High-fidelity Coupled Hydrostructural Optimizationof a 3-D HydrofoilNitin Garg1, Gaetan K. W. Kenway2, Joaquim R. R. A. Martins2, Yin L. Young1,*

SYM

POSI

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ON ROTATING MACHIN

ERY

ISROMAC 2016

InternationalSymposium on

TransportPhenomena and

Dynamics ofRotating

Machinery

Hawaii, Honolulu

April 10-15, 2016

AbstractWith recent advances in high performance computing, computational fluid dynamics (CFD) modelinghas become an integral part in the engineering analysis and design of marine vessels and propulsors. Apreviously developed 3-D compressible Reynolds-averaged Navier–Stokes (RANS) solver extended tosolve for nearly incompressible flows, coupled together with a structural finite element solver, is usedfor the high-fidelity hydrostructural optimization of a 3-D hydrofoil. The coupled hydrostructural solveris validated against the experimental measurements of a tapered hydrofoil from Zarruk et al. [1]. Thecoupled hydrostructural tool has been integrated with the adjoint method to enable efficient high-fidelity,optimization with large number of shape design variables. The coupled hydrostructural optimizationstudy for an unswept, cantilevered, tapered 3-D NACA 0009 hydrofoil made out of solid aluminum wascarried out. A total of 210 shape design variables were used in the optimization study, with constraintson the lift coefficient (CL), cavitation number, and maximum stress (based on the fatigue strength ofaluminum). The hydrostructurally optimized foil results in an increase in efficiency (CL{CD) of 12.4%,a reduction in foil mass of 7.3%, and an increase in cavitation inception speed by 45%, over the originalNACA 0009 hydrofoil, while satisfying the cavitation constraint and the maximum stress constraints.

Keywordshydrostructural design optimization, gradient-based optimization, cavitation, hydrofoil, propulsor.1Department of Naval Architecture and Marine Engineering, University of Michigan, MI 48109, USA2Department of Aerospace Engineering, University of Michigan, MI 48109, USA*Corresponding author: ylyoung@umich.edu

INTRODUCTIONThe shape of a vehicle that moves within a fluid (air or water)has a large impact on the vehicle performance. Typically, alow drag and mimimal weight are desired, and in many cases,a certain amount of lift is required. The development of com-putational fluid dynamics (CFD) based aerodynamic shapeoptimization over the last few decades has been motivated bythese goals, and has had a large influence on how aircraft aredesigned [2, 3]. However, these developments have not beenutilized to their full extent in maritime applications, such asmarine propulsors, turbines, hydrofoils and control surfaces.As 90% of the world trade still happens by maritime trans-portation, the energy-efficiency of maritime transport can becritical to the global sustainability as well as economic growthworldwide.

Over the last decade, significant advances have been madein the design and analysis of marine propulsors. However,most of the tools in the literature either only considered hydro-dynamic effects, use low-fidelity methods, or use high-fidelitymethods with few design variables (typically less than 15). Re-searchers have performed hydrodynamic shape optimizationof hydrofoils using potential flow methods without cavitationconsideration [4] and with cavitation consideration [5, 6, 7, 8].Few researchers also carried out high-fidelity hydrodynamicshape optimization for hydrofoils, naval vehicles, and catama-rans [9, 10, 11, 12]. They used gradient-free algorithms foroptimization, which limited the number of design variables toless than 15 due to the large number of function evaluationscompounded with the high computational cost of high-fidelitysolvers. The challenge of performing high-fidelity hydrody-

namic shape optimization with respect to large number ofdesign variables (¡ 200), with consideration for cavitation,has been tackled in Garg et al. [13, 14] through the use ofgradient-based algorithms together with the adjoint method toefficiently compute the required gradients. In Garg et al. [13],the authors concluded that an efficient high-fidelity hydrody-namic shape optimization method capable of handling largenumber of design variables is necessary to get a fully con-verged optimal design for highly loaded conditions, whereviscous effects may not be negligible.

In aircraft wing design, Kenway et al. [15] carried outcoupled aerostructural optimization of the Common ResearchModel (CRM) wing, with 472 geometric and structural vari-ables and 57 constraints. They minimized the fuel burn sub-ject to lift, pitch moment, geometric and structural constraints.The optimization minimized the fuel burn by up to 11.2%.While the aerostructural optimization has been successfullyapplied in aircraft wing design, maritime applications bringadditional challenges such as higher loading, stronger fluid-structure interaction, as well as the potential susceptibility tofree-surface, cavitation, and hydroelastic instabilities.

