## 2 Model and Methodology
### Concrete Compressive Strength Development over Time Model
In addition, \(m_{C}\) is the quantity of cement used in concrete, \(m_{SL}\) is the quantity of slag used in concrete, \(m_{FA}\) is the quantity of fly ash used in concrete, and \(m_{LF}\) is the quantity of limestone flour used in concrete. However, the correlation between the content and the 28-day compressive strength contribution of each gelled material is always nonlinear in practical engineering applications, so parameter \(i_{x}\) is introduced into the model. For instance, the 28-day compressive strength of concrete increases rapidly with increases in the cement content at first. However, when the content of cement reaches a certain level, the 28-day compressive strength of concrete increases slowly with an increase in the amount of cement. In such cases, the impact index of cement is greater than 0 and less than 1. The 28-day compressive strength of concrete increased slowly with increases in the fly ash content at first. However, when the content of fly ash reaches a certain level, the 28-day compressive strength of concrete increases quickly. In this case, the impact index of fly ash is always greater than 1. In order to measure the contribution of different gelled materials to the 28-day compressive strength of concrete, the 28-day compressive strength contribution coefficient of gelled material \(x\) is introduced into the model. \[S_{28x}=S_{x}q_{x}^{i_{x}}. \tag{7}\]  
where \(i_{x}\) is the impact index of gelled material \(x\). In order to compare the influence of each kind of gelled material on concrete, \(S_{x}\) is normalized as follows:  
\[ss_{x}=S_{x}/S_{C} \tag{8}\]  
where \(ss_{x}\) is the 28-day compressive strength contribution coefficient of gelled material \(x\). Substitute Equation (7) into (3) and we get Equation (9). \[S_{28}=S_{C}q_{C}^{i_{C}}+S_{SL}q_{SL}^{i_{SL}}+S_{FA}q_{FA}^{i_{ FA}}+S_{LF}q_{LF}^{i_{LF}}. \tag{9}\]  
where \(i_{C}\) is the impact index of cement, \(i_{SL}\) is the impact index of slag, \(i_{FA}\) is the impact index of fly ash and \(i_{LF}\) is the impact index of limestone flour. Just like the 28-day compressive strength of concrete, the factor of compressive strength in later periods is expressed as the sum of the contribution of each kind of gelled material. \[\lambda=\lambda_{C}q_{C}{}^{iC}+\lambda_{SL}q_{SL}{}^{i_{SL}}+\lambda_{FA}q_{FA}{ }^{i_{FA}}+\lambda_{LF}q_{LF}{}^{i_{LF}}. \tag{10}\]  
where \(\lambda_{C}\) is the factor of compressive strength in the later period of cement, \(\lambda_{SL}\) is the factor of the compressive strength in later periods of slag, \(\lambda_{FA}\) is the factor of the compressive strength in later periods of fly ash and \(\lambda_{LF}\) is the factor of the compressive strength in later periods of limestone flour. By using the compressive strength test results of concrete with various proportions of the same aggregates and gelled materials, the 12 parameters (\(S_{C}\), \(S_{SL}\), \(S_{FA}\), \(S_{LF}\), \(i_{C}\), \(i_{SL}\), \(i_{FA}\), \(\lambda_{C}\), \(\lambda_{SL}\), \(\lambda_{FA}\), \(\lambda_{LF}\)) of the model are inverted based on the optimization algorithm.