# Prediction of the extreme slow-drift response of moored floating structures using design waves

Athanasios Dermatis, Benjamin Bouscasse, Guillaume Ducrozet, Henrik Bredmose, Harry B Bingham

<figure><img src="image_1.png" /></figure>

## ► To cite this version:

Athanasios Dermatis, Benjamin Bouscasse, Guillaume Ducrozet, Henrik Bredmose, Harry B Bingham. Prediction of the extreme slow-drift response of moored floating structures using design waves. Ocean Engineering, 2025, 333, pp.121456. 10.1016/j.oceaneng.2025.121456. hal-05068363

HAL Id: hal-05068363

https://hal.science/hal-05068363v1

Submitted on 15 May 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_3.png" /></figure>

Research paper

# Prediction of the extreme slow-drift response of moored floating structures using design waves

Athanasios Dermatis<sup>a,†</sup>, Benjamin Bouscasse<sup>a</sup>, Guillaume Ducrozet<sup>a,∗</sup>, Henrik Bredmose<sup>b,†</sup>, Harry B. Bingham<sup>c</sup>

## ARTICLE INFO

**Keywords:**

Extreme response prediction

Second-order responses

Slow-drift motions

Response-conditioned waves

Design waves

## ABSTRACT

Design waves have been used in the past for the probabilistic assessment of wave-induced loads and responses of offshore structures. Various response-conditioning techniques have been employed to determine suitable wave episodes, typically based on linear response transfer functions. Nevertheless, extreme events are not always driven by linear phenomena but can be triggered by near-resonant effects, as in the case of the slow-drift motions of moored floating bodies. Limited research has been devoted to addressing this class of responses using response-conditioned waves (RCW). This paper presents a new approach for deriving RCWs that accounts for combined wave- and low-frequency responses. Both the response amplitude operator (RAO) and the quadratic transfer function (QTF) are employed in an iterative response-conditioning procedure. That permits the identification of appropriate short-duration wave episodes that excite resonant slow-drift motions. These wave episodes are then used in a two-step multi-fidelity design wave methodology for the probabilistic evaluation of the fully nonlinear extreme responses. The proposed approach is validated experimentally for predicting the surge excursions of a moored container ship, and good agreement is found against Monte Carlo results in irregular waves.

## 1. Introduction

An important aspect in the design of moored floating structures is the occurrence of large amplitude slow-drift motions in the horizontal plane. Due to the coupling of the structure with the mooring arrangement, the natural frequencies of these degrees of freedom are usually very low. Moreover, the absence of damping in this frequency range, results in considerable resonant motions under second-order difference-frequency loads. During the past decades, efforts have been made by researchers to accurately model these loads (Pinkster, 1980), as well as the associated motions in both the frequency- (Langley, 1987) and the time-domain (Langley, 1986; Emmerhoff and Sclavounos, 1996). Recently, the subject has gained new urgency in the offshore wind sector (Pegalajar-Jurado and Bredmose, 2019; Orszaghova et al., 2021), due to floating offshore wind turbines (FOWT) being installed in regions with harsh environmental conditions and the need to increase their size and capacity (Zhao et al., 2020).

When it comes to accurate estimation of design loads, industrial standards (Veritas, 2019; DNV, 2010) usually recommend the assessment of a large number of different sea states and operating conditions. For re-

sponses that are principally driven by wave-frequency excitation, and during the initial stages of the design procedure, linear potential-flow theory can be employed as an efficient approximation. For responses with extremes which are excited by near-resonant effects, such as the slowly varying motions of moored floating structures, second-order forces and motions must also be accounted for. Assuming a linear wave field, the free-surface elevation will be governed by Gaussian statistics, and for a linear dynamic system, the first-order response statistics will also follow the Gaussian distribution. Nevertheless, the statistics of the total first and second-order responses cannot be determined straightforwardly, due to the nonlinear relationship of the response with respect to the waves. On that subject, Neal (1974) used a second-order Volterra series to describe the total response of a moored vessel and determined the characteristic function of that response, applying the analysis of Kac and Siegert (1947). Given that the probability density function (PDF) and the characteristic function are Fourier Transform pairs, the former can be obtained through an inverse Fast Fourier Transform (FFT). However, Neal (1974) and Naess (1986) concluded that this PDF cannot be expressed explicitly and must be evaluated numerically, while Naess (1986) derived a closed-form expression for the upcrossing rate of pure

<footnote><sup>a</sup> Nantes Université, École Centrale Nantes, CNRS, LHEEA, UMR 6598, Nantes, France</footnote>

<footnote><sup>b</sup> Department of Wind and Energy Systems, Technical University of Denmark, Kgs. Lyngby, Denmark</footnote>

<footnote><sup>c</sup> Department of Civil and Mechanical Engineering, Technical University of Denmark, Kgs. Lyngby, Denmark</footnote>

<footnote>* Corresponding author.
E-mail address: guillaume.ducrozet@ec-nantes.fr (G. Ducrozet).</footnote>
second-order motion under a slow-drift approximation. On the statistics of the total second-order motion, Langley and McWilliam (1993) expressed the PDF of the total response in terms of a series, the first term of which corresponds to the assumption that the first and second-order responses are statistically independent. They found that for a system of linear oscillators, it was sufficient to consider only the first term, implying weak dependence between the linear and quadratic responses. Later, Naess (2001) and Naess et al. (2006) developed numerical procedures for the accurate estimation of the upcrossing rate of the total second-order response using the saddle point method. Nevertheless, the preceding analysis concerns a linear Single-Degree-of-Freedom (SDoF) system, subject to first- and second-order forces, and does not apply to highly nonlinear dynamical systems. Therefore, these approaches do not permit coupled motion analysis or the inclusion of nonlinear damping and restoring terms, which can significantly contribute to the resulting motions. Under these conditions, Naess et al. (2007) concluded that the most reliable approach for the combined response problem is the Monte Carlo approach and developed an extrapolation technique that significantly reduces the computational cost of that method.

In evaluating the extreme responses that lie in the tail of the short-term distributions, low-fidelity models are insufficient due to the significant approximations they introduce for the sake of efficiency. For that purpose, the most reliable approach is either model tests or high-fidelity numerical tools such as Computational Fluid Dynamics (CFD). To implement those tools within a Monte Carlo approach for each sea state, long irregular wave tests or simulations are required to obtain statistically sound results (van Essen et al., 2023). The required test duration is even longer for slow-drift motions with large periods. As a result, the application of the Monte Carlo method in model tests is very costly, and in the case of CFD, it might even be cost-prohibitive. As an alternative, for the accurate but efficient evaluation of the short-term extreme responses, the focus has lately shifted towards multi-fidelity approaches (van Essen and Seyffert, 2023). One such approach is the use of design waves, which have been lately investigated for various configurations (Drummen et al., 2009; Ripe and Lande-Sundall, 2023; Brown et al., 2023; Hansen et al., 2025b), due to their efficiency. These wave episodes are calculated using a low-fidelity/surrogate model within a response-conditioning procedure. They are thus also termed response-conditioned waves (RCW) and they can be reproduced with a high-fidelity model to obtain the fully nonlinear response distribution as a perturbation of the surrogate response distribution (Pastoor, 2002; Drummen et al., 2009).

The concept of conditional waves was first introduced by Tromans et al. (1991) with the NewWave model. A conditional mean process was used to describe the most probable shape around a wave crest, assuming that the free-surface elevation is a zero-mean stationary Gaussian process. However, as observed by many researchers (Hann et al., 2018; Brown et al., 2023; Ripe and Lande-Sundall, 2023) the use of such waves is not adequate when the memory effect of the structure is strong. In these cases, the most likely extreme response (MLER) method (Adegeest et al., 1998) or the most likely response wave (MLRW) (Dietz, 2004), are more appropriate since they utilize a linear motion response transfer function to identify the wave sequence that produces a specified response amplitude. Nevertheless, the use of linear transfer functions within the response-conditioning limits the applicability of these approaches to cases where extreme nonlinear responses are close to the linear responses. A different approach for finding conditional wave episodes is the First Order Reliability Method (FORM) (Der Kiureghian, 2000), which performs the response-conditioning by considering a limit-state function, as shown in Jensen and Capul (2006) and Jensen et al. (2014). Thus, given that no assumptions are made about the nature of the waves or the response, both nonlinear waves and nonlinear response models can be employed to determine the RCWs. Along these lines, Takami et al. (2023) used a nonlinear Higher-Order Spectral (HOS) wave model, coupled with FORM for the prediction of extreme wave-induced loads and motions using the up-crossing rate formulation of Jensen and Capul (2006). Closer to a design wave framework, Ghadirian and

Bredmose (2019) developed a response-conditioning technique based on the FORM, coupled with fully nonlinear wave kinematics using the OceanWave3D model (Engsig-Karup et al., 2009). In addition, Kim et al. (2022) used a similar iterative response-conditioning technique coupled with the nonlinear wave solver HOS-NWT (Ducrozet et al., 2012). The wave episodes obtained were implemented experimentally for the prediction of the wave crest distribution. The same method was extended for the prediction of the nonlinear vertical bending moment (VBM) of a rigid container ship in Kim (2022) and Dermatis et al. (2025), as well as for the nacelle acceleration of a SPAR-type FOWT in Dermatis et al. (2024). The RCWs in these studies were employed as design waves in a multi-fidelity methodology that consists of two steps. First, a simple and efficient surrogate response model is used to obtain an approximate initial response distribution. During the same step, a dedicated iterative procedure uses the same surrogate to determine appropriate conditional wave episodes. The output of this procedure is the wave-maker time history, which can be directly implemented in a physical or a CFD-based numerical wave tank. Therefore, during the second step, the wave episodes are reproduced with those high-fidelity tools and the fully nonlinear response is obtained. Eventually, the results of these short-duration experiments or simulations can be used to re-evaluate the preliminary surrogate response distribution.

