Climate change driven by global warming has gradually increased the intensity of recent typhoons and hurricanes, elevating the risk of large-scale damage and structural failure in floating offshore wind turbines. In response to such conditions, there is a growing demand for a new type of detachable mooring system that enables emergency detachment and reduces offshore maintenance costs.

The fairlead chain stopper (FCS) is a newly designed system that enables faster and safer attachment and detachment than existing detachable mooring systems for floating offshore structures. This system aims to enhance the efficiency of installation and decommissioning processes for floating offshore wind turbines. These turbines must meet the structural performance requirements set by classification societies and withstand high mooring loads. In addition, they require optimal structural designs that ensure structural integrity using high-strength materials and achieve weight reduction.

Research on the structural performance evaluation and optimization of detachable mooring systems for floating offshore wind turbines has been limited. Lee and Park (2024) proposed a structural strength evaluation method for the submersible mooring pulley (SMP), a key component of a new detachable mooring system designed to enhance the maintenance efficiency and economic feasibility of floating offshore wind farms. Their study conducted structural analyses accounting for mooring loads acting on the SMP and confirmed the validity of the design, providing fundamental data for future structural optimization. Lee and Song (2023) evaluated the structural safety of the FCS, a detachable mooring system for megawatt-class floating offshore wind turbines, by fabricating a scaled 3D-printed model and conducting structural tests. Based on the test results, the structural soundness of the FCS was verified, and the feasibility of full-scale application was qualitatively validated. Park and Song (2021) proposed a discrete design approximate optimization technique for minimizing the weight of an A60-class deck penetration piece by combining various metamodels with a multi-island genetic algorithm. The fire resistance performance was evaluated through transient heat transfer analysis, and the accuracy and efficiency of the proposed optimization approach were validated by comparing its results with actual analysis data. Benítez-Suárez et al. (2025) presented an efficient preliminary design strategy for reducing the cost of jacket substructures for NREL-5 MW offshore wind turbines using a particle swarm optimization-based approach. The convergence behavior and design diversity were improved compared to randomly initialized populations by introducing a precomputed swarm particle swarm optimization (PSO) algorithm, and the design suitability was numerically validated. As part of the development of a detachable mooring system, Yu et al. (2023) conducted a numerical evaluation of the structural and fatigue strength of an FCS designed for a 10 MW-class floating offshore wind turbine. The designed FCS was assessed for structural integrity and fatigue life based on the Det Norske Veritas (DNV) classification standards, and it met the structural and fatigue safety requirements. Ma et al. (2025) performed design optimization of the OC3 spar floater, accounting for rigid-body motion and hydroelastic responses. The optimal shape was derived by selecting the cylinder diameter, wall thickness, draft, and ballast height as design variables, achieving a 20%–40% reduction in surge, pitch, and hub motions. Lee et al. (2025) designed a mooring system for a combined wind and wave energy system operating in intermediate water depths and compared the performance of various mooring configurations, including chains, fiber ropes, and clump weights. The system stability, tension response, and installation cost were comprehensively evaluated by numerical simulations, and the most cost-effective mooring configuration was proposed. Yu et al. (2025) assessed the structural safety of an FCS used in the disconnectable mooring system of floating offshore wind turbines. The stress distribution under extreme and fatigue loading conditions was examined using finite element analysis, and the design adequacy and durability were confirmed. Im et al. (2025) numerically analyzed the dynamic response and tension variation caused by clump weights in contact with the seabed in the mooring system of floating offshore wind structures. They suggested that seabed contact increases the standard deviation of mooring line tension, which could affect the fatigue life, and proposed a strategy to distribute the clump weights to mitigate this effect. Li et al. (2024) evaluated the dynamic response of floating offshore wind turbine mooring systems under different inline tensioner configurations. Fully coupled load analyses were conducted to assess the tension variations, platform motion responses, and reverse tension generation, offering structural stability insights and design guidelines for each configuration. Chen et al. (2024) designed a mooring system for a 15 MW floating offshore wind turbine to be deployed in the South China Sea and evaluated the mooring line tension, breaking load, and fatigue damage by numerical analysis. A design solution that ensures structural stability and long-term fatigue life was proposed, combining the University of Maine design mooring (UDM) and VolturnUS mooring platforms. Brommundt et al. (2012) optimized a mooring system for floating offshore wind turbines using frequency-domain analysis. They derived an optimal configuration that minimized motion responses, considering the design parameters such as mooring line length, stiffness, and anchor location. Kim et al. (2013) designed a mooring system for a floating offshore wind turbine installed in the waters off Jeju Island. The analysis model evaluated the mooring line tension and motion response characteristics under environmental loading, and a suitable design configuration was proposed to ensure structural stability and optimal mooring layout. A review of previous studies revealed little optimal design research on detachable mooring systems, such as the FCS, for floating offshore wind turbines. Therefore, an optimization methodology was proposed for the structural design of the FCS using various surrogate models and a global exploration algorithm.

