## 2 Materials and Methods
### Computational Methods  
Smoothed-Particle Hydrodynamics (Figure 3) is a versatile computational method that was initially developed for astrophysical problems [35; 36] but has found applications in many fields such as ballistics, volcanology, and oceanography [37]. With its ability to simulate the mechanics of continuum media including solid mechanics and fluid flows, it is becoming increasingly popular among researchers interested in biomedical engineering [38]. One of the promising applications of SPH in biomedical engineering is the study of the mechanism of traumatic brain injuries [39; 40]. This study employed IMPETUS Afea Solver (IMPETUS Afea AS, Norway) to solve fluid motion and boundary interaction using SPH for the fluid domain and the high-order finite element method for the solid domain. The methodology has been described in detail in prior publications (e.g., [41]) which also validated the head model against cadaveric experimental data [25].  
The IMPETUS Afea \(\gamma\)SPH Solver and IMPETUS Afea Solver were used to handle fluid motion and boundary interaction calculations as well as large deformations in solid parts, respectively. The advanced next-generation SPH method with increased accuracy was utilized by the \(\gamma\)SPH solver, while both solvers employed a commodity graphics processing unit (GPU) for parallel processing. Additionally, fully integrated solid elements were applied to eliminate hourglass modes and element inversion, which are common issues found in classic under-integrated elements. An explicit integration scheme was implemented to solve both fluid and solid domains along with their interactions. The \(\gamma\)SPH method is well-suited for complex applications due to its ability to account for movement in any direction and accurately model particle-to-structure contact. Unlike finite element fluid solvers that involve complicated contact and require remeshing of the fluid domain  
Figure 3: Smoothed-Particle Hydrodynamics Kernel approximation.  
Figure 2: The model comprises different parts such as the skull, cerebrum, cerebellum, pituitary gland, and brain stem along with subarachnoid space and other cavities filled with fluid particles. Additionally, displayed is half of the skull to give better insight into its structure. The helmet included in the model has two layers; an outer hard-shell cover, colored blue, for protection, and inner padding, highlighted in yellow, for comfort purposes.  
during simulation, IMPETUS provides a high resolution in terms of particle density. This feature makes it an invaluable tool for simulating helmet impacts on brain tissue, especially given the intricate structure of this organ.  
The use of a Tesla K40 GPU with 12 GB of Graphic DDR memory and 2880 CUDA Cores has allowed for parallel acceleration in simulations. The resulting reduction in computation time is significant compared to traditional FSI techniques, taking only days instead of weeks. This advancement offers more efficient processing capabilities for complex engineering problems such as designing and analyzing safety equipment such as helmets for athletes, military and industrial workers.