Journal of Technology 2021, volume 6, issue 1, pp. 151 – 172. The Society of Naval Architects and Marine Engineers.

Combining Analytical Models and Mesh Morphing Based Optimization Techniques for the Design of Flying Appendages

Ubaldo Cella University of Rome “Tor Vergata” and Design Methods - Aerospace Engineering, Italy, [email protected].

Corrado Groth University of Rome “Tor Vergata” and RBF Morph, Italy.

Stefano Porziani Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 University of Rome “Tor Vergata” and RBF Morph, Italy.

Alberto Clarich ESTECO, Italy.

Francesco Franchini EnginSoft, Italy.

Marco Evangelos Biancolini University of Rome “Tor Vergata” and RBF Morph, Italy.

Manuscript received March 15, 2021; revision received May 18, 2021; accepted June 22, 2021.

Abstract. The fluid dynamic design of involves most of the typical difficulties of aeronautical wings design with additional complexities related to the design of a device operating in a multiphase environment. For this reason, “high fidelity” analysis solvers should be, in general, adopted also in the preliminary design phase. In the case of modern fast foiling sailing , the appendages accomplish both the task of lifting up the boat and to make possible upwind sailing by contributing balance to the side force and the heeling moment. Furthermore, their operative design conditions derive from the global equilibrium of forces and moments acting on the system which might vary in a very wide range of values. The result is a design problem defined by a large number of variables operating in a wide design space. In this scenario, the device performing in all conditions has to be identified as a trade-off among several conflicting requirements. One of the most efficient approaches to such a design challenge is to combine multi-objective optimization strategies with experienced aerodynamic design. This paper presents a numerical optimization procedure suitable for foiling multihulls. As a proof of concept, it reports, as an application, the foils design of an A-Class . The key point of the method is the combination of opportunely developed analytical models of the forces with high fidelity multiphase analyses in both upwind and downwind sailing conditions. The analytical formulations were tuned against a database of multiphase analyses of a reference demihull at several attitudes and displacements. An aspect that significantly contributes to both efficiency and robustness of the method is the approach adopted to the geometric parametrization of the foils which was implemented by a mesh morphing technique based on Radial Basis Functions.

Keywords: multiobjective optimization; mesh morphing; Radial Basis Functions; foiling ; aerodynamic design.

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NOMENCLATURE

퐶푓 Friction drag coefficient 퐶푇 Total hull resistance coefficient 퐶푤 Hull wave drag coefficient 퐷퐻 Bare hull drag [Kg] 퐹푁 Froude number 푮 known values at source points [m] 퐿퐻 Hull side force [Kg] 푁 Number of source points 푷 Constraint matrix 푅푒 Reynolds number 푠 Interpolation function

2 Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 푆퐻 Hull side force reference surface [m ] 2 푆푤푒푡 Hull wet surface [m ] 푼 Interpolation matrix 푉 Boat velocity [m s-1] 푊퐻 Operative displacement [Kg] 풙 Coordinates vector [m]

훽 Leeway angle [rad] 휸 Vector of radial function coefficients 휕퐶 퐿퐻 slope of the hull side force polar curve [rad-1] 휕훽 휼 Vector of polynomial coefficients -3 휌푤 Sea water density [kg m ] 휑 Radial function

6DoF Six Degree of Freedom CAD Computer Aided Engineering CFD Computational Fluid Dynamics RANS Reynolds-averaged Navier-Stokes RBF Radial Basis Functions VOF Volume Of Fluid VPP Velocity Prediction Program

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1 INTRODUCTION

Foiling is the modern term used to describe a sailing condition in which the boat is raised up from the water by lifting surfaces. It is not a new idea. The concept was first introduced, more than 100 years ago, by the Italian engineer, inventor and aeronautical pioneer, Enrico Forlanini. He had been studying the idea since 1898 and in 1905 he tested his first prototype equipped with a ladder system of foils under the hull and a 60 hp engine. During tests, the managed to reach a top speed of 37 knots flying half a metre above the water of Lake Maggiore (Calabrò, 2004). The first known sailing hydrofoil was produced by the Americans Robert Rowe Gilruth and Carl William Price who managed to achieve the same feat, albeit very slowly, under sail in 1938 (Sheahan, 2013). As often occurs for many innovative solutions, however, the efficient exploitation of the potentialities of adopting foils on sailing boats was related to the technological improvements in materials, manufacturing processes and design tools capability. The beginning of the modern era of foiling sailing boats (from the engineering point of view) might be identified with the 34th America's Cup. The foiling solutions developed for the AC72 catamarans stimulated the effort in adopting sophisticated engineering design approaches which had been limited to very high Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 technological environments such as aerospace. This view encouraged competing teams to enter into technical partnerships with companies specialized in aerospace technologies. An example of this synergy was the collaboration between AIRBUS and Oracle team USA for the 35th edition of the America's Cup. Airbus, more recently, has supported the team American Magic in the development of the AC75 for the edition 2021. The solutions adopted in the major classes also gave a strong impulse to the evolution of smaller classes which, furthermore, today can benefit from the availability of engineering services at a cost that begins to be compatible to their market requirements.

