Numerical Investigation on Mooring Line Configurations of a Semisubmersible Fish Farm for Global Performance

Abstract

The growth of the aquaculture industry has led to the development of innovative fish farming technologies, such as semisubmersible fish farms which offer significant advantages in terms of scalability and environmental sustainability. The study utilizes a new workflow based on an open-source finite element solver, Code_Aster, to calculate the hydrodynamic dynamic response of a fish farm. The mooring line configuration of a semisubmersible fish farm plays a crucial role in ensuring its stability and overall performance. This study presents a comprehensive numerical investigation aimed at evaluating the influence of different mooring line configurations (i.e., 4 × 1, 4 × 2, and 4 × 3) on the global performance of a semisubmersible fish farm under regular waves and irregular waves conditions. Increasing the number of mooring lines can reduce the mean and extreme tension in individual mooring lines and suppress the horizontal motions, but bring neglectable effects on the rotational responses. The findings from this research provide valuable insights into the optimal mooring line configuration for the global performance of semisubmersible fish farms.

1. Introduction

The aquaculture industry’s rapid expansion has created a demand for fish farming technologies that are both sustainable and efficient [1]. Traditional fish farms are located in sheltered, shallow, and nearshore waters for operational safety and easy access to service facilities such as feed, hatcheries, storage, maintenance, and processing areas for harvested fish [2]. However, due to the increasing need for higher production targets and cost-effectiveness, many nearshore sites have been fully utilized, leading to high fish stock densities in cage nets. The high density can increase the motility rate and threaten the fish welfare, and eventually damage the sustainability of the fish farm.

In order to address the issues of traditional fish farms, the fish farming industry has begun exploring alternative sites to ensure sustainable fish production and environmentally friendly operations. Offshore locations offer advantages by preventing the accumulation of fish waste (e.g., uneaten feed or feces) beneath cages, which helps to control the proliferation of parasites and diseases [3]. As a result, it has become evident that exploring offshore sites for fish farming is an essential choice to maintain sustainable and high-quality fish production. In offshore fish farms, aquaculture structures, including fish cages as well as mooring lines, play a pivotal role in providing a controlled environment for fish rearing in open water meeting the rising global demand for seafood [4].

Extensive studies have been conducted using experimental and numerical methods to explore the hydrodynamic characteristics of offshore aquaculture structures. Moe et al. [5] explored the deformation and cultivation volume of net cage under the pure current. Zhao et al. [6] studied the motion and the tension of a mooring line of a semisubmersible offshore fish farm in pure waves. Yu et al. [7] examined the nonlinear vertical accelerations of a semisubmersible offshore fish farm model under extreme conditions. Huang et al. [8] found that increasing the length of the mooring line could reduce the mooring force significantly under the current and wave. Cheng et al. [9] studied the dynamic responses of Ocean Farm 1 when subjected to irregular waves and current conditions. However, the hydrodynamic responses of the net, which is the main component in the structure design, mainly depend on gravity (buoyancy) and elastic and viscous phenomena, which cannot be simultaneously scaled using any scaling laws [10,11]. Numerical simulations have become important tools for analyzing the behavior of aquaculture structures and their mooring line configurations in various marine environments. Through numerical simulation, designers can have a detailed understanding of the dynamic behavior of aquaculture structures, including the deformations, stresses, and strains experienced by the mooring lines and the overall structure. Moreover, numerical simulations facilitate the evaluation of different mooring line configurations, offering opportunities for optimization to enhance stability and efficiency.

In recent years, a variety of specialized numerical codes have been created internally for the dynamic analysis of aquaculture structures, and many of them have been validated through experiments. Li et al. [12,13] utilized the SIMO program, based on linear potential theory, to study the dynamic response of offshore aquaculture structures. Tsukrov et al. [14] introduced the Aqua-FE computer program, successfully applying it to analyze the dynamics of net cages and mussel longlines. Zhao et al. [15] developed the DUT-FlexSim program and extensively validated their numerical models. To further evaluate the performance of the Aqua-FE and DUT-FlexSim programs, Zhao et al. [16] conducted a comparative study with experimental measurements and concluded that both programs offer sufficient accuracy for designing fish cages. Additionally, the collaborative efforts of researchers at SINTEF Ocean led to the successful development of the FhSim program, which has found applications in various scenarios, including fish cages in rough seas, trawl net systems, aquaculture operations, and structures in ice floes [17,18,19]. Despite the significant progress in the field and the multitude of published numerical codes, there is still room for further advancements to make these codes applicable to the dynamic analyses of offshore aquaculture structures. Furthermore, many of these codes remain either commercially available or limited to in-house use, posing challenges for users and researchers who wish to develop third-party modules for these structures.

