Parametric geometric model and hydrodynamic shape optimization of a flying-wing structure underwater glider

Combining high precision numerical analysis methods with optimization algorithms to make a systematic exploration of a design space has become an important topic in the modern design methods. During the design process of an underwater glider’s flying-wing structure, a surrogate model is introduced to decrease the computation time for a high 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, a surrogate model is constructed by adopting the design of experiment (DOE) theory to solve the multi-objects design optimization problem of the underwater glider. The procedure of a surrogate model construction is presented, and the Gaussian kernel function is specifically discussed. The Particle Swarm Optimization (PSO) algorithm is applied to hydrodynamic design optimization. The hydrodynamic performance of the optimized flying-wing structure underwater glider increases by 9.1%.


Introduction
The autonomous underwater glider (AUG) is a specially-designed autonomous underwater vehicle. It rises or sinks through the ocean by modulating its buoyancy and shifting the center of mass relative to the center of buoyancy to control the pitch and roll attitude. This concept of an underwater glider was initiated by Henry Stomme (Stommel et al., 1989). Seaglider (Eriksen et al., 2001), Slocum (Webb et al., 2001), and Spray (Sherman et al., 2001) are three currently well-known gliders that have slightly different designs from each other in terms of size, weight and configuration (Isa and Arshad, 2012). Advantages of underwater gliders in ocean sensing are as follows: longer duration (missions), greater operational flexibility and lowercost operations. Gliders are more mobile and flexible than fixed moorings, more maneuverable than drifters, of larger range than other AUVs, and do not need expensive support vessels .
Several research results have been reported in the literature regarding the optimization of the hydrodynamic performance of the underwater glider. Li et al. (2012) introduced the underwater glider in hydrodynamic shaping. Gu et al. (2009) solved the multi-objects design optimization problem of the underwater glider by constructing a surrogate model. These legacy gliders have a low life-drag ratio (L/D) and limited gliding efficiency because the hull barely generates any lift force . Underwater gliders have been well studied because of their low-power movement patterns. The flying-wing underwater glider is a tailless fixed-wing underwater glider. A flying-wing design significantly improves its maximum lift to drag ratio by eliminating the body of the glider (Graver, 2005). Developed by the U.S. Office of Naval Research (ONR, 2006), Liberdade XRAY was designed for applications in the shallow-water environment to obtain a greater payload carrying capability, cross-country speed, and the horizontal point-topoint transport efficiency than other existing underwater gliders.
The flying-wing structure of the underwater glider design is based on the air force's blended-wing-body (BWB) configuration. Compared with the classic wing-andtube fuselage configuration, the BWB has the following superior aerodynamic performances (Kroo, 2004;Qin et al., 2004). The reduction in the wetted area substantially re-duces the skin friction drag; the all-lifting design reduces swing loading and improves the spanwise lift distribution; and the smooth blended wing-center body intersection reduces the interference drag. Despite various aerodynamic benefits, the aerodynamic shape of the BWB also brings challenges to the design process. The complex shape of the BWB may cause difficulties during manufacturing. Without the conventional empennage, the chordwise lift distribution on the center body need to maintain a positive static margin. Jenkins et al. (2003) designed a 50 liter glider repackaged in the planform of a Horten H-11 flying wing glider using a thick low-Reynolds number section. Li et al. (2012) optimized the aerodynamic shape for the BWB transport. They put forward new design ideas for BWB. The cruise point, the maximum lift to drag point and the pitch trim point were the main reference indices. The Office of Naval Research (ONR, 2006) developed an advanced underwater glider based on the air force's BWB design of the ZRay. The ZRay exceeded 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 underwater glider known to the world.
To balance the contradiction between precision and efficiency, the researchers put forward a surrogate model based on the theory of design of experiments and approximate methods. In the last decade, the agent model in aviation, automobile, shipbuilding and other industries of structure design, fluid analysis and multidisciplinary design optimization has gained extensive applications and in-depth development. Chiplunkar et al. (2016) used multi-output Gaussian processes to optimize flight mechanics to flight loads. Lee et al. (2016) optimized the aerodynamic and structural analyses of multiple wing sails using surrogate models. Marrel et al. (2015) assessed sodium fast reactor accident by creating a surrogate model and sensitivity analysis methods.
In this paper, the study focuses on the effect of the geometric parameters on the hydrodynamic performance, and achieved the maximum lift to drag ratio and the minimum moment by a CFD-driven optimization. First, the parametric geometric model of the flying-wing structure underwater glider is established, and the complex hydrodynamic configuration is characterized by nine shape parameters. Second, the Gaussian kernel function algorithm is used to establish the surrogate model of the flying-wing structure underwater 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 by 9.1%. The relative sensitivity of each variable will then be analyzed. Aiming to address the problem of multiple design variables, the primary contribution of this paper is the establishment of the parametric geometric model of the flyingwing structure underwater glider. The Gaussian kernel function algorithm is innovative and is used to establish the surrogate model of the flying-wing structure underwater glider. Through analysis and optimization, the reasons for affecting the hydrodynamic performance of the flying-wing structure underwater glider are explored in this paper. This will provide the guidance for the shape design of the flying-wing structure underwater glider, and the basis for the multidisciplinary design and optimization of the flying-wing structure underwater glider.

