CFD-Powered Ship Trim Optimization: Integrating ANN for User-Friendly Software Tool Development

Abstract

This study presents a comprehensive approach to trim optimization as an energy efficiency improvement measure, focusing on reducing fuel consumption for one RO-RO car carrier. Utilizing Computational Fluid Dynamics (CFD) software, the methodology incorporates artificial neural networks (ANNs) to develop a mathematical model for estimating key parameters such as the brake power, daily fuel oil consumption (DFOC) and propeller speed. The complex ANN model is then integrated into a user-friendly software tool for practical engineering applications. The research outlines a seven-phase trim optimization process and discusses its potential extension to other types of ships, aiming to establish a universal methodology for CFD-based engineering analyses. Based on the trim optimization results, the biggest DFOC goes up to 10.5% at 7.5 m draft and up to 8% for higher drafts. Generally, in every considered case, it is recommended to sail with the trim towards the bow, meaning that the ship’s longitudinal center of gravity should be adjusted to tilt slightly forward.

1. Introduction

With the aim to reduce emissions of greenhouse gasses, in 2013, the International Maritime Organization (IMO) introduced the Energy Efficiency Design Index (EEDI) for new ships and later, in 2023, the Energy Efficiency for Existing Ships Index (EEXI). These new measures initiated changes in the shipbuilding industry. By implementing these regulations, it is anticipated to reduce global CO₂ emissions by 40% by the year 2030 and 70% (or at least 50%) by the year 2050, compared to the levels in 2008. All ships that fall under MARPOL Annex VI and have gross tonnage over 400 must meet the set criteria, which are based on nominal ship data. The IMO determined the parameters and procedure for calculating the required EEDI and EEXI, as well as the formula for calculating the attained EEDI/EEXI that must be lower than the required values. The required value for the EEDI is determined by the formula:

$$\mathrm{Required} \mathrm{EEDI}=a{\left(\mathrm{DWT}\right)}^{-c},$$

(1)

where parameters $a$ and $c$ are based on the regression curve fit [1] of data from ships that were built between 1999 and 2009, and also depend on the type of ship. The required EEDI decreases over the years according to the formula:

$$\mathrm{Attained} \mathrm{EEDI} \le \mathrm{Required} \mathrm{EEDI}=\left(1-X/100\right)\cdot \mathrm{Reference} \mathrm{line} \mathrm{value},$$

(2)

where $X$ is the specified reduction factor given in [2] through four phases. Values for the EEXI are based on the required EEDI, taken into account with the adequate reduction factor from [3]. The requirements for the EEXI are becoming slightly stricter with time, with the plan for the EEXI and EEDI to become equal in 2025. In addition to the two mentioned energy efficiency parameters, the IMO introduced operational efficiency indicators. In 2009 [4], the Energy Efficiency Operational Indicator (EEOI) was invented and is defined as the ratio of emitted CO₂ per unit of transport work. The EEOI never became mandatory, but it is a representative value of the ship’s operational energy efficiency level for a consistent period of time. Another indicator came into force in 2023, the Carbon Intensity Index (CII), which is calculated as the ratio of the total mass of emitted CO₂ to the total transport work undertaken in a specific calendar year, considering the traveled distance, time spent underway and the amount of consumed fuel over a period of one year. As well as for the EEDI and EEXI, the attained [5] and required [6] values are also calculated for the CII, where the attained CII has to be lower than the required, which decreases over time [7]. Based on the attained CII, the ship belongs to one of five energy efficiency categories (A, B, C, D, E). The boundaries between those categories and the calculation method for each ship type are defined in [8] and they depend on the ship’s DWT. Since the CII is a mandatory parameter, all ships of gross tonnage of 5000 or more are from 2019 obliged to record the relevant information (traveled distance, time spent underway and the amount of consumed fuel over a period of one year) under the Data Collection System (DCS) [9]. In the case that a ship achieved class E for at least one year, or class D for three years in a row, it must create a plan for reducing CO₂ emissions and satisfy the requirements for at least class C [10]. In 2013, the IMO introduced the Ship Energy Efficiency Management Plan (SEEMP), which consists of three parts: a ship management plan to improve energy efficiency, a ship fuel oil consumption data collection plan and a ship operational carbon intensity plan. Part I applies to all ships that fall under MARPOL Annex VI and have gross tonnage over 400, while Part II and Part III apply for ships that have gross tonnage over 5000 and fall under MARPOL Annex VI.

Various measures can be considered in order to meet the energy efficiency requirements developed by the International Maritime Organization and attain a reduction in CO₂ emissions. Those measures are divided into two categories: operational methods and technical methods. Operational methods include the following: improvement in voyage execution [11,12], reduction in auxiliary power consumption [11,13], weather routing [10,12], “just in time” voyage [13], optimum ballast [10,12], optimum cargo distribution [12], energy-saving utilities [14], optimum use of rudder and heading control systems [10], optimized hull and propeller maintenance [10], speed optimization [12,15], slow steaming [16,17] and trim optimization [12,18].

These measures can be applied for both existing ships as well as new ships. Technical measures are design-related and therefore more favorable for new ships. These measures refer to wind-assisted propulsion, fuel type change [10,16,19], waste heat recovery [12], upgrading and maintenance of propulsion system, hull retrofit (bulb and/or stern modification, installation of energy-saving devices [10], etc.

Optimum trim means that the angle of the trim for a specific operating condition, regarding the displacement and speed, provides minimum resistance, which directly implies the optimal efficiency level [12]. The resistance of the ship changes depending on the trim, although the displacement and speed stay the same [20]. The beneficial aspect of trim optimization is that neither a hull modification nor engine upgrade is needed. The ship is trimmed if the draught at the bow differs from the draught at the aft section of the ship. While a negative trim indicates that the draught at the bow is greater than the draught at the stern, a positive trim implies the opposite. Every vessel is optimized for a number of conditions (even keel at full load and design speed, ballast condition, etc.), but the actual operating conditions usually differ from the expected ones [21]. It was confirmed in [22] that the trim can affect the total resistance of a ship for various service speeds and the optimum trim for every speed is different. The trim optimization method is intended for minimizing the resistance in calm water and therefore minimizing fuel consumption, which can be accomplished with a specially developed program for the ship. The results of [18] indicate the possibility of reducing the total fuel consumption during a whole voyage by 1.2% by utilizing the calculated optimum trim for the whole voyage. A 4250 TEU container ship sea trial with the use of a trim optimizing program reported a main engine power reduction of 910 kW and energy-saving rate of 9.2% [23]. The trim of the ship can be influenced by the redistribution of ballast, fuel and/or load between tanks. Parameters that change when the ship is trimmed compared to an even-keel condition are a wetted surface area, the length of the waterline and a submerged hull form at the bow as well as at the stern [24]. Potential disadvantages of this method could be a reduction in visibility, reduced freeboard, emergence of the propeller [25], underkeel clearance, seakeeping, maneuverability [26], seakeeping [27] and shipped water on deck [28], and those should as well be taken into account while finding the optimum trim for an operating condition.

Calculating the ship resistance in calm water by conducting model tests and numerical simulations provides data about the resistance for different drafts and trims. As a result, a set of curves with highlighted lowest resistance for a specific draft are obtained, as in [22,29]. While onboard tools have this information at their disposal, model tests are still common and basins are equipped with traditional procedures and up-to-date insights to provide their proper execution, pursuing the measurement of small power variations within a foreseen range of 0 to 4% of the total installed power [30], or a 2% to 4% fuel consumption reduction [31]. With the advancement of technology and computers, CFD software calculations provide trim tables with an accuracy that can compete with the results gained from traditional model tests, with even less investments. Other methods that can be used to determine the optimum trim are sea trials and machine learning methods. The shortcomings of these methods are that sea trials are fuel- and time-consuming for determining the optimum ballast, while for obtaining information about the optimum trim with the machine learning method, a lot of data from a ship’s past voyages are needed [32]. In this study, the aim of this work is focused only on applying the CFD method on one RO-RO car carrier in order to optimize the trim for an energy-efficient voyage and exploitation.

The application of the CFD method for trim optimization for different ship types can be found in various existing studies. For example, in the study by [20], the effect of trim optimization for a container ship like MOERI (KCS) was a total resistance reduction of 2%, similar to the results of trim optimization for a US Navy ship [13]. A CFD investigation of the propulsion performance of a low-speed VLCC tanker at various initial trim angles was conducted by [33] and showed a 1.76–2.12% total resistance reduction. The authors of [34] and [35] report even greater profit, such as fuel savings of up to 5% [34] and a 6% reduction in delivered horsepower [35]. Since the trim conditions can vary significantly, so do the results from trim optimization. The authors of [36] highlight that in Series 60, the total resistance between the worst and best trims varies by up to 11%. The trim optimization of a bulk carrier in [37] results in a total resistance reduction possibility of up to 14%. Savings for RO-RO ships were studied by [12] and indicated a possibility of up to a 10.4% reduction in delivered power, or 1.2 t fuel per day. In addition to the calculation of the optimal trim, specialized software for input parameters that quickly provide information about the optimal trim has been mentioned [35,38].

