**4. Numerical Simulations**

#### *4.1. Simulation Scheme*

The structural schematic of the cone shaped floating body investigated in this study is shown in Figure 4.

The height of the cylinder part above the waterline is a constant, 0.5 m. In ANSYS Design Modeler, the 3D geometry with given parameters is created.

In this paper, ANSYS-AQWA, a commercial computation software based on potential flow theory, is utilized to calculate hydrodynamic parameters. The simulation process, including numerical modeling, parameters setting, mesh generation, and data post-processing, can be conducted in the graphical interface directly. The basic simulation steps for each sample are as follows:

1. The moment of inertia and center of mass of the floating body are calculated in Static Structural module;


**Figure 4.** Schematic of the floating body's structure.

For each simulation, the given frequency range is divided into 52 frequency points. The mean power for each sample at each frequency is calculated. The results of sample 1 and sample 2 are shown in Figure 5.

With the increase in wave frequency, the capture power rises firstly and then drops steadily. For each sample, there is a unique optimal frequency in which the capture power can reach the maximum. The 100 samples' capture power are calculated so that they can be used as training set and test set for BP neutral network. Only two samples' results are presented in this figure.

**Figure 5.** Mean capture power of sample 1 and sample 2.

*4.2. Theoretical Verification of Simulations*

Falnes [36] illustrated that the maximum power that a heaving axisymmetric body can absorb is

$$P\_{\text{max}} = \frac{J\lambda}{2\pi} \tag{23}$$

where *J* is the wave energy flux; *λ* is the wavelength. For deep-water waves, *λ* = g/2π. *J* is

$$J = \frac{\rho \text{g}^2 T H^2}{32\pi} \tag{24}$$

where *T* and *H* are wave period and height, respectively. Budal's upper bound [36] gave another upper limit power that a submerged body with given volume *V* can absorb. It is

$$P\_{\rm u} < \frac{\pi \rho g V H}{4T} \tag{25}$$

where *V* is the volume of the submerged part. The point of intersection of two theoretical curves can be defined as (Tc, Pc). *Pc* is

$$P\_{\mathbb{C}} = \frac{\rho g^2}{32\pi} \sqrt{2VH^3} \tag{26}$$

In this study, Equations (23) and (25) are used to verify the validity of simulations. To make comparisons, the results are normalized by dividing *Pc*. The three curves are shown in Figure 6.

It can be found that the capture power curves of two samples are in the area enclosed by curve *Pmax*, curve *Pu*, and coordinate axes, which means the simulation scheme is accurate and reliable. All the samples are verified successfully and only two of them are demonstrated in this section.

**Figure 6.** Power curves from simulations and theories. (**a**) results of sample 1; (**b**) results of sample 2.

#### **5. BP Neural Network**

The back propagation (BP) neural network is a kind of feedforward neural network trained by error back propagation algorithm. It is a most widely used form, and is composed of many nonlinear transformation units. This algorithm has a strong non-linear mapping ability and can simulate any nonlinear continuous functions with much higher accuracy theoretically. After the network is trained, the reflection between the inputs and outputs can be obtained and memorized. They are shown on the weights of each layer. BP neural network's structure is flexible, which means the number of layers and neurons can be changed according to research objectives. A BP neural network generally includes an input layer, one or two hidden layers, and an output layer. Full connections are applied between layers. More details about BP neural network can be seen in [37].
