Numerical Simulation and On-Site Measurement of Dynamic Response of Flexible Marine Aquaculture Cages

Abstract

Flexible cages are widely used in marine aquaculture, yet their mechanical features in extreme seas are still unclear. This study proposes a numerical algorithm to solve the coupled response of the multiple cage systems. The net and mooring lines are modeled using the lumped-mass model, while the flexible floating collar system is assessed with the large-deformation FEM model, and the two models are coupled through an iterative scheme. Sea trials are conducted, and the motion of the cage is obtained using an image processing technique, which validates the numerical algorithm. Using the proposed numerical algorithm, a series of simulations are performed to investigate the response of flexible cages in extreme seas. Motions, line tensions, and structural sectional forces are studied, and the effects of factors such as the wavelength of incident waves and the diameter of collar pipes are investigated.

1. Introduction

According to The State of World Fisheries and Aquaculture [1], published by the Food and Agriculture Organization of the United Nations in 2022, global fisheries and aquaculture have reached a record of 214 million tons, comprising 178 million tons of aquatic animals and 36 million tons of algae, which is largely due to the growth of marine aquaculture. Marine aquaculture is becoming increasingly important as it provides a solution to issues such as poverty and hunger, especially for countries with limited and barren land. To conduct aquaculture at sea, one requires safe and reliable equipment. The flexible cage depicted in Figure 1 is widely used in marine aquaculture, and its safety under common sea states has been proved after years of usage. However, in extreme sea states, the flexible cage may experience severe movement and structural failure, which can lead to the death of the cultured organisms and economic losses.

Typically, a flexible cage is composed of a floating collar system, a net, and mooring lines. Usually, research into the mechanical features of flexible cages mainly focuses on these three aspects. Research on the net started early. Various researchers studied the fluid load on the net, its deformation and motion, and the flow field around the net by means of theoretical analysis, numerical simulations, model tests, and on-site measurements. Guo et al. [2] presented a thorough review of the research methods and achievements concerning the nets of flexible cages. In terms of theoretical models, some researchers established the wave-net coupling equations in which the ocean waves were treated as small-amplitude linear waves and the net as the porous membrane structure, and analytical or semi-analytical solutions were derived using the expansion of the characteristic function [3,4,5]. In terms of model tests, the analogical law for the net was established under wave and current conditions. Some measurement methods for the deformation of the net were developed based on image processing technology, and extensive model tests were performed on various forms of nets [6,7,8]. In terms of numerical simulation, a method known as the lumped-mass model was proposed, which treated the net as lumped-mass nodes connected with zero-mass springs. Since a real net may contain millions of small lines, leading to many degrees of freedom (DOFs) during numerical simulation, a simplification method was proposed to reduce the number of DOFs by replacing multiple small meshes with a single large one [6,9]. As for the calculation of fluid load, the Morison Model and the Screen Model are the most widely used methods in the literature. The former separates the net into multiple lines and calculates the fluid load upon each line using the Morison Equation. The latter treats the net as an entire porous plane and calculates its fluid load by considering the influence of porosity. In recent years, a coupled CFD-porous-medium model has been increasingly adopted which solves the flow field using Computational Fluid Dynamics (CFD) technology and the net with a porous medium model [10,11]. In this model, the fluid load on the net is obtained through CFD simulation, while the shape of the porous medium is determined by calculating the deformation of the net using models such as the lumped-mass model. The CFD-porous-medium model is by far the most accurate method of net simulation as it considers the interactions between the fluid and the net. However, this model requires extensive computational efforts as iterations must be performed between the fluid solver and the net solver. Differing from the theoretical method, the numerical method, and the model test method, research concerning on-site observations remains rare. This is due to the complexity of the sea environment, and there is still a shortage of effective measurement methods for the motion and deformation of the nets. Recently, some researchers have used instruments such as Doppler acoustic sensors to conduct on-site measurements of the flow velocity distribution around the net and the net deformation [12,13,14]. However, more on-site data are needed to reveal the behavior of aquaculture cages under various sea conditions.

In recent years, with the advancement of numerical and test technology, research on the coupled systems comprising the floating collar structure, the net, and the mooring lines has been on the rise. Xu and Qin [15] provided a comprehensive review of the common research methods and recent findings. In these studies, the lumped-mass model is often used to solve the net and the mooring lines. Meanwhile, the Finite Element Method (FEM) is usually utilized to solve the flexible floating collar structure. Fluid loads are calculated using the time-domain/frequency-domain potential flow theory or the Morison Equation. The large-amplitude motion equations of rigid or elastic bodies are established and solved [16,17]. To improve the accuracy of fluid load calculation and to evaluate the influence of the cage on the flow field, some studies have established the coupled solution method of the CFD solver and the cage structural solver. This method evaluated the influence of the net upon the flow field based on a porous medium model and applied the obtained fluid load to the structural solver to solve the dynamic response of the cage [18]. Precise numerical simulations are challenging due to the complex turbulent flow behind the cage and the significantly nonlinear deformation of the net and the floating collar. Therefore, some researchers investigated the motion responses of the entire cage systems by conducting model tests and on-site measurements, whose data were meanwhile used to validate the accuracy of the numerical simulations [19,20,21]. Studies indicated that the load and damping effects of the net significantly influenced the motion responses and stress distributions of the cages [18,22,23]. The flexibility of the floating collar could reduce its stress and alter the distribution of the tension of mooring lines [16]. In addition, factors such as biological fouling, the focusing position of the wave train, and the fish distribution inside the cage also had an impact on the dynamic response of the cage [24]. It is worth mentioning that some researchers have developed stress and deformation prediction models by artificial neural networks and machine learning techniques [25], which offer a way of rapid prediction of dynamic responses of cages. Although existing research has achieved certain progress in numerical simulations and model tests, there are still challenges in simultaneously accounting for the multi-system coupling and the large-deformation effect of the flexible floating collars. Moreover, more on-site measured data are needed to study the dynamic features of the cages and verify the accuracy of the numerical methods.

This article considers the coupled effects of the net, the mooring lines, and the large-deformed floating collar and proposes a numerical algorithm solving the dynamic response of the coupled system based on the FEM of large-deformed beams and the lumped-mass model. Measurements of the motion of a flexible cage are conducted at sea. The measured results are compared against the numerical ones to validate the accuracy of the numerical method. Numerical simulations are performed to obtain the dynamic response of flexible cages under extreme sea conditions. The effects of the wavelength of incident waves, the stiffness of floating collars, and other factors on the deformation of the cage, the tension of mooring lines, and the sectional forces of the structures are discussed.