Up to now, artificial neural networks (ANNs) have found application in predicting fuel consumption, as demonstrated in the studies by [39] and [40], which focused on utilizing NOON reports. Furthermore, an ANN has been employed for forecasting the ship speed, as evidenced by the work of [41]. Moreover, research efforts have extended to utilizing ANNs for the joint prediction of the ship speed and fuel consumption, leveraging data from sails, particularly in the case of a barquentine, as explored by [42]. The study by [43] proposes a real-time hybrid electric ship energy efficiency optimization model considering time-varying environmental factors, aiming to optimize the EEOI under wind and wave conditions while maintaining speed limits, resulting in an average reduction in fuel consumption of 13.4% and real-time EEOI of 15.2%. In [44], a real-time prediction model of ship fuel consumption through BP neural network training-related data is presented and further used for ship speed optimization. The research of [45] developed an ANN model to predict the main engine power and pollutant emissions of 3020 container, cargo and tanker ships using 14 parameters, demonstrating its potential for use in future studies on fuel consumption and energy efficiency in maritime transport.

In today’s complex technological landscape, the integration of various disciplines is becoming increasingly vital for tackling engineering challenges. By bridging the fields of naval engineering, CFD, ANN and practical application development, this study not only underscores the importance of multidisciplinary association but also showcases its benefits in addressing real-world problems within the maritime industry. Through the synergy of these varied fields, novel solutions can be developed to optimize ship operations or design, enhance fluid flow analysis accuracy and streamline decision-making processes. Moreover, the utilization of an ANN enables the creation of sophisticated mathematical models that can effectively capture complex relationships and patterns within maritime systems, paving the way for more precise simulations and predictive analytics. Furthermore, the development of user-friendly applications facilitates the seamless implementation of these advanced methodologies, empowering stakeholders to leverage cutting-edge insights for improved vessel performance and operational efficiency. Essentially, this methodology not only promotes scientific comprehension but also encourages innovation and pragmatic breakthroughs in the field of maritime engineering, thus helping in the long-term growth of maritime technologies.

2. Methods

The conducted work consists of several phases: (1) 3D modeling; (2) first-level verification of the reliability of the digital twin (3D model); (3) open water test (OWT); (4) second-level verification of the reliability of the 3D model; (5) trim optimization; (6) determination of a mathematical model for assessing the optimal trim, ship speed, engine (brake) power, daily fuel consumption and propeller speed; (7) programming an application based on previously conducted analyses. In this study, various software has been used: StarCCM+ version 2021.3 for CFD analysis, aNETka for mathematical model determining and MATLAB (version 2020b) for application development. The goal of this paper, in addition to trim optimization, is to establish an initial procedure for setting up CFD simulations. Therefore, this chapter provides only a brief overview of the governing equations used to solve the problem of ship movement through two fluids. More attention is given to functions that are not predefined in the StarCCM+ software, which the user can define. Consequently, this paper includes formulas that can be found in various publications, ensuring that everything complies with the requirements and guidelines issued by relevant institutions in the maritime industry.

In

Table 1. Principal particulars.

Parameter *	Dimension	Value
$L_{pp}$	(m)	158
$B$	(m)	28
$H$	(m)	30.65
$T_s$	(m)	7.8

* $L_{pp}$—length between perpendiculars, $B$—breadth, $H$—depth, $T_s$—draft.

, the ship’s (RO-RO car carrier) principal particulars are given.

Propeller geometry characteristics are presented in

Table 2. Propeller characteristics.

Parameter *	Dimension	Value
$D$	(m)	5.5
$\mathrm{No}. \mathrm{of} \mathrm{blades}$	(-)	5
$\mathrm{BAR}$	(-)	0.83
$D_{\mathrm{hub}}$	(m)	0.98
$\mathrm{Chord} \mathrm{length} \mathrm{at} 0.7R_p$	(m)	2.0988
$P/D \mathrm{at} 0.7R_p$	(-)	0.9594

* $D$—diameter, $\mathrm{BAR}$—blade area ratio, $D_{\mathrm{hub}}$—hub diameter, $R_p$—propeller radius, $P$—pitch.

.

2.1. Three-Dimensional Modeling

Three-dimensional modeling is the first necessary step before conducting CFD simulations. Creating a 3D representation of the RO-RO vessel involved using existing documentation: hull construction drawings, rudder construction drawings and propeller drawings. This digital twin includes detailed modeling of the hull, rudder and propeller, with the goal of faithfully capturing the ship’s attributes and performance.

2.2. First-Level Verification

First-level verification involves comparing hydrostatic data such as volume displacement ($\nabla$), block coefficient ($C_b$), longitudinal center of buoyancy ($LCB$) and wetted surface ($WS$) of the 3D model with data given in the Trim & Stability booklet for various drafts. No parameter should deviate more than 1% from (as per authors’ experience, 1% deviation enables enough accuracy) its corresponding value given in the Stability Booklet.

2.3. Open Water Test

The OWT is a method for determining propeller characteristics such as thrust coefficient, torque coefficient and propeller efficiency. The procedure for determining these coefficients through model tests is defined in [46]. In this case, model testing was not conducted; instead, propeller parameters were determined through CFD simulation, following the procedure described in [47]. According to this document, the CFD results of the OWT are considered valid if they do not differ by more than 3% from the results obtained by model testing within the relevant propeller operating range. It is also emphasized that if the model test results of the propeller are not known, an additional OWT with the appropriate B-series propeller model should be conducted. An appropriate B-series propeller is considered one with the same geometric characteristics (diameter, pitch, ratio of characteristic areas) as the original propeller installed on the considered ship. The 3D model of the B-series propeller was obtained using the B-series Propeller Generator available online [48]. The OWT results of the B-series propeller should not deviate by more than 3% from the results obtained with the mathematical model for the same geometric parameters. If this condition is met, the same CFD calculation methodology is applied to the original propeller, and the results obtained this way are considered valid. A mathematical model can be found in [49].

The simulations are conducted using a steady-state numerical flow model, solved in the rotating frame of reference. The numerical method used is called Multiple Reference Frames (MRFs), where in one part of the domain, the equations are solved in a rotating frame of reference, while in the rest of the domain, a stationary frame of reference is used.

Figure 1 shows the domain of the open water test CFD setup. The outer cylinder shows the far-field boundary of the domain, while the inner, smaller cylinder represents the boundary between the rotational frame of reference and stationary frame of reference. The diameter and length of the small cylinder are 4D, while the diameter and length of the large cylinder are 10D. The propeller is positioned downstream from the shaft. All simulations are conducted in full scale.

The same grid refinement options were used for both propeller grids (B-series propeller and original propeller) and are presented in Figure 2.

Turbulence modeling is performed using the k-ω SST model as is required by [47]. The inflow turbulence intensity was set to 1% relative to the inflow speed. The inflow speed conditions and rotation rate of the propeller are selected to closely resemble the conditions expected at self-propulsion conditions.

2.4. Second-Level Verification

Second-level verification involves conducting a CFD simulation for a draft for which there are available results from the model tests or sea trials. The difference between the results obtained by the CFD analysis and model tests/sea trials should not be greater than 5% [47]. This kind of verification is typical in applied CFD analysis in the maritime industry [50,51].

The equations are discretized using the collocated Volume of Fluid (VOF) multiphase method implemented within the software. It is used for large-scale two-phase (in this case, sea water (${\rho }_w$ = 1.025 t/m³) and air (${\rho }_A$ = 1.212 kg/m³) at 18 °C) flows encountered in naval hydrodynamics. A two-phase, incompressible, turbulent and viscous flow model is employed, governed by the continuity and Navier–Stokes equations:

$$\nabla \cdot \left(\rho V\right)=0,$$

(3)

$${\displaystyle \frac{\partial \left(\rho V\right)}{\partial t}}+\nabla \cdot \left(\rho V\otimes V\right)=-\nabla \cdot \left(p\mathbf{I}\right)+\nabla \cdot {\mathbf{T}}_{\mathrm{v}}+f_b,$$

(4)

where $V$ stands for the velocity vector field, $p$ is the pressure, ${\mathbf{T}}_{\mathrm{v}}$ is the viscous stress tensor and $f_b$ is the resultant of the body forces.

The VOF multiphase Eulerian approach model implementation in Simcenter STAR-CCM+ belongs to the family of interface-capturing methods that predict the distribution and the movement of the interface of immiscible phases. This modeling approach assumes that the mesh resolution is sufficient to resolve the position and the shape of the interface between the phases. In this paper, the focus is not on elaborating the theoretical background of the functions in CFD software, but rather on presenting a methodology that can serve as a quick way to set up simulations. Therefore, more about the VOF method can be found in [52], where this concept is detailed.

An important quality of a system of immiscible phases (air and water) is that the fluids always remain separated by a sharp interface. The High-Resolution Interface Capturing (HRIC) scheme is used to mimic the convective transport of immiscible fluid components, resulting in a scheme that is suited for tracking sharp interfaces.

As in the OWT, the k-ω SST turbulence model was used here according to the requirements defined in [47].