The objective of this study was to reduce the weight of the FCS installed on the substructure of a floating offshore wind turbine by applying an approximate optimization technique based on a global exploration algorithm. A structural performance evaluation was carried out by defining the design load conditions according to the offshore classification rules provided by DNV and performing finite element method (FEM)-based structural analysis. Surrogate models used for approximate optimization were constructed using the design of experiments (DOE), and the response data were obtained in conjunction with the FEM results. The design variables were defined as the thicknesses of the major structural components of the FCS, and discrete design specifications feasible for manufacturing were considered. The objective function to be minimized was the weight, and the constraints were defined by applying 90% of the material yield strength as the allowable stress, in accordance with the DNV offshore classification rules. The DOE was structured using an orthogonal array design (OAD) to ensure solution diversity and analytical efficiency, accounting for variations in the design variables. The surrogate models constructed through this approach included the response surface model (RSM), Kriging, Chebyshev orthogonal polynomial (COP), and artificial neural network (ANN). The accuracy of each model in approximating the responses within the design space was evaluated. Furthermore, this study examined an optimal combination of design variables that minimizes the FCS weight while satisfying various constraints and compared the convergence characteristics of each surrogate model. The optimal designs derived from the surrogate models were validated by comparing their predicted responses with the actual responses obtained from full structural analysis and with the results based on continuous design variables, thereby examining the applicability and accuracy of each surrogate model.

This paper is structured as follows. Section 2 describes the design elements of the FCS, structural performance criteria, and the results of structural analysis. Section 3 discusses the DOE and the construction of each surrogate model, while Section 4 presents the results of the approximate design optimization. Finally, a comprehensive conclusion is provided.

Based on the information in Table 5, an approximate design optimization was performed to determine the structural design for the FCS, composed of discrete design variables, which satisfies the allowable stress constraints specified by the offshore classification rules while minimizing weight. A global exploration algorithm—PSO—was applied to accommodate both discrete and continuous design variables. The surrogate models used in the approximate design optimization were the RSM, Kriging, COP, and ANN models developed in Section 3. The optimization problem was formulated to identify the optimal design variables that minimize the weight of the FCS while satisfying the stress constraints under various design load conditions, including mooring and towing. The results of the approximate design optimization for each surrogate model were compared with the actual calculated results for the objective function and constraints, as well as the results obtained using continuous design variables. The most suitable approximate design optimization approach for minimizing the weight of the FCS was evaluated from this comparison. The formulation of the FCS optimization problem used in this study is as follows:

In Eq. (12), the upper limit values for the constraints under each DLC were set to 279 MPa, which corresponds to the minimum allowable stress among the materials applied to the FCS, based on the DNV offshore classification rules. The approach of using the classification-society-based allowable stress or a safety margin relative to the material yield strength as the structural strength constraint was adopted in several studies (Song et al., 2011; Lee et al., 2021; Song, 2025).

The PSO algorithm used for the optimal design search was a metaheuristic optimization technique inspired by the social behavior of swarming organisms, such as birds or fish searching for food. Within the design space, particles move as if flying, with a certain velocity and direction, and determine their next position by referring to their best-known position and the best-known position of the swarm or its neighbors. PSO is advantageous for global and discrete optimum searches because it considers multiple candidate solutions from the early stages of exploration and updates the search through iterative adjustments of velocity and position (Kennedy et al., 2001). When the PSO algorithm begins the search, the initial positions and velocities of the particles are generated randomly within the design range of the optimization variables. The velocity vector of each particle at step k + 1 is updated using the current position and velocity information at step k and the value of the objective function, as described in the following equation:

where r₁ and r₂ are random numbers between 0 and 1, and ω, c₁, and c₂ represent the inertia factor, self-confidence factor, and swarm-confidence factor, respectively. pki denotes the best position found so far by the i-th particle (cognitive contribution), while pkg represents the best position found by the swarm up to step k (social contribution). According to Eq. (13), the velocity vector of the i-th particle at step k + 1 is updated by incorporating its best position and the global best position of the swarm up to the previous step. The position of the i-th particle at step k + 1 is then updated to a new location using the velocity vector calculated in Eq. (13), as follows:

where Δt is considered as the unit time. Table 8 lists the PSO parameters used for the approximate design optimization of the FCS.