In this study, the structural responses of the new offshore semisubmersible fish cage, Ningde No. 1, are calculated using an open-source Finite Element solver, Code_Aster. This solver is further developed by Cheng [20] with an additional hydrodynamic library for aquaculture structures. Detailed information about Ningde No. 1 together with its mooring system are presented in Section 2. The numerical modeling methods for this structure are also explained in Section 2. Section 3 presents the results and discusses the effects of net, mooring line configurations on the structural responses, and Section 4 draws conclusions.

2. Materials and Methods

2.1. Description of the Offshore Aquaculture Structure

2.1.1. Main Framework

Ningde No. 1 is the first semisubmersible offshore aquaculture platform in China to be certified by the China Classification Society (CCS). The main dimensions of the studied fish farm are the following: 120 m in length, 56 m in width, and 12.5 m in height (total height including the central column is 32 m). The total cultivation volume is around 65,000 m³. Figure 1a shows the main framework structure during towing operation, and Figure 1b is the corresponding numerical model for the framework in the present study. The draft is adjustable according to weather conditions. The normal operation draft is 11.5 m, and the draft can be adjusted to 17.5 during an extreme storm and typhoon if necessary. The total mass is 4980 tons during normal operation conditions, and the total mass including increased ballast is 6013 tons during typhoon conditions. More detailed parameters for the main framework can be found in

Table 1. The main parameters of Ningde No. 1.

No.	Component	Length (m)	Diameter (mm)	Thickness (mm)
1	Bottom horizontal column	56.00	3000	12
2	Top horizontal column	56.00	2000	12
3	Bottom longitudinal column	120.00	3000	12
4	Top longitudinal column	120.00	2000	12
5	Vertical column	12.50	2000	12
6	Top horizontal beam	28.00	1000	10
7	Top diagonal beam	41.00	1000	10
8	Side diagonal beam 1	30.67	1000	10
9	Side diagonal beam 2	32.50	1000	10
10	Center column	14.50	5000	12

.

2.1.2. Net System

Ningde No. 1 employs netting crafted from ultra-high molecular weight polyethylene, with a mesh size of 60 mm and twine diameter of 4 mm. The solidity of the net is 0.13 when there is no biofouling. The solidity of the net is determined as the ratio between the area covered by the threads in a net panel and the outline area of the net panel. Nets are attached to the main framework through pre-tensioned cables. This is an engineering way to fasten the flexible net, and the level of the pretension is usually much smaller than the break strength. The pre-tensioned cables can prevent large deformations under waves and currents. The net is installed on the main framework and divided the whole aquaculture volume into eight chambers for fish farming.

2.1.3. Mooring System

Figure 2 shows the main frame from top and side views. In the present study, three different mooring configurations (including 4 × 1, 4 × 2, and 4 × 3 layouts) will be used to investigate the structural responses under extreme weather conditions. The last digit of the layout name represents the number of chains at each corner. Figure 2 shows the 4 × 2 mooring configuration, where mooring lines are connected to the four corners of the upper frames, and each corner has two mooring lines. The angle between the two mooring lines with the same attached location is 5°. The index numbers in Figure 2, following the counterclockwise direction, are the attached points on the framework. The horizontal distance between the anchor point and the attached point on the main framework is 295 m. The mooring system is an AM3-grade marine stud-link high-strength steel chain, the main properties of which are listed in

Table 2. Mooring line properties.

Property		Unit
Length	300	m
Nominal diameter	100	mm
Weight in the air	219	kg/m
Elasticity EA	62,000	kN
Breaking strength	7061	kN

. As shown in Figure 2, these mooring lines have a catenary shape under their own weight.

