Simulation-based multi-objective optimization of side-hull arrangement applied to an inverted-bow trimaran ship at cruise and sprint speeds

Numerical optimization of an inverted-bow trimaran is carried out through three simulation-based design (SBD) frameworks. Different positions of the trimaran’s side hull are investigated based on a computational fluid dynamics solver using the non-dominated sorting genetic algorithm-III (NSGA-III), simultaneous hybrid exploration that is robust, progressive and adaptive (SHERPA) and response surface (RS) multi-objective optimization for resistance at cruise and sprint speeds. The aims are to create and develop a convenient tool for optimization and investigate the appropriate position of the side hull. An automated, low-cost optimization platform is achieved that can be implemented in other maritime projects. A 10.5% drag reduction for cruise speed and 6.6% reduction for sprint are obtained, corresponding to lower longitudinal and large transversal distances of the side hull. SHERPA and NSGA-III produce the same results, but SHERPA is 2.5 times faster than NSGA-III. RS obtains less desirable results, but in the lowest central processing unit time.


Introduction
Trimaran vessels are a novel body form which, owing to their advantages over equivalent monohulls, have attracted the attention of many designers. Some of the most important advantages of this type of vessel are: lower resistance and consequently fuel consumption, especially at sprint speed; higher stability and seagoing performance; and larger deck area (Ghadimi, Nazemian, and Ghadimi 2019;Elcin 2003). The advantages of trimaran hull forms offer signature capability for warship designers. The design process of two-speed ships is more crucial and complicated. Therefore, design and optimization of trimaran ships leads to a multi-objective process. The combination of simulation-based design (SBD) and ship design has led to the definition of several optimization problems. However, for industrial applications, a low-cost optimization platform with appropriate accuracy is more popular.
In the present study, an optimized trimaran hull form based on the side-hull arrangement has been developed, and a consequence of this evolution is the introduction of a new hydrodynamic optimization process. The optimization process of a marine vehicle is related to flow around the vehicle. This has led to extensive research on the analysis of flow characteristics over the past few years (Nazemian and Ghadimi 2021a). Ackers et al. (1997) designed a body shape for a trimaran and tested it under different conditions, including different arrangements for the side hull and different displacements. Pattison and Zhang (1995) compared the resistance characteristics of trimarans with monohulls. They studied different concept designs of trimarans. However, different parametric investigations of the hull form have been manually performed without using a mathematical exploration approach. Javanmardi et al. (2008) computed the resistance and manoeuvring of a Wigley form trimaran hull using a computational fluid dynamics (CFD) code. The results showed that as the speed increases, the rate of total resistance growth of the aligned configuration of the trimaran bodies decreases. Research has also been conducted on the resistance of a trimaran and monohull at different speeds, and comparison of their performance in shallow waters (Lyakhovitsky 2007;Gligorov and Hofman 2001). Xu and Zou (2001) obtained the optimal position of trimaran outriggers to minimize wave-making resistance. In studying the wave-making resistance, other than the body shape, which causes the distortions in streamlines, another factor is the interference between the bodies. As a result, if the waves from different bodies are weakened by each other, the energy expended by the ship decreases and, consequently, the wave-making resistance decreases. In the field of resistance prediction, CFD solvers have appropriate capability for the simulation of fluid flow around the hulls. Most of the abovementioned research has been conducted through potential-based approaches that cannot predict the interactions of the hulls, changing the flow acceleration between hulls and turbulent transitions. Brizzolara, Bruzzone, and Tincani (2005) examined the effect of interference between the bodies on different configurations of the bodies. Yanuar et al. (2013) compared the effects of longitudinal and transverse side-hull arrangement and the interference of the flow waves between the bodies for a trimaran ship.
