## 3 Results
### Effect of Spatial Uncertainty on the Mechanical Behavior of Additively Manufactured Struts  
In the study, we consider simple struts with three different printing angles (90\({}^{\circ}\), 75\({}^{\circ}\), and 60\({}^{\circ}\)). In order to clearly visualize the effect from the spatial uncertainty, the struts in FE analysis are simplified as 2D struts. To investigate the effect of spatial uncertainty on the mechanical properties of additively manufactured struts, three computational models with different strut geometries are first prepared as follows: (1) _nominal strut_: desired CAD geometry with a smooth side surface (planned geometry without any staircase profile as well as spatial uncertainty), (2) _predicted AM strut_: geometry with predicted roughness in which the staircase profile and the spatial uncertainty are reproduced from our estimated Gaussian process model, and (3) _volume-matched strut_: strut also with a smooth side surface, having a thickness adjusted by smoothing the ridges and valleys of the rough side profile such that the volume is matched with the predicted AM strut. The analysis of (3) is necessary for fair comparison because the thickness of the manufactured struts is slightly lower than that of the planned geometry. By including (3), the difference in the mechanical properties that are caused by the volume change can be investigated. The thickness of the strut in the CAD model is 800 \(\upmu\)m, and the thicknesses of struts in the volume-matched geometries extracted from the sample images are approximately 754 \(\upmu\)m, 754 \(\upmu\)m, and 760 \(\upmu\)m for 90\({}^{\circ}\), 75\({}^{\circ}\), and 60\({}^{\circ}\), respectively. For the computational efficiency, one third of the height of printed strut (200 \(\upmu\)m) is modeled in the simulation. The three different aforementioned geometries are created for each printing angle, namely 90\({}^{\circ}\), 75\({}^{\circ}\), and 60\({}^{\circ}\). A linear static structural analysis was performed to calculate the mechanical property of struts in three different printing angles (90\({}^{\circ}\), 75\({}^{\circ}\), and 60\({}^{\circ}\)) under compressional loading conditions. A commercial FE software package, COMSOL MultiphysicsTM, running on the personal computer (Windows 10 64 bit, Intel i7-4790 CPU, 16 GB RAM), was used for all numerical simulations. The governing equation for the struts is as follows [34]:  
Figure 6: Twenty sampled realizations from the estimated Gaussian process model. \[0 = \
abla\cdot\left(\text{FS}\right)^{\text{T}}+\text{Fv}\] \[\sigma = \text{J}^{-1}\text{FS}^{\text{T}}\] \[\varepsilon = \tfrac{1}{2}\left(F^{T}F-I\right) \tag{10}\] \[\text{S} = \tfrac{\vartheta\text{W}_{\text{S}}}{\partial\varepsilon}\] \[\text{W}_{\text{S}} = \tfrac{\mu}{2}\left(\bar{\text{I}}_{1}-3\right)\]  
where F is the deformation gradient, S is the second Piola-Kirchhoff stress, v is the left stretch tensor. The Cauchy stress tensor, \(\sigma\), is the multiplication of F, S, the transpose of F, and the volume ratio, J, which is the determinant of F. The Lagragian-Green strain, \(\varepsilon\), is a function of F and the identity tensor, I. The second Piola-Kirchhoff stress, S, is the gradient of the strain energy-density function, \(\text{W}_{\text{S}}\) with respect to \(\varepsilon\). \(\bar{\text{I}}_{1}\) is the scalar invariant of the left Cauchy-Green deformation tensor. The struts are made of a photo-polymerizable poly(ethylene glycol) diacrylate (PEGDA) (molecular weight 250). PEGDA is assumed to be a hyperelastic, incompressible, and isotropic material and modeled by using neo-Hookean model [34]. The uniaxial compression test is performed to obtain the material property of PEGDA. The stress-strain curve obtained from the experiment is fitted to the neo-Hookean model to determine the shear modulus, \(\mu\). The fitted value of \(\mu\) is approximately 36.1 MPa for the PEGDA.