## 1 Introduction
This validated model is then integrated into a Kriging-based multi-objective genetic algorithm framework to systematically optimize the injector hole count and injection angle in a low-pressure direct injection (LPDI) configuration. The goal is to improve engine performance and reduce NO\({}_{\text{x}}\) emissions under lean hydrogen combustion conditions. Several previous studies have contributed valuable insights into hydrogen-fueled internal combustion engine development. For example, Sandia National Laboratories developed a single-cylinder hydrogen research engine in which nitrogen gas was injected to examine mixture formation without combustion. This study generated in-cylinder pressure and velocity vector data across crank angles using PIV measurements [6], and forms the experimental basis for the validation carried out in the present work. [PERSON] and [PERSON] et al. [7] focused on optimizing the injection system for a medium-speed, four-stroke hydrogen engine. Their research emphasized that turbulence intensity and injection penetration distance are critical for achieving homogeneous mixtures, particularly in large-bore DI hydrogen engines. Through systematic optimization, they proposed injection parameters 300 [bar] injection pressure, 10 [\({}^{\circ}\)CA] duration, and a 2.8 [mm] injector nozzle diameter that led to a significant reduction in NOx emissions (over 80%) while maintaining engine power above 80% of the original diesel engine benchmark. In the context of CFD validation methods, [PERSON] et al. [8] introduced several comparison metrics (e.g., weighted relevance index, magnitude index) to assess flow field accuracy between CFD predictions and PIV measurements. Their study demonstrated that early detection of non-uniformity in mixture distribution can significantly improve combustion modeling accuracy. However, slight overpredictions in intake jet behavior were observed during valve closure events. [PERSON] et al. [9] explored 1D-3D CFD coupling approaches using large eddy simulation (LES) for engine flow predictions. Their study compared 56 engine cycles between simulations and experiments, ultimately recommending a 1 mm mesh size for ports and runners to improve flow mixture representation. Further research by [PERSON] et al. [10] focused on simulating knock events in internal combustion engines. Their findings highlighted the role of temperature stratification and hotspot formation near the exhaust valve region in promoting knock, resulting in pressure oscillations exceeding 100 kPa following auto-ignition events. Experimental studies by [PERSON] et al. [11] demonstrated that hydrogen-air spark ignition engines can operate efficiently under low-load conditions while controlling NOx emissions. However, they also reported the emergence of abnormal combustion phenomena as the mixture approached stoichiometric conditions or full-load operation. Over the past decade, employing genetic algorithm techniques for the multi-objective optimization of combustion chamber design and engine parameters has been quite popular [12; 13]. In a recent study, [PERSON] et al. [12] implemented a comprehensive multi-objective optimization methodology by coupling the NSGA-II algorithm with CONVERGE CFD to evaluate the effects of operational parameters on a methane/diesel RCCI engine. Their results demonstrated that optimizing parameters such as injection timing, fuel mass, and engine speed led to a 27.8% reduction in ISFC and up to 99% lower CO emissions compared to conventional diesel combustion. The study also highlighted the existence of clear trade-offs between NOx and unburned hydrocarbon emissions across the Pareto-optimal solutions. [PERSON] et al. [13] conducted a shape optimization study to enhance combustion efficiency and reduce emissions in a compression-ignition engine. Using CFD-based simulations in CONVERGE and a multi-objective genetic algorithm in CAESES, they optimized key piston bowl and injection parameters.