### 1. Introduction

### 2. Configuration of OC4 Semi-Submersible FOWT Platform

### 3. Method

### 3.1 Morison Equation

*ρ*is the water density;

**C**,

_{I}**C**, and

_{A}**C**are the inertia, added mass, and drag coefficients, respectively;

_{D}**a**and

**v**are the acceleration and velocity of fluid particles at a geometric center;

**Ẍ**; and

**Ẋ**are the acceleration and velocity of the platform; and ∇ and

*A*are the displaced volume and drag area.

**C**,

_{I}**C**, and

_{A}**C**in the Morison equation are typically obtained experimentally and through analytical solutions and numerical simulations. This study adopted the Airy wave theory to obtain wave kinematics.

_{D}**F**, which is the first term in Eq. (1). Based on the configuration given in Fig. 1, the way to evaluate

_{I}**F**in the horizontal and vertical direction was different. In the case of the horizontal force, the column was first discretized into 20 elements along the vertical direction (1 m height for each element). The inertia force was calculated at each element with the element volume. The total force of each column was obtained by integrating the element forces. The volume exposed to water only needs to be considered in the case of the vertical force. A correction needs to be made if the top area is not exposed to water like OC4 semi-submersible because the Morison equation considers the body fully submerged in water. For Column 1 (the main column at the center), the inertia force was first estimated at its geometric center (i.e.,

_{I}*z*= −10 m) using the first term in Eq. (1). This portion was then corrected by deducting the FK force on the top surface because the top surface at the mean water level (i.e.,

*z*= 0 m) is not exposed to water. For Columns 2–4, the base columns are only exposed to water in the vertical direction. Therefore, the inertia force was obtained only from the base columns at its geometric center (i.e.,

*z*= −17 m). The FK force at the location at which the upper and base columns meet (i.e.,

*z*= −14 m) was partly eliminated with the area of the top column. The following equation can be used for horizontal and vertical inertia forces on the column (

**F**and

_{IH}**F**): where

_{IV}*e*is the element number in the vertical direction;

*P*is the incident wave pressure; and

_{I}*A*denotes the surface on the top section that is not exposed to water.

_{T}### 3.2 Potential Theory in Frequency Domain

*, Φ*

_{I}*, and Φ*

_{D}*are the first-order incident, diffracted, and radiated wave potentials. The first-order hydrodynamic pressure on the wetted body surface is given by the Bernoulli equation as follows:*

_{R}*P*and

_{I}*P*are the incident and diffracted wave pressures, and

_{D}*S*denotes the wetted surface. The 3D diffraction/radiation panel method was used to obtain the wave excitation forces and hydrodynamic coefficients (Lee, 1995). Fig. 2 presents the panel model used in this study. This study considered 7,181 panels below the mean water level.

_{b}### 3.3 Effective Inertia Coefficient in Morison Equation

**C**, is defined as 1+

_{I}**C**where one and

_{A}**C**are related to the contribution from the FK and diffraction forces. In a cylindrical structure,

_{A}**C**is typically set to one, meaning that the contribution from that FK force is the same as that from the diffraction force (Faltinsen, 1993). On the other hand, as described by Chakrabarti and Tam (1975), the total force is significantly affected by large diffraction in short waves, which requires some correction in the inertia coefficient to be well correlated with the wave excitation force. As a result, the EIC was introduced in this study. The horizontal and vertical EICs for each column in regular waves are defined by Eqs. (7)–(8), respectively: where

_{A}*ω*and

*β*are angular frequency and wave direction, respectively. In the case of random waves, depicting one EIC value is challenging, so a simple statistical approach was adopted. The time history of the wave excitation forces was first obtained by superposing the sinusoidal forces at different regular wave frequencies. The Pierson–Moskowitz (PM) spectrum was considered for fully developed seas. The root mean square error (RMSE) between the time histories of the wave excitation force and the Morison inertia force was calculated at different inertia coefficients. The inertia coefficient that gives the lowest RMSE was selected as EIC under random waves as follows: where

*t*and

*T*denote the time step and final time step,

*T*is the peak period, and

_{P}**F**is the Morison inertia force. The Morison equation may not be correlated well when diffracted waves significantly change the phase of the total force in very short waves because the Morison equation uses the phase of fluid acceleration only.

_{I}### 3.4 Time-Domain Equation of Motion for Platform

**M**is the mass matrix;

**A**(∞) is the added mass matrix at infinite frequency;

**C**is the external damping matrix;

**K**is the hydrostatic and gravitational restoring matrix;

**F**is the convolution-integral based radiation damping force; and

_{R}**F**is the Morison force as defined as Eq. (1).

_{M}**A**(∞) and

**F**can be obtained using the following equations: where

_{R}**A**(

*ω*) and

**B**(

*ω*) are the added mass and radiation damping matrices in the frequency domain and

**R**(

*t*) is the retardation function. A comparison of Eq. (10) with Eq. (11) showed that the heavy computation in the Cummins equation is due to

**F**. In the Morison equation, the radiation damping is excluded, and the last term of

_{R}**F**, which is the viscous drag force, is not considered. Fig. 3 shows the overall procedures to evaluate EICs for the irregular wave cases and run time-domain simulations from the Morison-equation-based model.

_{M}### 4. Results and Discussions

### 4.1 Wave Force Under Regular Waves

*z*= −10 m), which cannot represent vertical inertia force accurately. Nevertheless, the given method can still result in comparable results by introducing the concept of EIC.

### 4.2 Wave Force Under Random Waves

### 4.3 Dynamic Response of Platform Under Random Waves

*P*

_{I}*A*in Eq. (3), was inputted as an external force after precalculation. For a moving body, an added mass coefficient needs to be inputted, which was obtained by subtracting one from the obtained inertia coefficient (i.e., effective added mass coefficient). In the comparisons, only heave, roll, and pitch degree-of-freedom (DOF) motions were considered, while the other DOF motions were constrained because the mooring lines were not modeled. In this demonstration, 10% damping was considered in the heave, roll, and pitch DOFs, which excludes the influence of added mass because the effective added mass coefficient varies according to given environmental conditions. A time-domain simulation was conducted with a simulation time of one hour with a fixed time interval of 0.1 s. Random waves were generated by superposing 200 regular waves from the Jonswap wave spectrum with an enhancement parameter of two. In addition, the computational time of the Morison-equation-based model was approximately 25% faster than the Cummins-equation-based approach. On the other hand, it can vary depending on the division of the Morison element. In the present case, 1-m elements were used for Morison force calculations, which is considered acceptable. The Morison equation can be much more time-efficient if the proper element size is selected.

_{T}