Modeling and Information Processing

Response surface construction

Key words: response surface, neural networks, bootstrap, multi-objectives, selection of models

To reduce the calls to the computing code, it is replaced by a substitute model with a fast response time on which the optimization algorithm will be applied. Generally, this model may come from physics and when this is not possible regression models are used. This model, constructed from a database generated by the computing code, provides an approximate value of this code at any point as well as an estimate of the gradient. To improve the reduced model's accuracy, the database is iteratively enriched by adding, on the one hand, points in those places where the model is most poorly defined and, on the other, points obtained by minimization. A "bootstrap" type statistical method is used in order to limit the number of them by choosing those that will contribute the most information because they require new calculations.

The reduced models constructed from neural networks, validated in the framework of the INTELLECT D.M project, take discontinuities into account correctly and provide great flexibility for adapting to the different behaviors of the response surface. We developed an iterative enrichment algorithm under MATLAB that is independent of the reduced model. This enrichment process was successfully applied for the single objective and multi-objective optimization of a premix tube in the context of the European INTELLECT D.M. project using, on the on hand, the CEDRE code and, on the other, a neural network type reduced model.

It should be possible to construct several reduced models and compare them in order to define the response surface best adapted to the problem studied. Due to their specific construction, it seems likely that certain reduced models are better adapted than others for describing the behavior of the objective function to be optimized. Selection criteria have been set up for objectively comparing models and automating the process.

The results were presented during the SMSMEO06 conference at the Technical University of Denmark and to the 42nd Colloque d'Aérodynamique Appliquée in Nice in March 2007.

• Patricia Klotz, Nathalie Bartoli, Laurence Cornez, Manuel Samuelides and Philippe Villedieu, Database enrichment for optimization with artificial neural networks, Second International Workshop on Surrogate Modeling and Space Mapping for Engineering Optimization (SMSMEO-06), Technical University of Denmark, November 9-11, 2006 [PDF]

• Patricia Klotz, Nathalie Bartoli, Laurence Cornez, Renaud Lecourt and Nicolas Savary. Database enrichment for multi-objective optimization, AAAF, 42nd Colloque d'Aérodynamique Appliquée : Couplages et optimisation multidisciplinaires, Nice, March 19-21, 2007

Two articles stemming from these presentations have been submitted to OPTE and AST and are available in the publications.

•   P. Klotz, N. Bartoli, L. Cornez, R. Lecourt, and N. Savary. Database enrichment for multi-objective optimization, submitted to AST in June 2007. [PDF]
• P. Klotz, N. Bartoli, L. Cornez, M. Samuelides, and P. Villedieu. Database enrichment for optimization with artificial neural networks, submitted to OPTE in January 2007. [PDF]

Applications

Optimization of a system of injectors in a premix tube

The objective of this study is to define an optimization methodology for an injection system. The uniform evaporation rate on output and the mixture rate must be maximized as a function of the position of the kerosene injectors, the flow and the angle of the air inlets. To do this, a coupled problem, including a Navier Stokes part for the calculation of the gas and a lagrangian part for the calculation of the droplets, is resolved using the ONERA diphasic CEDRE code. A Latin Hypercube type design of experiments, associated with a neural network, is used to define a response surface on which a gradient type algorithm is applied. The Pareto front obtained defines the characteristics of an optimal injection system. The neural networks give a good representation of the surface.

The results obtained were presented during a meeting of the project in December. It was then decided that a new 3D case study supplied by TurboMéca should be validated using this approach.

Last Update: 7 April, 2008 - © ONERA 2009 - Terms of use