## 3 Method
### PDDO Implementation  
The PDDO (see Section 2.2) was applied to a regular square grid of PD points of both DIC Challenge [11] and in-house experiment virtual specimens surfaces, where displacements were measured from the images by DIC. For this purpose, the export data options _Pixel Grid_ and _Metric nodes_ in the DIC software VIC-2D and VIC-3D were used. These options enable to export points coordinates, their displacement and strain values from the DIC field over a selected regular square grid to.csv file. Then the exported DIC data from.csv file was directly imported to the PDDO code by ensuring the same location of the virtual and experimental specimens data points. When exporting the DIC displacement data at selected grid spacing to PDDO applications, the corresponding DIC step size adjustments are unnecessary. This is because only displacements are required for PDDO strain calculations and DIC VSG size has no effect on PDDO-calculated strains. Moreover, no additional data transformations (e.g., interpolation between the points, coordinates reposition) were applied in the PDDO code.  
In order to find the value of the PD grid spacing \(\Delta x\) (see Section 2.2) for which the DIC-PDDO strains converge to the DIC strains, in-house experimental DIC displacement measurements data (DIBOND composite specimen data) were exported at different grid spacing values from VIC-3D to PDDO applications. The lower bound of the PD grid spacing can be considered equal to one DIC camera pixel size (on the specimen surface), which can be calculated as:  
\[\begin{array}{c}\text{FOV}\\ \text{Camera sensor size}\end{array}\text{-Camera pixel size}=\frac{180\times 131}{14.1 \times 10.3}.\\ \\ 3.45\cdot 10^{-3}\times 3.45\cdot 10^{-3}=0.044\text{ mm}\times 0.044\text{ mm}.\end{array} \tag{5}\]  
To collect strain data only around the hole (high strain gradients can be captured in this region), a 30 mm \(\times\) 30 mm square area of the DIBOND composite specimen surface around the hole was used for the DIC-PDDO calculation (see Figure 6a).  
Figure 6: Computation scheme: (**a**) selected area of interest on the specimen surface for the DIC-PPDO calculation; (**b**) loading scheme of the specimen FE model.  
The PDDO based on Equation (3) adopted for the 2D plane stress case was implemented in MATLAB. Basic MATLAB code principles were followed according to [47]. In order to find the 2D strain (displacement gradient \(F(x_{j})\)) at point \(x_{j}\) from the discrete displacement data points, the integral Equation (3) is modified and discretized as summation of the \(n\) discrete data points over the integration domain:  
\[F\left(x_{j}\right)_{i}=\left[\sum_{k=1}^{n}\omega(|\xi_{k}|)f(x_{k}+\xi_{k}) \xi_{i}h\Delta x^{2}\right]A^{-1},\ i=1,\ 2, \tag{6}\]  
where \(\xi_{k}=x_{k}-x_{j}\), \(h\Delta x^{2}-2\) DD point volume. Substitution of the discretized form of \(A=\sum_{k=1}^{n}\omega(|\xi_{k}|)\begin{bmatrix}\xi_{1}^{2}&\xi_{1}\xi_{2}\\ \xi_{1}\xi_{2}&\xi_{2}^{2}\end{bmatrix}h\Delta x^{2}\) to Equations (2) and (3) finally results:  
\[F\big{(}x_{j}\big{)}=\left[\sum_{k=1}^{n}\omega(|\xi_{k}|)\big{(}(n_{k}+x_{k}) -\big{(}n_{j}+x_{j}\big{)}\big{)}\otimes\xi_{k}\right]\cdot\left[\sum_{k=1}^{ n}\omega(|\xi_{k}|)(\xi_{k}\otimes\xi_{k})\right]^{-1}, \tag{7}\]  
here \(u_{j}\), \(u_{k}\) and \(x_{j}\), \(x_{k}\) are displacement and position vectors associated with the discrete points \(x_{j}\), \(x_{k}\) (also see Figure 4). The MATLAB code calculates and plots the strain field using the DIC displacement field given at the regular grid of points on the specimen surface. The main parameters of the PDDO code are the horizon size \(\delta\) and the weight function \(\omega\big{(}|\xi_{j}|\big{)}\) which in our case is defined as a \(\omega\big{(}|\xi_{j}|\big{)}=\delta^{3}/|\xi|^{3}\).