In the field of wind power generation, as development progresses into deeper waters, the advantages of floating offshore wind turbines (FOWT) become increasingly apparent. Due to the large displacement of FOWTs, using the frequency domain Fast Fourier Transform (FFT) for wave load analysis can lead to significant errors. This is because FFT can only handle wave loads for fixed positions, making time-domain wave load analysis necessary. However, computational time becomes critical, particularly when employing second-order irregular wave theory. In the study of second-order irregular waves, Longuet-Higgins [1] proposed a method for calculating second-order wave interaction in infinite water depth in 1963. Dalzell [2] employed symbolic computations to handle the abundance of hyperbolic functions that promptly emerge in finite depth issues in Longuet-Higgins’ investigation. Johannessen and Swan [3] conducted a laboratory study where they focused on various waves at a specific point to create a significant transient wave group, comparing measurements of water-surface elevation and water-particle kinematics with both linear and second-order solutions based on interactions identified by Longuethiggins & Stewart [4]. Stansberg [5] further expanded the work of Longuethiggins & Stewart [4] by providing a hybrid model to estimate nonlinear interactions in random seas and validating second-order wave theory through model basin measurements. Bateman et al. [6] utilized the experimental results of Johannessen and Swan [3] as a validation of the numerical method. Additionally, Stansberg [7] investigated non-Gaussian extremes in numerically generated second-order random waves in deep water, offering insights into the statistical behavior of extreme wave crests under second-order wave theory. Zheng et al. [8] proposed a method for non-Gaussian random wave simulation using two-dimensional Fourier transforms, which allows efficient calculations of random wave interactions and their effects on structural responses. Building upon this, Zheng and Moan [9] explored the dynamics of freak waves within third-order models, further advancing the understanding of complex wave phenomena. Lee [10], in the development of the WAMIT utility, introduced the F2T module, which applies Fourier techniques to enhance the computational efficiency of second-order wave force calculations, particularly for floating structures. In 1981, Sharma and Dean [11] developed a theory to determine the second-order irregular waves and extended the calculations to intermediate water depths. Since then, numerous studies have been built upon this theory. Forristall [12] provided an accurate second-order superharmonic solution based on the theory of Sharma and Dean. Tromans and Vanderschuren [13] utilized the wave model introduced by Sharma and Dean to incorporate second-order nonlinearity in steep waves. They applied this model to calculate the probability of exceedance of crest elevation, and the outcomes demonstrated agreement with Forristall’s simulations. Toffoli et al. [14] explored the characteristics of second-order low-frequency response by adopting Sharma and Dean’s theory, substantiating their analysis with field measurements obtained from Lake George in water of finite depth. Ju et al. [15–17] also employed the theory of Sharma and Dean to analyze second-order irregular wave forces in FOWTs. Bredmose et al. [18] derived fully dispersive deterministic evolution equations and compared the second-order transfer functions to the governing equations provided by Sharma and Dean. From the above research review, it can be observed that the second-order irregular wave theory devised by Sharma and Dean has been extensively utilized and verified, affirming its stability and effectiveness.

In addition to employing the theory of Sharma and Dean, Forristall [19] formulated the kinematic boundary condition fit method to estimate the kinematics of irregular waves and determine a potential function. Schäffer [20] derived a comprehensive second-order mathematical model of irregular waves based on the potential flow hypothesis and experimentally validated the theory. Wang et al. [21] introduce an improved method for generating rogue waves by temporally and spatially focusing wave energy, enhancing the accuracy of wave formation in random seas by correcting spectral artifacts in previous approaches. In terms of time-domain wave force calculation, Xia et al. [22] developed an efficient nonlinear hydroelastic approach for analyzing wave- and slamming-induced vertical motions and structural responses of ships utilizing time-domain strip theory. Yang et al. [23] described the full second-order coupling theory of irregular waves in detail and conducted theoretical derivation, application evaluation, and experimental verification. Li [24] proposed a hybrid numerical wave absorption method to accurately simulate open sea conditions in fully nonlinear fluid-structure interactions, validating its efficiency through 3D numerical wave tank experiments and comparisons with physical tests. Xu et al. [25] proposed a time-domain second-order method to simulate three-dimensional wave-body interactions. There are also some studies related to frequency domain wave force analysis. By analyzing the second-order hydrodynamic forces generated through the interaction of first-order wave potentials, Chen et al. [26] formulated and solved hydro-elastic equations in the frequency domain. Based on the second-order Stokes theory and incorporating Rainey’s force model [27], Bredmose and Pegalajar-Jurado [28] developed an accelerated method for computing second-order monopile wave loads.