Based on the presented literature review, it is clear that most of the conducted research has focused on limited parametric studies because the research is time-consuming and expensive. Therefore, the need for an automated, fast and convenient tool for side-hull design of trimaran hulls is apparent. A complete evaluation of the side-hull arrangement requires multiple disciplines that should be integrated into an optimization platform. The multi-disciplinary design process involves parametric definition of the geometry, numerical simulation of flow around the geometry and an optimization algorithm Cui, Ölçer, and Turan 2009). CFD solvers predict the calm water resistance more accurately than other analytical or potential-based resistance calculation methods (Campana et al. 2009;Nazemian and Ghadimi 2021b). A CFD solver is an appropriate tool for the simulation of fluid flow around the trimaran hull, by considering interactions between the main hull and side hulls. Researchers have developed different software applications based on platform integrators related to the maritime industry. Abt et al. (2003) developed a hull form optimizer by combining the computer-aided design (CAD) platform FRIENDSHIP-Framework with the external solver. CAESES software has become a pervasive optimization tool. Tahara, Tohyama, and Katsui (2006) combined NAPA software with a CFD solver to optimize a container ship. They optimized the bow shape of a commercial ship with a bulbous bow through an SBD approach. A comprehensive study on geometry modification approaches was conducted by Brizzolara et al. (2015). They compared the full parametric approach and the free form deformation approach. Harries, Abt, and Hochkirch (2004) and Abt (2007) and Harries (1998) made other attempts at ship parameterization. They started from a set of form parameters and generated a new hull by setting up geometric entities that were associated with those parameters. Han, Lee, and Choi (2012) described how to develop hydrodynamic optimization of hull forms using parametric models. Vasudev, Sharma, and Bhattacharyya (2014) studied ship hull optimization using a multi-objective optimization platform. They applied two integrated SHIPFLOW software programs and the non-dominated sorting genetic algorithm-III (NSGA-II) for geometric optimization. Grigoropoulos and Chalkias (2010) integrated the CAD software Friendship-Modeler with the CFD software SHIPFLOW and studied ship form optimization. Gosain, Sharma, and Kim (2017) proposed a design optimization tool based on an evolutionary global optimization algorithm. They presented a semi-submersible model as a case study to validate the applicability of the design study.
Based on the surveyed literature, a side-hull arrangement improvement has not previously been applied for a wave-piercing bow trimaran hull. The basic aim of the present article is to develop a comprehensive optimization platform for multi-hull vessels. In addition, the applicability and validity of different optimization methods are investigated. Accordingly, different search-based global optimization algorithms are implemented for the capability assessment of optimization platforms. Using a search-based optimization algorithm requires an automated optimization cycle to control the design variables and obtain the final optimum values. Thus, the automation optimization approach without user intervention or user errors is the first achievement of the present article.
In the current study, the design variables and optimization response are defined in STAR-CCM+ software. The three defined variables are the longitudinal (stagger), transverse (clearance) and vertical distance of the side hull with respect to the centre hull. Ship resistance and its components are calculated by applying the Reynolds-averaged Navier-Stokes (RANS) solver. The third part of the optimization cycle comprises optimization algorithms. Two essential traits that need to be considered when choosing the best algorithm are the ease of implementation and the capability for global exploration. Many different optimization algorithms have been applied to hydrodynamic problems in recent years Serani et al. 2016;Chen et al. 2015). The present article offers an automated and low-cost method for improving the side-hull configuration of a proper trimaran hull form, which could be extended to other maritime projects. The optimization platform and investigation of widely used optimization algorithms accomplish this task with considerable savings in terms of time and cost.
Global optimization is performed through the multi-objective non-dominated sorting genetic algorithm-III (NSGA-III) algorithm, simultaneous hybrid exploration that is robust, progressive and adaptive (SHERPA) algorithm and response surface (RS) global optimization method to ensure that the global optimum point is found. These optimizers are applied after running a design of experiments (DoE) model for the system to investigate the accuracy and capability of the CFD solver and system behaviour. Accordingly, a two-level full factorial method is proposed to distribute the initial trial samples that represent the design space. The applied optimization tools are compared and introduced as an efficient and fast convergence application. To accomplish this task, a CAD-based translation is first applied for geometric parameterization. Secondly, initial CFD simulation is computed at the ship's cruise and sprint speeds. Then, three optimization cycles are employed to find the optimum design. The entire process automation is managed by HEEDS software package. Finally, the results of hull shape optimization are presented and analysed. The initial and optimized hulls are compared, and the optimization frameworks and their effectiveness are discussed.

