## 3 Meshless Approach Overview
### Methodology
#### 3.1.3 Algorithm
Discretization problems and numer

Figure 8: Rms error. Figure 7: \(L^{2}\) error. ical errors will undoubtedly lead to noisy reconstruction. As the number of projections increased, rapid convergence to the analytic solution was observed. The constrained optimization problem (23) can be solved using a variety of approaches. Our algorithm uses the \"lsqlin\" inbuild function in MATLAB. This convergence gives us confidence that in the face of real experimental uncertainties, the algorithm will converge to an analytic solution. ## 5 Experimental Demonstration

The rbf approximation methods presented above have been tested on synthetic and real data sets. We are now testing the algorithm on the experimental data previously studied for the offset ring-and-plug, and the crushed ring sample [26]. The crushed ring sample was 14 mm thick with 23 mm outer diameter and, 10 mm inner diameter. To test the algorithm for continuous and discontinuous stress fields, specific samples were selected. The ring-and-plug sample is designed to test the discontinuous strain field and the crushed ring for continuous strain field testing. The offset ring-and-plug sample geometry is shown in Figure 9, and the crushed ring is shown on the left side of Figure 10. The sample described included a total interference of \(40\pm 2\,\upmu\)m generated by cylindrical grinding. Although the plastic deformation of the crushed ring sample by 1.5 mm on the diameter was achieved using 8.4 kN rigid steel plates. A brief description of the model and experimental settings is provided below and can be found in [26]. Two EN26 steel bars have been heated to relieve stress and provide a uniform structure before assembly. The final test hardness of the sample was 290 HV for both models. The strain profile was calculated on raden along with an MCP detector (512 \(\times\) 512 pixels, 55 \(\upmu\)m per pixel) at 17.9 m from the beam source of power 409 kW (January 2018). In the Japan Proton Accelerator Research Complex j-parc, Japan, the neutron-energy-resolved imaging instrument was used to obtain the relative shifts of the Bragg edges corresponding to the lattice plane of the samples. Neutron strain scanning was performed on kowari, a residual stress diffractometer at the Australian Nuclear Society and Technology Organization (ANSTO), Australia, to verify our reconstruction independently. This strain-scan provide measurements of the three in-plane components of strain over a mesh of points within the sample using a \(0.5\times 0.5\times 14\) mm gauge volume. Sampling times on kowari were based on the providing of strain uncertainty around \(7\times 10^{-5}\), which required about 30 h of beam time per component. The raden sampling times, however, were based on the \(1\times 10^{-4}\) statistical uncertainty. A total of 50 profiles were measured at golden angle increments in the angle with a sampling time of two hours per projection. A two-dimensional scattered data point net is used to discretize the domain of the sample, as shown in Figure 9. Two different types of node discretization have been considered: uniform and non-uniform (random). Nodal set can be seen in Figure 11 and reconstructed strain fields can be seen with each discretization in Figures 12 and 13 respectively. As stated earlier, for the simulated problem, the reconstructed strain field did contain noise which can be from either experimental measurement or numerical discretizations. Table 1.0 Offset-Ring-and-Plug sample geometry. Table 1.1 Offset-Ring-and-Plug Sample. The figure on the left shows a uniform data points set containing \(N=260\). The figure on the right shows a scattered data points set containing \(N=261\) non-uniform spaced nodes on the sample geometry. Table 1.2 The reconstructed full-field two-dimensional strain components (\(\varepsilon_{xx},\varepsilon_{xy}\), and \(\varepsilon_{yy}\)) with non-uniform node distribution. Table 1.3 Crushed Ring Sample.