Probabilistic Structural Design of Detachable Mooring Apparatus for 10 MW Floating Offshore Wind Turbine Using Reliability-Based Robust Optimization

Abstract

This paper applies the reliability-based robust optimization (RBRO) technique to investigate the probabilistic structural design characteristics of the Fairlead Chain Stopper (FCS), a newly developed detachable mooring apparatus for installation on a 10 MW floating offshore wind turbine. The thickness dimensions of the FCS’s major structural members were considered as random design variables, including uncertainties such as manufacturing tolerances. The structural strength performance was defined as a probabilistic constraint function based on the allowable stresses specified by DNV classification rule. The structural strength performance of the FCS was evaluated through finite element analysis (FEA) using design load conditions for moored (LC1, LC2) and towed (LC3) conditions based on DNV classification rules. The RBRO design problem was formulated with weight minimization as the objective function, with probabilistic constraints on strength performance and 3-sigma robustness applied as side constraints. To evaluate reliability analysis methods suitable for probabilistic optimal design, the Mean Value Reliability Method (MVRM) and the Adaptive Importance Sampling Method (AISM) were applied during the RBRO process, and the results were compared and analyzed. The probabilistic optimal design using RBRO exhibited conservative design characteristics compared to the deterministic optimal design, ensuring robustness and reliability. After comprehensively considering the weight reduction rate and numerical computational cost (number of function evaluations), the RBRO method using MVRM was confirmed to be the most reasonable method for the probabilistic optimal structural design of the FCS.

1. Introduction

The recent increase in typhoon intensity driven by climate change poses a significant risk of structural damage and collapse to Floating Offshore Wind Turbines (FOWTs). Consequently, the development of a detachable mooring system that allows for evasion during severe weather conditions and reduces maintenance costs has emerged as an urgent priority. The Fairlead Chain Stopper (FCS) proposed in this study aims to maximize the efficiency of installation and dismantling processes by providing rapid and safe detachment and attachment performance compared to existing devices. In particular, since this system must satisfy classification society regulations while withstanding high mooring loads using high-strength materials, ensuring both structural stability and design robustness is essential.

Research applying probabilistic design and robust design to marine systems has been conducted across various fields. Patryniak et al. [1] proposed a design methodology that strategically places the instantaneous center of rotation to reduce dynamic loads and enhance the structural robustness of FOWTs. Kim et al. [2] comprehensively evaluated the structural integrity of IMO Type C cargo tanks for mid-sized LCO₂ carriers—next-generation eco-friendly vessels considering combined wave and thermal loads under ultimate, fatigue, and accidental limit states. To improve the survivability of structures in extreme marine environments, Wen and Ong [3] numerically investigated the dynamic behavior characteristics of a submersible gravity cage applying a single-point mooring system. Meanwhile, as probabilistic studies considering design uncertainties. With the increasing scale of offshore wind power systems and their expansion into deep-sea environments, ensuring the safety of support structures and mooring systems for FOWTs has become a paramount challenge. Given the stochastic nature of the marine environment, various design optimization techniques are being researched to guarantee structural reliability [4]. Traditional offshore structure designs have largely relied on Deterministic Design Optimization (DDO) based on safety factors. However, as noted by Chen and Kim [5], such deterministic approaches often fail to explicitly account for uncertainties in environmental loads—such as waves, wind, and currents—and material properties, potentially leading to inefficient over-design or unexpected structural failures. To overcome these limitations, Reliability-Based Design Optimization (RBDO), which defines design variables as random variables and treats failure probability as a constraint, has been actively introduced. Leimeister and Kolios [6] demonstrated that applying RBDO to spar-type floating structures can enhance structural efficiency while satisfying target reliability levels. Furthermore, Stieng and Muskulus [7] proposed an optimization methodology using analytical sensitivities to account for support structure uncertainties, thereby improving the precision of reliability analysis. Recent research trends are expanding beyond simple structural reliability to consider complex system interactions and multidisciplinary integrated design. Okoro et al. [8] analyzed the impact of dependencies in complex offshore structures on reliability optimization, while Chang and Peng [9] performed Multidisciplinary RBDO integrating FOWT support structures with vibration control devices to ensure dynamic stability in extreme environments. However, probabilistic approaches like RBDO entail high computational costs due to the requirement for repetitive Finite Element Analysis (FEA). To address this, the use of surrogate models such as Kriging or Radial Basis Functions (RBF) has emerged as a key solution to maximize optimization efficiency. Huang et al. [10] showed that RBF surrogate models could significantly reduce the runtime of RBDO for offshore wind support structures while maintaining analysis accuracy. Li et al. [11] conducted wind tunnel experiments and CFD simulations to analyze the dynamic response of a wind turbine tower with a high slenderness ratio and identified the effect of changes in aerodynamic characteristics according to the structural shape of the tower on the stability of the tower. Li et al. [12] conducted wind tunnel experiments and numerical simulations to determine the aeroelastic characteristics of a wind turbine system using a concrete-filled double tube (CFDST) structure and evaluated the vibration control performance and structural integrity of the system under wind load. As floating offshore wind turbine systems become larger, ensuring the safety of mooring systems becomes more important. In particular, for the newly developed detachable mooring device, the FCS, there is a lack of research on deriving a robust optimal design that satisfies both structural safety and economic feasibility by considering the uncertainty of design variables such as manufacturing tolerances. Therefore, reflecting these state-of-the-art trends, this study applies a Reliability-Based Robust Design Optimization (RBRO) technique to the structural design of an FCS for a 10 MW FOWT. By simultaneously considering the uncertainty and sensitivity of design variables, we aim to derive an optimal design and validate its effectiveness.

