**1. Introduction**

Low-power unmanned ocean device, such as unmanned surface/under water vehicles, ocean robots, and ocean buoys are widely used in unmanned combat, deep-sea exploration, and marine communications, and so on [1–5]. With the continuous development of deep sea and offshore strategies, the scope of work of low-power unmanned ocean devices continues to expand [6]. Long-term, stable, and reliable energy supply is the basis for the rapid development of low-power unmanned ocean device [7]. Currently, the common energy supply technology is a battery. However, its energy density is low and it is necessary to carry multiple batteries to meet the requirements. The device space is limited, which limits the amount of battery that is carried [8]. Therefore, it is difficult for the battery to meet the requirements of the long-range and long-life capability of the current low-power unmanned ocean device. Therefore, studying the energy supply technology of high energy density and improving

the long range and long battery life of the ocean device is the fundamental guarantee for effectively solving the application of low-power unmanned ocean device to the deep sea [9].

The ocean covers about 70% of the Earth's surface, making it the world's largest solar collector and energy storage system [10]. There are many types of renewable resources for ocean storage, such as Wave Power, Tidal Power and Tidal Current Energy, Ocean Thermal Energy Conversion (OTEC), and Salinity Gradient Energy. We can convert them into electricity by means of a power generator [11]. Wind blowing over the surface of the ocean creates waves that can be harvested for energy and creates uninterrupted, continuous wave energy in the Ocean's surface [12]. The amount of wave power that is stored by the ocean is enormous and high density. Theoretically, the global wave power resource is 2.11 ± 0.05 TW, of which 4.6% is extractable with the chosen WEC configuration [13,14]. Therefore, wave energy is the most ideal resource for a low-power unmanned ocean device.

The utilization of wave energy goes far back in time—people using wave push water mill taken place in the 1200 s. The world's first wave energy technology patent is filed by Frenchman Girard and his son in 1799 [15]. In 1910, the Frenchman Pocec-Plesic built the world's first wave energy converter (WEC) on the coast, namely air turbine private power station with a capacity of 1kW [10]. Salter of the University of Edinburgh successfully developed a duck-type WEC, and first published an article entitled "Wave Power" in Nature in 1974 [16]. However, due to the unsolved technical problems, such as the economy and stability of WEC, the research progress of wave energy technology was slower in the 1980s and early 1990s. In the last 20 years, with the emergence of problems, such as energy depletion, environmental pollution, and the greenhouse effect, the development of green and renewable energy has become the mainstream [7]. Wave energy has regained rapid growth under the guidance of the government, especially in some mariner countries, such as Ireland, Denmark, Norway, and UK [17–20].

WEC's are generally categorized by the method that is used to capture the energy of the waves, by location and by the power take-off (PTO) system [10,21]. Locations are shoreline, near shore, and offshore. Types of power take-off include: hydraulic ram, elastomeric hose pump, pump-to-shore, hydroelectric turbine, air turbine, and linear electrical generator. In addition, WEC'S can also be categorized by the energy transfer method, namely point absorber, surface attenuators, oscillating water columns, and overtopping devices.

Comparative and analytic research on the working principle of current WEC's, the point absorber is the best choice for the power supply of low-power unmanned ocean device [7]. However, the research on point absorber is mainly focused on large-scale grid-connected generation in shore and near shore, and the dimension of the device is larger, and the power generation operation is more complicated [10]. At present, typical point absorber includes Power Buoy [22], Wave bob [23], Sea Based [24], Fred Olsens Lifesaver [25], and Carnegies CETO [26]. The shape dimension of them is very large, such as Power Buoy, where the diameter of the buoy is about 3 m and the overall height about 14 m. As the characteristics, such as concealment, camouflage, small dimension, wide and far working area of low-power unmanned ocean device, and the huge dimension of the above point absorbers, makes it still difficult to provide a satisfactory power source for it.

Considering the technical requirements of the power supply for the low-power unmanned ocean device and the current research state of power generation technology, we developed a small novel heaving point absorber that is based on the counter-rotating self-adaptive mechanism, with the advantages of small space device and a stable and reliable energy conversion process [7]. It is an ideal WEC of power supply for the low-power unmanned ocean device, so as to increase their working hours and improve independent operation ability.

Previous work shown that the influence law of the blade angel on the impeller speed was obtained and the model prototype was trial produced base on the study of the blade angle and the relative speed of the upper and lower impellers for the Underwater PTO [7].The adjustment of the blade angle in impellers is passive action. The move displacement of the floating body and the Underwater PTO affects the deflection angle. The locking devices of the impellers limit the angle. In addition, previous works show that the maximum blade deflection angles between 35◦ and 55◦ are the best efficiency characteristics [7]. In order to increase the power production of the novel heaving point absorber, it is necessary to increase the elevation of the Underwater PTO in the vertical direction, that is, increase the heave amplitude of the floating body. Because the displacement of the body is performed by its immersion depth, shape, and dimension, so the geometric parameters of the body are optimized, which is of great significance for increasing the power production of the novel heaving point absorber.