In most practical applications that involve the assessment of low-frequency resonant-driven responses of offshore structures, such as the mooring design of FOWTs, the focus is on the mooring line tensions induced by the large-amplitude horizontal motions. A prerequisite for applying design wave approaches to such responses is to ensure that second-order motions can be properly accounted for. Despite the known limitation of the response-conditioning techniques that use linear transfer functions to address such responses (Tosdevin et al., 2022; Bouscasse et al., 2024), limited research has been carried out on that subject. Lim and Kim (2018) introduced a response-conditioning approach for pure second-order responses, providing multiple wave episodes instead of a single design wave. In the same study, the authors employed FORM for response-conditioning by using the slow-drift motion in the limit state function. Both approaches were used to predict the pure second-order motions of a semi-submersible FOWT and the design wave results compared well against a numerical Monte Carlo approach in irregular waves. Moreover, Lande-Sundall and Stansby (2024) introduced the concept of Low-frequency Extreme Response (LFER) design waves through appropriate tuning of the wave phases of an SDoF response model. Under these terms, the difference frequency components provide a maximum surge displacement led at a given instant in time. However, the wave episodes thus obtained led to a highly transient sub-harmonic forcing and the resonant slow-drift motion could not be attained by considering only a short wave episode. For that reason, the authors employed a genetic algorithm to optimally redistribute the energy of the single-wave episode to multiple consecutive wave groups. As a result, the short-term extreme responses were obtained with the use of several wave groups, the total duration of which was a fraction of the sea state duration, effectively reducing the computational time. It should be emphasised that this approach is classified here as a design wave approach in a broad sense, but does not involve the use of RCW, since the value of the generated response is not predetermined. Finally, Hansen et al. (2025a), derived design waves for the surge and pitch motions of a FOWT, by averaging the experimentally measured response signals and associated wave paddle motion from a random sea state realisation. The design waves were reproduced in the wave tank, and the resulting responses were compared against the irregular response signals. Close agreement was found for the wave-frequency surge and pitch, as well as the low-frequency pitch motions. Nevertheless, due to the transient nature of the design waves, they failed to reproduce the horizontal mean drift of the irregular signal. Moreover, the slow drift was developed within a wave group, and thus, the memory effects of the response were neglected, while the damping was unphysically low after the passage of the wave. The authors concluded that these features pose a limitation to the
direct use of design wave episodes for combined wave- and low-frequency responses.

The main novelty of the present work is that it provides a response-conditioning technique for combined wave- and low-frequency responses. Moreover, within this conditioning procedure, the waves propagate inside a nonlinear numerical wave tank. This allows for the identification of nonlinear design wave episodes, that can be straightforwardly reproduced in experiments or CFD. In continuation of the work of Kim et al. (2022) and Dermatis et al. (2025) the present paper aims at extending the multi-fidelity framework to responses that are principally driven by difference-frequency wave loads. The article proceeds with an overview of the design wave methodology in Section 2, followed by a description of the second-order surrogate response model for the total wave- and low-frequency motion in Section 3. The experimental setup and environmental conditions are outlined in Section 4, while Section 5 presents the results of the study and the relevant discussion. Finally, the conclusions of the present work are drawn in Section 6.

## 2. Multi-fidelity design wave methodology

The methodology consists of two steps, the first of which entails the use of an efficient low-fidelity surrogate model to give a preliminary estimate of the extreme response distribution. Within the same step, several response values are used to calculate the design wave through a response-conditioning procedure (HOS-RCW). The second step involves the evaluation of the fully nonlinear response of the structure to those design wave episodes using a high-fidelity model, which can be physical model tests or CFD simulations. Finally, the measured fully nonlinear responses are used to evaluate the preliminary response distribution.

### 2.1. High order spectral response-conditioned waves (HOS-RCW)

#### 2.1.1. Nonlinear numerical wave tank

The High-Order Spectral (HOS) method (Dommermuth and Yue, 1987; West et al., 1987) provides an efficient numerical approach for solving nonlinear wave propagation in an open fluid domain using potential flow theory. Ducrozet et al. (2012) further extended this approach to simulate wave generation and propagation in a numerical wave tank (NWT). In a full analogy with the features of an experimental wave tank (EWT), i) a wavemaker generates the waves, ii) the sidewalls are fully reflective, and iii) an absorbing beach is located opposite to the wavemaker to prevent wave reflections. A schematic depiction of the NWT concept is given in Fig. 1. The wave propagation is solved with respect to the global reference frame (O, e_x, e_y, e_z), where the origin O is defined at the mean location of the wavemaker and on the still water line. The position of a point in the tank is given in this frame by x = (x, y, z). The kinematic and dynamic boundary conditions are evaluated on the exact free surface z = η(x, y, t) for the velocity potential φ(x, t) = φ_{spec}(x, t) + φ_{add}(x, t). Introducing the free surface potential φ^s(x, y, t) = φ(x, y, z = η, t) the equations are,

$$ 
\begin{aligned}
\partial_t \eta &= (1 + |\nabla_h \eta|^2) \partial_z \phi - \nabla_h (\phi^s + \phi_{add}) \nabla_h \eta + \partial_z \phi_{add} \\
\partial_t \phi^s &= -\eta - \frac{1}{2} |\nabla_h \phi^s|^2 + \frac{1}{2} (1 + |\nabla_h \eta|^2) (\partial_z \phi)^2 \\
&\quad - \nabla_h \phi^s \cdot \nabla_h \phi_{add} - \frac{1}{2} |\nabla \phi_{add}|^2 - \partial_t \phi_{add} - \nu \partial_t \eta
\end{aligned} 
\quad (1) 
$$

where ∂ denotes derivative with respect to the subscript quantity, and ∇_h = (∂_x, ∂_y) is the horizontal gradient. Moreover, φ_{spec} is the potential that accounts for wave propagation in the absence of the wavemaker, while φ_{add} solves the influence of the wavemaker. Both potentials satisfy the Laplace equation, while no-flow conditions are applied on all wall boundaries. An appropriate condition is applied for φ_{add} on the wavemaker boundary to account for wave generation (Ducrozet et al., 2012). The vertical velocity ∂_z φ is the only term not defined explicitly on the free surface, and it is computed through the HOS scheme of West et al. (1987). Once this velocity is known, η and φ^s are time-marched through a 4th order Runge-Kutta time integration scheme with

<figure><img src="image_4.png" /><figcaption>Fig. 1. Numerical wave tank configuration.</figcaption></figure>

an adaptive timestep. The open-source HOS-NWT solver is used in the present work, which adopts a pseudo-spectral formulation for η and φ^s and permits the use of FFTs to evaluate the spatial derivatives in Eq. (1), resulting in relatively low computational cost.

The wavemaker position is defined as X(z, t) = X_{wm}(t)f(z) with f(z) describing its vertical shape (piston, flap-type, etc.). The motion of the numerical wavemaker inside the HOS-NWT is assumed to be a zero-mean Gaussian process given by,

$$ X_{wm}(t) = \sum_{i=1}^{N} \frac{\alpha_i}{L_i^{wm}} \cos(\omega_i t + \epsilon_i) = \sum_{i=1}^{N} (u_i c_i(t) + \bar{u}_i \bar{c}_i(t)) \quad (2) $$

where N is the total number of wave components generated at the wavemaker location, ω_i is the frequency of the ith component and α_i = √(2S_η(ω_i)dω) is the wave amplitude defined from the target wave spectrum S_η(ω_i). Moreover, L_i^{wm} is the magnitude of the wavemaker transfer function and ε_i the phase of this component. Finally, u_i, ū_i are the unknown uncorrelated standard normal variables for this problem, while the deterministic functions c_i(t) and c̄_i(t) follow

$$ 
\begin{aligned}
c_i(t) &= \sigma_i^{wm} \cos(\omega_i t) \\
\bar{c}_i(t) &= -\sigma_i^{wm} \sin(\omega_i t) \\
(\sigma_i^{wm})^2 &= \frac{S_\eta(\omega_i)}{(L_i^{wm})^2} d\omega_i
\end{aligned} 
\quad (3) 
$$

where (σ_i^{wm})^2 represents the variance of the wavemaker motion, while the amplitudes and phases are

$$ \frac{\alpha_i}{L_i^{wm}} = \sigma_i^{wm} \sqrt{u_i^2 + \bar{u}_i^2} \quad \epsilon_i = \tan^{-1} \left( \frac{\bar{u}_i}{u_i} \right) \quad (4) $$

