## 1 Introduction  
In modern industrial design, engineers frequently encounter optimization tasks involving computationally expensive black-box problems, such as finite element analysis models and computational fluid dynamics models. With the rapid advancement of the algorithm field, numerous derivative-free and derivative-based algorithms have been developed to address this challenge [1, 2]. Among them, traditional population-based algorithms such as particle swarm optimization, as well as emerging nature-inspired metaheuristic algorithms, are known to be highly reliable for optimizing black-box problems [3, 4]. However, the direct application of such algorithms typically requires a large number of evaluations of the objective function, which is impractical for time-sensitive industrial product design. To mitigate the high computational cost associated with directly applying intelligent algorithms to engineering optimization problems, surrogate-based optimization (SBO)methods have gained increasing attention over the past two decades [5]. Figure 1 illustrates the typical workflow of SBO methods. Leveraging the predictive capability of surrogate models and their low computational cost, the number of expensive objective function evaluations in the optimization process can be significantly reduced, thereby improving the overall efficiency of the optimization workflow. SBO methods have been extensively studied and applied to various problems, including aerodynamic performance optimization [6; 7] and structural optimization [8; 9]. The kriging-based efficient global optimization (EGO) algorithm is one of the most successful SBO methods [10; 11]. The kriging surrogate model not only provides predictions of the objective function but also quantifies the uncertainty of these predictions. Accordingly, the EGO algorithm incorporates an adaptive sampling criterion, known as expected improvement (EI), to guide the selection of new sample points. On one hand, the sampling criterion identifies candidate points with the highest probability of improvement; on the other hand, it enhances the predictive accuracy in the vicinity of the selected points. The optimization process terminates once the stopping criterion is met. The traditional EGO algorithm has been extensively studied and extended to address not only simple single-objective optimization problems [12; 13] but also complex challenges such as multi-objective optimization [14; 15; 16], numerical noise optimization [17; 18], constrained optimization [19; 20; 21], and gradient-based optimization [22]. However, a major limitation of the classical EGO algorithm is that it identifies only a single promising candidate point per optimization cycle. For automotive aerodynamic optimization problems, a complete point-filling process requires approximately 8 h of computational time [18], including 5 h for CFD numerical simulations on a 576-core supercomputer and 3 h for mesh preparation and queuing delays prior to evaluation [18]. Under such circumstances, when only one design sample point can be evaluated per optimization cycle, despite abundant available computing resources, the approach fails to effectively utilize the parallel computing capacity. This limitation substantially restricts the potential reduction in wall-clock time and consequently hinders optimization efficiency improvement. To overcome this limitation, researchers have explored various approaches to extend the classical serial EGO algorithm [23; 24; 25], enabling the generation of multiple candidate points within a single optimization cycle. As a result, the overall optimization process can be significantly accelerated by reducing the number of required optimization cycles. Among various multi-point sampling criteria, the pseudo expected improvement (PEI) criterion exhibits relatively high reliability. It employs additional influence functions (IFs) to approximate the impact of newly sampled points on the original EI criterion, thereby avoiding costly evaluations of new points and frequent updates of the kriging model.