Problem definition
The goal of the present article is to apply an optimization process to optimize the side-hull arrangement of a trimaran for drag reduction. The targeted objective is total resistance, but since optimization is executed at two different speeds, which presents two responses, the examined problem becomes a multi-objective problem ( in Equation 1). The present hull form optimization problem can be defined as follows: where f is the objective function, which is total resistance for the cruise speed and sprint speed. S ⊆ R N is the set of feasible solutions, while the feasible design space is limited by the constraint. X represents the vector of design variables, defined in terms of the parameters associated with the longitudinal, transverse and vertical distances of the side hull. A three-dimensional view of the trimaran and the Z-direction of the side-hull movement are shown in Figure 1. Design variables involved in the optimization process for the X-and Y-directions are shown in Figures 2 and 3. Shape modification and changes in the ship's hull are accomplished by the constraint in displacement of less than 1%.  Equation (2) presents the variation in ship's tonnage between the new and original hulls: In this study, a model of wave-piercing bow trimaran ships is studied. The dimensional characteristics of this ship and model (λ = 40) are shown in Table 1. The ship's bow has a wave-piercing form, and the lateral bodies are made in the form of a Wigley hull, as investigated by Akbari, Khedmati, and Seif (2014).
The definition and range of design variables are defined in Table 2. The longitudinal distance (Dx) represents the distance between the transom of the middle hull and the transom of the side hull from the vessel's stern, which is known as stagger, while the transversal distance (Dy) represents the distance between the centreline of the middle hull and the centreline of the side hull, which is known as clearance. Based on the general arrangement constraints in designing the spiral of the trimaran, ranges of design variables are defined. Side hulls are in the aft region because a vast deck area is established and the helideck is arranged in that area. In addition, the increasing transversal distance of the side hull causes a significant increase in the weight of the ship's hull structure. Some approximations of the weight distribution are considered in hydrostatic calculations for each generated hull because of the variation in the side-hull position. Accordingly, numerical simulation set-ups and ship attitudes are checked at the beginning of each CFD run during the optimization process.

Optimization framework
In this section, the overall optimization framework is introduced, followed by a description of its modules: design study, numerical simulation set-up and optimization algorithm. Total resistance is calculated by the RANSE-based CFD solver. Numerical simulation is performed using STAR-CCM+ software and three optimization methods are applied using HEEDS MDO software. Figure 4 shows the automated optimization process by the multi-disciplinary connection approach. The present optimization platform consists of four components, two of which are performed by STAR-CCM+ software and the other two by the HEEDS MDO package: • First, a hull geometry modelling and modification technique provides the necessary link between design variables (and their variations) and the geometric reconstruction. A non-uniform rational B-splines (NURBS) format (IGES file) of the ship model is imported to and exported from the CAD environment of STAR-CCM+ software. • Secondly, geometry generated from the CAD-based hull modification is introduced to the CFD solver to perform automated meshing and evaluate the objective(s). • Thirdly, an optimization technique is applied that can be used to minimize the objective function(s) under given constraints. • Finally, the design optimization framework is set up by integrating the above three components and holding the optimization loop.
To deploy an optimization algorithm, an automated process should be carried out. The automated process allows intensive design exploration to meet performance targets. Achieving a successful optimization, as well as its accuracy and computational time, depend on the performance of the optimizer, which is investigated in the remainder of this article.
No optimization method works best on all classes of engineering design problems because all of them have some limitations. Some methods work effectively when the design space is smooth and convex, and some methods work only with continuous or with discrete variables (but not both) and for relatively few variables. Some methods are inefficient, requiring a relatively large number of design evaluations to find an optimum. To choose the best method, the characteristics of the design space must be investigated. The following guideline is an efficient and useful approach in marine research on design optimization.

