## 3 Calculation of Mechanical Variables Using a Strong Form-Based Meshfree Method
### Calculation of Kinematic Variables Using PDM Approximation
\begin{array}{ccc}\phi_{I}^{[(1,0 )]}(\mathbf{x})&0\\ 0&\phi_{I}^{[(0,1)]}(\mathbf{x})\\ \phi_{I}^{[(0,1)]}(\mathbf{x})&\phi_{I}^{[(1,0)]}(\mathbf{x})\end{array} \right]. \tag{18}\]  
The scalar product of the matrices \(\mathbf{B}(\mathbf{x})\) and \(\mathbf{u}(\mathbf{x})\) in Equation (17) naturally involves a summation of the scalars for the nodes included in the influence domain. Thus, similar to Equation (12), Equation (18) is expressed as a summation. The strain is expressed in the matrix form, as follows:  
\[\boldsymbol{\varepsilon}\ =\ \mathbf{B}(\mathbf{x})\mathbf{u}. \tag{19}\]  
In fact, the calculation of the strain and stress using matrices is very similar to the FEM calculations. Given only the displacement values of the nodes in the PDM model, both the strain and stress can be easily calculated at any given location. However, if the number of TPs or PDM nodes is small, or the arrangement is highly uneven, the accuracy of the differential calculation such as that of the strain or stress calculation, could reduce. To avoid this problem, this study proposed a method to increase the number of TPs, and a sufficient number of nodes was properly distributed to ensure that the displacement field obtained by DIP for the TP displacement could be accurately reproduced over the entire area of the PDM model. In the FEM, because the analysis area is divided into element units, it is difficult to accurately reproduce the displacement value using the TP displacement in an arbitrary position within the entire FEM domain. The PDM, however, which uses only nodes, possesses the advantage of easily distributing the TP displacement value to the PDM node using interpolation- or extrapolation-type operations. A linear elastic analysis using PDM was considered to verify the accuracy of synthetic structural analysis presented in this paper. The [PERSON]'s equation for satisfying the equilibrium equation in solid mechanics is as follows:  
\[\mu\
abla^{2}\mathbf{u}+(\lambda+\mu)\
abla(\
abla\cdot\mathbf{u})\ =\ 0 \tag{20}\]  
Using the shape function for the first derivative approximation and the material property matrix, Equation (20) can be rewritten as below:  
\[\mathbf{B}^{T}\mathbf{D}\mathbf{B}\mathbf{u}\ =\ 0 \tag{21}\]  
This discretized [PERSON]'s equation was combined with traction and essential boundary conditions to construct a system equation. A detailed description of the PDM algorithm for solid mechanics can be found in previous studies [1; 2]. Synthetic Structural Analysis for Tracking Point (TP) Displacement Tracking and Kinematic Variable Computation  
To verify the accuracy of the kinematic variables obtained through the synthetic structural analysis with DIP-PDM, a three-point bending test was performed on a simply supported rubber beam. The shape of the specimen used in the experiment is illustrated in Figure 9; the elastic modulus and Poisson's ratio of rubber were taken as 2.0 MPa and 0.5, respectively, similar to those used in the study of [PERSON] [34]. A Zwick/Roell Z030 Universal Mechanical Tester, which can test the compressive and tensile strength in the loading range of 1-3000 N, was used as the loading equipment. For the digital image acquisition, a Canon 5D Mark IV full-frame camera with a resolution of 30.1 million pixels (6720 \(\times\) 4480 pixels) equipped with a Canon EF 24-70 F2.8L II lens was used. A tripod and a timer release were used to acquire high-quality images, and two indirect light sources were used to eliminate the effects of the natural light and shadows. Figure 10a shows the specimen used for the bending test. The targets were attached to one side of the specimen to acquire photographic images, and strain gauges were attached to the other side for measuring the local strain. Figure 10b shows the experimental setup consisting of the rubber beam, digital camera, lighting, datalogger, and a computer for data conversion.