2.4.1. Actuator Disk Propeller Model

The body force propeller method is used to model the effects of a propeller such as thrust and thereby creating propulsion without actually resolving the geometry of the propeller. The method employs a uniform volume force distribution over the cylindrical virtual disk. The volume force varies in the radial direction, so the total force can be calculated as the volume integral for the whole modeled cylinder. The radial distribution of the force components follows the Goldstein optimum [53] and is given by

$$f_{bx}=A_{x}r*\sqrt{1-r*},$$

(5)

where $r*$ is the normalized disk radius defined as

$$r*={\displaystyle \frac{r^{\prime }-r_h^{\prime }}{1-r_h^{\prime }}},$$

(6)

where $r^{\prime }=r/R_p$ and $r_h^{\prime }=R_h/R_p$. Volume force $f_{bx}$ is the body force component in the axial direction, $r$ is the radial coordinate, $R_h$ is the hub radius and $R_p$ is the propeller tip radius. Coefficient $A_x$ is defined as

$$A_x={\displaystyle \frac{105}{8\pi }}{\displaystyle \frac{T}{t_d\left(R_p-R_h\right)\left(3R_h+4R_p\right)}},$$

(7)

where $T$ stands for the propeller thrust and $t_d$ is the virtual disk thickness. More about this can be found in [54]. The simulation is performed for a certain operating point specified by the thrust ($T$). The advance ratio is calculated by solving the following equation numerically ($J$—advance coefficient, $D$—propeller diameter):

$$f\left(J\right)=K_T-K_T^{\prime }$$

(8)

where $K_T$ is evaluated from the propeller performance curve and $K_T^{\prime }$ is evaluated as

$$K_T^{\prime }={\displaystyle \frac{J^2{T}_{Operating point}}{{\rho }_w{V}_A^2{D}^2}}$$

(9)

With $K_T$ available, the thrust is calculated for the propeller:

$$T={\displaystyle \frac{K_T{\rho }_w{V}_A^2{D}^2}{J^2}}$$

(10)

With $T$($T_{Operating point}$) available, the axial body force component can be calculated. Inflow plane is the plane inside the actuator disk where the volume-averaged velocity and density are computed. The input thrust in this part of the simulation is equal to the total resistance that is evaluated as the sum of the pressure resistance and viscous resistance of the hull with the rudder included.

The wake fraction ($w$) is extracted at the propeller axis plane in a steady resistance simulation.

The thrust deduction factor is calculated as per

$$t=1-{\displaystyle \frac{R_T}{T}},$$

(11)

where $R_T$ is the total resistance of the ship evaluated in the steady resistance simulation.

2.4.2. Steady Resistance Simulations

Total resistance, including viscous and pressure resistance, is obtained by rerunning the self-propulsion simulation with a disabled actuator disk. This procedure implies the same mesh as was used earlier but only without an actuator disk.

2.4.3. Boundary Conditions

Fluid domain dimensions are defined as follows (from origin):

• Inlet: 2.5$L_{pp}$—defined as velocity inlet;
• Outlet: 3$L_{pp}$—defined as pressure outlet with wave damping boundary option included;
• Bottom: 1.5$L_{pp}$—defined as velocity inlet;
• Top: 1$L_{pp}$—defined as velocity inlet;
• Port Side: 2$L_{pp}$—defined as symmetry plane with wave damping boundary option included;
• Starboard Side: 2$L_{pp}$—defined as symmetry plane with wave damping boundary option included.

The wave damping length is defined as two wave lengths, where the wave length is calculated as per

$$2\pi \cdot {\mathrm{Fn}}_L^2\cdot L_{pp},$$

(12)

where ${\mathrm{Fn}}_L^2$ is the Froude number based on the length.

2.4.4. Rigid Body Motion

Two degrees of freedom are allowed for the motion for the vessel, namely pitch and heave, allowing for the dynamic trim and sinkage of the vessel.

2.4.5. Mesh

The triangle surface meshing method has been used to build the mesh. The prism-layer mesher generates prismatic cell layers next to the boundary layer. The number of these layers is calculated as per

$$m={\displaystyle \frac{\ln \left(1-\left(1-sf\right)\frac{\delta }{y_1}\right)}{\ln \left(sf\right)}},$$

(13)

where $sf=1.3$ is the stretch factor, $\delta$ is the boundary-layer thickness and $y_1$ is the first-layer thickness. These cells help to capture the viscous effects in the boundary layer correctly in the region where the thickness is calculated as per [55]

$$\delta ={\displaystyle \frac{0.16L_{wl}}{\sqrt[7]{{\mathrm{Re}}_{L_{wl}}}}},$$

(14)

where $L_{wl}$ is the length at the waterline of the ship and $\mathrm{Re}$ is the Reynolds number ($L_{wl}$ in subscription means that the Reynolds number is calculated with the waterline length as a reference value, otherwise $L_{pp}$ is used). The trimmed cell mesher creates a volume mesh by cutting a template mesh with the surface geometry. The first-layer thickness is calculated for $y^{+}$ = 150 by following the guidelines in [47] where it is emphasized that $y^{+}$ should be in a range from 30 to 300. This value (150) is used as the target factor to ensure that the actual $y^{+}$ values will fall within the required range.

$$y_1={\displaystyle \frac{150\nu }{u_{\tau }}},$$

(15)

where $\nu$ is the kinematic viscosity and $u_{\tau }$ is the shear velocity. The shear velocity or frictional velocity is a fictious quantity, it characterizes the shear at the boundary layer, and it is given by

$$u_{\tau }=\sqrt{{\displaystyle \frac{{\tau }_w}{{\rho }_w}}},$$

(16)

where ${\tau }_w$ is the wall shear stress. This presents a contradiction as follows: determining the thickness of the initial layer requires knowing the stress on the vessel’s hull, which in turn needs to be assessed using CFD. Hence, the recommendation from the ITTC [46,56] is utilized in this scenario to compute the wall shear stress for the hull.

$${\tau }_w=0.5\left(\left(1+k\right)C_{FS}+C_A\right)V_s^2{\rho }_w,$$

(17)

where $k$ is a form factor, $C_{FS}$ is the frictional resistance coefficient and $C_A$ is the correlation allowance.

$$C_{FS}={\displaystyle \frac{0.075}{{\left({\log }_{10}\mathrm{Re}-2\right)}^2}},$$

(18)

$$C_A=\left(5.68-0.6{\log }_{10}\mathrm{Re}\right)\cdot {10}^{-3}$$

(19)

The form factor is calculated as the mean value of the equations obtained by Grigson, Wright and Conn and Ferguson given in [57]:

$$k={\displaystyle \frac{0.028+3.3\left(\frac{WS}{L_{pp}^2}\sqrt{C_B\frac{B}{L_{pp}}}\right)+18.7{\left(C_B\frac{B}{L_{pp}}\right)}^2+2.48C_B^{0.1526}{\left(\frac{B}{T_s}\right)}^{0.0533}{\left(\frac{B}{L_{pp}}\right)}^{0.3856}-1}{3}}.$$

(20)

A grid convergence study is performed to assess the convergence of the results by systematically refining a specific input parameter in StarCCM+. In this study, three simulations are conducted, where the chosen input parameter is varied while keeping all other parameters constant. The refinement value used in this study aligns with the guidelines specified in [58] and is set to be uniform ($\sqrt{2})$. By systematically refining the input parameter and comparing the results from the different simulations, the grid convergence study aims to determine the level of convergence and establish the appropriate grid resolution for accurate and reliable results.

The convergence ratio is defined as

$$R={\displaystyle \frac{{\epsilon }_{21}}{{\epsilon }_{32}}},$$

(21)

where ${\epsilon }_{21}$ is the difference between the solution obtained using medium and fine mesh and ${\epsilon }_{32}$ is the difference between the solution obtained using coarse and medium mesh. $R$ is used for the estimation of the convergence conditions: monotonic convergence is achieved when $0<R<1$, oscillatory convergence is achieved when $-1<R<0$ and divergence is achieved when $\left|R\right|>1$. The numerical uncertainty and error for monotonic convergence condition is estimated using generalized Richardson extrapolation (RE). The order of accuracy is calculated as per

$$p={\displaystyle \frac{\ln \left({\epsilon }_{32}/{\epsilon }_{21}\right)}{\ln \sqrt{2}}},$$

(22)

$${\delta }_{RE}={\displaystyle \frac{{\epsilon }_{21}}{{\sqrt{2}}^p-1}},$$

(23)

The generalized RE solution (${\widehat{S}}_{RE}$) can be derived by

$${\widehat{S}}_{RE}={\widehat{S}}_1-\left|{\delta }_{RE}\right|.$$

(24)

To estimate the grid uncertainty $U_G$, a safety factor approach is employed, defining $U_G$ as

$$U_G=FS\cdot \left|{\delta }_{RE}\right|,$$

(25)

where $FS=1.25$ is a safety factor. The normalized uncertainties are calculated as follows:

$$\overline{U}={\displaystyle \frac{U_G}{{\widehat{S}}_{RE}}}\cdot 100\%.$$

(26)

2.4.6. Post-Processing

In the numerical simulations conducted for this study, the direct consideration of the roughness effects and air resistance resulting from the presence of a superstructure is not included. However, these effects are accounted for in the post-processing stage by following the recommended procedures and guidelines outlined in [59]. The ITTC guidelines provide specific methodologies for incorporating the effects of the roughness and air resistance into the analysis, allowing for an assessment of the overall performance and characteristics of the ship. The roughness allowance ($\mathsf{\Delta }{C}_F$) is calculated as per [60]

$$\mathsf{\Delta }{C}_F=0.044\left[{\left({\displaystyle \frac{k_s}{L_{WL}}}\right)}^{{\displaystyle \frac{1}{3}}}-10{\mathrm{Re}}^{-{\displaystyle \frac{1}{3}}}\right]+0.000125$$

(27)

and the air resistance coefficient ($C_{AAS}$) [54] is

$$C_{AAS}=0.8{\displaystyle \frac{{\rho }_A\cdot A_{VS}}{{\rho }_w\cdot WS}},$$

(28)

where $k_s$ indicates the roughness of the hull surface equals 150·10⁻⁶ m [58], and $A_{VS}$ is the projected area of the ship superstructure to the transverse plane. The new total resistance is calculated as

$$F_T^{\prime }={\displaystyle \frac{1}{2}}{V}_s^2\cdot WS\cdot {\rho }_w\cdot \left(C_T+\mathsf{\Delta }{C}_F+C_{AAS}\right),$$

(29)

where $C_T$ refers to the total resistance coefficient obtained as a result of the CFD simulation.