First step: understand and investigate
Design samples based on DoE techniques are generated to explore the sensitivity of and correlations between variables. This helps in understanding the behaviour of the system and finding out which variables can be omitted in the next step (to save computational time). The researcher should check whether there is a trend in database. Does it make sense from a physical point of view, or is there still possibly an error in the numerical solver? Is it a meshing or an analysis set-up? Sometimes large changes in the geometry or mesh deformation cause fatal errors or wrong outputs. For example, eight (2 3 = 8) design samples are generated in this study based on a two-level full factorial method, which is described in Section 2.3.

Second step: construct a surrogate model
The optimization process can be accelerated by surrogate model implementation. Surrogate models are intended to describe the relationships between the optimization target and adopted variables. The response of the system can be calculated from the model rather than through a full design evaluation, which can save a tremendous amount of time. Common types of surrogate models include the radial basis function (RBF) and kriging. Surrogate or metamodel construction can be accomplished simultaneously with the optimization process.

Third step: parameter optimization
Another method in optimization science is direct optimization. This means omitting the first and second steps and starting parameter optimization directly. Depending on how much time is devoted to the optimization phase of a project, parameter optimization and selection of the method can be different. There are five optimization methods (HEEDS MDO User Guide 2019; STAR-CCM+ User Guide 2020): • Local optimization: Local optimization methods are accurate and fast for local and single-objective optimization problems. The chance of achieving the global minimum is low. Local optimization techniques include gradient-based (gradient descent, Newton Raphson) and gradient-free methods (TSearch, Nelder-Mead simplex). • Local optimization multi-start: This strategy runs several local optimizations from different starting points at the same time. When a design space has numerous local minima, this method is more effective than the other local methods. • Global optimization: This is an appropriate method for multi-objective problems, although the number of evaluations is large and the simulation time is very long. It is therefore recommended that one learns as much as possible beforehand, with a minimum of expensive simulation runs. Examples of global optimization methods for multi-objective problems are NSGA-III, the multiobjective genetic algorithm and multi-objective simulated annealing. • Global optimization on response surface: This method is an adaptive interaction between a searchbased optimization method and a response surface method. The best design of the direct optimizer adds to the design points of the response surface, as in the kriging method, which is iteratively built up. A segment of the population is simulated by kriging evaluations to speed up the convergence until the best Pareto front is determined. A significant advantage of this combination is the reduced number of design evaluations that are necessary for the optimization; as a result, it can potentially save a lot of time. • SHERPA: The SHERPA optimization algorithm is a novel search-based hybrid and adaptive algorithm method that chooses the best attributes of each search method proposed by STAR-CCM+ (STAR-CCM+ User Guide 2020). This method is selected as a suitable optimization method during the process and needs lower design evaluation to reach the optimum design.
In the current study, three optimization methods are adopted and used for the trimaran hull optimization project: the NSGA-III, SHERPA algorithm and RS method. The results obtained from each of these methods are evaluated to ascertain the most efficient method. The important advantage of the proposed platform is that it saves the previous run history for the new initialization. This capability yields a faster convergence in every run, except for the first run (STAR-CCM+ User Guide 2020).

