RAO-based translation process for estimating extreme value distribution of nonlinear ship response in irregular waves

Abstract

The authors propose a new method named the RAO-based Translation Process (RTP) method for simply estimating the extreme value distribution of the nonlinear ship response in irregular waves based on the nonlinear response in regular waves. The proposed method combines the existing idea of the Nonlinear Correction (NLC) method, which is based on wave height-dependent RAO, with the theory of the translation process, and thus has both simplicity and a clear theoretical background. Furthermore, it is positioned as an extension of the Equivalent Design Wave (EDW) method used in conventional ship structural design. The validity of the proposed method is confirmed through a numerical study on the extreme value distribution of hull-girder sectional forces considering body nonlinear effects. The proposed method can be used for various purposes, and in particular, it can reduce the computational demand for strength evaluation by Direct Load and Structure Analysis (DLSA), which is a computationally extensive analysis method. Therefore, the proposed method is expected to make a significant contribution to structural design.

1 Introduction

The structural strength of ships is evaluated according to the rules of classification societies. In recent years, Direct Load and Structure Analysis (DLSA) has been recognized as an advanced alternative method [1,2,3]. The calculation procedure for estimating the maximum structural response during a service period by DLSA usually begins with the calculation of the maximum wave loads using a linear statistical prediction [4] based on the RAO of the loads, followed by consideration of nonlinear effects by a structural response analysis in Equivalent Design Waves (EDW, assumed to be regular waves in this paper).

However, it is known that the evaluation procedure using EDW (hereinafter referred to as “EDW method”) may overestimate or underestimate loads other than the focused dominant load parameter (DLP) and has non-negligible estimation errors (around 20%) for DLSA using whole ship structural models [5, 6]. Although the EDW method has the advantage of reducing the computational cost of structural analyses, it is also strongly implied that its use is unavoidable, because nonlinear effects and simultaneity cannot be taken into account in the usual statistical prediction. Under these circumstances, there is an urgent need for developing a practical method for statistical prediction that can be applied to DLSA and that takes nonlinearity into account.

Since DLSA is computationally demanding and the long-term statistical prediction method requires the calculation of ship responses under a large number of short-term sea states, the key point is how to minimize the computational volume. Among the various methods for statistical prediction of nonlinear response, the most practical and effective method for DLSA is considered to be either of the following.

• (A) Methods limiting to representative wave conditions

• (B) Frequency-domain calculation methods based on RAO.

The EDW method is included in (A). More advanced methods in (A) exist that use the design irregular wave [7], FORM [8], the most likely response wave (MLRW) [9], and the conditional random response wave (CRRW) [10]. These are suitable for considering nonlinearity but have the disadvantage that they require focusing on one DLP (or stress in a structural member) and may overestimate or underestimate other DLPs.

On the other hand, method (B) does not require the definition of DLP and can evaluate all structural members based on an analysis in regular waves, but has difficulty in considering nonlinearity. The higher order transfer function method [11,12,13] is a frequency-domain theory that considers nonlinearities. One of the authors examined the applicability of this method to nonlinear wave loads on ships, and confirmed that nonlinear effects are insufficient in the Quadratic Transfer Function (QTF)-based response for ships, such as container ships, which exhibit strong nonlinear effects [14]. In addition, the calculation of the QTF requires the square of the number of wave conditions required for the normal response function, making it difficult to adopt this method in practical design work.

A simple method that is categorized as (B) and can consider nonlinearity is the nonlinear correction (NLC) method proposed by Oka et al. for the long-term extreme value distribution (EVD) of sloshing loads [15]. The NLC method is based on linear theory and uses an RAO obtained under multiple wave heights to approximately account for nonlinearity in the long-term distribution, making it a good match for costly analysis methods such as DLSA. However, the NLC method lacks in the theoretical basis in the formulation, and it cannot define the spectrum or PDF of the nonlinear response.

