Hydrodynamic Optimization and Motion Stability Enhancement of Underwater Glider Combining CFD and MOPSO

Abstract

This study investigated the motion stability of underwater gliders and optimized their shape to enhance hydrodynamic performance. Given the critical role of stability in underwater operations, a multi-objective optimization framework was developed, focusing on the geometric configuration of hydrofoils. Computational fluid dynamics (CFD) simulations were employed, with stability assessed based on hydrodynamic moments in roll and pitch motions. A surrogate model was constructed using Kriging interpolation, leveraging Latin hypercube sampling (LHS) to generate 60 design points. Sensitivity analysis identified key shape parameters influencing stability, guiding a multi-objective particle swarm optimization (MOPSO) algorithm to explore optimal design configurations. Improvements of up to 68.91% in roll stability and 51.63% in pitch stability are achieved compared to the original model, which demonstrates the effectiveness of the proposed optimization approach. The findings provide valuable insights into the hydrodynamic design of underwater gliders, facilitating enhanced maneuverability and stability in complex marine environments.

1. Introduction

The ocean occupies 71% of the Earth’s surface and contains large amounts of ocean minerals and renewable energy. With the growing interest in ocean development, research on ocean exploration equipment has gradually become a topic of interest. Due to the complexity of tasks and variable ocean currents in exploration working process, efficiency and structural safety are the fundamental features of ocean exploration equipment design. The underwater glider, as a kind of simple-structured and highly efficient equipment, has been widely applied in ocean exploration tasks [1,2]. However, the structural design process needs to be reconsidered since the complexity of ocean current distribution often causes unsteady motion of the glider during its underwater operation. The design process for maintaining stable motion of an underwater glider is a multidisciplinary problem, which should take into account both the size of the main body and the dimension requirements of the monitoring equipment in the glider. Hydrodynamic design is the primary task during the development of underwater gliders. For a streamlined shape glider, a reasonable overall design, including the design of the main body dimension and layout of its attachments, can significantly improve the maneuverability and stability in the glider’s movement [3]. Optimizing the hydrodynamic design of underwater gliders is the key for enhancing their adaptability and operational performance in complex oceanic environments.

It is essential for an underwater glider to maintain its stable attitude in roll and pitch motions, since large deviations can degrade sensor accuracy, reduce data quality, and increase energy consumption, thereby shortening mission endurance. It is particularly critical because the roll and pitch of the glider are primarily induced by overturning moments generated by rotational forces, which directly affect the vehicle’s stability. In the case of external disturbances such as ocean turbulence, minimizing the overturning moment on an underwater vehicle is a fundamental way to maintain its overall stability [4]. Instability caused by a large overturning moment can lead to a cascade of problems affecting the glider’s pitch and roll motion and ultimately reducing its control ability and control effectiveness. The stability discussed above refers to hydrostatic stability, which means the glider’s inherent tendency to return to its original equilibrium position after being subjected to a small disturbance. Based on the principle of mechanics between motion stability of an underwater glider and the glider’s restoring moments, Leonard et al. [5] pointed out that the restoring moment generated by the center of buoyancy can balance the overturning moment. This study further discussed the effects of gravity center displacement on stability. Chen et al. [6] studied the relationship between the overturning motion on an autonomous underwater vehicle (AUV) and the motion stability by the shape optimization process on the vehicle and found that the overturning moment is an important index for measuring the deviation of the AUV from the equilibrium state after being disturbed; following the shape optimization process, the overturning moment was reduced by 14.08% compared to the original model. Lin et al. [7] used the overturning moment as an indicator to measure whether an autonomous underwater helicopter (AUH) can return to its equilibrium state when facing disturbances. In their research a fixed angle of attack (AOA) of vehicles in 10° was set, and it was pointed out that the overturning moment of model HG3 is reduced by 38.4% compared to model HG1, where HG1 and HG3 are the original and optimized model, respectively. They finally drew a conclusion that when HG3 is subjected to the same interference, its deviation from the equilibrium state is smaller, thus enhancing its stability. Wei et al. [8] pointed out that overturning moment is a key factor affecting the stability of a hybrid unmanned aerial underwater vehicle (HAUV) during its movement. This study further pointed out that the influence of the overturning moment on an underwater glider can be reduced through optimized design. In these studies, several factors affecting the overturning moments are discussed. However, few studies focus on the coupled effects of overturning moments on roll and pitch motions on a glider’s motion stability in multiple direction motions.

