## 3 Results
### Effect of Spatial Uncertainty on the Mechanical Behavior of Additively Manufactured Struts
The interfacial boundaries of each layer were merged together with its neighboring layers using \"union assembly\" feature in COMSOL software, assuming the perfect binding condition between layers. Zero-displacement was imposed on the bottom of the strut, and the controlled displacement was applied on the top surface of the strut in the vertical direction as described in Equation (11). \[\text{u}_{\text{bottom}} = 0\] \[\text{u}_{\text{top,vertical}} = \text{u}_{0} \tag{11}\]  
where u is the displacement vector in the computational domain and ubottom is the displacement vector of elements at the bottom of strut. \(\text{u}_{\text{top,vertical}}\) is the vertical displacement of elements at the top of the strut and u\({}_{0}\) is the controlled displacement that was applied as a boundary condition. Figure 7a shows the loading direction and the boundary condition in the three struts: nominal, predicted AM, and volume matched struts. The applied strain was limited up to 0.03 to avoid the nonlinear deformation of the strut. Since the strut is considered as the building block of unit cell structure, the strain is defined as the actual deformation of the strut divided by the height of the strut. With respect to the deformation of the strut, the resulting stress was calculated as the average of total pressure over the bottom surface of the strut. Triangular elements as well as the adaptive mesh size control were adopted to make sure that it captures all details of spatial uncertainties at the edge of struts. The mesh size convergence test was performed to ensure that the simulation result is not sensible to the mesh size. The number of elements in three different geometries, giving the consistent simulation results, are 10,040, 831,944, and 9474 for the nominal strut, predicted AM strut, and volume-matched strut, respectively. The number of elements for predicted AM strut is significantly higher than the others because of the geometrical details of spatial uncertainty at the edge of strut. Figure 7b shows the stress-strain curves of 9 different examples obtained from the FE simulation. The slope of the stress-strain curve denotes the stiffness of the strut. The slopes of all stress-strain curves are grouped into three categories based on its printing angle, and a slope deviation exists given the existence of the spatial deviation. The decrease in slope is qualitatively observed in the predicted AM strut and volume-matched strut. To quantitatively understand the effect of spatial deviation on the stiffness of the struts, the stiffness of the volume-matched and predicted AM struts are normalized with the stiffness of the nominal struts as shown in Figure 7c. The change in the effective stiffness of the volume-matched geometry is not significant (less than 7% of the stiffness of the nominal) when the printing angle changes from 90\({}^{\circ}\) to 60\({}^{\circ}\). However, when the geometry exhibits roughness including the staircase profile as well as the spatial uncertainty, the stiffness is reduced to 70% of  
the nominal geometry when the printing angle is 60\({}^{\circ}\). The reduction in stiffness is expected since the roughness decreases the cross-sectional area of each strut and decreases the stiffness of the strut. However, it is observed that the roughness effect is more pronounced when compared the reduction in volume-matched case to the predicted AM case. This result indicates that the necessity of details of surface roughness becomes more important as the printing angle reduces. Specifically, the angled strut is frequently used in the lattice structures, which is the assembly of multiple unit cells, consisting of angled struts. Therefore, modeling the surface roughness including spatial uncertainty is important for the accurate estimation of mechanical properties of the lattice structure. Additionally, we compare the maximum value of von Mises stress in the three cases.