Previously, some research has been carried out to opti-mize the design of composite marine propellers, using po-tential flow methods to compute hydrodynamic performanceand the Finite Element Analysis (FEA) to compute the struc-tural performance [16, 17]. Plucinski et al. [11] carried outoptimization of self-twisting marine propeller for many in-flow conditions, using a coupled Boundary Element Method(BEM)–Finite Element Method (FEM). However, as they used

High-fidelity Coupled Hydrostructural Optimization of a 3-D Hydrofoil — 2/9

gradient-free genetic algorithm as the optimizer, the numberof design variables were restricted to the ply angles and theirstacking sequence for the composite propellers. They alsodid not include any shape design variables. In [18], Mot-ley et al. carried out global performance-based design andanalysis for rigid and flexible marine propellers, also using3-D coupled boundary element method-finite element method(BEM-FEM). Thus, there is a need for an efficient, high-fidelity 3-D coupled hydrostructural design optimization toolthat can handle the large number of design variables and canenforce constraints on the maximum stress, as well as avoidor delay cavitation.

1. OBJECTIVESThe objective of this work is to present an efficient high-fidelity hydrostructural optimization tool for 3-D lifting sur-faces operating in viscous and nearly incompressible fluids,with consideration of cavitation and maximum allowablestress. An unswept, cantilevered, tapered NACA 0009 hy-drofoil is presented as a canonical representation of morecomplex lifting surfaces like propellers, turbines, and othercontrol surfaces.

2. METHODOLOGYThe tool used for optimization is modified from the Multi-disciplinary Design Optimization (MDO) of Aircraft Config-urations with High-fidelity (MACH) [19, 15]. The MACHframework has the capability of performing static aeroelas-tic (aerostructural) optimization that consists of aerodynamicshape optimization and structural optimization. In [13], theMACH framework was extended for hydrodynamic shapeoptimization of lifting surfaces in viscous and nearly incom-pressible flows, with consideration for cavitation. Fluids andstructures are tightly coupled disciplines in marine propulsordesign, as slight changes in the geometry can lead to signifi-cant changes in the hydrodynamic performance and internalstress distributions. The focus of this work is to present state-of-art high-fidelity hydrostructural optimization, with largenumber of design variables. The hydrostructural optimizationtool can be divided into following components: CFD solver,structural solver, geometric parametrization, mesh perturba-tion, load and displacement transfer algorithm, and optimiza-tion algorithm.

2.1 CFD SolverThe CFD solver used in this paper is Stanford UniversityUniversity of Michigan multi-block Automatic Differentia-tion solver (SUMad) [20]. SUMad is a finite-volume, cellcentered multi-block solver for the compressible flow equa-tions, and is already coupled with an adjoint solver (explainedin Section 2.6) for optimization studies [21]. The Jameson–Schmidt–Turkel [22] scheme (JST) augmented with artificialdissipation is used for spatial discretization. An explicit multi-stage Runge–Kutta method is used for the temporal discretiza-tion. The one-equation Spalart–Allmaras (SA) [23] turbulencemodel is used in conjunction with the adjoint optimizer. InGarg et al. [13, 14], a low-speed (LS) preconditioner wasimplemented in SUMad, to solve problems involving nearly

incompressible flows for Mach numbers as low as 0.01. TheLS SUMad RANS solver was validated against the experimen-tal measurements [1] and verified against commercial CFDsoftware (ANSYS CFX).

2.2 Structural SolverThe structural solver used in the MACH framework is theToolkit for the Analysis of Composite Structures (TACS) [24].TACS is a parallel, 3-D finite-element solver designed specif-ically for the analysis of stiffened, thin-walled, compositestructures using either static linear or geometrically nonlinearstrain relationship. In the current work, only linear structuralanalysis is used. For linear analysis, the governing equationfor structural discipline residual can be written as,

Spuoq � Kuo � F (1)

where S are the structural residuals, uo is the vector of struc-tural displacements, K is the linear stiffness matrix, and F isthe load vector. Similar to SUMad, the gradients in TACS arecomputed using an adjoint method, explained in Section 2.6,making the cost of the gradient calculation nearly independentof the number of structural design variables. The derivativeof the maximum von–Mises stress is analytically evaluatedusing the Kreisselmeir–Steinhauser (KS) aggregation func-tions [25, 26]. In the present research, as the foil is assumed tobe made of solid homogeneous isotropic material with linearelastic response. There is a constraint on the maximum allow-able stress, but there are no explicit structural design variables.However, the structure will undergo changes in shape as perthe shape variables in the hydrodynamic optimization.

