## 3 Results and Discussion
### Multi-Objective Optimization Results and Analysis
#### 3.4.3 Multi-Objective Optimization Results and Discussion  
The traditional multi-objective optimization method is to transform multi-objective optimization into single-objective optimization. Therefore, the traditional method has low computational efficiency, making it difficult to obtain a global optimization solution. So, a multi-objective genetic algorithm is proposed to solve practical problems. In multi-objective optimization, it is not easy to achieve the optimal solution among the objectives simultaneously, so multi-objective optimization often generates a series of effective solutions, also known as Pareto solutions. These solutions all meet the requirements, so we cannot compare them. In the multi-objective optimization problem, it is necessary to find as many Pareto solutions as possible that are unbiased and conform to the requirements. Then, according to the design requirements and practical engineering experience, the most satisfactory optimization results are objectively selected. NSGA-II is considered one of the most effective multi-objective genetic algorithms [26], and can reduce the computation scale and preserve the influential elite and population diversity. Since the optimization objectives of the direct-drive turntable are the total mass, the first-order natural frequency, and the maximum deformation, the multi-objective genetic algorithm is used to solve the mathematical model of multi-objective optimization. The initial population is 100, the number of samples for one iteration is 100, the maximum allowable Pareto percentage is 50, and the maximum number of iterations is 20. Other options adopt the system default, and 100 Pareto solutions are generated. The Pareto solutions between two of the three objective functions are:  
Figure 30: Normalized curve. are 0.1057 \(\upmu\)m, 0.055 \(\upmu\)m, and 0.081 \(\upmu\)m, and the prediction deviations are about 3.97%, 2.08%, and 3.01%, respectively. Compared with the predicting values and the simulation values of the first natural frequency, the absolute errors of the predicting values of the first, second, and third groups are 9.43 Hz, 2.18 Hz, and 9.25 Hz, and the prediction deviations are about 1.47%, 0.34%, and 1.45%, respectively. Compared with the predicting values and the simulation values of the total mass, the absolute errors of the predicting values of the first, second, and third groups are 1.086 kg, 0.1641 kg, and 3.11 kg, and the prediction deviations are about 1.05%, 0.16%, and 3.1%, respectively. It can be seen that there is a difference between the predicting value and the simulation value. The optimization objectives were to reduce the mass and deformation of the turntable and improve the first natural frequency. Under three sets of optimization values, the maximum deformation, the first-order natural frequency, and the total mass of the direct-drive turntable are optimized to a certain extent. Among the three optimization values, optimization value 1 has the greatest degree of optimization on the deformation of the direct-drive turntable. Optimization value 3 has the greatest degree of optimization on the total mass of the direct-drive turntable. The optimization degree of optimization value 3 on all the objectives is between optimization values 1 and 2. From the actual situation, the maximum deformation of the direct-drive turntable is far less than 10\(\upmu\)m, and the processing speed corresponding to frequency cannot reach the first natural frequency. Therefore, optimization value 3 is the optimal solution. Compared with before topology optimization, the mass of the direct-drive turntable is reduced by 9.02%, the first natural frequency is increased by 3.83%, and the deformation is decreased by 4.76%. Compared with the original model, the total mass of the direct-drive turntable decreases significantly, reaching 21.394%. To verify the effect of the optimization results on the machining conditions of the direct-drive turntable, we needed to extract and round the optimization data.