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Citation: Nazemian, A.; Boulougouris,
E.; Aung, M.Z. Utilizing Machine
Learning Tools for Calm Water
Resistance Prediction and Design
Optimization of a Fast Catamaran
Ferry. J. Mar. Sci. Eng. 2024,12, 216.
https://doi.org/10.3390/
jmse12020216
Academic Editor: Diego Villa
Received: 23 December 2023
Revised: 19 January 2024
Accepted: 22 January 2024
Published: 25 January 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Journal of
Marine Science
and Engineering
Article
Utilizing Machine Learning Tools for Calm Water Resistance
Prediction and Design Optimization of a Fast Catamaran Ferry
Amin Nazemian * , Evangelos Boulougouris and Myo Zin Aung
Maritime Safety Research Centre (MSRC), Department of Naval Architecture, Ocean and Marine Engineering,
University of Strathclyde, Glasgow G4 0LZ, UK; evangelos.boulougouris@strath.ac.uk (E.B.);
myo.aung@strath.ac.uk (M.Z.A.)
*Correspondence: amin.nazemian@strath.ac.uk
Abstract: The article aims to design a calm water resistance predictor based on Machine Learning
(ML) Tools and develop a systematic series for battery-driven catamaran hullforms. Additionally,
employing a machine learning predictor for design optimization through the utilization of a Genetic
Algorithm (GA) in an expedited manner. Regression Trees (RTs), Support Vector Machines (SVMs),
and Artificial Neural Network (ANN) regression models are applied for dataset training. A hullform
optimization was implemented for various catamarans, including dimensional and hull coefficient
parameters based on resistance, structural weight reduction, and battery performance improvement.
Design distribution based on Lackenby transformation fulfills all of the design space, and sequentially,
a novel self-blending method reconstructs new hullforms based on two parents blending. Finally, a
machine learning approach was conducted on the generated data of the case study. This study shows
that the ANN algorithm correlates well with the measured resistance. Accordingly, by choosing
any new design based on owner requirements, GA optimization obtained the final optimum design
by using an ML fast resistance calculator. The optimization process was conducted on a 40 m
passenger catamaran case study that achieved a 9.5% cost function improvement. Results show that
incorporating the ML tool into the GA optimization process accelerates the ship design process.
Keywords: systematic series; machine learning; lackenby variation method; self-blending method;
genetic algorithm
1. Introduction
The EU-funded project “TrAM-Transport: Advanced and Modular” develops battery-
driven zero-emission fast passenger vessels for coastal areas and inland waterways. Modu-
lar design and manufacturing methods are the focus of this project, with the objectives of
minimizing environmental impact and life cycle cost [
1
,
2
]. The development of a systematic
series of zero-emission catamaran hullforms for different displacement tonnage and ship
types can significantly help this process. Enormous catamaran hullforms will be generated
during the systematic series development, and resistance calculation takes time for each
design. An accurate and fast resistance predictor leads to a convenient tool for a class of
hullforms. Therefore, a new model for such diversity with appropriate generalization to
new predictions is desired in this field, leading to data mining approaches [
3
]. ML can be
combined with optimization algorithms to efficiently explore the design space and identify
optimal or near-optimal solutions. Genetic algorithms, particle swarm optimization, and
other optimization techniques can benefit from ML models to guide the search process
and converge to better solutions faster. By leveraging these techniques, naval architects
and ship designers can streamline the hullform optimization process, reduce the need for
time-consuming simulations, and ultimately arrive at more efficient and cost-effective ship
designs [4].
J. Mar. Sci. Eng. 2024,12, 216. https://doi.org/10.3390/jmse12020216 https://www.mdpi.com/journal/jmse
J. Mar. Sci. Eng. 2024,12, 216 2 of 24
2. Background
Resistance calculations in past decades have been implemented by model tests or sea
trial measurements. Classic regression models are limited to conventional vessels with
specified general particulars. Traditional methods of resistance prediction involve extensive
model testing, computational fluid dynamics (CFD) simulations, and empirical correlations.
However, the complexity of catamaran hullforms, coupled with the desire for enhanced
design efficiency, necessitates a paradigm shift in the approach to resistance prediction [
5
].
Enter machine learning—a transformative tool capable of extracting intricate patterns from
vast datasets, promising to revolutionize the maritime design landscape [
6
]. Accordingly,
the current study embarks on a comprehensive exploration of the systematic series devel-
opment and calm water resistance prediction for fast catamaran ferries. We leverage the
power of machine learning tools to augment traditional design methodologies, seeking to
enhance accuracy, efficiency, and the overall efficacy of the design process. Additionally, a
Genetic Algorithm optimization study will be implemented on a sample design by using a
machine learning resistance predictor as a fast approach objective calculation method.
Ship resistance optimization plays an important role in hullform development. Assess-
ing the ship resistance in the first stage of ship design allows the designer to analyze the
influence of different hullforms and parameters. Accordingly, different methods of geome-
try optimization and design study have been developed during past decades. Papanikolaou
et al. [
7
] implemented a global and local hullform optimization of the fast catamaran in two
design study scenarios. In the first stage of optimization, 1000 hullforms were elaborated
with a surrogate-based design study using a potential theory 3D panel code. After that,
the two most promising designs were selected as the initial hullform for local modification,
focusing on the stern region. However, a comprehensive design optimization might be pro-
posed according to balance accuracy and time [
8
]. An all-inclusive hullform optimization
in the field of ship design defines various hullforms with different geometrical parameters.
Accordingly, the marine industry needs an optimization platform to minimize the required
propulsion power according to various possibilities of hullform. Additionally, a systematic
series is developed on generated geometries to establish a resistance predictor.
Li et al. [
9
], by using Single-Parameter Lagrangian Support Vector Regression (SPL-
SVR), developed a metamodel on seakeeping data. A multidisciplinary design optimiza-
tion in the concept design stage of ships has been proposed. Recently, Fahrnholz and
Caprace [
10
] conducted a regression analysis on three sailboats’ systematic series. Based
on machine learning techniques, a resistance predictor was designed on resistance data.
Nazemian and Ghadimi [11], by using a D-optimal DoE study, investigated the resistance
performance of a trimaran hull series. A resistance analysis and its improvement were
encompassed to extract the optimum value of hull parameters and sidehull arrangement.
Machine learning techniques have commenced in the last decade in the field of ship de-
sign and hydrodynamics [
12
,
13
]. Resistance prediction has been developed and compared
with traditional approaches by Radojic et al. [
14
,
15
]. An Artificial Neural Network regres-
sion method was designed for planing boats at different series types. Machine learning
models can also be implemented on added resistance [
16
] and ice resistance [
17
]. Different
aspects of ship design targets can be considered in dataset analysis. Liu and Papaniko-
laou [
18
] developed a semi-empirical formula, approximating the added resistance of ships
in regular waves of arbitrary heading. The development of a catamaran class alongside
the optimization process has been considered in the current study with automatic design
generation.
The present paper is divided into three phases; systematic series development for
a fast passenger and freight zero-emission catamaran, applying machine learning on
generated data, and hullform optimization using a developed resistance predictor. Based
on surveyed literature, it can be concluded that a hullform optimization process needs to
be added to ship series. For each tonnage condition and ship type, a predictive machine
learning model is developed to calculate calm water resistance. Moreover, the final design
would be the best design with respect to the lowest resistance at multi-design speeds. A
J. Mar. Sci. Eng. 2024,12, 216 3 of 24
genetic algorithm optimization process connects to the resistance predictor to calculate
design objectives of the optimization process. In the frame of the TrAM project, various
optimized design options are prepared based on ship dimension and coefficient and
hullform alteration. Accordingly, the design study starts with numerous ship types and
tonnage and offers different possibilities of catamaran hullform as flexibility for the owner’s
selection. Owners can choose their optimized design based on their requirements. On the
other hand, by selecting design requirements like ship dimensions from the ship owner
and an ML fast-approach resistance predictor, an optimized design can be obtained from a
Genetic Algorithm in a fast approach.