2.3 Geometric ParametrizationThe free-form deformation (FFD) volume approach was usedto parametrize the geometry [19]. To get a more efficient andcompact set of geometric design variables, the FFD volumeapproach parametrizes the geometric changes rather than thegeometry itself. All the geometric changes are performed onthe outer boundary of the FFD volume. Any modificationof this outer boundary can be used to indirectly modify theembedded objects. Figure 4 displays an example of the FFDcontrol points used for optimization of the cantilevered, ta-pered NACA 0009 hydrofoil, which will be explained in detailin Section 4.2.4 Mesh PerturbationAs the geometry is modified during the optimization usingthe FFD volume approach, the mesh needs to be perturbed tocarry out the CFD analysis and FEA analysis for the modifiedgeometry. The mesh perturbation scheme is a hybridizationof algebraic and linear-elasticity methods [19]. In the hybridwarping scheme, a linear-elasticity-based warping schemeis used for a coarse approximation of the mesh to accountfor large, low-frequency perturbations; the algebraic warpingapproach is used to attenuate small, high-frequency perturba-tions. For the results shown in this paper, the hybrid schemeis not required, and the algebraic scheme is used becauseonly small mesh perturbations were needed to optimize thegeometry.

High-fidelity Coupled Hydrostructural Optimization of a 3-D Hydrofoil — 3/9

2.5 Load and Displacement Transfer Algo-rithm

The CFD and the structural solvers explained in Section 2.1and 2.2, respectively, are coupled to form a single hydrostruc-tural solver capable of predicting the coupled hydrodynamicand structural performance. In other words, hydrodynamicloads (pressure and shear stresses) computed by the SUMad(hydrodynamic solver) must be transferred to TACS (struc-tural solver), and displacements from TACS must be usedto deform the geometry. Kenway et al. [19] used the loadand transfer scheme from the work of Brown [27]. They con-structed rigid links between the CFD surface mesh points,and the points on the structural model lying closest to thisset of points. The rigid links are used to extrapolate the dis-placements from the structural surface to the CFD surface.The consistent force vector is determined by employing themethod of virtual work, ensuring that the force transfer isconservative. The integration of force is performed on theCFD mesh and is transferred to the structure through the rigidlinks. The two major advantage of using this scheme are that(1) it is consistent and conservative by construction, and (2)it may be used to transfer loads and displacements betweenCFD and structural meshes that are not coincident, such asthe cases with different fluid and structural meshes.

To solve the coupled hydrostructural equations, the hydro-dynamic analysis is first partially converged, and the forces areevaluated. These forces are then transferred to the structuralanalysis, and the corresponding displacements are computed.Thereafter, the displacements are transferred back to the hy-drodynamic analysis, the geometry and the correspondingmesh is deformed, and a new CFD solution is found. This it-erative loop continues until the coupled convergence criterion(25 iterations or when the relative decrease in both the aero-dynamic and structural residuals compared to the free-streamvalues is less than 1.0E-05) is met.

2.6 Optimization AlgorithmThe evaluation of the CFD solutions and FEA are the mostexpensive component of hydrostructural shape optimizationalgorithms, which can take up to several hours, days or evenmonths. Thus, for large-scale optimization problems, the chal-lenge is to solve the problem to an acceptable level of accuracywith as few CFD or FEA evaluations as possible. There aretwo broad categories of optimization, namely, gradient-freemethods and gradient-based methods. Gradient-free methods,such as genetic algorithms (GAs) and particle swarm opti-mization (PSO), have a higher probability of getting close tothe global minima for problems with multiple local minima.However, gradient-free methods can lead to slower conver-gence and require larger number of function calls, especiallywith large number of design variables (on the order of hun-dreds) [28]. To reduce the number of function evaluations forcases with large number of design variables, gradient-basedoptimization algorithm should be used. Efficient gradient-based optimization requires accurate and efficient gradientcalculations. There are some straight forward algorithms likefinite difference; they are neither accurate nor efficient [29].