In many numerical analyses of FOWTs, Morison’s equation [29] is commonly used to calculate wave loads acting on slender structures. Morison’s equation, introduced by Morison et al., is a semi-empirical equation designed to address the hydrodynamic loads experienced by slender structures situated at the region that may be hit by wave. The hydrodynamic forces in Morison’s equation consist of two components: the drag force caused by the water particle velocity and the inertia force caused by the water particle acceleration. This equation has found extensive application in calculating hydrodynamic loads on slender cylindrical bodies immersed in large bodies of water. However, due to the assumption of uniform flow acceleration, Morison’s equation has some limitations and is only accurately estimated when applied to compact bodies [30]. On the other hand, some researchers use quadratic transfer functions (QTFs) to calculate second-order wave forces [31,32]. The complete QTF method requires solving the second-order problem, which leads to an increase in computational time. Although Newman’s approximation method [33] can reduce the amount of calculation, it has accuracy issues in shallow water areas and low-frequency circumstances [34]. Another problem is that if FOWT requires large displacement finite element analysis, where structural positions change with time, Morison’s equation can be more flexible.

In research on offshore wind turbines (OWT) analysis, the benchmark often used for validation came from the FAST [35] or OpenFAST [36] program developed by the National Renewable Energy Laboratory (NREL). It is an aero-hydro-elastic coupled program that can perform numerical simulations of wind turbines. In the program, analysis in different fields is broken down into different calculation modules. Some researchers calculated wave forces using the HydroDyn module of FAST or OpenFAST [37–39]. The theoretical basis of this module operation is based on Duarte et al. [40]. Through experiments, Coulling et al. [41] proved the correctness of FAST in FOWT analysis. Other commercial simulation software such as OrcaFlex [42] and AQWA [43] are also robust tools for hydrodynamic analysis, commonly used in various OWT studies [44–48].

In terms of experiments on FOWTs, Hallak et al. [49] verified the accuracy of the numerical simulations through scaled experiments and found that in the simulation of irregular waves, the analysis results in the coupled time domain were relatively accurate. Coulling et al. [50] conducted scale tests and verified the measured data with the numerical simulations using FAST and showed that the simulations considering the effects of second-order waves on hydrodynamic forces and moments provide better accuracy close to experimental results. The influence of second-order wave force on FOWTs has also been studied by Duarte et al. [51], and the research results were applied to FAST. Zhang et al. [34] discussed and compared the effects of second-order wave loads on platform motions and mooring tensions by using different methods including Newman’s approximation and the full QTF method and found that the second-order wave loads may cause resonance and lead to structural failure. Mei and Xiong [52] also found through numerical simulations that the effect caused by the second-order wave cannot be ignored. Otherwise, the low-frequency response may be underestimated [53]. Additionally, some numerical studies have been conducted on irregular waves acting on OWTs through the frequency domain FFT method for fixed structures [54,55]. However, the computational cost in second-order FFT numerical calculations remains a severe problem.

Indeed, leveraging parallel computation can significantly mitigate the computational time issue associated with time-domain calculations, especially as modern computer Central Processing Units (CPU) feature an increasing number of cores. Therefore, this article will primarily explore using parallel operations to analyze the response of FOWTs subjected to irregular wave forces. Additionally, a method utilizing different random numbers between two FFT segments has been proposed to decrease CPU time to calculate second-order irregular wave loads substantially.

In this section, the irregular wave formulation from the reference [11] was first rewritten to the FFT form. Subsequently, this formulation was directly employed to obtain the time-domain result without necessitating the use of inverse FFT. Furthermore, the equations are split into dependent and independent terms of the x, y, and z coordinates, so computational efficiency can be performed for moving coordinate systems, such as large-displacement finite element analyses. If the time step is Δt and N_t time steps are used, the total time (T), maximum frequency (f_max), and the incremental frequency (Δf) of the simulated wave can be expressed as follows: (1)T=NtΔt,fmax=1/(2Δt),Δf=1/T=fmax/(Nt/2)=fmax/Nt2,where Nt2=Nt/2 is the number of required FFT terms for a real time-domain function due to the symmetric condition. The following FFT formulation can be used to determine the wave height, velocities, and accelerations at a specific location (x, y, z) at time t. (2)F(t)=∑j=0Nt−1(aj+bjic)exp⁡(2πfjt)ic=∑j=0Nt−1(aj+bjic)exp⁡(2πjk/Nt)icwhere ic=−1, aj and bj represent functions of x, y, z, and wave spectrum data, fj=jΔf=j/T, and t=kΔt=kT/Nt. It is evident that F(t) exhibits periodicity with a period T or N_t time steps. Moreover, for real F(t), the following symmetric form should be obeyed. (3)b0=bNt2=aNt2=0, and aNt−j−bNt−jic=aj+bjic forj=1 to Nt2−1.