The literature offers several references to hydrofoils application for fast mono and multi hull crafts (Besnard, et al., 1998; Prastowo, et al., 2016; Kandasamy, et al., 2011). An historical review of their application on several types of fast crafts, together with an experimental report of the advantages in several sailing conditions, is reported in (Migeotte, 2002). Sighard F. Hoerner is historically one of the most important contributors to this topic. His books represent one of the main references for hydrodynamic (Hoerner, et al., 1954) and aerodynamic (Hoerner & Borst, 1975) designers. The literature is, however, not so abundant on problems facing hydrofoil design for sailing boat applications. Reviews of America's Cup case studies (Cassio, 2016) or references to numerical analysis methodologies (Paulin, et al., 2015) are common. Apart from some recent valuable publications as (Graf, et al., 2021; Hagemeister & Flay, 2019), the same cannot be said in the case of foils design methodologies suitable to fast sailing crafts. This paper aspires to contribute to filling this gap.

This paper describes a foils design methodology, based on a numerical optimization environment able to evaluate the performance of a flying catamaran when sailing in both upwind and downwind conditions. The analysis of the surface piercing foils in flying configuration is performed by a two- phase CFD simulation. To reduce the computational resources, the foils forces in non-flying configuration are evaluated by a single phase CFD run in which the hull is present only as a small fixed portion of a rounded inviscid surface in the upper domain boundary. The contribution of the floating hull is integrated by opportunely developed analytical models. The foils geometric parametrization was implemented by a cutting-edge mesh morphing technology based on Radial Basis Functions.

To demonstrate the capabilities and the potentialities of the methodology, the design of A-Class catamaran foils was selected as a pilot study. The A-Cat test case represents a highly constrained problem that includes most of the typical complexities involved in flying boat appendages design. The work was previously reported in (Biancolini, et al., 2018). What follows integrates the presentation of the methodology with a more detailed description of the analytical formulations developed to model the hull hydrodynamic forces.

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The A-Class, born in the late 50s, is a small high-tech catamaran that is considered the fastest single-handed racing in the world. The rules are very simple and mainly constrain the minimum weight (75 Kg), the hulls length (18 ft) and the sail surface (150 ft2). In 2009, when the interest in flying configurations began to rise, new rules were added with the intention of preventing foiling for A-Cats. Given that the concept of hydrofoiling prohibition was ambiguous and difficult to define, an indirect approach was chosen. The idea was to introduce a set of constraints aimed at limiting the surfaces suitable for sustaining the boat so as to make a flying configuration unfavourable compared to a traditional one. The assumption adopted to the adjustment of the rules proved, however, to be conservative and was not able to prevent the development of very favourable flying configurations. Nevertheless, they made the foils dimensioning a complex and strongly constrained design problem that represents a challenging test case for engineering design tools and methodologies. The method here described, adopted for the A-Cat test case, is suitable for any class of flying boats of any dimension.

The description of the methodology and the work performed are reported in four sections. First a description of the parameterization strategy adopted, with the relative theoretical recall and setup Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 procedure, is provided. In the following section the A-Cat foils design problem, and its constraints, is introduced. The implementation of the procedure, the description of the hull analytical formulations and the setup of the foils parameterization for the optimization are detailed in section 4. The last section reports the solutions of the optimization and the verification of the selected optimum.

2 SHAPE PARAMETRIZATION BY MESH MORPHING

Geometric parametrization based on mesh morphing consists in implementing shape modifiers, amplified by parameters that constitute the variables of the design problem, directly on the computational domain. New geometric configurations are obtained by imposing the displacement of a set of mesh regions (e.g. wall boundaries or discrete points within the volume) by using algorithms able to smoothly propagate the prescribed displacement to the surrounding volume. The performances of the morphing action (in terms of quality of the morphed mesh and computational resources requirements) depend on the algorithm adopted to perform the smoothing of the grid. Among the several algorithms available in literature, Radial Basis Functions (RBF) are recognized to be one of the best mathematical frameworks to deal with the mesh morphing problem (Jakobsson & Amoignon, 2007). Its efficiency was successfully demonstrated in several engineering problems as shape optimization (Kapsoulis, et al., 2016), ice accretion (Groth, et al., 2019), biomedical engineering (Capellini, et al., 2020), static (Cella & Biancolini, 2012) and dynamic FSI analyses (Martinez-Pascual, et al., 2020) including application with (Viola, et al., 2015) and structural problems (Biancolini, 2014). It also offers great opportunities to improve the potentialities of analysis methodologies based on Reduced-Order Models (ROM) (Castronovo, et al., 2017).

Several advantages are related to the RBF mesh morphing approach:

• there is no need to regenerate the grid; • the robustness of the procedure is preserved; • its meshless nature allows to support any kind of mesh typology; • the smoothing process can be highly parallelizable; • the morphing action can be integrated in any solver.