The new thrust is calculated as per

$$T^{\prime }={\displaystyle \frac{R_T^{\prime }}{1-t^{\prime }}}$$

(30)

Together with the propeller performance data ($K_T$, $K_Q$, ${\eta }_0\left(J\right)$), the new operating point is obtained by the numerical solving of equation:

$${\displaystyle \frac{T^{\prime }}{{\rho }_w\cdot V_A^2\cdot D^2}}{J}^2-K_T\left(J\right)=0$$

(31)

Once the advance ratio coefficient is obtained, the propeller speed ($n$) and torque ($Q_0$) can be evaluated:

$$n={\displaystyle \frac{V_A}{JD}}={\displaystyle \frac{\left(1-w^{\prime }\right)V}{JD}},$$

(32)

$$Q_0=K_Q{\rho }_w{n}^2{D}^5,$$

(33)

where $K_Q$ is evaluated from the propeller performance curve and $Q_0$ is the torque in the open water condition. Hence, the actuator disk approach has been used in this case and the relative rotative efficiency is adopted to be 1, which means that the torque is calculated as

$$Q={\displaystyle \frac{Q_0}{{\eta }_R}}\approx Q_0$$

(34)

Finally, the brake power can be evaluated as

$$P_b={\displaystyle \frac{2n\pi Q}{{\eta }_S}},$$

(35)

where ${\eta }_S$ is the shaft efficiency, adopted to be 0.99. The hull efficiency is considered to be the same as in the steady resistance simulations, i.e., $w^{\prime }=w$ and $t^{\prime }=t$, where $w^{\prime }$ and $t^{\prime }$ are the wake fraction coefficient and thrust deduction factor in post-processing.

Calibration factors $C_n$ and $C_{Pb}$ are calculated as the average ration between the model test (MT) results and CFD results for the propeller speed ($n$) and brake power ($P_b$).

$$C_n={\displaystyle \frac{n_{MT}}{n_{CFD}}}$$

(36)

$$C_{Pb}={\displaystyle \frac{P_{b_{MT}}}{P_{b_{CFD}}}}$$

(37)

2.5. Trim Optimization

Trim optimization was conducted for drafts of 7.5 m, 8 m and 8.7 m at speeds of 12.5 kn, 15 kn and 18 kn at 7 different conditions (trims). The range of trims goes from −1.5 m to 1.5 m with a 0.5 m step. A negative sign means the trim by the bow and a positive sign, the trim by the stern. The outcome of trim optimization in CFD analysis typically involves determining the optimal trim settings for a vessel at considered drafts and speeds. This optimization aims to minimize resistance and ensure optimal performance under different operating conditions. By adjusting the vessel’s trim, operators can achieve better hydrodynamic characteristics, leading to improved overall performance and operational efficiency. With the available specific fuel oil consumption (SFOC) curve as a function of $P_b$, a possible reduction in fuel consumption was also calculated.

2.6. Mathematical Model for Assessing the Outcomes from Trim Optimization

The first outcomes from the CFD simulations are the brake power and propeller speed for one ship speed and draft, i.e., trim, i.e., displacement. The results are presented in the form of relative differences between the brake power of the trimmed ship and the ship on an even keel. A negative sign indicates that there is a saving, i.e., a reduction in the required brake power for that case. A graphical example of the results is shown in Figure 3a, where the relative brake power and fuel oil consumption savings are depicted as an area with the ship speed on the x-axis, trim on the y-axis and relative power/fuel oil consumption savings on the z-axis. The optimal (favorable) sailing zone is depicted in green on the same diagram, meaning that at the current trim and speed, less power or fuel oil consumption is required compared to when the ship is sailing on an even keel. The yellow zone represents the transitional zone where the change in the required brake power/fuel oil consumption is negligible, while the red zone is the zone to be avoided because more power/fuel oil is needed for the ship to sail at the same speed compared to sailing on an even keel. Each point on the surface in Figure 3a corresponds to a certain displacement, where it depends solely on the trim. For easier calculation, a surface $\nabla$ (trim, $V_s$) is formed from each line $\nabla$ (trim), and the surfaces for all three considered drafts are shown in Figure 3b. In order to have smooth surfaces, all values estimated from the CFD analysis are linearly interpolated. Therefore, the initial matrix 3 × 7 (3 speeds, 7 trims) is reshaped to 151 × 111 with a speed step of 0.05 kn and trim step of 0.02 m. The main parameters from the initial matrix are depicted with black lines in Figure 3a.

Additionally, values between drafts of 7.5 m and 8 m and from 8 m to 8.7 m are linearly interpolated with a step of 0.01 m. Thus, from the initial matrix of 3 × 7 × 3 (3 speeds, 7 trims, 3 drafts), matrices (${\mathsf{\Sigma }}_m$) of size 151 × 111 × 120 are obtained where $m$ is $\delta P_b$ or $\nabla$.

The daily fuel oil consumption (DFOC) is estimated based on the determined brake power from the CFD simulation and the available specific fuel oil consumption (SFOC) curve for the engine installed on the ship. The SFOC curve is approximated by a sixth-degree polynomial:

$$\mathrm{SFOC}=-5.54\cdot {10}^{-23}{P_b}^6+3.75\cdot {10}^{-18}{P_b}^5-8.28\cdot {10}^{-14}{P_b}^4+8.00\cdot {10}^{-10}{P_b}^3-3.11\cdot {10}^{-6}{P_b}^2-9.63\cdot {10}^{-4}{P}_b+209$$

(38)

The mentioned polynomial provides SFOC values with an average error of 0.02% compared to the exact available values from the reference document.

With the evaluated DFOC for each brake power, the same diagrams as shown in Figure 4 can be created (Figure 5).

In operations, after loading cargo onto the ship, only the displacement is known. The goal of the conducted trim optimization is to find the optimal position, i.e., trim, and recommend the sailing speed based on that single parameter. The optimal trim and recommended speed are those at which the greatest savings in brake power, and consequently fuel consumption, are achieved. The mathematical expression of the above statement is as follows:

$$\delta P_{b_{\min }}=\underset{ijk}{\min }\left(\delta {P_b}^{\left(ijk\right)}|\left|{\nabla }^{\left(ijk\right)}-{\nabla }_{LC}\right|\le \epsilon \cdot {\nabla }_{LC}\right),$$

(39)

where $\delta P_{b_{\min }}$ is the biggest brake power reduction where the corresponding displacement ${\nabla }^{\left(ijk\right)}$ is within a tolerance ($\epsilon$= 0.0001) of the target displacement (${\nabla }_{LC}$). Index $i$ represents the speed, $j$ represents the trim and $k$ represents the brake power reduction, i.e., displacement in matrix ${\mathsf{\Sigma }}_m$. When $\delta P_{b_{\min }}$ is found, it is easy to extract indices $i$, $j$ and $k$ therefore to derive the optimum trim and speed for the corresponding displacement, i.e., mean draft. In practice, there may be a need for speed as an input parameter, which depends on the operational requirements, and this case will be addressed in subsequent work.

Mathematical Model for Assessing Brake Power, DFOC and Propeller Speed

A sophisticated mathematical model utilizing artificial neural networks (ANNs) has been developed based on the CFD results. By specifically addressing the challenges posed by non-simulated loading conditions, this model offers a solution for estimating parameters such as the brake power, propeller speed and DFOC.

At its core, the model employs a feedforward artificial neural network (FFANN), a fundamental architecture in machine learning and deep learning. The FFANN’s structure comprises interconnected layers of neurons, including an input layer (white circles in Figure 6), one or more hidden layers (gray circles in Figure 6) and an output layer (black circles in Figure 6). Each neuron is equipped with adjustable parameters called biases, which introduce flexibility by allowing for offsets or shifts in output. During training, these biases are fine-tuned to better fit the data and ensure the accurate estimation of the desired parameters.

