Reducing the Number of Variables in a Response Surface Approximation: Application to Thermal Design

Abstract

Response surface approximations (RSA) are a common tool in engineering, often constructed based on finite element (FE) simulations. For some design problems, the FE models can involve a high number of parameters. However it is advantageous to construct the RSA as function of a small number of variables. The purpose of this paper is to demonstrate that a significant reduction in the number of variables needed for an RSA is possible through physical reasoning, dimensional analysis and global sensitivity analysis. This approach is demonstrated for a transient thermal problem, but it is applicable to any FE based surrogate model construction. The thermal problem considered is the design of an integrated thermal protection system (ITPS) for spacecraft reentry where an RSA of the maximum bottom face temperature was needed. The FE model used to evaluate the maximum temperature depended on 15 parameters of interest for the design: 9 thermal material properties and 6 geometric parameters of the ITPS panel. A small number of assumptions simplified the thermal equations allowing easy nondimensionalization, which together with a global sensitivity analysis showed that the maximum temperature mainly depends on only two nondimensional parameters. These were selected to be the design variables of the RSA for maximum temperature. The RSA was still fitted to the original non-simplified FE simulations. Having only two variables allowed a dense design of experiments thus providing a very good quality of fit. Consequently the major error remaining in the RSA is due to the fact that the two nondimensional variables account for only part (albeit the major part) of the dependence on the original 15 variables. This error was checked and good agreement was found. The two-dimensional nature of the RSA allowed graphical representation, which was used for material selection from among hundreds of possible materials for the design optimization of an ITPS panel.