The latter feature offers the very valuable capability to update the computational domain “on the fly” during the progress of computation. The main disadvantages of RBF mesh morphing methods are the requirement of a “back to CAD” procedure, some limitations in the model displacement amplitude, due to the distortion occurring after extreme morphing, and the high computational cost related to the solution of the RBF system that, if large computational domains are involved, imposes the implementation on HPC environments. 154

2.1 Radial Basis Functions

Radial Basis Functions are powerful mathematical functions able to interpolate, giving the exact values in the original points, functions defined at discrete points only (source points). The interpolation quality and its behaviour depend on the chosen RBF. A linear system (of order equal to the number of source points introduced) needs to be solved for coefficients calculation. Once the unknown coefficients are calculated, the motion of an arbitrary point inside or outside the domain is expressed as the summation of the radial contribution of each source point (if the point falls inside the influence domain). An interpolation function composed by a radial basis and a polynomial is defined as follows:

푁 ( ) ( ) 푠 풙 = ∑ 휸푖휑(‖풙 − 풙푘푖‖) + ℎ 풙 (1) 푖=1 Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 The minimal degree of polynomial ℎ depends on the choice of the basis function. A unique interpolant exists if the basis function is a conditionally positive definite function. If the basis functions are conditionally positive of order 푚 = 2, a linear polynomial can be used:

ℎ(풙) = 휼1 + 휼2푥 + 휼3푦 + 휼4푧 ( 2)

The values for the coefficients 휸 of RBF and the coefficients 휼 of the linear polynomial can be obtained by solving the system

푼 푷 휸 푮 ( ) ( ) = ( ) 푷푇 0 휼 0 (3) where 푮 are the known values at the source points, 푼 is the interpolation matrix defined calculating all the radial interactions between source points

푈푖푗 = 휑(‖풙푖 − 풙푗‖) 1 ≤ 푖 ≤ 푁, 1 ≤ 푗 ≤ 푁 ( 4) and 푷 is the constraint matrix

1 푥1 푦1 푧1 1 푥 푦 푧 푷 = ( 2 2 2 ) ⋮ ⋮ ⋮ ⋮ (5) 1 푥푁 푦푁 푧푁

The radial basis is a meshless method suitable for parallel implementation. In fact, once the solution is known and shared in the memory of each node of the cluster, each partition has the ability to smooth its nodes without taking care of what happens outside because the smoother is a global point function and the continuity at interfaces is implicitly guaranteed.

To deepen the mathematics behind Radial Basis Functions and to get an overview on their use in engineering applications the reader can refer to (Biancolini, 2017).

2.2 Mesh Morphing Setup

The mesh morphing software adopted to parametrize the geometry is RBF Morph (Biancolini, 2011). The tool offers large flexibility in implementing complex parametrizations for several applications (Cella, et al., 2017). The definition and the execution of a morphing action is completed by three steps:

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• Setup - consists in the manual definition of the domain boundaries within which the morphing action is limited, in the selection of the source points where fixed and moving mesh regions are imposed, and in the definition of the required movements of the points used to drive the shape deformation. • Fitting - is the action in which the RBF system, derived from each problem setup, is solved and stored into a file ready to be amplified. • Smoothing - is the morphing phase of surfaces and volumes according to arbitrary amplification factors. It is performed firstly applying the prescribed displacement to the grid surfaces and then smoothly propagating the deformation to the surrounding domain volume. It can be performed combining several RBF solutions, each one defined by a proper amplification factor, to constitute the parametric configuration of the computational domain.

Figure 1 reports an example of the setup of an RBF problem applied to a sail. A more detailed description of a setup applied to a can be found in (Biancolini, et al., 2014). In (Calì, et al., 2019) the use of RBF mesh morphing to reproduce the flying shape of a spinnaker is described. A Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 Fluid-Structure Interaction (FSI) analysis approached by RBF is reported in (Groth, et al., 2020).

Figure 1. Fixed and moving source points of an RBF setup.

3 FOILS DESIGN PROBLEM DESCRIPTION

Forces and moments that act on the whole system influence the sailing boats appendages operative conditions (Larsson & Eliasson, 1997). The sailing speed depends on the boat’s characteristics and on the sail’s performances with a complex mechanism that requires the several aspects of the involved physics to be taken into account. A component should then be designed within so-called Velocity Prediction Program (VPP) environments (Claughton, et al., 1998). Two approaches are in general followed to obtain the data required for VPPs: experimental activities (Day, et al., 2019) or CFD calculations. The use of numerical data can be expensive due to the cost of computations. In (Peart, et al., 2021) several approaches to reduce the calculation requirements are suggested. In (Cella, et al., 2016) it is possible to find an example on how sail trim optimization procedures and analytical VPPs can be coupled.

In the present work, some simplifications have been introduced in the definition of the design condition for the foils. It is assumed that the boat and the crew weight is the dominating vertical force. Furthermore, it is assumed that the modulus of the other components, derived from equilibrium, vary in a range that is much smaller than the global vertical forces magnitudes. With these assumptions, it is possible to consider as a constraint, in the foils optimization, a fixed vertical component of the lift. A set of similar considerations can be done for the side force which is strictly connected to the maximum righting moment generated by helmsman at the trapeze

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(knowing the height of the sail centre of effort). The final goal of the optimization is to determine the foils shape capable of generating the target lifting force, while minimizing the drag and respecting the optimization constraints.

