# A New Model Uncertainty Measure of Wave-Induced Motions and Loads on a Container Ship with Forward Speed

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Comparative Study

#### 2.1. Model Tests and Mass Distribution

_{pp}= 0.35 to 1.4 and a wave height of 1/60 of ship length, which is not very small compared to the draft [89]. Although the tests were conducted at various heading angles, only three headings are reported in this paper as μ = 45° quartering stern waves, 180° head waves, and 225° oblique bow waves. Heave and pitch motions at the centre of gravity, as well as vertical relative motions measured concerning the undisturbed wave at station 20 in the bow, are among the model test results used in this study. Furthermore, the strain gauges’ measurements of shear forces and bending moments in both vertical and horizontal planes, as well as torsional moments are used. Another experimental study for the same ship conducted by China Ship Scientific Research Centre (CSSRC) at a scale of 1:80, produced the results for heave and pitch motions at a lower speed that corresponds to Fr = 0.1 [90].

#### 2.2. Numerical Simulations

#### 2.2.1. PDSTRIP Simulations

#### 2.2.2. MAXSURF Simulations

#### 2.2.3. IST-CENTEC Simulations

#### 2.2.4. AQWA Simulations

#### 2.2.5. WASIM Simulations

_{R}is the rudder deflection, x

_{2}is the sway motion, x

_{6}is the yaw motion and ${\dot{x}}_{6}$ is the yaw angular velocity, k

_{1}= 1, k

_{2}= 100 and k

_{3}= 0.01.

#### 2.3. Uncertainty Measures

^{M}proposed by Kim and Kim [66] is then evaluated. Lastly, this study proposes a new uncertainty measure, modified total difference TD

^{E}, to quantify the uncertainty of individual seakeeping codes against available experimental results.

#### 2.3.1. Frequency-Independent Model Error

#### 2.3.2. Total Difference Approach

#### 2.3.3. The Modified Total Difference Approach

## 3. Results and Discussion

#### 3.1. Linear Response Comparisons

_{3}and vertical relative motions per wave amplitude ζ and pitch motion x

_{5}per wave slope kζ. Furthermore, sectional forces and moments are nondimensionalized by the terms ρgζBL and ρgζBL

^{2}, respectively, where ρ is the density of water, g acceleration of gravity, and B ship breadth. The nondimensional transfer functions are plotted versus the non-dimensional wavelength, defined as λ/L

_{pp}, where λ is the wavelength, while L

_{pp}is the ship length between perpendiculars. Figure 5 shows the non-dimensional heave, pitch, and vertical relative motions, whereas Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 show the non-dimensional sectional loads (VSF, HSF, VBM, HBM, and TM) at stations 5, 10, and 15.

#### 3.2. Comparisons of Uncertainty Measures

^{M}and TD

^{E}) in transfer functions computed using the aforementioned five numerical codes. Due to the limited number of applied codes utilized in the study, it is not possible to investigate the uncertainty in the transfer function based on groups employing the same seakeeping method or theory.

^{2}is greater than 0.9 in all cases except for VRM in quartering stern waves. Therefore, the bias measure $\widehat{a}$ is sufficient to represent the uncertainty in the transfer functions of motions. Pitch motion predictions are generally better than heave motion predictions. Table 3 and Table 4 show that bias and coefficient of determination for pitch motion are closer to unity than for heave and VRM.

^{M}vary between 2.5 to 23.2 % for heave motion, 3.2 to 29.7 % for pitch motion, and 3.1 to 37 % for VRM. On the other hand, the estimates of the total differences with respect to the experimental data (modified total difference) TD

^{E}vary between 7.4 to 32.7 % for heave motion, 7.9 to 53.9 % for pitch motion, and 8.1 to 57.3 % for VRM. It is noticed in many cases that the estimated total difference with respect to the experimental data TD

^{E}is higher than the estimated total difference with respect to the average of calculated results obtained by numerical codes TD

^{M}.

^{M}estimations may not reflect the numerical code’s accuracy. Table 3 shows, for example, that PDSTRIP predictions for heave motion in quartering oblique waves yield a TD

^{M}of 23.2% representing the highest uncertainty among all codes, whereas TD

^{E}is 7.4%, the lowest uncertainty among all codes. Therefore, when experimental measurements are available, it is more reasonable to evaluate the accuracy of numerical models based on comparisons with experiments than comparisons with the average of numerical results. In addition, estimates of the total difference with respect to the average of the computed results may be influenced by a variety of factors, including the number of numerical codes and frequencies used in the study, as well as the accuracy of each code.