Another approach to nonlinear stochastic processes is the translational process theory, which interprets non-Gaussian responses as being mathematically tractable maps of Gaussian processes [16, 17]. The main purpose of the theory is to characterize and approximate the statistical properties of a non-Gaussian process, which is different from the purpose of this study. However, the idea of mapping Gaussian to non-Gaussian processes can be applied to present problem as a simple way to consider nonlinearities.

Based on the above, this study proposes a new simple method for estimating the EVD of the nonlinear ship response in waves, which is categorized as (B). The proposed method is named the RAO-based Translation Process (RTP) method. This method is the same as the NLC method in the sense that both are based on a wave height-dependent RAO, but the RTP method incorporates the idea of the translation process in the consideration of nonlinearity.

In Sect. 2 of this paper, we first explain the proposed RTP method and confirm that the RTP method can be positioned as an extension of the EDW method. We also introduce the existing NLC method by Oka et al. Section 3 discusses the applicability of the proposed method through a comparison with the EVD calculated by direct time-domain computation for nonlinear hull-girder sectional forces (moments).

2 Simplified method for calculating extreme value distribution of nonlinear load

This section describes a simple method for obtaining the EVD of the nonlinear response during short-term sea states based only on the response calculations in regular waves.

The first assumption is that the RAO is obtained at several wave height $H_{w} ,$ which is denoted $\widehat{X}\left(\omega ,\beta ;{H}_{w}\right)$ (complex amplitude per unit wave amplitude) where $\omega$ is the wave frequency and $\beta$ is wave angle. Note that in the following, wave height ${H}_{w}$ means the wave height of the regular wave when obtaining the RAO $\hat{X}\left( {\omega ,\beta ;H_{w} } \right).$ Based on this and the wave spectrum $\Phi_{\zeta \zeta } \left( {\omega ,\beta } \right),$ we consider a method for estimating the EVD of the nonlinear response $X$ in irregular waves. The response spectrum is assumed to be narrowband. That is, if $X$ is a linear response, EVD ${Q}_{X}\left(x\right)$ follows a Rayleigh distribution, as is well known:

$${Q}_{X}\left(x\right)=\mathrm{exp}\left(-\frac{{x}^{2}}{2{\sigma }_{X}^{2}}\right),$$

(1)

where ${\sigma }_{X}$ is the standard deviation of $X$

$${\sigma }_{X}^{2}={\int }_{-\pi }^{\pi }{\int }_{0}^{\infty }{\left|\widehat{X}\left(\omega ,\beta \right)\right|}^{2}{\Phi }_{\zeta \zeta }\left(\omega ,\beta \right)\mathrm{d}\omega \mathrm{d}\beta .$$

(2)

There are several possible ways to define the wave height-dependent RAO $\hat{X}\left( {\omega ,\beta ;H_{w} } \right)$ from the time-series data of the nonlinear response in regular waves, such as considering the amplitude as a Fourier coefficient or peak to peak. In this study, positive and negative RAOs $\left( {\hat{X}^{ + } \left( {\omega ,\beta ;H_{w} } \right), \hat{X}^{ - } \left( {\omega ,\beta ;H_{w} } \right)} \right)$ are defined from the positive and negative peak values of the response time-series in regular waves, respectively, to make a conservative evaluation and handle the positive/negative asymmetry represented by the hogging/sagging asymmetry of the vertical bending moment (VBM). The respective RAOs are then used in the EVD of the maxima and minima. However, for simplicity, in the formulation in this section, we will not separate the positive and negative values and will formulate only one side.

