## 3 Numerical Simulation of CSAMT
### Coupled Finite Element-Infinite Element Method  
The coupled finite element-infinite element method divides the entire solution domain into a finite element region and an infinite element region, replacing the traditional outer boundary with the infinite element region. Finite element analysis and infinite element analysis are performed in the two regions, respectively, and they are combined through the assembly of the overall stiffness matrix to obtain the numerical solution. Figure 4 is a schematic diagram of the division of finite element and infinite element calculation regions. In the figure, the finite element region is the target region, which includes the field source, target body, measurement points, etc. ; the infinite element region extends from the boundary of the finite element region to infinity, serving as the boundary calculation region. Infinite element analysis involves using infinite element mapping and shape functions in a certain direction within this region to map the global coordinates to local coordinates. Its principle is the same as that of finite element analysis. 1. Finite Element Analysis  
When performing finite element analysis, the rectangular hexahedron is used for regional discretization, and the element node numbering and coordinates are shown in Figure 5.  
Figure 4: Calculation domain of the finite and infinite element. Figure 5: Mapping of the finite element. (**a**) Sub element; (**b**) parent element. In Figure 5, the corresponding relationship between the coordinates of the parent and child elements is as follows:  
\[x=x_{0}+\frac{a}{2}\xi\qquad y=y_{0}+\frac{b}{2}\eta\qquad z=z_{0}+\frac{c}{2}\zeta, \tag{10}\]  
where \(x_{0}\), \(y_{0}\), and \(z_{0}\) represent the midpoint of the child element and a, b, and c represent the three side lengths of the child element. The expression for the shape function of the rectangular hexahedron is as follows:  
\[N_{i}^{r}=\frac{1}{8}(1+\xi_{i}\xi)(1+\eta_{i}\eta)(1+\zeta_{i}\zeta) \tag{11}\]  
In the equation, \(\xi_{i}\), \(\eta_{i}\), and \(\zeta_{i}\) represent the coordinates of node i in the child element within the parent element. 2. Infinite Element Analysis  
When performing infinite element analysis, a three-dimensional eight-node Astley-type infinite element is used [29]. Figure 6 presents a schematic diagram of three-dimensional infinite element mapping, where nodes 1, 2, 3, 4, 5, 6, 7, and 8 are the basic elements of the infinite element. P is the coordinate origin, and nodes 1, 2, 3, and 4 are the four nodes of a finite element on the boundary. The distances from point P to nodes 5, 6, 7, and 8 are twice the distances from point P to nodes 1, 2, 3, and 4, respectively. Infinite element analysis involves mapping infinite coordinates to the local coordinates in Figure 6b through infinite element mapping. In Figure 6b, the four outermost nodes of the infinite element represent infinity, and their field values are zero. In Figure 6b, \(\xi\) represents the mapping direction of the infinite element. Within plane \(\zeta-\eta\), the infinite element and the finite element adopt the same mapping form and shape function. The coordinate mapping for the infinite element is as follows:  
\[y=\sum_{i=1}^{2}L_{i}y_{i}\qquad z=\sum_{i=1}^{2}L_{i}z_{i}, \tag{12}\]  
\(L_{i}\) represents the area of the quadrilateral, \(L_{i}=\frac{1}{4}(1+\eta_{i}\eta)(1+\zeta_{i}\zeta)\). Combining the coordinate mapping relationship in direction \(\zeta\), the coordinate mapping function for the eight-node infinite element can be obtained:  
\[x=\sum_{i=1}^{8}N_{i}x_{i}\quad y=\sum_{i=1}^{8}N_{i}y_{i}\quad z=\sum_{i=1}^{8 }N_{i}z_{i}, \tag{13}\]  
Figure 6: Mapping of the infinite element. **(a)** Sub element; **(b)** parent element. where  
\[\left\{ \begin{array}{l}N_{i}=L_{i}\frac{1-\zeta}{2}\frac{-(1-\zeta)}{2}\\ N_{i+4}=L_{i}\frac{1-\zeta}{2}\big{(}1-\zeta^{2}\big{)}\\ SNR=20*Log\frac{signal}{noise}\end{array}