^{E}(7.0 to 82.7 % for VSF, 7.5 to 46.8% for VBM, 13.7 to 33.7 % for HSF, 19.8 to 56.2 % for HBM and 9.8 to 33.4 % for TM transfer functions) are greater than the estimated total differences with respect to the average of results obtained by numerical codes TD

^{M}(8.1 to 40.9 % for VSF, 5.7 to 24.2 % for VBM, 9.1 to 9.6 % for HSF, to 15.3 % 17.2 for HBM and 10.0 to 14.2 % for TM). Since the uncertainty estimates of two codes with the total difference approach are identical and mainly dependent on the average, the newly proposed modified total difference approach has an additional advantage over the total difference measure.

^{2}close to one. In quartering stern waves, WASIM also provides the lowest uncertainty VBM transfer functions. In oblique waves, the AQWA code produces lower uncertainty in TM transfer functions versus WASIM and the opposite for HSF and HBM.

^{M}ranges between 4.2 to 17.6% for heave motion and between 1.8 to 26.2% for pitch motion. On the other hand, the modified total differences TD

^{E}are estimated to be between 7.4 and 23.4 % for the heave motion and 7.4 to 19.4 % for the pitch motion. It appears that the numerical codes may predict heave and pitch motions with less uncertainty at lower speeds, as shown by a comparison of the uncertainty estimates in the mean of numerical codes shown in Table 3 and Table 11.

## 4. Conclusions

^{M}estimations may not reflect the numerical code’s accuracy. It is more beneficial to evaluate the accuracy of numerical models based on comparisons with experiments (modified total difference) TD

^{E}than comparisons with the average of numerical results TD

^{M}. Since the uncertainty estimates of the total difference measure for only two codes are identical and mainly dependent on the average, the newly proposed modified total difference approach has an additional advantage over the total difference measure.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The body plan of the container ship Flokstra as reconstructed from data in reference [88].

**Figure 2.**Mass distribution of the Flokstra ship as reconstructed from data in reference [88].

**Figure 5.**Calculated and measured nondimensional motions (Heave, Pitch, and VRM) of the Flokstra ship at Fr = 0.245.

**Figure 6.**Calculated and measured nondimensional vertical shear forces (VSF) on the Flokstra container ship at Fr = 0.245.

**Figure 7.**Calculated and measured nondimensional horizontal shear forces (HSF) on the Flokstra ship at Fr = 0.245.

**Figure 8.**Calculated and measured nondimensional vertical bending moments (VBM) of the Flokstra ship at Fr = 0.245.

**Figure 9.**Calculated and measured nondimensional horizontal bending moments (HBM) of the Flokstra ship at Fr = 0.245.

**Figure 10.**Calculated and measured nondimensional torsional moments (TM) of the Flokstra container ship at Fr = 0.245.

**Figure 11.**Calculated and measured nondimensional motions (Heave and Pitch) of the Flokstra ship at Fr = 0.1.

**Figure 12.**Calculated nondimensional vertical bending moment (VBM) at station 10 in the Flokstra ship at Fr = 0.1.

Specifications | Units | Flokstra Ship (Full Scale) |
---|---|---|

Length (L_{pp}) | m | 270 |

Breadth (B) | m | 32.2 |

Draft (T) | m | 10.85 |

Displacement volume (∇) | m^{3} | 56,097 |

Longitudinal centre of gravity ^{1} (LCG) | m | 10.12 |

The vertical centre of gravity (KG) | m | 13.49 |

Metacentric height (GM_{T}) | m | 1.15 |

Pitch radius of gyration (K_{yy}) | m | 0.248 L_{pp} |

Roll radius of gyration (K_{xx}) | m | 0.375 B |

Natural heave period (T_{z}) | s | 8.7 |

Natural pitch period (T_{θ}) | s | 8.6 |

Natural roll period (T_{ϕ}) | s | 24.9 |

^{1}Measured aft of mid-ship.