2.2. Numerical Modeling Method

2.2.1. Structural Model

In this study, Code_Aster is employed as the structural solver for the dynamic analysis in the time domain. The netting, mooring lines, and pipes are divided into a set of line-type elements for calculating the structural responses. The equation governing the motions is:

$$\left[\mathit{M}\right]\ddot{\mathit{x}}+\left[\mathit{K}\right]\mathit{x}={\mathit{F}}_{\mathit{g}}+{\mathit{F}}_{\mathit{b}}+{\mathit{F}}_{\mathit{h}}$$

(1)

where M is the mass matrix; K is the stiffness matrix; $\mathit{x}$ is the displacements of nodes; F_g is the nodal gravitational force; F_b is the nodal buoyancy force; and F_h is the hydrodynamic loads. F_h and F_b are calculated by the UiS-Aqua library and assigned onto the nodes in the structural solver. Detailed verification and validation studies of UiS-Aqua can be found in [9,21].

The dynamic simulations are highly nonlinear. According to Antonutti et al. [22], the system nonlinearity can cause high-frequency oscillations and bring challenges for convergence. In the present solver, Equation (25) is solved based on the unconditionally stable Hilber–Hughes–Taylor–α (HHT–α) method, which introduces low numerical damping in the low-frequency band and high damping at the high-frequency band. The flowchart for the present simulation process is shown in Figure 3, where there is an additional module to calculate the seabed uplift resistance, compared to the flow chart in [9]. This module will be invoked every time interval. The present simulations process has been comprehensively validated through previous research [5,19].

2.2.2. Hydrodynamic Loads on Net

Hydrodynamic loads are usually complex and dependent on time, wave and current velocity, and wake effects. The hydrodynamic loads on nets are related to many parameters, such as the Reynolds number, solidity, attack angle, knot type, and twine construction [21,23]. However, a full characterization of all these parameters is impractical in numerical simulations for industrial usages. In the FEM analysis, these loads are usually determined by an additional module. The hydrodynamic loads on structures are calculated and transferred to the structural solver using the UiS-Aqua library in the present study [20].

Figure 4 shows the discretization of the whole net of Ningde No. 1 in the present study. Figure 5 shows one of the discretized net elements for calculating environmental loads. In the present study, the net is discretized into 2792 net panel elements, and the hydrodynamic loads are calculated using a screen model proposed by Kristiansen and Faltinsen [24]. This model considers the effect of the Reynolds number and solidity ratio on the force coefficients. The environmental loads are decomposed into drag force F_D and lift force F_L, as shown in Equations (5) and (6):

$${\mathit{F}}_{\mathit{D}}=\frac{1}{2}{C}_D{\rho }_w{A}_t{|\mathit{U}-\dot{\mathit{x}}|}^2{\mathit{i}}_{\mathit{D}}$$

(2)

$${\mathit{F}}_{\mathit{L}}=\frac{1}{2}{C}_L{\rho }_w{A}_t{|\mathit{U}-\dot{\mathit{x}}|}^2{\mathit{i}}_{\mathit{L}}$$

(3)

where ${\rho }_w$ is the fluid density; $A_t$ is the area of the net panel (the area of the triangular P1–P2–P3 in Figure 5); U is the velocity of the fluid at the centroid of the net panel (which is the vector sum of wave-particle velocity and current velocity); and $\dot{\mathit{x}}$ is the velocity of the structure. Furthermore, the unit vectors ${\mathit{i}}_{\mathit{D}}$ and ${\mathit{i}}_{\mathit{L}}$, used to identify the forces directions, are evaluated using the definition showed in Equations (7) and (8), respectively. $C_D$ and $C_L$ are the drag and lift coefficients. The values of these two force coefficients are calculated using Equations (7)–(14), which have been validated by previous studies [25,26].

$${\mathit{i}}_{\mathit{D}}=\frac{\mathit{U}-\dot{\mathit{x}}}{\left|\mathit{U}-\dot{\mathit{x}}\right|}$$

(4)

$${\mathit{i}}_{\mathit{L}}=\frac{{\mathit{i}}_{\mathit{D}}\times {\mathit{e}}_{\mathit{n}}\times {\mathit{i}}_{\mathit{D}}}{|{\mathit{i}}_{\mathit{D}}\times {\mathit{e}}_{\mathit{n}}\times {\mathit{i}}_{\mathit{D}}|}$$

