## 3 Model Reconstruction and Finite Element Analysis
### Structural Characterization and Reconstruction (SCR)
The Fourier transform is used to perform three-dimensional Gaussian smoothing operation to correct reconstructed 3D models, which is calculated as Equation (10). Finally, high-dimensional binary matrix is then obtained: \[\sum\limits_{1\leq i\leq M}I_{i,j}^{(V_{1})}\] \[porosity(\%)=\frac{1\leq j\leq N}{MN}\times 100\%,\] (9) \[f(x,y,z)=2\pi^{-\frac{3}{2\omega^{3}}}\exp\left(-\left(\frac{2x^{2}}{\sigma^ {2}}-\frac{2y^{2}}{\sigma^{2}}-\frac{2z^{2}}{\sigma^{2}}\right)\right).\] (10)
4. Kullback-Leibler (KL) divergence, also known as relative entropy, is a measure of the difference between two distributions P1 and P2 to evaluate the reconstruction quality, as shown in Equation (11). The KL divergence is calculated from 0 to \(+\infty\), indicating the similarity from the most to the least. \[KL(P_{1}\|P_{2})=\sum_{x\in X}P_{1}(x)\log\frac{P_{1}(x)}{P_{2}(x)}. \tag{11}\]  
Through the above process, a series of high-dimensional data equaling to the 3D geometry reconstructed model could be obtained and transferred into the Finite Element Analysis (FEA) model as below. The high-dimensional data can be discretized and reduced into a set of unit information, called volume data. These data logically form a 3D array space, and each array point stores volume location and feature information, called a voxel. The location of one voxel was determined by layer, row and column, as shown in Figure 7a. Voxels belonging to the medium are marked as v = 1, and those belonging to the pore are marked as v = 0. The thin layer of volume data is shown in Figure 7b. All voxels' information is stored in a TXT file and imported in the ANSYS Parametric Design Language (APDL) program to build the 3D entity in ANSYS Mechanical for FEA, which is a Boolean description of micron silver sintered material with voxels as units in space. The voxel is built by 8 key-points illustrated in Figure 7c, where \(i,j\) and \(k\) represent the voxel location coordinate, respectively. Without considering the effect of the grain boundary of fused micron silver particles, reconstructed models of joints can be obtained by reverse filling the pores with Boolean operation. Reconstructed pore visualization results and 3D FEA models corresponding to SEM images of micron silver sintered joints under different thermal shocks are shown in Figure 8.  
\[\Omega=\left\{(m,n)\big{|}m^{2}+n^{2}=r^{2},r\leq\left[\min\{M,N\}/2\right] \right\}.\]0 Indication of spatial data. (**a**) Spatial volume data; (**b**) A thin-layer model; (**c**) One voxel built in ANSYS Mechanical. \[\Omega=\left\{(m,n)\big{|}m^{2}+n^{2}=r^{2},r\leq\left[\min\{M,N\}/2\right] \right\}.\]1 Reconstructed pore topological structures and 3D Finite Element Analysis (FEA) models with mesh in a small area corresponding to SEM images of micron silver sintered joints under different thermal shocks: (**a**,**b**) 50; (**c**,**d**) 150; (**e**,**f**) 150; (**g**,**h**) 250 cycles. In 3D views of the left column, gray represents defined geometric boundaries, and blue is the iso-surface. Normalized autocorrelation functions of SEM images and 3D reconstruction models are plotted in Figure 9, where the size of reconstructed models (250 pixels) is much larger than the observed correlation length of SEM samples (the autocorrelation function asymptotic location), illustrating reconstructed models can represent the microstructure of sintered joints. The KL divergence values are calculated and listed in Table 2, which are within an acceptable range (less than 15%). Both qualitative observation and quantitative calculation show that the reconstructed models are consistent with the original images. \[\Omega=\left\{(m,n)\big{|}m^{2}+n^{2}=r^{2},r\leq\left[\min\{M,N\}/2\right] \right\}.\]2 Comparison of autocorrelation functions of SEM images and 3D reconstruction models with thermal shock cycles are (**a**) 50; (**b**) 100; (**c**) 150; (**d**) 250.