Vladimir Balabanov

Design and Analysis of Computer Experiments in Multidisciplinary Design Optimization: A Review of How Far We Have Come - Or Not

The use of metamodeling techniques in the design and analysis of computer experiments has progressed remarkably in the past two decades, but how far have we really come? This is the question that we investigate in this paper, namely, the extent to which the use of metamodeling techniques in multidisciplinary design optimization have evolved in the two decades since the seminal paper on Design and Analysis of Computer Experiments by Sacks et al. As part of this review, we examine the motivation for advancements in metamodeling techniques from both a historical perspective and the research itself. Based on current thrusts in the field, we emphasize multi-level/multi-fidelity approximations and ensembles of metamodels, as well as the availability of metamodels within commercial software and for design space exploration and visualization in this review. Our closing remarks offer insight into future research directions – nearly the same ones that have motivated us in the past.

Special Section on Multidisciplinary Design Optimization: Metamodeling in Multidisciplinary Design Optimization: How Far Have We Really Come?

The use of metamodeling techniques in the design and analysis of computer experiments has progressed remarkably in the past 25 years, but how far has the field really come? This is the question addressed in this paper, namely, the extent towhich the use ofmetamodeling techniques inmultidisciplinary design optimization have evolved in the 25 years since the seminal paper on design and analysis of computer experiments by Sacks et al. (“Design and Analysis of Computer Experiments,” Statistical Science, Vol. 4, No. 4, 1989, pp. 409–435). Rather than a technical review of the entire body of metamodeling literature, the focus is on the evolution and motivation for advancements in metamodeling with some discussion on the research itself; not surprisingly, much of the current research motivation is the same as it was in the past. Based on current research thrusts in the field, multifidelity approximations and ensembles (i.e., sets) of metamodels, as well as the availability of metamodels within commercial software, are emphasized. Design space exploration and visualization via metamodels are also presented as they rely heavily onmetamodels for rapid design evaluations during exploration. The closing remarks offer insight into future research directions, mostly motivated by the need for new capabilities and the ability to handle more complex simulations.

Multi-Fidelity Optimization with High-Fidelity Analysis and Low-Fidelity Gradients

The paper introduces a new approach to multi-fidelit y optimization. The approach employs gradient-based optimization, where the one-dimensional search points are evaluated using high-fidelity analysis, while the gradients are evaluated using low-fidelity analysis. Correlation between the results of the high- and low-fidelity analyses is not required. The approach is demonstrated using two example problems. Computational savings in terms of time and the number of high-fidelity analyses are discussed. I. Introduction NE of the obstacles in practical implementation of optimization in industry is a potential high computational cost. An analysis of a complex system may take several hours and even days to complete and optimization requires performing many of these analyses. The number of design variables in optimization directly affects the number of analyses: the more design variables in the problem, the more analyses should be performed. This is especially true for a gradient-based optimization, where the gradients are evaluated using finite-difference calculations. A partial answer to the computational cost problem is Response Surface optimization methods 1-7 ,12 , which do not require gradient information for optimization, thus reducing the required number of analyses. One difficulty with the Response Surface optimization methods is that their range of application is typically limited by about 20 design variables. Another approach to reducing the computational cost is multi-fidelity optimization methods 8-12 . These methods combine high and low-fidelity analyses. One example of employing multi-fidelity optimization is creating a response surface from a relatively small number of high-fidelity analyses, then performing low-fidelity analyses for the same points and creating a response surface for low-fidelity analyses. Next, a correction factor is introduced that helps converting low-fidelity analysis results into the high-fidelity analysis results. The correction may be done for the response surfaces or for the analysis results themselves. Finally, when optimization is performed using the lowfidelity analysis, the results of each low-fidelity analysis is updated using the obtained correction factor. At some intermediate stage of the optimization and at the optimum a high-fidelity analysis is performed to verify the results. If the correlation is not satisfactory, the response surfaces for high and low-fidelity analyses are recreated and the correction factor is reevaluated. The process may be repeated several times. And the correction factor itself may constitute a response surface 12 . One of the disadvantages of this approach is that the results of high and low-fidelity analyses have to be correlated periodically during the course of optimization. For a relatively large number of design variables and responses the correlation may become rather involved, particularly, if each response employs its own correction factor, bringing up the limitation in the number of design variables and responses used. The current paper proposes a modified approach to multi-fidelit y optimization, where the one-dimensional search points in gradient-base d optimization are evaluated using high-fidelity analysis and the finite difference gradient calculations are performed using low-fidelity analysis. One of the advantages of the proposed approach is that with the proper selection of high and low-fidelity analysis models there is no need to correlate the results of the two during optimization. Another advantage is that such an approach removes the potential limitation on the number of design variables and responses employed in response surface based multi-fidelity optimization.

VisualDOC: A Software System for General Purpose Integration and Design Optimization

The main purpose of this paper is to draw attention to existing commercial general-purpose optimization tools. The representative capabilities of such tools are discussed using VisualDOC by Vanderplaats Research and Development, Inc. as an example. The ease of use of VisualDOC allows a person without an optimization background to start applying optimization to his particular problem within a couple of hours after first encountering VisualDOC. This is emphasized by discussing main VisualDOC features. Particular attention is paid to several ways VisualDOC can be interfaced and/or integrated with almost any analysis program. Practical examples of applying VisualDOC to actual industrial problems are presented to emphasize the benefits of applying optimization in any field.

Combined Kriging and Gradient-Based Optimization Method

This paper presents a new Kriging-based optimization method. The goal of this research is to develop a practical and robust general-purpose Kriging-based optimization tool for general design problems. The proposed optimization method efficiently combines Kriging approximations with a gradient-based optimizer. The proposed method is applied to several test problems to examine its efficiency and versatility.

Response Surface Optimization with Discrete Variables

Recent advances in non-gradient based optimization methods (e.g., Genetic Algorithms, Particle Swarm Optimization) enhanced the abilities of discrete, integer, and mixed optimization problems. However, the very nature of non-gradient based algorithms is that the number of analyses required to get to an optimal solution is several orders of magnitude higher than for traditional gradient based optimization methods or response surface optimization methods, when considering continuous problems. Instead of these approaches we propose to use a response surface approximate optimization method modified to work with discrete design variables. In this case whenever it is required to perform the actual analysis of responses for the purpose of fitting a response surface approximation, the design variables will be converted to corresponding discrete values. Two discretization techniques are proposed. We demonstrate that although lacking global search properties like Genetic Algorithms and Particle Swarm Optimization, the discrete response surface optimization provides a computationally efficient way of improving an initial design and getting into a region of an optimum using only discrete points for analysis.

What-If Post-Processing of General-Purpose Optimization Results Using VisualDOC

This paper introduces and demonstrates a new post-processing capability that is available in Version 4.0 of VisualDOC. VisualDOC is a general-purpose optimization tool, available from Vanderplaats Research and Development, Inc. This new post-processing tool allows the user to perform real-time “What-If?” studies in the vicinity of the optimum point found by the optimizer. The tool is demonstrated using a stiffened panel that requires a non-linear analysis.