3.1 Geometric Constraints Definition

The rules governing the A-Class boats impose the foils installation from the top of the hull (to avoid the adoption of T-foils which are consequently possible only on the ) and a minimum distance between foils tips which must always be larger than 1.5 m (in order to constraint the span of the surfaces contributing to the vertical lift). The maximum beam of the boat, including appendages, must be lower than 2.3 m. Assuming that L-shaped foils are used, the cant angle 훿 (see Figure 2) has to comply with a minimum value to allow the foil insertion. Structural constraints for minimum foil thickness are also imposed to the design.

2.3 m Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021

1.5 m

훿

Figure 2. Scheme of A-Cat foils geometric constraints.

4 SETUP OF NUMERICAL MODEL

The optimization procedure was setup defining two objectives: the minimization of the hydro- dynamic drag of the boat, without considering the rudders, in both upwind and downwind conditions. For the downwind condition, the boat is expected to be completely lifted up from the water’s surface. If the candidate solution cannot guarantee a sufficient lift to respect this condition, it will be discarded. In upwind sailing condition it is assumed that the boat is only partially sustained by the foil.

4.1 CFD Configuration for the foils analysis

The aerodynamic analysis of appendages can be faced with methods of several orders of complexity, from empirical analytical models, very useful in the preliminary design stages, up to Navier-Stokes based codes. The solver to be selected depends on the foil configuration and on the physics to be captured. In case of surface piercing foils, phenomena as ventilation and free surface interference should be properly captured. It is then opportune to approach the foils analysis of flying boats with RANS solvers (although specialized panel solvers can be implemented to model phenomena as cavitation and ventilation (Gaggero & Brizzolara, 2009)).

In the downwind analysis, the air and water phases were modelled using the Volume Of Fluid (VOF) technique in a steady incompressible RANS analysis. The sailing speed was set to 15 knots and the heeling angle was fixed to 5 degrees. To set the attitude generating the required vertical force, the sinkage was trimmed iteratively defining, in the boundary conditions, the height of the free surface. The cavitation possibility was ignored. It was assumed the total displacement to be equal to 170 kg (boat and crew weight) but it was estimated that about 30% to be generated by the T-foils of the rudders. About 120 kg is then assumed to be generated by the foils.

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To determine the operative leeway angle, the global equilibrium of moments and forces should be considered. In the present work, however, it was assumed 훽 to be constant and equal to 3 degrees. This has made possible to reduce the computational efforts by avoiding the introduction of additional degrees of freedom. In the authors’ opinion, this assumption is justified when considering the balance between the impact of this simplification on the solutions and the additional computational efforts required for a more accurate simulation.

In the upwind analysis, the sailing speed was set to 10 knots and the attitude was set fixed in order to not change the computational domain; the heeling angle was set to 5 degrees in this condition too. When sailing upwind, only one hull is flying outside water while the other contributes to the boat sustainment. The analysis was then run with a single phase, with the top wall boundary condition set as inviscid. In order to take into account the hull/foil interferences at junction, the domain is shaped following the immersed hull profile. The free surface of the water is considered planar (see Figure 3). This simplified model neglects phenomena like ventilation, but such approximation is considered still sufficient to provide a numerical configuration that correctly indicate the optimization direction. The target, in fact, is to properly evaluate the drag difference Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 between the analysed configurations and not the accurate estimation of the total drag.

An analytical formulation was developed, comparing the CFD solutions obtained on an isolated demihull at different attitudes and displacements, to evaluate the hull forces. The drag resulting from the analytical model, which requires as input the lift generated by the foils, is added to the hydrodynamic drag to generate the objective function in upwind conditions.

Figure 3. Detail of the computational domain (medium mesh).

Since in upwind sailing the leeway angle is important, its value is tuned by modifying the inflow direction on the farfield boundaries. The value is evaluated by executing two simulations at two values of 훽 and linearly extrapolating the leeway angle that should generate the required target side force. In case the required side force is not generated at the extracted angle, the current solution is discarded since the foil operates outside the linear range of the aerodynamic lift polar. A target side force of 70 kg is evaluated from the equilibrium of moments around the sailing direction, assuming the sailing centre of efforts to be located at a height of 4 meters above the water.

4.1.1 Grid sensitivity analysis

The fluid dynamic domain was discretized with a multi-block structured hexahedral mesh extended 10 meters upstream and downstream the foils. It is 10 meters wide with a depth of 5 meters. In upwind conditions, the top boundary of the domain coincides with the water free surface. Three computational grids were generated with different mesh resolution to perform a mesh sensitivity analysis. Coarse, medium and fine meshes were made respectively of about 1, 7.5 and 25 million of elements. The sensitivity analysis was performed in downwind configuration (i.e. VOF with sinkage trimmed to keep constant the vertical lift component). In Figure 4 the foils drag results obtained are plotted. With respect to the fine mesh, the drag obtained with the coarse grid is 5% 158

higher, whilst the drag obtained with the medium mesh is 0.5% higher. The coarse mesh was selected to perform the foils optimization.

14.8

14.6 14.4 14.2 14

Foils drag [Kg] drag Foils 13.8 0 5 10 15 20 25 30 Mesh size [mill.]

