Parametric Geometric Model and Hydrodynamic Shape

Parametric Geometric Model and Hydrodynamic Shape Optimization of AFlying-Wing Structure Underwater GliderWANG Zhen-yua, b, YU Jian-chenga, *, ZHANG Ai-quna, WANG Ya-xinga, ZHAO Wen-taoa, baState Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016,China

bUniversity of Chinese Academy of Sciences, Beijing 100049, China

Received June 27, 2016; revised June 2, 2017; accepted August 10, 2017

©2017 Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature

AbstractCombining high precision numerical analysis methods with optimization algorithms to make a systematic explorationof a design space has become an important topic in the modern design methods. During the design process of anunderwater glider’s flying-wing structure, a surrogate model is introduced to decrease the computation time for ahigh precision analysis. By these means, the contradiction between precision and efficiency is solved effectively.Based on the parametric geometry modeling, mesh generation and computational fluid dynamics analysis, asurrogate model is constructed by adopting the design of experiment (DOE) theory to solve the multi-objects designoptimization problem of the underwater glider. The procedure of a surrogate model construction is presented, and theGaussian kernel function is specifically discussed. The Particle Swarm Optimization (PSO) algorithm is applied tohydrodynamic design optimization. The hydrodynamic performance of the optimized flying-wing structureunderwater glider increases by 9.1%.Key words: surrogate model, underwater glider, design optimization, blended-wing-body

Citation: Wang, Z. Y., Yu, J. C., Zhang, A. Q., Wang, Y. X., Zhao, W. T., 2017. Parametric geometric model and hydrodynamic shapeoptimization of a flying-wing structure underwater glider. China Ocean Eng., 31(6): 709–715, doi: 10.1007/s13344-017-0081-7

1 IntroductionThe autonomous underwater glider (AUG) is a spe-

cially-designed autonomous underwater vehicle. It rises orsinks through the ocean by modulating its buoyancy andshifting the center of mass relative to the center of buoy-ancy to control the pitch and roll attitude. This concept of anunderwater glider was initiated by Henry Stomme (Stom-mel et al., 1989). Seaglider (Eriksen et al., 2001), Slocum(Webb et al., 2001), and Spray (Sherman et al., 2001) arethree currently well-known gliders that have slightly differ-ent designs from each other in terms of size, weight andconfiguration (Isa and Arshad, 2012). Advantages of under-water gliders in ocean sensing are as follows: longer dura-tion (missions), greater operational flexibility and lower-cost operations. Gliders are more mobile and flexible thanfixed moorings, more maneuverable than drifters, of largerrange than other AUVs, and do not need expensive supportvessels (Graver et al., 2003).

Several research results have been reported in the literat-ure regarding the optimization of the hydrodynamic per-formance of the underwater glider. Li et al. (2012) intro-duced the underwater glider in hydrodynamic shaping. Gu

et al. (2009) solved the multi-objects design optimizationproblem of the underwater glider by constructing a surrog-ate model. These legacy gliders have a low life-drag ratio(L/D) and limited gliding efficiency because the hull barelygenerates any lift force (Wang et al., 2015). Underwatergliders have been well studied because of their low-powermovement patterns. The flying-wing underwater glider is atailless fixed-wing underwater glider. A flying-wing designsignificantly improves its maximum lift to drag ratio byeliminating the body of the glider (Graver, 2005). De-veloped by the U.S. Office of Naval Research (ONR, 2006),Liberdade XRAY was designed for applications in the shal-low-water environment to obtain a greater payload carryingcapability, cross-country speed, and the horizontal point-to-point transport efficiency than other existing underwatergliders.