Numerical Code | Method | Results |
---|---|---|

PDSTRIP | Strip Theory | motions |

IST-CENTEC | Strip Theory | motions and loads |

MAXSURF | Strip Theory | motions |

ANSYS-AQWA | 3D BEM (WGF) | motions and loads |

HydroD-WASIM | 3D BEM (Rankine) | motions and loads |

**Table 3.**Measures of uncertainty in heave and pitch transfer functions obtained by different codes at Fr = 0.245.

Measure | Heading | AQWA | WASIM | IST-CENTEC | PDSTRIP | MAXSURF | Mean of Codes | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Heave | Pitch | Heave | Pitch | Heave | Pitch | Heave | Pitch | Heave | Pitch | Heave | Pitch | ||

$\widehat{a}$ | 45° | 1.28 | 0.86 | 1.21 | 1.04 | 1.23 | 1.04 | 0.99 | 0.81 | 1.07 | 0.66 | 1.15 | 0.86 |

${R}^{2}$ | 0.93 | 0.99 | 0.96 | 0.98 | 0.97 | 0.96 | 0.99 | 0.99 | 0.98 | 0.97 | 0.99 | 0.98 | |

$T{D}^{E}$ | 32.77 | 16.57 | 23.82 | 7.97 | 24.64 | 11.00 | 7.40 | 27.02 | 9.98 | 53.97 | 17.83 | 19.17 | |

$T{D}^{M}$ | 17.52 | 6.86 | 10.08 | 16.75 | 9.40 | 16.11 | 23.20 | 7.02 | 12.62 | 29.74 | - | - | |

$\widehat{a}$ | 180° | 0.88 | 0.92 | 0.90 | 0.82 | 0.92 | 0.88 | 0.97 | 1.01 | 0.86 | 0.86 | 0.91 | 0.90 |

${R}^{2}$ | 0.96 | 0.98 | 0.95 | 1.00 | 0.96 | 0.97 | 0.96 | 0.98 | 0.95 | 0.97 | 0.96 | 0.98 | |

$T{D}^{E}$ | 17.49 | 14.27 | 17.80 | 24.39 | 16.85 | 20.74 | 15.75 | 12.36 | 20.86 | 22.18 | 15.42 | 17.43 | |

$T{D}^{M}$ | 4.36 | 3.24 | 4.75 | 12.12 | 5.50 | 3.55 | 12.08 | 12.37 | 7.84 | 4.86 | - | - | |

$\widehat{a}$ | 225° | 0.82 | 0.86 | 0.83 | 0.78 | 0.82 | 0.83 | 0.78 | 0.85 | 0.87 | 0.77 | 0.83 | 0.82 |

${R}^{2}$ | 0.95 | 0.98 | 0.96 | 0.99 | 0.96 | 0.96 | 0.98 | 0.95 | 0.98 | 0.95 | 0.97 | 0.97 | |

$T{D}^{E}$ | 24.24 | 18.72 | 21.06 | 29.48 | 24.77 | 24.83 | 24.90 | 24.28 | 16.08 | 35.12 | 21.79 | 26.42 | |

$T{D}^{M}$ | 3.31 | 6.13 | 2.54 | 8.77 | 4.32 | 3.24 | 7.53 | 6.87 | 5.19 | 6.99 | - | - | |

$\widehat{a}$ | Code Average Index | 0.99 | 0.88 | 0.98 | 0.88 | 0.99 | 0.92 | 0.91 | 0.89 | 0.93 | 0.76 | 0.96 | 0.86 |

${R}^{2}$ | 0.95 | 0.98 | 0.96 | 0.99 | 0.96 | 0.96 | 0.98 | 0.97 | 0.97 | 0.96 | 0.97 | 0.98 | |

$T{D}^{E}$ | 24.83 | 16.52 | 20.89 | 20.61 | 22.09 | 18.86 | 16.02 | 21.22 | 15.64 | 37.09 | 18.35 | 21.01 | |

$T{D}^{M}$ | 8.40 | 5.41 | 5.79 | 12.55 | 6.41 | 7.63 | 14.27 | 8.75 | 8.55 | 13.86 | - | - |

**Table 4.**Measures of uncertainty in vertical relative motion (VRM) obtained by different numerical codes at Fr = 0.245.

Measure | Heading | AQWA | WASIM | PDSTRIP | Mean of Codes |
---|---|---|---|---|---|

