## 2 ARBF-MLPA Method
### Multi-Objective Optimization Algorithm  
Genetic Algorithm (GA) is a very popular global optimization algorithm originated from natural evolution mechanism and principles of genetics. GA is better than many traditional optimization algorithms, as it avoids the problem of local minimal value while being able to search for optimal values. Non-dominated Sorting Genetic Algorithm (NSGA) is very effective in solving multi-objective problem. It arranges sequences applying non-dominated sorting solutions, and distributes fitness based on the sequences arranged. As an algorithm improved from NSGA, NSGA-II has been proved an effective multi-objective optimization algorithm in a lot of benchmark problems [19; 20]. NSGA-II is also a multi-objective optimization method discussed based on Pareto optimal front edge. The final objective of this paper is to find the optimal Pareto front edge of all multi-objective optimization problems, which exactly coincides with the aim of the paper to find the solution of optimization. In addition, what we are building is a double-objective optimization model, but NSGA-II algorithm is enough. See Figure 1 for flow chart of NSGA-II algorithm and specific optimization process is as follows:  
1. Firstly, the sample points obtained by OLHS method, and the response values obtained by finite element method, were used to construct the RBF metamodel;
2. In this algorithm, the design point is the first-generation population. The population is initialized, and the offspring is obtained through crossover, selection, mutation;
3. Merge the parents with the offspring and sort them non-dominated;
4. After sorting according to the rank, individuals are added to the next population in order to form a new parent population, and the remaining individuals will be eliminated after the number of the next generation arrives. This is the elite retention strategy based on the calculation of crowding degree;
5. If no new parent population is generated, the parent population of this generation will be sorted in a non-dominated way and a new parent generation will be formed from them. If a new parent population is created, the process of 2 to 4 is repeated until the termination condition is reached. Here, we set the maximum number of evolutionary generations as the termination condition;
6. Output the Pareto front of this optimization;
7. Using the MLPA method, select points in the Pareto front output from the last step. Additionally, the finite element method should be used to calculate the corresponding response values, so as to update the RBF model above. Adaptive Radial Basis Function Neural Network Combined with Machine Learning Point-Adding Algorithm (ARBF-MLPA)  
This paper proposes an adaptive radial basis function neural network combined with a machine learning point-adding method (ARBF-MLPA). The main idea of this method is: establish a mathematical surrogate model through ARBF neural network, obtain the optimal front edge through multi-objective optimization algorithm described above, find the original design sample points included in the optimal front edge, and find the original design sample points contained in the optimal front edge, add new sample points uniformly between the original sample points, and obtain the corresponding response values through calculation, add the newly obtained sample points to the original model, rebuild a new mathematical surrogate model, then continually update the surrogate model until the model converges. Combining the ARBF neural network with the MLPA algorithm makes model converge faster and continuously approach the optimal solution set. The point-adding method used in this paper mainly considers Max-Min Distance (MMD) point, Nearest Boundary (NB) point and the sample points selected by the ARBF neural network. As below is the algorithm description in details. Step 1: set initial parameters, e.g. : number, target, number of variable, and designed range of variable of initial samples. Step 2: generate initial sample point based on the OLHS DOE described above.