The content of the article is organized as follows. Section 2 proposes the numerical method, including the basic models of mooring lines, net, floating collar structures, and fluid load. Section 3 gives the details of the sea trial, which contains information on the measured cage, the equipment and the measurement method. In Section 4, the numerical model and the simulation cases are introduced, and the numerical method is verified against the on-site measurement. Section 5 gives the simulation results and analyzes the features of cage motion, line tension, and structural sectional force concerning the influence of the incident wavelength and the stiffness of floating collars. The main conclusions of the article are reported in Section 6.

2. Numerical Method

2.1. Basic Equations

An aquaculture cage is composed of the net, the mooring lines, the floating collar structures, etc. The numerical method in this article considers the coupling effects of these parts. The net and mooring lines are regarded as one-dimensional lines that only bear tension and have no bending stiffness. The stress state of the net and mooring lines is illustrated in Figure 2a. Assuming t represents time, and the arc length coordinate of the line is s with deformation and $\tilde{s}$ without deformation, then the motion of the lines satisfies Equations (1)–(3).

$${\displaystyle \frac{\partial }{\partial s}}\left(T_n{\displaystyle \frac{\partial {\mathit{r}}_n}{\partial s}}\right)+{\mathit{q}}_n={\rho }_n{\displaystyle \frac{{\partial }^2{\mathit{r}}_n}{\partial t^2}}$$

(1)

$${\displaystyle \frac{\partial {\mathit{r}}_n}{\partial s}}·{\displaystyle \frac{\partial {\mathit{r}}_n}{\partial s}}=1$$

(2)

$$\mathrm{d}s=\left(1+{\displaystyle \frac{T_n}{E{A}_n}}\right)\mathrm{d}\tilde{s}$$

(3)

where ${\mathit{r}}_n$ is the shape curve of the lines, $T_n$ is the sectional tension of the lines, ${\mathit{q}}_n$ is the distributed external load per unit length, ${\rho }_n$ is the density of the lines (i.e., mass per unit length), and $E{A}_n$ represents the sectional tensile stiffness of the lines. Equations (1)–(3) are the momentum equation, the geometric equation, and the constitutive equation, respectively. With a total of 5 unknowns (i.e., $T_n$, s, ${\mathit{r}}_n$) and 5 equations, the equations are closed and solvable.

The floating collars are made of high-density polyethylene (HDPE) pipes and, therefore, can be regarded as deformable curved beams capable of bearing section bending moment due to the good flexibility of the HDPE material. The deformed floating collar is separated into multiple small beams connected end to end, as shown in Figure 2b. A local coordinate system is established for each small beam, where coordinate axis 1 is along the axis direction of the beam and coordinate axes 2/3 are perpendicular to axis 1. Based on the Euler-Bernoulli beam theory, the following equations are established in the local coordinate system of each beam.

$${\rho }_p{\displaystyle \frac{{\partial }^2{\nu }_1}{\partial t^2}}=E{A}_p{\displaystyle \frac{{\partial }^2{\nu }_1}{\partial x^2}}+q_{p1}$$

(4)

$${\rho }_p{\displaystyle \frac{{\partial }^2{\nu }_2}{\partial t^2}}+E{I}_p{\displaystyle \frac{{\partial }^4{\nu }_2}{\partial x^4}}=q_{p2}$$

(5)

$${\rho }_p{\displaystyle \frac{{\partial }^2{\nu }_3}{\partial t^2}}+E{I}_p{\displaystyle \frac{{\partial }^4{\nu }_3}{\partial x^4}}=q_{p3}$$

(6)

$$I_R{\displaystyle \frac{{\partial }^2{\theta }_1}{\partial t^2}}=G{J}_p{\displaystyle \frac{{\partial }^2{\theta }_1}{\partial x^2}}$$

(7)

where ν = [ν₁, ν₂, ν₃]^T and θ = [θ₁, θ₂, θ₃]^T are the translational and rotational displacement of the small beam relative to the initial position, ${\rho }_p$ and $I_R$ are the linear density and the cross-section rotational moment of inertia of the small beam. $E{A}_p$, $E{I}_p$ and $G{J}_p$ are the tensile stiffness, bending stiffness, and torsional stiffness of the small beam, respectively (It should be noted that the floating collar has a tubular section with the same bending stiffness in all directions). q_p = [q_p1, q_p2, q_p3]^T represents the distributed external load on the small beam. Equation (4) governs the axial motion of the floating collar. Equations (5) and (6) denote the bending motion concerning localized y and z axes, and Equation (7) is the torsional motion equation.

2.2. Environmental Load

The net, mooring lines, and floating collars are subjected to static loads such as gravity and buoyancy, as well as dynamic loads from fluid. Since the net, mooring lines, and floating collars are all slender rod-shaped structures, the Morison Equation is employed to calculate the hydrodynamic load, which is expressed as follows.

$$F_n=\rho V{\dot{u}}_n+C_{an}\rho V\left({\dot{u}}_n-{\dot{u}}_{bn}\right)+{\displaystyle \frac{1}{2}}{C}_{dn}\rho A\left(u_n-u_{bn}\right)\left|u_n-u_{bn}\right|$$

(8a)

$$F_t=\rho V{\dot{u}}_t+C_{at}\rho V\left({\dot{u}}_t-{\dot{u}}_{bt}\right)+{\displaystyle \frac{1}{2}}{C}_{dt}\rho A\left(u_t-u_{bt}\right)\left|u_t-u_{bt}\right|$$

(8b)

where $F_n$ and $F_t$ are the hydrodynamic forces perpendicular to and tangent to the rod, respectively, ρ is the fluid density, V is the immerged volume, and A is the projected area along the flow direction. ${\dot{u}}_n$ and $u_n$ are the acceleration and velocity of the fluid in the direction perpendicular to the rod, ${\dot{u}}_{bn}$ and $u_{bn}$ are the acceleration and velocity of the rod in the direction perpendicular to itself, $C_{an}$ and $C_{dn}$ are the added-mass coefficient and drag coefficient in the direction perpendicular to the rod, respectively. Similarly, ${\dot{u}}_t$ and $u_t$ are the acceleration and velocity of the fluid in the direction tangent to the rod, ${\dot{u}}_{bt}$ and $u_{bt}$ are the acceleration and velocity of the rod in the direction tangent to itself, $C_{at}$ and $C_{dt}$ are the added-mass coefficient and drag coefficient in the direction tangent to the rod, respectively. By introducing added mass $m_{addn}=C_{an}\rho V$, $m_{addt}=C_{at}\rho V$ and added damping $c_{addn}={\displaystyle \frac{1}{2}}{C}_{dn}\rho A\left|u_n-u_{bn}\right|$, $c_{addt}={\displaystyle \frac{1}{2}}{C}_{dt}\rho A\left|u_t-u_{bt}\right|$, the Morison Equation can be transformed into

$$F_n=\left(1+C_{an}\right)\rho V{\dot{u}}_n+{\displaystyle \frac{1}{2}}{C}_{dn}\rho A\left|u_n-u_{bn}\right|u_n-m_{addn}{\dot{u}}_{bn}-c_{addn}{u}_{bn}$$

(9a)

$$F_t=\left(1+C_{at}\right)\rho V{\dot{u}}_t+{\displaystyle \frac{1}{2}}{C}_{dt}\rho A\left|u_t-u_{bt}\right|u_t-m_{addt}{\dot{u}}_{bt}-c_{addt}{u}_{bt}$$

(9b)

where the first two terms on the right side of the equation represent the forces exerted by fluid motion, while the last two terms represent the forces exerted by the motion of the rod itself (i.e., the added-mass force and added damping force).

