### 1. Introduction

### 2. Theoretical Background

### 2.1 Governing Equations for HydroQus

*M*is the mass of a floating body, and

_{ij}*ü*is the acceleration of a floating body. The subscript

_{j}*i*and

*j*represent 6 degrees of freedom, 1 = surge, 2 = sway, 3 = heave, 4 = roll, 5 = pitch, and 6 = yaw. The restoring force

*C*and the displacement of a floating body

_{ij}*u*as shown in Eq. (2). According to Cummin’s equation, the radiation force

_{j}*t*

_{1}should be long enough for the impulse response function

*K*to converge stably to zero. The wave damping coefficient

_{ij}*B*(

_{ij}*ω*) obtained through frequency response analysis is used to calculate the impulse response function as shown in Eq. (5). The added mass

*m*(∞) can be calculated by Eq. (4). The wave excitation force

_{ij}*F*is the response amplitude operator (RAO) of the wave excitation force obtained by frequency response analysis,

_{i}*H*is the wave height and

*ϕ*is the wave phase. The mooring force

*l*and

*h*denote the vertical and horizontal positions of the fairlead, respectively.

*H*and

*V*are the horizontal and vertical forces at the fairlead.

*EA*,

*W*, and

*L*are the properties of the mooring line: axial stiffness, weight per unit length, and unstretched length.

*C*is the friction coefficient between seabed and mooring line and

_{B}*L*is the laid length on seabed. By calculating Eq. (10) and Eq. (11), the location along the line segment

_{B}*s*is derived, which are used to model mooring lines.

### 2.2 Process of Fluid-structure Interaction

^{st}and 2

^{nd}order wave excitation forces. The hydrodynamic coefficients used in the calculations must be obtained in advance from a frequency response analysis. For the wave excitation force calculation, the wave height

*H*(or significant wave height

*H*) and wave period

_{s}*T*(or peak period

*T*) must be defined. HydroQus calculates the hydrodynamic forces using the displacement, velocity, and acceleration at the center of mass (CoM) of the FOWT at

_{p}*t*=

*t*

_{1}(current time) in Abaqus and gives them to Abaqus/Explicit. Abaqus solves the equations of motion for the FOWT and the ship, respectively, using the hydrodynamic forces. During this process, Abaqus is responsible for generating the drag-based mooring tension forces and collision forces. The displacements, velocities, and accelerations at the CoM obtained by solving the equations of motion are fed back to HydroQus and used to generate the hydrodynamic forces at

*t*=

*t*

_{2}(next time). This procedure is summarized in Fig. 1.

### 2.3 Flow Stress and Fracture Model

*A*,

*∊*

_{0},

*n*,

*k*

_{0},

*Q*, and

*β*through tensile tests and numerical analyses.

*η*is the stress triaxiality and

*θ̄*is the Lode angle. From Eq. (18) and Eq. (19)

*a*is the load angle sensitivity,

*b*is the fracture strain modulus,

*c*is the stress triaxiality sensitivity, and

*n*is the transformation exponent. The load angle parameter functions

_{f}*f*

_{1},

*f*

_{2}and

*f*

_{3}are calculated through Eqs. (20)–(22). To consider the stress path effect, the damage indicator

*D*in Eq. (23) is introduced. It is assumed that fracture initiates when the damage indicator reaches 1.0.

*d*is the Hosford exponent and

*p*is the transformation exponent. Stress triaxiality functions

_{f}*g*

_{1}and

*g*

_{2}are calculated using Eq. (25) and Eq. (26). A necking indicator

*N*is introduced to consider the stress path effect (see Eq. (27)). Fracture is considered to occur when

*N*reaches 1.0.

### 3. Setup of FOWT-tanker Collision Simulation

### 3.1 Frequency Response Analysis

^{st}order wave excitation force RAOs in the heave and pitch directions.

### 3.2 Finite Element Models for Collision Simulation

### 3.3 Material Properties for Finite Element Model

### 3.4 Collision Analysis Cases

*∊*. In this case,

_{f}*∊*was assumed to be 0.2, which has been the most widely used failure strain. Six cases were generated, which are summarized in Table 6. In all cases, the initial forward velocity of the tanker was assumed to be 5 knots (9.26 km/h). The collision analysis model is shown in Fig. 10.

_{f}