Finite difference methods are sensitive to the step-size. If thestep-size is higher or lower than the optimal step-size, theerrors in FD will increase, as noted by [29]. Complex-stepmethods yields accurate gradients, but are not efficient forlarge-scale optimization [29]. Thus, for gradient calculations,the adjoint method is used in this paper. The adjoint method isefficient as well as accurate, but is relatively more challengingto implement [21].

The optimization algorithm used in this paper is calledSNOPT (sparse nonlinear optimizer) [30]. SNOPT is a gradient-based optimizer that utilizes a sequential quadratic program-ming method. It is capable of solving large-scale nonlinearoptimization problems with thousands of constraints and de-sign variables.

3. MODEL SETUP AND VALIDATION3.1 Fluid ModelFor all the results presented in this paper, a cantilevered,unswept, tapered NACA 0009 hydrofoil was studied withan aspect ratio of 3.33 (a span length of 0.3 m and a meanchord length of 0.09 m, with maximum chord length of 0.12m at the root and a minimum chord length of 0.06 m at the tip),Reynolds Number, Re � 1.0 � 106, and V � 12.4 m/s. Inthe present work, Reynolds number (Re � V c{ν) is definedbased on the mid-span chord length, i.e. 0.09 m. The CFDmesh used for all the optimization and validation results isshown in Figure 1. The CFD mesh used is a structured O-gridwith 515,520 cells and a y� of 1.1. The domain size is 30chord lengths in all the directions. The tip of the hydrofoil ismodeled rounded, which helps in making the mesh continu-ous and smooth. The applied boundary conditions (B.C.) areshown in blue color in Figure 1.3.2 Structural ModelTo carry out the validation study of the hydrostructural solverand the coupled hydrostructural optimization, the cantilevered,tapered 3-D NACA 0009 hydrofoil (model shown in Sec-tion 3.1) was studied. Figure 2 shows the structural meshused for the computation. The mesh has 44,800 second or-der isoparametric 8-noded elements, with 7 elements alongthe thickness direction, 40 elements in the chordwise direc-tion, and 160 elements in the spanwise direction. A perfectlyclamped boundary condition is set at the root, i.e., the dis-placement and rotation is zero at the root.3.3 Eigenvalue Validation and VerificationTo validate the structural model, a 3-D cantilevered, taperedNACA 0009 hydrofoil (as shown in Section 3.1, 3.2), withRe � 1.0� 106 and V � 12.4 m/s was studied. The predic-tions were compared with experimental measurements con-ducted at the Cavitation Research Laboratory (CRL) variablepressure water tunnel at the University of Tasmania [31]. Theoperating velocity and pressure range in the tunnel was of 2 -12 m/s and 4 - 400 kPa, respectively. The tunnel test section is0.6 m square by 2.6 m long. They tested four foils of similargeometry but with different materials, namely, SS (stainlesssteel-316L), Al (Aluminum-6061T6), CFRP-00, and CFRP-30 (CFRP are Carbon Fiber Reinforced Polymer compositeswith the number accompanying them denoting the alignment

High-fidelity Coupled Hydrostructural Optimization of a 3-D Hydrofoil — 4/9

Figure 1. CFD Mesh (515,520 cells) used for the RANSoptimization of a cantilevered, tapered NACA 0009 hydrofoil atRe � 1� 106. a) Front view with the boxed in portion showing thelocation of the foil. b) Side view with boxed in portion showing thelocation of the foil. c) The zoomed-in view of the foil, showing themesh near the foil leading edge (LE) and trailing edge (TE). d)Geometry with the dimensions of the tapered NACA 0009 hydrofoil.

Figure 2. Mesh (44,800 elements) used for structural validationas well as hydrostructural optimization. The mesh has 44,800second order isoparametric 8-noded elements, with 7 elements alongthe thickness direction, 40 elements in the chordwise direction, and160 elements in the spanwise direction. A perfectly clampedboundary condition is set at the root, i.e., the displacement androtation is zero at the root. The red curve depicts the edges of thefluid model.

of unidirectional fibers). The geometry dimensions were se-lected such that confinement effects are negligible. Theyreported estimated measurement uncertainty of 0.5% in theforce measurement, of 0.001o in the angle of attack, and themaximum uncertainty of 13% for the tip bending deflections.

Table 1 shows the parameters used in the experiment.All the parameters (geometry and its dimensions, Reynoldsnumber, fluid density, velocity, material of the hydrofoil) fromthe experiments were matched in the numerical model, except

for the Mach number (defined as M � V {a, where V is theinflow velocity and a is the speed of sound). However, as thecompressibility effects are almost negligible for Mach numberless than 0.1, this discrepancy in Mach number (0.008 in theexperiments and 0.05 in the numerical model) should notaffect the solution. Table 2 displays the material propertiesof the Aluminum (Al) foil, which is used for validation andoptimization studies in this work.

