## 3 Many-Objective Joint Parallel Simulation Method
### Joint Parallel Simulation Method
* Taking \(S_{1}\) as the set of points on the boundary layer and defining \(S_{2}\) as the set of points on the inner layer, for each \(S^{\prime}_{ij}\in S_{2}\) and \(S_{ij}\in S_{1}\), the following exists: \[S^{\prime}_{ij}=\frac{1}{2}S_{ij}+\frac{1}{2M}\] (15) (e) Define the set of reference points as follows: \[S=S_{1}\cup S_{2}\] (16)
* Create a batch of \(n\) files based on the actual number of cores available on the computer. Each file contains an \(.m\) file that waits for parameter input and an \(.mph\) file that waits for the command execution. In addition, two \"sentry\" files are created, each of which also includes an \(.m\) file and an \(.mph\) file; this file is used to monitor the status of the \(.mph\) file, and if an execution exception is detected, it will replace the \(.mph\) file and the optimization process will be detected. \"Sentry\" operation process is shown in Figure 9.
* Population initialization. The primary function generates the first batch of population variables and stores them in a \(.txt\) file; the \(n\) MATLAB, with COMSOL running in the background, reads the \(.txt\) file and calls the corresponding COMSOL to start running, and then stores the result values in the corresponding \(.txt\) file after running; the fitness function in NSGA-III reads the result values in the \(.txt\) file, and returns to the primary function when all of the result values in the population have been read. * Determining constraint relationships and creating ideal points According to the non-linear inequality relationship used to solve the constraint relationship, and to extract the minimum value of population fitness, an ideal point is constructed, through which the objective function is transformed into an adaptive normalization function, with the following specific operation:
* The ideal point of the population is \(S_{t}\), which is defined as the minimum of \(\cup_{\tau|t=0}^{t}S_{\tau}\big{(}z_{i}^{min},i=1,2\ldots,M\big{)}\), and an ideal point \(\overline{z}=(z_{1}^{min},z_{1}^{min},\ldots,z_{M}^{min})\) is constructed. The objective function is transformed according to Equation (17) [19] as follows: \[f^{\prime}_{i}(x)=f_{i}(x)-z_{i}^{min}\] (17) (b) Find the additional points of each axis as follows: \[ASF(x,w)=\frac{max_{i=1}^{M}f^{\prime}_{i}(x)}{w^{i}},x\in S_{t},w^{i}=(\tau, \ldots,\tau),\tau=10^{-6}\] (18) (c) Construct the hyperplane using the extra points and find the intercept of the plane with the coordinate axis \(a_{i}\);
* Normalization of the objective function is as follows: \[f^{n}_{i}(x)=\frac{f^{\prime}_{i}(x)-z_{i}^{min}}{a_{i}-z_{i}^{min}}\ i=1,2, \ldots M\] (19)
* Following the termination determination of the population selection, crossover, and mutation steps were implemented to initiate population evolution. * Generating subpopulations. To avoid confusion between subpopulation and parent population parameters, the command to clear the parameter file and the result file in real-time is added to the subpopulation execution process. Subpopulations are then subjected to constraint determination and ideal point finding operations. * The parent and child populations are merged, and sequential operations, including non-dominated sorting, target space normalization, a calculation of reference lines,an association of individuals with reference points, and a selection of dominant populations are carried out. 8. If the termination condition is met, the optimal Pareto solution is output; otherwise, steps (5) through (8) are repeated.