The flying-wing structure of the underwater gliderdesign is based on the air force’s blended-wing-body(BWB) configuration. Compared with the classic wing-and-tube fuselage configuration, the BWB has the following su-perior aerodynamic performances (Kroo, 2004; Qin et al.,2004). The reduction in the wetted area substantially re-

China Ocean Eng., 2017, Vol. 31, No. 6, P. 709–715DOI: 10.1007/s13344-017-0081-7, ISSN 0890-5487http://www.chinaoceanengin.cn/ E-mail: [email protected]

Foundation item: This work was financially supported by the National Natural Science Foundation of China (Grant No. 61233013).*Corresponding author. E-mail: [email protected]

duces the skin friction drag; the all-lifting design reducesswing loading and improves the spanwise lift distribution;and the smooth blended wing-center body intersection re-duces the interference drag. Despite various aerodynamicbenefits, the aerodynamic shape of the BWB also bringschallenges to the design process. The complex shape of theBWB may cause difficulties during manufacturing. Withoutthe conventional empennage, the chordwise lift distributionon the center body need to maintain a positive static margin.Jenkins et al. (2003) designed a 50 liter glider repackaged inthe planform of a Horten H-11 flying wing glider using athick low-Reynolds number section. Li et al. (2012) optim-ized the aerodynamic shape for the BWB transport. Theyput forward new design ideas for BWB. The cruise point,the maximum lift to drag point and the pitch trim point werethe main reference indices. The Office of Naval Research(ONR, 2006) developed an advanced underwater gliderbased on the air force’s BWB design of the ZRay. The ZRayexceeded a 10 to 1 glider slope ratio. Hildebrand et al.(2001) developed the latest generation of “Liberdade ZRay”model, which is a BWB design and the largest underwaterglider known to the world.

To balance the contradiction between precision and effi-ciency, the researchers put forward a surrogate model basedon the theory of design of experiments and approximatemethods. In the last decade, the agent model in aviation,automobile, shipbuilding and other industries of structuredesign, fluid analysis and multidisciplinary design optimiza-tion has gained extensive applications and in-depth develop-ment. Chiplunkar et al. (2016) used multi-output Gaussianprocesses to optimize flight mechanics to flight loads. Lee etal. (2016) optimized the aerodynamic and structural ana-lyses of multiple wing sails using surrogate models. Marrelet al. (2015) assessed sodium fast reactor accident by creat-ing a surrogate model and sensitivity analysis methods.

In this paper, the study focuses on the effect of the geo-metric parameters on the hydrodynamic performance, andachieved the maximum lift to drag ratio and the minimummoment by a CFD-driven optimization. First, the paramet-ric geometric model of the flying-wing structure underwa-ter glider is established, and the complex hydrodynamicconfiguration is characterized by nine shape parameters.Second, the Gaussian kernel function algorithm is used toestablish the surrogate model of the flying-wing structureunderwater glider. Finally, the particle swarm optimization(PSO) algorithm is used for efficient global optimization.After the optimization, the hydrodynamic performance of

the flying-wing structure underwater glider increased by9.1%. The relative sensitivity of each variable will then beanalyzed. Aiming to address the problem of multiple designvariables, the primary contribution of this paper is the estab-lishment of the parametric geometric model of the flying-wing structure underwater glider. The Gaussian kernel func-tion algorithm is innovative and is used to establish the sur-rogate model of the flying-wing structure underwater glider.Through analysis and optimization, the reasons for affect-ing the hydrodynamic performance of the flying-wing struc-ture underwater glider are explored in this paper. This willprovide the guidance for the shape design of the flying-wingstructure underwater glider, and the basis for the mul-tidisciplinary design and optimization of the flying-wingstructure underwater glider.

2 Parametric geometric model and numerical simula-tionsThe wetted surface area is one of the most important

factors affecting the skin friction. Therefore, the wetted sur-face area became the origin of the design. Three canonicalforms are shown in Fig. 1, and each has nearly 0.025 L ofthe volume displacement. The sphere is not streamlined al-though it has the minimum surface area. The two canonicalstreamlined options include a conventional axisymmetricbody and a disk body. Compared with the conventionalaxisymmetric body, a disk body can realize an 11% reduc-tion of the wetted surface area. As shown in Fig. 2, the geo-metric profile of the flying-wing structure glider, similarlyto the disk body, is blended and smoothed into the wings.This construction reduces the friction dray and providesenough lift.

