# Optimization of Uplift Piles for a Base Plate Considering Local Anti-Floating Stability
## 2 Optimization Method
### BESO Process  
Initially, the BESO process takes all uplift piles as the design domain and discretizes the piles into pile elements, which is the step of establishing the numerical model and meshing the model. Pile length \(L_{i}\) is taken as the design variable, and the displacement of the pile end is the sensitivity coefficient. The constraints are considered in two aspects, as follows. i. Stability constraints: The structure meets the overall anti-floating stability requirements. ii. Volume constraints: The target total volume of piles after the final optimization is \(V^{*}\). The BESO process gradually reduces the total volume of pile elements until the volume constraints are met. The total volume of pile elements changed in a single iteration step is controlled by the volume evolution rate e and is calculated as follows:  
\[V^{k+1}=V^{k}(1-e) \tag{7}\]  
where \(V^{k}\) and \(V^{k+1}\) are the target volume in a single iteration step that needs to be reached for the \(k\)th iteration and the \(k+1\)th iteration, respectively. The replacement and addition process of pile elements in each iteration step is as follows: Suppose the current iteration step is \(k\)th. Firstly, traverse the vertical displacement \(u_{ei}\) of each pile end element in the numerical calculation result, and preliminarily determine the \(\alpha\) threshold as follows:  
\[\alpha_{th}=(u_{emax}+u_{emin})/2 \tag{8}\]  
where \(u_{emax}\) and \(u_{emin}\) are the maximum and minimum values of the pile end displacement, respectively. Secondly, comparing \(u_{ei}\) with \(\alpha_{th}\), the pile end elements with \(u_{ei}<\alpha_{th}\) are regarded as inefficient elements and replaced with soil elements; that is, the pile elements are replaced. The pile end elements with \(u_{ei}>\alpha_{th}\) are regarded as effective elements, and the soil elements below them are replaced with pile elements; that is, pile element are added. Thirdly, calculate the current pile elements volume \(V^{k}\), and judge whether it meets the target volume requirement of the single iteration step, namely, comparing \(V^{k^{\prime}}\) with \(V^{k}\). Before entering the next iteration step, the current pile elements volume \(V^{k^{\prime}}\) must be equal to the target volume of this iteration step (i.e., \(V^{k^{\prime}}=V^{k}\)), otherwise, \(\alpha_{th}\) will be updated based on the dichotomy method. For example, if \(V^{k^{\prime}}>V^{k}\), then by substituting \(u_{emin}=\alpha_{th}\) into Equation (8), a new \(\alpha_{th}\) can be obtained. This increases the threshold; thus, more pile elements are replaced by soil elements, and the total volume of pile elements is reduced. On the contrary, \(u_{emax}=\alpha_{th}\) can be substituted into Equation (8) to reduce the threshold and increase the amounts of pile elements. The above three steps are the process of increasing the length of the piles with insufficient pulling force and reducing the length of the piles with excessive pulling force. It should be noted that before each iteration, it is necessary to verify whether the current design meets the requirements of the overall anti-floating stability of the structure. If the requirements are not met, the initial pile length should be redesigned. However, the verification can be directly implemented in numerical calculation. If the buoyancy is greater than the total anti-floating force, the numerical calculation will not converge, and the vertical displacement of the base plate will become increasingly larger. Figure 3 shows the flow chart of the BESO process.