Table 1. Parameters used for an unswept, cantilevered, taperedNACA 0009 hydrofoil in the experimental results from [1].

Parameter Experiment [1]Geometry NACA 0009Aspect ratio 3.33Reynolds number 1� 106

ρf (fluid density) (Kg/m3) 999Mach number 0.008Vf (m/s) 12.4Material Aluminum (6061 T6)Fluid Water

Table 2. Structural properties of Aluminum(Al) used in thepresent research for validation and the optimization study, whichcorresponds to the values reported for the Al foil in [1].

Description Alρs (Kg/m3) Solid density 2700Es (GPa) Elastic modulus 69νs Poisson’s ratio 0.33σs,f (MPa) Fatigue strength 96.5

In Table 3, comparison of the predicted and measured firstmode frequency of the Al foil in air are presented. The pre-dictions from structural solver, TACS, were validated againstthe experimental measurements [1] and verified against pre-dictions from the commercial Finite Element Analysis (FEA)software (NASTRAN). TACS was found to under-predict thefirst mode in-air frequency by less than 4.0%, when comparedto the experimental measurements, and under-predict by lessthan 0.5%, when compared to the commercial FEA software.

Table 3. Comparison of first mode in-air frequency from thestructural solver, TACS, for an unswept, cantilevered, taperedNACA 0009 hydrofoil with the experimental results from [1] andthe commercial FEA software (NASTRAN). The structural mesh isshown in Figure 2.

Material Experiment [1] Nastran TACSAl (Hz) 100.0 98.5 98.4

3.4 Hydrostructural ValidationTo validate the coupled hydrostructural solver, a 3-D can-tilevered, tapered NACA 0009 hydrofoil (as shown in Sec-tion 3.1, 3.2), with Re � 1.0 � 106 and V � 12.4 m/s was

High-fidelity Coupled Hydrostructural Optimization of a 3-D Hydrofoil — 5/9

studied. In Figure 3, the predicted lift force (in Newtons)and the predicted tip bending deformation (in mm) at vari-ous angles of attack from modified MACH framework wascompared with the experimental measurements from [1]. Thepredictions were found to be within the experimental error(0.5% for the lift force and 13% for the bending deformation).For Re � 1.0� 106, experiments were only conducted untila maximum angle of attack of 6o, to avoid excessive forces onthe foil.

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Figure 3. Comparison between the predicted lift force (in N) andthe predicted tip bending deformation (in mm) at various angle ofattack (α) obtained using the modified MACH framework with theexperimental results from [1] for a cantilevered, tapered Al NACA0009 hydrofoil. The CFD mesh with 515,520 cells (y� � 1.1) forMACH framework is shown in Figure 1 and the structural mesh with44,800 8-noded elements is shown in Figure 2. The conditionsselected corresponds to Re � 1.0� 106 and V � 12.4 m/s, asreported in Table 1.

3.5 Grid Convergence StudyTo ensure that the structural results are independent of themesh size, the grid convergence was studied with four dif-ferent mesh sizes: 25,200 cells, 44,800 cells, 100,800 cells,and 179,200 cells for 3-D cantilevered, tapered NACA 0009hydrofoil (as shown in Section 3.1), with Re � 1.0 � 106,V � 12.4 m/s, and α � 6o. To be consistent, a medium-sizedCFD mesh of 515,520 cells was used for all the analyses.Grid convergence study of the CFD mesh has already beenpresented in [13]. Table 4 shows the comparison of lift co-efficient (defined as CL � Lift{0.5ρfV 2A), lift force (N),first mode in-air frequency (Hz), and tip bending deformation(mm) values from coupled hydrostructural simulation. Basedon the results and the computational time, the medium-sizedstructural mesh with 44,800 elements was selected to carryout the hydrostructural analysis and optimization.3.6 Optimization Problem FormulationTo demonstrate the advantages of hydrostructural shape op-timization, results are shown for the 3-D cantilevered, ta-pered NACA 0009 hydrofoil (as shown in section 3.1, 3.2),with Re � 1.0 � 106 and V � 12.4 m/s. The optimiza-tion problem setup is described in Table 5. The drag co-efficient, defined as CD � Drag force{0.5ρfV 2A, is mini-mized for a given CL, a given cavitation number, definedas σ � pPref � Pvapq{0.5ρfV

2, and the constraint on maxi-mum von-Mises stress (σv) of the optimized shape against the

fatigue strength (σf ) of Al with a factor of safety of 1.2. Con-straint on the minimum volume and minimum thickness arealso detailed in Table 5. C�L is the target CL, @ is the volumeof the optimized foil, @base is the volume of the original foil,tbase is the thickness of the original foil at a given section. Theleading edge of the hydrofoil is also constrained to deform toprevent any random behavior at the leading edge.