Numerical set-up
This article uses the RANS equation solver in STAR-CCM+ to simulate the trimaran hull form at a constant-velocity uniform flow. The resistance computation is carried out for the cruise speed of 16 knot (1.3 m/s for the model) and sprint speed of 25 knot (2.0333 m/s for the model) for a 3.1 m long trimaran model. The unsteady scheme with a physical time step of 0.01 s is used for temporal discretization. The applied turbulence model for the current study is a standard k-model, which has been extensively used for industrial applications and similar studies (STAR-CCM+ User Guide 2020; ITTC Recommendations 2011).
Following the International Towing Tank Conference (ITTC) recommendations as a guideline (ITTC Recommendations 2011, the dimensions of the computational domain are selected based on the ship's length. Figure 5 shows the computational fluid domain and location of the trimaran model and its boundaries. The volume of fraction (VOF) wave damping functionality is applied for the surrounding boundaries. The prism layers and additional wall functions are applied to resolve the boundary layer at the hull. The space of the first grid from the hull is y + ≈ 25. The ship's hull surfaces are defined as no-slip walls. At the inlet, top and bottom boundaries, velocity components and volume fractions are specified. At the outlet boundary, the outlet pressure boundary condition is defined, while at the symmetry plane and side boundary, a symmetry boundary condition is applied. To capture the free surface elevation and sharp corners of the hull, surface and volumetric refinements are applied to the unstructured trimmer mesh. A mesh study is implemented to select the appropriate base size of the mesh cell. Mesh refining and grid convergence is continued until the solutions become independent of the mesh size. The assumption of not considering the running attitudes is implemented in the optimization process. However, initial hydrostatic attitudes are calculated during the optimization and all generated designs are checked during the mesh generation and numerical set-ups.

Verification and validation of numerical simulation: KRISO container ship (KCS) model
Verification of CFD calculation of ship resistance is performed in this section for 7-meter long KCS model (Kim, Van, and Kim 2001;Larsson, Stern, and Visonneau 2014). The computational domain and boundary distance characteristics are considered identical to those in Section 2.2. To achieve  Note: CFD = computational fluid dynamics; EFD = experimental fluid dynamics.  convergence, the time step is reduced by a factor of 2 in every mesh refinement step. The time step is calculated based on the ITTC recommendation formula 0.005 ∼ 0.01Lpp/U. Details of the simulation conditions are shown in Table 3. The comparison of numerical and available experimental data for three different mesh plans is shown in Table 4. The difference between the CFD and experimental fluid dynamics (EFD) results is calculated by the following formula: Furthermore, the initial trim (τ 0 ) and sinkage (σ ) of simulations and their difference from the experimental data are summarized in Table 5.
Grid plan 1 is the fine mesh of the study with a difference of 2.15% in C T . The differences in trim and sinkage compared with the model test data are both below 1%. The above results and comparison efforts represent the accuracy and reliability of the numerical analysis strategy. Another graphical comparison is shown in Figure 6 for generated waves around the ship's hull for the present CFD analysis (grid 1) and available EFD data (Kim, Van, and Kim 2001;Larsson, Stern, and Visonneau 2014).
Simulations of three different grid plans, 1, 2 and 3, are considered to implement the uncertainty analysis of the simulation. Based on ITTC recommendations (ITTC Recommendations 2002, 2017, uncertainty analysis of iteration U I , grid U G , time step U T and other parameters U P are calculated to obtain uncertainty of the numerical simulation U SN .
For the present study, other parameters have not been considered and the uncertainties of grid convergence and time steps have been combined. This is due to the unsteady scheme for numerical solutions. Therefore, uncertainty analysis of discretization, U GT , is introduced. Moreover, the following formula is obtained:  The iterative uncertainty U I is calculated by two final minimum and maximum values of drag coefficient convergence: Simulation values of the total resistance coefficient are rewritten based on the ITTC indexing method in Table 6. The ratio of changes between simulations of different grid plans is defined as the convergence ratio (R G ): Values of S are outputs of the numerical simulation in the defined grid plan, which is the resistance coefficient. The R G (grid convergence factor) is less than 1, which means that the convergence of grid is monotonic. Therefore, grid uncertainty can be estimated by generalized Richardson extrapolation (RE) (ITTC Recommendations 2002, 2017: where P G and r are the order of accuracy and refinement ratio ( √ 2), respectively. The expression of P G is: The correction factor can be expressed as: where P G est is an estimate for limiting the order of accuracy and its value is considered to be 2 in the current work. All computed parameters are shown in Table 6. C G is sufficiently higher than 1 and P G is greater than P G est , which pertains to the solutions not being in the asymptotic range. The value of F S is calculated by the following equation: Uncertainty of the grid and time step U GT is calculated by: Finally, based on Equation (6), the numerical uncertainty U SN is 3.24% of the EFD data. Therefore, the uncertainty of simulation is accomplished, according to its small value. Another CFD verification is conducted for the Wigley trimaran hull, which is shown in Appendix 1 in the Supplementary file.

