Dimensionality Reduction Approach for Response Surface Approximations: Application to Thermal Design

Abstract

Response surface approximations are a common engineering tool, often constructed based on finite element simulations. For some design problems, the finite element models can involve a large number of parameters. However, it is advantageous to construct the response surface approximations as a function of the smallest possible number of variables. The purpose of this paper is to demonstrate that a significant reduction in the number of variables needed for a response surface approximation is possible through physical reasoning, dimensional analysis, and global sensitivity analysis. This approach is demonstrated for a transient thermal problem, but we also show how it can be applied to any finite-element-based surrogate model construction. The thermal problem considered is the design of an integrated thermal protection system for spacecraft reentry for which a response surface approximation of the maximum bottom face temperature is needed. The finite element model used to evaluate the maximum temperature depended on 15 parameters of interest for the design. 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 variables of the response surface approximation for maximum temperature, which was constructed using simulations from the original nonsimplified finite element model. The major error in the two-dimensional response surface approximation was found to be 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 reasonable agreement was found. The two-dimensional nature of the response surface approximations allowed graphical representation, which we used for material selection from among hundreds of possible materials for the design optimization of an integrated thermal protection system panel.

Including the Effect of a Future Test and Redesign in Reliability Calculations

“…(8), where M and N are the number of capacity and response samples, respectively: (10) Using the computational and experimental results, along with the corresponding error estimates for the test article, we are able to refine the calculated value and its error for any design described by the design variables d and random variables r. In this way, the result of the single test can be used to calibrate calculations for other designs. We examine two methods, which take different approaches in using the test as calibration.…”

“…Next, the experimentally measured temperature is found using Eq. (10). The choice can be made to calibrate by the correction factor approach or the Bayesian-updating approach, and, further, the choice of deterministic or probabilistic redesign can be made.…”

“…The maximum temperature of the bottom face sheet of the ITPS panel is calculated using the quadratic response surface developed by Villanueva et al [8] by a process similar to that of Gogu et al [10], using the MATLAB toolbox developed by Viana.…”

Thermal-Acoustic Analysis of a Metallic Integrated Thermal Protection System Structure

“…This simplified approach was earlier justified by comparing the results obtained from 1-D and 2-D heat transfer analyses by Bapanapalli [5]. The panel's core and webs were modeled as a thermally homogenous resultant material with its resultant conductivity and specific heat being determined through the rule of mixtures as [4][5] ( ) ) (…”

“…The above panel concept was studied extensively in the past for structural optimization under thermal loading only [2][3][4][5]. The geometric parameters explored in the optimization studies for fixed areal panel dimensions are illustrated in Figure 2.…”