Thermo-mechanical optimization of metallic thermal protection system under aerodynamic heating

Abstract

This paper details the application of the finite element model, Bayesian regularized neural network, and genetic algorithm for metallic thermal protection system parameter optimization. The object is to minimize the structure weight and satisfy multiple performance constraints which consist of the deformation of the top face sheet, the stress around the lug hole, the stress of the honeycomb core, and the inner temperature. Firstly, a high-fidelity thermo-mechanical coupled finite element model was established to investigate the thermal and mechanical properties of the metallic thermal protection system. Then, surrogate models were constructed based on the Bayesian regularized neural network which assigns a probabilistic nature to the network weights and biases and allows the network automatically and optimally penalize complex models. To guarantee full exploration of the design space, an optimal orthogonal-maximin Latin hypercube design was adopted and modified to generate preliminary sampling points. Finally, an approach for genetic algorithm constraint handling, stochastic ranking, was introduced and modified by multiple constraint ranking to handle constraints. In addition, a sensitivity analysis was performed to disclose the effects of individual design variables on the thermal and mechanical responses. The results indicate that the structure mass is decreased by 41.2 % when compared to the initial design and all the constraints are satisfied.

Appendices

Appendix A: Modified simulated annealing for optimal LHD

In the simulated annealing algorithm proposed by Morris and Mitchell (1995), a new design is generated by interchanging two randomly chosen elements within a randomly chosen column in L. In order to accommodate the special structures of SLHD, two simultaneous pair exchanges are made in a column to retain symmetry (Ye et al. 2000). For example, a column in a 6-row SLHD is (1, 2, 3, 4, 5, 6)^T. If element 1 is exchanged with i, element 6 must be exchanged with n + 1 − i to keep symmetry. Meanwhile, Ye et al. (2000) pointed out that the starting design plays an important role in the optimization procedure. Therefore, many SLHDs were generated randomly and compared according to ψ_q to obtain a good starting design. In the pseudocode, T₀ represents the initial value of annealing temperature parameter T, $I_{{\max \nolimits }}$ is the number of design perturbations the algorithm will try before going on to the next temperature and FAC_t is the temperature reduction factor.

Appendix B: Stochastic ranking

The pseudocode of stochastic ranking algorithm is provided here and in which λ is the population size, I_j denotes the j th individual, f denotes the fitness function, and ϕ represents the penalty function. The ranking is achieved by a bubble-sort-like procedure and can be regarded as the stochastic version of a classic bubble sort. In the ranking procedure, P_f determines whether the ranking is overpenalized or underpenalized. When P_f = 0, the ranking is an overpenalization, and for P_f = 1, the ranking is an underpenalization. The initial ranking is generated randomly. If P_f = 0.5, there will be an equal chance for a comparison to be made based on the objective function or the penalty function. Since we are only interested in feasible individuals as final solutions, P_f should be less than 0.5 so that there is a bias against infeasible solutions. Runarsson et al. (2000) chose P_f = 0.45 and obtained good results on benchmark functions. After stochastic ranking, a ranked set of individuals was obtained and then the individuals were selected using the ranked-based technique.

Appendix C: The final LHD

The design table of the final LHD is provided in Table 7.

Table 7 Sixty sampling points