The Airy wave theory is adopted for the determination of fluid velocity and acceleration. For a monochromatic wave, the flow velocity potential Φ and surface elevation η are

$$\mathsf{\Phi }=-a{\displaystyle \frac{ig}{\omega }}{\displaystyle \frac{\cosh k\left(z+h\right)}{\cosh kh}}{e}^{i\left(k_{x}x+k_{y}y-\omega t\right)}$$

(10)

$$\eta =a{e}^{i\left(k_{x}x+k_{y}y-\omega t\right)}$$

(11)

where $a$ is the wave amplitude, k is the wave number, k_x and k_y are the wave number components along the x-axis and y-axis, ω is the circular frequency, h is the water depth, $g$ is the gravitational acceleration, and i represents the imaginary unit.

For irregular waves, the sea surface is formed by the superposition of a series of monochromatic waves with random phases. The amplitude of the component waves is described by the wave spectrum function, and the wave surface expression is

$$\eta =\sum _j\sqrt{2S\left({\omega }_j\right)\mathrm{\Delta }\omega }{e}^{i\left(k_{xj}x+k_{yj}y-{\omega }_{j}t+{\theta }_j\right)}$$

(12)

where ${\omega }_j$ is the circular frequency of the j-th component wave, $k_{xj}$ and $k_{yj}$ are the wave number of the j-th component wave along the x-axis and y-axis, ${\theta }_j$ is the random phase of the j-th component wave. $S\left({\omega }_j\right)$ is the value of the spectral function corresponding to the j-th component wave, and $\Delta \omega$ is the circular-frequency interval of the wave components.

2.3. Environmental Model

The lumped-mass model is widely used to solve the equations of net and mooring lines, i.e., Equations (1)–(3). As is shown in Figure 3, the main idea of the lumped-mass model is to decompose one line into several elements. The mass and external force of each element is concentrated on its two nodes, and the elastic property of the element is represented by a zero-mass linear straight spring.

As the lumped-mass model is already widely used in the literature, the matrix equation of the model, i.e., Equation (13), is given without further discussion.

$$\left(\left[m_n\right]+\left[m_{nadd}\right]\right)\left\{{\ddot{\mathit{x}}}_n\right\}+\left(\left[c_n\right]+\left[c_{nadd}\right]\right)\left\{{\dot{\mathit{x}}}_n\right\}=\left\{{\mathit{F}}_{no}\right\}+\left\{{\mathit{F}}_{ni}\right\}$$

(13)

where $\left[m_n\right]$ is the nodal mass matrix, $\left[c_n\right]$ is the nodal damping matrix, $\left[m_{nadd}\right]$ and $\left[c_{nadd}\right]$ are the added mass and added damping matrix generated by the relative motion between fluid and lines, $\left\{{\mathit{x}}_n\right\}$ is the nodal position vector, $\left\{{\mathit{F}}_{no}\right\}$ is the nodal external load vector, and $\left\{{\mathit{F}}_{ni}\right\}$ is the internal force vector induced by internal line tensions.

The FEM is applied to solve the equation of the floating collar system. Within a FEM model, the matrix equation of the floating collar system is written as follows.

$$\left(\left[m_p\right]+\left[m_{padd}\right]\right)\left\{{\ddot{\mathit{x}}}_p\right\}+\left(\left[c_p\right]+\left[c_{padd}\right]\right)\left\{{\dot{\mathit{x}}}_p\right\}=\left\{{\mathit{F}}_{po}\right\}+\left\{{\mathit{F}}_{pi}\right\}$$

(14)

where $\left[m_p\right]$ is the nodal mass matrix, $\left[c_p\right]$ is the structural damping matrix, $\left[m_{padd}\right]$ and $\left[c_{padd}\right]$ are the added mass and damping matrix induced by the relative motion between the fluid and the floating collar system, $\left\{{\mathit{x}}_p\right\}$ represents the nodal displacement vector, $\left\{{\mathit{F}}_{po}\right\}$ is the external environmental load vector, and $\left\{{\mathit{F}}_{pi}\right\}$ represents the nodal internal force vector of nodes generated by the deformation of the floating collar (i.e., sectional force). Generally, in a conventional FEM matrix equation, there is no such internal force term on the right side but a stiffness term $\left[k_p\right]\left\{{\mathit{x}}_p\right\}$ on the left side, where $\left[k_p\right]$ is the stiffness matrix. Although Equation (14) (or Equation (13)) appears to be inconsistent with the typical FEM equation, they are essentially the same. This is because the internal force $\left\{{\mathit{F}}_{pi}\right\}$ equals $-\left[k_p\right]\left\{{\mathit{x}}_p\right\}$ within the linear elasticity theory. However, the large deformation of the net and the floating collar in this article leads to a nonlinear relationship between the internal sectional force and the nodal displacement. Therefore, it is more intuitive to use the internal force $\left\{{\mathit{F}}_{pi}\right\}$ instead of $\left[k_p\right]\left\{{\mathit{x}}_p\right\}$.

Equations (13) and (14) are in the same form and can be solved by the same numerical algorithm, which facilitates the development of computer programs. Taking Equation (14) as an example, a 4-order implicit Houbolt scheme is adopted for time integral, which can be written as

$$\left\{{\ddot{\mathit{x}}}_p^{t+\Delta t}\right\}={\alpha }_1\left(2\left\{{\mathit{x}}_p^{t+\Delta t}\right\}-5\left\{{\mathit{x}}_p^t\right\}+4\left\{{\mathit{x}}_p^{t-\Delta t}\right\}-\left\{{\mathit{x}}_p^{t-2\Delta t}\right\}\right), {\alpha }_1={\displaystyle \frac{1}{\Delta t^2}}$$

(15)

$$\left\{{\dot{\mathit{x}}}_p^{t+\Delta t}\right\}={\alpha }_2\left(11\left\{{\mathit{x}}_p^{t+\Delta t}\right\}-18\left\{{\mathit{x}}_p^t\right\}+9\left\{{\mathit{x}}_p^{t-\Delta t}\right\}-2\left\{{\mathit{x}}_p^{t-2\Delta t}\right\}\right), {\alpha }_2={\displaystyle \frac{1}{6\Delta t}}$$