One of the key features of the FFANN is its one-directional flow of information, from the input layer through the hidden layers to the output layer. This design simplifies the learning process and enables the efficient modeling of complex data relationships. Additionally, activation functions embedded within neurons introduce non-linearity, enabling the network to capture patterns present in real-world data.

Overall, this innovative approach represents a significant step forward in the advancement of engineering solutions. It not only addresses the inherent challenges of non-simulated loading conditions but also sets a precedent for the application of artificial intelligence in optimizing the performance of complex systems. As further research and development are conducted in this field, the potential for ANNs to revolutionize engineering practices and transportation systems continues to expand.

In the context of a feedforward artificial neural network (FFANN), the connections between neurons are governed by weights, which dictate the strength of the connections. These weights are adjusted during the training process using an algorithm called backpropagation. The name “backpropagation” stems from its mechanism of iteratively propagating errors backward through the network to adjust the weights and biases of neurons.

The training process involves two phases: the forward pass and the backward pass. During the forward pass, input data are fed into the network, and predictions are made based on the current weights and biases. These predictions are compared to the actual outputs, and the resulting error is calculated.

In the backward pass, this error is propagated backward through the network, layer by layer. The gradient of the error with respect to each weight and bias is computed using techniques like the chain rule of calculus. These gradients indicate the direction and magnitude of adjustment required to minimize the error. The weights and biases are then updated accordingly.

At each neuron, the weighted sum of the inputs is computed by multiplying each input value by its corresponding weight and summing the results. This sum is then combined with the neuron’s bias:

$$S_{x,i}=f\left({\displaystyle \sum _i{S}_{x-1,i}\cdot w_{x,i}+b_{x,i}}\right),$$

(40)

where $f$—activation function, $S_{x,i}$—output signal, of $i$th neuron in $x$th layer, $S_{x-1,i}$—output of $i$th neuron in $x-1$th layer, $w_{x,i}$—weights connected to the neuron and $b_{x,i}$—corresponding bias.

The resulting value is passed through an activation function, which introduces non-linearity into the network. Common activation functions include sigmoid, tanh and others, each serving different purposes in facilitating learning and modeling complex data relationships. In this case, the sigmoid function is used as follows:

$$f={\displaystyle \frac{1}{1+e^x}}$$

(41)

The general form of the function is

$$Y_u={\displaystyle \frac{f_p\left(p+{\displaystyle {\sum }_{w=1}^r\left(P_w\cdot f_{p-1}\left(\dots \cdot f_2\left(b_i+{\displaystyle {\sum }_{j=1}^m\left(B_{ij}\cdot f_1\left(a_j+{\displaystyle {\sum }_{k=1}^n\left(A_{jk}\cdot \left(P_k\cdot X_k+R_k\right)\right)}\right)\right)}\right)\dots \right)\right)}\right)-G_u}{L_u}},$$

(42)

where

• $X_k$—input parameters (number of neurons in input layer);
• $Y_u$—output parameters (number of neurons in output layer);
• $f_1,f_2,\dots ,f_{\left(p-1\right)},f_p$—activation function;
• $k,j,i,\dots ,v,w$—number of neurons in each layer;
• $A,B,\dots ,P$—weights between layers;
• $a,b,\dots ,p$—weights between neurons in each layer and bias.

The software that has been used for training the ANN in this study is called aNETka, developed in LabVIEW. The output file from this software is not a direct mathematical model, just weights coefficients and four additional parameters called Down offset Input ($D{I}_k$), Down offset Target ($D{T}_u$), Up offset Input ($U{I}_k$) and Up offset Target ($U{T}_u$).

$$D{I}_k=\min \left(X_k\right)$$

(43)

$$D{T}_u=\min \left(Y_u\right)$$

(44)

$$U{I}_k=\max \left(X_k-\min \left(X_k\right)\right)$$

(45)

$$U{T}_u=\max \left(Y_u-\min \left(Y_u\right)\right)$$

(46)

The rest of the parameters are calculated as follows:

$$P_k={\displaystyle \frac{0.9}{U{I}_k}}$$

(47)

$$R_k=-{\displaystyle \frac{0.9\cdot D{I}_k}{U{I}_k}}+0.05$$

(48)

$$L_u={\displaystyle \frac{0.9}{U{T}_u}}$$

(49)

$$G_u=-{\displaystyle \frac{0.9\cdot D{T}_u}{U{T}_u}}+0.05$$

(50)

In the trim optimization study, data obtained from CFD analysis, including brake power, DFOC and propeller speed, were utilized as the output layer for training the artificial neural network (ANN), while the draft, speed, trim and displacement are considered as the input layer.

The applicability range of the model is defined within specific boundaries for each parameter: draft ($T_s$) ranges from 7.5 to 8.7 m, speed ($V_s$) ranges from 12.5 to 18 kn, trim ranges from −1.5 to +1.5 m and displacement ($\nabla$) ranges from 18,079 to 22,884 m³. The input dataset for training consists of a 63 × 7 matrix, where number 63 represents the total number of CFD simulations conducted and number 7 denotes each parameter. Four parameters are designated as the input layer (draft, displacement, trim, speed), while the remaining three represent the output layer (propeller speed, brake power, DFOC).

Prior to training, all data were normalized within the range of 0.05 to 0.95 to ensure the stability of activation functions, which can be sensitive to extremely small or large input values. This normalization step enhances the convergence and effectiveness of the training process.

For testing purposes, 8% of the total input parameters (5 out of 63) were reserved, while the remaining data were allocated for training. The training process was programmed to iteratively adjust the weights and biases of the network, stopping when the Root Mean Square ($RMS$) percentage error reached the target threshold of 1%:

$$RMS=\sqrt{\left({\displaystyle \frac{{\displaystyle \sum \left[{\left(\frac{TV-OV}{TV}\right)}^2\right]}}{N}}\right)},$$

(51)

where $TV$—target values, $OV$—ANN output values, $N$—number of values.

When creating a neural network model, it is vital to carefully consider the number of neurons in each layer and the presence of hidden layers. These decisions must be well informed and tailored to match the complexity of the problem at hand and the specific attributes of the dataset.

In practice, just before leaving the port, the parameters that are categorized into the input layer are usually known, but until now, the fuel consumption could never be predicted in advance. It is very common that fuel consumption and the power engaged by the engine are not monitored during the journey, and the amount of fuel used was always determined retroactively. With such mathematical models and conducted CFD simulations for a larger number of loading conditions, it is possible not only to predict the fuel consumption in advance but also to determine the optimal speed at which the ship should sail to minimize consumption. Knowing the fuel consumption, it is easy to determine the amount of CO₂ emitted and predict the energy rating to which the ship will belong according to [8], which is a direct indicator of energy efficiency.

2.7. Trim Optimization Software Application

Formulas derived for a particular purpose may not always be user-friendly, leading to the creation of an application to facilitate their use. In this case, an application developed using MATLAB’s App Designer package offers a user-friendly interface for utilizing these formulas. The application allows users to input only one parameter, i.e., displacement and obtain corresponding output values efficiently: optimum trim, speed, propeller speed, brake power and DFOC. This enables users to quickly obtain results without the need for manual calculations.

3. Results

The results in this section will be presented in subsections in the same order, i.e., following the phases outlined in Section 2 (Methods).

3.1. Three-Dimensional Modeling

In Figure 7, a 3D model and characteristic sections of the waterlines (a), buttocks (b) and frames (c) are depicted in various projections obtained based on the construction drawings.

The propeller underwent a modeling process, aligning with the specifications outlined in the reference drawing. A comparison was conducted between the 3D model of the propeller and the reference drawing to ensure precise adherence to the intended design. The findings of this comparison are visually presented in Figure 8, offering a clear representation of the propeller’s details and shape. The 3D model of the propeller (Figure 9a) enables engineers and designers to perform CFD simulations and evaluate performance characteristics. The rudder was modeled to faithfully represent its original design but without any gaps between the rudder shaft and rudder shell (Figure 9b).

3.2. First-Level Verification

The reliability of the developed 3D model was assessed by comparing its calculated hydrostatic parameters against the corresponding ones given in the Trim & Stability booklet. The corresponding percentage deviations between $C_b$, $\nabla$, $LCB$ and $WS$ are listed in

Table 3. Comparison between hydrostatics data obtained on the basis of 3D model and Trim & Stability booklet.

${\mathit{T}}_{\mathit{s}}$ (m) (Trim = 0)	${\mathit{C}}_{\mathit{b}}$	$\mathbf{\nabla }$	$\mathit{L}\mathit{C}\mathit{B}$	$\mathit{W}\mathit{S}$
4	0.52%	0.88%	−0.03%	0.54%
5	0.43%	0.77%	−0.03%	0.74%
6	0.34%	0.62%	−0.01%	0.63%
7	0.27%	0.51%	0.01%	0.38%
7.5	0.22%	0.43%	0.03%	0.42%
8	0.21%	0.42%	0.03%	0.74%
8.7	0.23%	0.42%	0.03%	0.79%
9	0.31%	0.40%	0.03%	0.82%
10	0.14%	0.31%	0.03%	0.87%

.