Parametric geometric model and numerical simulations
The wetted surface area is one of the most important factors affecting the skin friction. Therefore, the wetted surface area became the origin of the design. Three canonical forms are shown in Fig. 1, and each has nearly 0.025 L of the volume displacement. The sphere is not streamlined although it has the minimum surface area. The two canonical streamlined options include a conventional axisymmetric body and a disk body. Compared with the conventional axisymmetric body, a disk body can realize an 11% reduction of the wetted surface area. As shown in Fig. 2, the geometric profile of the flying-wing structure glider, similarly to the disk body, is blended and smoothed into the wings. This construction reduces the friction dray and provides enough lift. Fig. 3 shows the flying-wing structure underwater glider shape parameterization of planform. The disk body is blended and smoothed into the wing by two arcs. The disk body includes a front round nose and a backround nose. A typical, symmetrical NACA 0012 airfoil is used as a baseline airfoil. The airfoil of the flying-wing structure underwater glider is designed with changing the thickness of NACA 0012 (=12.0%c, c is the chord length of the local section). The thickness is defined at the centerline (t 1 c), the merging point (t 2 c), and the section between the centerline and the merging point (t body c). t body is determined by the following equation.
To decrease the number of design variables, we define nine shape parameters as shown in Table 1.
The hydrodynamic numerical simulations of the flyingwing structure underwater glider presented in this paper were carried out using the commercial software CFX based on the finite volume method. The fluid material selected in this paper was liquid water, which was considered incompressible with a density of 998.2 kg/m 3 and a dynamic viscosity of 1.003×10 -3 Pa·s (Sun et al., 2015). The model selected the method of the standard wall function.
The governing equation for the cases considered in this paper is the RANS equation for incompressible viscous flow, and a statistically steady solution is found. The RANS equation is given below: where ρ is the density of the fluid, t is the time variable, U is the absolute velocity of the fluid, f is the unit mass of the volume force, p is the pressure, μ is the dynamic viscosity, δ is the unit matrix, is the Hamiltonian operator, is the tensor product and the superscript T is the matrix transpose.
The shape of the computational domain is a cuboid. By using the span length (b t ) as the reference length, the domain is with 2-2.5 million hexahedral cells. The flying-wing structure glider is placed at the origin of the coordinates. As shown in Fig. 4, the structured grids with the hexahedral cells are used in all the calculations presented here.
3 Surrogate model A surrogate model is an integrated modeling technique of the experimental design and approximate method. Currently, the response surface methodology (RSM), radical based functions (RBF) and neural networks are commonly used with surrogate models, but the number of parameters in these three models will linearly grow with the increase in the input variable dimension. The flying-wing structure underwater glider parameterized shape includes multiple design variables. A Gaussian kernel function is used in this paper to establish the surrogate model. In the surrogate model of the Gaussian kernel function, the number of parameters is determined only by the CFX simulation sample, but not the dimension of the input variables. The Gaussian kernel function expresses the surrogate function as: where is an n-dimensional vector (n is the number of design variables), is the model of the  c = (c 1 , c 2 , · · · , c n ) Gaussian kernel function, is the midvalue of each design variable, and h is the band width of the Gaussian kernel function.
To ensure the surrogate function best fits with the simulation date, a cost function is given below: is a m-dimensional vector (m is the number of CFX simulation examples), and y is the CFX simulation results.

J(θ)
The gradient descent algorithm is used to minimize the cost function and the equation is given below: From Eqs. (6) and (7), we can obtain: f θ (x) where θ j is the parameter of , α is the learning rate, := is the colon equal (a:=b means taking the value in b and use it to overwrite whatever the value of a), and superscript i is the number of simulation dates.
Feature scaling is used in this paper to ensure that the different design variables take on similar ranges of values; then, the gradient descents can converge more quickly, and the equation is given below: where c i is the average value of x i in the training sets, and s i is the standard deviation of x i . In the process of optimization, in addition to obtaining a lift-to-drag ratio that is as large as possible, an attempt should be made to reduce the moment. Based on the above described equations and the use of the weighted coefficient of multi-objective optimization problems, the objective function is expressed as follows: ω 1 and ω 2 where are the weighted coefficients, F L is the lift, F D is the drag, and M pitch is the torque.
The efficient global optimization is a PSO algorithm. The PSO is computed as follows. The PSO is initialized to a group of random particles (random solutions); then, the optimal solution is found through iterations. In each of iterations, the particles update themselves by tracking the two extreme values. The first value is the value of the particles to find the optimal solution, which is called the individual extreme value. The other extreme value is called the global extreme value, referring to the entire population being used to find the optimal solution. Assuming that the goal is a one dimensional search space, and there are N particles in a community, of which the i-th particle is expressed as a D-dimensional vector.
(11) The speed of the i-th particle is: (12) The i-th particle searches for the optimal position which is known as the individual extreme value: (13) The whole particle swarm searches for the optimal location of the global extreme value: , i = 1, 2, · · · , N.
(14) In order to find these two optimal values, the particles update their speed and position according to the following equation where c 1 and c 1 are the acceleration constants; r 1 and r 2 are the uniform random numbers in the range from 0 to 1; and ω is the inertia weight. The procedure of the PSO algorithm is shown in Fig. 5 as follows: 1. Initialize the particle swarm, including population size N, position X i and speed V i of each particle.  Y(x i ) 2. Calculate for each particle.
3. For each particle, compare its with , if , use to replace .
g best (i) 4. For each particle, compare its with , if , use to replace . 5. According to Eq. (15), update the particle's speed V i and position X i .
6. End the iteration if the end conditions are met or return to Step 2. Fig. 6 shows the optimization procedure of the flyingwing structure underwater glider surrogate model. First, N design samples are selected using Latin Hypercube Sampling (LHS). The hydrodynamic performance of N samples is evaluated by numerical simulations using the commercial software CFX. Second, the initial surrogate model is constructed based on N sample data. Then, using the gradient descent, 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 in one iteration. Finally, is found using the initial surrogate 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 numerical simulations.

