Hydrodynamic hull form optimization using parametric models
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Hydrodynamic optimizations of ship hull forms have been carried out employing parametric curves generated by fairness-optimized B-Spline form parameter curves, labeled as F-Spline. Two functionalities of the parametric geometry models are used in the present study: a constrained transformation function to account for hull form variations and a geometric entity used in full parametric hull form design. The present F-Spline based optimization procedure is applied to two distinct hydrodynamic hull form optimizations: the global shape optimization of an ultra-large container ship and the forebody hull form for the hydrodynamic optimization of an LPG carrier. Improvements of ship performance achieved by the proposed F-Spline procedure are demonstrated through numerical experiments and through correlations with experimental data. The ultra-large containership was built and delivered to the ship owner. The present study validates the effectiveness of the proposed hydrodynamic optimization procedure, ushering in process automation and performance improvement in practical ship design practices.
KeywordsFairness optimized B-Spline form parameter curve Constrained transformation function Full parametric hull form design Hydrodynamic optimization Fuel economic ship design Green ship technologies
List of symbols
Midship section area
- CB = ∇/(LPPBT)
Frictional resistance coefficient
- CM = AM/(BT)
Sectional area coefficient at midships
- CP=∇/(LPPAM) = CB/CM
Residual resistance coefficient
Total resistance coefficient in full scale
Frictional resistance coefficient by ITTC
Wave resistance coefficient from pressure integration
Length between perpendiculars
Length of SAC entrance
Length of parallel middle body
Length of run
Total resistance in model scale
Total resistance in full scale
Wave resistance from wave cut analysis
Frictional resistance by ITTC
Wetted surface area
Design draft at forward perpendicular
Design draft at aftward perpendicular
Volume of displacement
Computer aided design
Computer aided engineering
Computational fluid dynamics
Experimental fluid dynamics
Flat of bottom
Flat of side
Fairness-optimized B-Spline form parameter curve
International towing tank conference
Longitudinal center of buoyancy
Parallel middle body
Reynolds-averaged Navier–Stokes equations
Sectional area curve
Simulation based design
Green ship technologies are gaining in importance in diverse areas of ship design. Because CO2 emission levels are directly related to fossil fuel consumption, shipbuilding industries are focusing on developing new design concepts and technologies towards fuel economic ship design including the use of new propulsion devices and renewable energy sources. Hydrodynamic optimal shape designs are one component of the overall fuel economic design, although the percentage reduction of CO2 emission can be achieved 2–3% at most. However, given the prevailing ship construction process, it is an indispensable step, because hull form design is a starting point of the new shipbuilding process and it influences resistance and propulsion performance. In addition to its hydrodynamic effects, hull form design influences costs, the construction process and safety considerations in shipyards.
There are three core technologies used for hydrodynamic hull form optimization: geometric modeling, hydrodynamic analysis, and optimization technologies. Implementation of these technologies as tools for ship design requires a sufficient understanding of each technique, the use of practical design experience and methodologies in the optimization process, and the appropriate evaluations of the optimization results to verify improved ship operation performance. With the development of each technology, CFD simulation-based ship design (SBD) has continuously improved and become more appropriate for practical use [3, 4, 5, 6, 7, 8, 9, 16, 17, 26, 27, 30, 31, 32, 34]. The current CFD simulations make it possible to reliably predict and validate ship resistance, propulsion performance and seakeeping performance [4, 13, 29, 35]. Although ship design is generally a multi-objective optimization problem, objective function evaluations based on a RANS solver require effectiveness and efficiency. Effective global optimization algorithms such as variable fidelity models, filled function-based algorithms, and a particle swarm optimization have been introduced for SBD [3, 21, 24, 32].
In the field of ship geometry design, increasingly sophisticated design methodologies are continuously suggested and validated [6, 9, 15, 16, 17, 18, 19, 20, 22, 23, 26, 28, 32]. An SBD project entitled ‘FANTASTIC’, whose goal was to improve the functional design of ship hull shapes, was implemented over a period of 3 years by 14 European partners . As a result, three approaches to shape modeling emerged: ship parametric modeling using Friendship-modeler software, a template approach involving the use of NAPA , and shape transformation functions via GMS/Facet . Major progress in the optimization process has been obtained through the use of parametric modeling, which provides the most flexible ship design variations. Finally, geometric modeling for SBD addresses the problem of shape generation and transformation as necessary to generate design variations with the required flexibility and functionality. These efforts not only optimize ship hull performance but also ensure the effectiveness of the automated design process.
