## 4 Results and Discussion  
### Convergence of Beam-Based Model  
To check convergence of the model, the effect of time step and mesh density on both stresses and displacements must be examined. For vertical stresses, one area of interest was the corner where the lowest layer of the wall intersects the brim, because this is where the maximum stresses are located. These results and their location are depicted in Figure 8.  
As Figure 8 shows, the vertical corner stress is converging to a value between 5.2 and 5.3 MPa as the number of elements per layer is increased, while time step has virtually no effect on results. Based on this, 60 elements per layer, which exhibited a 4% difference in peak stress compared to the finest mesh, is a reasonable level of mesh refinement.  
Figure 6: Elastic and shear moduli.  
Figure 7: Thermal variation of coefficients of thermal expansion.  
In the case of displacements, no obvious point of interest was visible and maximum magnitude only varied 3% from the coarsest time step and mesh density to the finest. Therefore, convergence of change in average nodal displacement for the entire wall was examined and is shown in Figure 9, which indicates that average vertical displacements are converging with decreasing time step and increasing mesh density. Mesh density has a modest effect on convergence, mainly at larger element sizes. Time step has a much larger effect on the convergence of displacements. A time step of 20 s was chosen for subsequent simulations.  
Figure 10 shows the impact of time step and mesh density on model run time. As expected, simulation times grow significantly with decreasing time step and mesh density, with run times in Figure 10 varying by four orders of magnitude. For reference, the run times for the ABAQUS structural model are also given in Figure 10. The ABAQUS run times are significantly slower than all beam element model runs for the same time step, highlighting the gains in computational efficiency achievable with the beam element model. For example, the chosen configuration for the beam-based model of 60 elements per layer with a 20 s time step, shown with a red circle in Figure 10, ran in roughly six minutes, compared to 34 min and 3 h for the ABAQUS model runs with time steps of 100 and 10 s, respectively. This is due to the much larger mesh resulting from the 3D formulation with brick elements. It should be noted, however, that the ABAQUS mesh was chosen for simplicity and it is quite possible that fewer elements could be used while retaining acceptable accuracy, which could significantly reduce ABAQUS computation time. However, this is countered by the fact that the beam element model is implemented entirely in MATLAB, which is inherently slow due to the interpreted nature of the code; significant speedup would be achieved were the beam element model written in a compiled language such as FORTRAN or C. Further, while not explored here, we note that meshes with varying beam element lengths could be used to ensure convergence of peak stresses (e.g., vertical corner stress discussed above) while reducing computation costs. Overall, the time comparison indicates the promise of the beam element model for reducing run time for simulating residual stresses in large, 3D-printed polymeric parts.  
Figure 8: Final vertical stress at corner of wall versus time step and mesh density.