Fig. 3 shows the flying-wing structure underwater glider

Fig. 1. Effect of body type on surface area.

Fig. 2. Conceptual geometric profile of the flying-wing structure glider.

710 WANG Zhen-yu et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 709–715

shape parameterization of planform. The disk body is blen-ded and smoothed into the wing by two arcs. The disk bodyincludes a front round nose and a backround nose. A typic-al, symmetrical NACA 0012 airfoil is used as a baseline air-foil. The airfoil of the flying-wing structure underwaterglider is designed with changing the thickness of NACA0012 (=12.0%c, c is the chord length of the local section).The thickness is defined at the centerline (t1c), the mergingpoint (t2c), and the section between the centerline and themerging point (tbodyc). tbody is determined by the followingequation.

tbody = ωt1+ (1−ω)t2, 0 < ω < 1. (1)

To decrease the number of design variables, we definenine shape parameters as shown in Table 1.

The hydrodynamic numerical simulations of the flying-wing structure underwater glider presented in this paperwere carried out using the commercial software CFX basedon the finite volume method. The fluid material selected inthis paper was liquid water, which was considered incom-pressible with a density of 998.2 kg/m3 and a dynamic vis-cosity of 1.003×10–3 Pa·s (Sun et al., 2015). The model se-lected the method of the standard wall function.

The governing equation for the cases considered in thispaper is the RANS equation for incompressible viscousflow, and a statistically steady solution is found. The RANSequation is given below:

∂(ρU)∂t+∇ · (ρU⊗U) = ρ f

+∇ ·{−pδ+μ

[∇U+

13

(∇U)T]}−∇ · (ρU⊗U), (2)

where ρ is the density of the fluid, t is the time variable, U isthe absolute velocity of the fluid, f is the unit mass of thevolume force, p is the pressure, μ is the dynamic viscosity, δis the unit matrix, is the Hamiltonian operator, is the tensorproduct and the superscript T is the matrix transpose.

[−10bt,15bt]m× [−10bt,10bt]m× [−5bt,−5bt]m

The shape of the computational domain is a cuboid. Byusing the span length (bt) as the reference length, the do-m a i n i s with 2–2.5 million hexahedral cells. The flying-wing struc-ture glider is placed at the origin of the coordinates. Asshown in Fig. 4, the structured grids with the hexahedralcells are used in all the calculations presented here.

3 Surrogate model

fθ(x)

A surrogate model is an integrated modeling techniqueof the experimental design and approximate method. Cur-rently, the response surface methodology (RSM), radicalbased functions (RBF) and neural networks are commonlyused with surrogate models, but the number of parametersin these three models will linearly grow with the increase inthe input variable dimension. The flying-wing structure un-derwater glider parameterized shape includes multipledesign variables. A Gaussian kernel function is used in thispaper to establish the surrogate model. In the surrogatemodel of the Gaussian kernel function, the number of para-meters is determined only by the CFX simulation sample,but not the dimension of the input variables. The Gaussiankernel function expresses the surrogate function as:

fθ(x) =m∑

j=1

θ jK(x, c) (3)

with

K(x, c) = exp(−∥x− c∥

2h2

); (4)

∥x− c∥ =√

(x− c)T(x− c), (5)

x = (x1, x2, · · · , xn)K(x, c)

where is an n-dimensional vector (n isthe number of design variables), is the model of the

Table 1 Shape parameters of flying-wing structure underwater gliderParameter Explanation Equation

Planform

AR Aspect ratio AR=bt/ctGR Gradient radio GR=croot/ctipCR Camber radio CR=Rf/cbDL Relative distance of the wing’s tip from the front round nose DL=L/ctDS Relative spanwise of the front arcs from the span DS=b/bt

Section

A Sweep angle A=at1 Thickness rate of the centerlinet2 Thickness rate of merging pointω Growth rate

Fig. 3. Flying-wing structure underwater glider design

variables of planform.