Grid convergence assessment: trimaran model
To determine a finer mesh size with acceptable numerical accuracy and appropriate element number, mesh convergence studies are carried out based on the design speed. The mesh convergence study is conducted by changing the value of the total resistance coefficient, as shown in Figure 7. Three mesh plans are created according to the refinement ratio of √ 2. Uncertainty analyses are performed using the Richardson approach, as explained in Section 2.2.1. The iterative convergence is assessed by using the total resistance coefficient history for the last two periods of oscillation, which is about 0.57% S G1 (where S G1 is the drag value in grid plan 1). Mesh plans and uncertainty analysis of the simulations for the trimaran model are presented in Tables 7 and 8, respectively. Grid verification based on the drag coefficient is summarized in Table 9. The order of accuracy P G is 4.53, which is greater than the theoretical value (P G est = 2), and C G = 3.81 is sufficiently greater than 1, indicating that the solutions are not in the asymptotic range. As the simulation uncertainty is fairly small (1.56% S), it is feasible to capture the differences in C T between different ship hulls generated by the optimization algorithms in the rest of the article. Figure 8 shows the convergence history of the objective function for the trimaran model at cruise speed.
The ratio of the physical time step ( t) to the mesh convection time scale, which relates the mesh cell dimension x to the mesh flow speed U, is known as the Courant number (Courant-Friedrichs-Lewy), and is given as follows: The Courant number is typically calculated for each cell and should be less than or equal to 1 for numerical stability. Often, in implicit unsteady simulations, the time step is determined by the flow  Note: CFD = computational fluid dynamics. Table 8. Uncertainty analysis of the trimaran model.  Table 9. Grid verification.
properties, especially fluid flow characteristics in the free surface region. To gain a suitable level of accuracy within a reasonable running time, the initial value of the time step is approximated by the ITTC-recommended formula and a Courant number of 1 in the water level. The VOF contour and mesh characteristics are shown in Figure 9. The length of mesh size in the water level is about 0.25 of the base size. Accordingly, the graph of Courant number against time step is shown in Figure 10. In this plot, the range of Courant number for acceptable time steps is shown, and a time step of 0.01 s corresponds to a Courant number below 1.

Two-level full factorial design study
The two-level full factorial technique is adopted to distribute eight design points in the design space. Subsequently, the total resistance is calculated for these hull forms. Variable sensitivity and system behaviour are explored to gain an insight into the design. All combinations of the factors at two levels, lower bound and higher bound ([Dx + , Dx − ], [Dy + , Dy − ], [Dz + , Dz − ]), for the design variables are used to construct eight design samples. The graphical features of the study are shown in Figure 11. Smaller symbols indicate minimum resistance at cruise speed, and vice versa. The gray scale displays resistance at sprint speed. For example, the small white circle represents the minimum value for both drags, which corresponds to (Dx + , Dy + , Dz + ).