(5)

$$C_D=C_{D0}\left(0.9cos\theta +0.1cos3\theta \right)$$

(6)

$$C_L=C_{L0}(sin2\theta +0.1sin4\theta )$$

(7)

$$C_{D0}=C_{cylinder}\frac{Sn(2-Sn)}{2{(1-Sn)}^2};$$

(8)

$$C_{L0}=\frac{0.5C_{D0}-C_{L45}}{\sqrt{2}};$$

(9)

$$C_{L45}=\frac{\pi C_{N45}}{8+C_{N45}};$$

(10)

$$C_{N45}=C_{cylinder}\frac{Sn}{2{(1-Sn)}^2}$$

(11)

$$\begin{array}{ll}{C}_{cylinder}& =78.46675+254.73873{log}_{10}Re-327.8864{\left({log}_{10}Re\right)}^2\\ & -223.64577{\left({log}_{10}Re\right)}^3-87.92234{\left({log}_{10}Re\right)}^4+20.00769{\left({log}_{10}Re\right)}^5\\ & -2.44894{\left({log}_{10}Re\right)}^6+0.12479{\left({log}_{10}Re\right)}^7\end{array}$$

(12)

$$Re=\frac{d_{w0}(U-\dot{\mathit{x}})}{\nu \left(1-Sn\right)}$$

(13)

2.2.3. Hydrodynamic Loads on Mooring Lines and Framework

Figure 6 illustrates the hydrodynamic model that is used in the present study to calculate the environmental loads (F_h) on the line-type elements, such as the main frameworks and mooring lines.

In this hydrodynamic model, the environmental loads are decomposed into two: the normal drag force (${\mathit{F}}_{\mathit{n}}$, Equation (17)) and tangential drag force (${\mathit{F}}_{\mathit{t}}$, Equation (18)):

$${\mathit{F}}_{\mathit{n}}=\frac{1}{2}{C}_n{\rho }_{w}L{d}_w\left|{{\mathit{u}}^{\mathit{r}}}_{\mathit{n}}\right|{{\mathit{u}}^{\mathit{r}}}_{\mathit{n}}$$

(14)

$${\mathit{F}}_{\mathit{t}}=\frac{1}{2}{C}_t{\rho }_{w}L{d}_w\left|{{\mathit{u}}^{\mathit{r}}}_{\mathit{t}}\right|{{\mathit{u}}^{\mathit{r}}}_{\mathit{t}}$$

(15)

$$C_n=\left\{\begin{array}{cc}\frac{8\pi }{sRe}\left(1-0.87s^{-2}\right)& 0<Re<1\\ 1.45+8.55{Re}^{-0.9}& 1<Re<30\\ 1.1+4{Re}^{-0.5}& 30<Re<2.3\times {10}^5\\ -3.41\times {10}^{-6}\left(Re-5.78\times {10}^5\right)& 2.3\times {10}^5<Re<4.9\times {10}^5\\ 0.401\left(1-e^{\frac{-Re}{5.99\times {10}^5}}\right)& 4.9\times {10}^5<Re<1\times {10}^7\end{array}\right.$$

(16)

$$C_t=\pi \mu (0.55\sqrt{Re}+0.084{Re}^{2/3})$$

(17)

$$s=-0.077215665+\mathit{ln}\left(8/Re\right);$$

(18)

$$Re=\frac{d_w(U-\mathit{v})}{\nu }$$

(19)

where L is the length of an element; d_w is the section diameter; and ${\rho }_w$ is the fluid density. The normal and tangential fluid velocity relative to the structure are denoted as ${{\mathit{u}}^{\mathit{r}}}_{\mathit{n}}$ and ${{\mathit{u}}^{\mathit{r}}}_{\mathit{t}}$. $C_n$ and $C_t$ are the normal and tangential drag coefficients, respectively. The values of these two force coefficients are taken from DeCew et al. [27], as described in Equations (19)–(22). These force coefficients have been successfully applied to study the dynamic behavior of a single-point moored submersible fish cage under currents.