In this study, to strictly secure both structural safety and economic feasibility of the FCS at the design stage, an RBRO framework combining reliability analysis and robust design was introduced. Manufacturing tolerances present in the thickness dimensions of primary structural members were defined as uncertainties and modeled as random variables, while strength performance was formulated as a probabilistic constraint. The strength evaluation was performed based on the maximum stress calculated through FEA under load conditions reflecting actual operating environments and DNV classification rules. The optimization problem was set to minimize the structural weight while satisfying the probabilistic strength constraints, with a 3-sigma level applied to the constraints to ensure robustness. Specifically, to identify the optimal reliability analysis technique for FCS design, the Mean Value Reliability Method (MVRM) and the Adaptive Importance Sampling Method (AISM) were applied, respectively, and their convergence speeds, numerical computation characteristics, and reliability results were compared and analyzed. Furthermore, the validity of the probabilistic approach was verified through comparison with Deterministic Optimization results. The remainder of this paper is organized as follows: Section 2 covers the FEA-based strength evaluation; Section 3 discusses the theoretical background of RBRO and the comparative results of MVRM and AISM; and the conclusions summarize the research achievements.

2. Structural Strength Assessment of FCS Design

The FCS addressed in this study is a detachable device specialized for the transportation and mooring processes of FOWTs. It is designed to streamline the connection and disconnection operations between mooring lines and the floating platform. This system offers a significant functional advantage by enabling the rapid detachment and relocation of the floating platform in cases requiring large-scale maintenance or during emergency situations. The target application of this study is a 10 MW-class semi-submersible FOWT, which features a configuration consisting of three floating columns. Figure 1 schematically illustrates the overall configuration and arrangement of the FCS installed on the semi-submersible FOWT, while Figure 2 presents the detailed design geometry and a schematic of the major components constituting the FCS.