(16)

Introducing the transient stiffness matrix $\left[k_p^t\right]$ at time t, which represents the change rate of the internal force with respect to nodal displacement, then

$$\left\{{\mathit{F}}_{pi}^{t+\Delta t}\right\}=\left\{{\mathit{F}}_{pi}^t\right\}-\left[k_p^t\right]\left(\left\{{\mathit{x}}_p^{t+\Delta t}\right\}-\left\{{\mathit{x}}_p^t\right\}\right)$$

(17)

Substitute Equations (15)–(17) into Equation (14), we have

$$\left[A\right]\left\{{\mathit{x}}_p^{t+\Delta t}\right\}=\left\{\mathit{b}\right\}$$

(18)

where

$$\left[A\right]=2{\alpha }_1\left(\left[m_p\right]+\left[m_{padd}\right]\right)+11{\alpha }_2\left(\left[c_p\right]+\left[c_{padd}\right]\right)+\left[k_p^t\right]$$

(19)

$$\begin{array}{cc}\hfill \left\{\mathit{b}\right\}& ={\alpha }_1\left(\left[{\mathit{m}}_p\right]+\left[{\mathit{m}}_{padd}\right]\right)\left(5\left\{{\mathit{x}}_p^t\right\}-4\left\{{\mathit{x}}_p^{t-\Delta t}\right\}+\left\{{\mathit{x}}_p^{t-2\Delta t}\right\}\right)\hfill \\ & +{\alpha }_2\left(\left[c_p\right]+\left[c_{padd}\right]\right)\left(18\left\{{\mathit{x}}_p^t\right\}-9\left\{{\mathit{x}}_p^{t-\Delta t}\right\}+2\left\{{\mathit{x}}_p^{t-2\Delta t}\right\}\right)\hfill \\ & +\left\{{\mathit{F}}_{po}^{t+\Delta t}\right\}+\left\{{\mathit{F}}_{pi}^t\right\}+\left[k_p^t\right]\left\{{\mathit{x}}_p^t\right\}\hfill \end{array}$$

(20)

2.4. Coupling of Multiple Systems

As mentioned before, the net and mooring lines satisfy the same equations and hence can be solved in a single numerical solver, in which both the net and the mooring lines are treated as mass and springs within the framework of the lumped-mass model. However, the floating collar system is solved by a different solver which uses beam elements within the framework of FEM. In this article, the two solvers are coupled by an iteration scheme shown in Figure 4 to solve the entire problem. As is shown in Figure 4, the iteration scheme repeats, correcting the interaction force {${\mathit{F}}_{couple}$} between the net/mooring lines and the floating collar system until the displacements of the two systems at their connecting nodes (i.e., {${\mathit{x}}_{couple\_line}$} and {${\mathit{x}}_{couple\_beam}$}) equal each other. To ensure {${\mathit{x}}_{couple\_line}$} draws closer to {${\mathit{x}}_{couple\_beam}$} after each iteration, the correction of the interaction force (i.e., {$\mathsf{\Delta }{\mathit{F}}_{couple}$}) is determined such that {${\mathit{x}}_{couple\_line}$} or {${\mathit{x}}_{couple\_beam}$} becomes {${\mathit{x}}_{couple\_line}+{\mathit{x}}_{couple\_beam}$}/2 after a single iteration step.

With the coupled iteration algorithm, the entire solution scheme is illustrated in Figure 5. The entire numerical solution starts with the generation of nodes and elements from geometry data, which prepares basic variables for the lumped-mass model and the FEM model. In the second step, matrices and vectors such as $\left\{{\mathit{x}}_n\right\}$ and $\left\{{\mathit{x}}_p\right\}$ are initialized. In the third step, the environmental parameters and system status at time t are calculated, which involves the fluid velocity and acceleration, the submerged area and volume of the cage structure, etc. Then, the matrix equations of the lumped-mass model and the FEM model are established by computing matrices and vectors such as $\left[m_n\right]$, $\left[m_{nadd}\right]$, $\left[c_n\right]$, $\left[c_{nadd}\right]$, $\left[m_p\right]$, $\left[m_{padd}\right]$, $\left[c_p\right]$, $\left[c_{padd}\right]$, $\left\{{\mathit{F}}_{no}\right\}$, $\left\{{\mathit{F}}_{ni}\right\}$, $\left\{{\mathit{F}}_{po}\right\}$, $\left\{{\mathit{F}}_{pi}\right\}$, etc. The iteration for the multi-system solution shown in Figure 4 is performed in the next step, and the displacements $\left\{{\mathit{x}}_n\right\}$ and $\left\{{\mathit{x}}_p\right\}$ at time t + Δt are hence obtained. If the final time-step number is reached, then the solution is completed. Otherwise, it goes back to the third step and repeats the calculation.

3. Sea Trial

3.1. Tested Cage

A circular HDPE flexible cage shown in Figure 6a, which is widely used in marine aquaculture, is tested in the sea trial. The main dimensions of the cage are listed in

Table 1. Principal parameters of the cage.

Parameter	Value	Parameter	Value
Perimeter of the outer collar (m)	92	Width of collar connector (mm)	250
Perimeter of the inner collar (m)	88	Shell thickness of collar connector (mm)	20
Diameter of the collar pipe (mm)	315	Diameter of the baluster pipe (mm)	110
Thickness of the collar pipe (mm)	23.2	Thickness of the baluster pipe (mm)	10
Space between two collars (mm)	650	Diameter of the handrail pipe (mm)	95
Size of each web (mm)	35 × 35	Thickness of the handrail pipe (mm)	8.2
Diameter of the web line (mm)	2	Height of the handrail (mm)	800
Depth of the entire net (m)	8	Diameter of the mooring line (mm)	35
Material of the net	PE	Length of each mooring line (m)	64
Number of sinkers on the net	36	Material of the mooring line	PE
In-water weight of each sinker on the net (kg)	20	Length of the projection of a mooring line on the water plane (m)	55

. Figure 6b shows the arrangement of mooring lines with their numbers, where XOY is the water plane and the x-axis is along the main flow direction (i.e., the current direction). A total number of 10 mooring lines are used, with 8 lines arranged forward and backward in the main flow direction and 2 lines arranged laterally. The angle between Line 1 and the x-axis is 30 deg, and that between Line 2 and the x-axis is 10 deg. The direction of other mooring lines can be easily obtained by noting that the mooring system is symmetric with respect to both the x-axis and y-axis.