The simplest and, therefore, most widely used method of generating hull shape parameters is the fitting of discrete ship offset data with parametric representations such as polynomials, cubics, and Bezier and B-Spline curves or surfaces. The discrete data obtained, including hull offset data and control points, are directly used as design variables to modify the shape. This approach provides flexibility in controlling each control point but results in less functionality because the high degree of freedom in control points variations can cause deviations from the desired shape [10, 26, 33].
In a more sophisticated method, the ship shape is partitioned into several sections, defined within several boxes and controlled by using a small number of control points as design variables. An effective grouping of control points can yield sufficiently large and realistic changes in the hull form [14, 16, 31]. If the hull shape is generated using form parametric data and a fairing process, the shape can be generated using a highly sophisticated fully parametric model. This fully parametric model generates the entire hull surface from a limited parameter set based on higher-level geometrical properties [1, 6, 18, 19, 20, 28].
A set of transformation functions can be defined by types of data including analytical curves, surface data, and discrete data. They can then be used in any type of geometry, offset data, or functional model. Although the formulation of the transformation function is a critical issue in geometric variation, it is mainly dependent on geometric modeling and the optimization problem [2, 12, 26, 27].
From a practical viewpoint, the geometric modeling process for ships is expected to be simple and similar to the design approaches with which ship designers are already familiar. The principal methods of ship design include the distortion transformation of a given hull form and the ab initio design of a new hull form using form parameters. In the present study, distortion transformation and form parameter design based on the F-Spline technique were investigated [6, 7]. Two hull form optimization approaches were used with practical ship designs. The applicability and subsequent improvements in hydrodynamic performance of these techniques are discussed in this article.
The commercial software SHIPFLOW  was used as a CFD solver, and FS-Framework [1, 2, 6, 7, 8, 9] was used as a geometric modeler and optimizer. The resistance and propulsion performance tests were performed at the Hamburg ship Model Basin (HSVA, Germany) and Maritime and Ocean Engineering Research Institute (MOERI, Korea), respectively.
2 Parametric hull form design
The process of curve generation begins with the complete mathematical definition of the curve from a set of given data elements of the curve. Since the data elements usually describe only a subset of the properties of the curve shape, the results of the curve generation process may vary. These results depend on the choice of mathematical curve representations and the properties of the generation process [18, 19, 20]. In this study, fairness-optimized parametric curves with constraints were used. These curves were represented by B-Spline curves and generated using the fairness optimization process with sets of form parameters as constraints. These fairness-optimized parametric curves have been used as unconstrained transformation functions for distortion transformation and as geometric entities to generate new hull form for the form parameter design.
The general formulation of the fairness-optimized parametric curve with constraints follows below:
Distance constraints: The Euclidean distance between the given data points Pi and the resulting r(t), taken at the associated parameter knot ti, weighted by wi, and squared, must be no greater than the error tolerance ε. In normalized form for n + 1 data points,
End constraints: For the first and last points on the curve, tangent vectors Qi and curvature vectors Ki,i = 0, or n, may be given as follows:
Area constraints: The actual area under a curve, S, shall match a given area value, S0:
Other constraints: Many other types of constraints can be imposed in equality or inequality form.
Form parameters describing a planar curve
i = 0
i = 1
i = 0
i = 1
i = 0
i = 1
Position Xi, Yi (m)
Tangent angle Qi (°)
Area S (m2)
1.25 × π/4
Centroid XC, YC (m)
2.2 Distortion transformation based on fairness-optimized B-Spline form parameter curves (F-Spline) with constraints
Distortion is the transformation of a given hull form via transformation functions to obtain new hull form data. A transformation function is a mathematical formulation that defines the degree of transformation of a given geometry necessary to achieve a certain new shape. In ship design, it is conventionally used to transform basic ship curves. The flexibility and variability of the distortion transformation depend on the geometrical properties and applicability of the transformation function. There are two types of transformation functions: unconstrained and constrained.
The unconstrained transformation function is a mathematical formulation used to define the degree of transformation of the given geometry without constraints. It can be defined in any type of function and applied to any part of the geometry for a complex and flexible transformation. This function is mostly used in hydrodynamic hull form optimization because it is easy to implement. However, an additional iteration is required to ensure that the design requirements are met and the target geometry is achieved. This function is, therefore, typically used for small local variations [16, 26, 27, 30].