As shown in Figure 1, an FCS is mounted on each column of the floating platform, and a Submersible Mooring Pulley (SMP) is installed at the junction between the mooring line connected via the FCS and the mooring line connected to the anchor. The FCS is designed with a length of approximately 7.2 m and a weight of about 35 tons. As depicted in Figure 2a–c, the FCS is connected to the mooring chain above the sea level and is designed to ensure structural safety under conditions of maximum tensile force. The FCS is primarily composed of an Arm system (Figure 2b, Arm components) and a Housing system (Figure 2c, Housing components). The Arm system consists of nine major components: the Guide plate, Main chain stopper block, Chain stopper, Upper chain stopper, Upper chain stopper block, Hydraulic cylinder support, Wall plate, Arm pin plate, and Arm pin. The Housing system comprises ten major components: the 5-Pocket chain wheel, Chain wheel pin, Chain wheel pin bushing, Arm pin bushing, Housing wall plate, Arm pin bearing plate, Main pin, Base plate, Top plate, and Flange bushing. The 5-Pocket chain wheel guides the movement of the mooring chain during installation or disconnection. The Arm guides the chain movement and transfers the support load of the Chain stopper to the FCS structure. The Chain stopper and Upper chain stopper support the chain loads generated during the operation, installation, or disconnection of the FOWT. The Housing serves to connect the FCS to the substructure. The Hydraulic cylinder support connects the hydraulic unit controlling the operation of the Upper chain stopper. The major components of the FCS are connected by pins to allow for rotational operation. As shown in Figure 1 and Figure 2, the FCS has a structure that engages with the mooring chain above the water surface and is designed to maintain structural integrity even under extreme external forces equivalent to the Minimum Breaking Load (MBL) applied to the mooring lines. To obtain design certification for such a mooring device, it is mandatory to verify satisfaction of strength performance under mooring conditions specified by classification societies. Furthermore, load evaluation for towing conditions must be included in the scope of strength analysis, considering various situations that may occur during actual operation.

In this study, design load conditions for strength performance evaluation were established in accordance with DNV-OS-E301 [13], DNV-ST-0119 [14], and DNV-OS-C101 [15]. Since the FCS has structural characteristics of being directly installed on the semi-submersible FOWT platform, the Design Working Range (DWR) in the horizontal plane relative to the sea level and the Design Inlet Angle (DIA) in the vertical plane must be determined within the ranges defined by DNV regulations. The applicable ranges of DWR and DIA are schematically illustrated in Figure 3.

As shown in Figure 3, the DWR was fixed at 0°. For the DIA, 10° and 29°—the angle at which the maximum tensile force occurs according to the integrated load analysis results [16]—were selected as the primary input conditions for strength evaluation. The load applied to the Chain stopper under mooring conditions was applied based on the MBL in accordance with DNV regulations; the MBL of the 147 mm studless chain for the 10 MW class adopted in this study is 21,179 kN. Meanwhile, in the towing condition, the towing force of the semi-submersible FOWT is transferred to the FCS. Based on the integrated load analysis results [16], a load of 3434 kN was calculated for the Upper stopper and Chain wheel. These design load conditions are summarized in

Table 1. Design load conditions (according to DNV rule [13,14,15]).

# of Load Case	DWR	DIA	Force	Operation Condition
LC1	0°	10°	21,179 kN	Mooring
LC2	0°	29°	21,179 kN	Mooring
LC3	0°	46°	3434 kN	Towing

. Based on this, FEA-based structural analysis was performed by classifying the conditions into LC1 and LC2 for verifying structural safety during mooring, and LC3 for evaluating the structural integrity during towing.

Mooring systems for offshore structures, such as FCS, are so large that direct verification through structural testing is difficult. Therefore, structural analysis must be performed based on the design load conditions and allowable stress regulations provided by classification societies. Classification societies ensure design safety by applying a safety factor to the allowable stresses, taking into account potential errors in the structural analysis process [13]. Examples of structural analysis of mooring systems for offshore structures based on classification society regulations have been documented in several previous studies [17,18,19]. Additionally, the authors performed structural strength tests on the initial FCS design using a 3-D printed scale model and verified the validity of the numerical analysis model through comparison with the structural analysis model [20].

Finite element modeling and result analysis for FEA were performed using HyperWorks 2021 software [21]. As shown in Figure 4, the mesh was generated with a mesh size of 25 mm, resulting in a total model consisting of 498,879 elements and 384,260 nodes. Considering the efficiency and accuracy of the analysis, the modeling utilized a mixture of shell and solid elements and contact conditions capable of simulating actual physical phenomena were applied to the interfaces between components.

Since FEA-based structural analysis results can be influenced by the finite element size, a convergence test for mesh size is necessary. As shown in Figure 5, the mesh size of the FCS was varied from 20 mm to 50 mm in 5 mm intervals to observe changes in stress results in regions with low stress gradients.

As indicated in Figure 5, the mesh size of the FCS was found to converge at 25 mm, and this element size was applied.