It should be mentioned that the net in our sea trial consists of two parts: the above-water part and the underwater part. The underwater part, which is the most important part of the net, is installed on the inner collar of the cage and offers the volume for fishing. The above-water part, whose function is to prevent fish from jumping out, is installed between the inner collar and the handrail. Since the underwater part is the dominant load-bearing component, only the underwater net is considered in the numerical model. The site of the sea trial is located at the northern sea of Nanri Island, Fujian Province, China, with a water depth of 16 m. The sea trial was conducted on the afternoon of 8 December 2021. Red and green balls are placed on the cage as track points, which are shown in Figure 6a. A long-focus camera was set up on the shore to record the motion of the track points. The trajectories of the track points were obtained based on image processing technology. The locations of the cage and the camera are shown in Figure 7a, and the numbering of track points is shown in Figure 7b.

3.2. Instruments

The acoustic wave and current profiler (AWAC) was applied to record wave and current data of the sea. The main parameters of the AWAC are presented in

Table 2. Parameters of the acoustic wave and current profiler.

Picture	Parameter	Value
	Acoustic frequency	≥400 KHz
	Acoustic beams	one vertical, three slanted at 25°
	Beam opening angle	1.7°
	Maximum depth	100 m
	Wave height range	−15 to +15 m
	Wave accuracy (Hs)	<1% of measured value/1cm
	Wave accuracy (Dir)	2°
	Wave period range	0.5 to 30 s
	Current velocity range	±10 m/s horizontal, ±5 m/s along beam
	Current accuracy	1% of measured value ±0.5 cm/s

.

The motion of track points was recorded by a long-focus camera whose main parameters are shown in

Table 3. Parameters of the long-focus camera.

Picture	Parameter	Value
	Pixel	2 to 5 million
	Camera components	1/3″ CMOS
	Rotation angle	horizontal 355°, vertical 105°
	Infrared distance	60~120 m
	Optical zoom	30×
	Focal length	4~96 mm
	WIFI	Support, 802.11b/g/n
	Low illumination	0.09 and 0.5Lux@F1.6
	Electronic shutter	1/50~1/10,000s

.

3.3. Measurement Method

Figure 8 illustrates the processing procedure of the videos recorded by the long-focus camera. Since the frame rate of the videos is 25.0 frames per second, hence the same sampling frequency is used in the motion curves. First, an image processing program is applied to the original video to capture the positions of all track points, resulting in a series of raw pixel data in the stepped format. Second, a moving average filter is used to switch the stepped pixel data into smoothed curves. The expression of the filter algorithm is $X_{js}=\left({{\displaystyle \sum }}_{i=1}^b{X}_{j-i}+X_j+{{\displaystyle \sum }}_{i=1}^b{X}_{j+i}\right)/\left(2b+1\right)$, where $X_j$ is the raw data to be filtered, b is the half span of the filtering which is taken as 5 during the video processing. Third, the smoothed pixel curves are converted into the actual positions based on the mapping relationship between image coordinates and world coordinates. It can be found that the resulting curves shift downward or upward as time progresses. This is due to tidal level fluctuations causing the cage to shift up or down in the video graph. Therefore, by performing mean position compensation upon the original curve according to the sea-level shift caused by the tide, the final motion curves of track points are obtained in the last step. It should be mentioned that the tidal data were extracted from the observations of a nearby tide station.

4. Numerical Simulation

4.1. Simulation Model

A simulation model is established based on the circular HDPE cage used in the sea trial. The principal parameters of the cage can be found in Table 1. Figure 9 shows the simulation model of the floating collar system and the net. For the FEM model of the floating collar system, the element number is 288, and the node number is 216, resulting in a total DOF of 216 × 6 = 1296. There are 10 mooring lines, each of which is divided into 20 elements (21 nodes). The total number of mooring line elements is 10 × 20 = 200, with a DOF number of 21 × 10 × 3 = 630. As mentioned before, the number of lines in an actual net is enormous, and involving all lines of the actual net within the numerical model will lead to massive DOFs and make the simulation extremely time-consuming. Therefore, this article adopts a widely used technique known as clustering to reduce the scale of the solution. The idea of clustering is to use a large mesh to replace a set of small meshes. The principle is that the net before and after clustering are equal in mass, buoyancy, drag force, etc. Since the validity and effectiveness of clustering have already been proved by many works (a review of these works is given by Fan et al. [26]), the details of clustering are not introduced here. Anyway, after clustering, there are 544 lines and 277 knots in the net model. With each line divided into 4 elements, the total element number of the net model is 544 × 4 = 2176, and the DOF number is (277 + 544 × 3) × 3 = 5727.

The elongation rate ε, diameter d and tension T of net lines (mooring lines) satisfy the relationship $T=C_1{d}^2{\epsilon }^{C_2}$ [27]. According to Klust’s experimental results [28], the elastic coefficients are C₁ = 345.37 × 10⁶ N/m² and C₂ = 1.0121 for PE lines. Here C₂ ≈ 1 is used for the sake of convenience. According to these formulas, the tensile stiffness of mooring lines and net lines are 4.476 × 10⁵ N and 1.382 × 10³ N, respectively. Other parameters are listed in Table 1. The normal drag coefficient of the net line is taken as 1.2 based on the study of Zhan et al. [29] and Bessonneau and Marichal [30]. The tangent drag coefficient of the net line is taken as 0.1, according to Bessonneau and Marichal [30] and Zhu et al. [31]. Alternatively, for Reynolds numbers within the range of 30 < Re < 2.33 × 10⁵, the formulas proposed by DeCew et al. [32] can be used, although not employed in this paper: $C_{dn}=1.1+4R{e}^{-0.5}$, $C_{dt}=\pi \mu \left(0.55R{e}^{1/2}+0.084R{e}^{2/3}\right)$. Mousselli [33] provided the drag coefficient of pipes at different Reynolds numbers, which is listed in

Table 4. Drag coefficient of pipes varying with Reynolds number.

Re	<5 × 103	5 × 103 < Re < 1 × 105	1 × 105 < Re < 2.5 × 105	2.5 × 105 < Re < 5 × 105	>5 × 105
Cd	1.3	1.2	1.53-Re/(3 × 105)	0.7	0.7

. As the Reynolds number of the floating collar is greater than 2.5 × 10⁵ and the Reynolds number of the mooring line is within the range of 5 × 10³ < Re < 1 × 10⁵, the normal drag coefficients of the two components are 0.7 and 1.2, respectively. The tangent drag coefficient is 0.4 for the floating collar [34] and 0.045 for the mooring line [31]. The normal added-mass coefficient is 0.2 for the floating collar [34] and 1.0 for both the net line and the mooring line [31]. The tangent added-mass coefficient of all three components (i.e., the net line, the mooring line, and the floating collar) is 0. Studies have shown that the front meshes can provide a shielding effect to the rear ones. The shielding coefficient is set to be 0.8 according to the research of Su and Zhan [35]. The time interval length of the numerical simulation is 0.02 s. It should be noted that a convergence test is performed with smaller time intervals (0.015 s and 0.01 s). The deviations (in both force and displacement) between results calculated using different time intervals are smaller than 0.5%, which exhibits a good convergence behavior.