$\widehat{a}$ | 45° | 1.04 | 1.02 | 0.67 | 0.95 |

${R}^{2}$ | 0.76 | 0.81 | −0.25 | 0.01 | |

$T{D}^{E}$ | 8.11 | 8.69 | 57.32 | 19.69 | |

$T{D}^{M}$ | 19.91 | 17.16 | 37.08 | - | |

$\widehat{a}$ | 180° | 0.84 | 0.73 | 0.93 | 0.83 |

${R}^{2}$ | 0.97 | 0.99 | 0.84 | 0.98 | |

$T{D}^{E}$ | 19.60 | 32.64 | 18.56 | 21.58 | |

$T{D}^{M}$ | 3.11 | 12.58 | 13.04 | - | |

$\widehat{a}$ | 225° | 0.85 | 0.75 | 0.85 | 0.82 |

${R}^{2}$ | 0.98 | 0.91 | 0.97 | 0.98 | |

$T{D}^{E}$ | 15.93 | 30.41 | 18.95 | 21.76 | |

$T{D}^{M}$ | 5.28 | 10.62 | 8.12 | - | |

$\widehat{a}$ | Code Average Index | 0.91 | 0.83 | 0.82 | 0.87 |

${R}^{2}$ | 0.90 | 0.90 | 0.52 | 0.66 | |

$T{D}^{E}$ | 14.55 | 23.91 | 31.61 | 21.01 | |

$T{D}^{M}$ | 9.43 | 13.45 | 19.41 | - |

**Table 5.**Measures of uncertainty in VSF and VBM transfer functions obtained by different codes at station 5 and Fr = 0.245.

Measure | Heading | AQWA | WASIM | IST-CENTEC | Mean of Codes | ||||
---|---|---|---|---|---|---|---|---|---|

VSF | VBM | VSF | VBM | VSF | VBM | VSF | VBM | ||

$\widehat{a}$ | 45° | 0.73 | 0.71 | 0.84 | 0.88 | 0.70 | 1.11 | 0.77 | 0.88 |

${R}^{2}$ | 0.03 | 0.60 | 0.88 | 0.84 | 0.55 | 0.96 | 0.84 | 0.85 | |

$T{D}^{E}$ | 35.74 | 41.46 | 19.22 | 15.79 | 39.66 | 10.18 | 30.13 | 15.99 | |

$T{D}^{M}$ | 19.03 | 21.96 | 9.46 | 2.90 | 16.81 | 21.78 | - | - | |

$\widehat{a}$ | 180° | 0.95 | 1.19 | 1.20 | 1.12 | 0.92 | 1.38 | 1.04 | 1.28 |

${R}^{2}$ | 0.35 | 0.56 | 0.84 | 0.14 | 0.31 | 0.50 | 0.51 | 0.74 | |

$T{D}^{E}$ | 37.80 | 26.64 | 17.82 | 16.32 | 30.15 | 40.65 | 21.05 | 24.19 | |

$T{D}^{M}$ | 18.79 | 10.25 | 22.93 | 27.93 | 13.39 | 23.66 | - | - | |

$\widehat{a}$ | 225° | 0.68 | 0.99 | 1.06 | 1.11 | 1.15 | 1.22 | 0.97 | 1.14 |

${R}^{2}$ | −0.56 | 0.72 | 0.56 | 0.15 | 0.93 | 0.46 | 0.17 | 0.79 | |

$T{D}^{E}$ | 56.60 | 16.05 | 13.98 | 18.68 | 14.99 | 29.52 | 19.60 | 15.09 | |

$T{D}^{M}$ | 38.53 | 14.34 | 21.69 | 23.26 | 18.00 | 22.48 | - | - | |

$\widehat{a}$ | Code Average Index | 1.18 | 1.45 | 1.55 | 1.55 | 1.38 | 1.85 | 1.39 | 1.65 |

${R}^{2}$ | −0.09 | 0.94 | 1.13 | 0.56 | 0.90 | 0.96 | 0.76 | 1.19 | |

$T{D}^{E}$ | 65.07 | 42.07 | 25.52 | 25.39 | 42.41 | 40.18 | 35.39 | 27.64 | |

$T{D}^{M}$ | 38.17 | 23.27 | 27.04 | 27.05 | 24.10 | 33.96 | - | - |

**Table 6.**Measures of uncertainty in VSF and VBM transfer functions obtained by different codes at station 10 and Fr = 0.245.

Measure | Heading | AQWA | WASIM | IST-CENTEC | Mean of Codes | ||||
---|---|---|---|---|---|---|---|---|---|