Table 2. Design load conditions.

Material Name	Density [Ton/mm3]	Elastic Modulus [MPa]	Poisson’s Ratio	Yield Stress [MPa]
A694F70 (A694)	7.85 × 10−9	209,000	0.3	485
DH36 (DH)	7.85 × 10−9	209,000	0.3	310
SCM440 (SCM)	7.85 × 10−9	209,000	0.3	834
A148	7.85 × 10−9	209,000	0.3	585
OILESS500-ABR(ABR)	7.4 × 10−9	126,000	0.3	617

presents the mechanical properties of each FCS component selected considering the extreme marine environment. Specifically, SCM440 was applied to the Stopper and Pins, OILESS500-ABR to the Bushings, A148 to the Chain wheel, and DH36 and A694F70 to the main structural parts, respectively. These high-strength alloy steels guarantee excellent mechanical performance through heat treatment, although they are characterized by higher material costs compared to general steel.

The degrees of freedom (DOF) and boundary conditions applied to the FEA model are shown in Figure 6. Boundary conditions were set based on the Main pin connecting to the semi-submersible FOWT platform, constraining all DOFs except for rotational DOF in the direction of gravity to simulate actual behavior. Additionally, physical contact conditions were assigned between the Main pin and the Flange bushing. Particularly for regions where heterogeneous elements—shell and solid elements—are combined, the contact area with shell elements was effectively configured by defining the surface of solid elements as shells. Symmetry conditions were applied to the entire model to maximize numerical computation efficiency.

Load conditions were applied as shown in Figure 7 based on Table 1. In the mooring state, loads were input in the form of bearing loads reflecting the surface contact characteristics between the chain and the Stopper. In the towing state, the analysis was performed by applying distributed loads to the surfaces of the Upper stopper and Chain wheel where the towing force acts.

FEA-based structural analysis was performed using Abaqus/Implicit 2020 [22]. The structural safety review was conducted based on the von Mises equivalent stress, and the acceptance criterion was defined as 90% of the yield strength of each material in accordance with DNV-OS-E301 [13]. Accordingly, the allowable limits for each material were set as follows: 436.5 MPa for A694F70, 279 MPa for DH36, 750.6 MPa for SCM440, 526.5 MPa for A148, and 555.3 MPa for OILESS500-ABR.

Table 3. Structure analysis results.

# of Load Case	Max. Von-Mises Stress (MPa)					Structure Safety
	A694	DH36	SCM	A148	ABR
LC1	291.5	220.5	725.6	7.9	220.8	OK
	(Part 15)	(Part 7)	(Part 3)	(Part 10)	(Part 13)
LC2	290.7	221.3	725.8	8.8	199.6	OK
	(Part 15)	(Part 7)	(Part 3)	(Part 10)	(Part 13)
LC3	120.6	122.7	406.8	168.2	54.5	OK
	(Part 18)	(Part 14)	(Part 11)	(Part 10)	(Part 12)

shows the analysis results for each load condition based on these criteria. The evaluation confirmed that the maximum stresses generated under all load conditions were distributed within the allowable stress limits of each material. This verified that the structural safety of the FCS was sufficiently secured, with the highest stress level observed in the SCM440 component under the LC2 condition. The overall stress distribution behavior of the FCS under design load conditions is illustrated in Figure 8.

As shown in Figure 8a,b, the highest stress concentration area appeared at the Chain stopper in the mooring state of the FCS, and relatively high stresses also occurred at the Arm pin bearing plate and Arm wall plate. As shown in Figure 8c, in the towing state of the FCS, the highest stress concentration area appeared at the Chain wheel pin where the towing force is concentrated.