The constrained transformation function is a mathematical formulation used to define the degree of transformation of the given geometry with design constraints. This function should be derived so as to satisfy design requirements before it is applied to hull form variation. The essential prerequisite to define the mathematical relationship between the degree of local transformation and the design requirements makes the application complicated. Therefore, the functions employed so far generally feature a simple form and fewer degrees of freedom [6, 12].
The Lackenby transformation  is a classical distortion based on a constrained transformation function expressed by a first or second degree polynomial. These polynomial-based transformation functions are analytically formulated and applied to SAC variation to derive a new required SAC. Because it yields robust and effective hull form distortions, this process is often used in the preliminary ship design phase. However, the Lackenby transformation has limited flexibility because of its simple polynomial-based transformation function. The function is expressed by a few form parameters, such as CP, LCB, LR and LE.
Define an open uniform third degree B-Spline curve with n + 1 vertex points.
Minimize the second order fairness criterion:
Form parameter constraints for SAC transformation function
Form parameter constraints FPi
Beginning tB = X0R
tE = X1R
tB = X0E,
tE = X1E
Centroid of area
This yields a nonlinear system of equations for the unknown, vertex points and Lagrange multipliers that can be solved numerically.
The ability to control the angle parameter of each part of the SAC using the proposed approach (10) is the critical functionality that distinguishes it from the Lackenby transformation. The differential form parameters expressed as Δα in Table 2 have a significant influence on resistance performance, transforming the entrance angle and stern profile shape. Figure 2 shows an example of a SAC distortion based on F-Spline transformation function
2.3 Form parameter design based on F-Spline with constraints
An arbitrary ship hull form can be fully parametrically expressed by suitably combining the form parameters constituting the basic curves and hull section frames. When the parametric design of the longitudinal basic curves is generated using fairness-optimized parametric curves, transversal basic curves can be generated. Similarly, if the transversal sections are derived by parametric modeling, a set of surfaces that interpolate or closely approximate the design sections is generated. Thus, efficient and effective form variations based on the parametrically generated hull form surfaces are possible [6, 18, 19, 20]. However, since there are many form parameters that make up each form parameter basic curve, and since there are possible correlations among the form parameters, it is not straightforward to determine the parametric input data sets. The increasing flexibility of the basic curve designs makes the design process increasingly complex.
Basic curves can be parameterized using the F-Splines. Since the curves are geometrically optimized for fairness to meet the least number of curve requirements, the use of F-Spline reduces the complexity arising from the parameter input sets, while increasing flexibility and improved shape quality [1, 2, 6, 7].
Basic curves describing a hull form
Flat of side curve
Center plane curve
Flat of bottom curve
Tangent angle at beginning and end
Curvature at beginning and end
Sectional area curve
Centroid of area
Vertical moments of sectional area
Lateral moments of sectional area
The parametric modeling was based on the given shape by importing the IGES file. Based on the IGES surface, a dense offset distribution for the forebody and aftbody was generated as offset data. The form parameter value describing each basic curve of the initial hull form was extracted from the given initial hull form to begin the parametric modeling. The number of form parameters is strongly dependent on the shape variation strategy. To fully exploit the F-Spline optimization and develop a completely new type of forebody hull form, approximately 80 form parameters were used for the parametric modeling. The relationships among the parameters were investigated in depth and sharing points were identified. A well-developed structure of parameter dependencies can help to avoid unnecessary increases in parameter inputs and degeneration in hull form variation.
3 SAC optimization of an ultra-large containership based on a constrained F-Spline transformation function
The size of containerships has been continuously increasing. With the expansion plan for the new Panamax, the largest size of operating containerships will reach around 14,000 TEU. Containerships of approximately 18,000 TEU are also currently under development. The block coefficient CB reaches 0.7, and the design speed range decreases to approximately 0.2 in FN or even lower. The design concept for ultra-large containerships tends to make them larger, fuller and slower. They can facilitate improvements in ship fuel economy and help to create “greener” ships.
Another design trend to improve the fuel economy is the multi-objective shape optimization based on the life cycle operating conditions. The speed performance in the ballast draft condition has become increasingly important. However, it has still been a standard process for containership design in ship yards that the hull form design is conducted only for the performance in the design draft condition, and then the speed performances are generally simulated and tested at two drafts, design and ballast conditions. This section describes a procedure used to optimize the hydrodynamic hull form for an ultra-large containership. Optimal hull form has been developed only for the performance in the design draft condition.