VSF | VBM | VSF | VBM | VSF | VBM | VSF | VBM | ||

$\widehat{a}$ | 45° | 0.81 | 0.98 | 0.95 | 1.07 | 0.53 | 1.00 | 0.75 | 1.03 |

${R}^{2}$ | −0.30 | 0.85 | 0.98 | 0.97 | 0.73 | 0.67 | 0.84 | 0.95 | |

$T{D}^{E}$ | 33.23 | 15.29 | 7.02 | 7.46 | 71.09 | 14.26 | 33.34 | 7.44 | |

$T{D}^{M}$ | 27.98 | 13.07 | 20.26 | 5.74 | 37.10 | 12.89 | - | - | |

$\widehat{a}$ | 180° | 0.86 | 1.69 | 0.89 | 1.51 | 0.83 | 1.05 | 0.87 | 1.38 |

${R}^{2}$ | 0.05 | 0.81 | 0.97 | 0.79 | 0.76 | 0.89 | 0.83 | 0.91 | |

$T{D}^{E}$ | 20.90 | 46.83 | 12.94 | 35.82 | 23.46 | 12.03 | 17.69 | 29.89 | |

$T{D}^{M}$ | 15.19 | 24.16 | 8.06 | 9.34 | 11.11 | 32.63 | - | - | |

$\widehat{a}$ | 225° | 0.61 | 1.49 | 0.96 | 1.56 | 0.94 | 1.47 | 0.91 | 1.55 |

${R}^{2}$ | −2.73 | 0.67 | 0.94 | 0.60 | 0.21 | 0.80 | −4.09 | 0.85 | |

$T{D}^{E}$ | 82.76 | 36.05 | 10.07 | 37.25 | 25.98 | 35.16 | 37.01 | 36.15 | |

$T{D}^{M}$ | 40.99 | 9.24 | 31.11 | 19.19 | 16.17 | 13.85 | - | - | |

$\widehat{a}$ | Code Average Index | 0.76 | 1.39 | 0.93 | 1.38 | 0.77 | 1.17 | 0.84 | 1.32 |

${R}^{2}$ | −0.99 | 0.78 | 0.96 | 0.79 | 0.57 | 0.79 | −0.81 | 0.90 | |

$T{D}^{E}$ | 45.63 | 32.72 | 10.01 | 26.84 | 40.18 | 20.48 | 29.35 | 24.49 | |

$T{D}^{M}$ | 28.05 | 15.49 | 19.81 | 11.42 | 21.46 | 19.79 | - | - |

**Table 7.**Measures of uncertainty in VSF and VBM transfer functions obtained by different codes at station 15 and Fr = 0.245.

Measure | Heading | AQWA | WASIM | IST-CENTEC | Mean of Codes | ||||
---|---|---|---|---|---|---|---|---|---|

VSF | VBM | VSF | VBM | VSF | VBM | VSF | VBM | ||

$\widehat{a}$ | 45° | 1.12 | 1.11 | 0.97 | 1.31 | 0.62 | 0.71 | 0.90 | 0.99 |

${R}^{2}$ | 0.91 | 0.06 | 0.88 | 0.85 | −0.60 | 0.26 | 0.47 | 0.50 | |

$T{D}^{E}$ | 11.87 | 18.47 | 10.92 | 24.18 | 50.00 | 42.57 | 16.56 | 14.80 | |

$T{D}^{M}$ | 20.28 | 14.55 | 14.97 | 25.92 | 30.78 | 39.31 | - | - | |

$\widehat{a}$ | 180° | 1.64 | 1.15 | 1.16 | 1.68 | 0.72 | 0.70 | 1.07 | 1.09 |

${R}^{2}$ | 0.78 | 0.57 | 0.92 | 0.01 | 0.83 | 0.72 | 0.94 | 0.73 | |

$T{D}^{E}$ | 43.47 | 35.40 | 14.53 | 44.61 | 40.14 | 37.31 | 9.13 | 20.75 | |

$T{D}^{M}$ | 37.80 | 23.82 | 11.16 | 35.36 | 46.97 | 52.12 | - | - | |

$\widehat{a}$ | 225° | 1.73 | 0.90 | 1.24 | 1.53 | 1.20 | 0.98 | 1.36 | 1.16 |

${R}^{2}$ | 0.52 | 0.56 | 0.67 | 0.62 | 0.88 | 0.39 | 0.78 | 0.85 | |

$T{D}^{E}$ | 41.23 | 25.64 | 21.29 | 38.03 | 17.25 | 20.45 | 25.75 | 16.66 | |

$T{D}^{M}$ | 21.62 | 32.00 | 9.54 | 27.99 | 12.28 | 24.86 | - | - | |

$\widehat{a}$ | Code Average Index | 2.25 | 1.58 | 1.69 | 2.26 | 1.27 | 1.20 | 1.66 | 1.62 |

${R}^{2}$ | 1.10 | 0.59 | 1.24 | 0.74 | 0.55 | 0.69 | 1.09 | 1.04 | |

$T{D}^{E}$ | 48.29 | 39.75 | 23.37 | 53.41 | 53.70 | 50.17 | 25.71 | 26.10 | |

$T{D}^{M}$ | 39.85 | 35.19 | 17.83 | 44.63 | 45.01 | 58.14 | - | - |

**Table 8.**Measures of uncertainty in HSF, HBM, and TM transfer functions by different codes at station 5 and Fr = 0.245.