3. Reliability-Based Robust Design Optimization

3.1. Theoretical Background

The formulation of RBRO is defined as follows [23]:

$$\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e} f\left({\mu }_y\left(x\right),{\sigma }_y\left(x\right)\right)$$

(1)

$$\mathrm{s}.\mathrm{t}. G_i\left({\mu }_y\left(x\right),{\sigma }_y\left(x\right)\right) \le 0$$

$$x_L+n{\sigma }_x \le {\mu }_x \le x_U - n{\sigma }_x$$

In Equation (1), f and G represent the objective function and the probabilistic constraint function, respectively, while μ and σ denote the mean and standard deviation of the random variables. Additionally, x_L and x_U refer to the lower and upper bounds of the design variables, and n indicates the sigma level of quality reliability to be achieved. As presented in Equation (1), RBRO simultaneously considers the uncertainties inherent in design factors and the quality target value n. In other words, it is a technique that searches for an optimal design solution that minimizes the objective function within the range that satisfies both probabilistic constraints and robustness side constraints. To efficiently explore the optimal solution, the Sequential Approximate Optimization (SAO) technique [24], widely used for constrained optimization problems, was adopted.

During the optimization process, the probabilistic constraint function must be quantitatively evaluated through reliability analysis. In this study, the MVRM and the AISM were applied and compared to perform this evaluation.

MVRM is a technique that approximates the mean and standard deviation of the probabilistic constraint function using Taylor series expansion. The precision of reliability analysis in this method depends on the order and method of the series expansion. The reliability index by MVRM is calculated using the following equation, as presented below [25]:

$$\beta = {\displaystyle \frac{{\mu }_x}{{\sigma }_x}}$$

(2)

The reliability index calculated from Equation (2) is converted into the failure probability P_f and reliability R through the following equations. This allows for the quantitative evaluation of the statistical safety of the designed structure.

$$p_f = \Phi \left(-\beta \right)$$

(3)

$$R = PG>0 = 1-p_f$$

(4)

Here, Φ is the standard normal distribution function.

In the AISM, the failure region for the constraint function is approximated as a second-order polynomial using principal curvature, as shown in Figure 9. The curvature value is repeatedly updated to encompass the failure region based on the number of samples existing within it. In Figure 9, the reliability index is denoted as β [26,27].

Specifically, AISM performs a procedure of iteratively updating the curvature value based on the number of samples extracted within the failure region. Through this iterative process, the approximate model is guided to gradually encompass the actual failure region.

3.2. RBRO Results and Discussion

The formulation of RBRO for the structural design of the FCS was defined as follows:

$$\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e} W\left({\mu }_y\left(X_i\right),{\sigma }_y\left(X_i\right)\right), i = 1,2,3,4,5,6,7,8,9$$

(5)

$$\mathrm{s}.\mathrm{t}. G_i\left({\mu }_y\left(X_i\right),{\sigma }_y\left(X_i\right)\right) \le 279 \mathrm{M}\mathrm{P}\mathrm{a}, j=1,2,3$$

$$\begin{gathered} \mathbf{24.0}+\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{1}}} \le {\mathit{\mu }}_{{\mathit{X}}_{\mathbf{1}}} \le \mathbf{36.0}-\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{1}}} \\ \mathbf{240.0}+\mathbf{3}{\mathit{\sigma }}_{\mathbf{2}} \le {\mathit{\mu }}_{{\mathit{X}}_{\mathbf{2}}} \le \mathbf{360.0}-\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{2}}} \\ \mathbf{88.0}+\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{3}}} \le {\mathit{\mu }}_{{\mathit{X}}_{\mathbf{3}}} \le \mathbf{132.0}-\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{3}}} \\ \mathbf{80.0}+\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{4}}} \le {\mathit{\mu }}_{{\mathit{X}}_{\mathbf{4}}} \le \mathbf{120.0}-\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{4}}} \\ \mathbf{80.0}+\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{5}}} \le {\mathit{\mu }}_{{\mathit{X}}_{\mathbf{5}}} \le \mathbf{120.0}-\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{5}}} \\ \mathbf{80.0}+\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{6}}} \le {\mathit{\mu }}_{{\mathit{X}}_{\mathbf{6}}} \le \mathbf{120.0}-\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{6}}} \\ \mathbf{80.0}+\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{7}}} \le {\mathit{\mu }}_{{\mathit{X}}_{\mathbf{7}}} \le \mathbf{120.0}-\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{7}}} \\ \mathbf{104.0}+\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{8}}} \le {\mathit{\mu }}_{{\mathit{X}}_{\mathbf{8}}} \le \mathbf{156.0}-\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{8}}} \\ \mathbf{104.0}+\mathbf{3}{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{9}}} \le {\mathit{\mu }}_{{\mathit{X}}_{\mathbf{9}}} \le \mathbf{156.0}-3{\mathit{\sigma }}_{{\mathit{X}}_{\mathbf{9}}} \end{gathered}$$

As presented in Equation (5), this study applied a 3σ level, corresponding to a reliability of 99.73%, to the side constraints to ensure robustness. Furthermore, to reflect the uncertainties inherent in the FCS design factors, the manufacturing tolerances of major components were defined as uncertainty elements. The detailed settings of the random variables are summarized in

Table 4. Definition of random variables.