2.1 Translational process

As a method for estimating the EVD of a non-Gaussian process $X(t)$, a method for approximating $X(t)$ by the following nonlinear transformation of the Gaussian process $U\left(t\right)$ is known as a translation process [16, 18]:

$$X\left(t\right)=g\left(U\left(t\right)\right).$$

(3)

When $U\sim \mathcal{N}\left(0,{\sigma }_{U}\right)$ and $g$ is a monotonically increasing function, the PDF ${f}_{X}\left(x\right)$ and EVD ${Q}_{X}\left(x\right)$ of $X$ are as follows:

$${f}_{X}\left(x\right)=\frac{\mathrm{d}{g}^{-1}\left(x\right)}{\mathrm{d}x}\frac{1}{\sqrt{2\pi }{\sigma }_{U}}\mathrm{exp}\left(-\frac{{\left\{{g}^{-1}\left(x\right)\right\}}^{2}}{2{\sigma }_{U}^{2}}\right)$$

(4)

$${Q}_{X}\left(x\right)=\mathrm{exp}\left(-\frac{{\left\{{g}^{-1}\left(x\right)\right\}}^{2}}{2{\sigma }_{U}^{2}}\right).$$

(5)

The key point in this translation process is how to find an appropriate inverse transformation ${g}^{-1}$ that maps $X$ to $U$. Wint´erstein have shown how to approximate ${g}^{-1}$ by the Hermite polynomial from the moments of the PDF ${f}_{X}\left(x\right)$ up to the fourth order [18], and this method has been applied to wind and wave loads [13, 19]. The translation process is also used to generate non-Gaussian processes with targeted PDF and spectra [17, 20].

In general, the Gaussian process $U$ used in the context of the translation process is a mathematical one that has no physical meaning, but in this study, we assume a situation where a linear RAO can be used, so the linear response obtained from it can be regarded as $U$ as it is. Therefore, the nonlinear response in irregular waves is $X$, the linear response in irregular waves is $U$, and their transformation $g$ can be approximated using the wave height-dependent RAO $\widehat{X}\left(\omega ,\beta ;{H}_{w}\right)$.

2.2 Proposal for a new method: RTP (RAO-based translation process) method

In this study, we propose a method in which ${g}^{-1}\left(x\right)$ in Eqs. 4, 5 is defined using the wave height-dependent RAO $\widehat{X}\left(\omega ,\beta ;{H}_{w}\right)$ in the wave condition where the response spectrum peaks. We call this method the RAO-based Translation Process (RTP) method. The RTP method defines ${g}^{-1}\left(x\right)$ as follows, with the wave height ${H}_{w}$ as an independent variable:

$${g}^{-1}\left(x\left({H}_{w}\right)\right)=\frac{{H}_{w}}{2}\left|\widehat{U}\left({\omega }_{pk},{\beta }_{pk}\right)\right|$$

(6)

$$x\left({H}_{w}\right)=\frac{{H}_{w}}{2}\left|\widehat{X}\left({\omega }_{pk},{\beta }_{pk};{H}_{w}\right)\right|,$$

(7)

where ${\omega }_{pk}\left({H}_{w}\right)$ and ${\beta }_{pk}\left({H}_{w}\right)$ are the wave frequency and wave direction that maximize the wave height-dependent response spectrum ${\Phi }_{xx}\left(\omega ,\beta ;{H}_{w}\right)$. That is

$$\left( {\omega_{pk} \left( {H_{w} } \right), \beta_{pk} \left( {H_{w} } \right)} \right)\equiv \mathop {\text{arg max}}\limits_{{\left( {\omega ,\beta } \right)}} \Phi_{xx} \left( {\omega ,\beta ;H_{w} } \right),$$

(8)

$${\text{where }}\Phi_{xx} \left( {\omega ,\beta ;H_{w} } \right)\equiv \left| {\hat{X}\left( {\omega ,\beta ;H_{w} } \right)} \right|^{2} \Phi_{\zeta \zeta } \left( {\omega ,\beta } \right).$$

(9)

$\widehat{U}\left(\omega ,\beta \right)$ in Eq. 6 is a linear RAO and can be defined as the response at a sufficiently small wave height ${H}_{w}=\varepsilon$ as follows:

$$\widehat{U}\left(\omega ,\beta \right)=\widehat{X}\left(\omega ,\beta ;\varepsilon \right);$$

