# Full-Field Strain Determination for Additively Manufactured Parts Using Radial Basis Functions
## 3 Evaluation of Strain Determination Using RBFs
### Verification Examples of WAAM-Produced Cylindrical Components
#### 3.1.2. Tension and Torsion of a Tube with Rough Surface
In our example, we assume \(\hat{r}(R)=\overline{r}(Z)=\lambda^{-1/2}\hat{R}(Z)\) and \(\hat{z}(Z)=\lambda Z\)--similar to the classical tension-torsion example, where \(\lambda\) represents the axial stretch--with \(\hat{R}(Z)\) of Equation (41). As our geometrical setting, \(R_{0}=15\,\mathrm{mm}\), \(D_{S}=0.5\,\mathrm{mm}\), \(H=8\,\mathrm{mm}\), \(D_{W}=0.3\,\mathrm{mm}\), \(c=20\,\mathrm{mm}^{-1}\), \(\delta=0.2\,\mathrm{mm}\), \(n_{\mathrm{ws}}=10\), and \(n_{\mathrm{wb}}=2\) are defined. Moreover, equidistant data points between \(-45^{\circ}\leq\Phi\leq 45^{\circ}\) and \(20\,\mathrm{mm}\leq Z\leq 60\,\mathrm{mm}\) are generated. The surface contour \(\hat{R}(Z)\) is shown in Figure 7.  
Figure 8 (left) shows the maximum and minimum principal stretches over the axial direction \(Z\) evaluated at \(\Phi=0^{\circ}\). For the triangulation concept, this is only approximately the case, \(\Phi\approx 0^{\circ}\)). Obviously, there is a strong stretch inhomogeneity generated by the surface. This has an essential influence on the accuracy of the stretch results. In Figure 8 (right), the relative error using the classical triangulation concept is plotted using varying dot distances (\(\Delta d=(0.15,0.25,0.35,0.45)\) mm). Especially in the areas of notches, the errors are very large. This is nearly independent of the point distances and obvious against the background that hardly any points enter the notch. However, even between the notches, an error of less than 0.1% can be seen in the main extensions due to the roughness. Figure 7: Surface contour of verification example with rough surface at time \(t=0\). When applying the Wendland and inverse multi-quadrics (IMQ) functions with the respective use of monomials, the selection of appropriate shape parameters is needed. In this regard, see Figure 3 (right). For the following, \(\beta_{W}=10\), i.e., \(R_{0}=\beta_{W}\Delta d\) are chosen for the Wendland parameter, and \(\beta_{I}=10c_{\text{IMQ}}\) for the IMQ, with \(\epsilon_{\text{IMQ}}\) from Equation (A4) in Appendix A. Here, the factors \(c_{\text{IMQ}}=\{0.3,0.5,0.7,0.9\}\)mm are obtained for \(\Delta d=\{0.15,0.25,0.35,0.45\}\)mm. When considering the same behavior of the RBFs in Figure 9, one can see that the RBF-approach using the Wendland function \(\hat{m}_{W}(\rho)\) with monomials yields smaller errors between the notches than the triangulation concept. Across the notches, relative errors similar to those obtained with the triangulation concept are observable. For smaller point distances, however, the relative errors are smaller for the Wendland ansatz. The results are even better for the inverse multi-quadrics functions \(\hat{m}_{I}(\rho)\) with monomials, where the shape parameter is chosen according to Equation (A4). The condition numbers are in all cases around \(10^{8}\)-\(10^{9}\). In conclusion, the stretch (strain) results must be interpreted carefully for real applications in regions of strong changes in the surface geometry. The documented errors are in the order of 20% close to notches--and this only includes errors that result from  
Figure 8: (**left**) Minimum and maximum principal stretches for verification examplewith rough surface, (**right**) relative error of maximum principal stretches using triangulation with different point distances \(\Delta d=(0.15,0.25,0.35,0.45)\) in mm. Figure 9: Rough surface verification example for different point distances \(\Delta d=(0.15,0.25,0.35,0.45)\) in mm and adapted shape parameters: (**left**) Wendland function with monomials (\(R_{0}=10\Delta d\)), (**right**) inverse multi-quadrics with monomials (\(\epsilon_{\text{IMQ}}(\Delta d)\) in Equation (A4)). any of the interpolation concepts to determine the in-plane stretches (strains). Further, larger point distances can lead to relative errors of the stretches in the order of 0.1% in the \"smoother\" regions.