# Full-Field Strain Determination for Additively Manufactured Parts Using Radial Basis Functions
## 3 Evaluation of Strain Determination Using RBFs
### Discussion  
It turns out that Wendland and inverse multi-quadrics radial basis functions are appropriate to determine the strains (or stretches) in the surface of curvilinear material bodies. This totally new global interpolation approach to evaluate DIC-data for strain analysis has the advantage that the accuracy can be increased--in comparison to local strain determination approaches--as shown using analytical verification examples concerning the deformation of smooth and non-smooth curvilinear surfaces. For future applications, the Wendland function might be an interesting ansatz, since it yields sparse matrices which are required to reduce the computational costs for a larger number of points. Even more, a concept is provided, see Appendix A, on how to adapt the shape factor of the RBFs to the point distances of the DIC-patterns. However, there are also some disadvantages which have to be treated very carefully. In the investigations of the real samples for curvilinear surfaces, there might be non-unique mappings between the surface coordinates \((\Theta^{1},\Theta^{2})=(Y,Z)\) and the real surface description. This is caused by scattered data and by boundary points that are very close to each other in the mapped data (this might result in singular covariant metric matrices). This can be circumvented using other surface parameters, for example \((\Theta^{1},\Theta^{2})=(\Phi,Z)\) in the case of our applications. Furthermore, the so-called boundary effect--i.e., weakly oscillating courses at the boundary of the region under consideration--imply a stronger deviation of the gradients. However, the gradients have a direct impact on the strain determination. One possibility to circumvent this problem is to cut off the edge region after the interpolation step. For the Wendland function, where a particular radius in the RBF is given, the multiple of the radius can be chosen. For smaller point distances, this concept of choosing the multiple of some geometrical measure seems to be sufficient, although  
Figure 16: Distribution of principal strain directions associated to the maximum principal strain. The bounding box shows the projected region of the DIC-data: (**left**) 9 px facet size, (**right**) 19 px facet size. it is a non-objective choice. This approach, however, is difficult to use for the inverse multi-quadrics function, since this type of function has a longer distance of influence. An alternative approach might be to extrapolate the data of the boundary region, to interpolate all data (real data and the generated synthetic data), and to show only the original region. However, both approaches require detailed investigations. Instead of the interpolation concept (center points equal data points), a regression approach is possible. This allows to reduce the resulting system of linear equations and to smooth the strain distribution, which raises the question whether adaptive schemes are suitable to automatically determine the center points. With regard to the interpretation of the results for additively manufactured components, where the surface has a higher roughness, it should be noted that special care has to be taken regarding accuracy--for all strain calculation tools--especially if there are small notches in the specimen's surface due to the production process. This is particularly the case for products manufactured by WAAM processes. Furthermore, the small notches have an essential influence on the strain distribution in the components. Apart from this, we observed that the produced specimens show misalignments due to the difficult manufacturing process. This leads to additional shear and twist deformation modes, which essentially have significant influence on the magnitude of the expected strains. In this context, the deviation of the principal strain orientation between loading direction and the axial axis is different so that it is hard to compare it to some global strain measure.