Update Schemes for Global Approximations in Robust Design Optimization
Florian Jurecka, Kai‐Uwe Bletzinger, M. Ganser · mediaTUM – the media and publications repository of the Technical University Munich (Technical University Munich) · 2005
1. AbstractThe goal of robust design optimization is to improve the quality of a product or process by minimizingthe deterioratinge ects ofvariableoruncertain parameters. This robustnesscan be achievedby di erentapproaches using formulations of statistical decision theory which all require many function evaluationsthroughout the optimization. In cases where no closed-form descriptions of the objective are availableand pointwise solutions are expensive to evaluate, metamodelling techniques based on design of exper-iments are used to replace the actual numerical analysis codes. In this paper we present a well-suitedupdate criterion for global approximation models which have a local measure for the model variance to nd a robust optimal design up to the desired precision both in the objective and the robustness.2. Keywords: robust design, surrogate models, sequential approximate optimization.3. IntroductionIn many engineering disciplines variations and uncertainties are encountered in design parameters orenvironmental variables e.g. in form of dimensions of structural members, material properties or appliedloads. For reasons of simpli cation and reduced numerical e ort their e ects are reduced to meansor fractiles in most of today’s engineering practice. Genichi Taguchi introduced robust (parameter)design in the 1980s to describe a method to reduce product or process variation by choosing levels ofcontrol factors that make the system insensitive to changes in noise factors which represent the source ofvariation. This approach is not dealing with the possibilities to reduce the variance of the noise variablesbut it focuses on reducing the e ects of variations and uncertainties existing in the input parameterson the system performance [1, 2]. There are several methods available for achieving this goal of qualityengineering [3, 4].For the task of robust design optimization we must distinguish between two di erent types of inputparameters. The design variables (sometimes also called control parameters) can be chosen or controlledby the designer during the design process and the optimization e.g. cross-sections of trusses and beamsor other design dimensions. Noise parameters are subject to variations that cannot be controlled butmight be known to the designer and describable as by probability density functions. Sometimes theonly information available about noise parameters is an interval of possible occurrences. In the commoncase where design variables also include stochastic properties, they are split and treated as two separatesystem parameters: a deterministic value e.g. the mean as design variable and the deviation from thatvalue as noise parameter.Taguchi’s contributions started a process which made aware the importance of parameter variationsto many design engineers and statisticians, thus his methods were reviewed, criticized and enhancedthroughout the years [6]. The original concept was based on classical experimental design with alldesign variables being varied according to an orthogonal array (inner array).At each design variable setting the noise variables are varied according to a second orthogonal ar-ray and the response data gained at the di erent replications are used to estimate a signal-to-noise-ratio (SNR). The resulting array of estimated SNRs is used to perform a standard analysis of variance(ANOVA) to identify design variable settings that yield a robust performance.In recent publications di erent mathematical formulations for quantifying quality loss due to vari-ations in system performance were introduced as enhancement to the SNR concept of Taguchi [3].Furthermore statistical design of experiments o ers a broad variety of experimental designs which arein many cases more ecient than orthogonal arrays [7, 8, 9]. All of these formulations have in commonthat they need substantially more (numerical) experiment evaluations compared to deterministic opti-mization and thus the numerical e ort is signi cantly higher. In many cases this leads to the need forsurrogate models (also called metamodels) to considerably reduce computing time [10, 11, 12].1