## 3 Calculation of Mechanical Variables Using a Strong Form-Based Meshfree Method
### Synthetic Structural Analysis Using Expanded TP (ETP)
(**a**) Conventional radius of influence domain (_r_) optimized for 42 TPs, and (**b**) doubling the radius of domain of influence (2_r_) at \(t=60\) s.  
Figure 15: Arrangements of original points for DIP-based displacement tracking: (**a**) TP (42 points), and (**b**) expanded TP (280 points). The relative errors were analyzed to evaluate the mathematical robustness of the synthetic structural analysis in performing accurate kinematic variable calculations. The error was calculated using the relative L\({}_{2}\) error norm such as Equation (22), which [PERSON] and [PERSON] [29; 30] used to evaluate the error in the strong form-based meshfree method. \[E = \sqrt{\frac{\sum\limits_{I\in A}\left(\varepsilon_{\text{exp}}- \varepsilon_{SSA}\right)^{2}}{\sum\limits_{I\in A}\left(\varepsilon_{\text{ exp}}\right)^{2}}}, \tag{22}\]  
where \(\varepsilon_{\text{exp}}\) denotes the strain measured directly through the experiment, and \(\varepsilon_{SSA}\) denotes the strain calculated through the synthetic structural analysis. To calculate the error according to the number of TPs, the number of TPs in the analysis was increased to 42 (TP), 84 (2 TP), 168 (4 TP), and 280 (ETP), while the number of nodes in the PDM analysis model was fixed at 671. The result of the error calculation is shown in Figure 17, wherein the \(x\)-axis represents the number of TPs and the \(y\)-axis the magnitude of the error. As the number of applied TPs increases, the relative error decreases; this implies that the numerical scheme presented in this study has a convergence property, indicating that the synthetic structural analysis is mathematically robust as a numerical technique. Figure 18 shows the computational time consumed in the PDM analysis when using the TP and the expanded TP. The PDM simulation was performed using MATLAB, and the average computational time was calculated by repeating the simulation five times under the same conditions. In the synthetic structural analysis, the computational time required for the PDM simulation was 97.02 s for the TP and 102.33 s for the expanded TP. Although the expanded TP, with more than six times the number of TPs than that in TP, took longer to compute, the difference was not significant. Therefore, the use of the expanded TP is better for improving the accuracy of the kinematic variable calculation without sacrificing computation efficiency. The PDM analysis duration differed by approximately 5 s between the TP and the expanded TP, indicating that the DIP and the labeling process occupied most of the duration of the synthetic structural analysis. In summary, the use of the expanded TP is economical for improving the analysis accuracy, which is important to overcome the disadvantage of actual experiments that require many targets attached onto the test specimen. Figure 17: Convergence study of the synthetic structural analysis using the L\({}_{2}\) relative error norm. In addition, the error analysis was performed in the same manner as before, while changing the number of nodes in the PDM model when using the TP and the expanded TP. Figure 19a-c demonstrates three PDM models with 186 (\(31\times 6\)), 671 (\(61\times 11\)), and 2541 nodes (\(121\times 21\)) used for the error analysis. Figure 20 shows the calculation result of the relative error, with a fixed number of TPs and an increasing number of nodes used in the PDM model. As the number of nodes in the PDM model increases, the relative error decreases. Figure 17 revealed the same result even when the error evaluation was performed with a fixed number of nodes for the PDM model and an increased number of TPs. This shows again that the synthetic structural analysis exhibits convergence properties, indicating its robustness as a numerical analysis. Figure 19: Nodal arrangements of the PDM models for the error analysis: (**a**) 186-node model (\(31\times 6\)), (**b**) 671-node model (\(61\times 11\)), and (**c**) 2541-node model (\(121\times 21\)). Figure 18: Computational time comparison for the PDM simulations with TP and expanded TP.