Optimization process with NSGA-III
Various research in the field of marine studies has explained the pros and cons of different optimization methods Campana et al. 2009;Kuhn et al. 2007;Nazemian and Ghadimi 2020). The NSGA-III global optimization method is suitable for several, often conflicting, objectives that need to be minimized (Campos et al. 2016). This method is also suitable for trade-off study in conducting multi-objective optimization. Therefore, a hydrodynamic optimization problem for multi-design speed ships leads to a multi-objective optimization problem with mutual conflicting objectives. The NSGA-III implements the non-dominated sorting-based multi-objective evolutionary algorithm. The set of designs in every generation that is not dominated by any other determined designs is called the Pareto set. The number of Pareto sets is identified by population size, which is 16 in the current study (HEEDS MDO User Guide 2019). The set of designs that is not dominated by any designs in the entire search space is known as the Pareto front. As the run progresses, the Pareto sets  continue to approach the Pareto front (the set of ideal solutions). Ten generations continue the optimization study. Finally, 160 new designs are generated during the progress of NSGA-III. Figure 12 displays the distribution of the designs. The baseline design and the optimal trade-off Pareto front point, as the optimum design, are illustrated in Figure 12.
An optimal design that reduces the total resistance by about 10.39% for the cruise speed and 6.5% for the sprint speed is obtained and compared with the original design (Table 10)    14 show the geometry and wave generation of the free surface to compare the original and optimal designs, respectively. It is indicated that larger clearance is generally advantageous for the trimaran ship design. The optimum value for the side-hull stagger is determined to be slightly behind the main hull.
A parallel data plot is displayed in Figure 15, which presents a high-level graphical view of the relationships between multiple variables and responses at the same time. This enables the identification of minima for comparing the design aspects of different sets of designs. The best design is identified as a continuous black line in Figure 15.

Optimization process with SHERPA
The SHERPA optimization method is a hybrid and adaptive algorithm that chooses the best virtues of other search-based methods (HEEDS MDO User Guide 2019; STAR-CCM+ User Guide 2020). SHERPA employs multiple search strategies adaptively, based on learning from the design space. To identify optimized designs, SHERPA requires fewer model evaluations than other leading methods. Two values are assigned for the optimization process: archive size and number of evaluations. A recommended value of 20 for the archive size based on the reference (HEEDS MDO User Guide 2019) and 60 design cases for the evaluation number is selected. If no significant variation is observed, the process ends. Accordingly, 57 design points are produced in the current optimization problem. Figure 16 shows the distribution of design samples. The final optimum design is acquired in less computational time than by the NSGA-III method. Figure 17 displays the parallel data for design variables and output total resistance. Similar behaviour of the parallel data is observed between the other two studied methods.
Initial and optimal designs and corresponding parameter values are shown in Table 11. The obtained variables and final ship resistances are similar to the results from NSGA-III. One may therefore conclude that SHERPA is an efficient method with lower computational time and effort. Total resistance is reduced by 10.52% at cruise speed and 6.6% at sprint speed. Figure 18 illustrates a comparison of the geometry between the original and optimal designs. Wave generation around the original and optimum designs is displayed in Figure 19. The results of the side-hull position in SHERPA method are similar to those in the NSGA-III method. These results include larger clearance and backward shift of stagger towards the transom.

Global optimization on response surface
A significant advantage of using response surfaces is that the actual objective function, which may be very time consuming to compute, is not evaluated in searching for better points on the response surface. This potentially saves a lot of time. Even though it searches different regions in the design space, there is no guarantee that a minimum will be found on a response surface that will correspond to the     main target. This is because this method may concentrate on a single region in the design space. The central composite design (CCD) method is selected as the sampling method and the RBF method is chosen as the response surface model in the postprocessing step. The number of design samples, based on CCD, is defined as 15 and the total number of evaluations at the end of the optimization process is 20 runs. The behaviour of the design parameters and the corresponding responses determine that 20 runs are sufficient to construct an appropriate response surface and find an optimum design. The RS method is the fastest global optimization algorithm, compared to the other studied algorithms. However, it uses discrete variables in the process and does not find the optimum point. Figure 20 shows the distribution of design samples for 20 design points, of which three design points are infeasible. This infeasibility is due to the displacement constraint and intertwining of the side hull and main hull.
The results of the RBF model for Dx and Dy parameters are presented in Figure 21. Two optimum regions are found, as seen in blue. The results obtained from the RS method are slightly different from those of the previous two methods. In this method, two minimum points are obtained, one of which is the location of the side hull in the area around the midship of the middle hull. The RS method is not a recommended method for global optimization on its own, but it can be implemented along with another optimization algorithm to investigate the whole design space. Searching hidden regions of the design space is one advantage of the RS method. Overall, two optimum designs and their corresponding design variables are presented in Table 12.

