## 4 Numerical Simulation Applied to the FFF Process
### Melt Flow  
The melt flow behavior in FFF significantly influences the extrusion stability, the deposition accuracy, and the interlayer adhesion, which directly affect the mechanical properties and dimensional precision. Understanding polymer melt dynamics within the hot end, during extrusion, and after deposition enables control over print quality and the optimization of process parameters. CFD has advanced predictions of melt flow, specifically addressing viscosity effects, shear-induced deformation, and heat transfer processes. Temperature and shear rate, closely related to extrusion speed, primarily govern the melt flow of thermoplastic materials during deposition. Due to the low pressures in FFF, these rheological factors dominate flow dynamics. [PERSON] et al. (2018) [39] developed a CFD model that considered polymer flow as an isothermal Newtonian fluid, demonstrating how filament geometry depends on nozzle-to-substrate gap and extruder-to-substrate velocity ratio. Their findings emphasize the precise control of deposition parameters to achieve optimal filament shape and interlayer adhesion. Figure 8 shows how the strand shape (W/H) and density (A/(W\(\cdot\)H)) change based on the distance from the nozzle to the surface (g/D) and the speed of the material flow (V/U). As g/D increases, the aspect ratio decreases, indicating a wider and flatter strand profile, while the compactness is maximized at intermediate V/U values, reflecting the interplay between shear rate and material deposition behavior. Building on their earlier findings, [PERSON] et al. (2022) [40] developed an advanced CFD model, incorporating viscoelastic rheology to simulate the non-isothermal polymer flow more accurately through the hot end in filament-based additive manufacturing. Unlike previous models that assumed fully liquefied flow or employed generalized Newtonian approximations, this approach accounted for shear and elongational stresses, offering improved predictive accuracy for filament feeding forces across varying process conditions. The model was experimentally validated and demonstrated strong alignment with measured data, particularly under high flow rates. Furthermore, it enabled the parametric optimization of key hot-end geometries, such as liquefier length, capillary diameter, and inlet angle, highlighting their impact on flow stability and extrusion performance. These  
Figure 8: Aspect ratio (W/H) and compactness (A/(W\(\cdot\)H)) of the deposited strand as functions of the normalized nozzle gap (g/D) and velocity ratio (V/U). Adapted with permission from [39]. findings reinforce the earlier conclusions by [PERSON] et al. (2020) [41], who identified recirculation regions at the solid-melt interface as critical factors that influence temperature distribution and pressure buildup within the nozzle, underscoring the importance of physically representative rheological modeling (Figure 9). Nozzle design plays a fundamental role in controlling melt flow dynamics in extrusion-based additive manufacturing. [PERSON] et al. (2021) developed a nozzle modification strategy aimed at controlling the orientation of filler particles during extrusion [42]. By embedding an orifice geometry into the nozzle, they induced the perpendicular alignment of anisotropic reinforcements, which directly influenced the mechanical anisotropy of printed parts. Through combined experimental and numerical analysis, the study demonstrated how nozzle-induced flow fields affect particle orientation, highlighting the importance of microstructural control in melt flow modeling and the functional optimization of fiber-reinforced filaments [42]. Building on this notion of nozzle-function tailoring, [PERSON] et al. (2024) [43] employed CFD simulations coupled with global optimization algorithms to improve nozzle geometry for high-speed FFF applications.