A modified multi-objective sorting particle swarm optimization and its application to the design of the nose shape of a high-speed train

Based on the concepts of niche count and crowding distance, a modified multi-objective particle swarm optimization (MPSO) is introduced. The niche count and crowding distance are used to determine the globally best particle across four test cases using an external file. A comparative analysis was carried out between MPSO and non-dominated sorting multi-objective adaptive genetic algorithms, both real-coded and binary-coded. The results show that MPSO based on the crowding distance is best for getting the Pareto front, especially for problems with high-dimensional and non-continuous Pareto fronts. In order to verify the efficiency of MPSO in solving engineering problems, the optimal design of the aerodynamic nose shape of high-speed trains was undertaken using a modified vehicle modeling function (MVMF) parametric method. Taking the aerodynamic drag of the whole train (Cd) and the aerodynamic lift of the trailer car (Cl) as the optimization goals, the Kriging surrogate model was introduced to reduce the computational time, and the MPSO based on crowding distance was used to find the Pareto front. The optimization results show that MPSO is efficient at getting the Pareto front; compared to the original shape, the Cd and Cl of the optimal shape are reduced by 1.6% and 29.74%, respectively.