The computational fluid dynamics (CFD) technique, which is widely used in underwater flow field simulation, can be used to assess the forces and moments acting on a glider’s structure during its motion. The stability of an underwater glider strongly correlates with the forces acting on its main body. Consequently, the stability of the underwater vehicle can be evaluated using the CFD technique. Phillips et al. [9] conducted a study on the dynamic stability of an underwater vehicle, named AutoSub, using the CFD technique to simulate the oblique towing and planar motion mechanism experiments to obtain its steady-state hydrodynamic derivatives. The numerical results for dynamic stability were compared with the experimental data, and the effectiveness of CFD simulation in evaluating the motion stability of underwater vehicles was demonstrated. Lin et al. [7] used the CFD software ANSYS-Fluent to simulate two groups of AUH models, HG1 and HG3, with different angles of attack. The stability indices in the horizontal and vertical planes were evaluated using the Routh stability criterion (RSC). Their study revealed that the main factor affecting the motion stability of different models was their difference in shape. To address the issues of depth control and stable motion in an underwater towing system during the towing process, Yuan and Liang [10] investigated a method by adjusting the cable length to maintain the towed vehicle in a given depth. The numerical simulation results showed that the method can be used to keep the vehicle stable at a designed depth during towing operation. However, effective implementation of vehicle stability requires integration with intelligent winch systems. Saout and Ananthakrishnan [11] used a boundary integral method based on the hybrid Lagrangian–Eulerian formulation to simulate the planar motion mechanism test motion of an underwater vehicle to obtain the hydrodynamic derivatives. The study suggested that the factors affecting the stability of underwater vehicle include the position of its center of gravity and the layout of its external attachment. Park et al. [12] investigated the towing stability of an underwater towed object by modeling experiments. The experiments employed a 1/4 scale model, and sets of towing cables made of different materials were selected. The motion behavior was measured using an inertial measurement system. The results were used to evaluate the effects of towing cable type, gravity position, and attachment site on towing stability based on the external moments acting on the vehicle.

Moreover, it has been found that the most efficient way to improve the pitching moments on an underwater vehicle is to adjust the gravity position of the vehicle to reduce the effects of fluid disturbances during its motion. Yang et al. [13] simulated the motion of a towing system consisting of a towed underwater vehicle and an unmanned surface vehicle (USV). The results showed that adjusting the center of gravity can improve the motion stability of an underwater vehicle. In addition to adjusting the center of gravity and the shape of the main body, redesigning the position and geometric form of the hydrofoil is an important tool. This is because the hydrofoil is an essential attachment for manipulating the attitude of an underwater vehicle, and it has been widely used as trajectory and attitude control mechanism in an underwater vehicles [14,15]. Minowa and Toda [16] used an adjustable hydrofoil as the primary maneuvering device to enhance the motion stability of an underwater vehicle and proposed a stability analysis method for the motion control system using a high-gain observer. In these works, motion stability mostly relies on the active feedback control of vehicles. Building on these insights, this study adopts a control strategy for a glider attitude control based on the principle of self-stability, whereby the glider depends on its optimized hydrodynamic design. Specifically, the overturning moments generated by hydrodynamic disturbances can be reduced by adjusting the configuration of components or the overall shape.

In the design optimization of underwater gliders, an effective solution must balance the trade-offs among conflicting optimization objectives. Due to their ability to handle multiple conflicting objectives, multi-objective optimization algorithms have many applications in the design process of underwater vehicles. Fu et al. [17] conducted optimization research on the shape of an autonomous underwater glider based on non-dominated sorting genetic algorithm-II (NSGA-II). With the elongated ellipsoid as the reference model, the optimization efficiency of the glider with the same wing configuration was evaluated in terms of drag reduction. This study showed that, compared with the original model, the optimized model reduced its total drag by 26.7% and its form drag by 47.8% at a speed of 0.5 knots. Wu et al. [18], on the other hand, performed a multi-objective optimization of a variable airfoil configuration for an underwater glider, in which the optimized parameters were chosen as the wingspan and the swept-back angle of the hydrofoil. A surrogate model was used to improve the efficiency of the optimization process. NSGA-II was used to further obtain the Pareto optimal solution set, and four sets of optimal hydrofoil configuration parameters were selected. Wu et al. [19] optimized the shape of an underwater glider to improve its glide range and at the same time to offset energy consumption uncertainty. Two surrogate models, polynomial response surface (PRS) and an artificial neural network (ANN), were adopted and compared in the optimization process. Chen et al. [6] optimized the shape parameters of an AUV using the NSGA-II algorithm to minimize drag and enhance motion stability of the vehicle. Li et al. [20], who also used the NSGA-II algorithm, conducted an optimization study on the hydrofoil structure of an adjustable hydrofoil deep-sea towed body, aiming to improve towing stability. The optimization process established a two-way Gaussian process (GP) prediction model for the hydrofoil angle of attack and tuning variables. The results of the model showed high prediction accuracy and strong generalization ability, with a prediction error of less than 4%. Existing stability optimization methods often suffer from computational inefficiency and limited adaptability to complex hydrodynamic conditions.

In the current work, motion stability evaluation criteria and methods for enhancing stability using CFD techniques were discussed. However, due to the complexity of combining stability evaluation criteria with CFD techniques and multi-objective optimization algorithms, further work is necessary. Furthermore, research exploring the optimization of appendage attachments to refine the overall shape and improve the stability of underwater gliders remains scarce. Additionally, limited attention has been given to stability optimization under flow disturbance conditions in both pitch and roll motions. To address these gaps, this paper proposes a novel optimization framework which integrates CFD techniques, Kriging surrogate modeling, and multi-objective particle swarm optimization (MOPSO) to enhance the motion stability of underwater gliders. This approach enhances computational efficiency without compromising accuracy. By using Latin hypercube sampling (LHS), 60 sample sets were generated to construct a surrogate model. The MOPSO was then applied to identify optimal configurations. The proposed method effectively enhances the ability to maintain the stability of the underwater glider under various flow disturbance conditions. And the results provide valuable insights and a practical optimization framework for the underwater glider design process.