4.2. Validation

Based on the sea trial state and the numerical method proposed in Section 2, a numerical simulation is performed to obtain the motion of the cage. Figure 10 shows the measured and numerical results of the motion curves of all track points. It is found that the numerical results agree well with those measured during the sea trial.

Figure 11 employs wavelet transform to derive the time–frequency energy spectrums of all track points. The wavelet transform used here, also known as Gabor Transformation [36], is written as $G\left(t,\omega \right)={\displaystyle \int }f\left(t^{*}\right)g\left(t^{*}-t\right)e^{-i\omega t^{*}}\mathrm{d}{t}^{*}$ where the window function is $g\left(t\right)=1/\sqrt{4\pi \alpha }{e}^{-t^2/4\alpha }$ and the window width parameter $\alpha$ equals 3 here. It is evident that the energy peaks appear intermittently and randomly, yet the frequencies concerning the energy peaks remain stable at around 1 rad/s. The numerical results are in good agreement with the measured ones, which substantiates the validity of the numerical method.

Table 5. The average values Z_mean, significant values Z_1/3, and 1/10 maximum values Z_1/10 of the displacement oscillation amplitudes of all track points.

Track Point No.		1	2	3	4	5	6	7
Sea trial (m)	Zmean	0.23	0.22	0.19	0.19	0.20	0.20	0.19
	Z1/3	0.35	0.34	0.31	0.30	0.31	0.32	0.31
	Z1/10	0.45	0.44	0.40	0.39	0.40	0.40	0.40
Numerical simulation (m)	Zmean	0.24	0.21	0.19	0.19	0.21	0.20	0.18
	Z1/3	0.37	0.33	0.29	0.30	0.32	0.31	0.29
	Z1/10	0.48	0.43	0.36	0.38	0.39	0.38	0.36
relative deviation	Zmean	0.026	−0.060	0.001	0.026	0.036	0.002	−0.049
	Z1/3	0.071	−0.029	−0.072	−0.030	0.014	−0.041	−0.073
	Z1/10	0.076	−0.026	−0.095	−0.046	−0.031	−0.056	−0.097

presents the mean values Z_mean, significant values Z_1/3, and 1/10 maximum values Z_1/10 of the oscillation amplitudes of curves in Figure 10. It can be found in Table 5 that the numerical results are close to those measured during the sea trial, with a difference of less than 10%.

Figure 12 illustrates the probability density function (PDF) of the Z displacements of all track points, which conform to the normal distributions. Again, the numerical results agree well with the measured ones. One may find that the PDFs of numerical simulation show means which are slightly smaller than 0. This is because the mean heights of track points are slightly smaller than their initial heights in still water. In a sea state with currents and waves, the mooring lines must offer enough tension to overcome the drag force induced by currents and the drift force by waves. Since the tension of mooring lines has a negative z-directional component, the draft of the cage is enlarged, and therefore, the vertical positions of track points descend (The same phenomenon can also be found in Section 5.1). Although this effect exists in both sea trials and numerical simulation, one cannot determine the initial positions of track points in a sea trial. Specifically, it is difficult to determine the vertical shifts of track points undercurrents and waves. In fact, the mean vertical positions of track points are manually set to be 0 in the image processing program, which means all PDFs of the sea record in Figure 12 have mean values of 0.

4.3. Simulation Cases

To study the response of the flexible cage under the worst sea conditions that may happen, the largest current and wave, which are obtained by long-term observation data at the sea trial site, are assumed to be in the same direction. Meanwhile, three different sizes of floating collars are considered to investigate the influence of structure flexibility. The simulation cases are listed in

Table 6. Parameters of all simulation cases.

Case No.	Current	Wave	Wavelength	Floating Collar Size
1	Velocity = 0.7 m/s Uniform current Direction = 0deg	Wave height = 5 m Wave direction = 0 deg Monochromatic regular wave Water depth = 16 m	40	D = 315 mm
2			60
3			80
4			100
5			120
6			140
7			160
8			180
9			200
10			80	D = 280 mm
11				D = 250 mm

Note on physical parameters of the double floating collars: (1) D = 315 mm, t = 23.2 mm, M = 3636 kg, d = 180 mm, $Z_{cg}$ = 158 mm, (2) D = 280 mm, t = 20.6 mm, M = 2870 kg, d = 131 mm, $Z_{cg}$ = 140 mm, (3) D = 250 mm, t = 18.4 mm, M = 2289 kg, d = 126 mm, $Z_{cg}$ = 125 mm, $R_x$ = 10.13 m, $R_y$ = 10.13 m, $R_z$ = 14.33 m for (1–3), where D is outer pipe diameter, t is pipe thickness, M is mass, d is draught, $Z_{cg}$ is the vertical position of the gravity center, $R_x$, $R_y$ and $R_z$ are inertial radii with respect to the X, Y, and Z axes, respectively.

.

5. Results and Discussion

5.1. Motion

Figure 13 shows the response of the flexible cage under Case 3. It can be seen that the upstream mooring lines (Lines 6 to 9) are taut, while the others are slack. The cage deforms and moves with the wave contours. The upstream side of the cage is submerged in water by the drag of mooring lines. The net experiences significant deformation under waves and currents. Figure 14 shows the trajectories of 5 typical points (Point A to E) of the floating collar. Each point presents irregular trajectories, which exhibit significant nonlinear characteristics being affected by the constraint of mooring lines. The trajectories appear like mangos with different positions. To be specific, trajectories on the upstream side (Points D and E) seem like mangos with roots on the top and rotating counterclockwise from the upright position, and those on the downstream side (Points A and B) like mangos with roots on the bottom and rotating clockwise from the upright position, the trajectory at the center of the cage (Point C) is basically like an upright mango with its root on the bottom. The highest points of the trajectories of Points E and D are lower than the wave amplitude (2.5 m), which shows the water-entry phenomenon on the upstream side of the cage when the wave crest approaches. By contrast, the lowest point of each trajectory is close to −2.5 m, indicating that no water-entry happens when the wave trough comes. The periodic water entry and water exit can induce extreme slamming load upon the floating collar system, which may destroy the entire cage.