3. Methodology
The present design study code capabilities, allowing any type of hullform to be
modelled in case of different ship design targets, offer scope for the creation of a wide
range of hullforms and provide an optional selection for owners. Combined with the
built-in resistance, structure weight, and battery-driven system performance calculations,
you have the tools to experiment with shapes and explore design parameters. Accordingly,
an extensive fast catamaran series has been created, and a resistance predictor model has
been developed on the generated dataset. The case study is a catamaran hull [
1
,
2
] as an
initial design for database production. The database consists of three tonnages (
∆1=
75,
∆2=
80,
∆3=
85) tons. Two types of passenger and freight catamaran boats are defined as
the initial hullform. The general arrangement of the under-studied catamarans is depicted
in Figure 1.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 3 of 24
learning model is developed to calculate calm water resistance. Moreover, the final design
would be the best design with respect to the lowest resistance at multi-design speeds. A
genetic algorithm optimization process connects to the resistance predictor to calculate
design objectives of the optimization process. In the frame of the TrAM project, various
optimized design options are prepared based on ship dimension and coefficient and hull-
form alteration. Accordingly, the design study starts with numerous ship types and ton-
nage and offers different possibilities of catamaran hullform as flexibility for the owners
selection. Owners can choose their optimized design based on their requirements. On the
other hand, by selecting design requirements like ship dimensions from the ship owner
and an ML fast-approach resistance predictor, an optimized design can be obtained from
a Genetic Algorithm in a fast approach.
3. Methodology
The present design study code capabilities, allowing any type of hullform to be mod-
elled in case of different ship design targets, offer scope for the creation of a wide range
of hullforms and provide an optional selection for owners. Combined with the built-in
resistance, structure weight, and baery-driven system performance calculations, you
have the tools to experiment with shapes and explore design parameters. Accordingly, an
extensive fast catamaran series has been created, and a resistance predictor model has
been developed on the generated dataset. The case study is a catamaran hull [1,2] as an
initial design for database production. The database consists of three tonnages (∆=75,
∆=80, ∆=85) tons. Two types of passenger and freight catamaran boats are defined as
the initial hullform. The general arrangement of the under-studied catamarans is depicted
in Figure 1.
Figure 1. General arrangement plan of passenger and freight catamaran boats.
Firstly, a Machine Learning dataset is generated using shift transformation and self-
blending methods. After that, total resistance is calculated for all generated hullforms via
the slender body method. The structural weight of each design is estimated by a regres-
sion formula and shell expansion of the hull surface. The propulsion system of the vessel
works with electrically powered baery spares. Performance and baery weight are com-
puted based on resistance and consequently, the break power of the catamaran [19,20].
The framework of the design study and machine learning is illustrated in Figure 2.
For each tonnage, ship geometry is designed and distributed on the design space accord-
ing to a multi-level combination of design variables. Performing parametric transfor-
mations and self-blending methods creates a series of hullforms with systematically var-
ying parameters, which have been coded in the MATLAB program (R2022b) [21]. Para-
metric transformation by moving ship sections and self-blending by moving Control
Points implement parametric transformations to create new hulls. Accordingly, the design
Figure 1. General arrangement plan of passenger and freight catamaran boats.
Firstly, a Machine Learning dataset is generated using shift transformation and self-
blending methods. After that, total resistance is calculated for all generated hullforms via
the slender body method. The structural weight of each design is estimated by a regression
formula and shell expansion of the hull surface. The propulsion system of the vessel works
with electrically powered battery spares. Performance and battery weight are computed
based on resistance and consequently, the break power of the catamaran [19,20].
The framework of the design study and machine learning is illustrated in Figure 2. For
each tonnage, ship geometry is designed and distributed on the design space according to
a multi-level combination of design variables. Performing parametric transformations and
self-blending methods creates a series of hullforms with systematically varying parameters,
which have been coded in the MATLAB program (R2022b) [
21
]. Parametric transformation
by moving ship sections and self-blending by moving Control Points implement parametric
transformations to create new hulls. Accordingly, the design dataset will be prepared for
the machine learning process. Outputs of the design study are total resistance, structure
weight, and battery weight calculated for each design.
J. Mar. Sci. Eng. 2024,12, 216 4 of 24
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 4 of 24
dataset will be prepared for the machine learning process. Outputs of the design study
are total resistance, structure weight, and baery weight calculated for each design.
Figure 2. Framework of machine learning and design optimization methodology.
Pre-processing progress is applied to obtained data to define different regression
schemes. Herein, Regression Tree (RT), Support Vector Regression (SVR), and Artificial
Neural Network (ANN) methods are used to predict other interesting designs and find a
resistance predictive model. Then, a fast-approach resistance predictor will be applied in
optimization process, in which the Genetic Algorithm (GA) method will be used, which is
shown in the second stage of Figure 2.
The case study catamaran is the prototype hullform, which is designed and built in
frame of the Horizon 2020 European Research project TrAM [22]. The main purpose of
this effort is to replicate this hullform based on small modifications. The optimization pro-
cess is conducted on six design variables and two constraints that are shown in Table 1.
The output of the optimization process is resistance at 12 knots and resistance at 22 knots,
which is represented by a weighting cost function:
Figure 2. Framework of machine learning and design optimization methodology.
Pre-processing progress is applied to obtained data to define different regression
schemes. Herein, Regression Tree (RT), Support Vector Regression (SVR), and Artificial
Neural Network (ANN) methods are used to predict other interesting designs and find a
resistance predictive model. Then, a fast-approach resistance predictor will be applied in
optimization process, in which the Genetic Algorithm (GA) method will be used, which is
shown in the second stage of Figure 2.
The case study catamaran is the prototype hullform, which is designed and built in
frame of the Horizon 2020 European Research project TrAM [
22
]. The main purpose of this
effort is to replicate this hullform based on small modifications. The optimization process
is conducted on six design variables and two constraints that are shown in Table 1. The
output of the optimization process is resistance at 12 knots and resistance at 22 knots, which
is represented by a weighting cost function:
Cos t f uncti on = RtLowFn
RtLowFn0×WtLowFn + RtHigh Fn
RtHighFn0!×W tHi ghF n + (Weight/
Weight0)×WtWei ght !×Disp0
Dis p_get ×1
1000 ,
(1)
J. Mar. Sci. Eng. 2024,12, 216 5 of 24
Table 1. Design parameters of the catamaran case study for design optimization.
Optimization Parameter Symbol Specifications
Design Variable Lwl Waterline length (m)
Design Variable B Demi hull Beam (m)
Design Variable T Draft (m)
Design Variable DT Demi hull transverse distance (m)
Free Variable Cb Block coefficient
Design Variable Cm Max section area coefficient
Design Variable LCB (% of Lwl) Longitudinal Center of Buoyancy
Constraint ∇Displacement (ton)
Constraint (DT ×2) + B Total Beam (m)
The resistance at a low Froude number signifies as Rt
LowFn
, indicating resistance at
12 knots. Rt
HighFn
corresponds to the resistance at a high Froude number, contributing to
resistance at 22 knots. Weight pertains to the lightweight design of the ship, nondimen-
sionalized against the initial design weight. Disp0 and Disp_get respectively represent the
displacements of the initial hull and the displacement of the new hull, calculated from
hydrostatic data. Alterations in the shape and geometry of the ship hull are achieved by
imposing a constraint that maintains the difference in displacement under 1%. If displace-
ment is the sole constraint, the length, breadth, and draft are automatically adjusted to
accommodate the hull geometry redistributed volume. Consequently, the displacement
constraint is defined as follows:
∆new −∆org
∆org
≤0.01, (2)
Another constraint of the present study is the total beam of the catamaran to satisfy
port requirements, therefore:
2×Demi hul l o f f set +demi hull beam ≤9, (3)
Demi hull offset is the distance between centerline of each demi hull.