It can be easily observed that deviations between the calculated and reference values of the hydrostatic data are less than 1%, especially for the range of interest (drafts 7.5 m, 8.0 m and 8.7 m).

3.3. Open Water Test

A graphical representation of the CFD results ($K_T$, $K_Q$, ${\eta }_0$) from the conducted OWT with the B-series propeller with data obtained with the mathematical model is presented in Figure 10a, while the relative differences between each evaluated parameter are presented in

Table 4. Relative differences between data obtained with mathematical model and results from CFD.

$\mathit{J}$	$\mathit{\delta }{\mathit{K}}_{\mathit{T}}$	$\mathit{\delta }{\mathit{K}}_{\mathit{Q}}$	$\mathit{\delta }{\mathit{\eta }}_0$
0.5	1.9%	−2.3%	2.2%
0.6	2.9%	−0.8%	2.3%
0.7	2.8%	1.1%	1.2%
0.8	2.7%	2.4%	0.2%
0.9	2.2%	0.6%	1.0%

.

As the results for each of the advance coefficients are within the prescribed 3%, the CFD calculation methodology is considered valid; therefore, with the same setup, the original propeller OWT was conducted, and the results are presented in Figure 10b.

3.4. Second-Level Verification

The obtained results from the CFD OWT simulation with the original propeller are used as the input propeller characteristics data in self-propulsion simulations. The first group of simulations were conducted at a design draft (7.8 m) and speeds of 17 kn, 19 kn and 21 kn because for these draft and speeds model tests, data are available.

In Figure 11, the meshed domain is shown. For a better evaluation of the free surface, additional refinement zones are defined in wave field zones and near the hull (Figure 12a,b).

The results from the completed self-propulsion simulations and steady resistance simulations by following the procedure described in Section 2.4 are presented in

Table 5. Results from CFD analysis for design draft.

${\mathit{V}}_{\mathit{s}}$ (kn)	${\mathit{R}}_{\mathit{p}}$ (kN)	${\mathit{R}}_{\mathit{v}}$ (kN)	${\mathit{R}}_{\mathit{T}\prime }$ (kN)	$\mathit{t}$ (-)	$\mathit{w}$ (-)	$\mathit{T}\prime$ (kN)	$\mathit{Q}$ (kNm)	${\mathit{\eta }}_0$ (-)	$\mathit{n}$ (rpm)	${\mathit{P}}_{\mathit{b}}$ (kW)
17	125.454	292.216	475.936	0.1746	0.2487	576.644	466.316	0.6736	103.1	5083
19	178.852	366.869	622.133	0.1579	0.2479	738.831	595.864	0.6721	116.0	7309
21	321.762	438.076	857.035	0.1600	0.2458	1020.295	820.407	0.6656	132.8	11,523

, where $R_p$ is the pressure resistance and $R_v$ is the viscous resistance.

The results from the CFD analysis and model test data are graphically presented in Figure 13. In Figure 13a, the brake power as a function of the ship speed curve is depicted while in Figure 13b, the brake power as a function of the propeller speed is presented.

The calibration factors are $C_n$ = 1.05 and $C_{P_b}$ = 1.04, which means that a 5% deviation requirement in the results compared to the model test is met. Differences in the results obtained from the CFD analysis and model testing can arise from several factors, such as the reliability of the model testing results, the reliability of the method used to extrapolate the results from the model to full scale, as well as the CFD analysis itself. A portion of the difference undoubtedly stems from the differences in the OWT results presented in Table 4. Due to all the aforementioned reasons, an acceptable margin of 5% exists according to [47].

In Figure 14, the y+ values for the underwater part of the hull are presented and the average y+ value is pointed out (average $y^{+}$ = 70.61). A particular case is extracted from the steady resistance simulation at a speed of 18 kn at 7.5 m draft. For all other speeds, the $y^{+}$ values do not deviate from those aforementioned.

Three different meshes were considered for the grid uncertainty study with mesh sizes of 1.852 million cells (coarse), 3.628 million cells (medium) and 7.454 million cells (fine). The study was conducted at a speed of 18 knots at a 7.5 m draft (even keel). The results of the study, including the grid uncertainties for thrust, are presented in

Table 6. Grid uncertainty results for thrust.

Mesh Quality	$\mathit{T}$ (kN)
Coarse	544.445
Medium	539.287
Fine	535.586
$\overline{U}$	2.2%

.

3.5. Trim Optimization

The trim optimization CFD study was performed at a 7.5 m, 8 m and 8.7 m draft for a range of speeds, 12.5 kn, 15 kn and 18 kn, at seven different loading (trim) conditions and coarse mesh. Therefore, the results obtained for the propeller speed, brake power and estimated DFOC are presented in

Table 7. Propeller speed ($n$ [rpm]): CFD analysis results.

		Trim by Bow (m)			Even Keel	Trim by Stern (m)
	${\mathit{V}}_{\mathit{s}}$ (kn)	−1.5	−1	−0.5	0	0.5	1	1.5
7.5 m	12.5	82.7	83.6	84.2	85.0	86.6	87.9	88.9
	15	97.2	98.3	99.3	100.9	102.2	103.5	105.3
	18	116.8	117.9	119.0	119.7	121.1	122.6	125.1
8.0 m	12.5	82.4	83.1	84.0	84.7	85.7	87.7	89.5
	15	98.3	99.3	100.2	100.7	102.3	103.5	105.8
	18	120.4	120.2	121.2	122.3	123.5	126.1	127.4
8.7 m	12.5	84.4	85.0	85.7	86.4	87.3	89.1	91.2
	15	101.9	102.3	103.1	104.0	105.3	107.1	108.9
	18	125.1	125.6	126.0	127.4	129.1	131.3	132.9

,

Table 8. Brake power ($P_b$ [kW]): CFD analysis results.

		Trim by Bow (m)			Even Keel	Trim by Stern (m)
	${\mathit{V}}_{\mathit{s}}$ (kn)	−1.5	−1	−0.5	0	−1.5	−1	−0.5
7.5 m	12.5	2248	2335	2370	2447	2601	2739	2837
	15	3509	3665	3772	3965	4130	4414	4642
	18	5957	6187	6345	6441	6667	7124	7480
8.0 m	12.5	2155	2221	2285	2357	2450	2664	2882
	15	3587	3719	3818	3869	4073	4352	4639
	18	6595	6567	6750	6942	7120	7879	7991
8.7 m	12.5	2293	2353	2403	2478	2564	2771	3048
	15	4018	4072	4181	4303	4504	4890	5113
	18	7531	7663	7707	8003	8405	9143	9361

and

Table 9. DFOC [t fuel/day]: CFD analysis results.

		Trim by Bow (m)			Even Keel	Trim by Stern (m)
	${\mathit{V}}_{\mathit{s}}$ (kn)	−1.5	−1	−0.5	0	−1.5	−1	−0.5
7.5 m	12.5	10.70	11.08	11.24	11.57	12.25	12.84	13.26
	15	16.11	16.75	17.20	18.00	18.68	19.85	20.79
	18	26.22	27.18	27.83	28.23	29.18	31.09	32.58
8.0 m	12.5	10.29	10.58	10.86	11.18	11.59	12.52	13.46
	15	16.43	16.98	17.39	17.60	18.45	19.60	20.78
	18	28.88	28.76	29.53	30.33	31.08	34.26	34.73
8.7 m	12.5	10.90	11.16	11.38	11.71	12.08	12.98	14.16
	15	18.22	18.44	18.89	19.39	20.22	21.81	22.73
	18	32.80	33.35	33.53	34.78	36.46	39.56	40.48

, respectively.

3.6. Mathematical Model for Assessing the Outcomes from Trim Optimization

Brake power reduction and DFOC reduction are linearly interpolated for additional trims and speeds (values considered in CFD analysis) and the results are graphically presented for the three considered drafts in Figure 15 and Figure 16. The optimum trim and speed are also depicted with red points in Figure 16. The black line in Figure 15 and Figure 16 represents a neutral line, i.e., the state of the even keel against which the brake power savings and DFOC savings are calculated.

The biggest reduction in the DFOC can be achieved at a 1.5 m trim by the bow, 7.5 m draft and speed of 15.17 kn (10.5%), while for the 8 m and 8.7 m draft, the biggest reduction in brake power is at 12.5 kn (8%, 7%, respectively). Sailing at 12.5 kn instead of 15 kn or 18 kn, and maintaining an even keel, can result in greater fuel savings for higher drafts. Although greater fuel savings might be achievable with a greater trim by the bow, any condition more than −1.5 m trim by the bow cannot be attained on the current ship.

In Figure 17, wave fields are depicted for the case of $V_s$ = 15 kn, $T_s$ = 7.5 m at trim = 0 m (Figure 17a) and at trim = −1.5 m (Figure 17b). It is evident that the total wave amplitude when trimmed towards the bow by 1.5 m is reduced by 0.25 m, indicating a decrease in the pressure resistance–displacement ratio of 15% (see

Table 10. Pressure resistance decrease.

${\mathit{V}}_{\mathit{s}}$ = $15 \mathbf{kn}, {\mathit{T}}_{\mathit{s}}$ = 7.5 m	Trim = 0 m	Trim = −1.5 m
$R_p/\nabla \cdot {10}^3$ (-)	0.6039	0.5127

). The volume displacement is multiplied by the gravity constant (9.81 m/s²) and sea water density.