#### 2.1.2. Response-conditioning procedure

Using the wavemaker motion described by Eq. (2) and assuming unidirectional waves, HOS-NWT calculates the nonlinear wave propagation inside the NWT and provides the incident free-surface elevation at the location of the structure of interest η_{HOS}(x_0, t). Using this free-surface elevation, the surrogate model response χ(x_0, t|u) can be obtained, where u is the vector of uncorrelated normal variables, {u_i, ū_i} = {u_1, ū_1, ..., u_N, ū_N}. This wave elevation is thus dependent on the realisation of u and the corresponding wavemaker motion. Since the objective of this study is to generalise the design wave framework to total first- and second-order responses, the use of linear transfer functions as in Adegeest et al. (1998) or Dietz (2004) is not sufficient. Instead, by considering a FORM-based limit state function, the response is not obtained through a closed-form expression. Instead, as suggested by Jensen and Capul (2006), it can include some nonlinear aspects by considering higher-order terms or by performing time-domain simulations. This limit state function is defined as the difference between the predetermined target response χ_{tar} and the response of the surrogate model at the conditioning time t_0 as χ(x_0, t_0|u),

$$ G(\mathbf{u}) = \chi_{\text{tar}} - \chi(x_0, t_0 | \mathbf{u}) = 0 \quad (5) $$

Due to the definition of this function, as well as of Eq. (2), in the standard normal space, the point u^* that has the shortest distance from the origin will correspond to the outcome with the highest probability of occurrence. Thus, according to the approach of Jensen and Capul (2006),
it is termed as the Most Probable Point (MPP) and it corresponds to a single wave episode, interpreted as the RCW. In the NWT formulation of this problem, the MPP will correspond to a realisation of the wave-maker motion that will generate the desired RCW episode at the location of interest $x_0$. To determine $u^*$, the optimisation problem in Eq. (6) needs to be solved through a dedicated iterative algorithm, known as the Modified Hasofer and Lind with Goldstein-Armijo rule (MHLGA) (Santos et al., 2012),

$$ \min \sqrt{\sum_{i=1}^{N} (u_i^2 + \bar{u}_i^2)} \quad \text{subject to} \quad G(\mathbf{u}) = 0. \qquad (6) $$

The detailed description regarding the implementation of the algorithm in the present procedure can be found in Kim et al. (2022). Given that Eq. (6) is solved iteratively, the choice of the initial point is significant for the computational performance of the algorithm. In Kim et al. (2022), where the wave propagation problem was addressed, the NewWave was chosen for this purpose. Analogously, in Dermatis et al. (2025) the MLER (Adegeest et al., 1998) wave episode was set as the initial solution. For the present problem, given that there is no such analytical solution, a different strategy was devised. The procedure starts with the solution of Eq. (6) using the same surrogate model applied on linear waves, instead of the HOS-NWT solver. At this stage, all vector $\mathbf{u}$ elements are set equal to zero as an initial guess, similar to what is done in Jensen and Capul (2006). Given the high efficiency of the algorithm with linear waves, the solution is found in a matter of seconds. The output of this preliminary step is the linear wave elevation at the location $x_0$ that produces the target response $\chi_{tar}$ at $t_0$. Back-propagation of this wave to the location of the wavemaker provides the wavemaker time history that will generate this linear RCW. This serves as the initial solution and the iterative procedure with HOS-NWT then begins. Once the MPP has been identified, the Gaussian wavemaker motion can be determined using Eq. (2), enabling straightforward reproduction of the wave episode in an experimental or numerical wave tank.

## 2.2. High-fidelity response evaluation

In the second step of the multi-fidelity methodology, the design waves obtained by the HOS-RCW approach are reproduced in a high-fidelity model, which consists of model tests in the present study. Given that the HOS-RCW procedure delivers the wavemaker motion $X_{iwm}(t)$ of the NWT, the reproduction of those waves can be performed directly. Thus, no additional phase correction (Fernández et al., 2014) is required in contrast with previous similar studies (Takami et al., 2020; Tosdevin et al., 2022), where deviations from the target wave profile may still be present. Instead, as demonstrated by Dermatis et al. (2025), the nonlinear wave episodes extracted from the HOS-RCW approach can be exactly reproduced up to severe sea states.

Finally, during the high-fidelity implementation of the wave episodes, the maximum response $\chi_{NL}$ is measured and used to correct the preliminary surrogate response distribution. It should be emphasized here that a necessary condition to apply this perturbation of the low-fidelity response distribution, according to Pastoor (2002), is the existence of a monotonically increasing relationship between the two responses of different fidelity. This can be physically interpreted as increasing nonlinearity depending on the level of the response. Denoting the surrogate response as $\chi$ and the fully nonlinear response as $\chi_{NL}$, and assuming that such a relationship $h()$ exists, it is expressed as

$$ \chi_{NL} = h(\chi) \quad \text{with } h \text{ monotonously increasing} \qquad (7) $$

Then it can be easily proven that,

$$ \begin{aligned} F_{X_{NL}}(\chi_{NL}) &= P[h(X) \le \chi_{NL}] \\ &= P[X \le h^{-1}(\chi_{NL})] \\ &= F_X(h^{-1}(\chi_{NL})) \end{aligned} \qquad (8) $$

where $F$ is the cumulative distribution function (CDF), $P[]$ denotes probability, capital symbols denote random variables and lowercase symbols

denote fixed threshold values. Therefore, it is shown that the CDF of the fully nonlinear response can be evaluated through the surrogate model distribution, by a re-definition of the threshold according to the high-fidelity response to the RCW.

An overview of the proposed multi-fidelity approach is presented as a flowchart in Fig. 2, where the initial inputs are given in green boxes and the final output, which is the fully nonlinear response evaluation, is given in red. Moreover, an illustrative depiction of the approach is given on the right plot of the figure to demonstrate the correction of the response at a certain return period level. Since this study investigates the extreme values of a response that is driven by phenomena with different frequency ranges, the distributions are expressed in terms of the return period. The return period expresses the average interval between two occurrences of a given extreme event, in the present case the surge excursions of the container ship. It is defined as the duration for which the non-exceedance probability of a response level $\chi$ will be $P_c(\chi, RP) = 1/e$ (Veritas, 2019). Under these terms, it is related to the CDF of the response by,

$$ \mathrm{RP}(\chi) = \frac{T}{N \log(F_X(\chi))} \qquad (9) $$

where $T$ is the observation duration and $N$ is the number of observed peaks.

## 3. Second-order surrogate response model

The surrogate model for the response conditioning process should be representative of the main dynamics of the system and also be able to indicate the occurrence of the nonlinear maxima. According to many researchers (Naess, 1986; Langley, 1986) in the past, but also by Lim and Kim (2018) and Lande-Sudall and Stansby (2024) for the design wave problem, a linear SDOF model under first- and second-order wave loads is sufficient to describe the horizontal motions of floating moored structures.

To define the translational motions of the vessel, a body-fixed reference frame ($O_b$, $e_{xb}$, $e_{yb}$, $e_{zb}$) is considered at the body's centre of gravity (COG). Considering the location of the body at rest as $\mathbf{x}_0 = [x_0, y_0, z_0]$ and the distance vector from the origin of the global reference frame as $\overline{\mathbf{O}_b}$, the translations are expressed as $\xi = [\xi_1, \xi_2, \xi_3] = \overline{\mathbf{O}_b} - \mathbf{x}_0$. For the surge motion $\xi_1$ that is of interest in this study, it is assumed to be uncoupled from the other DoFs, and the equation of motion can be written in the frequency domain and solved separately for the first- and second-order as

$$ \begin{aligned} &[-\omega^2(M + M_A) + i\omega(B + B_{add}) + (K_H + K_M)]\Xi_1^{(1)} = F_1^{(1)} \\ &[-(\Delta\omega)^2(M + M_A) + i\Delta\omega(B + B_{add}) + (K_H + K_M)]\Xi_1^{(2)} = F_1^{(2)} \end{aligned} \qquad (10) $$

where $M$ is the body mass, $M_A$ and $B$ are the frequency-dependent added mass and radiation damping coefficients in surge, while $B_{add}$ is the additional linear damping, $K_H$ is the hydrostatic stiffness and $K_M$ is the equivalent linearized mooring stiffness. The horizontal wave-induced loads of first order are denoted as $F_1^{(1)}$ and of second order as $F_1^{(2)}$. As a simplification, the slow-drift approximation is adopted, meaning that $F_1^{(2)}$ corresponds only to the difference-frequency loads. The linear and difference-frequency quadratic transfer functions of the surge motion are denoted as $\Xi_1^{(1)}$ and $\Xi_1^{(2)}$ respectively. Upon application of these transfer functions to the wave elevation, the total surge motion can be reconstructed in the time domain as the superposition of the two terms,

$$ \begin{aligned} \xi_1(t) &= \xi_1^{(1)}(t) + \xi_1^{(2)}(t) \\ &= \sum_{i=1}^{N} \Xi_1^{(1)}(\omega_i) A_i e^{i(\omega_i t)} + \sum_{i=1}^{N} \sum_{j=1}^{N} \Xi_1^{(2)}(\omega_i, \omega_j) A_i A_j^* e^{i(\omega_i - \omega_j)t} \end{aligned} \qquad (11) $$

where $A_i$ are the complex wave amplitudes at $x_0$ obtained through an FFT of the time-history of the free-surface elevation $\eta_{HOS}(x_0, t)$ and $A^*$ denotes the complex conjugate.
<figure><img src="image_5.png" /><figcaption>Fig. 2. Flowchart of the design wave methodology.</figcaption></figure>