Measure | Heading | AQWA | WASIM | Mean of Codes | ||||||
---|---|---|---|---|---|---|---|---|---|---|

HSF | HBM | TM | HSF | HBM | TM | HSF | HBM | TM | ||

$\widehat{a}$ | 45° | 0.87 | 0.67 | 0.91 | 1.03 | 0.74 | 0.90 | 0.97 | 0.73 | 0.91 |

${R}^{2}$ | −0.31 | −21.24 | 0.80 | 0.77 | −0.38 | 0.76 | 0.22 | −4.32 | 0.82 | |

$T{D}^{E}$ | 38.40 | 50.16 | 10.11 | 15.34 | 43.86 | 12.55 | 25.33 | 40.39 | 10.91 | |

$T{D}^{M}$ | 13.87 | 13.85 | 3.91 | 13.87 | 13.85 | 3.91 | - | - | - | |

$\widehat{a}$ | 225° | 0.87 | 1.06 | 1.87 | 1.06 | 1.54 | 1.03 | 0.97 | 1.30 | 1.35 |

${R}^{2}$ | 0.22 | −0.82 | 0.56 | 0.92 | 0.75 | −0.53 | 0.70 | 0.29 | 0.21 | |

$T{D}^{E}$ | 32.15 | 34.32 | 47.79 | 11.63 | 37.64 | 20.31 | 20.43 | 31.07 | 27.63 | |

$T{D}^{M}$ | 14.47 | 21.81 | 30.21 | 14.47 | 21.81 | 30.21 | - | - | - | |

$\widehat{a}$ | Code Average Index | 0.87 | 0.87 | 1.39 | 1.04 | 1.14 | 0.97 | 0.97 | 1.02 | 1.13 |

${R}^{2}$ | −0.05 | −11.03 | 0.68 | 0.85 | 0.19 | 0.12 | 0.46 | −2.01 | 0.52 | |

$T{D}^{E}$ | 35.28 | 42.24 | 28.95 | 13.48 | 40.75 | 16.43 | 22.88 | 35.73 | 19.27 | |

$T{D}^{M}$ | 14.17 | 17.83 | 17.06 | 14.17 | 17.83 | 17.06 | - | - | - |

**Table 9.**Measures of uncertainty in HSF, HBM, and TM transfer functions by different codes at station 10 and Fr = 0.245.

Measure | Heading | AQWA | WASIM | Mean of Codes | ||||||
---|---|---|---|---|---|---|---|---|---|---|

HSF | HBM | TM | HSF | HBM | TM | HSF | HBM | TM | ||

$\widehat{a}$ | 45° | 1.10 | 0.69 | 0.99 | 0.94 | 1.03 | 0.75 | 1.02 | 0.84 | 0.85 |

${R}^{2}$ | −0.25 | −0.66 | −0.15 | 0.41 | 0.19 | −0.42 | 0.17 | −0.41 | −0.29 | |

$T{D}^{E}$ | 15.44 | 56.22 | 18.48 | 13.67 | 20.89 | 33.37 | 10.07 | 38.07 | 20.82 | |

$T{D}^{M}$ | 9.60 | 17.02 | 14.16 | 9.60 | 17.02 | 14.16 | - | - | - | |

$\widehat{a}$ | 225° | 1.30 | 0.92 | 1.08 | 1.20 | 1.22 | 0.91 | 1.26 | 1.06 | 0.99 |

${R}^{2}$ | 0.75 | 0.70 | 0.95 | 0.87 | 0.95 | 0.91 | 0.83 | 0.88 | 0.95 | |

$T{D}^{E}$ | 33.66 | 24.04 | 9.85 | 26.87 | 19.80 | 14.30 | 29.49 | 15.08 | 10.29 | |

$T{D}^{M}$ | 9.10 | 15.31 | 10.04 | 9.10 | 15.31 | 10.04 | - | - | - | |

$\widehat{a}$ | Code Average Index | 1.20 | 0.81 | 1.04 | 1.07 | 1.13 | 0.83 | 1.14 | 0.95 | 0.92 |

${R}^{2}$ | 0.25 | 0.02 | 0.40 | 0.64 | 0.57 | 0.25 | 0.50 | 0.24 | 0.33 | |

$T{D}^{E}$ | 24.55 | 40.13 | 14.17 | 20.27 | 20.35 | 23.84 | 19.78 | 26.58 | 15.56 | |

$T{D}^{M}$ | 9.35 | 16.17 | 12.10 | 9.35 | 16.17 | 12.10 | - | - | - |

**Table 10.**Measures of uncertainty in HSF, HBM, and TM transfer functions by different codes at station 15 and Fr = 0.245.

