# 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
#### 3.1.1 Tension and Torsion of a Cylindrical Tube
Since there is the boundary effect of the RBF, see Figure 2, the following evaluations will only consider points in the region \(-25^{\circ}\leq\Phi\leq 25^{\circ},30\,\text{mm}\leq Z\leq 50\,\text{mm}\). **Remark 1**. : _The region of interest is chosen in an arbitrary manner to circumvent the influence of the individual shape parameters of each RBF, and to avoid the boundary effects that lead to a further misrepresentation of the convergence behavior. Not only the RBFs show such boundary effects. On a closer look at the triangulation concept, all triangles are omitted which have angles below a permissible critical angle. This serves to also exclude highly inaccurate solutions that occur due to the coordinate transformation between global and surface coordinates. To circumvent this discussion for both approaches, we restrict ourselves to the seemingly arbitrary center region._  
To compare the new approach, the classical triangulation concept is considered as well. Here, geometry and deformation are interpolated with shape functions for triangles, similar to the finite element method. The triangulation is completed with the method of [27, 28]. Within each triangle, the in-plane stretches (or for practical applications the principal in-plane strains, can be determined, see [4]). This is followed by a calculation of the mean value of the principal stretches of triangles adjacent to a data point (this implies an additional interpolation). For our investigations, the point distance \(\Delta d=\{0.15,0.25,0.35,0.45\}\) is varied and the mean relative error (37) of the different methods is analyzed, see Figure 3 (left). Figure 2: Relative error of maximum principal stretch (Wendland function withmonomials, \(R_{0}\leavevmode\
obreak\ =\leavevmode\
obreak\ 1.5\,\text{mm}\), \(\Delta d=0.15\,\text{mm}\)) showing the boundary region effect of RBFs; (**left**) 3D representation, (**right**) 2D projection. To allow for a comparison of the functions \(\hat{m}_{G}(\rho)\), \(\hat{m}_{W}(\rho)\), and \(\hat{m}_{I}(\rho)\), the shape factors are chosen to be constant, but in such a manner that the functions look quite similar (\(R_{0}=1.5\,\)mm, \(c_{G}=2.7\), \(c_{I}=0.2\)), see Appendix A. The Gaussian-RBF \(\hat{m}_{G}(\rho)\) with monomials (Gau + Mon) show very good results in comparison to the triangulation concept. However, the condition number of the resulting systems of linear Equations (23) and (25) are too large for particular point distances so that the systems are not solvable for smaller values of the point distances \(\Delta d\), see \[\hat{r}(R)=\lambda^{-1/2}R,\quad\hat{\varphi}(\Phi,Z)=\Phi+DZ,\quad\hat{z}(Z)= \lambda Z, \tag{32}\]0  
Thus, the relation between the point distance and the shape factor \(c_{G}\) is essential--as it is known from the literature. The Wendland function \(\hat{m}_{W}(\rho)\) shows much better properties. Since there is an intersection of the pure RBF approach (Wen) with the extension of monomials (Wen + Mon), there cannot be made an accuracy statement whether linear polynomials are essential. Looking only at the condition numbers, the consideration of monomials increases the condition numbers again. The inverse multi-quadrics approach (IMQ) and (IMQ + Mon) yields similar results regarding the aspect of conditioning but better results in the accuracy. The latter can be interpreted by the course of the function, see Figure A1, which shows a larger region until the values of the function becomes very small. Thus, linear functions seem to be better approximated--leading to the property that the monomial approach implies more linear dependencies of the column vectors in the linear system. In the next study, the point distance is fixed by \(\Delta d=0.25\,\)mm. Then, the normalized shape factor \(\beta_{W}=R_{0}/\Delta d=R_{0}/0.25\) and that of the inverse multi-quadrics function, \(c_{\text{IMQ}}:=c_{I}R_{0}=\beta_{I}/10\), are varied, see Figure 3 (right). The factor \(1/10\) is chosen for a better presentation in Figure 3 (right). The inverse multi-quadrics converges very fast for