Finally, as mentioned in Section 2, a preliminary response distribu-
tion needs to be constructed using the surrogate model. The Rayleigh
distribution of the first-order response was shown in Bouscasse et al.
(2024) to be unable to serve as a sufficiently accurate approximate dis-
tribution for this purpose. Numerical estimation of the total first- and
second-order surrogate response distribution, as in Naess (2001), would
be optimal in terms of efficiency to statistically account for the large
amplitude slow-drift motions. However, as discussed in the introduc-
tion, this would not apply to nonlinear, non-Gaussian waves or nonlin-
ear dynamical systems. Therefore, a preliminary Monte Carlo approach
in irregular waves is used to obtain the surrogate response distribu-
tion. To this end, HOS-NWT is employed, and numerous realisations
of each state are simulated to numerically evaluate the wave eleva-
tion at the structure location x₀. Then, the complex wave amplitudes
in the frequency domain are obtained through an FFT of the numeri-
cal probe signal. Once the wave amplitudes have been determined, lin-
ear and quadratic transfer functions of the motion are applied accord-
ing to Eq. (11). Using the resulting response time series, the empirical
surrogate distribution can be constructed according to the Monte Carlo
approach.

4. Experiments

4.1. *Experimental setup*

Experiments were carried out in the Ocean Engineering Wave Tank of École Centrale Nantes using the "CN101" 1/65 scale rigid model of a 6750-TEU container ship. The geometry of the model is identical to the one used in the ITTC-ISSC benchmark study of Kim and Kim (2016), and it has been used by the authors in previous studies (Bouscasse et al., 2022; Kim et al., 2024; Dermatis et al., 2025). The principal dimensions of the vessel are listed in Table 1, while a picture of the model inside the wave tank is shown in Fig. 3, along with an overview of the experimental setup. The model kinematics, including the surge motion, are measured through a Qualisys optical motion tracking system. To measure the free-surface elevation, 8 resistive wave gauges (WG) are installed at several locations along the wave tank. Among these, WG2 is aligned with the COG of the ship in calm water, and the incident wave field at the location of the structure is monitored. A more detailed description of the model construction, calibration, and measuring devices can be found in Bouscasse et al. (2022). In the same study, the surge natural period
<figure><img src="image_6.png" /><figcaption>Fig. 3. 6750-TEU Container ship model and experimental setup.</figcaption></figure>

<table><caption>**Table 1**

Principal dimensions of the 6750-TEU container ship.</caption><thead><tr><th></th><th>Full scale</th><th>Model scale</th></tr></thead><tbody><tr><td>Scale</td><td>1/1</td><td>1/65</td></tr><tr><td>LBP (m)</td><td>286.6</td><td>4.409</td></tr><tr><td>Breadth (m)</td><td>40</td><td>0.615</td></tr><tr><td>Draft (m)</td><td>11.98</td><td>0.188</td></tr><tr><td>Displacement (kg)</td><td>85,663,776</td><td>311.93</td></tr><tr><td>KM (m)</td><td>18.662</td><td>0.287</td></tr><tr><td>GM (m)</td><td>2.10</td><td>0.032</td></tr><tr><td>KG (m)</td><td>16.562</td><td>0.257</td></tr><tr><td>LCG from AP (m)</td><td>139.56</td><td>2.147</td></tr><tr><td>kxx (m)</td><td>14.4</td><td>0.222</td></tr><tr><td>kyy (m)</td><td>71.5</td><td>1.109</td></tr><tr><td>kzz (m)</td><td>71.4</td><td>1.106</td></tr></tbody></table>

<table><caption>**Table 2**

Description of environmental conditions (full scale).</caption><thead><tr><th>Case</th><th>H<sub>x</sub> (m)</th><th>T<sub>p</sub> (s)</th><th>γ</th><th>Seeds</th></tr></thead><tbody><tr><td>SS6</td><td>6</td><td>12.25</td><td>1</td><td>8</td></tr><tr><td>SS8</td><td>8.3</td><td>14</td><td>1.5</td><td>12</td></tr><tr><td>SS10</td><td>10</td><td>14</td><td>1.5</td><td>10</td></tr><tr><td>SS12</td><td>12</td><td>14</td><td>1.5</td><td>8</td></tr><tr><td>SS17</td><td>17</td><td>15.5</td><td>2.6</td><td>34</td></tr></tbody></table>

was measured through decay tests as $T_N = 11.52$ s (92.9 s in full scale). The mooring arrangement of the problem consists of four springs with a stiffness of $k = 58$ N/m and a pretension of $T_0 = 17$ N. Lines 1 and 2 have a length of 21.9 m, while lines 3 and 4 have a length of 24.7 m, with all lines having an angle of 45° with the $e_x$ axis.

## 4.2. Environmental conditions

This study examines five sea states defined by the JONSWAP spectrum which are detailed in Table 2. Experiments with irregular waves (IW) were performed to derive the reference distribution through a Monte Carlo approach. To ensure statistically convergent results, each sea state was tested multiple times, with different sets of wave phases. The duration of each test was approximately 20 minutes at model scale (equivalent to 2 hours and 40 minutes at full scale), with the first and last minute excluded from the analysis to remove transient effects.

In the experimental analysis of RCW, four target surge values were selected for each sea state and they are shown in Table 3. It is noted that the four levels correspond to multiples of the standard deviation $\sigma$ of the surrogate response spectrum for each sea state. Given that a preliminary response distribution is computed in the first step of the methodology, these target surge values can be associated with a return

<figure><img src="image_7.png" /><figcaption>**Table 3**

Surge target values for RCW analysis at model scale (m).</figcaption></figure>

<table><thead><tr><th>Target</th><th>SS6</th><th>SS8</th><th>SS10</th><th>SS12</th><th>SS17</th></tr></thead><tbody><tr><td>1σ</td><td>-0.012</td><td>-0.021</td><td>-0.032</td><td>-0.045</td><td>-0.078</td></tr><tr><td>2σ</td><td>-0.024</td><td>-0.043</td><td>-0.064</td><td>-0.090</td><td>-0.156</td></tr><tr><td>3σ</td><td>-0.036</td><td>-0.064</td><td>-0.096</td><td>-0.134</td><td>-0.234</td></tr><tr><td>4σ</td><td>-0.048</td><td>-0.085</td><td>-0.128</td><td>-0.179</td><td>-0.312</td></tr></tbody></table>

period through interpolation in that distribution. Additionally, all calculated RCWs were designed to produce the maximum response at $t_0 = 45$ s and $x_0 = 18.2$ m, which is aligned with the location of the vessel's COG in calm water. The effect of the target time instant $t_0$ was investigated, but it was found that the same wave episodes were obtained for larger $t_0$. Therefore, it was set to 45 s to minimise the time for the optimisation procedure of the HOS-RCW method.

## 5. Results and discussion

This section begins with outlining the development of the surrogate response model for the present configuration. Furthermore, the results of the experimental investigation of the multi-fidelity design wave methodology are presented. For the five sea states described in Table 3, the wave episodes were reproduced in the EWT within the second step of the multi-fidelity methodology. The analysis begins with a comparison between the experimental and numerical results for the free-surface elevation and the surge motion. Next, the experimentally measured wave episodes and induced surge motions are compared with similar events from the IW tests. In addition, the technique of harmonic separation is applied to isolate the contribution of the wave- and low-frequency components to the total surge motion. Finally, the ability of the two-step methodology to predict the short-term extreme responses is demonstrated by comparing the experimental Monte Carlo distribution with the design wave results.