Figure 4. Sensitivity of the solution to the grid dimension. Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 4.2 Analytical model for the hull forces

The analytical formulations used to model the hull forces were both found on literature and opportunely developed tuning them by a comparison with the CFD database obtained for an isolated hull. The reference solutions were generated at several velocities, attitudes, displacement, and leeway angles by running multiphase RANS analyses with numerical configurations similar to the downwind configuration setup for the present work. The geometry adopted for the model tuning is a Flyer S (2009) A-Class catamaran demihull with no nor appendage.

The two formulations adopted to model the side force and drag are described in the following sections.

4.2.1 Hull Side Force Formulation

The hull is modelled as a lifting surface. The side force (force laying in a plane parallel to the water plane and normal to the sailing direction) is modelled as:

1 휕퐶퐿 퐿 = 휌 푉2푆 퐻 훽 퐻 2 푤 퐻 휕훽 (6)

휕퐶 where 휌 is the sea water density, 푉 is the boat velocity, 훽 is the leeway angle and 퐿퐻 is the 푤 휕훽 slope of the side force polar curve. The adopted reference surface 푆퐻 is the side projection on the symmetry plane of the submerged part of the demihull. It changes with displacement and is estimated by CAD. For a typical A-Class hull shape, its relation with the displacement can be split in two regions: an exponential grow starting from zero displacement and a linear behaviour after a defined value of displacement 푊퐻0 :

휏 푊 푆퐻 (푘 푊 + 푘 ) ( 퐻 ) , 푊 < 푊 푆퐻1 퐻0 푆퐻2 퐻 퐻0 푆퐻 = { 푊퐻0 ( 7) 푘 푊 + 푘 , 푊 ≥ 푊 푆퐻1 퐻 푆퐻2 퐻 퐻0

The coefficients 푘 and 푘 are computed knowing two surface values in the linear region, 휏 is 푆퐻1 푆퐻2 푆퐻 estimated adding a known value in the exponential region.

The slope of the side force polar curve was assumed to linearly change with the displacement. Nevertheless, also a non-linear relation with velocity (Reynolds effect) and leeway angle was

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observed. To account for these effects the developed formulation contains also exponential expressions of the two parameters. It then assumes the form:

휕퐶 퐿퐻 휏 휏 = 푉 퐻1 훽 퐻2 (푘 푊 + 푘 ) 휕훽 퐻1 퐻 퐻2 (8)

The parameters , , and are tuned by a comparison with the known solutions of the 푘퐻1 푘퐻2 휏퐻1 휏퐻2 isolated hull. The graphs in Figure 5 compare the analytical solutions at two velocities and three leeway angles with the reference CFD solutions obtained on the reference isolated demihull. Figure 6 reports the same comparison for three values of displacements.

The observed non-linearity with the leeway angle is, as already introduced, modelled in the slope of the side force polar curve with the exponent 휏퐻2 (which was evaluated to be about 0.2). The adopted side force formulation ((6), in fact, models the hull as it was a lifting surface. The aspect ratio of the hull is, nevertheless, extremely low and, consequently, responds with a non-linear Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 aerodynamic. The non-linearity with the leeway angle of equation (8 is then justifiable by having forced such a similitude.

V = 10 knots V = 14 knots 35 35 CFD ( β = 1 deg) 30 30

CFD ( β = 2 deg)

25 CFD ( β = 3 deg) 25 Analitic. ( β = 1 deg) 20 Analitic. ( β = 2 deg) 20 Analitic. ( β = 3 deg)

15 15 Side force [Kg] force Side Side force [Kg] force Side 10 10

5 5

0 0 100 150 200 250 100 150 200 250 Displacement [Kg] Displacement [Kg]

Figure 5. Demihull side forces for three sideway angles (Flyer S A-Cat 2009).

V = 10 knots V = 14 knots 35 35 CFD (Disp. = 150 Kg) 30 CFD (Disp. = 170 Kg) 30 25 CFD (Disp. = 190 Kg) 25 Analitic. (Disp. = 150 Kg) 20 Analitic. (Disp. = 170 Kg) 20 Analitic. (Disp. = 190 Kg)

15 15 Side force [Kg] force Side Side force [Kg] force Side 10 10

5 5

0 0 0 2 4 0 2 4 β [deg] β [deg]

Figure 6. Demihull side forces for three displacements (Flyer S A-Cat 2009).

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4.2.2 Hull Drag Formulation

The total hull resistance is expressed by:

1 퐷 = 휌 푉2푆 퐶 ( 9) 퐻 2 푤 푤푒푡 푇

The hull wet surface 푆푤푒푡 grows with displacement. Its values are evaluated by CAD assuming the submerged volume to be determined cutting the hull with a plane representing the water surface. The formulation adopted is similar to the one used to model 푆퐻:

휏 푊 푆푤 (푘 푊 + 푘 ) ( 퐻 ) , 푊 < 푊 푆푤1 퐻0 푆푤2 퐻 퐻0 푆푤푒푡 = { 푊퐻0 ( 10) 푘 푊 + 푘 , 푊 ≥ 푊 Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 푆푤1 퐻 푆푤2 퐻 퐻0

The value of displacement 푊퐻0 from which the curve begins to be linear depends on the geometry and has to be evaluated by CAD. Its value is in general close to the value at which the 푆퐻 curve turn linear. It can then, as first approximation, assume the same value. The terms 푘 , 푘 and 푆푤1 푆푤2

휏푆푤 are found knowing the hull wet surface at two displacement values in the linear and one in the exponential region. Figure 7 reports the comparison between the wet surface modelled by the developed analytical formulation and the values computed by CAD.

