Thermo-structural optimization of integrated thermal protection panels with one-layer and two-layer corrugated cores based on simulated annealing algorithm

Abstract

Toexplore weight saving potential capability, a multidisciplinary optimization procedure based on simulated annealing algorithm was proposed to unveil the minimum weight design for integrated thermal protection system subjected to in-service thermal and mechanical loads. The panel configurations with one-layer and two-layer corrugated cores are considered for comparison. Heat transfer and structural field analysis for each panel configuration were performed to obtain the temperature, buckling, stress and deflection responses for structural components of interest, which were then considered as critical constraints of the optimization problem. Sensitivity analysis was performed to disclose the effect of individual design variables on the thermo-structural extreme responses, and the designed thermal protection system performance and weight for the two configurations were discussed. The results demonstrated that the two-layer structure provides superior structural efficiency and performance to resist thermal buckling deformation in comparison with the one-layer panel. Its area-specific weight is reduced by more than 14–29 % with respect to the one-layer panel design, and 30–50 % weight efficient can be implemented at higher thermal buckling constraint levels, while keeping considerable temperature, stress and deflection margins.

Appendices

Appendix A: Heat transfer models of ITPS (Bapanapalli et al. 2006)

The ITPS panel is subject to an incident heat flux. Re-radiation condition is applied on the top surface with an emissivity of 0.8. Convective heat loss is also applied on the top surface, after aerodynamic heating is ceased. The bottom surface is assumed to be conservative adiabatic boundary condition. It is also assumed that there is no thermal contact resistance at the interface between the face sheets (and webs) and the insulation materials. The properties of the sandwich core are calculated using the rule of mixtures formula given below:

$$ \rho_{C} =\frac{\rho_{W} t_{W} +\rho_{S} \left( {p\sin \theta -t_{W} } \right)}{p\sin \theta } $$

(7)

$$ C_{C} =\frac{C_{W} \rho_{W} t_{W} +C_{S} \rho_{S} \left( {p\sin \theta -t_{W} } \right)}{\rho_{W} t_{W} +\rho_{S} \left( {p\sin \theta -t_{W} } \right)} $$

(8)

$$ k_{C} =\frac{k_{W} t_{W} +k_{S} \left( {p\sin \theta -t_{W} } \right)}{p\sin \theta } $$

(9)

where C for specific heat, k for conductivity, θ for the corrugation angle. The subscripts C and W represent the homogenized core and structural web material.

The heat transfer problem is modeled as a one-dimensional thermal analysis. The governing equations and boundary conditions can be written as following.

Heat transfer equation for ITPS:

$$\begin{array}{@{}rcl@{}} k_{T,C,(M),B} \frac{\partial^{2}T\left( {x,\tau } \right)}{\partial x^{2}}&=&\rho_{T,C,(M),B} C_{T,C,(M),B} \frac{\partial T\left( {x,\tau } \right)}{\partial T}\\ && 0<t<t_{end} \end{array} $$

(10)

Initial condition:

$$ T\left( {x,\tau =0} \right)=T_{i} $$

(11)

Boundary conditions:

$$ q_{out} =\left. {-k_{B} \frac{\partial T\left( {x,\tau } \right)}{\partial x}} \right|_{x=h+(t_{T} +t_{B} )/2} =0 $$

(12a)

$$\begin{array}{@{}rcl@{}} q_{in} &=&\left. {-k_{T} \frac{\partial T\left( {x,\tau } \right)}{\partial x}} \right|_{x=0} =q_{i} \left( \tau \right)-\varepsilon \sigma T\left( {0,\tau } \right)^{4}\\ &&-h\left( \tau \right)T\left( {0,\tau } \right) \end{array} $$

(12b)

Where T is temperature, τ is time, x is location along the thickness direction, T _i is the initial temperature of the panel before atmospheric reentry, ε the emissivity of the top surface, σ is the Stefan-Boltzmann constant, q _i (τ) is the heat influx, and h(τ) is the convection coefficient at the top surface. The subscripts T, C, M and B represent the top face sheet, homogenized core, medium sheet and bottom face sheet of the ITPS panel, respectively.

Appendix B: Response surface methodology

The response surface method fits a function to a set of experimentally or numerically evaluated design data points. In this approach, an appropriate polynomial is fitted to a set of data points by least squares method.

For a quadratic response surface approximation, the polynomial is formulated as:

$$ Y=\beta_{0} +\sum\limits_{i=1}^{n} {\beta_{i} X_{i} } +\sum\limits_{i=1}^{n} {\beta_{ii} X_{_{i} }^{2} } +\sum\limits_{j=1}^{n} {\sum\limits_{i=1}^{j-1} {\beta_{ii} X_{i} } } X_{j} $$

(13)

Where β is regression coefficient and X _i represents the ith design variables, n is the number of the design variables.

One of the most frequently used parameter to access the quality of the fitted response surface is the coefficient of multiple determination, which measures the fraction of variation in data captured by the response surface. The remaining variation is attributed to random noise. The coefficient of multiple determination R ² is defined as

$$ R^{2}=\frac{\sum\limits_{i=1}^{n} {(\widehat{Y_{i} }-\overline Y } )^{2}}{\sum\limits_{i=1}^{n} {(Y_{i} -\overline Y } )^{2}} $$

(14)

Where Y _i is the actual value of the response at the design point, $\widehat {Y_{i} }$is the predicted value, and $\overline Y$is the average value of Y _i . If the R value is closer to 1, the polynomial fitted model gives a good fit. More details on the polynomial response surface approximation have been provided in Ref. Poteet et al. (2004).

To evaluate the accuracy of response surface approximation, two data sets and various statistical criteria were used. The design data set includes all data points used in constructing response surface. The test data set contains n _t (n _t =50 here) randomly selected points$Y_{i}^{\ast }$. The total root mean square error (RMSE) σof response surface in the design data set is defined as

$$ \sigma =\sqrt {\frac{1}{n}\sum\limits_{i=1}^{n} {(Y_{i} -\overline {Y_{i} } )^{2}} } $$

(15)

The percentage total RMSE for the design points, designated as ‘RMSE-D %’, is obtained as the ratio of σ/$\overline Y$. The root mean square error in the test data (probability calculation) εis given by the expression

$$ \varepsilon =\sqrt {\frac{1}{n_{t} }\sum\limits_{i=1}^{n_{t} } {(Y_{i}^{\ast } -\overline {Y_{i} } )^{2}} } $$

(16)

The percentage total RMSE for the test points, designated as ‘RMSE-T %’, is obtained as the ratio of ε/$\overline Y$. If this value is close to zero then the model performs well.

Appendix C: Correlation coefficient method

The correlation coefficient r is given by

$$ r=\frac{\sum\limits_{i=1}^{n} {(X_{i} -\overline X )(Y_{i} -\overline Y )} }{\sqrt {\sum\limits_{i=1}^{n} {(Y_{i} -\overline Y )^{2}} } \sqrt {\sum\limits_{i=1}^{n} {(Y_{i} -\overline Y )^{2}} } } $$

(17)

Where $\overline X$ represents the mean value of input variables. The correlation coefficient can be interpreted as the fractional contribution to the uncertainty in the output due to uncertainty in a given input parameter. The magnitude of the correlation coefficient provides a way to rank the importance of the individual physical variables. If the absolute value of correlation coefficient is close to unity, the parameter strongly affects the response. On the other hand, if the correlation coefficient is close to zero, the parameter has little contribution to the response. Evaluating the correlation coefficients can therefore offer good insight into the mechanism of the response.