2.2.4. Wake Effect

When the current passes one net panel, the current velocity will be reduced by the friction from the twines of the nets. In order to calculate the forces on downstream net panels, it is necessary to know how much the velocity is reduced. Since the velocity is squared in hydrodynamic models, as shown in Equations (5) and (6) and (17) and (18), it significantly contributes to the hydrodynamic force. In practice, a flow reduction factor (r) is adopted to represent this current velocity reduction. The fluid velocity at the downstream structures U_downstream can be expressed by Equation (23), where U_c represents the undistributed current velocity and U_w represents the wave particle velocity, which is based on the potential flow theory. In this study, the values of r are calculated using Equation (24), which is originally proposed and validated in [9,21]:

$${\mathit{U}}_{downstream}=r{\mathit{U}}_c+{\mathit{U}}_w\hspace{1em}(0<\mathrm{r}<1)$$

(20)

$$r=\mathrm{m}\mathrm{a}\mathrm{x}(0,\hspace{1em}\frac{\cos \theta +0.05-0.38Sn}{\cos \theta +0.05})$$

(21)

2.2.5. Seabed Model

In the present study, the mooring line is a catenary shape (Figure 7a) rather than a taut shape (Figure 7b). Hereby, the interaction of a mooring line with the seabed should be considered in the numerical simulations. The contact problem between the mooring line and seabed is a complex subject which interfaces structural and geotechnical engineering. Currently, the most common method suggests a combination of two nonlinear dissipative mechanisms to model the reactive soil forces: lateral friction and uplift resistance [22]. In the present study, a simple seabed model is further implemented in UiS-Aqua. The friction factor between the mooring line and the seabed is chosen as 0.1. The rigid seabed introduces a vertical stiffness (K_b) between the cable and the seabed. When one of the cable elements penetrates through the seabed, the vertical stiffness will introduce upwards forces to support this structure lying on the seabed, as shown in Figure 8.

2.3. Environmental Conditions

The water depth in the operation site is 22 m. The wave and current conditions are estimated based on the historical weather data from 1949 to approximately 2019. Figure 9 shows the significant wave height (Hs) and the spectral peak period (Tp) with a probability of exceedance corresponding to a return period of 50 years. The long axis of Ningde No. 1 is along with the direction of the most extreme wave conditions. At this wave direction, Hs has the maximum value of 7.64 m and Tp has the maximum value of 12.13 s.

In present dynamic analysis, the structural responses will be first calculated under 27 different regular wave conditions to study the hydrodynamic characteristics with respect to wave heights, wave periods, and wave directions. These conditions are listed in

Table 3. Regular wave conditions.

NO.	Wave Height [m]	Wave Period [s]	Wave Direction [°]
N1	5.00	9.693	0.00
N2	4.00	9.693	0.00
N3	3.00	9.693	0.00
N4	5.00	7.390	0.00
N5	4.00	7.390	0.00
N6	3.00	7.390	0.00
N7	5.00	6.263	0.00
N8	4.00	6.263	0.00
N9	3.00	6.263	0.00
N10	5.00	9.693	45.00
N11	4.00	9.693	45.00
N12	3.00	9.693	45.00
N13	5.00	7.390	45.00
N14	4.00	7.390	45.00
N15	3.00	7.390	45.00
N16	5.00	6.263	45.00
N17	4.00	6.263	45.00
N18	3.00	6.263	45.00
N19	5.00	9.693	90.00
N20	4.00	9.693	90.00
N21	3.00	9.693	90.00
N22	5.00	7.390	90.00
N23	4.00	7.390	90.00
N24	3.00	7.390	90.00
N25	5.00	6.263	90.00
N26	4.00	6.263	90.00
N27	3.00	6.263	90.00

. The condition with a 5 m wave height and 6.263 s wave period might need a higher order Stokes wave theory for a more accurate result. However, the higher-order wave theory has not been fully implemented and validated in the employed library (Cheng et al., 2023 [9]). Thus, this study is based on the Second-order Stokes wave theory for convenience and consistency. The Second-order Stokes wave theory is employed for the wave generation in this study, where the wave elevation profiles are given by:

$$\eta \left(\phi \right)=\frac{H}{2}\cos \phi +\frac{\pi H^2}{8\lambda }\frac{\cosh kd}{{sinh}^{3}kd}(2+\cosh 2kd)\mathrm{c}\mathrm{o}\mathrm{s}2\phi$$

(22)

$$\phi =k\left(xcos\beta +ysin\beta \right)-\omega t$$

(23)

where H is the wave height, λ is the wavelength, d is the water depth, and k is the wave number.

Then, the extreme weather conditions, as shown in

Table 4. Environmental conditions for dynamic analysis.

