Optimized design of sandwich panels for integral thermal protection systems

Abstract

This paper proposes a new ITPS panel with special corrugated-core webs which are designed with cutouts for weight saving. The structural design problem was formulated with mass per unit area of the ITPS as the objective function and some functional requirements as constraints. We developed the optimizer fulfilling both thermal and structural functions for minimal areal density. The optimization problem was solved by interpolating the residual error of response surface approximation (RSA) with Radial Basis Function (RBF) to establish the improved RSA (IRSA). The 400 preliminary design points were obtained using Latin Hypercube Sampling method. The quadratic polynomial RSA of the ITPS sandwich panel performance was generated by the least squares method (LSM) based on finite element results and IRSA was used to optimize the constraints. Transient heat transfer, stress and buckling analyses were conducted using finite element method (FEM). Finally, a new ITPS panel with optimal dimensions was obtained. The optimization results show that the areal density of the new ITPS panel decreases by 26.27 % compared with the previous research, which proves the potential of this new design optimization method for the future spacecraft vehicles.

Appendix A: The response surface approximation

A quadratic response surface approximation in 9 variables has 55 coefficients. Therefore, 55 function evaluations are needed. To obtain an accurate response surface approximation, the number of function evaluations, N, is twice the number of coefficient. As for this paper design, N has been typically found to be greater than 200 in order to obtain sufficiently accurate response surfaces. Latin Hypercube Sampling (LHS) is used for the design of experiments. In this paper a total of 400 LHS points is selected. Using N combinations of geometric variables, N heat transfer finite element analyses were carried out. The maximum bottom face sheet temperature for each of these N experiments is obtained from the analyses. In this way, N function values for N different combinations of the variables were obtained. Then, a complete quadratic polynomial in 9 variables was obtained by the Least Squares Approximation (LSA) method.

The response surface approximation fits a function by a set of experimental and numerical evaluated design data points. In this paper, in order to improve the approximate accuracy, the improved RSA (IRSA) by interpolating the residual error of RSA with RBF is established.

The RSA is written by f _k (x) in the text. The error between the RSA and the real result in the sample points is marked as R. R represents the residual matrix. The residual matrix and Neural Network Toolbox in MATLAB are combined to get the RBF interpolating function.

To evaluate the accuracy of the IRSA, The total root mean square error (RMSE) σ of the response surface is used to access the quality of the fitted improved response surface approximation in the designed date set. σ is defined as

$$ \sigma =\sqrt{\frac{1}{n}{\displaystyle \sum_{i=1}^n{\left({x}_i-\overline{x_i}\right)}^2}} $$

(6)

Where x _i is the actual value of the response at the design point and $\overline{x_i}$ is the average value of x _i . The percentage of the total RMSE for the design points, designated as RMSE %.

The coefficient of multiple optimization R² is another most frequently used parameter to access the quality of the fitted response surface. It measures the fraction of variation in data captured by the response surface. R² is define as

$$ {R}^2=\frac{{\displaystyle \sum_{i-1}^n{\left({\widehat{\mathrm{X}}}_{\mathrm{i}}-{\overline{\mathrm{X}}}_{\mathrm{i}}\right)}^2}}{{\displaystyle \sum_{i-1}^n{\left({\mathrm{X}}_{\mathrm{i}}-{\overline{\mathrm{X}}}_{\mathrm{i}}\right)}^2}} $$

(7)

Where X _i is the actual value of the response at the design point. ${\widehat{X}}_i$ is the predicted value, and ${\overline{X}}_i$ is the average value of X _i . If the R value is closer to 1, the polynomial fitted model gives a good fit.

PRESS is an acronym for Predicted Error Sum of Squares. It is a better indicator of response surface accuracy than the root mean square error. The procedure for calculating the PRESS error is as follows. Consider that there are n response values from which the response surface approximation is calculated. First calculate the response surface approximation R₁ using responses 2 to (n-1) by leaving out the first response. Then determine the error of R₁ by substituting the variable values corresponding to response 1 into this response surface approximation. Similarly calculate the error of R ₂ to R _n. The root mean square of the errors from R ₁ to R _n is the PRESS error for the response surface approximation.