The floating body dynamics of the WEC have been investigated by many researchers. For example, Black et al. [27] comparatively analyzed the wave radiation forces and scattering forces for horizontal rectangular and vertical circular cylinders using the Haskind's theorem. Mohapatra and Guedes Soares [28] studied the wave forces on a two-dimensional rectangular floating structure based on linearized Boussinesq. Rodriguez et al. [29] investigated the numerical nonlinear heave response of a rectangular box concerning the importance of the relative body dimensions. Islam et al. [30] analyzed the wave radiation of a heaving box-type floating structure based on CFD simulations with a volume of fluid method. Yeung et al. [31] and Sabuncu et al. [32] discussed the added mass and damping of a vertical cylinder in finite depth water. Calisal et al. [33] presented an efficient method of hydrodynamic coefficients calculation for vertical composite cylinders at finite depth. Mansour et al. [34] analyzed the diffraction of linear waves by a uniform vertical cylinder with cosine-type radial perturbations. Bhata et al. [35] studied scattering and radiation problems for a cylinder on nonlinear wave loading at finite depth. Kim [36] researched the hydrodynamic coefficients of the floater with elliptical cylinder and ellipsoid on a free surface. Bihs et al. [37] simulated a horizontal cylinder in heave motion and the motion of a freely floating rectangular barge in waves using the CFD model and compared the results with experimental data. Koh et al. [38] used Matched Eigenfunction Expansion Method (MEEM) to solving the radiation problem of the heaving circular cylinder in the context of linear potential theory. Wang sheng [39] discussed the added mass and damping of an ellipsoid in infinite and finite depth water. Finnegan et al. [40] determined an analytical approximation for the wave excitation forces on a floating truncated vertical cylinder in water of infinite depth and solved the appropriate boundary value using the method of separation of variables. Ghadimi et al. [41] presented a detailed analytical solution for the boundary value problem to evaluate the wave loads for the cylinder with heave and pitch motions in water of finite depth in the presence of an incident wave. Although the floating body dynamics of the WEC are more researched, the energy efficiency of multi-type floating bodies in the heaving wave energy point absorber is poorly studied. Furthermore, in the above paper, the separation variable method and the eigenfunction expansion method is usually used for the calculation of the radiation force and diffraction force of the buoy base on the potential flow theory. However, it is difficult to solve their analytical solutions and is often time consuming to calculate.

Under the Airy's wave theory, the Froude-Krylov approximation method is implemented to solve the wave force when the device dimension is considerably smaller than the wave length [42]. Froude-Krylov approximation method [33–45] is assumed that the original wave pressure distribution of the incident wave does not change due to the presence of the floating body. Therefore, wave excitation force is the product of the force of the undisturbed incident wave pressure on the floating body (dynamic Froude-Krylov force) and the diffraction correction coefficient. The diffraction coefficients reflect the attached mass effect and diffraction effects, which is determined by the model test. Although this method is an approximation method, it is simple to calculate and it is a very practical method for estimating the wave force. Moreover, because the method is built on model tests, accurate calculations can be obtained [42]. Falnes and Perlin [46,47] analyze the oscillating bodies in low-amplitude waves and obtain that the diffraction is negligible when the device dimension is considerably smaller than the wave length. Clement and Ferrant [48] described a method for the computation of free surface flows generated by submerged bodies, and obtain that radiation nonlinearities are negligible for floating bodies that are small as compared to the wave length. Merigaud et al. [43] added specific nonlinear terms to hydrodynamic models for wave energy devices,

to improve the validity of such models across the full operational spectrum, showing that the response of the device is mainly affected by nonlinear FK forces, while nonlinear radiation and diffraction forces have minor effects on the system dynamics.

The focuses of the paper is on the novel heaving point absorber operating in the power production region, using the Froude-Krylov method to optimal shape design and maximize energy capture of the floating body for the power supply of the low-power unmanned ocean device. The remainder of this paper is organized as follows: Section 2 introduces the structure model of the novel heaving point absorber. Section 3 gives the mathematical model of the floating body of the novel absorber. Section 4 presents the algebraic solution of the wave excitation force in the vertical direction of floating bodies and nonlinear Froude-Krylov force integral. The numerical simulation and simulation analysis for the wave force, heaving velocity, heaving displacement, and capture width ratio of the multi-type floating bodies are in Sections 5 and 6. Some conclusions and final remarks are presented in Section 7.