WANG Zhen-yu et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 709–715 711

c = (c1,c2, · · · ,cn)Gaussian kernel function, is the mid-value of each design variable, and h is the band width of theGaussian kernel function.

fθ(x)To ensure the surrogate function best fits with thesimulation date, a cost function is given below:

J(θ) =1

2m

m∑i=1

[fθ(x(i))− y(i)

]2, (6)

θ = (θ1,θ2, · · ·θ j, · · · ,θm)where is a m-dimensional vector (mis the number of CFX simulation examples), and y is theCFX simulation results.

J(θ)The gradient descent algorithm is used to minimize the

cost function and the equation is given below:

θ j := θ j−α∂∂θ j

J(θ). (7)

From Eqs. (6) and (7), we can obtain:

θ j := θ j−α1m

m∑i=1

[fθ(x(i))− y(i)

]x(i)

j , (8)

fθ(x)where θj is the parameter of , α is the learning rate, :=is the colon equal (a:=b means taking the value in b and useit to overwrite whatever the value of a), and superscript i isthe number of simulation dates.

Feature scaling is used in this paper to ensure that thedifferent design variables take on similar ranges of values;then, the gradient descents can converge more quickly, andthe equation is given below:

xi :=xi− ci

si, (9)

where ci is the average value of xi in the training sets, and siis the standard deviation of xi.

In the process of optimization, in addition to obtaining alift-to-drag ratio that is as large as possible, an attemptshould be made to reduce the moment. Based on the abovedescribed equations and the use of the weighted coefficientof multi-objective optimization problems, the objectivefunction is expressed as follows:

Y(x) = ω1FL

FD+ω2

1∣∣∣Mpitch∣∣∣ , (10)

ω1 and ω2where are the weighted coefficients, FL is the lift,FD is the drag, and Mpitch is the torque.

The efficient global optimization is a PSO algorithm.The PSO is computed as follows. The PSO is initialized to agroup of random particles (random solutions); then, the op-timal solution is found through iterations. In each of itera-tions, the particles update themselves by tracking the twoextreme values. The first value is the value of the particlesto find the optimal solution, which is called the individualextreme value. The other extreme value is called the globalextreme value, referring to the entire population being usedto find the optimal solution. Assuming that the goal is a onedimensional search space, and there are N particles in acommunity, of which the i-th particle is expressed as a D-di-mensional vector.Xi = (xi1, xi2, · · · , xiD) , i = 1,2, · · · ,N. (11)

The speed of the i-th particle is:Vi = (vi1,vi2, · · · ,viD), i = 1,2, · · · ,N. (12)

The i-th particle searches for the optimal position whichis known as the individual extreme value:pbest(i) = (pi1, pi2, · · · , piD), i = 1,2, · · · ,N. (13)

The whole particle swarm searches for the optimal loca-tion of the global extreme value:gbest(i) = (gi1,gi2, · · · ,giD), , i = 1,2, · · · ,N. (14)

In order to find these two optimal values, the particlesupdate their speed and position according to the followingequation.{ vid = ωvid + c1r1(pid − xid)+ c2r2(gid − xid)

xid = xid + vid(15)

where c1 and c1 are the acceleration constants; r1 and r2 arethe uniform random numbers in the range from 0 to 1; andω is the inertia weight. The procedure of the PSO algorithmis shown in Fig. 5 as follows:

1. Initialize the particle swarm, including populationsize N, position Xi and speed Vi of each particle.

Fig. 4. 3D structured grids.

Fig. 5. Procedure of the PSO algorithm.