(10)

${\sigma }_{U}$ in Eqs. 4 and 5 can be obtained by the following equation:

$${\sigma }_{U}^{2}={\int }_{-\pi }^{\pi }{\int }_{0}^{\infty }{\left|\widehat{U}\left(\omega ,\beta \right)\right|}^{2}{\Phi }_{\zeta \zeta }\left(\omega ,\beta \right)\mathrm{d}\omega \mathrm{d}\beta .$$

(11)

The function ${g}^{-1}$ in Eq. 6 can be approximated by a polynomial or other function or by a multilinear (piecewise linear) approximation from the values of $x\left({H}_{w}\right)$ and ${g}^{-1}\left(x\left({H}_{w}\right)\right)$ obtained at each wave height ${H}_{w}$. In this study, a multilinear approximation is employed, because some cases could not be accurately approximated by polynomial functions. Approximating ${g}^{-1}$ by a multilinear line is equivalent to calculating $x\left({H}_{w}\right)$ and ${Q}_{X}\left(x\left({H}_{w}\right)\right)$ and then linearly interpolating or extrapolating on the $x$-$\sqrt{-\mathrm{log}{Q}_{X}}$ plane.

A schematic representation of the above method is shown in Fig. 1. The RTP method is clear in that it is based on the following assumptions and approximations.

• The assumption that the instantaneous values of the linear and nonlinear responses correspond one-to-one by the nonlinear transformation $g$.

• The approximation that ${g}^{-1}$ is determined by the response in regular waves.

Fig. 1

Schema of the procedure to obtain the extreme distribution of nonlinear response based on the RTP method

The former assumption is not necessarily valid for instantaneous response values, but it is not unreasonable for estimating EVD, since simultaneity need not be considered, as will be discussed at the end of Sect. 3. The latter approximation, on the other hand, approximates the magnitude of nonlinear effects in irregular waves to that in regular waves, which is likely to lead to errors.

The RTP method is extremely computationally inexpensive for a nonlinear statistical theory. Denoting the number of cases of wave frequency, wave direction, and wave height as ${n}_{\omega }$, ${n}_{\beta }$, and ${n}_{Hw}$, respectively, the number of regular waves required for the RTP method is ${n}_{\omega }\times {n}_{\beta }\times {n}_{Hw}$, which is about ${n}_{Hw}$ times larger than that of the linear method. On the other hand, the QTF-based method requires ${\left({n}_{\omega }\times {n}_{\beta }\right)}^{2}$ cases. In addition, time-domain calculations with irregular waves require a long computation time for each sea state, which is incomparably longer.

2.3 Characteristics of the RTP method in terms of differences from the EDW method

The proposed RTP method is essentially the same as the EDW method commonly used in structural design in that it approximates the nonlinear effects of the response in irregular waves with those in regular waves. The EDW method starts from the horizontal axis (probability of exceedance) on the right in Fig. 1, while the RTP method starts from the vertical axis on the left in Fig. 1, because $X$ is treated as a random variable. However, the proposed method differs from the conventional EDW method in the following points.

• The EDW method is applied to the long-term maximum response, while the RTP method is applied for the short-term maximum response. Therefore, the RTP method is expected to have a smaller error in the approximation representing the regular wave.

• In the EDW method, the RAO is used to determine the wave condition of the EDW. Since the RAO does not include wave information, it is often necessary to make judgments such as eliminating non-dominant conditions. On the other hand, since the RTP method uses the wave condition of the peak of the response spectrum, which can generally be considered the dominant condition for the response, it allows for a uniform procedure that eliminates arbitrariness.

• The EDW method [including other design wave methods categorized as (A) in the Introduction] yields a response with a certain probability of exceedance, while the RTP method yields a probability distribution.

