### 1. Introduction

### 2. Numerical Analysis Method

### 2.1 Higher-order Boundary Element Method

*S*and

_{B}*S*represent the boundaries of the floating body surface and the sea bottom, respectively.

_{W}*ϕ*is the linear velocity potential for the flow.

*g*denotes the gravitational acceleration, and

*ζ*denotes the wave height.

*n*and

*v*are the normal vector and normal velocity defined on the floating body surface or the sea bottom, respectively.

_{n}*z*denotes the coordinate in the vertical direction with respect to the free surface.

*k*and

*r*are the wave number and the diameter of the boundary, respectively. Eq. (1) is the Laplace equation, which is the governing equation for the fluid domain based on the potential flow theory. Eq. (2) indicates the dynamical and kinematic boundary conditions on the free surface, respectively. Eqs. (3) and (4) are the non-penetration boundary conditions for the floating body surface and sea bottom, respectively. Eq. (5) corresponds to the radiation condition at the far field.

*k*

_{0}(=ω

^{2}/

*g*) is the wave number at infinite depth and

*ω*is the wave frequency.

*∊*is a parameter that controls the degree of numerical damping. In this study, 0.02 was applied for

*∊*.

### 2.2 Evaluation of Stiffness Matrix

*J*

_{1}and

*J*

_{2}can be represented as a local stiffness matrix ([

*K*]

^{e}_{3×6}) as follows: where

*F*

_{n}_{+ 1},

*F*

_{n}_{+ 2}, and

*F*

_{n}_{+ 3}are the restoring forces acting on the module

*J*

_{1}, while

*F*

_{m}_{+ 1},

*F*

_{m}_{+ 2}, and

*F*

_{m}_{+ 3}are the restoring forces acting on the module

*J*

_{2}.

*n*= 6×(

*J*

_{1}− 1) and

*m*= 6×(

*J*

_{2}− 1) are integers for the degree of freedom of the two modules, respectively.

*s*is the local motion of

_{n}*n*th degree of freedom at the connector position, and (

*k*

_{x}*,k*

_{y}*,k*) are the components of the spring stiffness. Then, the local stiffness matrix can be converted into a global stiffness matrix using a transformation matrix ([

_{z}*T*]

^{e}_{6×12}) as follows:

##### (9)

*x*

_{1}

*,y*

_{1}

*,z*

_{1}) and (

*x*

_{2}

*,y*

_{2}

*,z*

_{2}) correspond to the positions where the connectors of modules

*J*

_{1}and

*J*

_{2}are attached. Considering the rotation matrix ([

*R*]

^{e}_{3×3}) for the moment of rotational motion, the stiffness matrix ([

*K*]) by the connector can be calculated as follows:

##### (10)

### 3. Validation of Numerical Method

### 3.1 Module Specifications

### 3.2 Connectors Between Modules

### 3.3 Validation

*λ*/

*L*= 3.34), whereas the AFT module showed a smaller heave RAO than the other modules at all frequencies. The small motion response of the AFT module can be interpreted as an insignificant wave excitation effect due to its relatively large size and location away from the outside. In higher frequencies than 0.3 rad/s, both numerical and experimental results showed a decreasing tendency of the heave and pitch RAOs as the wave frequency increased. Looking at the module's pitch RAO, the closer the module is to the incident wave direction, the larger the peak response occurs at a higher wave frequency. From the results of Fig. 3, it can be confirmed that, even though the FORE module and MID module are the same size, the magnitude of the response and the peak period differ based on the location and connection method.

_{x,small}### 4. Results and Discussions

### 4.1 Effect of Tail Module

*ω*= 0.2

*rad*/

*s*, there is a slight difference in heave RAO due to the tail module, but overall, it can be seen that the effect of tail module on the motion RAOs of the outer modules is not significant. This indicates that the interaction effect is not strong owing to the large separation distance between the tail module and the front outer modules, and the effect of tail module on the overall motion response characteristics of the floating island is quite small. When comparing the motion response characteristics of the two outer modules at frequencies higher than 0.36 rad/s, the HEAD module consistently showed larger heave and pitch motions than the MID module. This means that when a high-frequency short wave is incident, only the outermost modules on the weather side are mainly excited, and the motions of the modules positioned farther within are reduced. In this case, most of the incident waves are reflected, creating high waves in front of the outer modules.

*ω*= 0.32, 0.36, and 0.40 rad/s). First, both floating island models show that the vertical motion of the central modules is relatively less than that of the adjacent outer modules. This is because the outer modules move by absorbing the wave energy, and the wave exciting forces acting on the central module are significantly reduced. Another reason is that the central module has about 4 times inertia of the outer module. Another pattern in the contour plots is that the vertical motion increases intensively in the outer modules on the weather side (right side of the figure) under the short wave conditions and rapidly decreases on the opposite side. This is consistent with the numerical analysis results of Otto et al. (2019a), which show that at the peak frequency where the maximum motion of the modular floating island occurs, the modules on the lee side exhibit a very small motion, while the modules on the weather side experience a large motion. This feature indicates that the modules on the weather side absorb the majority of the wave kinetic energy under the short wave conditions, which leads to reduced motion responses of the other modules. When comparing the motion responses of two floating islands at the three frequencies, the MARIN model and Floater A show very similar patterns of vertical motion response distribution. This indicates that there is only a small effect of the tail module on the vertical motion of the floating island since it is placed in the opposite direction of the incident wave.

### 4.2 Effect of Central Modules

*λ*/2

*L*= 1.17). Floater B shows the maximum pitch RAO at the wave frequency of 0.2 rad/s. Here, the length ratio between the transverse length of the central hexagon module and the half of the incident wavelength is approximately 0.9 (

_{x,large}*λ*/2

*L*= 0.9). This indicates that the peak period of pitch motion can change depending on whether the central module is divided, and the pitch RAO of the central module has the maximum value under the condition that the wavelength of the incident wave corresponds to about twice the length of the central module. In this section, it was found that when using a single central module, not only heave and pitch motions are reduced, but also the peak frequency of pitch motion shifts to a lower frequency than when using divided central modules. Therefore, a large single central module may be a good design alternative to reduce the wave-induced motion of modular floating island.

_{x,hexagon}### 4.3 Effect of Outer Modules

*λ*/

*L*= 3.34). In other words, Floater C showed smaller heave and pitch response than Floater B at frequencies lower than 0.3 rad/s, whereas the opposite pattern was seen at frequencies higher than 0.3 rad/s. To examine the detailed motion characteristics, Fig. 12(c) compares the local vertical motion RAO of the outer modules of Floater B and Floater C. Since the motion response of the outer modules varies depending on the position of the module, the local vertical motion was examined for the outer end position of the outer modules (red dot position in Fig. 11), which exhibits a large vertical motion by directly facing the wave. The vertical motion response of the outer modules likewise altered at a frequency of 0.3 rad/s and shows the opposite tendency to the motion of the central modules. At frequencies below 0.3 rad/s and above 0.3 rad/s, the local vertical motion of the outer modules of Floaters B and C was smaller, respectively.

_{x,small}*H*) of the irregular wave was set at 11.5 m, and the peak period (

_{s}*T*) was changed to 14.5 s, 16.0 s, 17.5 s, and 19.0 s. This allowed for the consideration of a total of four irregular wave conditions. Fig. 13 shows the four irregular wave spectra considered in the motion performance evaluation of this study.

_{p}*S*and

_{ζ}*S*are the response spectra of the wave spectrum and the

_{j}*j*th motion mode of the floating island, respectively. Furthermore,

*j*th motion mode, and

*ξ*denotes the single significant response for the

_{j}*j*th motion mode.