### 5.1. Surrogate model validation

As a first step, the development and initial validation of the surrogate model are outlined. For the calculation of the coefficients, $M_A(\omega)$, $B(\omega)$ and $K_H$, as well as the linear and quadratic transfer functions, the frequency domain 3D-BEM solver Hydrostar (Chen, 2007) was used. The mesh consisted of approximately 2500 panels for the submerged part of the half hull. A frequency range of 0.1–10 rad/s was considered with a constant interval of 0.1 rad/s for the radiation-diffraction analysis. The QTF computation was performed using the near-field approach (Pinkster, 1980) for the QTF component that involves quadratic
<figure><img src="image_8.png" /><figcaption>Fig. 4. Surge motion RAO (left) and difference-frequency motion QTF (right).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Time history comparison between HOS-NWT and experimental wave elevation (left) and between surrogate model and experimental surge motion (right) for SS10.</figcaption></figure>

products of first-order quantities. The second-order potential force con-
tribution was evaluated using the indirect approach for the diffraction
potential (Molin, 1979) and by neglecting the free-surface integral. This
is a reasonable approximation for the present case study since only the
difference-frequency loads are of interest (de Hauteclocque et al., 2012).
The linearized mooring stiffness in the $e_x$ direction was $K_M = 121.8$ N/m
according to the mooring arrangement described in Section 4, while the
linear added damping coefficient was set equal to $B_{add} = 30$ N/m. The
calibration of this parameter was based on a comparison of the surge
time series of the surrogate model against experiments for SS6. This is
the only parameter of the surrogate model not explicitly provided from
the potential flow analysis, and it may significantly influence the ampli-
tude of the slow-drift motions (Pegalajar-Jurado and Bredmose, 2019).
For this reason, further investigation on the impact of the damping on
the surrogate modelling and on the performance of the design wave
methodology is presented in Section 5.6. The transfer functions for the
surge motion are shown in Fig. 4 where the peak of the RAO indicates a
natural period of $T_N = 11.6$ s, similar to the experimentally determined
value. Furthermore, it is observed that all the energy of the motion QTF
is concentrated in the diagonals that correspond to difference-frequency
values close to the natural frequency.

An indicative comparison between the numerical and experimental
time history for a realisation of SS10 is shown in Fig. 5. On the left plot,
the wave elevation obtained with HOS-NWT is compared with the mea-
surements of the WG2 probe, showing overall good agreement, consid-
ering the reflections on the absorbing beach and wave maker that occur
after such a long duration. On the right, the surge motion obtained by
application of Eq. (11) is plotted against the experimentally measured
response. This comparison serves as an initial verification that the sur-

rogate model can indicate the extreme events since the time instants
of the peaks from the surrogate model and experimental surge nearly
coincide.

5.2. *Analysis of RCW and RCW-induced motions*

The numerically determined RCW episodes were compared with the
waves measured by WG2 during the experimental RCW tests. Fig. 6
presents the free-surface elevation of SS10 and SS17 for all wave
episodes. Given that the relevant figures for all the other sea states do
not provide significant additional insight to this study, they are omit-
ted but are available upon request. It can be seen that the HOS-RCW
episodes are accurately reproduced without the need for frequency do-
main phase/amplitude corrections, as more extensively discussed in
Dermatis et al. (2025). Even for the most extreme test case of SS17-
4, the experimental deviations are negligible and only observed after
the wave’s passage. They can be potentially attributed to diffracted and
radiated waves from the vessel that interact with the incident wavefield
at WG2. The design wave consists of two large crests before the focusing
time of $t_0$, and counter-intuitively, a long wave group is not required to
build the necessary slow-drift. This can be attributed to the process of
response conditioning, which tunes the phases of the wave to induce
a prescribed response level, thus guaranteeing the development of the
slow-drift excursion.

The comparison between the numerical second-order surge and the
fully nonlinear experimental surge of the vessel under the RCW episodes
is depicted in Fig. 7 for the same two sea states. It is demonstrated that
the proposed surrogate response model, despite its relative simplicity,
can successfully indicate the nonlinear extreme responses since the two
<figure><img src="image_10.png" /><figcaption>Fig. 6. Experimentally reproduced wave elevation for SS10 (top row) and SS17 (bottom row).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7. Experimentally reproduced surge motion for SS10 (top row) and SS17 (bottom row).</figcaption></figure>

events coincide in time and differ only in magnitude. As expected, an increasing difference between the second-order and the fully nonlinear maxima is observed, with an increase in the target response level. Hence, the condition of a monotonically increasing relationship, imposed by Eq. (7), between the low and high-fidelity response models is preserved. This fact suggests that the SDoF response to first- and second-order wave loads is a sufficiently effective surrogate model. Therefore, in the present configuration, it can be used for predicting the extreme short-term responses within a multi-fidelity design wave approach. A phase offset between the surrogate model and the experimentally measured surge can also be observed in Fig. 7, with an increase in the target response level, especially for SS17. This offset can be potentially attributed to the significant change in the position at which the ship encounters the incident waves in the experiments. More specifically, due to the slow-drift motion of the vessel, its location is shifted, and the incident waves encounter the ship at a location different than $x_0 = 18.2$ m. As demonstrated in Kim (2022), this causes a temporal offset in the response, which becomes more apparent with increasing drift motion, as shown in Fig. 7. This discrepancy is not expected to affect the final results since it is incorporated in the surrogate model.

### 5.3. Comparison between IW and RCW

The resemblance of the RCW profiles with similar events from the IW tests was then investigated to assess whether the HOS-RCW episodes are representative of the whole sea state. Even though, strictly speaking, the ensemble average is not identical to the most probable event, it is a good indicator of the similarity of the RCW to the most frequent events within a sea state. For each experimentally measured peak surge value during the RCW tests, the IW time history was screened to determine events during which a ±5% surge excursion was measured. For the wave elevation, the signal measured at WG2 during those events was extracted and compared with the RCW episodes. The results are presented in Figs. 8 and 9, respectively. The RCW wave profiles follow the ensemble of the IW profiles but with significantly larger crests and troughs. This difference in the free-surface elevation can be explained by the fact that the vessel, within an irregular sea state, is oscillating around the mean drift position. However, during the design wave experiments, the body starts from rest, thus requiring more impactful waves to reach the same excursion level within one response cycle. This exact effect is illustrated in the case of the surge motion, for which the time frame is
<figure><img src="image_12.png" /><figcaption>Fig. 8. Comparison between experimentally measured IW and RCW free-surface elevation.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9. Comparison between experimentally measured IW and RCW surge motion.</figcaption></figure>

increased to cover 4 natural periods. Furthermore, as also observed by Hansen et al. (2025a), the RCW-induced response is impulsive and thus non-symmetric with respect to $t_0$, while the oscillations after the peak response are of larger amplitude.

In Fig. 10, the same results for the wave elevation and surge motion of SS10 are shown, but after subtraction of the mean offset. This filtering is performed before the identification of the irregular wave events, thus corresponding to a total shift of the response around $x_0$. In that case, the design events are representative of the irregular wave events, and this favourable comparison is expected to occur when the change in the mean position does not significantly alter the mooring and hydrodynamic forces. Overall, despite the proposed approach triggering the slow-drift motions with a single wave episode, the initial conditions are not comparable between RCW and IW, due to the mean offset. The inclusion of the mean drift in the design waves needs to be further investigated and can be potentially accounted for through constrained wave episodes (Dietz et al., 2004) or through imposing appropriate initial conditions in the surrogate model and in the high-fidelity RCW tests. Nevertheless, as explained in Section 5.5, the effect of the mean drift on the probabilistic assessment is taken into account in the overall design wave methodology, through the surrogate response distribution.

## 5.4. Harmonic separation

The source of the nonlinearity was investigated by the harmonic separation method (Walker et al., 2004; Fitzgerald et al., 2014; Orszaghova et al., 2021). More specifically, the wave phase of each RCW test was shifted by 180° and a new test was performed with the opposite free-surface elevation. Denoting the measured surge motions for the 0° and 180° tests as $\xi_{1,NL}$ and $\xi'_{1,NL}$ respectively, the odd and even harmonics can be obtained via subtraction and addition of the two signals, as shown below,

$$ \begin{array}{l} \text{odd:} \\ \frac{1}{2} (\xi_{1,NL} - \xi'_{1,NL}) = \xi_{1,NL}^{(1)} + O(A^3) \\ \text{even:} \\ \frac{1}{2} (\xi_{1,NL} + \xi'_{1,NL}) = \xi_{1,NL}^{(0)} + \xi_{1,NL}^{(2)} + O(A^4). \end{array} \qquad (12) $$

Regarding the surrogate response model, the odd harmonics will correspond to the $\xi_{1}^{(1)}$ term of Eq. (11), while the even harmonics to the $\xi_{1}^{(2)}$ term, since it is truncated after second order. Therefore, a comparison between the experimental and numerical harmonic contributions is possible. The results of this harmonic separation procedure are demonstrated in Fig. 11. This shows that the odd harmonic results corresponding to the wave-frequency response are reproduced very well by the
<figure><img src="image_14.png" /><figcaption>Fig. 10. Comparison between experimentally measured IW and RCW free-surface elevation (top row) and surge motion (bottom row), with filtering of the mean drift.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 11. Odd and even harmonic contribution to total surge motion.</figcaption></figure>

surrogate model, indicating that this part of the surge motion is almost linear and uncoupled from the low-frequency motion. Regarding the even harmonic term, its amplitude is accurately captured for the mild events. With an increase in the target surge levels, a growing discrepancy between the surrogate and experimental low-frequency response is observed. It is therefore shown that the functional relationship between the low- and the high-fidelity model, described by Eq. (7), can be mostly attributed to the contribution of the low-frequency response. This finding is a formal confirmation of what has been observed by previous researchers (Tosdevin et al., 2022; Ripe and Lande-Sundall, 2023; Bouscasse et al., 2024) that RCWs based solely on the linear transfer function are not able to account for second-order difference-frequency surge motions. The sources of discrepancy between the low- and high-fidelity responses can be various and are mostly related to imperfections in the modelling of the dynamical system. In the present case, the principal reason is identified as the linearization of the damping and restoring terms, as well as the use of a sea state independent empirical coefficient $B_{add}$ for the former.