Random Variable	RSD [mm]	Distribution Type	Design Variable	Mean [mm]
X1	0.55	Normal	t1	30
X2	2.1	Normal	t2	300
X3	1.7	Normal	t3	110
X4	1.3	Normal	t4	100
X5	1.5	Normal	t5	100
X6	1.5	Normal	t6	100
X7	1.7	Normal	t7	100
X8	1.3	Normal	t8	130
X9	1.3	Normal	t9	130

. As shown in Table 4, the mean values were set to the design thicknesses shown in Figure 10.

The Relative Standard Deviation (RSD) was calculated with reference to the allowable manufacturing tolerances for thickness and width of steel materials specified in KS D 3500 [28], and the probability distribution function was assumed to follow a normal distribution. This assumption best represents the variability in strength performance that may occur due to manufacturing tolerances under normal operating conditions. The probabilistic constraint function was set not to exceed the allowable stress of 279 MPa for DH36 material under three design load conditions, and MVRM and AISM were applied as reliability analysis techniques.

Additionally, to verify the effectiveness of the probabilistic optimal design derived through RBRO, Deterministic Optimization (DO) was performed. The formulation process for DO is as follows:

$$\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e} W\left(t_i\right), i = 1,2,3,4,5,6,7,8,9$$

(6)

$$\mathrm{s}.\mathrm{t}. g_j\left(t_i\right) \le 279 \mathrm{M}\mathrm{P}\mathrm{a}, j=1,2,3$$

$$\begin{gathered} 24.0 \le t_1 \le 36.0 \\ 240.0 \le t_2 \le 360.0 \\ 88.0 \le t_3 \le 132.0 \\ 80.0 \le t_4 \le 120.0 \\ 80.0 \le t_5 \le 120.0 \\ 80.0 \le t_6 \le 120.0 \\ 80.0 \le t_7 \le 120.0 \\ 104.0 \le t_8 \le 156.0 \\ 104.0 \le t_9 \le 156.0 \end{gathered}$$

The DO model presented in Equation (6) was formulated to search for the optimal solution using the SAO algorithm, excluding reliability and robustness constraints. In this study, this deterministic design was used as a baseline for comparison with the probabilistic optimal design results of RBRO according to the applied reliability analysis techniques. Specific comparative figures and results are detailed in

Table 5. Comparison of design optimum (unit: mm).

Random Variable	RBRO (Sigma Level)		DO
	MVRM	AISM	SAO
X1	28.8 (3.0)	29.9 (3.0)	25.0
X2	299.2 (3.1)	304.2 (3.0)	284.5
X3	88.4 (3.0)	88.9 (3.0)	88.0
X4	94.2 (3.2)	94.9 (3.1)	87.5
X5	80.0 (3.1)	80.0 (3.1)	80.0
X6	80.0 (3.0)	81.1 (3.2)	80.0
X7	80.9 (3.0)	81.2 (3.0)	80.0
X8	126.7 (3.0)	126.8 (3.4)	104.0
X9	122.2 (3.2)	125.4 (3.0)	104.7

and

Table 6. Comparison of objective and constraint function.

Method	Constraints (MPa)			Obj. [Ton]	# of F.E (1)
	LC1	LC2	LC3
RBRO (MVRM)	262.0	266.3	220.7	33.4	7630
RBRO (AISM)	258.4	262.2	220.1	33.6	14,497
DO (SAO)	274.3	278.9	219.0	33.0	401

⁽¹⁾ FE: function evaluation.

.