Nine design parameters of the catamaran ship are selected as input data for the
regression learner. The total resistance value is the output parameter of the study, which
is calculated through the slender body method [
23
,
24
]. Attribute selection is depicted in
Table 2. Regression models are implemented for each ship speed [12, 13.25, 14.5, 15.75, 17,
18.25, 19.5, 20.75, 22] knots. Finally, a comprehensive regression is applied to all generated
hulls at different drafts and dimensions to generalize the systematic series.
Table 2. Selected attributes for data mining with their respective statistical values.
Specifications Symbol Min Max Mean Deviation
Ship speed [kn] V 12 22 17 3.4232
Waterline length (m) Lwl 28 36 32 2.2633
Demi hull Beam (m) B 2.0985 2.2065 2.1407 0.0352
Draft (m) T 1.183 1.386 1.283 0.0518
Demi hull transverse distance (m) DT 3.35 3.424 3.3889 0.0239
Block coefficient Cb 0.4349 0.5062 0.4663 0.0144
Max section area coefficient Cm 0.7091 0.7610 0.7293 0.0135
Longitudinal Center of Buoyancy (% of L) LCB 0.5346 0.5549 0.5463 0.0047
Waterplane area (m2)Aw 96.235 104.339 100.488 2.9536
Maximum section area (m2)Ax 3.923 4.153 4.094 0.0582
3.1. Database Generation of Catamaran Case Study
In the process of geometry generation and resistance calculation, the entirety of the
design procedures is coded using the MATLAB (R2022b) programming tool and Maxsurf
J. Mar. Sci. Eng. 2024,12, 216 6 of 24
(V23) software. The MATLAB script includes six sequential sections, beginning with the
interaction between Maxsurf and MATLAB using the Component Object Model (COM).
The design dataset will automatically be pre-processed and imported into MATLAB Re-
gression Learner. The hullform alterations were made by a combination of the Lackenby
variation and a novel self-blending method in order to prepare the dataset of catamaran
hullforms [
21
]. Shift transformation is a geometry modification technique used to modify or
move hull sections with respect to the main axes of x, y, and z. The geometry parametriza-
tion method used in the current research is the Lackenby variation method, also known as
the Lackenby shift transformation [
25
]. The Lackenby variation approach entails creating a
collection of related hullforms by gradually altering the design specifications of the initial
hullform. During the transformation, the positions of the stations are shifted forward and
backward until the desired parameter specifications are achieved. In connection with the
section transformation, the relation
∆
y =
∆
x(x) is established. Essentially, this means that at
a given x-coordinate along the hull, a geometric shift is made in the x-direction using the
corresponding y-value. A positive y-value indicates a forward shift, while a negative value
indicates a backward section movement. A notable feature of this function is its exceptional
ability to maintain hull fairness to an extremely high degree throughout the transformation
process [21,26].
In the blending technique, on the other hand, the geometry is changed by manipulating
the control points. A parametric transformation is performed to generate a hull design
derived from blending two superior hull shapes. In this approach, two hulls are combined
by a blending function, resulting in a new hull shape. The coordinates of the control points
for the blended hull are determined proportionally based on a blending ratio (
α
) that ranges
from 0 to 1, where 0 corresponds to complete hull number 1 and 1 corresponds to complete
hull number 2. The expression for the blending function in three directions is as follows:
CPnew(xnew,ynew,znew)=α×CPi(xi,yi,zi)+(1−α)×CPjxj,yj,zj, (4)
In this context, CP denotes the control point,
α
represents the blending ratio, while i
and jrefer to the indices of the hulls being blended. To determine the updated positions
of control points for the newly created hulls, relation (4) is employed. Figure 3presents
a schematic illustration featuring four distinct control point distribution patterns. These
patterns are delineated as follows: (a) Control points located at the YZ intersection of
the waterline and the outline of the cross-section; (b) Control points positioned at the
YZ intersection of the buttock line and the outline of the cross-section; (c) Control points
situated at the YZ intersections of the cross-section outline, originating from the intersection
of the deck line and the centerline; (d) Control points employed for section blending in the
X direction. As a result, diverse hull configurations are reconstituted during the application
of the self-blending method within the process of ship design [
21
]. A sample structure of
the Design Dataset is depicted in Table 3.
Table 3. Design dataset preparation.
Design_ID
Lwl B T Cb Cm Cp
LCB
Aw Ax
Demi_offset
Disp Cost Rt_Low Rt_High
0
29.92
2.20 1.34 0.45 0.74 0.62 0.55 100.24 4.10 3.40 79.99 1.00 12.75 51.27
1
29.00
2.10 1.33 0.49 0.72 0.69 0.54 97.86 4.03 3.35 79.85 0.96 11.95 50.39
2
31.00
2.13 1.31 0.46 0.73 0.63 0.54 102.34 4.12 3.38 79.84 0.97 12.35 50.13
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
9775
36.00
2.09 1.19 0.50 0.74 0.71 0.54 104.21 4.15 3.4 84.65 1.03 14.33 47.63
J. Mar. Sci. Eng. 2024,12, 216 7 of 24
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 7 of 24
Table 3. Design dataset preparation.
Design_ID Lwl B T Cb Cm Cp LCB Aw Ax Demi_offset Disp Cost Rt_Low Rt_High
0 29.92 2.20 1.34 0.45 0.74 0.62 0.55 100.24 4.10 3.40 79.99 1.00 12.75 51.27
1 29.00 2.10 1.33 0.49 0.72 0.69 0.54 97.86 4.03 3.35 79.85 0.96 11.95 50.39
2 31.00 2.13 1.31 0.46 0.73 0.63 0.54 102.34 4.12 3.38 79.84 0.97 12.35 50.13
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
9775 36.00 2.09 1.19 0.50 0.74 0.71 0.54 104.21 4.15 3.4 84.65 1.03 14.33 47.63
(a) (b) (c)
(d)
Figure 3. Self-blending of section control points: (a) YZ waterline intersection, (b) YZ Buock line
intersection, (c) YZ diagonal intersection, (d) X direction section blending.
3.2. Machine Learning Training
As shown in Figure 2, the dataset collection based on geometry parameters input and
resistance output on different ship hullforms is the first step of the machine learning pro-
cess. The combination of geometry construction methods presented in the previous sec-
tion ensures that the dataset is diverse and representative of the range of hullforms. Re-
gression learners are applied to diverse design configurations, resulting in 9775 designs.
The pre-processing procedure reforms the database for the application of machine learn-
ing techniques. Linear normalization is implemented on each parameter according to
Equation (5):
Figure 3. Self-blending of section control points: (a) YZ waterline intersection, (b) YZ Buttock line
intersection, (c) YZ diagonal intersection, (d) X direction section blending.
3.2. Machine Learning Training
As shown in Figure 2, the dataset collection based on geometry parameters input
and resistance output on different ship hullforms is the first step of the machine learning
process. The combination of geometry construction methods presented in the previous
section ensures that the dataset is diverse and representative of the range of hullforms.
Regression learners are applied to diverse design configurations, resulting in 9775 designs.
The pre-processing procedure reforms the database for the application of machine learn-
ing techniques. Linear normalization is implemented on each parameter according to
Equation (5):
parameternormalized =parameterori ginal −parametermin.value
parametermax.v alue −parametermin.value
, (5)
Another step of preprocessing is using a principal component analysis technique
(PCA) and outlier detection using the Hotellings T2 test [
27
]. Selecting the outliers can be
useful for removing them from the dataset or for deeper investigation. Dimensionality
reduction is applied to the inputs to project data into a space of lower dimension while
preserving a maximum of information. The number of data reduces from 9775 to 8745
records according to PCA and outlier detection with a confidence interval of 0.05 [
28
,
29
].
