## 1 Introduction  
The optimization of the powertrain mount design holds significant engineering importance, as these mounts play a crucial role in reducing vibration and noise in both conventional fuel vehicles and electric vehicles, thereby directly impacting the NVH (Noise, Vibration, Harshness) performance experienced by vehicle users [1, 2]. Previous studies on powertrain mount optimization have encompassed diverse methodologies. [PERSON] et al. [3] introduced the concept of generalized force transmissibility (GFT) and the summation of GFT integrals as appropriate indicators for assessing the vibration isolation performance of powertrain mounting systems. Their proposed design optimization methodology incorporated multiple design variables, rigid-body modal frequency constraints, and the objective of minimizing the sum of GFT integrals. [PERSON] et al. [4] presented a design rationale and computational approach for determining the stiffness and damping of a powertrain mounting system by considering the reduction in vehicle vibration and noise attributed to the mounts. [PERSON] et al. [5] developed a dynamic model of a powertrain mounting system to investigate the multi-dimensional coupling vibration characteristics. They computed the natural frequencies with six degrees of freedom and validated the decoupling conditions of the most significant vibration modesthrough the design and verification of a powertrain mounting system for a heavy truck. These studies primarily treated relevant parameters as deterministic values, yielding fixed, single values for optimization outcomes [6; 7; 8; 9; 10; 11; 12]. However, during the mount manufacturing process, various factors can influence the actual stiffness of a mount, deviating from its theoretical design value. Neglecting the concept of uncertainty in optimization design can impede the attainment of the desired vibration isolation performance. Consequently, the research literature has witnessed an upsurge in studies focused on the optimization of mounting systems under conditions of uncertainty. One prevalent approach in mount optimization under conditions of uncertainty is robust design, which relies on a probabilistic model that considers the variation in mounting system stiffness with parameters exhibiting specific distribution characteristics. Monte Carlo methods are frequently employed to evaluate the robustness of the system response [13; 14]. For example, [PERSON] et al. [15] developed a six-degree-of-freedom dynamics model for the powertrain mount system of vehicles, and they utilized the Monte Carlo method to analyze the system's robustness. Their investigation focused on the effects of design value variations on the objective function. Similarly, [PERSON] et al. [16] established a dynamic analysis model for the powertrain mount system of engines. They employed Latin hypercube sampling to fit a RSM approximation model and applied a multi-island genetic algorithm for optimization, aiming to identify the optimal solution with constraints and analyze its robustness. Other approaches, such as 6\(\sigma\) theory and genetic algorithms, have also been employed for robustness optimization design, as demonstrated by [PERSON] et al. in their study on decoupling design, which is based on torque roll axis decoupling theory [17; 18; 19]. [PERSON] et al. [20] proposed a robust multi-objective optimization scheme for powertrain mount systems in electric vehicles, utilizing genetic algorithms for global optimization and Latin hypercube sampling to achieve optimal robustness of dynamic reaction force. Another category of optimization design under uncertainty involves non-probabilistic models, such as interval models, fuzzy models, and evidence-theoretic models. These models are particularly valuable when the available stiffness sample data of a mounting system is insufficient or unreliable. For instance, [PERSON] et al. [21] proposed a robust design method that employed interval analysis and Particle Swarm Optimization (PSO) to optimize a Powertrain Mounting System (PMS).