1 Copyright © 2002 by ASME Proceedings of DETC’02 ASME 2002 Design Engineering Technical Conferences And Computers and Information in Engineering Conference MONTREAL, Canada, September 29-October 2, 2002 DETC2002/DAC-34092 ON SEQUENTIAL SAMPLING FOR GLOBAL METAMODELING IN ENGINEERING DESIGN Ruichen Jin Wei Chen* Dept. of Mechanical and Industrial Engineering University of Illinois at Chicago Agus Sudjianto V-Engine Engineering Analytical Powertrain Ford Motor Company *Corresponding Author Mechanical and Industrial Engineering (M/C 251) 842 W. Taylor St., University of Illinois at Chicago, Chicago IL 60607-7022 Phone: (312) 996-6072; Fax: (312) 413-0447; E-mail: weichen1@uic.edu ABSTRACT Approximation models (also known as metamodels) have been widely used in engineering design to facilitate analysis and optimization of complex systems that involve computationally expensive simulation programs. The accuracy of metamodels is directly related to the sampling strategies used. Our goal in this paper is to investigate the general applicability of sequential sampling for creating global metamodels. Various sequential sampling approaches are reviewed and new approaches are proposed. The performances of these approaches are investigated against that of the one- stage approach using a set of test problems with a variety of features. The potential usages of sequential sampling strategies are also discussed. NOMENCLATURE d Distance between two sample points d s Scaled distance between two sample points k Number of input variables n Number of sample points l Number of sample points generated at all the previous sampling stages m Number of sample points generated at the new sampling stage X D A sample set with n sample points Dn D D x x x ,..., , 2 1 X P A sample set with all l previous sample points Pl P P x x x ,..., , 2 1 X C A sample set with m new sample points Cm C C x x x ,..., , 2 1 R Correlation matrix INTRODUCTION Mathematical models have been widely used to simulate and analyze complex real world systems in the area of engineering design. These mathematical models, often implemented by computer codes (e.g., Computational Fluid Dynamics and Finite Element Analysis), could be computationally expensive. For example, one run of a finite element model for vehicle crashworthiness can take several hours. While the capacity of computer keeps increasing, to capture the real world systems more accurately, today’s simulation codes are even getting much more complex and unavoidably more expensive. The multidisciplinary nature of design and the need for incorporating uncertainty in design optimization have posed additional challenges. A widely used strategy is to utilize approximation models which are often referred to as metamodels as they provide a model of the model [1], replacing the expensive simulation model during the process. Recent studies on using metamodels in design applications include [2, 3, 4, 5, 6], etc. For dealing with multidisciplinary systems, Meckesheimer, et al. [7] presented a generic integration framework to integrate metamodels from multiple subsystems. An important research issue related to metamodeling is how to achieve a good accuracy of a metamodel with a reasonable number of sample points. While the accuracy of a metamodel is directly related to the metamodeling technique used and the properties of a problem itself, the types of sampling approaches also have direct influences on the performance of a metamodel. Koehler and Owen [8] provided a good review on various sampling approaches for computer experiments. Simpson, et al. [9] compared five sampling strategies and four metamodeling approaches in terms of their