J. Mar. Sci. Eng. 2024,12, 216 8 of 24
Finally, the database is randomly split into a learning set and test set, which contain 70%
and 30% of the records, respectively.
Regression trees (RTs), support vector machines (SVMs), and artificial neural network
(ANN) regression models are applied for dataset training based on nine predictors and
one response. The regression tree is a supervised learning algorithm with tree-structured
classification. There is a decision-related algorithm for each node based on the attributes.
Each step in a prediction involves checking the value of one predictor variable to determine
whether an attribute is larger than, smaller than, or equal to a value of the following
branch. The response value is contained in the last node, which is known as the leaf
node. The second supervised regression tool is linear epsilon-insensitive SVM regression.
This method disregards prediction errors that are less than some fixed hyperplane. Data
points include the support vectors that have errors larger than the admissible error of the
model. The function the SVM uses to predict new values depends only on the support
vectors to minimize the error. Box constraint, Epsilon value, and Kernel scale parameter
are set to automatic mode, and the application uses a heuristic procedure to select an
appropriate value.
The artificial neural network consists of interconnected neurons organized in layers.
An ANN algorithm works based on the human neuron system, which consists of a number
of layers, the kind of neural synapses, and the learning algorithm [
10
,
30
]. The artificial
neural network is herein applied to the dataset using multilayer feedforward networks.
Ship hull parameters define the first fully connected layer, and each subsequent layer has a
connection from the previous layer. The weight matrix multiplies to each fully connected
layer. Weight intensity iteratively changes aiming to decrease the final error. The number
of layers and their neurons are selected by the Bayesian optimizable algorithm [31].
Internal parameters of regression model can be chosen manually; however, the opti-
mized regression methods can select optimized internal values by using hyperparameter
optimization. Some of these options can strongly affect the regression method’s perfor-
mance. Accordingly, Optimizable Regression Tree, Optimizable SVM, and Optimizable
ANN methods are applied herein to automate the selection of hyperparameter values [
32
].
The Bayesian optimization technique has been used to tune hyperparameters in terms of
the mean squared error (MSE) as an objective function. Model evaluation is implemented
by statistical parameters and test datasets. The coefficient of model determination consists
of R-squared (R
2
), mean squared error (MSE), mean absolute error (MAE), and root mean
square error (RMSE):
R2=1−∑i(yi−xi)2
∑i(yi−y)2, (6)
y=1
n∑n
i=1yi, (7)
MSE =1
n∑n
i=1(yi−xi)2, (8)
MAE =1
n∑n
i=1|yi−xi|, (9)
RMSE =r1
n∑n
i=1(yi−xi)2, (10)
where
yi
is predicted resistance of the record i,
xi
is the calculated resistance from the
dataset, and n is the number of samples.
3.3. Genetic Algorithm Optimization
The Genetic Algorithm (GA) of the MATLAB optimization toolbox was used for di-
rect optimization using the obtained resistance predictor. The genetic algorithm (GA) is
an approach to optimization problems based on the mechanism of natural selection that
underlies biological evolution. This algorithm iteratively adjusts a collection of individual
J. Mar. Sci. Eng. 2024,12, 216 9 of 24
solutions. At each iteration, the genetic algorithm selects specific individuals from the
existing population to serve as parents, from which it generates offspring for the next
generation. In subsequent iterations, the population undergoes an “evolutionary” develop-
ment and converges to an optimal solution. The population size was selected as 40 with
100 Max Generations. Algorithm’s stopping or termination criterion is regulated via Max
Generations and Max Stall Generations. The variable “Max Generations” denotes the upper
limit of generations that process will go through during the optimization. The variable
“Max Stall Genera-tions” represents the maximum number of consecutive generations
in which there is no improvement in the best fitness value of the population. Max Stall
Generations and Function tolerance are set to 50 and 0.005, respectively, to determine the
stop condition. The algorithm terminates when the average relative change in the fitness
function value over Max Stall Generations becomes smaller than the Function tolerance.
In the GA solver domain, there is a different treatment of linear constraints and
bounds compared to nonlinear constraints. The optimization process ensures that all linear
constraints and bounds are consistently satisfied. Nevertheless, it is important to note that
the satisfaction of all nonlinear constraints may not be achieved in every generation during
the GA operation.
In the case that the GA converges to a solution, the resulting solution is guaranteed
to satisfy the requirements of the nonlinear constraints. In the genetic algorithm, three
primary rule categories are employed during each iteration to generate the subsequent
generation based on the existing population [33]:
■
Selection rules select the individuals, called parents, that contribute to the population
of the next generation. The selection depends on the individuals’ scores.
■Crossover rules combine two parents to form children for the next generation.
■Mutation rules apply random changes to individual parents to form children.
4. Results
Three regression models have been developed according to internal parameter selec-
tion to minimize the MSE value. The PCA dimensionality reduction reduces the number
of features from nine to six features. Table 4presents evaluation results of the model
performance and internal obtained parameters of regression models.
Table 4. Internal parameters of optimum regression models.
Optimizable Regression Tree
Optimizable Neural Network
Optimizable SVM
RMSE: 0.1043 RMSE: 0.03037 RMSE: 0.1168
R2: 0.98 R2: 1 R2: 0.97
MSE: 0.01088 MSE: 0.000922 MSE: 0.01365
MAE: 0.057334 MAE: 0.020429 MAE: 0.06614
Minimum leaf size: 3
Num. of layers: 2
Activation: Sigmoid
Lambda: 1.5276 ×10−8
First Layer size: 26
Second Layer size: 77
Box constraint: 17.0223
Kernel scale: 8.5763
Epsilon: 8.17 ×10−4
Kernel function: Gaussian
Regression evaluation results depict that the model developed using the artificial
neural networks algorithm has been fitted more suitable than other implemented models.
This model has an R-squared determination equal to 1, while the errors and dispersion mea-
surements are minimal. Figure 4illustrates the history of the MSE parameter minimization
for three applied methods. The dark blue point corresponds to observed minimum MSE,
and the light blue one represents the estimated minimum MSE. The number of iterations is
considered 30, with the best point of MSE value shown in red color.
J. Mar. Sci. Eng. 2024,12, 216 10 of 24
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 10 of 24
Lambda: 1.5276 × 10
−8
First Layer size: 26
Second Layer size: 77
Epsilon: 8.17 × 10
−4
Kernel function: Gaussian
Regression evaluation results depict that the model developed using the artificial
neural networks algorithm has been fied more suitable than other implemented models.
This model has an R-squared determination equal to 1, while the errors and dispersion
measurements are minimal. Figure 4 illustrates the history of the MSE parameter minimi-
zation for three applied methods. The dark blue point corresponds to observed minimum
MSE, and the light blue one represents the estimated minimum MSE. The number of iter-
ations is considered 30, with the best point of MSE value shown in red color.
(a)
(b)
Figure 4. Cont.
J. Mar. Sci. Eng. 2024,12, 216 11 of 24
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 11 of 24
(c)
Figure 4. MSE history reduction through the optimizable regression process: (a) Regression Tree,
(b) Regression SVM, (c) Regression ANN.
The response plot presented in Figure 5 shows the main and predicted responses
versus the record number. Additionally, predicted vs. actual and residual plots are shown
in Figure 6 for each regression model. These plots help us understand how well the re-
gression model makes predictions for different response values. It can be indicated that
the ANN method can predict responses close to the actual ones due to well-scaered sam-
ples along the diagonal line. Additionally, the residual plot depicts the difference between
the predicted and true responses, which can be interpreted as a clear distribution around
zero for the ANN regression method. In order to evaluate overfiing, 15% of samples were
applied during regression modelling, and the RMSE of the validation value under training
results is compared to the RMSE of the test value under test results, comprising 15% of
the total samples. The assessment of response plots and modelling summary represents
the appropriate performance of the ANN method against other implemented methods. In
addition, the test RMSE is higher than the validation RMSE, which indicates that this
model can be an appropriate resistance prediction model for the rest of the design study.