Thus, the above equation changes to: (4)F(t)=a0+Re(2∑j=1Nt2−1(aj+bjic)exp⁡(2πjkic/Nt))

According to the reference, the FFT parameters aj and bj for the first-order and second-order formulations related to wave height, velocities, and accelerations can be determined in the subsequent sections, where those formulations were arranged to be suitable for the time-domain calculation. This section also defines various variables utilized in the subsequent sections. Throughout the paper, i and j represent the indices of wave frequency, while m and n denote the indices of wave direction, with k uniformly used to represent the number of time steps. For a wave direction characterized by an angle θm measured from the X axis, the wave number k→pq is defined as below, where p is index i or j, and q is index m or n. (5)k→pq=(kpqx,kpqy)=(|k→p|cos⁡θq,|k→p|sin⁡θq), (6)Rp=|k→p|tanh⁡(|k→p|d),and (7)Sp=|k→p|2−Rp2 (8)where|k→p|=2π/λp forawave frequency fp,and DNV-RP-C205 was used to determine λp, requiring only f_p and the water depth (d). It is noted that Eqs. (5) to (8) depend solely on the wave frequency and direction, being independent of the position.

The frequency-domain method, employing the inverse FFT, is efficient. This is because the FFT terms of nodal wave loads, as presented in Section 2.1, only need to be calculated once, and the inverse FFT is subsequently employed to obtain all the time-history wave loads. Conversely, the time-domain method requires the computation of FFT terms at each time step. However, the frequency-domain method is applicable only for small-displacement finite element analysis, since the nodal coordinates x, y, and z in the equations of Section 2.1 must remain fixed for the inverse FFT to be valid. Therefore, for the wave loads of FOWT support structures with a large displacement effect, the time-domain method is the preferred choice. In this paper, the following traditional schemes were proposed to enhance the computational efficiency of the time-domain method:

(1) Eliminating high-frequency terms

The second order model has demonstrated a good fit to experimental data when a cut-off frequency (fcut−off) equal to 2g/Hs/2π is applied [58]. This criterion was empirical rather than theoretical, and the recommended practice from Det Norske Veritas (DNV) [58] suggests this cut-off frequency equal to 4 times the spectral peak frequency. Despite its empirical nature, employing a cut-off frequency can significantly reduce the computational cost of second-order irregular wave loads. For example, if N_t2 is set to 8192 with fcut−off=2g/Hs/2π, the number of required frequencies is only 264. Consequently, the second-order irregular wave load calculation can be significantly accelerated, approximately (8192/264)² = 963 times.

(2) Employing buffers to store the calculation results independent of x, y, and z coordinates

From the equations in Section 2.1, one can clearly understand the variable dependence. Variables with i and j are contingent on frequencies, those with m and n depend on wave direction, and the bolded equations with x, y, and z are linked to nodal coordinates. Variables independent of coordinates can be computed once and stored in a buffer for all the nodes and time steps. Subsequently, for each time step, the coordinate-dependent variables are calculated to determine those FFT parameters. For example, Eq. (28) can be modified to: al=∑l=1NbufferE(imjn)lcosh⁡kimjn−(d+z)cos⁡(rp−ijnm)where E(imjn)l=gs8[cicjDnDm1cosh⁡kimjn−d⋅Dimjn−(2π/Ti−2π/Tj)(kimx−kjnx)]l, and Nbuffer is the total number of iterations for i−j=l and i≥j under l=1 to Nt2. It is noted that variables E(imjn)l are computed only once for all the nodes and time steps.

(3) Parallel computation of element wave forces

The equations presented in Section 2.1 are not interdependent across distinct nodal coordinates, making parallel computation efficient at each node. Given that OWT structural analyses frequently demand the calculation of numerous nodal wave loads at each time step, complete parallelization proves to be highly efficient.

(4) Extrapolation of element wave forces over a defined number of time steps (N_extra)

Although wave loads are low-frequency, the time step length must still be sufficiently small to accommodate other types of loads. Therefore, an alternative is the extrapolation of wave loads over a certain number of time steps (N_extra), providing a balance between computational efficiency and accuracy. In this study, a quadratic exploitation using the wave loads from the three previous time steps was employed to determine the current wave loads for the next time step. For example, with two extrapolations and a calculation, the data from time steps n, n + 3, and n + 6 are used to extrapolate the data for steps n + 7 and n + 8, after which the data for step n + 9 is subsequently calculated.