The influence of the number of design variables was stud-ied in [13], where the authors concluded at least 200 FFDcontrol points are necessary to achieve a converged optimalhydrodynamic design for the same unswept cantilevered, ta-pered hydrofoil. Thus, for the results shown in Section 4, atotal of 210 design variables were used with 200 FFD controlpoints (10 spanwise � 10 chordwise � 2 thickness) and 10spanwise twist design variables.

Figure 4 shows a pictorial description of the design vari-ables with 200 FFD control points and 10 twist design vari-ables on the right side. On the left side in the Figure 4, theoriginal NACA 0009 hydrofoil with the deflected shape atCL � 0.65 is displayed.

Figure 4. Pictorial description of the problem setup used for theoptimization study. Original cantilevered Al NACA 0009 hydrofoilwith the initial and the deflected shape is displayed on the left sideand the shape design variables; 200 FFD control points (10 spanwise� 10 chordwise � 2 thickness, as indicated by the red circles), andtwist design variables on the right side.

4. RESULTS AND DISCUSSIONIn this section, the single-point RANS-based coupled hy-drostructural optimization results are presented for a can-tilevered, unswept, tapered aluminum (6061 T6) NACA 0009hydrofoil at Re � 1.0 � 106 and V � 12.4 m/s. The foilwas optimized to achieve the minimum drag coefficient (CD)for a target lift coefficient (CL), a cavitation number (σ) of1.6, and a constraint on the material stress against the fatiguestrength of Al with a factor of safety of 1.2. The optimizationwas carried out with 210 design variables.

Figure 5 depicts the difference in 3-D geometry betweenthe cantilevered, tapered NACA 0009 hydrofoil and the single-point hydrostructural optimized foil at CL � 0.65. In Fig-ure 6, the stress constraint, i.e., σv ¤ σf {1.2, of the originalNACA 0009 hydrofoil and the hydrostructural optimized foilat CL � 0.65, at various CL values is presented. The blackdotted line represents the stress constraint, i.e. σv � σf {1.2.While the original tapered NACA 0009 hydrofoil violates thestress constraint forCL ¡ 0.35, the hydrostructural optimizedfoil satisfies the stress constraint for CL values up to 0.65,

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Table 4. Comparison of CL, lift force (N), first mode in-air frequency (Hz), and tip bending deformation (mm) values from the coupledhydrostructural simulation for the cantilevered, tapered Al NACA 0009 hydrofoil at Re � 1.0� 106, V � 12.4 m/s, and α � 6o usingdifferent structural mesh with the modified MACH framework. The same CFD mesh of 515,520 cells was used for all the analyses.

Mesh Mesh Size CL Lift Force (N) Frequency (Hz) Tip bending deformation (mm)Coarse 25,200 0.4980 1034.8 99.53 12.46Medium 44,800 0.4987 1035.2 98.43 13.02Fine 100,800 0.4990 1035.9 97.58 13.06Finest 179,200 0.4991 1036.1 97.26 13.36

Table 5. Optimization problem for the cantilevered, tapered Al NACA 0009 hydrofoil (shown in Section 3.1, 3.2).

Function variables Description Qty.minimize CD Drag coefficient 1

Design variables f FFD control points 200Twist design variables 10

Constraint C�L Lift coefficient constraint 10.5� tbase ¤ ti ¤ 1.5� tbase Minimum thickness constraint 400@ ¥ @base Minimum volume constraint 1

Fixed leading edge constraint 10�Cp   σ Cavitation number constraint 1σv ¤ σf {1.2 Stress constraint 1

the design point. In Figure 6, the predicted non-dimensionalbending deformation, i.e., the fraction of the maximum tipbending deformation to the tip chord length (0.06 m), of theNACA 0009 hydrofoil and the hydrostructural optimized foilat CL � 0.65, at various CL values is also presented. Whilethe original tapered NACA 0009 hydrofoil has the maximumnon-dimensional bending deformation of 0.20, the optimizedfoil at CL � 0.65 has the maximum non-dimensional bendingdeformation of 0.16.

