###### Abstract  
Due to the inherent uncertainties in material properties, loads, geometric dimensions, et al., the uncertainty-based optimization design method has become increasingly important for the design of the thermal protection system (TPS) by carefully considering the influences of uncertainties. In this study, an uncertainty-based comprehensive optimization design method, which sequentially performs the robust design of aerodynamic shape and structure size for the TPS of a hypersonic wing is proposed, on the presence of uncertain-but-bounded parameters. The robust design of the TPS's aerodynamic shape is firstly carried out. The results show that the proposed method decreases the fluctuation of the lift-to-drag ratio by 5.7%, with a small increase of heat flux on the stagnation point by only 0.13% when compared with the conventional deterministic optimization method. After that, based on the optimized aerodynamic shape and heating loads, the robust design of the multilayer TPS tile is conducted. The results show that the mass of the TPS tile efficiently deceased from 2.713 kg to 2.445 kg by 9.89%, and the robustness of the optimized design is better than the initial design. Finally, the effectiveness of the proposed optimization method is validated by the heat insulting experiment of the typical multilayer TPS tiles. uncertainty-based optimization design; uncertainty propagation analysis; aerodynamic shape; multilayer TPS tile; hypersonic wing +
Footnote †: journal: Applied Sciences  
[PERSON], [PERSON], [PERSON]. Footnote †: journal: Applied Sciences  
[PERSON], [PERSON]. Footnote †: journal: Applied Sciences  
and thermal protection material responses with the Monte Carlo method. Thus, the uncertainty is an important issue that should be comprehensively taken into account in the design phase of the TPS. Over the past few years, the uncertainty-based optimization design for the TPS of hypersonic vehicles have attracted much attention. [PERSON] [16] estimated thermal reliability from an assessment of TPS uncertainties. [PERSON] [17] investigated the probabilistic optimization of integrated TPS, that combines the thermal protection function with the structural load carrying function. [PERSON] [18] presented a statistical methodology to conduct a size optimization for the TPS of atmospheric entry vehicles. [PERSON] et al. [19] established a six sigma robust design optimization based on a surrogate model for the TPS, in order to improve the reliability and robustness while considering the uncertainties. [PERSON] and [PERSON] [20] applied the Monte Carlo Simulation technique to determine the TPS margins with probabilistic design. [PERSON] et al. [21] developed this technique to the thermochemical uncertainty and sensitivity analysis of an ablative TPS during a Titan atmospheric entry. [PERSON] et al. [22] identified the multi-sources of uncertainties in the diverse TPS materials and aerothermal environments, and then performed the TPS thickness optimization. Nevertheless, according to the current research state described above, there are still some issues that need to be further investigated. (1) Most researches on the uncertainty-based optimization design of the TPS consider the structural size optimization design with the assumed aerodynamic shape and aerodynamic heating loads. It means that the structural optimization and aerodynamic shape optimization of the TPS are separated. However, the structural optimization of the TPS is highly dependent on the aerodynamic shape and the corresponding aerodynamic loads, which act as the geometric boundary constrains and thermal loads of the TPS, respectively. Consequently, an uncertainty-based comprehensive optimization design, integrating the aerodynamic shape optimization and the TPS structural optimization, is more meaningful than any individual optimization design in practical engineering. (2) The optimization design of the TPS in most literature is based on the stochastic models, which always need sufficient information to determine the density function.