• The EDW method (including other design wave methods) requires first determining the DLP and recalculating the response in the EDW corresponding to the DLP, whereas the RTP method allows uniform calculations based on a wave height-dependent RAO, so it is possible to target all structural member stresses without determining the DLP.

Considering the above, it can be understood that the RTP method can be positioned as an extended and systematized method for the EDW method.

2.4 NLC (nonlinear correction) method

This section introduces the NLC method, which is a simple method proposed by Oka et al. [15] This method uses the wave height-dependent RAO $\widehat{X}\left(\omega ,\beta ;{H}_{w}\right)$ to obtain the long-term EVD for each wave height based on linear theory, and then determines their applicability limits by the peak values of the RAO at each wave height. In this paper, the NLC method is applied to the short-term EVD. The procedure is as follows.

First, the wave height-dependent standard deviation should be determined as follows:

$${\sigma }_{X}^{2}\left({H}_{w}\right)={\int }_{-\pi }^{\pi }{\int }_{0}^{\infty }{\left|\widehat{X}\left(\omega ,\beta ;{H}_{w}\right)\right|}^{2}{\Phi }_{\zeta \zeta }\left(\omega ,\beta \right)\mathrm{d}\omega \mathrm{d}\beta .$$

(12)

As the EVD corresponding to these standard deviations, we define the following wave height-dependent Rayleigh distribution assuming a linear and narrowband response:

$${Q}_{X}^{R}\left(x;{H}_{w}\right)=\mathrm{exp}\left(-\frac{{x}^{2}}{2{\sigma }_{X}^{2}\left({H}_{w}\right)}\right).$$

(13)

Then, the range of the Rayleigh distribution ${Q}_{X}^{R}\left(x;{H}_{w}\right)$ for each wave height is limited by the maximum response amplitude $({H}_{w}/2)\left|\widehat{X}\left({\omega }_{pk},{\beta }_{pk};{H}_{w}\right)\right|$ for each wave height. This means that the EVD is defined as follows, with ${H}_{w}$ as the independent variable:

$${Q}_{X}\left(x\left({H}_{w}\right)\right)=\mathrm{exp}\left(-\frac{{x}^{2}\left({H}_{w}\right)}{2{\sigma }_{X}^{2}\left({H}_{w}\right)}\right)$$

(14)

$$x\left({H}_{w}\right)=\frac{{H}_{w}}{2}\left|\widehat{X}\left({\omega }_{pk},{\beta }_{pk};{H}_{w}\right)\right|.$$

(15)

Figure 2 is a schematic representation of the above derivation of the nonlinear EVD ${Q}_{X}\left(x\right)$. The nonlinear EVD is the curve obtained by connecting the intersection of the nonlinear response amplitude level in regular waves and the Rayleigh distribution ${Q}_{X}^{R}\left(x;{H}_{w}\right)$ at the same wave height.

Fig. 2

Schema of the procedure for obtaining the extreme distribution of nonlinear response based on the NLC method

Although Oka et al. defined ${\omega }_{pk}$ and ${\beta }_{pk}$ simply as the wave condition where the RAO peaks, this definition cannot be applied to responses such as surge, where the RAO does not have a clear peak, and it is not an appropriate approximation when the wave condition where the RAO peaks is not in the main frequency band of the wave spectrum. Therefore, in this study, as in the RTP method, ${\omega }_{pk}\left({H}_{w}\right)$ and ${\beta }_{pk}\left({H}_{w}\right)$ are defined as the wave frequency and direction at which the response spectrum (Eq. 9) for each wave height is maximum.

3 Numerical validation for hull-girder sectional forces considering body nonlinear effects

Here, the applicability of the EVD based on the simplified method described in Sect. 2 is verified for hull-girder sectional forces (moments) in short-crested irregular waves. Figure 3 shows the definitions of the wave direction and directions of the sectional forces.