Based on the discrete design variable space defined in the formulation of the FCS approximate design optimization, the RSM, Kriging, COP, and ANN models were applied, and the PSO algorithm was used to derive the approximate optimal solutions. The results obtained by applying the optimal design from the approximate design optimization were compared with the actual calculated results for each objective function and constraint to evaluate the accuracy of the optimization results for each surrogate model. Table 9 lists the results of the approximate design optimization. All results from the approximate optimization yielded design optima that satisfied the constraints and minimized the weight, regardless of the type of surrogate model used (Table 9). On the other hand, the actual values of g₁ and g₂ did not satisfy the constraint upper limit of 279 MPa when the COP model was applied. The average approximation accuracy of the COP model was 0.88, the lowest among the four surrogate models, as shown in Table 7. Consequently, the use of the COP model in the FCS approximate design optimization yielded optimal design results that violated the constraints. Although the design obtained using the COP model showed the greatest weight reduction, it cannot be considered a valid solution because of the constraint violation. In contrast, the ANN model, which showed the highest approximation accuracy, produced response function results with the lowest error ratio compared to the actual response functions and achieved a 3.3% reduction in weight relative to the initial design weight of 17 tons, confirming it as the best-performing model. Therefore, the ANN model can be identified as the most suitable surrogate model for the approximate design optimization of the FCS structural design. A comparison of the RSM and Kriging models showed that although Kriging exhibited higher approximation accuracy than RSM and achieved greater weight reduction, its error ratios in the response functions were higher than those of the RSM. These results suggest that the performance of approximate design optimization is influenced by the approximation accuracy of the surrogate model and the characteristics of each model.

Fig. 10 shows the convergence behavior of the optimization process, in terms of the objective function.

The convergence characteristics of the objective function varied according to the surrogate model (Fig. 10), and the ANN model showed the widest search range during the optimization process. The results of the discrete design optimization were compared with those of the optimization using continuous design variables by converting the discrete range of the design variables in Eq. (12) to a continuous range and performing the approximate design optimization under the same conditions:

Table 10 compares the results of the approximate design optimization using continuous design variables with those of the discrete design optimization.

A comparison of the results of continuous and discrete design optimization showed that the differences in design variables for each surrogate model were relatively small. Given the practical applicability of the optimal design to actual manufacturing, the approximate design optimization using discrete design variables is considered more Surrogate model Optimum design (mm) Approximate response functions Actual response functions Error ratio (Average) appropriate. Therefore, for structural designs such as the FCS, the characteristics and approximation accuracy of the surrogate model are strongly correlated with the optimal design outcome, whereas the format of the design variables has a relatively minor impact.

This study applied a PSO-based approximate design optimization approach to reduce the weight of the FCS mounted on the substructure of a 10 MW-class floating offshore wind turbine and identified the most suitable surrogate model for discrete design optimization.

A structural performance evaluation was conducted by defining the design load conditions based on the offshore classification rules provided by DNV, and FEA was performed. Under all design load conditions, the maximum calculated stress did not exceed the allowable stress defined by DNV rules, confirming the structural integrity of the FCS. Among the components, the maximum stress in the part made of DH36 steel was closest to the allowable limit.

The surrogate models for approximate design optimization were constructed using DOE, and the response function data regarding the FCS weight and maximum stresses under different load conditions were obtained by linking with the FEA results. The design variables and their variation levels were defined as the thickness dimensions of the major components of the FCS at three levels, reflecting discrete design specifications feasible for manufacturing. An OAD was used to design 128 experiments to ensure solution diversity and analytical efficiency. Four surrogate models (RSM, Kriging, COP, and ANN) were developed. The response approximation accuracy of each model within the design space was evaluated. Except for the COP model, all surrogate models achieved high approximation accuracy with average R₂ values exceeding 0.93 for the objective function and constraints. In particular, the ANN model achieved an average R₂ value of 1.0, showing excellent performance.

In the approximate design optimization, each surrogate model (RSM, Kriging, COP, and ANN) was applied. The optimization problem was formulated to identify the optimal design variables that minimize the weight of the FCS while satisfying the stress constraints under various mooring and towing load conditions. After applying PSO to the approximate design optimization, all surrogate models produced design optima that satisfied the constraints and minimized the weight. Nevertheless, the actual stress constraints under mooring conditions exceeded the upper allowable limit when using the COP model. On the other hand, when the ANN model showing the highest approximation accuracy was applied, the results of the response functions had the lowest error ratios compared to actual response values, and the weight was reduced most significantly (3.3% from the initial design of 17 t). A comparison of the results of continuous and discrete design optimization showed that the variation in design variables across surrogate models was relatively minor. Considering the practical applicability of the optimal design, approximate design optimization using discrete design variables was deemed more suitable. This study confirms that for structural design problems such as the FCS, the characteristics and approximation accuracy of the surrogate model play a more critical role in determining the optimal outcomes than the type of design variable used. Future studies will extend this research to include reliability-based design optimization that accounts for design uncertainties.