Relative sensitivity analysis
In this paper, we determine the sensitivity of each variable by establishing the covariance matrix of each variable and Y(x). Eq. (16) is established according to the surrogate model of the flying-wing structure underwater glider.
The covariance matrix of Eq. (16) is: The covariance between two random variables is defined as: is the expectance of x i . For the standard transform of Eq. (18), the sensitivity of Y(x) for each variable is: , i = 1, 2, · · · , 11.

Results and discussion
According to the parametric design of the flying-wing structure underwater glider, the parameter ranges of the design space are shown in Table 2, which are determined by avoiding abnormal shapes.
In the CFX calculation process, the inlet velocity at the front of the fluid domain was set to be equivalent to the glider velocity, 1 m/s. The pressure at the end outlet of the fluid domain was set to be 0 Pa. Meshing was used to strengthen the wall function, and the boundary layer number was set to be 12. The first element height of the cells adjacent to the body was set to be smaller than 5y+ (Mulvany et al., 2004) units for the compatibility with the turbulence model. An attack angle of 3° was used for each model performance evaluation. When searching J(θ) min , γ was set as 10 -3 . The tests showed that the X-ray has better hydro-  (ONR, 2006); thus, in this paper, the Xray was used as the initial design for optimization. The number and distribution of sample points are important factors that affect the accuracy of the surrogate model. The number of sample points should be sufficient so that these points can be uniformly spread over the design space. In this paper, the initial 50 samples were selected by the LHS, and the performance of the 50 samples was evaluated using the commercial software CFX. The initial surrogate model was then constructed based on this sample data. As previously discussed, the objective functions are expressed as follows:

J(θ) J(θ) Y(x)
After 57 iterations, decreases smaller than γ. Fig. 7 shows the convergence of the gradient descent of . as an additional sample was evaluated by using CFX, and the new surrogate model was built with the data from 50 samples. After 13 iterations, the error was smaller than 2%. The results of the optimization of the surrogate model based on a Gaussian kernel function and RSM are shown in Table  3. The error of the first-order and second-order RSM surrogate model is too large. The error of the Gaussian kernel function surrogate model is close to the error of the secondorder RSM surrogate model, but its polynomial number is reduced by 2/3. This can largely reduce the computing time.
When searching , the parameters of Eq. (15) are expressed as follows: The optimization results of the flying-wing structure underwater glider are shown in Table 4. Fig. 8 shows the pressure contour of the optimum and initial design. The optimized and the initial designs have a similar pressure distribution in that the negative pressure area appears around the leading edge of the wing, and the positive pressure area appears at the nose of the fuselage and the trailing edge of the wing. The initial design has clustered pressure contour lines near the wing kink, which indicates the presence of a shock. The optimal design shows the parallel pressure contour lines with roughly equal spacing, indicating a shock-free solution. Fig. 9 shows the relative sensitivity of each variable. According to the sensitivities, Y(x) is sensitive to the changes in and a, and is insensitive to the changes in other design variables. The trailing edge radius R b has a great influence on the wet surface area and the mass distribution of the model, and it influences Y(x). The thickness rate of the merging point t 2 and the length of the airfoil root c root determine the relative thickness and the wet surface area of the airfoil, which influence the friction of the model. The sweep angle 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 airfoil achieves a higher hydrodynamic performance.

Conclusions
In this paper, the shape of a new underwater glider was designed with a BWB configuration, and the parametric geometric model of this underwater glider has been built with nine design variables. The hydrodynamic numerical simulation has been performed using the commercial software CFX with the incompressible RANS equations. The Gaussian kernel function algorithm, a gradient descent algorithm, is used in the hydrodynamic shape optimization of the flying-wing structure underwater glider. The results of this study suggest that compared with a conventional underwater glider, the flying-wing structure underwater glider has a better hydrodynamic performance. After the optimization, the hydrodynamic performance increased by 9.1%. The sensitivity of Y(x) to each of the design variables was analyzed. According to the results, it was found that the parameters of the wing shape and the shape of the trailing edge are important factors affecting the hydrodynamic performance.