## 3 Model Reconstruction and Finite Element Analysis  
### Structural Characterization and Reconstruction (SCR)  
The Joshi Quiblier Adler (JQA) method [23] is a morphology autocorrelation-function-based tool for reconstructing porous media. To study the relationship between microstructure and performance  
Figure 5: SEM images of micron silver sintered joints under thermal shocks for: (**a**) 0; (**b**) 50; (**c**) 100; (**d**) 150; and (**e**) 250 cycles. of micron silver sintered joints from a microscopic perspective, the method was simplified and used to characterize 2D cross sections and reconstruct a two-phase heterogeneous 3D model to provide the high-dimensional point cloud data required for simulation. During reconstruction, a Gaussian random field was used to generate spatial media, and the morphology and dispersion of two-phase interface were described by autocorrelation function. A region-based image segmentation technique iteratively solved the porosity of 3D models which was consistent with original images, and these models were corrected by smoothing operation. The whole flow is illustrated in Figure 6.  
There are four main steps required to reconstruct the randomly distributed medium from an SEM picture using SCR, which are given below:  
1. Denoising, threshold segmentation and binarization processing are applied on the original image to get the two-phase random medium, which is expressed as \(V(\omega)\in\mathbb{R}^{3}\), a spatial domain. Where \(\omega\) is the domain intercepted from the probability space of volume \(V\), including two parts: the pore volume fraction \(\varphi_{1}\) in the region \(V_{1}\) and the volume fraction of micron silver particles \(\varphi_{2}\) in the region \(V_{2}\). Binary porous media may be represented by an indicator function \(I(x)\), as defined below: \[I(x)=\left\{ \begin{array}{l}1,x\in V_{1}\\ 0,x\in V_{2}\end{array} \right..\] (1)
2. The two-point autocorrelation function S2(r) is used to describe the morphology as shown in Equations (2) and (3). S2(r) is defined as follows: two arbitrary points, x1 and x2 of the distance r,  
Figure 6: The Flow chart of reconstructing stochastically equivalent 3D morphology of sintered micron silver joints. are selected in an observation region, and the probability that both points are in one phase is S2(r), which is illustrated in Equation (4). For the isotropic material in this study, it can be calculated by bilinear interpolation [24]. \[S_{2}^{i}(r)=\left\langle I^{i}(x_{1})I^{i}(x_{2})\right\rangle=P\big{[}I^{i}(x_{ 1})=1,I^{i}(x_{2})=1\big{]},\] (2) \[\left\{ \begin{array}{l}S_{2}^{i}(r)=\phi_{i},r=0\\ \lim\limits_{r\rightarrow\infty}S_{2}^{i}(r)=\phi_{i}^{\,2}\end{array} \right.,\] (3) \[S_{2}(r)=\frac{\sum\limits_{(m,n)\in\Omega}\left[\sum\limits_{i=1}^{M}\sum \limits_{j=1}^{N}I_{i_{j}}I_{i+m,j+n}\right]}{\omega MN},\] (4) where \(\omega\) is the number of elements in the set \(\Omega\) which is calculated in Equation (5): \[\Omega=\left\{(m,n)\big{|}m^{2}+n^{2}=r^{2},r\leq\left[\min\{M,N\}/2\right] \right\}.\] (5) The normalized autocorrelation function in the spherical coordinate, as Equation (6), is used as the filtering function of normally distributed noise: \[F(R)=\frac{E\big{[}I^{i}(x+R)-\phi_{i}\big{]}\big{[}I^{i}(x)-\phi_{i}\big{]} \big{]}}{E\big{[}\big{[}I^{i}(x)-\phi_{i}\big{]}^{2}\big{]}}.\] (6)
3. X1 and X2 are uniformly distributed random numbers. Based on Box-Muller, these two random numbers can be used to generate Gaussian-distributed noise N efficiently, with a mean of 0 and a variance of 1, as shown in Equation (7). \[N=\sqrt{-2\ln X_{1}}\cos(2\pi X_{2}),X_{1}\sim U(0,1),X_{2}\sim U(0,1).\] (7) As shown in Equation (8), the initial 3D image with Gaussian noise can be obtained: \[I(i,j,k)=\sum\limits_{r,s,t}N(r,s,t)\times F(i+r,j+s,k+t).\] (8) According to Equation (9), iterative threshold segmentation is performed on 3D images to match the porosity of SEM pictures.