Y(xi)2. Calculate for each particle.Y(xi) pbest(i)

Y(xi) > pbest(i) Y(xi) pbest(i)3. For each particle, compare its with , if

, use to replace .Y(xi) gbest(i)

Y(xi) > gbest(i) Y(xi) gbest(i)4. For each particle, compare its with , if

, use to replace .5. According to Eq. (15), update the particle’s speed Vi

and position Xi.6. End the iteration if the end conditions are met or re-

turn to Step 2.

J(θ)

J(θ)

Fig. 6 shows the optimization procedure of the flying-wing structure underwater glider surrogate model. First, Ndesign samples are selected using Latin Hypercube Samp-ling (LHS). The hydrodynamic performance of N samples isevaluated by numerical simulations using the commercialsoftware CFX. Second, the initial surrogate model is con-structed based on N sample data. Then, using the gradientdescent, it searches the parameters of the cost function based on the predicted value by the initial surrogate model.

is expected to decrease after every iteration, and the

J(θ)Y(x)

convergence is declared if decreases smaller than γ inone iteration. Finally, is found using the initial surrog-ate model and is evaluated by new numerical simulations,and the surrogate model is updated with N+1 sample dates.δ is the error between the surrogate model and the new nu-merical simulations.

4 Relative sensitivity analysisIn this paper, we determine the sensitivity of each vari-

able by establishing the covariance matrix of each variableand Y(x). Eq. (16) is established according to the surrogatemodel of the flying-wing structure underwater glider.

T =

x11 x1

2 · · · x111 Y(x1)

x21 x2

2 · · · x111 Y(x2)

....... . .

......

xn1 xn

2 · · · xn11 Y(xn)

(16)

The covariance matrix of Eq. (16) is:

Σ =

cov(x1, x1) cov(x1, x2) · · · cov(x1, x11) cov(x1,Y(x))cov(x2, x1) cov(x2, x2) · · · cov(x2, x11) cov(x2,Y(x))

......

. . ....

...

cov(x11, x1) cov(x11, x2) · · · cov(x11, x11) cov(x11,Y(x))cov(Y(x), x1) cov(Y(x), x1) · · · cov(Y(x), x11) cov(Y(x),Y(x))

(17)

The covariance between two random variables isdefined as:

cov(xi, x j)= E{(xi−E[xi])(x j−E[x j])

}, i, j = 1,2, · · · ,11,

(18)E[xi]where is the expectance of xi.

For the standard transform of Eq. (18), the sensitivity ofY(x) for each variable is:

R(xi,Y(x))=

∣∣∣∣∣∣ cov(xi,Y(x))√

cov(xi, xi) · cov(Y(x),Y(x))

∣∣∣∣∣∣ , i = 1,2, · · · ,11.

(19)

5 Results and discussionAccording to the parametric design of the flying-wing

structure underwater glider, the parameter ranges of thedesign space are shown in Table 2, which are determined byavoiding abnormal shapes.

k− ε

In the CFX calculation process, the inlet velocity at thefront of the fluid domain was set to be equivalent to theglider velocity, 1 m/s. The pressure at the end outlet of thefluid domain was set to be 0 Pa. Meshing was used tostrengthen the wall function, and the boundary layer num-ber was set to be 12. The first element height of the cells ad-jacent to the body was set to be smaller than 5y+ (Mulvanyet al., 2004) units for the compatibility with the turbu-lence model. An attack angle of 3° was used for each modelperformance evaluation. When searching J(θ)min, γ was setas 10–3. The tests showed that the X-ray has better hydro-

Table 2 Parameter ranges of the design spaceParameter Lower bound Upper bound

AR 1 1.7GR 2.5 10CR 0.14 1.2DL 0.5 1.2DS 0.17 0.33A 30° 50°t1 0.1 0.3t2 0.2 0.3ω 0.1 0.4

Fig. 6. Optimization procedure of the flying-wing structure underwaterglider surrogate model.


dynamic efficiency (ONR, 2006); thus, in this paper, the X-ray was used as the initial design for optimization.