In Figure 18, the free surfaces in the center line and along the hull are presented at two different conditions: trim = 0 m (white hull) and trim = −1.5 m (yellow hull). The black line corresponds to the free-surface level at a zero trim condition while the red line corresponds to the free-surface level at a 1.5 m trim by the bow. In Figure 18a, the bow waves are presented, while in Figure 18b, the stern waves are presented. The peak-to-peak amplitude of the bow wave is reduced by 0.64 m while the stern wave is reduced by nearly 1 m when the ship is trimmed towards the bow by 1.5 m.

For other drafts and trims, the impact of the bow trim remains similar; nevertheless, there is a less noticeable reduction in the pressure resistance. It seems that this kind of bulb has a better effect when it is more submerged into the water at lower drafts. This contradicts what is highlighted in numerous literature sources, including [61,62] but it is not the first time that the same conclusion with a fully submerged bulb has been set [63,64].

Mathematical Model for Assessing Brake Power, DFOC and Propeller Speed

Unfortunately, a single mathematical model did not attain the desired RMS within a reasonable training timeframe and iteration count. As a result, two separate mathematical models were devised as follows: one focusing on the brake power and DFOC and the other dedicated to propeller speed prediction. The formula for brake power follows a general form:

$$P_b={\displaystyle \frac{f_3\left(c_{P_b}+{\displaystyle {\sum }_{i=1}^7\left(C_{{iP}_b}\cdot f_2\left(b_i+{\displaystyle {\sum }_{j=1}^{10}\left(B_{ij}\cdot f_1\left(a_j+{\displaystyle {\sum }_{k=1}^4\left(A_{jk}\cdot \left(P_k\cdot X_k+R_k\right)\right)}\right)\right)}\right)\right)}\right)-G_{P_b}}{L_{P_b}}}.$$

(52)

The general form of the function for DFOC is

$$DFOC={\displaystyle \frac{f_3\left(c_{DFOC}+{\displaystyle {\sum }_{i=1}^7\left(C_{iDFOC}\cdot f_2\left(b_i+{\displaystyle {\sum }_{j=1}^{10}\left(B_{ij}\cdot f_1\left(a_j+{\displaystyle {\sum }_{k=1}^4\left(A_{jk}\cdot \left(P_k\cdot X_k+R_k\right)\right)}\right)\right)}\right)\right)}\right)-G_{DFOC}}{L_{DFOC}}}.$$

(53)

The general form of the function for propeller speed is

$$n={\displaystyle \frac{f_3\left(c_n+{\displaystyle {\sum }_{l=1}^5\left(C_{ln}\cdot f_2\left(b_l+{\displaystyle {\sum }_{m=1}^8\left(B_{lm}\cdot f_1\left(a_m+{\displaystyle {\sum }_{s=1}^4\left(A_{ms}\cdot \left(P_s\cdot X_s+R_s\right)\right)}\right)\right)}\right)\right)}\right)-G_n}{L_n}}.$$

(54)

Configurations of the neurons for both mathematical models are presented in Figure 19a,b.

The obtained coefficients for Equations (52) and (53) are given in

Table 11. Coefficients $A$ and $a$ for Equations (52) and (53).

${\mathit{A}}_{\mathit{j}1}$	${\mathit{A}}_{\mathit{j}2}$	${\mathit{A}}_{\mathit{j}3}$	${\mathit{A}}_{\mathit{j}4}$	${\mathit{a}}_{\mathit{j}}$
1.39544	4.80982	−1.76190	−1.30478	−0.75004
0.22339	3.25659	−0.04068	2.23871	−1.54100
−2.10995	3.49346	−0.21458	1.21715	−1.63827
3.03760	4.14154	−2.40221	1.04647	−6.47006
−0.04455	0.71422	1.58515	−0.98086	−0.02750
−1.50944	0.27102	−1.41296	1.52488	1.06833
−0.97408	−0.21472	−0.34123	−0.58728	0.38679
1.83711	2.00358	−9.92587	−1.27416	3.94278
0.08588	0.39723	1.06558	−0.68159	−0.95401
−1.45797	−0.49383	3.67284	−2.41111	0.23743

,

Table 12. Coefficients $B$ and $b$ for Equations (52) and (53).

${\mathit{B}}_{\mathit{i}1}$	${\mathit{B}}_{\mathit{i}2}$	${\mathit{B}}_{\mathit{i}3}$	${\mathit{B}}_{\mathit{i}4}$	${\mathit{B}}_{\mathit{i}5}$	${\mathit{B}}_{\mathit{i}6}$	${\mathit{B}}_{\mathit{i}7}$	${\mathit{B}}_{\mathit{i}8}$	${\mathit{B}}_{\mathit{i}9}$	${\mathit{B}}_{\mathit{i}10}$	${\mathit{b}}_{\mathit{i}}$
−3.94741	−1.10676	−1.51183	−3.90542	1.28454	1.37817	1.67905	10.13879	0.82581	2.39590	3.05951
−0.83685	−0.80410	−0.84916	−0.74439	−0.53931	1.20460	0.17759	−0.26222	−0.48840	−0.58512	0.76405
−1.21006	−1.28977	−1.53713	−0.74023	−1.04296	0.87026	0.00654	0.35832	−0.85771	−1.31206	0.45529
0.79825	−0.92382	−0.16195	−3.22868	0.48528	0.41493	0.29515	−2.64142	0.12449	1.60271	0.59712
−0.83157	−1.72121	−1.74315	−1.78391	−0.13521	2.35956	0.85288	1.08050	−0.61823	−0.66071	2.10411
−1.03458	−1.10428	−1.25646	−0.56187	−0.81983	0.76972	−0.00495	0.29741	−0.77011	−1.13999	0.35562
−0.48851	−0.49214	−0.56036	−0.16494	−0.22853	0.26762	−0.48143	1.41729	−0.31051	−1.40889	0.36591

,

Table 13. Coefficients $C_{P_b}$ and $c_{P_b}$ for Equations (52) and (53).

$${\mathit{C}}_{{\mathit{P}}_{\mathit{b}}1}$$	$${\mathit{C}}_{{\mathit{P}}_{\mathit{b}}2}$$	$${\mathit{C}}_{{\mathit{P}}_{\mathit{b}}3}$$	$${\mathit{C}}_{{\mathit{P}}_{\mathit{b}}4}$$	$${\mathit{C}}_{{\mathit{P}}_{\mathit{b}}5}$$	$${\mathit{C}}_{{\mathit{P}}_{\mathit{b}}6}$$	$${\mathit{C}}_{{\mathit{P}}_{\mathit{b}}7}$$	$${\mathit{c}}_{{\mathit{P}}_{\mathit{b}}}$$
−3.60944	−1.69300	−1.92004	−2.15282	−2.51012	−1.80109	−2.08429	7.38079

,

Table 14. Coefficients $C_{\mathrm{DFOC}}$ and $c_{\mathrm{DFOC}}$ for Equations (52) and (53).

${\mathit{C}}_{\mathit{D}\mathit{F}\mathit{O}\mathit{C}1}$	${\mathit{C}}_{\mathit{D}\mathit{F}\mathit{O}\mathit{C}2}$	${\mathit{C}}_{\mathit{D}\mathit{F}\mathit{O}\mathit{C}3}$	${\mathit{C}}_{\mathit{D}\mathit{F}\mathit{O}\mathit{C}4}$	${\mathit{C}}_{\mathit{D}\mathit{F}\mathit{O}\mathit{C}5}$	${\mathit{C}}_{\mathit{D}\mathit{F}\mathit{O}\mathit{C}6}$	${\mathit{C}}_{\mathit{D}\mathit{F}\mathit{O}\mathit{C}7}$	${\mathit{c}}_{\mathit{D}\mathit{F}\mathit{O}\mathit{C}}$
−3.60911	−1.64363	−2.01584	−2.11789	−2.49256	−1.76852	−2.02170	7.32870

and

Table 15. Coefficients $P_k$, $R_k$, $G_{P_b}$, $L_{P_b}$, $G_{\mathrm{DFOC}}$ and $L_{\mathrm{DFOC}}$ for Equations (52) and (53).

${\mathit{P}}_{\mathit{k}}$	${\mathit{R}}_{\mathit{k}}$	${\mathit{G}}_{{\mathit{P}}_{\mathit{b}}}$	${\mathit{L}}_{{\mathit{P}}_{\mathit{b}}}$	${\mathit{G}}_{\mathbf{DFOC}}$	${\mathit{L}}_{\mathbf{DFOC}}$
0.75000	−5.57500	−0.21915	0.00012	−0.25636	0.02980
0.16364	−1.99545
0.30000	0.50000
0.00019	−3.33586

, while the coefficients for Equation (54) are given in

Table 16. Coefficients $A$ and $a$ for Equation (54).