(a)
Figure 4. MSE history reduction through the optimizable regression process: (a) Regression Tree,
(b) Regression SVM, (c) Regression ANN.
The response plot presented in Figure 5shows the main and predicted responses
versus the record number. Additionally, predicted vs. actual and residual plots are shown
in Figure 6for each regression model. These plots help us understand how well the regres-
sion model makes predictions for different response values. It can be indicated that the
ANN method can predict responses close to the actual ones due to well-scattered samples
along the diagonal line. Additionally, the residual plot depicts the difference between the
predicted and true responses, which can be interpreted as a clear distribution around zero
for the ANN regression method. In order to evaluate overfitting, 15% of samples were
applied during regression modelling, and the RMSE of the validation value under training
results is compared to the RMSE of the test value under test results, comprising 15% of
the total samples. The assessment of response plots and modelling summary represents
the appropriate performance of the ANN method against other implemented methods.
In addition, the test RMSE is higher than the validation RMSE, which indicates that this
model can be an appropriate resistance prediction model for the rest of the design study.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 11 of 24
(c)
Figure 4. MSE history reduction through the optimizable regression process: (a) Regression Tree,
(b) Regression SVM, (c) Regression ANN.
The response plot presented in Figure 5 shows the main and predicted responses
versus the record number. Additionally, predicted vs. actual and residual plots are shown
in Figure 6 for each regression model. These plots help us understand how well the re-
gression model makes predictions for different response values. It can be indicated that
the ANN method can predict responses close to the actual ones due to well-scaered sam-
ples along the diagonal line. Additionally, the residual plot depicts the difference between
the predicted and true responses, which can be interpreted as a clear distribution around
zero for the ANN regression method. In order to evaluate overfiing, 15% of samples were
applied during regression modelling, and the RMSE of the validation value under training
results is compared to the RMSE of the test value under test results, comprising 15% of
the total samples. The assessment of response plots and modelling summary represents
the appropriate performance of the ANN method against other implemented methods. In
addition, the test RMSE is higher than the validation RMSE, which indicates that this
model can be an appropriate resistance prediction model for the rest of the design study.
(a)
Figure 5. Cont.
J. Mar. Sci. Eng. 2024,12, 216 12 of 24
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 12 of 24
(b)
(c)
Figure 5. Prediction vs. true design comparison through the optimizable regression process: (a) Re-
gression Tree, (b) Regression SVM, (c) Regression ANN.
(a)
Figure 5. Prediction vs. true design comparison through the optimizable regression process: (a) Re-
gression Tree, (b) Regression SVM, (c) Regression ANN.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 12 of 24
(b)
(c)
Figure 5. Prediction vs. true design comparison through the optimizable regression process: (a) Re-
gression Tree, (b) Regression SVM, (c) Regression ANN.
(a)
Figure 6. Cont.
J. Mar. Sci. Eng. 2024,12, 216 13 of 24
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 13 of 24
(b)
(c)
Figure 6. Residual plot comparison through the optimizable regression process: (a) Regression Tree,
(b) Regression SVM, (c) Regression ANN.
4.1. Regression Model Evaluation
4.1.1. Dataset Test Cases
A comparison is conducted between RT, SVM, and ANN methods for evaluating re-
sistance predictors. Two designs from the dataset have been selected randomly for evalu-
ation in this subsection. Figure 7a shows the results for a random hull in the hullform
series. In addition, Figure 7b depicts the results for a random catamaran hullform for the
85-ton series.
The proposed models fit well the observed data for test cases within the dataset.
However, small underestimate values can be indicated at speeds 15 to 18 knots. R-square
and RMSE values for Figure 7 (random design test model 1 and 2) are presented in Table
5. The artificial neural networks algorithm fits the observed data effectively according to
lower values of prediction parameters.
Table 5. Prediction parameters of the model test for dataset designs.
Test Model 1 Test Model 2
RT RMSE: 0.6051
R
2
: 0.9991
RMSE: 1.4895
R
2
: 0.9934
SVM RMSE: 0.3185 RMSE: 1.1625
Figure 6. Residual plot comparison through the optimizable regression process: (a) Regression Tree,
(b) Regression SVM, (c) Regression ANN.
4.1. Regression Model Evaluation
4.1.1. Dataset Test Cases
A comparison is conducted between RT, SVM, and ANN methods for evaluating
resistance predictors. Two designs from the dataset have been selected randomly for
evaluation in this subsection. Figure 7a shows the results for a random hull in the hullform
series. In addition, Figure 7b depicts the results for a random catamaran hullform for the
85-ton series.
The proposed models fit well the observed data for test cases within the dataset.
However, small underestimate values can be indicated at speeds 15 to 18 knots. R-square
and RMSE values for Figure 7(random design test model 1 and 2) are presented in Table 5.
The artificial neural networks algorithm fits the observed data effectively according to
lower values of prediction parameters.
J. Mar. Sci. Eng. 2024,12, 216 14 of 24
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R2: 0.9996 R2: 0.9971
ANN RMSE: 0.8083
R2: 0.9997
RMSE: 0.3606
R2: 0.9994
(a)
(b)
Figure 7. Model comparison between RT, SVM, and ANN methods for (a) random design Test
model 1 and (b) random design Test model 2.
4.1.2. Interpolation Test Cases
In this section, two interpolated designs based on ship tonnage have been imported
into regression models. Catamaran hullforms of 77.5 ton and 82.5 tons are designed for
regression model evaluation. Three implemented regression models are adjusted for the
77.5-ton hullform (Figure 8a) and the 82.5-ton hullform (Figure 8b). Regression data are
well adjusted using the ANN method for both designs according to predictor parameters
presented in Table 6. However, a slight difference can be observed at higher speeds of the
82.5-ton case, which is slightly superior.
10
15
20
25
30
35
40
45
50
55
11 12 13 14 15 16 17 18 19 20 21 22 23
Resistance [KN]
Speed [kn]
SBM results
Regression Tree
SVM
ANN
10
15
20
25
30
35
40
45
50
55
60
11 12 13 14 15 16 17 18 19 20 21 22 23
Resistance [KN]
Speed [kn]
SBM Results
Regression Tree
SVM
ANN
Figure 7. Model comparison between RT, SVM, and ANN methods for (a) random design Test
model 1 and (b) random design Test model 2.
Table 5. Prediction parameters of the model test for dataset designs.
Test Model 1 Test Model 2
RT RMSE: 0.6051
R2: 0.9991
RMSE: 1.4895
R2: 0.9934
SVM RMSE: 0.3185
R2: 0.9996
RMSE: 1.1625
R2: 0.9971
ANN RMSE: 0.8083
R2: 0.9997
RMSE: 0.3606
R2: 0.9994
4.1.2. Interpolation Test Cases
In this section, two interpolated designs based on ship tonnage have been imported
into regression models. Catamaran hullforms of 77.5 ton and 82.5 tons are designed for
regression model evaluation. Three implemented regression models are adjusted for the
77.5-ton hullform (Figure 8a) and the 82.5-ton hullform (Figure 8b). Regression data are
well adjusted using the ANN method for both designs according to predictor parameters
J. Mar. Sci. Eng. 2024,12, 216 15 of 24
presented in Table 6. However, a slight difference can be observed at higher speeds of the
82.5-ton case, which is slightly superior.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 15 of 24
Table 6. Prediction parameters of the model test for interpolation designs.
77.5 ton 82.5 ton
RT RMSE: 0.7597
R2: 0.9965
RMSE: 1.4029
R2: 0.9911
SVM RMSE: 1.0426
R2: 0.9976
RMSE: 1.4938
R2: 0.9906
ANN RMSE: 0.4677
R2: 0.9988
RMSE: 0.8633
R2: 0.9977
(a)
(b)
Figure 8. Model comparison between RT, SVM, and ANN methods for (a) interpolation design, 77.5
tons and (b) interpolation design, 82.5 tons.