Optimization methods: comparison and validation
Computational times of the three methods are compared in this section. CFD simulation and software integration are the same for all methods. The optimization is performed in a personal computer cluster environment with 8 Intel Core TM (64-bit, 2.8 GHz up to 3.8). A comparison of central processing unit (CPU) times and number of runs for the methods is illustrated in Figure 22. The comparative analysis of the CPU time requirements of the three methods shows that the SHERPA method has  a lower CPU time than the NSGA-III method, and the RS method is faster than the other methods, but the results of the RS method are not as desirable as those of the other methods. Meanwhile, SHERPA is shown to require less computation than NSGA-III for obtaining the optimum design, and is almost 2.5 times faster. Comparison of the optimization results for total resistance between the three optimization is algorithms presented in Table 13. It can be seen from Table 13 that the reduction in resistance for the NSGA-III and SHERPA algorithms is, respectively, 10.39% and 10.52% for the cruise speed and 6.5% and 6.6% for the sprint speed, and these values are almost the same. This reduction is about 7.15% and 3.76% for low and high speeds, respectively. The RS global optimization method is different from the other methods. However, the CPU time for the RS method is better than for the other techniques. Overall, considering the reduction in resistance and CPU time, one may conclude that the SHERPA scheme is more appropriate for simulation-based optimization studies. To verify the optimization process, validation of the optimization method through the side-hull arrangement of the Wigley hull is presented in Appendix 2 in the Supplementary file.

Conclusion
Parametric study and shape optimization based on CFD solutions are extended in marine and maritime industries. The characteristics of the design space is different for every project and solving the problem depends on the nonlinearity, modality and smoothness of the study. Three global optimization algorithms are adopted in the current study for multi-objective side-hull arrangement optimization applied on a wave-piercing bow trimaran ship. The total resistance of the trimaran hull is investigated at cruise and sprint speeds, as the objectives, and the optimum design is selected based on the appropriate longitudinal, transverse and vertical positions of the side hull. After a two-level full factorial DoE study, NSGA-III, SHERPA and RS global optimization algorithms are used for a tradeoff study. This option is beneficial in parameter optimization studies to obtain a better understanding of the involved trade-offs and to determine the best solution for a problem with multiple objectives. The HEEDS software manager tool is implemented to execute the optimization cycle, which plays the role of the discipline connector. The RANS equation with a k-turbulent model and VOF scheme is applied in the numerical simulation. The results obtained by the SHERPA and NSGA-III algorithms are similar. The optimization results achieve a 10.5% drag reduction at cruise speed and a 6.6% reduction at sprint speed. The longitudinal position of the trimaran's side hull is attained slightly behind the main hull and the transverse distance is obtained about at 0.1 m for the trimaran model. The results of the RS method are different from those of the other methods. However, the CPU time for this method is better than for the other employed techniques. The authors recommend the SHERPA method, based on the lower CPU time and achievement of successful optimization. Although using SHERPA alone may lose some hidden regions of the design space, the global optimization on RS resolves this problem. The automated optimization loop without user intervention and its errors are important characteristics of the current optimization platform. Furthermore, the obtained results illustrate a successful optimization with less computational time and effort. The proposed method is also validated by a Wigley trimaran model. Accordingly, the side-hull position of the Wigley trimaran is optimized by the suggested algorithm.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This research received no specific grants from any funding agency in the public, commercial or not-for-profit sectors.