Fig. 3

Definition of the wave angle $\beta$ and directions of the hull-girder sectional forces

The ships to be used are the container ship with L = 284 m and C_B = 0.63 and the bulk carrier with L = 278 m and C_B = 0.84 shown in Fig. 4. In this study, the ship’s speed is uniformly 5 knots based on the common structural rules [21].

Fig. 4

Body plan with S.S. 0.5 intervals and the waterline of the target ships. Left: container ship (L = 284 m, C_B = 0.63); right: bulk carrier (L = 278 m, C_B = 0.84)

3.1 Calculation method of nonlinear sectional forces and their RAO

The nonlinear sectional forces in regular and irregular waves were calculated using a linear 3D-Green’s function method program, SPREME [22]. For ship motion, the results of the linear theory were used, but the sectional forces considering the so-called body nonlinear effects (includes Froude Krylov and restoring forces nonlinearity) [23, 24] were calculated by the following procedure:

• i) Convert the wave pressure distribution on the hull surface in the frequency-domain to the time-domain data by the inverse Fourier transform. Here, the pressure at the waterline is extrapolated to freeboard by linear extrapolation, and the negative pressure is set to zero.

• ii) Calculate the inertia force balanced by the fluid force obtained by integrating the pressure in (i).

• iii) Integrate the pressure and inertia force distribution to obtain the nonlinear sectional forces.

The sectional forces obtained by this method are equivalent to those obtained when inertial relief option is applied in the structural analysis in DLSA using a whole ship FEM model.

The positive and negative sides of the RAO ${\widehat{X}}^{+}\left(\omega ,\beta ;{H}_{w}\right), {\widehat{X}}^{-}\left(\omega ,\beta ;{H}_{w}\right)$ were created from the maximum and minimum values of the time-series data of the sectional forces in regular waves obtained by the above method. Wave heights ${H}_{w}$ were selected in increments of 2 m, and the small wave height $\varepsilon$ was set at 0.1 m.

Figure 5 shows the RAO of the vertical bending moment (VBM) amidship in head sea ($\beta =180^\circ$) at a ship speed of 5 knots. The RAO of the horizontal bending moment (HBM) amidship and the torsional moment (TM) around the keel at station 2.5 in oblique waves ($\beta =60^\circ$) are shown in Figs. 6 and 7, respectively. Figure 5 shows that the hogging moment becomes smaller and the sagging moment becomes larger as the wave height increases, as is well known as a trend of body nonlinear effects [23, 24].

Fig. 5

Wave height-dependent RAO of VBM in head sea, 5 kt. Top: container ship; bottom: bulk carrier

Fig. 6

Wave height-dependent RAO of HBM in oblique sea ($\beta =60^\circ$), 5 kt. Top: container ship; bottom: bulk carrier

Fig. 7

Wave height-dependent RAO of TM in oblique sea ($\beta =60^\circ$), 5 kt. Top: container ship; bottom: bulk Carrier

3.2 Comparison of extreme value distributions in irregular waves

To verify the RTP method and NLC method, we calculated the EVD of the sectional forces in short-crested irregular waves for the container ship and the bulk carrier, respectively. For VBM, irregular waves with the mean wave direction $\chi =180^\circ$, significant wave height ${H}_{s}$=12 m, and mean wave period ${T}_{z}=\sqrt{2\pi L/g}$ (mean wavelength corresponds to $L$) were used. For HBM and TM, the mean wave direction $\chi =60^\circ$, significant wave height ${H}_{s}$=12 m, and mean wave period ${T}_{z}=\sqrt{\pi L/g}$ (mean wavelength corresponds to $L/2$) were used. The ISSC wave spectrum was used, and the cos² distribution was used for the directional distribution function [25].