4.1.3. Extrapolation Test Cases
Extrapolation designs define hullforms outside the displacement bound of the da-
taset. Considering the displacement of all designs from the dataset are designed between
75 to 85 tons. two catamaran hullforms of 71.5 tons and 88.5 tons are considered for re-
gression model evaluation. The purpose of the extrapolation test is the assessment of re-
gression models for out-of-boundary catamarans. Figure 9a,b shows resistance values
against speed for Slender Body Method results and fied regressions for the 71.5-ton
10
15
20
25
30
35
40
45
50
55
11 12 13 14 15 16 17 18 19 20 21 22 23
Resistance [KN]
Speed [kn]
SBM results
Regression Tree
SVM
ANN
10
15
20
25
30
35
40
45
50
55
11 12 13 14 15 16 17 18 19 20 21 22 23
Resistance [KN]
Speed [kn]
SBM results
Regression Tree
SVM
ANN
Figure 8. Model comparison between RT, SVM, and ANN methods for (a) interpolation design,
77.5 tons and (b) interpolation design, 82.5 tons.
Table 6. Prediction parameters of the model test for interpolation designs.
77.5 ton 82.5 ton
RT RMSE: 0.7597
R2: 0.9965
RMSE: 1.4029
R2: 0.9911
SVM RMSE: 1.0426
R2: 0.9976
RMSE: 1.4938
R2: 0.9906
ANN RMSE: 0.4677
R2: 0.9988
RMSE: 0.8633
R2: 0.9977
4.1.3. Extrapolation Test Cases
Extrapolation designs define hullforms outside the displacement bound of the dataset.
Considering the displacement of all designs from the dataset are designed between 75 to
85 tons. two catamaran hullforms of 71.5 tons and 88.5 tons are considered for regression
J. Mar. Sci. Eng. 2024,12, 216 16 of 24
model evaluation. The purpose of the extrapolation test is the assessment of regression
models for out-of-boundary catamarans. Figure 9a,b shows resistance values against speed
for Slender Body Method results and fitted regressions for the 71.5-ton design and 88.5-ton
design, respectively. In Figure 9a, all regression models estimate resistance higher than
actual values. On the contrary, the proposed models are inferior to SBM results in Figure 9b.
In the transition to high speeds, the models become less accurate. In addition, Table 7
presents prediction values of fitting quality, which depicts that regressions are more precise
in the lower displacement design than in the higher one.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 16 of 24
design and 88.5-ton design, respectively. In Figure 9a, all regression models estimate re-
sistance higher than actual values. On the contrary, the proposed models are inferior to
SBM results in Figure 9b. In the transition to high speeds, the models become less accurate.
In addition, Table 7 presents prediction values of fiing quality, which depicts that regres-
sions are more precise in the lower displacement design than in the higher one.
Table 7. Prediction parameters of the model test for extrapolation designs.
71.5 ton 88.5 ton
RT RMSE: 1.8147
R2: 0.9964
RMSE: 2.4631
R2: 0.9975
SVM RMSE: 1.6215
R2: 0.9965
RMSE: 2.7815
R2: 0.9975
ANN RMSE: 1.3860
R2: 0.9968
RMSE: 2.2180
R2: 0.9983
(a)
(b)
Figure 9. Model comparison between RT, SVM, and ANN methods for (a) extrapolation design, 71.5
tons and (b) extrapolation design 88.5, tons.
10
15
20
25
30
35
40
45
50
55
11 12 13 14 15 16 17 18 19 20 21 22 23
Resistance [KN]
Speed [kn]
SBM results
Regression Tree
SVM
ANN
10
15
20
25
30
35
40
45
50
55
60
11 12 13 14 15 16 17 18 19 20 21 22 23
Resistance [KN]
Speed [kn]
SBM results
Regression Tree
SVR
ANN
Figure 9. Model comparison between RT, SVM, and ANN methods for (a) extrapolation design,
71.5 tons and (b) extrapolation design 88.5, tons.
Table 7. Prediction parameters of the model test for extrapolation designs.
71.5 ton 88.5 ton
RT RMSE: 1.8147
R2: 0.9964
RMSE: 2.4631
R2: 0.9975
SVM RMSE: 1.6215
R2: 0.9965
RMSE: 2.7815
R2: 0.9975
ANN RMSE: 1.3860
R2: 0.9968
RMSE: 2.2180
R2: 0.9983
J. Mar. Sci. Eng. 2024,12, 216 17 of 24
4.2. Genetic Algorithm Optimization
The optimization process is conducted for a 40 m catamaran to obtain the best design
based on the defined cost function. The developed geometry reconstruction model offers
the designer the possibility to control/specify the main particulars of the demi hull along
with the hullform details within a reasonable range of variation of the defined design
variables, while at the same time, ensuring adequate quality of fairness and smoothness
of the hull. The Genetic algorithm parameters have been set up based on settings in
Section 3.3.
Figure 10 illustrates a three-dimensional perspective of the TrAM catamaran hull, rep-
resenting the initial hullform for the design optimization procedure. The design parameters
for this investigation are presented in Table 8, along with their corresponding ranges.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 17 of 24
4.2. Genetic Algorithm Optimization
The optimization process is conducted for a 40 m catamaran to obtain the best design
based on the defined cost function. The developed geometry reconstruction model offers
the designer the possibility to control/specify the main particulars of the demi hull along
with the hullform details within a reasonable range of variation of the defined design vari-
ables, while at the same time, ensuring adequate quality of fairness and smoothness of the
hull. The Genetic algorithm parameters have been set up based on settings in Section 3.3.
Figure 10 illustrates a three-dimensional perspective of the TrAM catamaran hull,
representing the initial hullform for the design optimization procedure. The design pa-
rameters for this investigation are presented in Table 8, along with their corresponding
ranges.
Figure 10. 3D view of the passenger catamaran.
Table 8. Design variables of the study.
Optimization Design Variables Symbol Min Bound Max Bound
Waterline length (m) Lwl 38 41
Demi hull Beam (m) B 1.9 2.1
Draft (m) T 1.1 1.15
Demi hull transverse distance (m) DT 3.3 3.7
Block coefficient Cb 0.49 0.53
Max section area coefficien
t
Cm 0.72 0.77
Prismatic coefficient Cp 0.66 0.71
Longitudinal Center of Buoyancy (% of L) LCB 0.51 0.56
Objectives & Constraints Symbol Min bound Max bound
Resistance at two Ship speeds [kn] V 12 22
Ship Light Weight (ton) W 62.23
Ship Displacement (ton) ∆ 90 ± 1%
After 51 iterations the optimization terminates. The Genetic Algorithm convergence
curve of the current study is shown in Figure 11. The best fit is the best design of each
generation, and the mean fit is the average cost function of the population in each gener-
ation. Accordingly, 51 × 50 = 2550 designs have been generated in the GA optimization
process by using a fast approach ML resistance predictor, which reduces computation cost
to a few minutes.
Figure 10. 3D view of the passenger catamaran.
Table 8. Design variables of the study.
Optimization Design Variables Symbol Min Bound Max Bound
Waterline length (m) Lwl 38 41
Demi hull Beam (m) B 1.9 2.1
Draft (m) T 1.1 1.15
Demi hull transverse distance (m) DT 3.3 3.7
Block coefficient Cb 0.49 0.53
Max section area coefficient Cm 0.72 0.77
Prismatic coefficient Cp 0.66 0.71
Longitudinal Center of Buoyancy (% of L)
LCB 0.51 0.56
Objectives & Constraints Symbol Min bound Max bound
Resistance at two Ship speeds [kn] V 12 22
Ship Light Weight (ton) W 62.23
Ship Displacement (ton) ∆90 ±1%
After 51 iterations the optimization terminates. The Genetic Algorithm convergence
curve of the current study is shown in Figure 11. The best fit is the best design of each
generation, and the mean fit is the average cost function of the population in each generation.