Measure | Heading | AQWA | WASIM | Mean of Codes | ||||||
---|---|---|---|---|---|---|---|---|---|---|

HSF | HBM | TM | HSF | HBM | TM | HSF | HBM | TM | ||

$\widehat{a}$ | 45° | 0.87 | 0.85 | 1.00 | 0.96 | 1.31 | 0.69 | 0.93 | 1.06 | 0.82 |

${R}^{2}$ | 0.25 | −2.16 | 0.96 | 0.93 | 0.93 | 0.70 | 0.75 | 0.15 | 0.86 | |

$T{D}^{E}$ | 29.95 | 44.14 | 3.33 | 10.03 | 23.98 | 43.40 | 17.79 | 27.49 | 21.47 | |

$T{D}^{M}$ | 13.53 | 21.79 | 18.05 | 13.53 | 21.79 | 18.05 | - | - | - | |

$\widehat{a}$ | 225° | 1.01 | 0.91 | 0.79 | 1.21 | 1.29 | 0.84 | 1.13 | 1.08 | 0.82 |

${R}^{2}$ | −1.37 | 0.14 | 0.87 | 0.96 | 0.92 | 0.91 | 0.48 | 0.75 | 0.93 | |

$T{D}^{E}$ | 37.66 | 22.82 | 26.81 | 18.51 | 21.96 | 19.52 | 26.44 | 16.86 | 22.37 | |

$T{D}^{M}$ | 15.62 | 18.75 | 10.09 | 15.62 | 18.75 | 10.09 | - | - | - | |

$\widehat{a}$ | Code Average Index | 0.94 | 0.88 | 0.90 | 1.09 | 1.30 | 0.77 | 1.03 | 1.07 | 0.82 |

${R}^{2}$ | −0.56 | −1.01 | 0.91 | 0.94 | 0.93 | 0.81 | 0.61 | 0.45 | 0.89 | |

$T{D}^{E}$ | 33.81 | 33.48 | 15.07 | 14.27 | 22.97 | 31.46 | 22.12 | 22.18 | 21.92 | |

$T{D}^{M}$ | 14.58 | 20.27 | 14.07 | 14.58 | 20.27 | 14.07 | - | - | - |

**Table 11.**Measures of uncertainty in heave and pitch transfer functions obtained by different numerical codes at Fr = 0.1.

Measure | Heading | AQWA | WASIM | IST-CENTEC | PDSTRIP | MAXSURF | Mean of Codes | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Heave | Pitch | Heave | Pitch | Heave | Pitch | Heave | Pitch | Heave | Pitch | Heave | Pitch | ||

$T{D}^{M}$ | 45° | 17.60 | 9.85 | 4.49 | 6.86 | 14.57 | 12.43 | 14.82 | 2.18 | 14.37 | 26.23 | - | - |

$\widehat{a}$ | 180° | 0.96 | 0.99 | 1.00 | 0.95 | 1.03 | 0.96 | 1.02 | 1.01 | 1.00 | 0.94 | 1.01 | 0.97 |

${R}^{2}$ | 0.85 | 0.98 | 0.83 | 0.99 | 0.90 | 0.97 | 0.67 | 0.99 | 0.83 | 0.97 | 0.84 | 0.98 | |

$T{D}^{E}$ | 16.31 | 10.75 | 16.07 | 9.06 | 14.80 | 13.02 | 23.36 | 9.54 | 15.12 | 14.06 | 14.06 | 10.79 | |

$T{D}^{M}$ | 5.91 | 2.63 | 7.81 | 3.98 | 10.97 | 3.15 | 11.09 | 5.89 | 5.35 | 4.97 | - | - | |

$\widehat{a}$ | 225° | 0.89 | 0.96 | 0.88 | 0.91 | 0.88 | 0.95 | 0.89 | 0.93 | 1.02 | 0.87 | 0.91 | 0.92 |