During simulation, it is found that the flexible cage deforms evidently along the X and Y directions. Taking Case 3 as an example, Figure 15 shows two typical deformation states of the cage, as well as the change in cage dimensions over time (the location of Points A, C, E, and G can also be found in Figure 14). According to the loading status shown in Figure 15c, the deformation of the cage during simulation can be divided into three stages. The first stage is from 0 to 60 s, and only a 0.7 m/s sea current is applied in this stage. At t = 60 s, the deformation ratio |AE|/|CG| reaches a constant value of 1.04, which indicates the deformation is not obvious undersea current only. The second stage is from 60 to 100 s, during which a wave is involved in the simulation apart from the current. In the second stage, the wave height increases gradually from 0 (t = 60 s) to 5 m (t = 100 s), making the deformation oscillation of the cage stronger. The third stage is t > 100 s, during which the current and wave keep steady (current velocity = 0.7 m/s, wave height = 5 m), and the cage undergoes stable periodic deformation. In the third stage, |AE| and |CG| change significantly, with the range of |AE|/|CG| lying from 0.96 to 1.19. Compared with the constant value of 1.04 at t = 60 s, the deformation in the third stage is 5 times bigger than that in the first stage. Another phenomenon is that the variation of |AE| and |CG| show opposite trends.

Figure 16 shows the maximum Z coordinate of each observation point (as shown in Figure 14) changing with wavelength of incident waves and the diameter of the collar pipes. As is depicted in Figure 16a, when the wavelength is less than 4 times the cage diameter (the mean of the outer collar diameter and the inner collar diameter, i.e., 90 m in our simulation), the maximum Z coordinates increase significantly with the growth of wavelength. This indicates that the water-entry phenomenon is more likely to happen in short waves. When the wavelength is greater than 4 times the cage diameter, the maximum Z coordinates are no longer affected by the changing of wavelength. As is shown in Figure 16b, the maximum Z coordinate of each observation point reaches its largest value when the diameter of collar pipes is 280 mm. Both large collars (the diameter of collar pipes = 315 mm) and small collars (the diameter of collar pipes = 250 mm) can reduce the Z coordinates of the observation points. This indicates that a moderate collar is required if one wants to reduce the water entry of the cage.

Figure 17 shows the variation of |AE|/|CG| with respect to the wavelength of incident waves and the diameter of collar pipes. As is shown in Figure 17a, when the wavelength is less than 4 times the cage diameter, maximum |AE|/|CG| shows a decreasing trend with the increase of wavelength. This indicates that the global deformation of the cage is decreasing. Yet when the wavelength is greater than 4 times the cage diameter, the deformation of the cage is no longer affected by the wavelength. In Figure 17b, it is obvious that the deformation of the cage decreases when the collar pipe size becomes larger.

One may note that both the floating collar and the net show large deformation in Figure 13, leading to a high reduction of the net volume. Using the Gauss Theorem, the net volume of any moment is written as $V_t=\iiint \mathrm{d}v=\iiint \nabla ·\left(0,y,0\right)\mathrm{d}v=∯\left(0,y,0\right)·\overrightarrow{n}\mathrm{d}s={\iint }_{net surface}y{n}_y\mathrm{d}s+{\iint }_{sea surface}y{n}_y\mathrm{d}s$. Since the wave direction is 0 deg during simulation, namely the y component of the normal sea surface is zero, hence $V_t={\iint }_{net surface}y{n}_y\mathrm{d}s$ which can be easily computed numerically. The volume ratio is thus defined as $C_v=V_t/V_0$, with $V_0$ denoting the original net volume without deformation. Figure 18 illustrates the variation of $C_v$ concerning the wavelength of incident waves and the diameter of collar pipes. Both minimum $C_v$ and maximum $C_v$ increase with the growth of wavelength, yet the change in minimum $C_v$ is little when the wavelength is larger than 4 times the cage diameter. In Figure 18b, the volume reduction becomes severe when the collar pipe diameter increases from 250 mm to 280 mm, yet changes little when the collar pipe diameter increases from 280 mm to 315 mm. As high volume reduction can cause the loss of fish, heavier sinkers, together with wave and current dissipation devices, can be used during practical operations to lower net deformation.

5.2. Tension of Mooring Lines and Net Lines

Figure 19 shows the tension curves of typical mooring lines under Case 3. Similar to Figure 15c, the tension curves can also be divided into three stages. At the end of the first stage (t = 60 s), the tension reaches a constant value denoted by T_static, which is induced by the steady current. In the second and third stages, an additional line tension induced by waves is involved apart from T_static. One can find that the wave-induced tension is composed of two parts. One is the mean tension increment T_drift generated by the constant wave drift force. The other is the dynamic oscillatory tension T_dyna generated by the periodically fluctuating wave force. The meaning of T_static, T_drift, and T_dyna is illustrated in Figure 19.

Figure 20 shows the static tension T_static and the maximum tension T_max of each mooring line and net line (Lines 1 to 10 are mooring lines, and lines with indices greater than 10 are net lines). It can be seen that Lines 6 to 9 are the dominant load-bearing lines, and the tension of other mooring lines is, by contrast, very small. Similarly, the tension of net lines also shows uneven distribution. It should be noted that the dynamic-load effect upon the present cage is significant. For instance, for the mooring line (Line 7) and net line (Line 63), which bears the largest tension, their T_max are 3 times and 7 times their T_static, respectively. Another function of the dynamic-load effect is that it can weaken the unevenness of tension distribution among different lines. For example, the T_static of Line 7 is 3.7 times that of Line 6 in Figure 20a, whereas the T_max of Line 7 is only 1.3 times that of Line 6 in Figure 20b.

Figure 21 illustrates how the wave-induced tensions (T_drift and T_dyna) and the maximum tension (T_max) of mooring lines vary with the wavelength of incident waves and the diameter of collar pipes. As is depicted in Figure 21a, T_dyna gradually increases with the growth of wavelength. By contrast, T_drift declines with the growth of wavelength, yet the declining rate becomes very slow when the wavelength exceeds 4 times the cage. As a result, the total tension T_max initially decreases and then rises with the growth of wavelength. T_max reaches its minimum when the wavelength is nearly 3.4 times the cage diameter. When the wavelength is less than 3.4 times the cage diameter, both T_drift and T_dyna of Line 6 are close to those of Line 7, and T_drift is greater than T_dyna. When the wavelength becomes larger, the T_drift of Line 6 becomes greater than that of Line 7 yet the situation for T_dyna is exactly the opposite, and T_drift turns to be smaller than T_dyna. In summary, the wave-induced tension T_drift + T_dyna of Line 6 and Line 7 is close to each other, and the tension difference between the two lines primarily lies in T_static, which explains why the difference of T_max between the two lines seems to be constant in Figure 21a. In Figure 21b, T_max and T_drift increase with the growth of the collar pipe size, yet the increment is not obvious, which indicates that the strength of the collar has little impact on the line tension.