### 5.5. Extreme short-term response prediction

The design wave methodology outlined in Section 2 was employed for the probabilistic assessment of the surge motions of the container ship vessel. In this study, the wavemaker motion of the experiments was used as an input for HOS-NWT to obtain the irregular wave database for the surrogate model distribution. However, potential differences in the IW seeds are not expected to generally influence the results, as long as adequate statistical convergence is ensured. As already mentioned, the return period was used in the present work to represent the response distribution. In particular, given a response time history of duration $T$ and an observed number of peaks $N$ that are sorted in increasing order, the empirical non-exceedance probability and the return period are given by

$$ 
\begin{aligned}
P_{\text{MCS}} &= \frac{i}{N+1} \\
RP_{\text{MCS}} &= \frac{T}{N \log(P_{\text{MCS}})}
\end{aligned} 
\qquad (13) $$
<figure><img src="image_16.png" /><figcaption>**Fig. 12.** Experimental surge return periods for different block sizes (full scale).</figcaption></figure>

where *i* is the index of the sorted peaks.

To identify the peaks within the time series of the surge response and use them in Eq. (13), the simplest approach would be a level-crossing analysis of the motion time series. Under these terms, there would be no treatment regarding statistical independence. As indicated by Eq. (8), this choice of simple level-crossing analysis does not pose any problem for the conclusions of the present study, as long as the surrogate and experimental response distributions are similarly treated. However, due to wave grouping phenomena and the coexistence of two response mechanisms of different frequencies, extreme events are anticipated to appear in clusters. To remove the dependence of the response distribution on consecutive extreme events, the Block Maxima (BM) approach was used. According to the BM method (Ferreira and Haan, 2013), the response time history is divided into non-overlapping blocks of equal size, and the maximum/minimum response value of each block is considered. By taking the extreme responses during these disjoint time blocks, de-clustering of the extreme response distribution is performed. It should be noted that the approach here is limited to the identification of extreme events and does not imply fitting a Generalised Extreme Value (GEV) distribution to the data.

For the results to be of direct practical interest, all return period plots beyond this point are presented in full scale, according to Froude scaling. Unless indicated otherwise, all other results of this section are presented in model scale. Moreover, Eq. (13) considers only the negative peaks, with surge excursion in the direction of wave propagation; however, the results are presented in absolute values. The duration of both the surrogate MCS and the experimental MCS is taken as $N_s \cdot 2\hbar40'$ in full scale, where $N_s$ is the number of realisations for each sea state (Table 2). However, the number of observed peaks may differ between the two approaches. Moreover, the choice of the block size is known to be significant for the application of the BM method, as it is the principal parameter regarding the level of de-clustering. In the present configuration, the slow-drift motions are dominant, driving the vessel to oscillate with a period close to the surge natural period $T_N$, as also observed in Fig. 5. Therefore, the block size is associated with this value and in Fig. 12, the return period for several experimental surge levels in SS10 is plotted against the block size. It is observed that after around $1.5T_N$, the return period values tend to converge, indicating a sufficient degree of statistical independence of the extremes. This can be explained by the fact that this block size is large enough to eliminate memory effects in the statistical analysis. Taking this into account, the block size was set equal to $2T_N$ for both the surrogate and the experimental surge distribution. This sensitivity study also indicates that the memory effects in the present configuration are limited.

Fig. 13 presents the short-term distributions of the surge motion of the container ship over all 5 sea states at full scale. The fully-nonlinear surge distribution obtained by the experimental Monte Carlo approach

is plotted in blue, and the surrogate distribution is shown in green. The latter serves as the starting point for the high-fidelity evaluation through interpolation of the target surge values of Table 3 in this distribution. Next, the peak surge excursion measured throughout the individual RCW tests is plotted in red for the same return period values. To assess the level of statistical variability (Brown et al., 2001), the Jeffrey 95 % confidence interval is also shown. The estimates obtained from the high-fidelity reproduction of the RCW are in close agreement with the experimental Monte Carlo approach. This indicates that short design wave episodes can potentially replace the long-duration Monte Carlo for near-resonant low-frequency responses. It is also noticed that there are no events in the high-probability region of the distribution. This can be explained because small peaks that would be attributed to wave-frequency motions are omitted within the BM approach since only the global maxima/minima are considered within each block.

As discussed in Section 5.3, an important aspect that should be considered when using design waves for low-frequency responses is related to their transient nature. More specifically, as illustrated in Fig. 9, they cannot replicate the mean drift conditions that occurs within an irregular sea state. The present methodology treats this problem by incorporating the mean drift in the surrogate MCS distribution. More specifically, the surrogate response model is applied to the irregular wave HOS-NWT simulations, allowing for the mean offset to be numerically determined in that step. Therefore, the design wave results are used to re-evaluate the total surrogate response distribution, which includes the mean drift. It should be emphasised that the problem is partially tackled, and it also depends on how accurately the mean drift is captured by the surrogate model. That can possibly explain some slight discrepancies between the RCW and experimental Monte Carlo results in Fig. 13. To further treat this issue, the initial conditions of the surrogate model need to be set to the numerically predicted offset, which should also be artificially imposed in the experimental RCW tests.

Finally, it is important to consider that in the design wave approach, the experimental duration is independent of the return period level. Even for the tail of the distribution, where the required test duration increases drastically, the duration of the design wave tests is around one minute in model scale. At the same time, despite the considerable number of seeds, the experimental MCS results are statistically sound up to return periods of four hours at full scale. Therefore, the design wave approach is highly efficient for low probability levels, since long-duration IW tests are avoided. For the number of sea state realisations investigated experimentally in the present work and considering that four RCW tests were carried out per sea state, the experimental duration of the design wave methodology was less than 3 % of the full MCS approach.

### 5.6. Effect of added damping

Recall that the present surrogate response model involves the use of an empirical linear additional damping $B_{add}$. The standard practice for the choice of this parameter is to assume that it is constant and obtain the appropriate value by calibration with decay tests. Lately, it has been observed that a more realistic approach is to find a sea state-dependent value, and more sophisticated methods have been developed for this purpose (Pegalajar-Jurado and Bredmose, 2019). Given the sensitivity of the surrogate model to $B_{add}$, as well as the fact that it practically defines the amplitude of the slow-drift motion, the effect of this parameter on the results of the extreme response prediction was investigated. More specifically, for SS10 two further series of RCW tests were performed, with damping coefficients of $B_{add} = 15$ N/m and $B_{add} = 50$ N/m respectively. To be able to compare the results, the conditional-response levels were kept the same as the default case of $B_{add} = 30$ N/m.

An example of the comparison regarding the numerically determined RCW episodes of the more lightly- and heavily-damped surrogate models is shown on the left plot of Fig. 14 for SS10-4. As expected, the wave episode for the former is less extreme than the default case since less
<figure><img src="image_17.png" /><figcaption>**Fig. 13.** Surge response distribution in terms of return period for all sea states (full scale).</figcaption></figure>

effort is necessary to excite the same response level. The opposite can be observed for the case of the latter, considering that the system requires larger waves for the same slow-drift response to develop. In the right plot of the same figure, the surrogate surge responses for the same $B_{add}$ are shown, along with the corresponding experimental responses with dotted lines. It can be seen that for the lightly damped case, where as mentioned above the wave is milder, the experimental surge response is also lower. On the contrary, for the heavily damped surrogate, the experimental response reaches higher levels than the default case, which is in accord with the profile of the determined wave episode.

Fig. 15 shows the return period curves for SS10 at full scale, using the two linear damping coefficients. It can be seen that the results are not influenced by the choice of the added damping within the surrogate model. In the lightly damped scenario, the surrogate response overestimates the experimental Monte Carlo distribution. However, as already mentioned, the wave episodes determined through HOS-RCW are milder and therefore excite lower fully nonlinear responses. The exact opposite is observed in the case of the heavily damped scenario, where the surro-

gate model significantly underestimates the fully nonlinear response distribution. Nevertheless, since larger waves are calculated, re-evaluating the fully nonlinear response through RCWs still gives accurate results. This finding indicates the robustness of the present methodology and it proves that the influence of the linearised viscous damping will be incorporated into the functional relationship of Eq. (7) and further portrayed in the RCW episodes. As a consequence, and given that the preliminary response distribution is built using the same surrogate, the prediction of the extreme responses will not be significantly influenced by differences in the level of linear damping. It should be emphasised that this conclusion holds as long as the surrogate model is capable of predicting the extreme surge events and Eq. (7) is not violated. For instance, with very large damping coefficients, the slow-drift motion would be entirely damped in the surrogate, and the effect of the difference-frequency motion would not be properly taken into account. Similarly, in the absence of added damping, the low-frequency oscillations would be unphysically large, and the nonlinear maxima would not coincide with the surrogate maxima.
<figure><img src="image_18.png" /><figcaption>Fig. 14. Comparison between the wave elevation (left) and the corresponding surge motion (right) of SS10-4 for different $B_{add}$ values.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 15. Surge return periods with $B_{add}$ = 15 N/m (left) and $B_{add}$ = 50 N/m (right) for SS10 (full scale).</figcaption></figure>

## 6. Conclusions

The present study demonstrates a response-conditioning technique that accounts for combined wave- and low-frequency responses, such as the extreme horizontal motions of moored floating structures. The distinctive feature of the proposed approach compared to traditional RCW methods is that, in addition to the linear response transfer function, it also considers the QTF within a refined surrogate response model. Moreover, a nonlinear wave model is used within this surrogate, permitting the identification of appropriate wave episodes that trigger resonant large amplitude slow-drift motions associated with difference-frequency loads. The proposed procedure is nested in a two-step multi-fidelity framework to predict the extreme surge motions of a moored container ship using short-duration wave episodes. As a first step, the surrogate model described above is employed to obtain a preliminary response distribution and several response thresholds are used for the conditioning process. The output of this procedure is the wavemaker motion signal that will yield the nonlinear RCW episodes. During the second step of the methodology, the waves are reproduced in model tests to obtain the fully nonlinear response that is associated with certain return period levels. Finally, these results are used to correct the surrogate response distribution.