${\mathit{A}}_{\mathit{m}1}$	${\mathit{A}}_{\mathit{m}2}$	${\mathit{A}}_{\mathit{m}3}$	${\mathit{A}}_{\mathit{m}4}$	${\mathit{a}}_{\mathit{m}}$
1.60283	1.91017	−1.71984	0.23942	−1.93017
1.62613	2.44722	−0.17892	−1.00908	−3.09041
1.09543	1.09342	2.08237	0.00351	−0.00032
−0.17611	2.37242	0.49469	1.22030	−2.13783
−1.12935	5.25606	−0.88296	−1.08825	−0.79561
−1.10799	−0.07232	−2.85448	0.21127	3.80378
−2.76944	2.04227	4.00000	−0.90497	−1.59122
−0.09699	1.92632	1.05398	−0.48593	−0.22092

,

Table 17. Coefficients $B$ and $b$ for Equation (54).

${\mathit{B}}_{\mathit{l}1}$	${\mathit{B}}_{\mathit{l}2}$	${\mathit{B}}_{\mathit{l}3}$	${\mathit{B}}_{\mathit{l}4}$	${\mathit{B}}_{\mathit{l}5}$	${\mathit{B}}_{\mathit{l}6}$	${\mathit{B}}_{\mathit{l}7}$	${\mathit{B}}_{\mathit{l}8}$	${\mathit{b}}_{\mathit{l}}$
−0.85076	−0.86022	−0.92193	−1.21839	−1.28573	0.12384	−0.42344	−0.84941	0.72250
−2.21759	−3.16121	1.26550	−1.20003	−2.78787	4.04679	−1.95753	0.95956	3.10950
−1.04299	−0.99883	−1.19340	−1.29826	−1.50607	0.14846	−0.49616	−1.15469	1.08981
−1.11884	−0.96068	−1.18134	−1.19153	−1.62624	0.20455	−0.46174	−1.00754	1.02576
−0.43893	−1.02698	−1.08333	−1.41701	−0.49332	1.67727	0.33505	−0.39926	1.34012

,

Table 18. Coefficients $C$ and $c$ for Equation (54).

${\mathit{C}}_{\mathit{n}1}$	${\mathit{C}}_{\mathit{n}2}$	${\mathit{C}}_{\mathit{n}3}$	${\mathit{C}}_{\mathit{n}4}$	${\mathit{C}}_{\mathit{n}5}$	${\mathit{c}}_{\mathit{n}}$
−2.44334	−5.02627	−2.91111	−2.88484	−2.67661	6.86984

and

Table 19. Coefficients $P_s$, $R_s$, $G_n$ and $L_n$ for Equation (54).

${\mathit{P}}_{\mathit{s}}$	${\mathit{R}}_{\mathit{s}}$	${\mathit{G}}_{\mathit{n}}$	${\mathit{L}}_{\mathit{n}}$
0.75000	−5.57500	−1.41852	0.01782
0.16364	−1.99545
0.30000	0.50000
0.00019	−3.33987

.

The standard deviations of the relative differences in the brake power, DFOC and propeller speed calculated the values using Equations (52)–(54), and the results obtained from the CFD analysis for the same three parameters are 0.6%, 0.6% and 0.2%, respectively.

3.7. Trim Optimization Software Application

The development of software focusing on trim optimization is another step forward that demonstrates how engineering and technical practice can be more effectively connected. The equations obtained (52), (53) and (54) are very complex and not suitable for manual calculation. Additionally, the matrix obtained through interpolation of the CFD analysis results contains over 2 million elements (a 151 × 111 × 120 matrix), meaning tabular or graphical representation of the results will be impractical for the average user. Therefore, the most elegant solution is to develop an application with a graphical interface. Today, there are many software tools that operate based on writing in various programming languages, but this task is addressed to other engineering disciplines. This paper demonstrates the development of a software tool within MATLAB App Designer. MATLAB App Designer is highly suitable for the rapid development of simple applications as it has certain functions predefined within the Component Library. In this specific case, only three predefined functions were used: “Button”, “Edit Field” and “Label”. The interface design is depicted in Figure 20:

In the “Input displacement” label, the user should insert the current displacement in cubic meters in the range from 18,078 to 22,884 m³. The estimation of the optimal trim, corresponding mean draft, estimated speed, brake power, DFOC and propeller speed is performed by simply clicking the “Calculate” button. In case the user inputs a displacement below or above the specified values, despite the stated applicability limits of the mathematical model, and initiates the calculation, a message will appear on the screen: “Displacement is out of range” (see Figure 21).

If a displacement within the specified limits is entered, and the “Calculate” button is clicked, all the data are instantly displayed (see Figure 22).

The application provides results based on previously obtained mathematical models for estimating the optimal trim, speed, corresponding mean draft, brake power, DFOC and propeller speed. There is also a note that sign “−” in the Optimum Trim label corresponds to the trim by the bow. Due to the specified project task, which required trim optimization at certain speeds, trims and drafts rather than displacements, it is clear that for the same mean draft and different trims, the vessel has different displacements (due to the different position of the ship, the $C_b$ is different and therefore displacement). Considering the upper limit of the performed calculations with a draft of 8.7 m and taking into account various trims, accordingly, the upper limit of displacements for which the calculations were performed at this draft (8.7 m) is variable (refer to Figure 4b). Considering the obtained results, there is a displacement range where the optimal trim is 0 m, meaning that sailing on an even keel is recommended (see Figure 23). This zone starts from the displacement corresponding to a draft of 8.7 m and even-keel position.

The mathematical model does not take draft as an input parameter but determines the mean draft by seeking the optimal trim for a given displacement. It is logical to apply the recommendation of the optimal trim obtained from the mathematical model only through the additional ballasting of the vessel if achieving the optimal trim cannot be attained with the existing ballast water in the tanks (if any). This is manifested by further increasing the displacement. Therefore, it is necessary to rerun the application with the new displacement. Through a few iterations, the optimal trim can be achieved, thus obtaining the estimated ship speed, propeller speed, brake power and DFOC in a reliable manner. It should be noted that the application is not part of the Loading Computer nor does it track stability data for a specific loading condition for a recommended optimal trim. However, it can be integrated into the Loading Computer system to enhance its functionality.

4. Discussion and Conclusions

In this work, not only trim optimization aiming to improve the energy efficiency of the ship, but specifically the CII parameter by reducing the fuel consumption, was presented. The methodology for approaching this issue using CFD software, the application of results obtained from CFD analysis, i.e., using an ANN to develop a mathematical model that provides estimates of the parameters (brake power, DFOC and propeller speed) for all conditions for which CFD calculations were performed, as well as for all intermediate conditions, was also introduced. Since the use of an ANN resulted in a complex mathematical model that is not straightforward to use in engineering practice, it went a step further, incorporating it into a simple software tool (application).

The entire trim optimization process is described through seven phases that should be applied to other types of ships in future research to develop a universal methodology, not only for trim optimization but also for CFD calculations based on which various engineering analyses such as the optimization of the bow or stern, determination of $V_{ref}$ for EEXI, efficiency of energy-saving devices, etc., can be conducted. For now, the applied methodology, which includes 3D modeling of a digital twin whose hydrostatic parameters differ by up to 1% when comparing the same parameters with data from the Trim & Stability booklet, has proven to be sufficiently good. With additional assumptions about determining the boundary-layer thickness and shear wall stress, i.e., shear velocity, satisfactory accuracy of the y+ parameters, which are very significant for obtaining reliable results, was achieved. The obtained values of $y^{+}$(approximately 70–72) differ from the initial target y+ of 150, which is to be expected because the equations used to estimate both the boundary-layer thickness and wall shear stress are not the only ones that can be applied and are very general. For example, the formula for the boundary-layer thickness applies to a flat plate, but in calculating the shear stress, a form factor is included, which is itself obtained based on empirical expressions. Therefore, these formulas cannot provide exact parameter values because these parameters are actually calculated by CFD analysis. However, they can be used to determine enough fine mesh (parameters) so that the results obtained by CFD analysis fall within prescribed limits, according to currently published requirements (up to 5% difference in power brake and shaft speed) for a specific ship case.

The application of an ANN to develop a mathematical model has proven to be very reliable. Generally speaking, artificial intelligence (AI) is increasingly finding applications in various engineering industries, where new design approaches can be developed, and existing ones significantly accelerated based on machine learning. It should be noted that the obtained mathematical model applies to navigation in calm waters, which is practically an almost impossible scenario. This approach, which encompasses multidisciplinarity (CFD analysis, ANN application, software tool programming), can lay the foundation for the further development of the shipbuilding industry from an engineering perspective. In the future, research should further focus on the impact of trim optimization on maneuverability and seakeeping, as these characteristics are crucial for safe navigation. Currently, these aspects are often overlooked in trim optimization projects, as they are not typically included in the project assignments. Addressing this gap will enhance the overall safety and performance of maritime ships.

The further idea is to collect real (measured) data on fuel consumption, engine power, ship speed and navigation conditions during voyages. Monitoring and recording some of these parameters have already become mandatory through [10]. Based on the stored parameters and the formed database, and with the help of machine learning and AI, it will be possible in the future to predict in advance to which energy class the ship belongs and what can be changed during real-time operation to make the ship “green”. This can lead to a global reduction in exhaust gas emissions instantly, without ship owners being exposed to penalties imposed by the system because the CII is determined retroactively.