For verification, the EVD obtained by direct numerical calculation in irregular waves was used as the correct distribution. The wave spectrum was divided into 100 equal area segments, and the top 20% of absolute values out of a total of 20,000 peak values (2000 waves × 10 runs) were used in Weibull fitting to generate the EVD. An example of a time-series of the linear and nonlinear VBM of the container ship in irregular waves is shown in Fig. 8. The linear VBM in Fig. 8 is the result of multiplying the results for small significant wave heights (${H}_{s}$=0.1 m).

Fig. 8

Example of the time-series of the linear and nonlinear VBM in irregular waves for a container ship

The positive and negative sides of the EVD of VBM, HBM, and TM for the container ship and the bulk carrier in irregular waves are shown in Figs. 9, 10 and 11, comparing the distributions obtained by direct calculation (time-series results) and the RTP and NLC simplified methods. The Rayleigh distribution using a linear RAO is also shown in the same figures.

Fig. 9

Comparison of the extreme value distribution of the VBM in a short-crested irregular wave (${H}_{s}$=12 m, $U$=5 kt, $\chi =180^\circ$). Left: container ship; right: bulk carrier

Fig. 10

Comparison of the extreme value distribution of the HBM in a short-crested irregular wave (${H}_{s}$=12 m, $U$=5 kt, $\chi =60^\circ$). Left: container ship; right: bulk carrier

Fig. 11

Comparison of the extreme value distribution of the TM in a short-crested irregular wave (${H}_{s}$=12 m, $U$=5 kt, $\chi =60^\circ$). Left: container ship; right: bulk carrier

From Fig. 9, as is the well-known trend, the hogging moment is not significantly different from the linear results for both ship types, while the sagging moment shows a significant nonlinear effect, especially for the container ship. Both of the two simplified methods capture the same trend, with the RTP method in particular showing a high degree of agreement with the time-series results. The NLC method tends to estimate nonlinear effects slightly smaller than the RTP method. This is thought to be because the RPT method is representative of the nonlinear effects under conditions where the RAO peaks, while the NLC method considers nonlinear effects based on the standard deviation, which also affects the contribution of frequency components other than the peak.

The HBM and TM shown in Figs. 10 and 11 also show stronger nonlinearities on the negative side, which is affected by body nonlinearity on the weather side. Both simplified methods reproduce this trend but evaluate the nonlinear effect slightly smaller on the negative side. This may simply be due to the fact that the nonlinear effect is stronger in irregular waves than in regular waves due to the contribution of component waves other than the dominant frequency.

The two simplified methods give generally similar results, but the RTP method tends to be more consistent with the time-series results. In addition to accuracy, the RTP method is considered more suitable for practical use in that it has a clear theoretical background for approximation and is positioned as an extension of the conventional EDW method.

Note that the symbols indicating the simplified methods ($*$, $\mathrm{o}$) in Figs. 9, 10, 11 are the values obtained from the RAO of the wave height at 2 m increments. As seen in the figure, when the wave height increments are coarser than 2 m, the interpolation error becomes larger when the plots of $x$ and $-\mathrm{log}{Q}_{X}$ are connected by a straight line. Therefore, when interpolating EVD ${Q}_{X}\left(x\right)$, it is preferable to perform linear interpolation on the plot with $x$ and $\sqrt{-\mathrm{log}{Q}_{X}}$ on the axes.

3.3 Considerations on the RTP method

The RTP method assumes a one-to-one correspondence between the linear response $U$ and the nonlinear response $X$ in irregular waves. To confirm the validity of this assumption, Fig. 12 shows the relationship between the peak values of the linear and nonlinear VBM time-series for the container ship and the bulk carrier during a short-term sea state.