E1	Uc = 0.46 m/s, Hs = 7.64 m, Tp = 12.13 s, θw = 0°
E2	Uc = 0.46 m/s, Hs = 6.36 m, Tp = 11.06 s, θw = 45°
E3	Uc = 0.46 m/s, Hs = 4.57 m, Tp = 8.10 s, θw = 90°

, are applied to study the effects of different mooring line configurations. Among these, working condition E1 is the maximum survival condition of the aquaculture platform, which represents the highest sea state that the aquaculture platform is designed to endure. In extreme weather conditions, the flow velocity Uc = 0.46 m/s, which corresponds to a return period of 50 years. The waves and currents are assumed to occur simultaneously and have the same direction. The Hs and Tp at different θw are from Figure 3. Since the offshore aquaculture structure is symmetric in the X and Y directions, the selected environmental load directions for calculations are 0°, 45°, and 90°. In the present study, the irregular waves are generated using the JONSWAP spectrum with the given Hs and Tp in Table 4. The peak enhancement factor (γ) is adapted according to whether Hs and Tp are applied in the simulations [28]. The equation for the JONSWAP spectrum is shown as Equation (3):

$$S(\mathsf{\omega })=(1-0.287\mathrm{l}\mathrm{n}(\mathsf{\gamma }))\frac{5}{16}\frac{{\mathsf{\omega }}_p^4}{{\mathsf{\omega }}^5}{H}_S^2\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{5}{4}\frac{{\mathsf{\omega }}_p^4}{{\mathsf{\omega }}^4}\right){\mathsf{\gamma }}^{\mathrm{e}\mathrm{x}\mathrm{p}\left[-0.5{\left(\frac{\mathsf{\omega }-{\mathsf{\omega }}_p}{\mathsf{\sigma }{\mathsf{\omega }}_p}\right)}^2\right]}$$

(24)

where ω_p = 2π/Tp is the angular spectral peak frequency, and σ = 0.07 when ω ≤ ω_p and 0.09 when ω > ω_p. The value of γ is determined according to Equation (4). The wave spectrum corresponding to E1–E3 is shown in Figure 10.

$$\gamma =\left\{\begin{array}{cc}5& Tp/\sqrt{Hs}\le 3.6\\ e^{(5.75-1.15Tp/\sqrt{Hs})}& 3.6<Tp/\sqrt{Hs}<5\\ 1& 5\le Tp/\sqrt{Hs}\end{array}\right.$$

(25)

3. Results and Discussion

3.1. Motion Responses under Regular Wave Conditions

3.1.1. When the Wave Heading Direction Is 0°

In regular wave conditions, the mooring line has a 4 × 1 configuration. The motion responses are calculated under 27 different regular wave conditions to study the hydrodynamic characteristics with respective wave heights, wave periods, and wave directions. In addition, the effect of nets is also investigated. In each condition, the responses are calculated with a duration of 30 wave cycles in order to reach the steady state. Figure 11 shows the time-series results for the motion responses in the six degrees of freedom of the main framework without net. As the wave heading angle is 0°, the motion in the sway, roll, and yaw is relatively small, compared to these in the other three directions. The time-series results for the heave motion seem a non-zero equilibrium, and this downwards motion in the vertical direction might be due to the combination of the wave drift force and the mooring forces. This motion can also be observed from the work by Cheng et al. (2023) [9].

Figure 12 shows the motion responses of the studied structures in the surge, heave, and pitch directions. The height of bar represents the mean value of the motion responses over the last five wave cycles. The error bars in the results represent the dynamic range of motion responses. Bars with lighter colors and red edges represent the case without nets, and bars with darker colors and black edges represent the case with nets. The same processing method is also used for the analysis in the following text.

According to Figure 12, the surge motions of the main framework without net are related to the wave height and wave period. The mean values of surge increase with the increasing wave height as well as the wave period. Regarding the dynamic range, the wave height has a negative effect, but the wave period can significantly affect the dynamic range. When the wave period is 7.39 s, the dynamic range has the smallest range compared to the other two wave periods. When nets are included in the dynamic analysis, the mean surge motion is significantly increased compared to the main framework under the same wave conditions. This is because the wave draft force increases significantly due to the net.

Additionally, the heave motion is also affected by the wave height and wave period. The mean values of heave motions are always negative due to the downwards force from the heave mooring lines. The mean heave motions of the main framework have a positive correlation with the wave height but are insensitive to the wave period. Meanwhile, the dynamic ranges of the heave motion have a positive correlation with the wave period but are insensitive to the weave height. When nets are included in the dynamic analysis, the dynamic range gets smaller compared to those of the main framework without net, but the mean heave motion has insignificant changes.