To verify whether the design variations yielded by the F-Spline transformation functions were geometrically flexible and functional and could be conveniently integrated into the classical design practice.
To verify whether the design variations yielded by the F-Spline transformation functions were acceptable in terms of resistance performance.
3.1 Design configurations
Principal particulars of an ultra-large containership
3.2 CFD simulation
The wave profiles and wave making resistance associated with alternative ship hull designs were simulated using SHIPFLOW XPAN by FLOWTECH. SHIPFLOW provides a non-linear potential flow code that uses Rankine panels to discretize the hull and free surface. The nonlinear boundary condition was iteratively solved. Trim and sink were allowed during the simulation.
The number of panels on the hull and free surface depends on the fullness and complexity of the hull geometry and ship speeds mainly. A detailed verification process including grid sensitivity analysis and error estimation for the numerical predictions were undertaken for a wide range of ship series and ship speeds using SHIPFLOW. In this thesis, sensitivity analyses for panel convergence, computational range convergence and iterative convergence were performed . The numerical errors that propagated in the wave resistance coefficient were estimated. The panel convergence analysis determined hull panel variation in terms of size, aspect ratio and distribution as well as free surface panel variation in terms of size per wavelength, computation region on the free surface, aspect ratio, panel distribution near the hull. The results indicated that there exist systematic error trends between simulations and experiment as a function of ship geometry and ship speeds. Using a potential code based on the SHIPFLOW analysis, a simulation-based design can be created using a comparison of the performance of alternative ship designs of the numerical configuration is carefully determined. The degree of numerical error based on the potential code is somewhat small and randomly distributed in comparison with the effect of geometric ship variation.
The hull surface was discretized using around 2,000 panels. On the longitudinal plane, a resolution of 25 panels per fundamental wavelength was utilized. The computation region of the free surface extended from x/L = −0.5 to 2.0 upwards and downwards, respectively, and from y/L = 0.0 to 0.7 from side to side. To reduce oscillation, the aspect ratio of free surface panel was maintained at levels larger than 1.5.
3.3 Sensitivity analysis of F-Spline form parameters
A systematic investigation of the SAC transformation was performed with respect to the form parameters for the F-Spline transformation functions for the wave resistance performance. Wave resistance is not representative of overall ship performance. However, because the sensitivity analysis including further formal optimization was based on the generalized Lackenby method, the shape topology and characteristics of the ship were not significantly changed during the process. Thus, it was possible to compare wave resistance levels only during the optimization process. However, the wetted surface area is generally monitored during the entire simulation-based optimization process.
3.3.1 The effect of the parallel middle body (PMB) on the wave resistance
Variation cases for PMB
Start X0R − ΔX0R (station)
End X0E + ΔX0E (station)
Length LM (% LPP)
3.3.2 The effect of the differential form parameters on wave resistance
Form parameters used for F-Spline variation
F-Spline form parameter
ΔX0E, ΔX1E (station)
Δα0E, Δα1E (°)
ΔX0R, ΔX1R (station)
Δα0R, Δα1R (°)
3.4 SAC optimization
An optimal design was achieved after 37 iterations within the given convergence criterion. The wave resistance performance was slightly improved but remained similar to that of the hull form in the second baseline design, with a smaller entrance angle and small bulb. The PMB of the optimal design increase slightly. In the low speed full containership design range, the effect of the LCB position was relatively small within a certain variation range. A reduction in the entrance angle of the SAC resulted in a very narrow waterline angle at the FP. The entrance angle of the optimum SAC was mainly influenced by the incident angle of the design waterline when a bulb shape was given.
During the optimization process, the major part of the improvement was achieved during the first step via systematic parameter variation. This trend depended on the set up of the optimization problem. In this investigation, the first step was to conduct a Pareto search for optimal designs based on a small number of systematic analyses in a wide range of design spaces. Generally, this step is formulated by optimization strategy such as DoE. The selected designs were also used as a starting point for deterministic optimization in the second step. The final optimum design was associated with a relatively small Pareto range.
In the present study, an ultra-large containership hull design was developed using an SAC optimization based on an F-Spline transformation function. This hull form showed good resistance performance although limited longitudinal shifting was possible in each section; local variations in the section frame and variations in the bulb shape were not possible. Nevertheless, the simplicity and efficiency of this process are sufficiently valuable to support the use of this technique in the preliminary hull form design stage.
