## 1 Introduction  
In response to the global climate crisis and the need to combat global warming, the European Union officially signed the Green Deal Agreement in 2020. This accord aims to achieve a 55% reduction in carbon dioxide (CO2) emissions by 2030 compared to 1990 levels [1], with the ultimate goal of reaching net-zero emissions by 2040. Consequently, significant initiatives have been undertaken across the energy and automotive sectors to meet these ambitious targets. Within the European Union, the transportation sector accounts for approximately 32% of total CO2 emissions, with road transportation contributing 73% of this share. Heavy-duty vehicles alone produce 6% of global CO2 emissions, accounting for 26% of emissions from road transport [2]. In response, manufacturers are increasingly shifting towards battery-electric propulsion for light-duty and passenger vehicles. Although regional delivery trucks with battery-electric capabilities have emerged, the practical implementation of battery-electric systems for long-distance heavy-duty trucks remains challenging due to limited range and extended charging times. Hydrogen fuel is thus anticipated to provide a viable solution for heavy-duty transportation applications [3]. Two primary hydrogen powertrain configurations have been proposed. The first involves the use of fuel cells to power electric motors and batteries. However, fuel cells are sensitive to dusty environments, require high-purity hydrogen, and demand precise thermal management. Moreover, their reliance on rare materials [4], susceptibility to efficiency loss under high loads, limited operational lifespan, and high costs [5] present significant drawbacks. Conversely, hydrogen-fueled internal combustion engines (H\({}_{2}\)-ICEs) offer a more economical, durable, and sustainable alternative. Unlike fuel cells, H\({}_{2}\)-ICEs can operate effectively without requiring high-purity hydrogen. Nevertheless, due to the high diffusivity of gaseous hydrogen, ensuring a homogeneous air-fuel mixture and maintaining stable combustion within the cylinder remains a critical challenge. Effective management of hydrogen combustion is essential to avoid knocking phenomena and preserve engine performance, particularly for heavy-duty engines requiring substantial injection quantities. In this study, injector hole design parameters were optimized to improve air-fuel mixture homogeneity and combustion stability through hydrogen gas injection into the cylinder. Given the importance of model accuracy in achieving reliable optimization results, the CFD model employed in this work was first validated against experimental data obtained from a single-cylinder research engine with available Particle Image Velocimetry (PIV) measurements [6]. Building upon this validated framework, the central research question of this study is whether optimizing the injector hole design, specifically the number of holes and their injection angles, can improve combustion stability, enhance engine performance, and reduce NO\({}_{\text{x}}\) emissions in a lean hydrogen-fueled heavy-duty engine. The main hypothesis is that adjusting these parameters leads to better in-cylinder air-fuel mixing, which increases indicated mean effective pressure (IMEP), while reducing the likelihood of misfire, knock, and excessive NO\({}_{\text{x}}\) formation, without the need for high injection pressures or complex fuel systems. While several studies have addressed hydrogen injector design using either experimental or CFD methods, most have focused on fixed geometries or did not include experimental flow field validation. Moreover, there remains limited work combining CFD-validated flow structures with surrogate-assisted optimization techniques under lean hydrogen conditions. To address this gap, the present study uses a CFD model that was previously validated through Particle Image Velocimetry (PIV) comparisons.