## 3 Calculation of Mechanical Variables Using a Strong Form-Based Meshfree Method
### Synthetic Structural Analysis Using Expanded TP (ETP)  
This section presents a method for using the expanded TP to improve the accuracy of the calculation of the kinematic variables. When the DIP and the PDM are combined, the number of TPs affects the accuracy of the synthetic structural analysis. In other words, if the number of TPs is large, the resolution of the construction of node displacement in the PDM model increases. This is advantageous for calculating kinetic or kinematic variables such as strain, stress, and force, which require differential displacement calculations. As explained in Section 2.2, it is possible to set the expanded TP using a given image set when performing DIP. On the other hand, the number of TPs and the number of nodes in the PDM model can improve the accuracy of the calculation of the kinematic variables. In the synthetic structural analysis, the number of nodes in the PDM model can be increased without restrictions, even in the PDM application stage, regardless of the use of the expanded TP. However, note that it is difficult to reconstruct the mesh, and remap and interpolate the nodal displacement using the newly constructed shape function when the number of nodes increases because of the constraints on the mesh structure during the FEM analysis. In addition, even if the arrangement of the TPs in the PDM nodes is irregular, bringing the TP displacement value to the PDM nodes through PDM approximation is not a problem, unlike FEM because the nodal arrangement in the PDM model can be configured regardless of the TP arrangement. Thus, the number of nodes can also be increased without restrictions, if necessary. Figure 15a illustrates the distribution of the initial TPs. Figure 15b illustrates the expanded TP distribution achieved by applying the TP expansion method suggested in Section 2.2 to the initial TP distribution. In this case, the 42 initial TPs expanded to 280 by a factor larger than six, without additional photographing or actual target adjustments. The increased number of TPs naturally increased the resolution of the nodal displacement configuration of the PDM model. Figure 14: Analysis results at t = 60 s: (**a**) displacement in the x-direction, (**c**) displacement in the y-direction, and (**e**) von Mises stress by the synthetic structural analysis; and (**b**) displacement in the x-direction, (**d**) displacement in the y-direction, and (**f**) von Mises stress by the simulation using only PDM. Figure 16a,b displays the result of the von Mises stress calculation obtained by applying the expanded TP. When the expanded TP was not used, 42 TP displacement values were distributed between 671 PDM nodes. However, when the expanded TP was applied, 280 TP displacement values were distributed between 671 PDM nodes. Figure 16a shows the von Mises stress calculation results when the expanded TP was used, but the domain of influence of the PDM approximation function optimized over 42 TPs was used as is. In this case, the analysis result was generally similar to that of the previous analysis, but the stress increased abnormally at the boundary with some oscillation. This is because the radius of the domain of influence of the PDM approximation is maintained without considering the density of the expanded TPs. In this study, this problem was resolved by doubling the radius of the domain of influence, and the improved results are shown in Figure 16b. In practice, determining the radius of the domain of influence is important in all types of meshfree methods [21; 29; 30; 32; 33]. This study has the advantage of easily increasing the number of TPs and achieving a high accuracy of kinematic variable calculation by applying the expanded TP generation method. However, the appropriate number of nodes with the proper radius of the domain of influence should be determined by considering the number of TPs and an expected resolution level in this process. Figure 16: Von Mises stress contour according to the corrected influence domain radius.