The main findings of this study indicate that the surge motion of a linear SDoF oscillator, subject to first- and second-order wave loads, can serve as an appropriate surrogate response model to identify critical wave episodes that trigger near-resonant motions. This conclusion applies to the present case study but can potentially be generalised to most non-slender structures with relatively simple mooring arrangements. For more complex configurations or responses, strong coupling between dif-

ferent degrees of freedom might be necessary. This would require the transition to time-domain surrogate models in the response conditioning and will be investigated in future work.

Furthermore, a comparison between the design wave episodes and similar events from irregular time series indicates considerable discrepancies, due to the difference in the initial conditions of the body, associated with the mean drift. After filtering the mean drift from the irregular response time series, reasonable agreement between the irregular wave and design wave events is achieved. This is therefore identified as the principal limitation in the use of design waves for resonant slow-drift motions. In the present work, this issue is partially treated by applying the design wave results to re-evaluate a surrogate model distribution that already includes the mean drift. However, towards a more radical solution, the numerically determined offset should be imposed in the surrogate model and taken properly into account in the high-fidelity evaluation of the design waves. Under these terms, the initial conditions of the structure in irregular wave tests can be assimilated, and the body dynamics in the displaced mean position can be captured.

Via the technique of harmonic separation, the odd and even harmonics of the experimentally measured surge motions are compared with the linear and quadratic terms of the surrogate model, respectively. It is found that the discrepancy between the fully nonlinear response and the surrogate can be attributed solely to the even-order terms. Lastly, the short-term extreme surge motion of the vessel is estimated very efficiently by considering only short design wave tests whose experimental duration is lower than 5% of the irregular wave tests. A comparison against the experimental Monte Carlo approach under irregular waves shows that the design wave results are accurate throughout the whole range of sea states. Finally, given that the added damping in the lin-
earised SDoF system representing the model is a sensitive parameter and influences the amplitude of the slow-drift motions, different values of this parameter are examined. For two cases of more lightly or more heavily damped surrogate models, it is found that the performance of the design wave methodology is not noticeably influenced, indicating the robustness of the approach.

Overall, from an experimental point of view, short-duration design wave tests for wave-frequency responses have been proven to effectively replace time-consuming irregular wave Monte Carlo tests (Drummen et al., 2009; Tosdevin et al., 2022; Dermatis et al., 2025). In the evaluation of low-frequency driven responses, the required duration to obtain statistically convergent results using the Monte Carlo approach increases significantly. Therefore, using design waves to predict extreme low-frequency responses is an interesting alternative. The present study serves as an initial verification that design waves can be used for low-frequency responses and aims at identifying some principal limitations, such as the problem associated with the mean drift. To render this methodology pertinent for practical applications where the mooring line tensions are sought, the total second-order motions need to be coupled with a mooring dynamics solver, within the response-conditioning procedure. This will permit the application of the design wave methodology for the prediction of mooring loads and is considered the main direction of future work.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgement

The work presented in this paper was supported by Centrale Nantes - Bureau Veritas research chair and by the WASANO project, funded by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" Programme (ANR-16-IDEX-0007).

## CRediT authorship contribution statement

**Athanasios Dermatis:** Writing - review & editing, Writing - original draft, Methodology, Investigation, Formal analysis, Conceptualization; **Benjamin Bouscasse:** Writing - review & editing, Supervision, Methodology, Data curation, Conceptualization; **Guillaume Ducrozet:** Writing - review & editing, Supervision, Methodology, Data curation, Conceptualization; **Henrik Bredmose:** Writing - review & editing, Validation, Supervision, Methodology; **Harry B. Bingham:** Writing - review & editing, Validation, Supervision, Methodology.

## References

Adegeest, L., Braathen, A., Vada, T., 1998. Evaluation of methods for estimation of extreme nonlinear ship responses based on numerical simulations and model tests. In: Proc. 22nd Symposium on Naval Hydrodynamics, Washington DC, pp. 70-84.

Bouscasse, B., Dermatis, A., Ducrozet, G., de Hauteclocque, G., Lasbleis, M., 2024. Assessment of nonlinear loads and motions through response-conditioned waves: practical applications and limits. In: 43rd International Conference on Ocean, Offshore and Arc-tic Engineering, organization = American Society of Mechanical Engineers.

Bouscasse, B., Merrien, A., Horel, B., de Hauteclocque, G., 2022. Experimental analysis of wave-induced vertical bending moment in steep regular waves. J. Fluids Struct. 111, 103547. https://doi.org/10.1016/j.jfluidstructs.2022.103547.

Brown, L.D., Cai, T.T., DasGupta, A., 2001. Interval estimation for a binomial proportion. Stat. Sci. 16, 101-133. https://doi.org/10.1214/ss/1009213286

Brown, S., Tosdevin, T., Jin, S., Hann, M., Simmonds, D., Greaves, D., 2023. On the selection of design waves for predicting extreme motions of a floating offshore wind turbine. Ocean Eng. 290, 116400. https://doi.org/10.1016/j.oceaneng.2023.116400

Chen, X., 2007. Middle-field formulation for the computation of wave-drift loads. J. Eng. Math. 59, 61-82. https://doi.org/10.1007/s10665-006-9074-x

Der Kiureghian, A., 2000. The geometry of random vibrations and solutions by FORM and SORM. Probab. Eng. Mech. 15, 81-90.

Dermatis, A., Lasbleis, M., Kim, S., De Hauteclocque, G., Bouscasse, B., Ducrozet, G., 2025. A multi-fidelity approach for the evaluation of extreme wave loads using nonlinear response-conditioned waves. Ocean Eng. 316, 119919. https://doi.org/10.1016/j.oceaneng.2024.119919

Dermatis, A., Leroy, V., Ducrozet, G., Bouscasse, B., Delacroix, S., Bredmose, H., Bingham, H., 2024. Stochastic prediction of the extreme responses of a floating offshore wind turbine using response-conditioned waves. In: ISOPE 2024 34th International Ocean and Polar Engineering Conference.

Dietz, J., 2004. Application of Conditional Waves as Critical Wave Episodes for Extreme Loads on Marine Structures. Ph.D. thesis. Technical University of Denmark.

Dietz, J.S., Friis-Hansen, P., Jensen, J.J., 2004. Most likely response waves for estimation of extreme value ship response statistics. Seehafen Verlag. 9th International Symposium on Practical Design of Ships and Other Floating Structures, 286-293.

DNV, 2010. DNV-RP-c205 environmental conditions and environmental loads.

Dommermuth, D.G., Yue, D.K., 1987. A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267-288. https://doi.org/10.1017/S002211208700288X.

Drummen, I., Wu, M., Moan, T., 2009. Numerical and experimental investigations into the application of response conditioned waves for long-term nonlinear analyses. Mar. Struct. 22, 576-593. https://doi.org/10.1016/j.marstruc.2008.12.002

Ducrozet, G., Bonnefoy, F., Le Touzé, D., Ferrant, P., 2012. A modified high-order spectral method for wavemaker modeling in a numerical wave tank. Eur. J. Mech. B Fluids 34, 19-34.

Emmerhoff, O., Sclavounos, P., 1996. The simulation of slow-drift motions of offshore structures. Appl. Ocean Res. 18, 55-64. https://doi.org/10.1016/0141-1187(96)00015-6.

Engsig-Karup, A., Bingham, H., Lindberg, O., 2009. An efficient flexible-order model for 3D nonlinear water waves. J. Comput. Phys. 228, 2100-2118. https://doi.org/10.1016/j.jcp.2008.11.028.

van Essen, S., Seyffert, H., 2023. Finding dangerous waves-review of methods to obtain wave impact design loads for marine structures. J. Offshore Mech. Arct. Eng. 145, 060801. https://doi.org/10.1115/1.4056888.

van Essen, S.M., Scharnke, J., Seyffert, H.C., 2023. Required test durations for converged short-term wave and impact extreme value statistics - part 1: ferry dataset. Mar. struct. 90, 103410. https://doi.org/10.1016/j.marstruc.2023.103410.