Figure 12 also shows the effect of the wave height and wave period on the pitch motion when the wave direction is 0°. The mean pitch motion of the main framework is insensitive to the wave height, and has a positive correlation with the wave period. The dynamic range of the pitch motion reduces with the decreasing wave height and wave period. When nets are included in the dynamic analysis, the mean and the dynamic range of the pitch motion get larger than the main framework without net.

3.1.2. When the Wave Heading Direction Is 90°

Figure 13 shows the effect of the wave height and wave period on the sway, heave, and roll responses when the wave direction is 90°. The main finds are similar to the results when the wave direction is 0° because of the geometrical properties. However, there are some exceptions, which will be discussed as follows. The mean values of the horizontal motions of the main framework, which is the sway motion when the wave direction is 90° and the surge motion when the wave direction is 0°, are similar under the same wave height and wave period. However, when nets are included, the increments of the mean horizontal motion are smaller when the wave direction is 90°. These smaller increments may be due to the smaller projected net area when the wave direction is 90°. The mean values of the heave motions of the main framework with and without nets have similar mean values when the wave directions are 90° and 0°. However, the dynamic ranges of the heave motions are slightly smaller when the wave directions change from 0° to 90°. The results of the roll motion in Figure 13 are also similar to those in Figure 12, but the mean value of the roll motion is positive rather than negative. In addition, the mean roll motion of the main framework increases with the increasing wave height when the wave period is 9.693 s.

3.1.3. When the Wave Heading Direction Is 45°

Figure 14 shows the motion responses in the six directions when the wave direction is 45°. The motions in the surge, sway, and heave directions are sensitive to the wave height. The mean values of these three translational motions increase with the increasing wave height. But the dynamic range almost keeps the same under different wave heights. Including nets in the dynamic analysis can significantly increase the mean horizontal motions, but will reduce the mean vertical motion. The dynamic range of the translational motions is not sensitive to the wave height but can be significantly affected by the wave periods. A shorter wave period can reduce the dynamic range of the translational motions. Including nets in the dynamic analysis can also reduce the dynamic range of the translational motions. This is because the nets can act as a viscous damper to the system, which will dissipate the wave excitation energy.

The mean roll and pitch motions are relatively insensitive to the wave height, but sensitive to the wave period. The mean yaw motion has a strong positive correlation with both the wave height and wave period. The dynamic range of the three rotational motions are insensitive to the wave height, but have a strong positive correlation with the wave period. Including nets in the dynamic analysis will increase the mean rotation motions as well as the dynamic range of the three rotational motions.

3.2. Tension Force under Regular Wave Conditions

3.2.1. When the Wave Heading Direction Is 0°

Figure 15 shows the time-series results of the mooring line tension in the four mooring lines under the N1 environmental condition. As the wave heading angle is 0°, the windward mooring lines (M1 and M2) clearly have a larger mean tension than the leeward mooring lines (M3 and M4).

Figure 16, Figure 17 and Figure 18 show the tension forces in the four mooring lines under different wave conditions. The height of bar represents the mean value of the motion responses over the last five wave cycles. The error bars in the results represent the dynamic range of tension forces. Bars with lighter colors and red edges represent the case without nets, and bars with darker colors and black edges represent the case with nets. Figure 16 shows that when the wave heading direction is 0°, the windward mooring lines (M1 and M2) are the main load carrier. The mean tensions in these two mooring lines are significantly larger than those in M3 and M4. When the net is not included in the dynamic analysis, the mean tensions in these windward mooring lines have a clear positive correlation with the wave height, but are insensitive to the wave period. Including nets in the dynamic analysis will significantly bring additional loads on the mooring lines, and the increments of the mean tension are sensitive to the wave periods. However, including nets can reduce the dynamic range of tension. This can also be explained by the viscous damper effect of the net structures.

3.2.2. When the Wave Heading Direction Is 90°

Figure 17 shows when the wave heading direction is 90°. At this direction, the windward mooring lines are M2 and M3. The main findings in Figure 17 are similar to those in Figure 16. However, the mean tension in the two windward mooring lines becomes larger when the wave heading direction changes from 0°to 90°. This is because when the wave heading direction is 90°, the projected area on the wave direction becomes larger than that of 0°. The larger projected area brings larger wave loads, and hereby larger tension on the windward mooring lines.

