# Artificial Intelligence Algorithm-Based Arrangement Optimization of Structural Isolation Bearings
## 3 Optimal Arrangement Method of Isolation Bearings
### HPO Algorithm  
In 2021, [PERSON] [37] proposed a new optimization algorithm to simulate the hunting process of predatory animals, which has good convergence speed and optimization capability. Its main algorithm flow is as follows:  
Firstly, the predator or prey population randomly initializes its position through the upper and lower boundary:  
\[X_{1},\,X_{2},\,\ldots,\,X_{\rm k}\,(\rm k=1,\,2,\,\ldots\,\rm N)=lb+\rm RAND\, (0,\,1)\,(ub-lb) \tag{4}\]  
where: RAND (0, 1) is a random number between (0, 1), \(lb\), \(ub\) are respectively the lower and upper boundary, which are determined by the specific problem investigated.  
Secondly, the predator searches the prey population then attacks an individual far away from the prey population, and updates its position according to the position of the prey and the best position through the fitness mechanism of objective function in the process of predation. During the search process of predators, an adaptive parameter \(Z\) is applied to adjust the range of position and updated to ensure search efficiency:  
\[X_{\rm j,k+1}=X_{\rm k}+0.5[(2 CZP_{\rm posj}-X_{\rm j,k})+(2(1-C)Z\mu-X_{\rm j, k})] \tag{5}\]  
where: \(P_{\rm posj}\) is the position of prey in \(k\)th iteration. \(X_{\rm j,k+1}\), \(X_{\rm j,k}\) are the positions of predators in iteration \(k+1\) and \(k\), respectively. \(\mu\) is the average position of all predators, and the adaptive parameter \(Z\) is adjusted by Formula (6):  
\[Z=RAND(0,1)index(R_{1})+R_{3}index(R_{1}) \tag{6}\]  
\[P=R_{1}<C \tag{7}\]  
\[C=1-iter\frac{0.98}{maxiter} \tag{8}\]  
where: \(R_{1}\) and \(R_{3}\) are random vectors between \([0,1]\). \(C\) is a gradually decreasing balance parameter that enables predators to choose their exploration and development state. _Iter_ and _maxiter_ are current and total iteration numbers, respectively, while \(P\) is the index value of the random vector when \(R_{1}<C\), and index (\(R_{1}\)) is the index value when random vector \(R_{1}\) meets \(P=0\).  
During hunting, through searching the prey with the largest distance to average position of all prey, the predator determines corresponding prey to be hunted.  
\[P_{pos,j}=X_{Dmax} \tag{9}\]  
\[D=\left(\sum_{j=1}^{m}\bigl{(}x_{ij}-\mu_{j}\bigr{)}^{2}\right)^{\frac{1}{2}} \tag{10}\]  
\[\mu=\frac{1}{n}\sum_{i=1}^{n}x_{i} \tag{11}\]  
where: \(D\) represents distance of all predators to the average position of \(\mu\), which is the average position of all predators. \(X_{Dmax}\) is the individual that is farthest from the population.  
After a hunt is finished, the predator updates a new hunting target. Initially \(C=1\) and \(kbest=N\); as the number of iterations rises, the balance parameters \(C\) and \(kbest\) gradually decrease, ensuring better convergence accuracy in the latter period.  
\[P_{pos,j}=X_{D,kbest} \tag{12}\]  
\[kbest=round(C\times N) \tag{13}\]If prey successfully escapes or find safe shelter, the predators have no opportunity to hunt. In order to improve the efficiency of hunting, with the updating of prey, the predators will also update their positions:  
\[\text{X}_{j,k+1}=P_{\text{best}}+C\text{Zcos}(2\pi\text{R}_{4})\left(P_{\text{ best}}-X_{j,k}\right)] \tag{14}\]  
where: \(P_{\text{best}}\) is the global optimal location of prey, while \(R_{4}\) is a random number between \([-1,1]\).  
Based on the fast convergence speed of the predator algorithm, the sizes of isolation bearings can be optimized through this algorithm because the size of an isolation bearing is fixed within a predetermined range, which could be substituted into the Formula (4). The size of the isolation bearing can be considered as each individual of prey, while the iterative optimization of multiple populations will be simultaneously executed to obtain the size groups of isolation bearings that minimize the fitness value of objective function.