Fernández, H., Sriram, V., Schimmels, S., Oumeraci, H., 2014. Extreme wave generation using self correcting method - revisited. Coastal Eng. 93, 15-31. https://doi.org/10.1016/j.coastaleng.2014.07.003.

Ferreira, A., Haan, L., 2013. On the block maxima method in extreme value theory: PWM estimators. Ann. Stat. 43. https://doi.org/10.1214/14-AOS1280.

Fitzgerald, C.J., Taylor, P.H., Taylor, R.E., Grice, J., Zang, J., 2014. Phase manipulation and the harmonic components of ringing forces on a surface-piercing column. Proc. Royal Soc. A 470, 20130847. https://doi.org/10.1098/rspa.2013.0847.

Ghadirian, A., Bredmose, H., 2019. Investigation of the effect of the bed slope on extreme waves using first order reliability method. Mar. struct. 67, 102627. https://doi.org/10.1016/j.marstruc.2019.05.005

Hann, M., Greaves, D., Raby, A., Howey, B., 2018. Use of constrained focused waves to measure extreme loading of a taut moored floating wave energy converter. Ocean Eng. 148, 33-42. https://doi.org/10.1016/j.oceaneng.2017.10.024

Hansen, C.L., Bredmose, H., Moalemi, A., Wolgamot, H., Orszaghova, J., Taylor, P.H., 2025a. Design waves and low-frequency responses of a semi-sub floating wind turbine in spread seas. Renew. Energy.

Hansen, C.L., Wolgamot, H., Taylor, P.H., Kurniawan, A., Orszaghova, J., Bredmose, H., 2025b. Design waves and extreme responses for an m4 floating, hinged wave energy converter. J. Fluids Struct. 133, 104253. https://doi.org/10.1016/j.jfluidstructs.2024.104253.

de Hauteclocque, G., Rezende, F., Waals, O., Chen, X., 2012. Review of approximations to evaluate second-order low-frequency load. https://doi.org/10.1115/OMAE2012-83407.

Jensen, J.J., Andersen, I.M.V., Seng, S., 2014. Stochastic procedures for extreme wave induced responses in flexible ships. Int. J. Naval Archit. Ocean Eng. 6, 1148-1159. https://doi.org/10.2478/LJNAOE-2013-0236.

Jensen, J.J., Capul, J., 2006. Extreme response predictions for jack-up units in second order stochastic waves by FORM. Probab. Eng. Mech. 21, 330-337. https://doi.org/10.1016/j.probengmech.2005.11.007.

Kac, M., Siegert, A.J.F., 1947. An explicit representation of a stationary gaussian process. Ann. Math. Stat. 18, 438 - 442. https://doi.org/10.1214/aoms/1177730391.

Kim, S., 2022. Experimental Study on Wave Bending Moments of a Zero-Speed Rigid Containership Model in Regular, Irregular, and Equivalent Design Waves. Ph.D. thesis. École centrale de Nantes.

Kim, S., Bouscasse, B., Ducrozet, G., Canard, M., De Hauteclocque, G., Ouled Housseine, C., Ferrant, P., 2022. Numerical and experimental study of a FORM-based design wave applying the HOS-NWT nonlinear wave solver. Ocean Eng. 263, 112287. https://doi.org/10.1016/j.oceaneng.2022.112287.

Kim, S., De Hauteclocque, G., Bouscasse, B., Lasbleis, M., Ducrozet, G., 2024. Experimental analysis of extreme wave loads on a containership. Ocean Eng. https://doi.org/10.1016/j.oceaneng.2024.118031.

Kim, Y., Kim, J.H., 2016. Benchmark study on motions and loads of a 6750-TEU containership. Ocean Eng. 119, 262-273. https://doi.org/10.1016/j.oceaneng.2016.04.015.

Lande-Sudall, D., Stansby, P., 2024. Short-duration design waves for modelling of extreme second-order surge response with spar substructure test case. Appl. Ocean Res. 153, 104232. https://doi.org/10.1016/j.apor.2024.104232.

Langley, R., 1986. On the time domain simulation of second order wave forces and induced responses. Appl. Ocean Res. 8, 134-143. https://doi.org/10.1016/S0141-1187(86)80012-8.
Langley, R., 1987. Second order frequency domain analysis of moored vessels. Appl. Ocean Res. 9, 7–18.

Langley, R., McWilliam, S., 1993. A statistical analysis of first and second order vessel motions induced by waves and wind gusts. Appl. Ocean Res. 15, 13–23. https://doi.org/10.1016/0141-1187(93)90029-W.

Lim, D.H., Kim, Y., 2018. Design wave method for the extreme horizontal slow-drift motion of moored floating platforms. Appl. Ocean Res. 71, 48–58.

Molin, B., 1979. Second-order diffraction loads upon three-dimensional bodies. Appl. Ocean Res. 1, 197–202. https://doi.org/10.1016/0141-1187(79)90027-0.

Naess, A., 1986. The statistical distribution of second-order slowly-varying forces and motions. Appl. Ocean Res. 8, 110–118. https://doi.org/10.1016/S0141-1187(86) 80006-2.

Naess, A., 2001. Crossing rate statistics of quadratic transformations of gaussian processes. Probab. Eng. Mech. 16, 209–217. https://doi.org/10.1016/S0266-8920(01)00003-0.

Naess, A., Gaidai, O., Teigen, P., 2007. Extreme response prediction for nonlinear floating offshore structures by Monte Carlo simulation. Appl. Ocean Res. 29, 221–230. https://doi.org/10.1016/j.apor.2007.12.001.

Naess, A., Karlsen, H., Teigen, P., 2006. Numerical methods for calculating the crossing rate of high and extreme response levels of compliant offshore structures subjected to random waves. Appl. Ocean Res. 28, 1–8. https://doi.org/10.1016/j.apor.2006.04. 001.

Neal, E., 1974. Second order hydrodynamic forces due to stochastic excitation. In: 10th ONR Symposium.

Orszaghova, J., Taylor, P., Wolgamot, H., Madsen, F., Pegalajar-Jurado, A., Bredmose, H., 2021. Wave- and drag-driven subharmonic responses of a floating wind turbine. J. Fluid Mech. 929. https://doi.org/10.1017/jfm.2021.874.

Pastoor, L.W., 2002. On the Assessment of Nonlinear Ship Motions and Loads. Ph.D. thesis.
Delft University of Technology.

Pegalajar-Jurado, A., Bredmose, H., 2019. Reproduction of slow-drift motions of a floating wind turbine using second-order hydrodynamics and operational modal analysis. Mar. Struct. 66, 178–196. https://doi.org/10.1016/j.marstruc.2019.02.008.

Pinkster, J.A., 1980. Low Frequency Second Order Wave Exciting forces on Floating Structures. Ph.D. thesis. Mechanical, Maritime and Materials Engineering, TU Delft.

Ripe, A.M.T., Lande-Sundall, D., 2023. A comparison of extreme mooring loads and response of a spar-buoy wind turbine using conditional waves. J. Phys. Confer. Ser. 2626. https://doi.org/10.1088/1742-6596/2626/1/012062.

Santos, S., Matioli, L., Beck, A., 2012. New optimization algorithms for structural reliability analysis. Comput. Model. Eng. Sci. 83, 23–55.

Takami, T., Fujimoto, W., Houtani, H., Matsui, S., 2023. Combination of HOSM and FORM for extreme wave-induced response prediction of a ship in nonlinear waves. Ocean Eng. 286, 115643. https://doi.org/10.1016/j.oceaneng.2023.115643.

Takami, T., Komoriyama, Y., Ando, T., Ozeki, S., Iijima, K., 2020. Efficient FORM-based extreme value prediction of nonlinear ship loads with an application of reduced-order model for coupled CFD and FEA. J. Mar. Sci. Technol. 25, 327–345. https://doi.org/10.1007/s00773-019-00667-8.

Tosdevin, T., Jin, S., Simmonds, D., Hann, M., Greaves, D., 2022. On the use of constrained focused waves for characteristic load prediction. In: Trends in Renewable Energies Offshore. CRC Press, London, 1 edition. pp. 609–617. https://doi.org/10.1201/9781003360773-69.

Tromans, P.S., Anaturk, A.R., Hagemeijer, P., 1991. A new model for the kinematics of large ocean waves-application as a design wave. In: The First International Offshore and Polar Engineering Conference.

Veritas, B., 2019. Guidance for long-term hydro-structure calculations.

Walker, D.A., Taylor, P.H., Taylor, R.E., 2004. The shape of large surface waves on the open sea and the draupner new year wave. Appl. Ocean Res. 26, 73–83. https://doi.org/10.1016/j.apor.2005.02.001.

West, B.J., Brueckner, K.A., Janda, R.S., Milder, D.M., Milton, R.L., 1987. A new numerical method for surface hydrodynamics. J. Geophys. Res. 92, 11803–11824.

Zhao, Z., Wang, W., Shi, W., Li, X., 2020. Effects of second-order hydrodynamics on an ultra-large semi-submersible floating offshore wind turbine. Structures 28, 2260–2275. https://doi.org/10.1016/j.istruc.2020.10.058.