3.5

3

] 2 2.5

2

1.5 et surface [m surface et 1 Measured

Analitic. exp. Hull w Hull 0.5 Analitic. linear.

0 0 50 100 150 200 250 300 Displacement [Kg]

Figure 7. Reference wet surface of demihull (Flyer S A-Cat 2009).

The total resistance coefficient 퐶푇 can be modelled by a combination of a friction and a wave drag component (Insel & Molland, 1992) as:

퐶푇 = (1 + 푘)퐶푓 + 퐶푤 ( 11)

The form factor (1 + 푘) accounts for the over velocity generated by the thick shape of the body (Hoerner, 1965). Its value must be evaluated from literature or from a known bare hull drag value. The skin friction coefficient 퐶푓 is estimated according to the ITTC-57 friction line expression (ITTC, 2002):

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0.075 퐶푓 = 2 ( 12) (log10 푅푒 − 2) where 푅푒 is the Reynolds number referred to the hull waterline length. This formula provides a good correspondence with the viscous drag estimated by the CFD computations (Figure 8).

0.006

0.005 CFD

0.004 ITTC-57

f

C 0.003 Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 0.002

0.001

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

FN

Figure 8. Skin friction coefficient of the hull (Flyer S A-Cat 2009).

A significant simplification was chosen to model the wave drag coefficient. A combination of two quadratic formulations, for speeds lower and higher than a critical value, and function of the Froude number, was adopted. The proposed formulations are:

푘푤 푘푤 (푊 − 푤 ) (푘 + 2 + 3 ) 퐹 2, 퐹 < 퐹 퐻 푤 푤1 퐹 2 푁 푁 푁푐푟 퐶푤 = { 푁푐푟 퐹푁푐푟 ( 13) ( ) 2 푊퐻 − 푤푤 (푘푤1 퐹푁 + 푘푤2 퐹푁 + 푘푤3 ), 퐹푁 ≥ 퐹푁푐푟

The factor (푊퐻 − 푤푤) accounts for the dependency from the displacement and is tuned by the term 푤푤. The values of 푤푤, 푘푤1 , 푘푤2 and 푘푤3 are to be tuned against the matrix of the known demihull solutions and were identified by a numerical optimization procedure that converges toward the combination of values that minimize the absolute difference between the analytical formulation and the computed CFD values. To best match the data, the boundary Froude number might differ from the theoretical critical value of 0.4 referred to the waterline length.

The Figure 9 compares the solutions of the analytical wave drag model with the CFD solutions at two displacement values while the Figure 10 compares the computed and the modelled total viscous and wave drag.

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0.0016 CFD (Disp. = 150 Kg) 0.0014 CFD (Disp. = 190 Kg) 0.0012 Analitic. (Disp. = 150 Kg)

0.001 Analitic. (Disp. = 190 Kg)

w 0.0008 C 0.0006

0.0004

0.0002

0 0 0.5 1 1.5 2 Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 FN

Figure 9. Wave drag coefficient (Flyer S A-Cat 2009).

Viscous drag Wave drag 40 10 CFD (Disp. = 150 Kg) 9 35 CFD (Disp. = 190 Kg) 8 Analitic. (Disp. = 150 Kg) 30

7 Analitic. (Disp. = 190 Kg)

25 6 20 5 4

Drag [Kg] Drag 15 Drag[Kg] 3 10 2 5 1 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Speed [knt] Speed [knt]

Figure 10. Computed and modelled hull resistance breakdown (Flyer S A-Cat 2009).

An additional factor that accounts for the drag increment due to the leeway angle was also added. Such dependency was assumed to be quadratic with leeway angle. From the CFD computations it was also observed to be linearly dependent to displacement and exponentially to velocity. The proposed factor to be included is:

휏훽 2 1 + 푘훽푉 (푊퐻 + 푤훽)훽 (14)

Also in this case, the terms 푘훽, 휏훽 and 푤훽 are tuned against the CFD database. The final analytical drag formulation assumes then the form:

1 퐷 = 휌 푉2푆 [(1 + 푘)퐶 + 퐶 ][1 + 푘 푉휏훽 (푊 + 푤 )훽2] (15) 퐻 2 푤 푤푒푡 푓 푤 훽 퐻 훽

Figure 11 compares the modelled hull drag increase due to the leeway angle with the CFD computations at two velocities.