Figure 17. The effect of wave height and wave period on the tension responses when the wave direction is 90°.

Figure 17. The effect of wave height and wave period on the tension responses when the wave direction is 90°.

3.2.3. When the Wave Heading Direction Is 45°

Figure 18 shows when the wave heading direction is 45°. At this direction, the windward mooring lines are M1, M2, and M3. Because M2 is in line with the wave heading direction and located in the very front of the structure, the mean tension on M2 is much larger than the other two windward mooring lines. The mean tensions in M3 are slightly larger than those in M1, because the projected area of the structure has a larger portion on M3. The dynamic range of the tension decreases with the decreasing wave period, but is not sensitive to the wave height. Including nets can reduce the dynamic range of the tension, but can increase the mean value of tension.

Figure 18. The effect of wave height and wave period on the tension responses when the wave direction is 45°.

Figure 18. The effect of wave height and wave period on the tension responses when the wave direction is 45°.

3.3. Structural Responses under Extreme Weather Conditions

3.3.1. Motion Responses

Figure 19 shows the selected time-series results for the motion responses of Ningde No. 1 using the 4 × 2 mooring configuration under extreme weather condition E1. The zoom-in subplots give a closer look at the wave elevation, surge, heave, and pitch motions with a time duration of 500 s. Due to the current loads, a mean drift is observed in the surge direction. Figure 20 shows one instantaneous motion status of Ningde No. 1 under the N1 condition using the 4 × 2 mooring configuration. As observed from this figure, the main structure has a certain yaw motion to reduce the project area to the current and wave. The vertical distance between the lowest point of the framework and the seabed is more than 10 m, which is a safe margin to avoid the collision to the seabed.

Figure 21 shows the motion responses in the six degrees of freedom under different extreme weather conditions using different mooring configurations. In general, increasing the number of mooring lines can clearly suppress the mean horizontal motion, but have an insignificant effect on the vertical motion. This is because the vertical motion is mainly dominated by the water plane area and the mass of the structure; however, these two factors do not have significant changes with the number of mooring lines. The pitch and roll motions are also insensitive to the number of mooring lines, as the restoring moment is mainly from the buoyancy forces.

3.3.2. Tension Force in Mooring Lines

Figure 22 shows the effects of different mooring configurations on tension responses under different extreme weather conditions. The results show that the maximum value in the mooring lines can significantly reduce with increasing the number of mooring lines, especially when the mooring system changes from 4 × 1 to 4 × 2. However, when the mooring line continues to increase to 4 × 3, the extreme tension does not have clear reductions. Moreover, as discussed in Section 3.3.1, increasing the number of mooring chains brings little depression on the rotational motions and the heave motion. Thus, with the considerations of cost, the 4 × 2 mooring configuration might be the most cost-effective design for controlling the motions of Ningde No. 1.

4. Conclusions

A new seabed contact module is implemented in the open-source numerical library for modeling the global responses of offshore aquaculture structures with centenary mooring lines subjected to waves and currents in the present study. The dynamic responses of the semisubmersible offshore aquaculture structure Ningde No. 1 are investigated thoroughly under regular waves and combined wave–current conditions.

This study shows that it is feasible to use Code_Aster to study the hydrodynamic responses of large offshore aquaculture structures together with mooring systems. Based on numerical results, the following conclusions are drawn for this study:

(1)

When nets are included in the analysis, the mean motion response will increase, but the dynamic ranges of response are reduced.

(2)

When the wave heading angle is 0°and 90°, the horizontal motion (sway and surge) of the main framework is positively correlated with the wave height and wave period.

(3)

The heave motion of the main frame is positively correlated with the wave height, but is insensitive to the wave period.

(4)

The rotational motion (roll, pitch, and yaw) of the main frame is not sensitive to the wave height but is sensitive to the wave period.

(5)

The presence of fishing nets significantly increases the average values of the horizontal and rotational motion of the main frame but decreases the average value of its heave motion.

(6)

Increasing the number of mooring lines can reduce the mean and extreme tension in individual mooring lines and suppress the horizontal motions, but bring neglectable effects on the rotational responses.