any other general design concept that exhibit improvement in all design objectives. Further, we say that the set of s-Pareto design alternatives comprises the s-Pareto frontier. Under the proposed approach the s-Pareto frontier plays a paramount role in the concept selection process, as it is used to dee ne and classify concept dominance. Thes-Pareto frontier-based concept selection method can becharacterized as onethat capitalizes on the benee ts of computational optimization during the conceptual phase of design, before a general design concept has been chosen. An introduction to s-Pareto optimality and a method for generating s-Pareto frontiersaredeveloped.Anapproachforusing s-Paretofrontierstoperformconceptselectionisalsopresented.The methods proposed can effectively aid in the elimination of dominated design concepts, keep competitive concepts, and ultimately choose a specie cdesign alternative from theselected design concept. A truss design problem is used to illustrate the usefulness of the method. Nomenclature g = vector of inequality constraints h = vector of equality constraints J = aggregate objective function mi = number of points along Ni Ni = ith vector dee ning the utopia plane n = number of design objectives nx = number of design variables T k = relaxation/slack variable for concept k X P = generic point on the utopia plane x = vector of design variables ± = increment by which feasible space is reduced π = vector of design objectives (or design metrics) π i¤ = ith anchor point π ¤k = optimum design objective value for concept k π si¤ = s-anchor point for the ith objective

Exploration of the Effectiveness of Physical Programming in Robust Design

Computational optimization for design is effective only to the extent that the aggregate objective function adequately captures designers preference. Physical programming is an optimization method that captures the designers physical understanding of the desired design outcome in forming the aggregate objective function. Furthermore, to be useful, a resulting optimal design must be sufficiently robust/insensitive to known and unknown variations that to different degrees affect the designs performance. This paper explores the effectiveness of the physical programming approach in explicitly addressing the issue of design robustness. Specifically, we synergistically integrate methods that had previously and independently been developed by the authors, thereby leading to optimal-robust-designs. We show how the physical programming method can be used to effectively exploit designer preference in making tradeoffs between the mean and variation of performance, by solving a bi-objective robust design problem. The work documented in this paper establishes the general superiority of physical programming over other conventional methods (e.g., weighted sum) in solving multiobjective optimization problems. It also illustrates that the physical programming method is among the most effective multicriteria mathematical programming techniques for the generation of Pareto solutions that belong to both convex and non-convex efficient frontiers.

Ability of Objective Functions to Generate Points on Nonconvex Pareto Frontiers

New ground is broken in our understanding of objective functions ability to capture Pareto solutions for multi-objective design optimization problems. It is explained why widely used objective functions fail to capture Pareto solutions when the Pareto frontier is not convex in objective space, and the means to avoid this limitation, when possible, is provided. These conditions are developed and presented in the general context ofn-dimensional objective space, and numerical examples are provided. An important point is that most objective function structures can be made to generate nonconvex Pareto frontier solutions if the curvature of the objective function can be varied by setting one or more parameters. Because the occurrence of nonconvex efficient frontiers is common in practice, the results are of direct practical usefulness.

Visualizing the Optimization Process in Real-Time Using Physical Programming

Visualizing the optimization process in real-time can play a critical role in the human-computer infrastructure of the optimization process. If effectively exploited, visualizing the optimization process in real-lime can greatly increase the effectiveness of practical engineering optimization. The recent exponential growth of computational power has made it possible to begin making the related seminal steps. While the traditional archival literature is nearly void of directly related work, there is currently a flurry of recent activities on the world wide web that point to growing interest in this area. In this paper, (i) we discuss the practical need to develop methods for visualizing the optimization process, (ii) we briefly present the state-of-the-art in this field, which is at its very infancy, and (iii) we propose a new optimization visualization approach that exploits the formulation paradigm of the physical programming method. We importantly note that the visualization method presented here, while inspired by physical programming, is applicable in any optimization framework. Through generic numerical examples, we show that this new method beneficially impacts the optimization process.

Control-structure integrated design with closed-form design metrics using physical programming

The technological area of Control-Struc ture Integrated Design (CSID) has witnessed significant evolution over the past decade and a half. This paper builds on recent developments in the area of computational optimization, and on new analytical methods developed herein, to make three distinct but synergistic contributions. First, this paper explores the application of the Physical Programming optimization method to the CSID problem. Second, a new approach is developed for the closed-form evaluation of time-domain performance metrics for highorder systems (e.g., settling-time). Third, a method for design-variable-linking that uses splines is proposed and is shown to be computationally beneficial while also leading to more physically meaningful structural designs. The numerical results obtained for a controlled idealized rotating structure suggest that the analytical developments of this paper might have practical applicability

A new family of convex splines for data interpolation

Abstract This paper develops a new family of convexity-preserving splines of order n , hereby entitled the CP n -spline, that preserves convexity when derivatives at the data points satisfy some reasonable conditions. The spline comprises four components: a constant term, a first order term, and two n th order binomials. A slope-averaging-method is proposed for the general implementation of the new spline. Numerical results that allow for an assessment of the new spline are provided. In particular, a comparative analysis of the CP n -spline, the cubic spline, and of the Carnicer 92 spline is performed. By varying two parameters, the spline shape can be controlled at the local level, while other conventional means can be used to control the shape at the global level. The CP n -spline has no singularities in the case where inflection points are present. Additionally, a less general form of the CP n -spline that applies to most practical cases can be implemented with extreme ease.

Optimal Design of a Vibration Isolation Mount Using Physical Programming

Vibration isolation tables find application in diverse production and research environments. The structure of a table is such that a designer is forced to compromise among design metrics such as transmissibility, force-disturbance rejection, and controller effort. As both structural and controller parameters impact these design metrics, we chose to perform simultaneous control-structure integrated design (CSID) to optimize the overall performance. We employed a new model that enhances physical insight and used physical programming as the optimization framework. We explored several design scenarios and effectively uncovered the appropriate compromises among the competing objectives. The results highlight the utility of the new model in this design context and the usefulness of physical programming in performing simultaneous CSID.