## 3 Meshless Approach Overview
### Methodology
#### 3.1.3 Algorithm  
Here, we discuss the difference between two methods; namely fem and rbf. Detailed pseudo-code algorithm is given below Algorithms 1 and 2 for both the methods, respectively. ``` Result: Reconstructed tensor strain field (\(R^{*}\)) Initialization: Number of nodes (NoD) and elements (elem) (Triangular/Quadrilateral), Smoothing parameter (\(\lambda\)) for Tikhonov regularization, Number of projections per direction (NoR); Data: Generate nodal and element data set in the domain \(\Omega\) and projections passing through sample from different orientations (9 projection in each orientation from 0 to 180\({}^{\circ}\)); for i = 1 : NoRdo  Find intersection with the sample geometry and each element discretization (entry and exit points in each Quadrilateral/Triangle); for j = 1 : elem do \([\Gamma]\leftarrow\) Solve LRT for each projection in an element; \([S][e]=[\Gamma]\leftarrow\) Assemble all the matrices for each element (Note: at least one node will always be shared between elements neighborhood); \([Eq]=[Eq1;Eq2]\leftarrow\) use equilibrium equations and assemble; \(R\leftarrow\)\(\mathit{lsqlin}\)\((S,\Gamma,\,\,\,\,\,\,\,\,\



simulations. In the next section, we talk about this difference between methods with a simulated example. ## 4 Cantilevered Beam

Here we illustrate the practical performance of the proposed algorithm, starting with the well-known two-dimensional cantilevered beam problem that was previously successfully determined by other authors [9]. In this case, we take the \([0,20]\times[-5,5]\) a two-dimensional strain domain into account with a 2 kN load over the beam. The beam's material properties reflect common steel, while other parameters are listed below. The data consist of 9 projections evenly spaced in \([0,170]^{\circ}\) with 200 lines each, yielding a total of 1800 measurements. Figure 4 shows two types of nodal mesh (uniform and non-uniform) used for beam strain field reconstruction using rbf. Both nodal mesh consist of 400 nodes including boundary points. Figure 5 shows the reconstruction of strain field using rbf, whereas Figure 6 shows the analytical solution of the cantilevered beam. The plane stress Saint-Venant approximation to the strain field is [42]:

\[\varepsilon_{xx}=\frac{P(L-x)y}{EI},\ \ \ \varepsilon_{xy}=-\frac{(1+v)P}{2 EI} \Big{(}\frac{h^{2}}{y}-y^{2}\Big{)},\ \ \ \varepsilon_{yy}=-\frac{vP(L-x)y}{EI},\]

where \(I\) represents the second moment of area, \(L\) the beam length, \(h\) the beam height, \(t\) is the thickness, \(E\) is [PERSON]'s modulus, \(v\) is the Poisson ratio, and the dimensions are shown in Table 1.

Figure 4: Uniform node set containing \(N=400\) versus non-uniform nodal domain containing \(N=400\) randomly spaced nodes. Figure 5: rbf Beam solution (Reconstructed solution \(\varepsilon_{xx},\varepsilon_{xy},\varepsilon_{yy}\)). _Error Analysis through Simulation Results for rbf Methods_

A \(L^{2}\) error analysis of strain field associated with both fem and rbf methods is shown in Figure 7. Started with 9 projections evenly spaced in \([0,170]^{\circ}\) with 10 lines each and ended with 700 lines each 9 projections. Both rbf case results shows better agreement than fem in the final stage. However, faster convergence is seen in the case of uniform rbf. Figure 8 shows the rms error between the reconstructed and analytical strain field as we change Gaussian function shaping parameter value. The numerical rms error with different shaping parameter values are calculated using the root-mean-square error and shown in Figure 8

\[\text{Error}=\sqrt{\frac{\sum_{i=1}^{N}(T_{i}-R_{i})^{2}}{N}},\]

where \(N\) is the total number of nodes, \(T_{i}\) is the true strain value and \(R_{i}\) is the reconstructed strain value at the particular node \(i\). We found out the proposed reconstruction algorithm extremely useful for the reconstruction of the strain field. The simulation results imply that the reconstruction algorithm will converge to an appropriate reconstruction provided measurements over \(180^{\circ}\) of a sample are calculated.