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V = 10 knots V = 14 knots 16 30

15 28

14 26

13 24

12 22

Total[Kg] drag Total[Kg] drag

11 20

10 18 0 1 2 3 4 0 1 2 3 4 β [deg] β [deg] Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021

CFD (Disp. = 150 Kg) CFD (Disp. = 190 Kg) Analitic. (Disp. = 150 Kg) Analitic. (Disp. = 190 Kg)

Figure 11. Hull resistance rise due to leeway angle (Flyer S A-Cat 2009).

4.3 Implementation of Foils Shape Parametrization

Several geometric parameters play crucial roles in the optimization of the foils. Some simplifications were, however, adopted and some important variables neglected. The optimization of the airfoil, for instance, is an important parameter that significantly affects performance. Nevertheless, it was decided not to include it as variable of design to reduce the burden of the computations and because it can be deferred to a parallel stage once defined its operative working range at the several stations (Cella, et al., 2010). The foil twist was ignored. Even though this parameter is very important its inclusion would have required the setup to account for the several geometrical complexities related to the installation of the foils. The foils were analysed by fully turbulent RANS computations with the idea of demanding the verification of the laminar stability to a final post design analysis. The laminar flow, in fact, when the airfoil operates in the aerodynamic linear region, mainly affects the drag with a gain roughly proportional to the laminar flow percentage. It mildly affects the lift curve (Cella, et al., 2005). The correctness of the optimum search direction should be then maintained. In the view that this application serves as a proof of concept, such a set of simplifications should be considered reasonable.

The chosen foil geometry is composed by two straight portions blended in their junction area. To offer a slightly higher rolling stability, foils were connected to the external part of the hull with both segments – inner and outer – leaning inboard (Roskam, 1997). The hull-foil junction location was not assumed as a design variable. To obtain the foil segments, a straight extrusion of the NACA 63-412 laminar aerofoil was employed, with a constant section for the inner portion and a tapered chord for the outer part.

The design variables were:

1. total foil draft; 2. cant angle of the outer segment (angle 훿 on Figure 2); 3. angle between boat symmetry plane and inner segment; 4. chord of the inner segment (maintaining fixed the thickness of the foil); 5. taper ratio for the outer portion; 6. sweep angle of the foils. 164

Foil sweep angle was conceived as a trim, more than a shape parameter, having direct effect on the horizontal angle of incidence. Operatively this parameter was introduced in the optimisation as a foil rotation applied around an axis near its hull junction and perpendicular to the boat symmetry plane.

A total number of seven shape parameters were implemented using RBF morphing techniques: four, shown in Figure 12, to change the length and the angles of both wing portions; one to define the inner segment chord; one to control the taper ratio of the outer part and one to change the sweep angle. The morphing setup was conceived to respect the constraints defined by the rules of the class for all the possible amplification factors employed. To obtain this, all the shape modifiers were coupled with a logic based on geometrical relations, adjusting the amplification factors when needed (for instance, when modifying the cant angle, if the tip falls outside the allowed range, the outer section it is properly scaled to recover the limits shown in Figure 2). The shape parameters were activated in sequence. The morphing action was limited to a volume surrounding the foils by means of a bounding box. Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021

Figure 12. Foil’s front shape modifiers.

4.3.1 Integration in the Optimization Environment

In Figure 13 the optimisation workflow implemented is descripted. The foils shape of the numerical domain is modified by the RBF mesh morphing action at each cycle according to the suggestions of the optimisation algorithm. The candidates evaluation is managed by script procedures which runs the two analyses in sequence. The upwind analysis is run only in case of a successful downwind analysis.

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Starting geometry

Update domain

Downwind analysis Starting sinkage  maximum draft

Update sinkage CFD analysis in downwind conditions

Target lift Lift < target obtained within & max draft? No No tolerance? Upwind analysis Yes Yes Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021 Reject design

Run number Run = 1 Run = 3 Run = 2 Leeway angle = linear Leeway angle = 1 deg. Leeway angle = 2 deg. extrapolation of Run 1 and 2

CFD analysis in upwind conditions

Foils lift Target side force Run = 3? within tolerance? No No Yes Analytical hull drag model Yes

Reject design Obj. Func. 1: foils/hull drag upwind Obj. Func. 2: foils drag downwind

Decision making criterion

New shape parameters Ending criteria meet? Pareto solution No Yes

Figure 13. Flowchart of the optimization procedure.

The downwind analysis is first carried out at maximum draft, with the hull flying at 15 cm from water surface. If the generated lift is lower than the target the design point is discarded, otherwise the sinkage is updated. Once the equilibrium is achieved and the downwind solution generated, the upwind analysis is carried out. Three calculations are run to select the leeway angle that generates the needed side force. The candidate is discarded if the foils perform outside the linear aerodynamic polar region.

The software modeFrontier was employed to implement a multiobjective optimisation procedure (Aittokoski & Miettinen, 2008) setup to minimise the drag at both upwind and downwind conditions. The chosen optimisation algorithm, named MOGA-II (Poloni & Pediroda, 1998), is based on a genetic algorithm search criterion (Quagliarella, et al., 1997). The analytical model used for the hull forces calculation in upwind condition was implemented in Scilab.

