# Full-Field Strain Determination for Additively Manufactured Parts Using Radial Basis Functions
## 3 Evaluation of Strain Determination Using RBFs
### Real Application Example of a Tube under Tension and Torsion
#### 3.2.3 Smoothing Data  
The final issue we would like to address is the application of the linear least-square procedure, see Equation (27). The linear system (27) is solved using a singular value decomposition scheme to circumvent the ill-conditioned coefficient matrices in some of the computations. Here, it is assumed that the number of center (evaluation) points \(n_{\text{cp}}\) is not identical to the number of data points \(n_{\text{D}}\) of the DIC-system. Again, the IMQ-method with monomials is used (\(c_{\text{IMQ}}=0.3\,\text{mm}\)). The number of data points is \(n_{\text{D}}=66,013\) for the tensile problem with a smooth outer surface, resulting from the chosen facet size. Center points are generated for the first calculation, making sure that the number of center points corresponds to approximately 50% of the number of data points, i.e., 98 in circumferential and 335 in  
Figure 14: Comparison of rough (**top**) and smooth (machined off; (**bottom**)) outer surface for torsion (IMQ + Mon) showing principal maximum Green strain. Figure 13: Comparison of rough and smooth (machined off) outer surface for tension (IMQ + Mon) showing principal maximum Green strain \(\epsilon_{\text{max}}^{(2)}\). longitudinal direction (\(n_{\text{cp}}=32\),\(830\)). In a second calculation, we use only (approximately) a number of center points, which is around 25% of the number of data points (69 points in circumferential and 236 points in longitudinal direction, \(n_{\text{cp}}=16\),\(284\)). The number of data points remains the same. We expect a smoothing of the strains. In Figure 15, four evaluations of the maximum principal strains \(\varepsilon_{\text{max}}^{(2)}\) are compiled, where the RBFs are evaluated for the plotting routine at the data points (Figure 15 (top right) is equivalent to Figure 13 (right)). For comparative purposes, we start off with compiling the classical triangulation concept (top left figure), followed by the IMQ + Mon-approach (\(n_{\text{D}}=n_{\text{cp}}\); top right). Again, the RBF-approach resolves finer the strains, when the data points are equivalent to the center points. The bottom left figure shows the result of the strains from the computation with 50% center points. Obviously, the strains become more homogeneously distributed (blue regions) due to the regression property of using less center points, i.e., RBFs, than data points. This behavior increases significantly with the calculation using the 25% center point computation. For future investigations, it must be asked whether error-controlled, adaptive schemes yield an optimized number of center points. **Remark 2**. : _If one considers the displacements horizontally and in the viewing direction of the reader (\(X\)-direction see Figure 10), which is not shown here for reasons of brevity, it turns out that the specimens are not optimally aligned. In other words, they are bent and twisted if tension (or compression) is applied. This does not have very much influence when twisting the specimens by pure torsion, when comparing the order of magnitude to the analytical results in Section 3.1.1, but it does if they are drawn or pressed. Both the machine stiffness and the misalignment have a significant influence on the strain distribution--as well as curved specimens and ones that are not exactly aligned. Overall, this affects the magnitude of the strains, which are larger than the expected strains on the view side. The specimen elongation is approximately \(0.1\,\mathrm{mm}\) after subtracting the rigid body motion (resulting from the testing machine's stiffness). Therefore, special care must be taken when fabricating specimens for the purpose of identifying their material properties. They should turn out as flawless as possible.