## 3 Calculation of Mechanical Variables Using a Strong Form-Based Meshfree Method
### Calculation of Kinematic Variables Using PDM Approximation  
The synthetic structural analysis method calculates the strain and stress by obtaining the displacement of the TP from the image acquired in the experiment and substituting it into the PDM approximation. First, we describe the method of distributing the TP displacement obtained using DIP on the node of the PDM model. Basically, because the arrangements of the TP and the nodes in the PDM model do not match, the displacement obtained from the TP is converted into the PDM node value using the following equation:  
\[\mathbf{u}^{PDM}(\mathbf{x})\ =\ \mathbf{I}(\mathbf{x})\mathbf{u}^{TP}, \tag{11}\]  
where \(\mathbf{u}^{PDM}(\mathbf{x})\) denotes the displacement at position \(\mathbf{x}\) of the PDM model, and \(\mathbf{u}^{TP}\) is the vector containing the \(TP\) displacement values included in the influence domain of \(\mathbf{u}^{PDM}(\mathbf{x})\). The \(\mathbf{I}(\mathbf{x})\) matrix contains the zero-th order shape function in the diagonal and is defined at the nodal positions of the PDM model as follows:  
\[\mathbf{I}(\mathbf{x})\ =\ \sum_{I\ =\ 1}^{N}\left[ \begin{array}{cc}\phi_{I}^{[(0,0 )]}(\mathbf{x})&0\\ 0&\phi_{I}^{[(0,0)]}(\mathbf{x})\end{array} \right]. \tag{12}\]Although the matrix calculation of \(\mathbf{u}^{TP}\) and \(\mathbf{I}(\mathbf{x})\) in Equation (11) is a scalar product, note that the summation symbol \(\Sigma\) indicates the sum of calculations for the nodes included in the influence domain. In this process, the Jacobian need not be considered. This is a type of interpolation method, and the solving of Equation (11) between any two domains is not a mapping. Note that the PDM approximation is extremely suitable for this type of calculation. To calculate the stress from the displacement values distributed between the nodes of the PDM model, a constitutive equation is required. The following constitutive equation for the elastic materials was used:  
\[\boldsymbol{\sigma}\ =\ 2\mu\boldsymbol{\varepsilon}+\lambda\text{tr}(\boldsymbol {\varepsilon})1, \tag{13}\]  
where \(\boldsymbol{\sigma}\), \(\boldsymbol{\varepsilon}\), \(\text{tr}(\boldsymbol{\varepsilon})\), and \(1\ =\ \delta_{ij}\boldsymbol{\varepsilon}_{i}\otimes\boldsymbol{\varepsilon}_{i}\) denote the stress tensor, strain tensor, strain trace, and second unit tensor, respectively, and \(\lambda\) and \(\mu\) are the Lame constants. If the constitutive equation is expressed in the matrix form, it can be written as Equation (14), and the component of matrix \(\mathbf{D}\) representing the material properties for a plane strain case can be found in Equation (15). \[\boldsymbol{\sigma}\ =\ \mathbf{D}\boldsymbol{\varepsilon}, \tag{14}\]  
\[\mathbf{D}\ =\ \left[ \begin{array}{ccc}2\mu+\lambda&\lambda&0\\ \lambda&2\mu+\lambda&0\\ 0&0&\mu\end{array} \right]. \tag{15}\]  
In a small deformation problem, the strain is expressed by the symmetric part of the displacement gradient. Therefore, the strain was calculated using the following compatibility equation, which expresses the relationship between the strain and displacement:  
\[\boldsymbol{\varepsilon}\ =\ \frac{1}{2}\big{(}\
abla\mathbf{u}+\
abla\mathbf{u}^{T} \big{)}, \tag{16}\]  
where \(\
abla\) is the differential operator. As Equation (13) expresses the stress as a function of strain, if Equation (16) is substituted in Equation (13), the constitutive equation expressed by the displacement function can be obtained in the matrix form, as follows:  
\[\boldsymbol{\sigma}(\mathbf{x})\ =\ \mathbf{D}\mathbf{B}(\mathbf{x})\mathbf{u}( \mathbf{x}), \tag{17}\]  
where \(\mathbf{u}(\mathbf{x})\ =\ (u_{\text{II}}(\mathbf{x}),u_{\text{II}}(\mathbf{x}))^{T}\) is applicable for a 2D problem, and the matrix \(\mathbf{B}(\mathbf{x})\) is constructed using the PDM shape function for the first derivative approximation, as follows:  
\[\mathbf{B}(\mathbf{x})\ =\ \sum_{I=1}^{N}\left[