Fig. 12

Correspondence of the peak values of the linear and nonlinear VBM by the time-series and RAO. Top: container ship; bottom: bulk carrier

The plus symbols ( +) indicate the corresponding linear and nonlinear VBM peak values during the same zero-upcrossing interval. That is, these symbols represent the linear and nonlinear VBM at the same time. From the data, an overall strong correlation of the peak values of the linear and nonlinear VBM was confirmed, but significant variation was observed on the sagging side for the container ship, which means that the assumption of a one-to-one correspondence is not valid. This violation of the assumption of one-to-one correspondence affects the case of multivariate simultaneity, but does not affect the extreme value distribution of a single variable, as in the present case. This is because EVD do not consider simultaneity, but rather whether or not a threshold value is exceeded, i.e., a distribution sorted by value.

The circular symbols (o) are the peak values of the linear and nonlinear VBM sorted in decreasing order of absolute value, and hence on a single curve. In other words, they correspond to the same probability of exceedance. Since each of the circular symbols corresponds to the same exceedance probability, this curve represents the correspondence between the linear and nonlinear EVD. Therefore, if this curve can be approximated satisfactorily by the RAO $\widehat{X}\left(\omega ,\beta ;{H}_{w}\right)$, this means that the proposed method has good accuracy.

The asterisk symbols (*) indicate the relation between the wave height-dependent RAO and the linear RAO in the peak wave condition of the response spectrum, i.e., $({H}_{w}/2)\left|\widehat{X}\left({\omega }_{pk},{\beta }_{pk};{H}_{w}\right)\right|$ and $({H}_{w}/2)\left|\widehat{U}\left({\omega }_{pk},{\beta }_{pk}\right)\right|$. The curve (multilinear) connected by the asterisks denotes the nonlinear transformation $X=g(U)$ in the proposed method. Since these symbols coincide closely with the curve of the circular symbols, it can be understood that the proposed method provides a good approximation of VBM to the degree seen in Fig. 9.

However, since the RTP method (and also the NLC method) represents nonlinear effects by the response in regular waves at a single point in $(\omega ,\beta )$ space, it does not reflect changes in the nonlinear effects caused using wave spectra with different shapes. Therefore, it should be noted that if the wave or response spectrum is non-narrowband, and especially if it is a multimodal spectrum, the present method may give a different trend from the results of the validation performed in this paper.

4 Conclusion

In this study, the authors newly proposed the RAO-based Translation Process (RTP) method, which is a simple method for obtaining the EVD of the nonlinear response in irregular waves from the nonlinear response in regular waves. The applicability of the proposed method is examined, together with that of the existing simple method, the Nonlinear Correction (NLC) method, for wave-induced hull-girder sectional forces considering body nonlinear effects.

Overall, both the RTP and NLC methods showed generally similar results and agreed with the EVD obtained from direct nonlinear simulations with practical accuracy. The results confirm that the RTP method tends to evaluate nonlinear effects more strongly than the NLC method and agrees well with the results of the direct nonlinear simulations.

In addition, the RTP method has a clear theoretical background for approximation and is characterized by the ability to obtain not only the EVD but also the PDF of instantaneous values. Furthermore, it is positioned as an extension of the EDW method, which has been used conventionally for structural strength evaluation, and is therefore considered to be a suitable method for application in design.

While the RTP method can target relatively strong nonlinear effects such as wave-induced VBM in slender ships if the bandwidth is narrow enough, it cannot handle nonlinear phenomena with periods outside of the wave period such as whipping and sprigging. It also cannot handle higher order phenomena with long periods that do not show up in the response in regular waves, and is therefore not suitable for estimating phenomena such as drift forces.

The proposed method is applicable to various purposes of computing nonlinear EVD, and in particular, it is effective for statistical prediction of the structural response in DLSA. In the conventional DLSA, it is necessary to consider the nonlinearity of wave loads by performing a structural analysis under EDW, after first calculating a linear long-term prediction using the RAO of the DLP or stress. In contrast, the present method enables directly short- and long-term prediction calculations that take into account the nonlinearity of wave loads from the wave height-dependent RAO of stresses, without setting up the DLP or EDW. The proposed RTP method is expected to make a substantial contribution to improving the reliability of strength evaluation by DLSA.