5.3. Structural Sectional Force

Figure 22 shows the distribution of static (t = 60 s) sectional load (including axial force F_x, torque moment M_x, horizontal bending moment M_z, and vertical bending moment M_y) of the floating collar system under sea current only. Figure 23 shows the distribution of the maximum and minimum dynamic sectional force of the floating collar system under Case 3.

Table 7. Key features of extreme sectional force (Case 3).

Force	Type	Value	Belonging Structure	Position θ
Axial force Fx (kN)	Max. dynamic Fxmax	171	Outer floating collar	180 deg
	Min. dynamic Fxmin	−163	Inner floating collar	180 deg
	Max. Static Fxmax_static	43.2	Outer floating collar	180 deg
	Min. Static Fxmin_static	−41.7	Inner floating collar	180 deg
Torsional moment Mx (kNm)	Max. dynamic Mxmax	2.56	Inner floating collar	145 deg
	Min. dynamic Mxmin	−2.56	Inner floating collar	−145 deg
	Max. Static Mxmax_static	0.261	Inner floating collar	145 deg
	Min. Static Mxmin_static	−0.261	Inner floating collar	−145 deg
Vertical bending moment My (kNm)	Max. dynamic Mymax	5.88	Inner floating collar	180 deg
	Min. dynamic Mymin	−6.98	Inner floating collar	0 deg
	Max. Static Mymax_static	0.237	Outer floating collar	−145 deg
	Min. Static Mymin_static	−0.309	Outer floating collar	170 deg
Horizontal bending moment Mz (kNm)	Max. dynamic Mzmax	11.1	Collar connector	−40 deg
	Min. dynamic Mzmin	−11.1	Collar connector	40 deg
	Max. Static Mzmax_static	2.51	Inner floating collar	170 deg
	Min. Static Mzmin_static	−1.38	Outer floating collar	145 deg
	Note: to describe the location where extreme sectional force occurs, the position of any point P is described by angle θ.		Note: the inner and outer floating collars form a composite beam structure by the collar connectors.

lists the key characteristics of various sectional loads under Case 3. According to the above-mentioned figures and table, one can observe that the axial force is much greater than other sectional force types. The inner and outer collars are the main load-bearing structures of the cage, while the collar connectors, handrails, and balusters bear the little load.

The maximum (tension) and minimum (compression) axial forces of the collars are mainly distributed in four intervals: $\theta \in$ [−20°, 20°], $\theta \in$ [70°, 110°], $\theta \in$ [160°, 200°] and $\theta \in$ [250°, 290°] (refer to Table 7 for the definition of θ). The tension and pressure alternate in different intervals. To be specific, in the inner floating collar, if compression is shown in $\theta \in$ [−20°, 20°], then the signs of axial forces within other intervals are tension in $\theta \in$ [70°, 110°], compression in $\theta \in$ [160°, 200°] and tension in $\theta \in$ [250°, 290°]. The outer collar also exhibits a similar pattern. Another phenomenon is that the load signs of the inner and outer collars are always opposite. For example, if compression is shown in $\theta \in$ [−20°, 20°] of the inner floating collar, then the outer collar will experience tension at the same interval. This is because the inner and outer collars are fixed by the connectors and form some composite beam (see the note in Table 7). When the cage bends and deforms in the horizontal plane, the directions of axial forces of the inner and outer collar are opposite as they are located on opposite sides of the neutral axis of the composite beam. It can also be observed in Table 7 that the magnitude of tension and compression are almost equal. This is because the inner and outer collars have the same distance measured from the neutral axis of the composite beam.

Figure 24 shows the variation of different sectional forces concerning the wavelength of incident waves and the diameter of collar pipes. As is shown in Figure 24a, all kinds of sectional forces decrease obviously with the growth of wavelength when the wavelength is less than 4 times the cage diameter. However, when the wavelength is greater than 4 times the cage diameter, the sectional forces are almost unaffected by wavelength except for the negative vertical bending moment (−M_ymin), which rises with the growth of the wavelength. The magnitude of sectional forces can reflect the status of the cage. For instance, F_xmax is slightly larger than −F_xmin, indicating that the cage is mainly in a tensile state. −M_ymin is slightly larger than M_ymax, indicating that the cage is mainly in a sagging state. In Figure 24b, it can be seen that as the collar pipes become stronger, the axial force F_x and horizontal bending moment M_z increase significantly. Since axial force is the dominant sectional force of the cage, it can be concluded that increasing the size of collar pipes will increase their inner forces.

6. Conclusions

A coupled numerical algorithm is established based on the FEM theory of large-deformation beams and the lumped-mass model to solve the dynamic response of multiple systems of flexible marine aquaculture cages. The numerical algorithm is verified against the sea trial records. The response of the cage subject to extreme sea conditions is studied using the proposed numerical algorithm. Motions, line tensions, and structural sectional forces are studied, and the effects of factors such as the wavelength and the collar pipe size are investigated. The following conclusions are drawn.

• The motion curves, time–frequency energy spectrums, and PDFs of the track points obtained by numerical simulation are in good agreement with those by the sea trial. The validity of the numerical method proposed in this article is therefore proved.
• Water entry may occur on the upstream side of the cage in extreme seas. When the wavelength is less than 4 times the cage diameter, a smaller wavelength can cause severe water entry. Excessive or insufficient stiffness of the cage can also exacerbate the water entry.
• The cage undergoes global deformation in extreme seas, which stretches along the current/wave direction and leads to an apparent reduction of the net volume. When the wavelength is less than 4 times the cage diameter, a smaller wavelength can cause severe deformation of the cage and a reduction of the net volume. As the stiffness of the cage decreases, the overall deformation of the cage becomes more pronounced, yet the reduction of the net volume declines.
• As wavelength increases, T_drift gradually decreases, T_dyna gradually increases, and T_max decreases first and then increases, reaching its minimum when the wavelength equals 3.4 times the cage diameter. The wave-induced line tension T_drift + T_dyna barely changes with wavelength, and the difference in tension between different lines mainly comes from the difference in their static tension T_static. As the stiffness of the cage increases, the tension of the mooring lines also increases, but the magnitude of the change is not significant.
• Collars are the primary load-bearing structures of the cage, and the axial force is the dominant sectional load of the collars. Extreme axial forces occur at the east, west, south, and north positions of the cage, with alternating tension and compression at adjacent positions. The inner and outer collars form an entire composite beam by the collar connectors, and the signs of the axial forces of the inner and outer collars are always opposite. When the wavelength is less than 4 times the cage diameter, a smaller wavelength can cause greater sectional loads. Increasing the stiffness of the cage can significantly increase the sectional force of the cage.
• When the wavelength is greater than 4 times the cage diameter, the water-entry phenomenon, overall deformation, and sectional force of the cage are barely affected by the wavelength.