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5 SOLUTIONS

The complete evaluation of a single candidate required, adopting the coarse mesh, between 15 and 20 minutes using a workstation with 2 Intel Xeon E5-2680 processors at 2.8 GHz (for a total of 20 parallel physical processors). 2 minutes were consumed by the mesh morphing process. The solution reported here is the result of three-day calculation with a design space filled by 400 candidates. Almost 160 solutions were discarded because they did not fulfil the requirements in terms of target lift in the downwind analysis. The Pareto solution of the optimisation is shown in Figure 14.

19

18

17 Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021

16

15

14

13 Total[Kg] downwind drag

12 14 15 16 17 18 19 20 21 22 Total drag upwind [Kg]

Figure 14. Pareto solution of the final two-objectives optimization.

The optimal solution, extracted from the Pareto front, is highlighted in green. This point was considered as the best compromise in terms of upwind and downwind drag improvement. The baseline geometry, inspired by existing designs, is reported in the plot with a red dot. The drag reduction in upwind conditions, taking into account both the foils and the hull, is 7% while the reduction in downwind sailing is equal to 7.9%.

5.1 Post Design Verification

Thanks to the meshless nature of the RBF method, the same RBF solutions that were setup on the coarse mesh were applied to the fine mesh employed in the grid sensitivity analysis, to verify the performances of the obtained optimized solution. The results of this comparison are shown in Table 1 for both downwind and upwind sailing conditions.

Table 1. Foils drag solutions.

Mesh Baseline Optimized Drag reduction Kg Kg % Coarse 14.7 13.54 7.89 Downwind Fine 13.99 12.92 7.65 Coarse 16.55 15.4 6.96 Upwind Fine 16.85 16.5 2.08

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The results showed that in downwind conditions the drag reduction, computed with the coarse mesh, was overestimated of only the 0.24% with respect to the same configuration employing the fine grid. This result confirms that the solutions of the optimisation procedure can rely on the coarse mesh. The drag reduction in upwind condition is, conversely, significantly overestimated adopting the coarse mesh. The difference in this prediction can be ascribed to a separation in the hull-foil junction, present in both the baseline and optimised configurations (Figure 15), that was not captured by the coarse mesh. The difference, however, does not affect the search direction of the optimisation procedure, being the phenomenon expected to be present in all candidates. Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021

Figure 15. Flow separation in the hull/foil junction computed adopting a fine mesh.

The separation observed adopting the fine grid, causing the performance reduction, is generated as interference between the hull wall and the foil and is a typical occurrence in aircraft design when dealing with wall junctions. The impact of the flow separation can be reduced by an approach based on local shape modification as shown in a similar case in (Biancolini, et al., 2016) where an improvement of about 35% in efficiency was achieved on a manoeuvring glider experiencing flow separation in the wing-fuselage junction area.

In Figure 16 the wake on the free surface of the optimised geometry in downwind sailing conditions is visualized.

Figure 16. Free surface in downwind sailing generated by the optimized solution.

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6 CONCLUSIONS

In this paper a workflow for the multiobjective optimisation of flying multihulls appendages was described. To demonstrate the performance of the method, it was applied to a highly constrained design problem: the design of the foils of an A-Class catamaran. The test case is adopted as proof of concept with some simplifications but the method can be applied to several types of flying sailing boat configurations. The flexibility of the adopted geometric parameterization strategy coupled with the RANS approach selected for the analysis module offers the designer the possibility to implement an optimization environment whose level of accuracy is limited only by the computational resources available.

The optimisation procedure is based on a meshless parameterisation of the numerical domain by means of RBF mesh morphing, coupled with a genetic-based searching criterion to reduce the drag in both upwind and downwind sailing conditions. The workflow consists of three main blocks that perform the following tasks: Downloaded from http://onepetro.org/JST/article-pdf/6/01/151/2478351/sname-jst-2021-09.pdf by guest on 24 September 2021

• generate of the candidates by means of geometrical variations driven by RBF mesh morphing; • perform the computations in upwind and downwind sailing (setting the vertical and horizontal equilibrium by sinkage and leeway angle update) • compute the objective functions.

To evaluate the total drag in upwind conditions, in which the boat is not fully flying, an analytical model able to estimate the hull forces was employed to integrate the result of the CFD computation. The hull drag is provided using a function implemented in Scilab.

The results of the optimisation procedure allowed for the identification of a solution whose drag was estimated 7% in upwind and 7.9% in downwind conditions lower than the baseline geometry. To validate this result, the computation was repeated adopting a very fine mesh. The performance in downwind conditions was confirmed with an error of 0.24% while in upwind sailing the performance improvement was significantly downsized. The coarse mesh, in fact, did not allow the capture of a separation region in the junction area between the hull and the foil that affected the results. These solutions, nevertheless, confirm that the optimal descent direction is correctly evaluated independently from the mesh adopted.

RBF mesh morphing is thus demonstrated to be a powerful shape parameterization approach to be adopted in such complex strongly constrained optimisation environments. Within the advantages of RBF mesh morphing over standard methods, we can mention the robustness of the procedure and the possibility of being strongly parallelized. The ability to take advantage of HPC environments makes this approach very powerful for heavy high-fidelity calculations with large computational domains. The workflow here described can be adapted to the design of any sailing boat of any class and dimension.

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