## 4 Numerical Simulation Applied to the FFF Process
### Thermal-Mechanical  
Thermal-mechanical behavior in FFF involves the interaction between thermal gradients, material shrinkage, and mechanical constraints imposed by the layered manufacturing process. These interactions lead to the development of residual stresses and strain localization, which can result in defects such as warping, delamination, or dimensional instability. Numerical approaches typically address these effects by coupling transient thermal simulations with mechanical models that incorporate temperature-dependent material properties and layer-by-layer deposition sequences. One of the earliest mathematical models for warp deformation in FFF was developed by [PERSON] et al. (2007) [53], who analyzed thermal stresses induced by material contraction during cooling. Their study examined the influence of deposition layer count, section length, chamber temperature, and material shrinkage rate on deformation. They found that higher chamber temperatures reduce warp deformation, while larger stacking sections and higher shrinkage rates increase it. Their model provided valuable insights into process optimization, emphasizing that adjusting raster angles and partitioning deposition areas  
Figure 13: Finite element simulation of heat transfer within the liquefier: (**a**) diagram of the three geometric regions—heater tube, constriction, and nozzle; (**b**) temperature distribution at volumetric flow rates of 30, 60, and 90 cm\({}^{3}\)/h, showing reduced thermal penetration at higher rates; (**c**) core temperature as a function of build rate at various axial distances from the inlet. Adapted with permission from [50]. can significantly improve dimensional accuracy. However, the model does not fully account for residual stress evolution during cooling or the effects of complex geometries, limiting its applicability for intricate parts. To expand simulation capabilities and reduce computational demands, [PERSON] and [PERSON] (2020) [54] introduced a discrete-event thermal model capable of capturing transient heat transfer behavior during the printing process. Their approach demonstrated good agreement with more computationally intensive methods, making it suitable for efficient thermal prediction across various geometries and parameter settings. More recently, [PERSON] (2023) [55] developed a high-fidelity thermal simulation framework specifically aimed at improving the prediction of temperature fields and guiding the process optimization of printed polymer and composite parts. The study emphasized the importance of accurately capturing spatiotemporal temperature gradients to mitigate stress concentrations and improve structural performance. Building on earlier findings, advanced finite element approaches have been developed to simulate stress accumulation and deformation in FFF. [PERSON] et al. (2023) [56] developed a 3D thermo-mechanical finite element model to predict dimensional accuracy, defect formation, and residual stresses in wood/PLA biocomposites fabricated via FFF. Their study identified uneven filler distribution and high hydrophilicity as major challenges, influencing printability and final part stability. The model, validated experimentally, demonstrated that residual stresses are sensitive to layer thickness and extrusion temperature. As shown in Figure 14, residual stress increased significantly with thicker layers, with values peaking at 48 MPa for 0.25 mm layer thickness. Lower nozzle temperatures and thinner layers resulted in reduced thermal gradients and lower residual stress values, especially in the lower portions of the parts, which remained more constrained by platform adhesion. These findings reinforce the need to optimize process parameters, especially for heterogeneous biocomposites, and highlight current limitations in capturing anisotropic thermal-mechanical interactions in such materials. Another finite element modeling approach was developed by [PERSON] et al. (2018) [57], who used ABAQUS to predict distortions in FFF with a sequential element activation method.