# Multi-Point Deformation Prediction Model for Concrete Dams Based on Spatial Feature Vector
## 2 2D-1D-CNN Deformation Prediction Model for Concrete Dams  
### Multi-Point Model of Concrete Dams Incorporating Spatial Coordinates  
Based on the existing theory of dam construction [1], this paper mainly considers the deformation of the dam body caused by water pressure, temperature, and aging. Amongthem, the deformation caused by water pressure and temperature is reversible elastic deformation, and the deformation caused by aging factors is irreversible deformation caused by the deterioration of the dam material. Therefore, the concrete dam deformation can be divided into the water pressure component \(\delta_{H}\), the temperature component \(\delta_{T}\) and the aging component \(\delta_{\theta}\), as shown in Equation (1):  
\[\delta=\delta_{H}+\delta_{T}+\delta_{\theta} \tag{1}\]  
For concrete dams in which the heat of hydration has been completely dissipated during the operation period, the harmonic factor can be chosen as the temperature factor. Therefore, Equation (2) can be used as the expression of the concrete dam deformation model [8].  
\[\delta=a_{0}+\sum_{i=1}^{4}a_{i}H^{i}+\sum_{i=1}^{2}\biggl{(}b_{1i}\sin\frac{2 \pi i\Delta t}{365}\biggr{)}+\sum_{i=1}^{2}\biggl{(}b_{2i}\cos\frac{2\pi i \Delta t}{365}\biggr{)}+c_{1}\theta+c_{2}\ln\theta \tag{2}\]  
where \(H\) is the upstream water level height; \(\Delta t\) is the cumulative number of days from the deformation monitoring date to the starting monitoring date; \(\theta=\Delta t/100\); \(a_{0}\) is a constant term; \(a_{i}\), \(b_{1i}\), \(b_{2i}\) and \(c_{1}\), \(c_{2}\) are all fitting coefficients.  
Since the deformation of concrete dams exists in both time and space dimensions, the traditional single-point statistical model only models a single-point, which has difficulty in accurately reflecting the overall deformation state of the dam. The multi-point model of concrete dams is based on the single-point model, and the spatial coordinates of the measuring points are combined with the spatial correlation between the deformation sequences of measuring points, so it can better reflect the overall deformation of the dam. By incorporating the spatial coordinates of the measuring points \((x,y,z)\) into the single-point model of the concrete dam, the expression of the multi measuring point model of the concrete dam can be derived, as shown in Equations (3) and (4) [9].  
\[\delta{=}f_{1}[(f(H),f(x,y,z)]+f_{2}[(f(T),f(x,y,z)]+f_{3}[(f(\theta),f(x,y,z)] \tag{3}\]  
where \(H\), \(T\), \(\theta\) are water pressure, temperature, and aging factor, respectively.  
\[f(x,y,z)=\sum_{l,m,n=0}^{3}A_{lmm}x^{l}y^{m}z^{n} \tag{4}\]  
Jointly with (2), the expression of the concrete dam multi-point model can be expressed as  
\[\delta=\sum_{i=0}^{4}\sum_{l,m,n=0}^{3}A_{ilmn}H^{i}x^{l}y^{m}z^{n}+ \tag{5}\] \[\sum_{i,j=0}^{1}\sum_{l,m,n=0}^{3}B_{ilmn}\sin\frac{2\pi i\Delta t }{365}\cdot\cos\frac{2\pi j\Delta t}{365}x^{l}y^{m}z^{n}+\] \[\sum_{i,j=0}^{1}\sum_{l,m,n=0}^{3}C_{ilmn}\theta_{i}\ln\theta_{j} x^{l}y^{m}z^{n}\]  
where \(A_{lmm}\), \(B_{lmn}\) and \(C_{lmn}\) are the fitting coefficients.