## 1 Introduction
Given the three-dimensional environment, especially when there exists water flow and obstacles underwater, it is necessary to consider the optimal coverage scheme of underwater nodes in three-dimensional space under the condition of balanced network coverage performance. Intelligence optimization algorithms have prompted researchers to propose innovative techniques capable of improving coverage performance [15]. After transforming node coverage and energy problems into octahedral task allocation problems, [PERSON] et al. proposed a vampire bat optimizer method, which enhanced coverage efficiency and reduced energy consumption [16]. As the water moves, the nodes change position, making UWSNs prone to dynamic topology drifts. To solve the coverage problem in an underwater dynamic environment and with depth information as the preferred selection condition, the optimal deployment was selected using a strategy of obtaining the optimal solution step by step; this was also effective for irregularly distributed underwater nodes [17]. To directly improve the global search capability for obtaining global optimal coverage, [PERSON] et al. utilized an enhanced fruit fly optimization algorithm to reasonably adjust node positions [18]. [PERSON] et al. combined the advantages of adaptive multi-parent crossover and fuzzy dominance to balance UWSN performance, including coverage rate [19]. However, it is difficult for the UWSNs to achieve a balance in coverage, node energy consumption, and execution time. [PERSON] et al. adopted a bald eagle search (BES) to optimize the entire UWSNperformance [20]. Node battery replacement is challenging and demanding in changing underwater environments. We need to pay attention to the loss of energy generated by nodes in the networks in the flow to maintain its stability. Overly complex optimization algorithms are prone to falling into local optima and cannot maximize coverage rates. Many scholars regard underwater nodes as particles and adjust their locations by calculating the forces between particles, thereby reducing the coverage holes. Considering the limited underwater node resources, [PERSON] et al. conducted a three-dimensional redeployment of underwater mobile nodes based on virtual forces between nodes and utilized water flow forces to reduce energy consumption [21]. Unlike the conventional calculation of attraction and repulsion between nodes, some scholars proposed a three-dimensional improved virtual forces algorithm (3D-IVFA) by introducing central gravity and equilibrium force. The algorithm was proved to be effective in terms of coverage and uniformity [22]. One NP problem for achieving optimal network coverage performance with fewer nodes is the \(K\)-coverage problem. [PERSON] et al. propose an improved virtual force algorithm (VFA) combined with the radius method of the same utility to solve a variety of \(K\)-cover optimization problems with varying coverage densities [23]. Due to the inability to predict underwater node locations, centralized node deployment makes it difficult to achieve effective coverage. A distributed algorithm based on virtual forces (DABVFs) was utilized to enhance the coverage rate of regions of interest, while the issue of coverage holes caused by underwater faulty nodes was also considered [24]. Of course, some scholars have also focused on the issue of a relatively small number of underwater nodes covering key areas of interest and proposed the focus virtual force field method [25]. Some scholars tried to combine particle swarm optimization algorithms with virtual force to enhance UWSN coverage performance. [PERSON] et al. used the virtual forces between particles to guide particle optimization, accelerating particle convergence to the best overall solution and achieving the best coverage value [26]. All the above methods can improve the UWSN coverage, but these algorithms are prone to oscillations in the later stages and uneven coverage due to poor stability.