## 2 Materials and Methods
### Multi-Objective Optimization
#### 2.6.3 Multi-Objective Optimization Based on Genetic Algorithm  
We used the Design Exploration module of ANSYS Workbench to determine the test point via experimental design within the range of multiple design variables. The experimental data were obtained via simulation and calculation. The response surface model of the objective function was constructed by fitting the multiple sets of data, and the appropriate optimization algorithm was selected to process the response surface to obtain the comprehensive optimal target. A mass change in the direct-drive turntable leads to a change in the turntable structure. It may increase the deformation of the workpiece during the machining process of the turntable and affect the machining accuracy. Therefore, it is necessary to minimize the deformation of the direct-drive turntable. The closing of the machining frequency and the natural frequency may lead to a decline in machining accuracy and even damage the direct-drive turntable. Therefore, the first natural frequency is as large as possible according to the actual situation. Combined with the parameters obtained from the statics and structural dynamics analysis, the mathematical optimization model is obtained:  
\[\left\{ \begin{array}{c}\min(y_{3}(x))\\ \min(y_{4}(x))\\ \max(y_{5}(x))\\ X=[x_{1},x_{2},x_{3},x_{4},x_{5},x_{7}]\\ X_{iL}<x_{i}<x_{iL}\end{array} \right. \tag{9}\]  
In this formula: \(y_{3}(x)\) is the total mass of the turntable and workpiece, \(y_{4}(x)\) is the maximum deformation of the turntable and workpiece, \(y_{5}(x)\) is the first natural frequency of the direct-drive turntable, \(y\) is the objective function, \(X\) is all the design variables, \(x\) is a design variable, \(x_{iL}\) is the lower limit of the design variable, \(x_{iL}\) is the upper limit of the design variables, and \(x_{iL}\) and \(x_{iL}\) are the constraint conditions. After multi-objective optimization, a series of optimization solutions that meet the constraint conditions will be generated, and these solutions are called Pareto solutions. To improve the solution precision, finding the solution that meets the requirements as much as possible is necessary. The most suitable optimization result is selected from all the solutions. [PERSON] proposes a genetic algorithm under the influence of biological evolution theory. It is an optimization method that uses the concept of animal genetics to solve engineering problems [26]. The optimization exploration module of ANSYS Workbench can carry out multi-objective optimization based on a genetic algorithm. So, a multi-objective genetic algorithm is used to optimize the mathematical model (Formula (9)). The NSGA-II (non-dominated sorting genetic algorithm) is used in this paper and is one of the most effective multi-objective genetic algorithms. According to the non-dominated principle, NSGA-II takes constraints as the processing goal, which can quickly sort non-dominated solutions and maintain high precision and population diversity. If the solution of the genetic algorithm is required to have a global property and not to converge too fast, population uniformity needs to be maintained. The shifted Hammersley sequence sampling method and trade-off function method can meet the above conditions. The Hammersley sequence sampling technique is a quasi-random number-sampling technique which generates uniformly distributed sample points in the n-dimensional feasible solution region based on the Hammersley algorithm [27]. Hammersley sample points are more concentrated in the starting point area in the K-dimensional hypercube. Shifted Hammersley sampling technology [28] is adopted to make up for this deviation, which moves the sample points based on Hammersley sequence sample points. The sample points are more evenly distributed. Shifted Hammersley sequence points have a low minimum bias [29]. Here, the Hammersley sample points are offset \(\Delta=\text{N}/2\) to make them uniformly concentrated in the central area of the design space.