Accordingly, 51
×
50 = 2550 designs have been generated in the GA optimization process
by using a fast approach ML resistance predictor, which reduces computation cost to a
few minutes.
J. Mar. Sci. Eng. 2024,12, 216 18 of 24
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 18 of 24
Figure 11. Genetic Algorithm convergence curve.
The resistance value of the baseline design at a speed of 12 knots is 14.97 kN, and at
speed 22 knots, it is 48.228 kN. According to Figure 12, the results of the optimization
process on a 40 m passenger catamaran achieved a 12.2% resistance reduction at cruise
speed and a 7.1% resistance reduction at sprint speed. In this Figure, each asterisk point
represent best design of each generation during GA optimization process. By using Ge-
netic Algorithm on a sample catamaran ship, it resulted in a 9.5% cost function improve-
ment. Therefore, one can conclude that the developed in-house resistance predictor, com-
bined with hullform optimization software, provides a superior and cost-effective solu-
tion for ship design. Resistance and cost function values and their corresponding improve-
ments are presented in Table 9.
Figure 12. Genetic Algorithm best design distribution of 51 generations.
Table 9. GA optimization results.
Design Study Symbol Original Catamaran
Resistance [kN]
Optimized Catamaran
Resistance [kN] Improvement %
Genetic Algorithm 𝑅𝑡 [kN] 14.97 13.183 12.20
𝑅𝑡 [kN] 48.228 44.84 7.10
CF 1 0.905 9.5
CF: Cost Function.
Figure 11. Genetic Algorithm convergence curve.
The resistance value of the baseline design at a speed of 12 knots is 14.97 kN, and
at speed 22 knots, it is 48.228 kN. According to Figure 12, the results of the optimization
process on a 40 m passenger catamaran achieved a 12.2% resistance reduction at cruise
speed and a 7.1% resistance reduction at sprint speed. In this Figure, each asterisk point
represent best design of each generation during GA optimization process. By using Genetic
Algorithm on a sample catamaran ship, it resulted in a 9.5% cost function improvement.
Therefore, one can conclude that the developed in-house resistance predictor, combined
with hullform optimization software, provides a superior and cost-effective solution for
ship design. Resistance and cost function values and their corresponding improvements
are presented in Table 9.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 18 of 24
Figure 11. Genetic Algorithm convergence curve.
The resistance value of the baseline design at a speed of 12 knots is 14.97 kN, and at
speed 22 knots, it is 48.228 kN. According to Figure 12, the results of the optimization
process on a 40 m passenger catamaran achieved a 12.2% resistance reduction at cruise
speed and a 7.1% resistance reduction at sprint speed. In this Figure, each asterisk point
represent best design of each generation during GA optimization process. By using Ge-
netic Algorithm on a sample catamaran ship, it resulted in a 9.5% cost function improve-
ment. Therefore, one can conclude that the developed in-house resistance predictor, com-
bined with hullform optimization software, provides a superior and cost-effective solu-
tion for ship design. Resistance and cost function values and their corresponding improve-
ments are presented in Table 9.
Figure 12. Genetic Algorithm best design distribution of 51 generations.
Table 9. GA optimization results.
Design Study Symbol Original Catamaran
Resistance [kN]
Optimized Catamaran
Resistance [kN] Improvement %
Genetic Algorithm 𝑅𝑡 [kN] 14.97 13.183 12.20
𝑅𝑡 [kN] 48.228 44.84 7.10
CF 1 0.905 9.5
CF: Cost Function.
Figure 12. Genetic Algorithm best design distribution of 51 generations.
Table 9. GA optimization results.
Design Study Symbol Original Catamaran
Resistance [kN]
Optimized Catamaran
Resistance [kN] Improvement %
Genetic Algorithm
RtLow [kN] 14.97 13.183 12.20
RtH igh [kN] 48.228 44.84 7.10
CF 1 0.905 9.5
CF: Cost Function.
J. Mar. Sci. Eng. 2024,12, 216 19 of 24
The final optimized catamaran from GA optimization has been obtained. A compar-
ison of the initial and optimized values of ship attributes is illustrated in Table 10. It is
indicated that the lengthening of the ship improves resistance and block coefficient, and the
midship coefficient was reduced in the optimized design. The value for LCB is decreased,
which can be inferred that a backward longitudinal displacement of the center of buoyancy
position leads to a beneficial impact. Contrary to the fact, the demi hull distance was
reduced from 3.5 to 3.326; one may conclude that the distance between demi hulls depends
on the general configurator and ship hullform.
Table 10. Principal dimensions of the initial and optimized catamaran hull.
Ship Principal Parameters Symbol Initial Design Optimized Design
Waterline Length (m) Lwl 39.80 41.0
Total demi hull Beam (m) B 2.000 2.078
Draft (m) T 1.100 1.110
Block coefficient Cb 0.515 0.504
Midship coefficient Cm 0.754 0.721
Prismatic coefficient Cp 0.683 0.699
Demi Hull Distance (m) DT 3.500 3.326
Longitudinal Centre of
Buoyancy (m) LCB (% of L) 46.59 44.52
Total breath (m) (DT ×2) + B 9.00 8.73
Displacement (ton) ∇90.00 89.23
Cp=Cb
Cm.
Figure 13 displays the body plan, and Figure 14 shows a perspective view of the initial
and optimized hull. Evidently, it can be noticed that the length increases in the bow region
of the optimized design. The obtained optimization results illustrate that the forward
movement of the start point of the stern bottom improves hydrodynamic performance. The
position of the profile view in the bow portion has been elevated, resulting in an alteration
of the bow wave pattern.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 19 of 24
The final optimized catamaran from GA optimization has been obtained. A compar-
ison of the initial and optimized values of ship aributes is illustrated in Table 10. It is
indicated that the lengthening of the ship improves resistance and block coefficient, and
the midship coefficient was reduced in the optimized design. The value for LCB is de-
creased, which can be inferred that a backward longitudinal displacement of the center of
buoyancy position leads to a beneficial impact. Contrary to the fact, the demi hull distance
was reduced from 3.5 to 3.326; one may conclude that the distance between demi hulls
depends on the general configurator and ship hullform.
Table 10. Principal dimensions of the initial and optimized catamaran hull.
Ship Principal Parameters Symbol Initial Design Optimized Design
Waterline Length (m) Lwl 39.80 41.0
Total demi hull Beam (m) B 2.000 2.078
Draft (m) T 1.100 1.110
Block coefficient Cb 0.515 0.504
Midship coefficient Cm 0.754 0.721
Prismatic coefficient Cp 0.683 0.699
Demi Hull Distance (m) DT 3.500 3.326
Longitudinal Centre of
Buoyancy (m) LCB (% of L) 46.59 44.52
Total breath (m) (DT × 2) + B 9.00 8.73
Displacement (ton) ∇ 90.00 89.23
𝐶=
.
Figure 13 displays the body plan, and Figure 14 shows a perspective view of the ini-
tial and optimized hull. Evidently, it can be noticed that the length increases in the bow
region of the optimized design. The obtained optimization results illustrate that the for-
ward movement of the start point of the stern boom improves hydrodynamic perfor-
mance. The position of the profile view in the bow portion has been elevated, resulting in
an alteration of the bow wave paern.
Figure 13. Body plan comparison of the initial and optimized hulls.
Figure 13. Body plan comparison of the initial and optimized hulls.
J. Mar. Sci. Eng. 2024,12, 216 20 of 24
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 20 of 24
Figure 14. 3D view comparison of the initial and optimized hulls.
