# Adaptive Optimization Method for Piled Raft Foundations Based on Variable Pile Spacing
## 3 Adaptive Optimization Method
### Calculate \(V_{t}\) and Divide It into Equal Parts  
It is difficult to express the settlement surface of the raft using a continuous function (such as \(z(x,y)\)), and then Equation (1) cannot be used directly. Therefore, a discrete function is adopted instead of the continuous function, using the physical meaning of the integral to solve Equation (1). The settlement surface of the raft is meshed using the interpolation method to form a two-dimensional surface composed of \(N_{t}\) discrete nodes. These nodes  
Figure 1: Two-dimensional plane problem of piled raft foundations: (**a**) piled raft with one pile; (**b**) piled raft with two piles; (**c**) piled raft with N piles. are evenly distributed on the raft surface, assuming each node supports the same size of the raft area, expressed in \(a_{0}\), in which \(a_{0}=A_{\text{raft}}/N_{t}\). Then, \(x\), \(y\), and \(z\) of each node are represented by matrix \([X]\), \([Y]\), and \([Z]\), where \([X]\) and \([Y]\) represent the x and y co-ordinates of the nodes, and \([Z]\) represents the settlement (z) of the nodes. Finally, Equation (1) can be expressed as  
\[V_{T}=\sum_{j=1}^{N_{t}}v_{j}=\sum_{j=1}^{N_{t}}Z_{j}\times a_{0} \tag{3}\]  
where \(Z_{j}\) is the settlement of node \(j\) in the matrix \([Z]\), \(v_{j}\) represents the volume supported by point \(j\), and \(v_{j}=Z_{j}\times a_{0}\). Equation (2) can be expressed as  
\[V_{i}=V_{avg}=V_{T}/N,\quad i=1,2,3\ldots N \tag{4}\]  
where \(V_{avg}\) represents the average value of \(N\) parts. There are many ways to divide \(V_{T}\) into equal parts. The key is that the shape of \(V_{i}\) needs to be restricted. If the size difference of \(V_{i}\) in the x and y directions is too large, it is easy to cause the centroid spacing of the two adjacent \(V_{i}\) (such as \(V_{i}\) and \(V_{i+n}\)) to be too small or even overlapping; that is, the pile spacing is too small or even overlapping, and this does not meet the specification requirements [30]. Therefore, it is necessary to find a search method to limit the shape of \(V_{i}\) in the \(x\) and \(y\) planes to avoid the situation where the pile spacing is too small or the pile overlap cannot be applied to the actual project. The central search method (CSM) is applied in AOM to divide \(V_{T}\). We determine a central point \(P_{i}\), add \(v_{j}\) of point \(j\) (sub-point \(P_{j}\)), which is the nearest to \(P_{i}\), to \(v_{i}\) continuously, until \(v_{i}+v_{1}+v_{2}+\ldots+v_{j}\) is no less than \(V_{avg}\) (in Figure 2a). If there are no less than two points with equal distances from \(P_{i}\), the point with the smallest \(Z_{j}\) has priority. During this process, the central point and its sub-point will be removed from \([Z]\) one after another, and then we go to the next central point. In this way, the sub-points will be spread layer by layer around the central point, which can avoid the issue of the size differences of \(V_{i}\) in the \(x\) and \(y\) directions being too large. The flow chart of CSM is shown in Figure 2b. Figure 2: Central search method: (a) divide \(V_{T}\); (b) flowchart of CSM. During the use of CSM, the way to determine a central point is another key point, which can affect the accuracy of search results. For a raft, if the central points are simply determined one by one from one side to the other (for example, from top to bottom, or from left to right), it is easy to leave some points with a large dispersion, or even discontinuous points (call them \"leftover material\", as shown in Figure 3) along the edge of one side. The \(V_{i}\) formed by these points will result in large different sizes of the \(x\) and \(y\) directions of \(V_{i}\), and its centroid position will be very close to or even overlap the existing piles, which will eventually result in the ineffective location of pile \(i\). If the centre point is searched from the centre to the surrounding area across the raft, the problem of fragmentation of the remaining points will still exist, and then more discontinuous \"leftover material\" will be generated.