As can be confirmed from the results in Table 5 and Table 6, both MVRM and AISM-based RBRO yielded more conservative design results compared to DO. While MVRM and AISM consider the sigma level of design factors, DO does not. As a result, it is analyzed that DO shows a distinct difference from MVRM and AISM-based methods in terms of constraint proximity of the main response and convergence characteristics of the objective function. As shown in Figure 2, based on an initial weight of 36.4 tons, DO achieved the greatest weight reduction of 9.5%, while MVRM-based RBRO and AISM-based RBRO achieved weight reductions of 8.0% and 7.6%, respectively. Notably, the deterministic design exhibited behavior close to the constraints under LC1 and LC2 conditions, while the RBRO results secured a safety margin of more than 4%, demonstrating robustness against uncertainty.

Comparison between the two reliability analysis techniques revealed that AISM exhibited a more conservative tendency than MVRM. The design variables for the Arm section (t1~t6) were at similar levels, but distinct differences occurred in the Housing section (t7~t9). Specifically, when AISM was applied, the sigma level of t8 increased, whereas t9 showed a decrease of about 6%, suggesting that the robustness allocation for individual design variables may vary depending on the reliability analysis technique. Meanwhile, the computational cost (number of function evaluations) was highest for AISM. This is attributed to the algorithmic characteristic of AISM, which requires repetitive sampling to explore the failure region during the convergence process, unlike MVRM which obtains an approximate solution via Taylor series expansion.

The optimization convergence characteristics are compared and plotted in Figure 11.

Examining the convergence history in Figure 11, RBRO shows distinctly different convergence characteristics from DO. This is because a reliability analysis loop is included in the optimization search process of RBRO. Comparing the two reliability analysis techniques, AISM showed a more conservative convergence trend across the objective function and constraint functions than MVRM. In particular, relatively large fluctuations were observed during the convergence process of AISM, which is analyzed to be due to the expansion of the design search region as the number of samplings increased with the progress of convergence steps. On the other hand, DO, which did not consider uncertainty, showed a result where the constraint of LC2 reached the structural safety limit at the optimal point.

In conclusion, considering both weight reduction efficiency and computational cost, the MVRM-based RBRO technique is judged to be the most suitable for the probabilistic optimal design of the FCS in this study.

4. Conclusions

This study proposed a RBRO framework for the FCS of a 10 MW FOWT, incorporating uncertainties in manufacturing tolerances. The probabilistic design performance was evaluated using the MVRM and the AISM. The key conclusions are as follows:

• While Deterministic Optimization (DO) achieved the highest weight reduction of approximately 9.5% compared to the initial design, it was confirmed to have a high risk of exceeding structural safety limits under specific load conditions (LC2) due to the lack of uncertainty consideration. In contrast, the RBRO technique proposed in this study successfully derived a design that guarantees stable performance even under the variability of design variables by setting a target reliability (β = 3.0) for strength and a 3-sigma level of robustness as constraints.
• Comparing the optimization results according to reliability analysis techniques, the MVRM-based RBRO achieved an 8.0% weight reduction, while the AISM-based RBRO achieved a 7.6% reduction. Although AISM presented the most conservative design due to its high accuracy in tail distribution, the MVRM-based design also sufficiently satisfied the target reliability while achieving a lightweight effect close to the DO result. This suggests that a probabilistic approach is essential to maximize economic feasibility while ensuring structural safety.
• Analyzing the trade-off between computational cost and accuracy, this study concludes that the MVRM-based RBRO technique is the most rational and practically efficient design strategy for large-scale offshore structures, including FOWT components: compared to the high-precision but computationally intensive AISM, MVRM offers substantially higher numerical efficiency while maintaining comparable reliability, making it a scalable and universally applicable methodology for reducing time and cost in the initial design phase of complex offshore systems.
• This study focused on structural response analysis under static load conditions. However, in real-world marine environments, dynamic behavior and fatigue damage due to wave and wind loads are key design factors. Although this paper does not directly address these dynamic and fatigue behaviors, future research will extend this research to ensure structural safety throughout the lifecycle of floating offshore wind turbine (FOWT) mooring systems by performing fatigue reliability-based optimal design using a surrogate model coupled with time-domain dynamic analysis.