Figure 15a illustrates the resistance plot corresponding to the ships velocity for both
the initial and optimized hull configurations. Moreover, the graph displays the overall
resistance, as well as the coefficients of wave-making and viscous resistance in Figure 15b,
Figure 15c, and Figure 15d, respectively. These plots depict lower resistance and its com-
ponent at different speeds except at speeds between 12.7 knots to 14 knots. The lower
value of the wave-making resistance coefficient at hollow and hump regions can be con-
cluded in Figure 15c, and these phenomena occur at speeds higher than initial design
speeds.
(a) (b)
Figure 14. 3D view comparison of the initial and optimized hulls.
Figure 15a illustrates the resistance plot corresponding to the ship’s velocity for both
the initial and optimized hull configurations. Moreover, the graph displays the overall
resistance, as well as the coefficients of wave-making and viscous resistance in Figure 15b,
Figure 15c, and Figure 15d, respectively. These plots depict lower resistance and its
component at different speeds except at speeds between 12.7 knots to 14 knots. The
lower value of the wave-making resistance coefficient at hollow and hump regions can
be concluded in Figure 15c, and these phenomena occur at speeds higher than initial
design speeds.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 20 of 24
Figure 14. 3D view comparison of the initial and optimized hulls.
Figure 15a illustrates the resistance plot corresponding to the ships velocity for both
the initial and optimized hull configurations. Moreover, the graph displays the overall
resistance, as well as the coefficients of wave-making and viscous resistance in Figure 15b,
Figure 15c, and Figure 15d, respectively. These plots depict lower resistance and its com-
ponent at different speeds except at speeds between 12.7 knots to 14 knots. The lower
value of the wave-making resistance coefficient at hollow and hump regions can be con-
cluded in Figure 15c, and these phenomena occur at speeds higher than initial design
speeds.
(a) (b)
Figure 15. Cont.
J. Mar. Sci. Eng. 2024,12, 216 21 of 24
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(c) (d)
Figure 15. Comparison of total resistance (a), total resistance coefficient (b), wave-making resistance
coefficient (c), and viscous resistance coefficient (d) at different speeds between the initial and opti-
mized design.
As depicted in Figure 16a for the wave paern surrounding the initial vessel and
Figure 16b for the optimized vessel travelling at 12 knots, the calculated wave height
around the hulls reflects the decrease in resistance. The optimized hull design results in
diminished wave heights at the specified design velocities. The predominant alterations
are enacted in the bow shape and the boom of the stern region, leading to a decrease in
wave heights in the middle and aft portions of the hull, as illustrated in Figure 16.
(a) Initial design at 12 kn (b) Optimized design at 12 kn
(c) Initial design at 22 kn (d) Optimized design at 22 kn
Figure 15. Comparison of total resistance (a), total resistance coefficient (b), wave-making resistance
coefficient (c), and viscous resistance coefficient (d) at different speeds between the initial and
optimized design.
As depicted in Figure 16a for the wave pattern surrounding the initial vessel and
Figure 16b for the optimized vessel travelling at 12 knots, the calculated wave height
around the hulls reflects the decrease in resistance. The optimized hull design results in
diminished wave heights at the specified design velocities. The predominant alterations
are enacted in the bow shape and the bottom of the stern region, leading to a decrease in
wave heights in the middle and aft portions of the hull, as illustrated in Figure 16.
J. Mar. Sci. Eng. 2024, 12, x FOR PEER REVIEW 21 of 24
(c) (d)
Figure 15. Comparison of total resistance (a), total resistance coefficient (b), wave-making resistance
coefficient (c), and viscous resistance coefficient (d) at different speeds between the initial and opti-
mized design.
As depicted in Figure 16a for the wave paern surrounding the initial vessel and
Figure 16b for the optimized vessel travelling at 12 knots, the calculated wave height
around the hulls reflects the decrease in resistance. The optimized hull design results in
diminished wave heights at the specified design velocities. The predominant alterations
are enacted in the bow shape and the boom of the stern region, leading to a decrease in
wave heights in the middle and aft portions of the hull, as illustrated in Figure 16.
(a) Initial design at 12 kn (b) Optimized design at 12 kn
(c) Initial design at 22 kn (d) Optimized design at 22 kn
Figure 16. Comparison of wave pattern at (a) Initial design at 12 knots, (b) Optimized design at
12 knots, (c) Initial design at 22 knots, (d) Optimized design at 22 knots.
J. Mar. Sci. Eng. 2024,12, 216 22 of 24
5. Conclusions
A systematic series of novel catamaran ships has been developed for two types of
passenger and freight boats. Three different ship tonnages: 75, 80, and 85 tons, are con-
sidered to produce new designs. A shift transformation and self-blending method are
sequentially applied to generate different hullforms. Three different supervised machine
learning methods have been applied to the generated dataset of catamarans to predict
resistance at different ship speeds. Accordingly, 9775 catamaran hullforms have been
produced to create a vast optional condition for ship owners and provide this dataset for
a machine learning resistance prediction model. Using machine learning algorithms, it
is worth developing a continuous total resistance predictor well-fitted to the database of
ship series. A highly significant concern is the huge time and cost of the optimization
process, which herein we develop a machine learning resistance predictor that facilitates
Genetic Algorithm optimization in an expedited manner. Three regression algorithms:
Regression Tree, Support Vector Machine, and Artificial Neural Network approaches, are
applied to the dataset. Regression estimation has good compliance with results of the SBM
method at a wide range of speeds. However, RT and SVM methods have some differences
in higher speed. The ANN approach depicts well-adjusted regression on the data. The
validation of fitting methods was evaluated by case test of the dataset, interpolation, and
extrapolation of catamarans. Accordingly, a general and unique tool is proposed to pre-
dict the resistance of the series at different displacements and hullforms. The proposed
model is a valuable tool to assess the resistance of catamaran hulls during the early design
stages. Finally, a sophisticated ANN model is proposed by exploring different features and
training/optimization algorithms. A direct optimization algorithm (Genetic Algorithm)
is applied for the optimization study. Waterline length, Demi hull breadth, ship draft,
Demi hull offset distance, block coefficient, midship coefficient, prismatic coefficient, and
longitudinal centre of buoyancy are design variables of the optimization study, considering
total width and ship displacement as constraints of optimization. Total resistance at cruise
and sprint speed and its light weight are the objectives of this study. The optimization
process was conducted on a 40 m passenger catamaran that achieved a 12.2% resistance
reduction at cruise speed and a 7.1% resistance reduction at sprint speed. The best approach
is the Genetic Algorithm, which results in the highest resistance reduction, a 9.5% cost
function improvement. For the optimized configuration, a reduction in wave amplitudes
was observed at different design speeds, indicating the effects of the changes to the bow
shape and lower stern area. Hence, it can be inferred that utilizing an internally developed
resistance predictor in conjunction with hullform optimization software offers a superior
and cost-efficient solution for ship design. The current approach can be extended to include
additional objectives such as stability, seakeeping, and general arrangement. In addition,
some non-linear design blending can be applied to hullform development to diversify the
geometry of the ship, which can be carried out for future works.
Author Contributions: Conceptualization, A.N. and M.Z.A.; methodology, A.N.; software, A.N. and
M.Z.A.; validation, A.N.; formal analysis, A.N.; investigation, A.N. and M.Z.A.; writing—original
draft preparation, A.N., M.Z.A. and E.B.; supervision, E.B. All authors have read and agreed to the
published version of the manuscript.
Funding: The TrAM project has received funding from the European Union’s Horizon 2020 research
and innovation program under grant agreement No 769303. https://tramproject.eu/ (accessed on
23 January 2024).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are contained within the article.
Acknowledgments: Maritime Safety Research Centre (MSRC) at Strathclyde University is an industry–
University partnership involving Strathclyde’s Department of Naval Architecture, Ocean & Marine
J. Mar. Sci. Eng. 2024,12, 216 23 of 24
Engineering, and sponsors of the Royal Caribbean Group and DNV Classification Society. The
opinions expressed herein are those of the authors and do not reflect the views of DNV and RCG.
Conflicts of Interest: The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported in this paper.
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