For models with large numerical simulation costs, such as high-speed trains, using as few samples as possible to construct a high-precision surrogate model during aerodynamic multi-objective optimization is critical to improving optimization efficiency. This study proposes a sequential infill criterion (SIC) appropriate for the Kriging surrogate model to address this issue. Three multi-objective functions are employed to test the feasibility of constructing a surrogate model based on SIC, and the SIC surrogate model then performs multi-objective aerodynamic optimizations on the high-speed train. The findings indicate that the expected improvement infill criterion (EIC) in the first stage can enhance the global prediction accuracy of the SIC. An infill criterion based on EIC that fuses gradient information (PGEIC) in the second stage is proposed to seek samples in the Pareto front. The PGEIC surrogate model achieves the lowest generational distance and prediction error. The performance of EIC for global search, EIC for Pareto front search, and infill criterion for Pareto front search using only gradient information is poor. The final PGEIC–SIC surrogate model of train aerodynamics has less than 1% prediction error for the three optimization objectives. The optimal solution reduces the aerodynamic drag force of the head car and the aerodynamic drag and lift force of the tail car by 4.15%, 3.21%, and 3.56%, respectively, compared with the original model. Furthermore, sensitivity analysis of key parameters revealed that the nose height v1, cab window height v3, and lower contour line have a greater impact on aerodynamic forces. Moreover, the nose and cab window heights of the optimal model have been reduced, and the lower contour line is concave. Correspondingly, the streamlined shape appears more rounded and slender.

中文翻译：

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基于克里金法的高速列车空气动力学多目标优化，使用带有梯度信息的顺序填充准则

对于高速列车等数值模拟成本较大的模型，在气动多目标优化过程中使用尽可能少的样本构建高精度代理模型对于提高优化效率至关重要。本研究提出了适合克里金代理模型的顺序填充准则（SIC）来解决这个问题。采用三个多目标函数来测试基于SIC构建代理模型的可行性，然后SIC代理模型对高速列车进行多目标气动优化。研究结果表明，第一阶段的预期改进填充准则（EIC）可以提高 SIC 的全局预测精度。提出了一种基于 EIC 的填充准则，在第二阶段融合梯度信息（PGEIC），以在 Pareto 前沿中寻找样本。 PGEIC代理模型实现了最低的代距和预测误差。用于全局搜索的 EIC、用于 Pareto 前沿搜索的 EIC 以及仅使用梯度信息的 Pareto 前沿搜索的填充准则的性能较差。最终的列车空气动力学 PGEIC-SIC 替代模型对于三个优化目标的预测误差均小于 1%。最优解与原模型相比，头车气动阻力和尾车气动阻力和升力分别降低了4.15%、3.21%和3.56%。此外，关键参数的敏感性分析表明，机头高度v1、驾驶室车窗高度v3和下轮廓线对气动力的影响较大。而且，优化车型的机头和驾驶室车窗高度均有所降低，下部轮廓线呈凹形。相应地，流线型的造型显得更加圆润修长。