4 Full parametric hull form optimization of an LPG carrier
An LPG carrier is a tanker ship that is designed to transport liquefied petroleum gas. The hydrodynamic hull form design of LPG carriers has generally been considered to involve particular requirements because of the operational peculiarities and geometric limitations of these ships. One such peculiarity is that an LPG carrier operates evenly in the three draft conditions, i.e., the design, ballast, and scantling conditions. Therefore, the speed performance under all draft conditions should be considered at the same time. Similarly, one geometric limitation of the design of these ships is that the LPG tank must be positioned in the most forward part of the ship. This constraint makes it difficult to design a hydrodynamically superior waterline shape and results in increase of wave resistance.
Principal particulars for the LPG carrier
4.1 Problem statement for forebody hull form optimization
Inequality constraint: RT in the design draft condition
Equality constraint: Keeping breadth at four frames
The form factor k of the baseline hull form was estimated using the correlation between that by Holtrop method  and measurement tested by MOERI in the design and ballast draft conditions. It yielded (1 + k) = 1.229 for ballast and (1 + k) = 1.209 for design draft of the baseline hull form. During the optimization process, the form factor was recomputed for each design variant based on the empirical form factor  that is a function of LPP, B, T, LR, CB, CP and LCB, which yielded small adjustments. RA is a correlation allowance, ΔRT denotes an additional resistance component, and RAA represents the wind resistance. The additional resistance components RA, ΔRT, and RAA were estimated at each draft conditions based on the experience with similar ship hulls. These components were applied to each design variant as a function of wetted and exposed surface area respectively.
For the final deterministic optimizations, a mixed objective function was introduced that represented the weighted sum of the total resistance ratios in the ballast and design draft conditions.
δHPi denotes the shortest 1-norm distance from the frame to its respective hard-point i, and mHP is a user defined positive scaling factor. A penalty becomes active only if the respective frame lies inside its hard-point definition. To amplify the penalty in the infeasible domain close to the constraint bound, a constant zero offset HPzeroOffset was added to the penalties if and only if one or more hard-point constraints actually became active: i.e., δHPi < 0 (for i = 1,…, 9). Finally, all hard-point penalties were added together and added to the mixed objective.
4.2 CFD simulation
4.3 Optimization procedure
fi denotes the value of the free variable i mapped into the normalized domain of its lower and upper bound [0..1] .
4.4 Optimum hull form and performance
Improvements in performance characteristics
Design condition (% diff.)
Ballast condition (% diff.)
A comparison the total resistance achieved via the CFD simulation with that indicated in the EFD results yields some discrepancies. The improvement in the resistance performance of the optimal ship is much greater in the EFD case than in the CFD case. This still holds given the experimental and computational uncertainties involved. The new bulb may result in favorable wave-viscous interaction not only by reducing the breaking waves but also by influencing the stern wave and boundary. The scale effect extrapolated from model to ship, in which the form factor must be used, should be carefully considered. In any case, this type of forebody hull form is favorable for propulsion and resistance performance.
Since F-Spline is automatically generated via the optimization process for fairness to satisfy the given requirements, a fair curve is always generated even with a few subsets of form parameters available to satisfy the given requirements. This curve can, therefore, be easily used in the complicated form generation and form transformation processes.
Hull form distortion based on the constrained transformation function F-Spline is simple, flexible, and well suited to the optimization process. In particular, the implementation of differential form parameters for the SAC transformation remarkably improved the availability of the hull form distortion by transforming the entrance angle and stern profile shape, which are significant factors in resistance performance. An ultra-large container ship optimized using this process showed excellent resistance performance at the design speed of 24 knots.
The fully parametric ship hull design offers the flexibility to generate new types of hull forms. This holds even if it is still difficult to model and control the ship hull because of its complex structure. This design approach was used in forebody hull form optimization for an LPG carrier. The optimal ship with a completely different bulb shape was successfully validated, showing a 5.7% improvement in total resistance and a 7.8% improvement in delivery power.
The flexibility and functionality of geometric modeling constitutes an important technology in hydrodynamic optimization, not only because it is directly related to the performance of the optimal hull form but also because it is related to the effectiveness and automation of the optimization process.
This research was supported by World Class University (WCU) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-2008-000-10045-0). The author would like to thank Prof. K.C. Park of University of Colorado at Boulder for his sincere advice and encouragement and Dr. Stefan Harries of Friendship Systems GmbH for his long and constant friendship.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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