## 3 Calculation of Mechanical Variables Using a Strong Form-Based Meshfree Method  
### Particle Difference Method (PDM) Approximation  
The basic concept of the PDM revolves around developing a Taylor series by using the moving least-squares method based on a set of nodes. The PDM obtains a numerical solution by discretizing the given governing equation as is. As the Taylor series contains the differentials of the function at the position that forms the equation, the strain and stress including the first derivative of displacement can be calculated by developing the Taylor series for the displacement at each node in the PDM model. In the PDM, the approximation function is naturally derived in the series construction process by using the moving least squares method. Figure 6: TP label mismatch. Figure 7: Correction of TP label mismatching. In this study, the arrangement of the TP using DIP was not the same as the node arrangement of the PDM model. In general, the kinematic variables are calculated with a fixed structural model such as the mesh of the FEM or grid of the finite difference method. FEM can only obtain the stress at the Gaussian point, while PDM can calculate the stress at any location. However, accurate kinematic variables can be calculated without causing problems because the PDM consists of an approximate function only with nodes. Even if the arrangement of the TPs is irregular or the arrangement changes during the analysis, no further difficulties are caused. Figure 8 shows a comparison of the PDM process between the elastic analysis and synthetic structural analysis. Synthetic structural analysis PDM analyzes the stress using the displacement extracted from the image, so the analysis procedure is relatively simple. In particular, there is no need for treatment of the boundary condition and discretization of the governing equations. To discretize the governing equation by using the displacement for analyzing a solid mechanics problem in a strong formulation, the second order derivatives of the displacement are required [33]. However, a new PDM that can use the strong formulation without the second differential was used in this study [32]. This method exhibits excellent calculation efficiency because only the first differential is used to discretize the divergence of stress. As discussed earlier, many strong form-based meshfree methods utilize the so-called Navier's equation, which involves second order differentials [1; 2; 23; 24; 30; 33]. In this section, the process presented by [PERSON] et al. [33] for constructing an approximate function in the PDM is introduced briefly. When constructing the approximate function, the Taylor polynomial at an arbitrary point \(\mathbf{x}\) relative to the reference point \(\mathbf{y}\) is given as follows:  
\[u_{i}(\mathbf{x};\mathbf{y})\ =\ \left(\frac{(\mathbf{x}-\mathbf{y})^{ \boldsymbol{\alpha}_{1}}}{\boldsymbol{\alpha}_{1}! },\,\cdots,\,\frac{( \mathbf{x}-\mathbf{y})^{\boldsymbol{\alpha}_{K}}}{\boldsymbol{\alpha}_{K}!} \right)\cdot\left( \begin{array}{c}D_{\mathbf{x}}^{\boldsymbol{\alpha}_{1}}u _{i}(\mathbf{y})\\ \vdots\\ D_{\mathbf{x}}^{\boldsymbol{\alpha}_{K}}u_{i}(\mathbf{y})\end{array} \right)\ =\ \mathbf{p}_{m}^{T}( \mathbf{x},\mathbf{y})\,\boldsymbol{\mathbf{c}}_{i}(\mathbf{y}), \tag{7}\]  
where \(\mathbf{p}_{m}^{T}(\mathbf{x};\mathbf{y})\) denotes the \(m\)-th order polynomial vector and \(\boldsymbol{\mathbf{c}}_{i}(\mathbf{y})\) denotes the derivative coefficient vector. \(\boldsymbol{\alpha}_{1}\ =\ (0,\cdots,0)\), \(\boldsymbol{\alpha}_{K}\ =\ (0,\cdots,m)\) in the framework of multi-index. \(K\ =\ \frac{(m+n)!}{m!n! }\) where \(n\) denotes the space dimension and \(m\) denotes the order of highest derivative. Figure 8: Comparison of the particle difference method (PDM) process between the elastic analysis and synthetic structural analysis.