Figure 5. Figure showing the comparison of 3-D geometrybetween the original tapered NACA 0009 hydrofoil and thehydrostructural optimized foil at CL � 0.65. The computationswere carried out for solid Aluminum (Al) hydrofoil atRe � 1.0� 106 and V � 12.4 m/s.

Figure 7 shows the comparison of efficiency (i.e.,CL{CD)at various CL values for the NACA 0009 hydrofoil and thesingle-point optimized foil at CL � 0.65. As can be observedfrom the figure, the hydrostructural optimized foil performsbetter than the original NACA 0009 foil for CL ¥ 0.3. BelowCL value of 0.3, the optimized foil performance becomes

worse than the original NACA 0009 hydrofoil, which is ex-pected as the foil was optimized for the CL � 0.65 conditiononly. Thus, there is a need to carry out probabilistic multipointoptimization, as shown in [13], to achieve a design capableof performing well under the full range of expected operatingconditions.

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Figure 6. Comparison between the predicted stress constraint((1.2� σvq{σs,f ) and the non-dimensional bending deformation(δmax{Ctip) of the original NACA 0009 hydrofoil and thehydrostructural optimized foil at CL � 0.65, at various CL values.The black dotted line represents the stress constraint, i.e.σv � σs,f {1.2. The computations were carried out for solidAluminum (Al) hydrofoil at Re � 1.0� 106 and V � 12.4 m/s.

Figure 8 shows a comparison between the NACA 0009 hy-drofoil and the hydrostructural optimized hydrofoil at CL �

High-fidelity Coupled Hydrostructural Optimization of a 3-D Hydrofoil — 7/9

0.65. For a given CL of 0.65, the hydrostructural optimizedfoil leads to a 11.3% reduction in CD when compared tothe original NACA 0009 hydrofoil. Figure 8c displays theskin friction (Cf � wall shear stress{p0.5ρfV 2)) contourplot for the original tapered NACA 0009 hydrofoil and theoptimized foil at CL � 0.65. To meet the stress constraint(σv ¤ σf {1.2), the optimal foil needed to be thicker nearthe root, where the stress is the highest, and is balanced bythe thinner sections at the tip, as shown in the sectional ge-ometry in Figure 8d. In Figure 8b, the difference betweenthe stress constraint contours, i.e. 1.2� σv{σf of Al, of theoriginal NACA 0009 hydrofoil and the coupled hydrostruc-tural optimization is presented. The red region, indicatingwhere the stress constraint is violated, can be noticed near theroot of the original NACA 0009 hydrofoil in the Figure 8b.While meeting the maximum stress constraint, the mass ofthe optimized foil is also reduced by 7.3%. The mass reduc-tion while satisfying the structural constraints is particularlyimpressive considering the NACA 0009 hydrofoil is a verythin and efficient foil to start with. The optimized designhas a significant camber resulting in higher moment of inertia,which makes the structure more rigid. Thus, the optimized foilhas much higher stiffness and lower stress, compared to theoriginal NACA 0009 hydrofoil. Figure 8a indicate that partialleading edge cavitation (as indicated by gray contour regionwith �Cp ¥ σ) will develop around the original NACA 0009hydrofoil at CL � 0.65 and σ � 1.6, but no cavitation isobserved for the optimized foil. The maximum negative pres-sure coefficient, �Cp, reduces from 2.1 for the NACA 0009hydrofoil to 1.0 for the optimized foil, as shown in Figure 8d,which will help to significantly delay cavitation inception. Inother words, the cavitation inception speed for the coupledhydrostructural optimized foil will increase from 10.2 m/s to14.8 m/s, for an assumed submergence depth of 1 m.

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12.4 % increase

Figure 7. Figure showing the comparison of CL{CD versus CL

for the original tapered NACA 0009 hydrofoil and thehydrostructural optimized foil at CL � 0.65, at various CL values.Note that the performance of the optimized foil exceeds that of theNACA 0009 not only at the design point (CL � 0.65), but also forall lift coefficients above 0.25. All the results were obtained usingcoupled hydrostructural solver for solid Aluminum (Al) hydrofoil atRe � 1.0� 106 and V � 12.4 m/s.