Schemes (1) and (2) are essential procedures, and this study will exclusively examine the efficiency of schemes (3) and (4) by using a FOWT support structure. A semi-submersible platform was selected for this investigation because it features a high number of structural degrees of freedom and nodes, which leads to a heavier computational burden. This characteristic makes it an ideal case to highlight the benefits of the proposed acceleration techniques. In Fig. 4, the setup employed for this investigation involves the integration of the IEA 15 MW wind turbine [59] onto a DeepCwind semi-submersible platform [60] with four columns, positioned in waters with a depth of 100 m. The central column provides support for the tower, while the remaining three floating columns contribute additional stability. Structural integrity is preserved by interconnecting the pontoons through a system of 15 braces. To model the columns and braces, two-node Timoshenko beam elements were employed. These elements encompass functions that represent the floating stiffness at the sea surface, water-effective mass, and the load on the floating element. Key dimensions of both the tower and platform are specified in Fig. 4. A loading condition of IEC 61400-3-1 [61] Design Load Cases (DLC) 6.2 (parked (standing still or idling)) was employed, including a significant wave height (H_s) of 10 m, a wave direction from the X-axis (θ_wave) at 30 degrees, an average hub wind speed (V_hub) of 57 m/s, wind direction from the X-axis (θ_wind) at 30 degrees, and a misalignment of the yaw system (θ_yaw) at 90 degrees. These directions are illustrated in Fig. 4. The Newmark’s integration method was employed to conduct the large-displacement dynamic finite element analysis, utilizing a time step length of 0.05 s and a total time of 700 s. The following two ratios were defined to assess accuracy and CPU time under various conditions. (38)Accuracy ratio=∑k=1all time steps|Fk(Nextra=1)−Fk|/∑k=1all time steps|Fk(Nextra=1)| (39)CPU ratio =CPU time of current condition/CPU time with Nextra=1and all CPU threadswhere F_k is the x-direction total wave force from all the members at time step k.

The values of these two ratios under various conditions are presented in Table 1, illustrating the following characteristics:

(1)

Regarding parallel computing, the number of CPU threads presented in Table 1 is only specified for wave load calculations, with other components being fully parallelized across all CPU threads. The efficiency of parallel computing is evident from Table 1, demonstrating that a threefold speedup can be achieved on a personal computer. Additionally, the table highlights that the computation of wave loads constitutes a significant portion of the CPU time, underscoring the necessity of parallel computation. The parallel procedure executed during nodal wave load calculation is the most effective approach.

This study collected and presented the explicit FFT parameters for second-order irregular wave height and loads from the research results of Sharma and Dean and applied them to both frequency-domain and time-domain methods. Given that FOWT support structures experience significant displacement effects under wave loads, the time-domain wave method becomes suitable for this problem. However, the time-domain method entails higher CPU time requirements than the frequency-domain method. To address this issue, the paper employs several numerical schemes, including efficient parallel computing and the extrapolation of element wave forces during specific time steps. While those schemes prove effective, the computational challenges persist due to the CPU time requirements in the time-domain wave method. Therefore, this study proposes a straightforward approach utilizing different random numbers in distinct FFT segments, aiming to significantly reduce the computation of second-order wave loads. Furthermore, this random number interpolation facilitates the creation of a continuous curve between two FFT segments. The paper also recommends interpolating near the end of the first FFT segment to minimize errors. Numerical studies demonstrate a significant reduction in the CPU times of FOWT structural analyses through the proposed method. Moreover, the steel design results do not vary excessively. Implementing this method, which involves using different random numbers between two FFT segments, is not complicated, as it only requires a simple subprogram using Eqs. (40) and (41) to modify a traditional wave load computer program. Furthermore, the proposed scheme can be used for time-domain methods for any physical problem that is not suitable for the use of IFFT. While the method was demonstrated using a semi-submersible platform with high structural complexity, it is also applicable to other FOWT support structures. However, the efficiency of the proposed extrapolation and segmentation techniques may be influenced by the specific dynamic behaviors and structural layouts of different platforms, and this warrants further study. Despite the advantages of the proposed method, several limitations should be noted. First, the time-domain wave load calculation remains computationally more expensive than frequency-domain approaches using IFFT, particularly for large-scale simulations. Second, although dividing the simulation into multiple FFT segments with interpolated random numbers can significantly reduce computation time, each segment still requires a non-trivial amount of computation. If the number of time steps per segment is too large, the efficiency gain diminishes. In this study, a segment size of 16,384 time steps was found to be a suitable balance between accuracy and computational cost. In addition, future extensions of this work may consider incorporating wave–structure interactions among multiple FOWTs in wind farms, which would allow further evaluation of the method’s scalability and applicability to cluster-level simulations.