## 3 Meshless Approach Overview
### Methodology
#### 3.1.3 Algorithm
The figure on the left shows the sample geometry for crushed ring steel sample before and after applying force (all dimensions are in mm). The figure on the right shows a node set containing \(N=\) non-uniform spaced nodes on the crushed ring sample. To summarize, we proposed a meshless approach using rbf. They are the natural generalization in a multivariate sense of univariate polynomial splines. It works well for high-dimensional arbitrary structures, and no mesh requirement is the main benefit of this form of approximation. A rbf is a function, the value of which depends only on the distance between the points. Due to the use of distance functions, one can conveniently deploy rbfs to recreate the surface using data points in 2D, 3D or higher-dimensional spaces. The Least square solution was found for 3790 nodes for the non-uniform and 3795 nodes for uniform ring-and-plug discretization from 22,678 projections. Figures 12 and 13 shows the reconstructed field of ring-and-plug with uniform and non-uniform node distribution respectively in terms of three Cartesian strain components namely; \(\varepsilon_{xx},\varepsilon_{xy}\) and \(\varepsilon_{yy}\). The reconstructed strain field was found for 1352 nodes and 1086 quadrilateral elements via fem approach and 1352 nodes for crushed ring discretization via rbf approach from 20,664 projections. Figures 14 and 15 shows the reconstructed strain field of crushed ring with fem and rbf respectively in terms of three Cartesian strain components. Figures 16 and 17 show the corresponding validated strain field from kowari data. The effectiveness of the suggested approach is not based solely on sample structures and can, thus, be extended to three-dimensional sample bodies. Due to radial basis function computation, the real challenge is the computational cost and time since the size of the problem is quite large. Table 1.4 The reconstructed full-field two-dimensional strain components (\(\varepsilon_{xx},\varepsilon_{xy}\), and \(\varepsilon_{yy}\)) with fem. Table 1.5 The reconstructed full-field two-dimensional strain components (\(\varepsilon_{xx},\varepsilon_{xy}\), and \(\varepsilon_{yy}\)) with uniform node distribution. ## 6 Conclusions

From Bragg-edge neutron images, an elastic strain tensor field reconstruction algorithm has been presented. Unlike previous algorithms such as fem, our suggested methodology is capable of reconstructing residual strain field as no elastic strain compatibility is assumed. Simulated and experimental data from kowari and raden have shown this process. The analysis revealed a strong agreement with the strain maps measured using the constant diffractometer Kowari. A complete strain field tomography can

Table 1.6 Two-dimensional strain components \(\varepsilon_{xx},\varepsilon_{xy},\varepsilon_{yy}\) respectively with kowari Data. Table 1.7 Two-dimensional strain components \(\varepsilon_{xx},\varepsilon_{xy},\varepsilon_{yy}\) respectively with kowari Data. Table 1.8 The reconstructed full-field two-dimensional strain components (\(\varepsilon_{xx},\varepsilon_{xy}\), and \(\varepsilon_{yy}\)) with rbf. now be achieved using physical constraints as an equilibrium without thinking about two-dimensional mesh. This approach opens up more study, including three dimensions, for future studies. The framework taken allows us to concentrate on the highly stressed region of the sample using adaptive node discretization, which can be accomplished by calculating the gradient over the sample. [PERSON] assisted with the methodology, conceptualization. [PERSON] helped with the numerical part, validation and coding. [PERSON] developed software, algorithm, validation and led the write-up. [PERSON] conceived the study idea and led the experimental program. All authors have read and agreed to the published version of the manuscript. This research was funded by Australian Institute of Nuclear Science and Engineering; Australian Research Council, Grant No. DP170102324; Japan Proton Accelerator Research Complex, Grant No. 2017L0101. Not applicable.