# Dynamic Inversion Method for Concrete Gravity Dam on Soft Rock Foundation
## 2 Inversion Methods for Mechanical Parameters of Concrete Gravity Dam  
The estimation of mechanical parameters of dam concrete and bedrock usually relies on laboratory test data. However, such experimentally determined parameters often demonstrate significant discrepancies with in situ conditions due to scale effects and long-term environmental exposure. To address this limitation, inversion analysis methodology integrates numerical modeling with prototype monitoring data to accurately identify the time-dependent mechanical properties of dam-foundation systems [21,22]. The basic principles and methods for identifying the elastic modulus are as follows:  
Divide the dam and dam base into two areas \(\Omega_{1},\Omega_{2}\). Define the average elastic modulus and Poisson's ratio of the materials in the two regions as \(E_{c},E_{r},\mu_{r},\mu_{c}\), respectively. In the elastic phase, the equilibrium equations of the whole structure are shown in Equation (1):  
\[[K]\{\delta_{H}\}=\{R_{H}\} \tag{1}\]  
\([K]\) is the overall strength matrix, which can be written as Equations (2) and (3):  
\[[K]=\sum_{\epsilon_{j}\in\Omega_{1}}[C]_{\epsilon_{j}}^{T}[K]_{\epsilon_{j}}[C ]_{\epsilon_{j}}+\sum_{\epsilon_{j}\in\Omega_{2}}[C]_{\epsilon_{j}}^{T}[k]_{ \epsilon_{j}}[C]_{\epsilon_{j}} \tag{2}\]  
\[[k]_{\epsilon_{j}}=\iint\limits_{\Omega_{j}}[B]^{T}[D][B]d\Omega=E\iint\limits _{\Omega_{j}}[B]^{T}f(\mu)[B]d\Omega=E\Big{[}\vec{k}\Big{]}_{\epsilon_{j}} \tag{3}\]  
where \([D]=Ef(\mu)\) is the elasticity matrix; \([B]\) is the cell geometry matrix; \([C]_{\epsilon_{j}}\) is the strength transformation matrix of cell \(\epsilon_{j}\). Since \(E= \begin{cases}E_{c},&\epsilon_{j}\in\Omega_{1}\\ E_{r},&\epsilon_{j}\in\Omega_{2}\end{cases} \), Equation (2) can be written in the following form:  
\[ \begin{split}&[K]=E_{c}\Bigg{\{}\sum\limits_{\epsilon_{j}\in \Omega_{1}}[C]_{\epsilon_{j}}^{T}\Big{[}\vec{k}\Big{]}_{\epsilon_{j}}[C]_{ \epsilon_{j}}+R\sum\limits_{\epsilon_{j}\in\Omega_{2}}[C]_{\epsilon_{j}}^{T} \Big{[}\vec{k}\Big{]}_{\epsilon_{j}}[C]_{\epsilon_{j}}\Bigg{\}}=\\ & E_{c}\cdot f(L,\mu_{r},\mu_{c},\frac{E_{r}}{E_{c}})\end{split}  \tag{4}\]  
where \(R=\frac{E_{r}}{E_{c}}\); \(L\) is the geometry of the dam. If \(\frac{E_{r}}{E_{c}}\) is certain, \(\mu_{r}\) and \(\mu_{c}\) are unchanged, then \([\overline{K}]=f(L,\mu_{r},\mu_{c},\frac{E_{r}}{E_{c}})\). Equation (4) can be rewritten as Equation (5):  
\[[K]=E_{c}[\overline{K}] \tag{5}\]Simplification is given as Equation (6):  
\[\left\{\delta_{H}\right\}=\left\{\overline{\delta}_{H}\right\}/E_{c} \tag{6}\]  
where \(\left\{\overline{\delta}_{H}\right\}=\left[\overline{K}\right]^{-1}\{R_{H}\}\). From Equation (6), the displacement of the dam body \(\left\{\delta_{H}\right\}\) induced by the reservoir water pressure is inversely proportional to \(E_{c}\) when the ratio of the concrete and bedrock modulus of the dam body is a certain value under the reservoir water force. The dam's safety monitoring displacement is used as the target variable for inversion analysis. The objective function of the parameter inversion optimization problem is defined as the weighted sum of squares of the residuals between the finite element computed displacement and the measured displacement at the monitoring points. This approach aims to determine the basic parameter that most closely reflects actual conditions. The flowchart of the parameter inversion process is shown in Figure 1. This flowchart systematically outlines the iterative optimization framework, beginning with the initialization of observed data and finite element inputs, followed by stochastic parameter generation and displacement computation. The adaptive feedback loop, driven by the objective function evaluation and termination criteria, ensures continuous refinement of parameters until convergence. Notably, the integration of real-time monitoring data with numerical simulations highlights the algorithm's capability to bridge empirical observations and theoretical models.