The number and distribution of sample points are im-portant factors that affect the accuracy of the surrogate mod-el. The number of sample points should be sufficient so thatthese points can be uniformly spread over the design space.In this paper, the initial 50 samples were selected by theLHS, and the performance of the 50 samples was evaluatedusing the commercial software CFX. The initial surrogatemodel was then constructed based on this sample data. Aspreviously discussed, the objective functions are expressedas follows:

Y(x) = 0.6FL

FD+0.4

1∣∣∣Mpitch∣∣∣ . (20)

J(θ)J(θ) Y(x)

After 57 iterations, decreases smaller than γ. Fig. 7shows the convergence of the gradient descent of . as an additional sample was evaluated by using CFX, andthe new surrogate model was built with the data from 50samples. After 13 iterations, the error was smaller than 2%.The results of the optimization of the surrogate model basedon a Gaussian kernel function and RSM are shown in Table3. The error of the first-order and second-order RSM sur-rogate model is too large. The error of the Gaussian kernelfunction surrogate model is close to the error of the second-order RSM surrogate model, but its polynomial number isreduced by 2/3. This can largely reduce the computing time.

Y(x)maxWhen searching , the parameters of Eq. (15) areexpressed as follows:{

vid = 0.6vid +2r1(pid − xid)+2r2(gid − xid)xid = xid + vid

(21)

The optimization results of the flying-wing structure un-derwater glider are shown in Table 4.

Fig. 8 shows the pressure contour of the optimum andinitial design. The optimized and the initial designs have asimilar pressure distribution in that the negative pressurearea appears around the leading edge of the wing, and thepositive pressure area appears at the nose of the fuselageand the trailing edge of the wing. The initial design hasclustered pressure contour lines near the wing kink, whichindicates the presence of a shock. The optimal design showsthe parallel pressure contour lines with roughly equal spa-cing, indicating a shock-free solution.

Rb, t2,croot

Fig. 9 shows the relative sensitivity of each variable. Accor-ding to the sensitivities, Y(x) is sensitive to the changes in

and a, and is insensitive to the changes in otherdesign variables. The trailing edge radius Rb has a great in-fluence on the wet surface area and the mass distribution ofthe model, and it influences Y(x). The thickness rate of themerging point t2 and the length of the airfoil root croot de-termine the relative thickness and the wet surface area of theairfoil, which influence the friction of the model. The sweepangle a determines the mass distribution of the airfoil,which influences the torque of the model. From the above,the design with a sharper trailing edge and a thinner airfoilachieves a higher hydrodynamic performance.

Table 3 Results of the optimization of the surrogate modelSurrogate model Polynomial number Error (%)First-order RSM 10 24.95Second-order RSM 55 11.73Third-order RSM 184 1.83

Gaussian kernel function 63 1.78

Table 4 Hydrodynamic shape optimization result of the flying-wingstructure underwater glider

Initial design Optimal design Increase rateY(x) 7.08 7.73 9.1%

Fig. 7. Convergence of gradient descent of J(θ).

Fig. 8. Pressure distributions around the optimum and initial design.

Fig. 9. Sensitivity of Y(x) for each variable.


6 ConclusionsIn this paper, the shape of a new underwater glider was

designed with a BWB configuration, and the parametricgeometric model of this underwater glider has been builtwith nine design variables. The hydrodynamic numericalsimulation has been performed using the commercial soft-ware CFX with the incompressible RANS equations. TheGaussian kernel function algorithm, a gradient descent al-gorithm, is used in the hydrodynamic shape optimization ofthe flying-wing structure underwater glider. The results ofthis study suggest that compared with a conventional under-water glider, the flying-wing structure underwater glider hasa better hydrodynamic performance. After the optimization,the hydrodynamic performance increased by 9.1%. Thesensitivity of Y(x) to each of the design variables was ana-lyzed. According to the results, it was found that the para-meters of the wing shape and the shape of the trailing edgeare important factors affecting the hydrodynamic perform-ance.

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Parametric Geometric Model and Hydrodynamic Shape