## 4 Numerical Simulation Applied to the FFF Process
### Material Property Characterization  
The mechanical and thermal behavior of materials in FFF depends on process parameters, material composition, and reinforcement strategies [36]. Factors such as layer thickness, infill density, raster orientation, and cooling rate influence strength, ductility, and failure mechanisms in printed parts [10]. To optimize material performance and structural integrity, researchers have combined experimental testing, numerical modeling, and real-time defect detection to evaluate how printing parameters affect mechanical behavior. The influence of printing parameters on mechanical properties has been studied. [PERSON] et al. (2013) [59] and [PERSON] et al. (2015) [60] investigated the mechanical behavior of parts produced via FFF, focusing on ABS-M30 and Polycarbonate (PC), respectively. Both studies concluded that strength and stiffness were highest when filament layers were aligned with the loading direction, whereas perpendicular loading resulted in weaker performance due to reduced interlayer bonding. Analytical and finite element models developed by these authors closely matched experimental results, achieving average deviations below 10%. However, both studies emphasized the need for improved modeling to better capture plastic deformation and failure behavior beyond the elastic limit. The experimental and numerical configurations used by [PERSON] et al. are illustrated in Figure 15.  
Figure 15: Experimental setup for the physical test (**left**) and ANSYS® simulation schematic (**right**). Small displacement views (**a,b**), and images at 35 mm of deformation (**c,d**). All dimensions in millimeters. Adapted from [60]. In addition to mechanical performance, geometric accuracy and surface quality are crucial for material characterization. [PERSON] et al. (2019) [61] studied how build orientation, layer thickness, and feed rate affect PLA part dimensions, flatness, and surface texture. Thinner layers improved dimensional accuracy, while feed rate had minimal impact. Flatness errors primarily resulted from variations in layer thickness and extruder acceleration/deceleration, with higher feed rates increasing material accumulation. Surface roughness increased with thicker layers due to the stair-stepping effect inherent in layered manufacturing. The study highlighted the importance of precise extruder control and optimized parameters for better surface quality and geometric accuracy. A simplified numerical model was proposed by [PERSON] et al. (2021) [62] to predict the mechanical behavior of FFF 3D printing parts that are built from an ABS pre-structured material. The model effectively addresses the anisotropic build configuration by assigning local material reference frames to individual mesh elements. It treats the material as homogeneous and employs specific material constants to introduce anisotropy associated with intricate deposition trajectories. Their results verify that the material exhibits quasi-isotropic elastic behavior, as the longitudinal and transversal Young's modulus are approximately equivalent. The strength anisotropy in tensile specimens with varying raster angles has been determined by combining [PERSON]'s criterion with the filament's plastic flow curve. The longitudinal direction possesses the highest yield strength and a ductile behavior, whereas the transversal direction is the weakest and demonstrates fragile behavior at the yielding point. These effects are quantitatively illustrated in the comparison between simulated and experimental load-displacement curves shown in Figure 16.  
Advancing such models requires precise experimental characterization to obtain reliable and process-relevant material data. Dynamic Mechanical Analysis (DMA) provides viscoelastic properties, such as storage and loss moduli across a range of temperatures, that are commonly used to model stress relaxation and residual stress evolution [63].