## 4 Numerical Simulation Applied to the FFF Process
### Cooling and Solidification  
The cooling and solidification phase in FFF is designed to achieve optimized interlayer adhesion, mechanical performance, and dimensional accuracy ([PERSON] et al., 2021) [44]. Removing heat properly during the polymer deposition process helps to create strong bonds between layers by allowing effective molecular diffusion. Therefore, it is necessary to control the heat transfer mechanisms (conduction, convection, and radiation) to maintain structural integrity and ensure high-quality printed components. Among these mechanisms, conduction has been identified as the dominant heat transfer mode. [PERSON] et al. (2014) [45] confirmed that conduction between deposited filaments and the build plate significantly outweighs convection and radiation in cooling efficiency. They recommended strategies such as preheating the build surface or controlling chamber temperatures to effectively reduce thermal gradients, thereby minimizing warping and delamination. Figure 11: Pressure drops of PLA as a function of shear rate, measured using capillary and orifice dies. Adapted with permission from [29]. Heat transfer also determines the quality of filament bonding. [PERSON] et al. (2017) [46] developed a 3D transient thermal model and found that reheating from newly deposited layers improved molecular entanglement, whereas rapid cooling negatively impacted interlayer strength. In addition to the thermal findings, their study employed the _Element Birth and Death_ technique in ANSYS to simulate the sequential nature of layer deposition. This approach allowed the model to dynamically activate elements as new material was added, accurately capturing the temporal evolution of the thermal field and heat accumulation across layers. As a result, the simulation provided enhanced insight into local reheating effects, cooling gradients, and the influence of deposition timing on interlayer temperature history. Similarly, [PERSON] et al. (2016) [47] used FEA, which considers how materials behave at different temperatures, and found that quick cooling is harmful because it limits how materials can mix, weakens their stickiness, and raises internal pressures. To optimize bonding quality, they recommended controlling cooling rates through adjustments in build plate temperature and using enclosed printing environments. Further deepening the understanding of adhesion, [PERSON] and [PERSON] (2017) [48] developed a diffusion-based healing model, emphasizing maintaining interfacial temperatures above the polymer's glass transition temperature (Tg) to enhance polymer chain diffusion effectively. Similarly, [PERSON] et al. (2020) [49] utilized infrared thermography and thermal simulations to explore heat transfer and adhesion during FFF. They demonstrated that filament adhesion depends heavily on coalescence and polymer chain reptation, the thermal motion of polymer chains that allows them to entangle and effectively bond. Both phenomena are driven by the thermal history experienced during the process. Their findings indicate that precise thermal management, particularly accurate control over the cooling rate, significantly enhances mechanical performance, especially in high-performance thermoplastics such as PEKK (see Figure 12). Addressing practical constraints, [PERSON] et al. (2017) [50] identified key limitations such as extrusion force, heat transfer efficiency, and printhead motion in FFF. Their results indicated that elevated nozzle temperatures enhance extrusion speeds but are limited by insufficient thermal conduction within the filament core, increasing extrusion forces and the risk of filament buckling. To address this, they suggested longer liquefiers to enhance heating efficiency, enabling higher flow rates without excessive force. These thermal limitations are further illustrated in Figure 13, where finite element simulations show that, as flow rate increases, heat penetration into the filament core decreases significantly, especially near the liquefier inlet.