5. CONCLUSIONWe presented an efficient high-fidelity coupled hydrostructuraloptimization capable of handling a large number of designvariables, with constraints on cavitation and maximum stress.The hydrostructural solver was validated against experimentalmeasurements [1] for the hydrodynamic load coefficients,tip bending deformations, and in-air natural frequencies forthe case of a tapered, cantilevered, solid aluminum NACA0009 hydrofoil. Good agreements were observed between thenumerical predictions and experimental measurements. Themaximum difference in the force predictions is 3.5% and inthe tip bending deflections is 7.0%.

To provide a canonical representation of a complex hy-drodynamic lifting surface,we performed a RANS-based hy-drostructural optimization using the adjoint method for thesame NACA 0009 hydrofoil at Re � 1.0�106 and V � 12.4m/s.

The single-point high-fidelity hydrostructural optimiza-tion was performed at CL � 0.65 with 210 design variables.The optimized hydrofoil resulted in increase in CL{CD of12.4%, a mass reduction of 7.3%, and an increase in the cavi-tation inception speed by 45%, when compared to the originalNACA 0009 hydrofoil. While the original NACA 0009 hydro-foil fails the maximum stress constraint (i.e., σv ¡ σf {1.2) forlift coefficients greater than 0.35, the optimized foil was ableto meet the stress constraint up to the design lift coefficient of0.65.

A systematic study of the design space of marine propul-sors using the present state-of-art high-fidelity coupled hy-drostructural optimization has the potential to drastically im-prove propulsors’s performance. Improved propulsor per-formance can include improved fuel efficiency and hence areduction in the operational cost and reduced negative envi-ronmental impact, enhanced agility, and reduced structuralweight, while ensuring the structural integrity and delayingcavitation inception.6. FUTURE WORKThe paper presents a state-of-art high-fidelity coupled hy-drostructural design optimization framework, capable of han-dling large number of design variables with consideration forcavitation and maximum stress. While the optimization toolis demonstrated for hydrofoils, it can be easily extended toother lifting bodies, such as marine propellers and turbines,and to optimize the material properties and stacking sequenceof composite propulsors. As shown previously by the au-thors [13], multipoint optimization results in a much moreefficient design over the wide range of operating conditions,when compared to the single-point optimized design. Thus,coupled hydrostructural multipoint optimization will be car-ried out to take full advantage of this framework.ACKNOWLEDGMENTSSupport for this research was provided by the U.S. Officeof Naval Research (Contract N00014–13–1–0763), managedby Ms. Kelly Cooper. The authors would like to thank Prof.Paul Brandner of the Australian Maritime College for provid-ing the experimental data and answering questions about theexperimental setup.

High-fidelity Coupled Hydrostructural Optimization of a 3-D Hydrofoil — 8/9

Figure 8. Figure showing comparison of the performance of the original NACA 0009 hydrofoil and the single-point hydrostructuraloptimized hydrofoil at CL � 0.65, Re � 1.0� 106, and V � 12.4 m/s. A reduction in CD of 11.3% is noted for the optimized foil. Goingfrom top to bottom and from left to right. a) Cp (pressure coefficient) contours plot on the suction side are displayed for NACA 0009hydrofoil and the optimized foil. Gray contour region along the leading edge of the tapered NACA 0009 foil displays the area with �Cp ¥ σ.b) Comparative study of the stress coefficient contours (1.2� σv{σf ) of the NACA 0009 hydrofoil and hydrostructural optimization ispresented. Excessive stress (σv ¡ σf {1.2) region can be noticed near the root for the original NACA 0009 hydrofoil. Reduction in mass of7.3% for the optimized foil can be noted. c) Displays the skin friction (Cf ) contour plot for the original tapered NACA 0009 hydrofoil andthe optimized foil at CL � 0.65. In general, the optimized foil has a lower skin friction, and hence lower drag coefficient. d) Figures showthe sectional �Cp plots and the geometry profile of the foil at 3 sections along the span of the hydrofoil. Blue solid line represents the NACA0009 hydrofoil and the red solid line corresponds to the hydrostructural optimized foil. Grey horizontal line represents the constraint oncavitation number (when �Cp � σ � 1.6). As noted, there is significant decrease in maximum �Cp from the NACA 0009 to the optimizedhydrofoil. Difference in the sectional shape as fraction of the span (z{s) between the original and optimized foil are also shown in the bottomof each subplot.

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