${R}^{2}$ | 0.92 | 0.98 | 0.95 | 0.99 | 0.85 | 0.98 | 0.95 | 0.99 | 0.94 | 0.96 | 0.96 | 0.98 | |

$T{D}^{E}$ | 14.60 | 7.37 | 15.95 | 12.35 | 18.43 | 7.97 | 15.42 | 9.34 | 7.43 | 19.45 | 11.20 | 11.09 | |

$T{D}^{M}$ | 4.88 | 4.15 | 5.18 | 1.86 | 7.64 | 2.93 | 4.17 | 2.53 | 16.21 | 7.53 | - | - | |

$\widehat{a}$ | Code Average Index | 0.93 | 0.98 | 0.94 | 0.93 | 0.95 | 0.96 | 0.95 | 0.97 | 1.01 | 0.91 | 0.96 | 0.95 |

${R}^{2}$ | 0.89 | 0.98 | 0.89 | 0.99 | 0.87 | 0.98 | 0.81 | 0.99 | 0.89 | 0.97 | 0.90 | 0.98 | |

$T{D}^{E}$ | 15.45 | 9.06 | 16.01 | 10.70 | 16.61 | 10.49 | 19.39 | 9.44 | 11.27 | 16.75 | 12.63 | 10.94 | |

$T{D}^{M}$ | 9.46 | 5.54 | 5.82 | 4.23 | 11.06 | 6.17 | 10.03 | 3.53 | 11.98 | 12.91 | - | - |

**Table 12.**Measures of uncertainty in VBM transfer functions obtained by different numerical codes at Fr = 0.1.

Measure | Heading | AQWA | WASIM | IST-CENTEC |
---|---|---|---|---|

VBM | VBM | VBM | ||

$T{D}^{M}$ | 45° | 17.40 | 5.40 | 13.10 |

180° | 17.04 | 6.85 | 12.63 | |

225° | 32.50 | 8.30 | 35.23 | |

Code Average Index | 22.31 | 6.85 | 20.32 |

Fr = 0.245 | ||||
---|---|---|---|---|

Result | Location | Heading | ||

45° | 180° | 225° | ||

Heave | CG | PDSTRIP | PDSTRIP | MAXSURF |

Pitch | CG | WASIM | PDSTRIP | AQWA |

VRM | Bow | WASIM | PDSTRIP | AQWA |

VSF | St.5 | WASIM | WASIM | IST-CENTEC |

St.10 | WASIM | WASIM | WASIM | |

St.15 | WASIM | WASIM | IST-CENTEC | |

HSF | St.5 | WASIM | - | WASIM |

St.10 | WASIM | - | Non | |

St.15 | WASIM | - | WASIM | |

VBM | St.5 | IST-CENTEC | WASIM | AQWA |

St.10 | WASIM | IST-CENTEC | Non | |

St.15 | Non | Non | IST-CENTEC | |

HBM | St.5 | Non | - | Non |

St.10 | WASIM | - | WASIM | |

St.15 | Non | - | Non | |

TM | St.5 | AQWA | - | Non |

St.10 | AQWA | - | AQWA | |

St.15 | AQWA | - | WASIM | |

Fr = 0.1 | ||||

Result | Location | Heading | ||

45° | 180° | 225° | ||

Heave | CG | WASIM | IST-CENTEC | MAXSURF |

Pitch | CG | PDSTRIP | PDSTRIP | AQWA |

VBM | St.10 | WASIM | WASIM | WASIM |

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## Share and Cite

**MDPI and ACS Style**

Abdelwahab, H.S.; Wang, S.; Parunov, J.; Guedes Soares, C.
A New Model Uncertainty Measure of Wave-Induced Motions and Loads on a Container Ship with Forward Speed. *J. Mar. Sci. Eng.* **2023**, *11*, 1042.
https://doi.org/10.3390/jmse11051042

**AMA Style**

Abdelwahab HS, Wang S, Parunov J, Guedes Soares C.
A New Model Uncertainty Measure of Wave-Induced Motions and Loads on a Container Ship with Forward Speed. *Journal of Marine Science and Engineering*. 2023; 11(5):1042.
https://doi.org/10.3390/jmse11051042

**Chicago/Turabian Style**

Abdelwahab, Hossam S., Shan Wang, Josko Parunov, and C. Guedes Soares.
2023. "A New Model Uncertainty Measure of Wave-Induced Motions and Loads on a Container Ship with Forward Speed" *Journal of Marine Science and Engineering* 11, no. 5: 1042.
https://doi.org/10.3390/jmse11051042