2. Optimized Geometric Model and Numerical Method

2.1. Optimized Geometric Model

The underwater glider discussed in this paper is a type of underwater apparatus which is widely used in oceanographic research. The power source of the glider is a buoyancy engine driving the glider by changing its gravity center and buoyancy to adjust the attack angle of hydrofoil. With the adjustment of the AOA, the driven power for the glider forward movement is obtained through a conversion from its vertical motion.

The underwater glider to be optimized here consists of two main kinds of components: the main body and the hydrofoils. The main body serves as the primary structure for carrying underwater detection devices, and the hydrofoil acts as control mechanism for the motion and attitude of the glider.

The main body of the underwater glider is modeled as a rotating circular cylinder, which can be divided into three sections: the forebody, the parallel midbody, and the trailing part. Its shape follows the recommended design in [21], where the geometric profiles of the forebody and trailing section are defined by the Myring linear functions. These functions are recognized as a reliable theoretical model for describing streamlined axisymmetric bodies and are widely used as a baseline in the geometric modeling of underwater vehicles. By employing the Myring linear functions, the modeled geometry achieves smooth forebody and afterbody profiles while allowing flexible adjustment of body shape through a set of parameters. This capability makes them particularly suitable for hydrodynamic design studies focused on stability enhancement. The Myring linear functions are defined as

$$r_{forbody}\left(x\right)={\displaystyle \frac{1}{2}}d{[1-{\left({\displaystyle \frac{x-a}{a}}\right)}^2]}^{{\displaystyle \frac{1}{n}}}$$

(1)

$$r_{tailing}\left(x\right)={\displaystyle \frac{1}{2}}d-({\displaystyle \frac{3d}{2c^2}}-{\displaystyle \frac{tan\theta }{c}}){(x-a-b)}^2+({\displaystyle \frac{d}{c^3}}-{\displaystyle \frac{tan\theta }{c^2}}){(x-a-b)}^3$$

(2)

where x is the distance between sideline to median of the main-body. a, b, and c represent the length of forebody, parallel midbody, and trailing part, respectively. d is the diameter of the cross-section of the parallel midbody. n is the shape factor of the forebody part. $\theta$ represents the indentation angle of the trailing part. Figure 1 shows the shape of main-body according to the Myring linear profile equations. The values of a, b, c, d, n, and $\theta$ in the research are given as 0.3 m, 0.8 m, 0.4 m, 0.2 m, 2.0, and 36°, respectively. And the total length is 1.5 m, and the parallel midbody length is 0.8 m.

The cross-sectional profiles of all the foils in this underwater glider are NACA-0012. The foils are classified as main foils and tail foils as shown in Figure 2.

As shown in Figure 2, all the foils are mounted on the main body. During its motion, the hydrodynamic performance of the underwater glider is directly affected by the characteristics of the main foils and tail foils. In the optimization process, motion stability is primarily evaluated in terms of roll and pitch. Due to the significant effect of the foils on stability, the selected optimization parameters focus on their key characteristics. These include the chord length (CLength) and wingspan of the main foils (SLength), the wingspan of the tail foils (TWLength), the distance from the leading edge of the main body to the main foils (WHLength), and the distance from the main body’s tailing edge to the tail foils (WTLength). A set of parameters was selected as the original model, and

Table 1. Optimization parameters, initial values (m), and ranges (m) for underwater glider hydrofoils.

Symbol	Parameter	Initial Value	Range
CLength	Chord length of main foil	0.21	[0.10, 0.30]
SLength	Wingspan of main foil	1.19	[1.00, 1.50]
TWLength	Wingspan of tail foil	0.56	[0.40, 0.60]
WHLength	The distance from main body’s leading edge to main foil	0.32	[0.30, 0.40]
WTLength	The distance from main body’s tailing edge to tail foil	0.26	[0.20, 0.30]

shows the optimization parameters, initial values, and ranges for underwater glider hydrofoils.

2.2. Forces and Moments Acting on the Underwater Glider

Figure 3 illustrates the forces and moments analysis of the underwater glider under flow disturbance conditions. Figure 3a shows the forces and moments acting on the model when the glider is under the heel condition. The forces acting on the body include gravity, buoyancy, and hydrodynamic pressure induced by the flow disturbance. The restoring moment is generated by the distance separation between gravity and buoyancy forces, while the overturning moment arises from the hydrodynamic pressure induced by flow disturbance. The hydrodynamic pressure can be decomposed into components parallel and vertical to the glider’s surface, with the vertical component contributing to the overturning moment. Figure 3b presents the forces and moments acting on the model when the glider is under the trim condition. The generation of forces and moments is essentially the same as in the heel condition, with hydrodynamic disturbances producing overturning moments and the interaction between gravity and buoyancy providing restoring moments.

In this design scheme, the fact that the overturning moment to the glider should be less than its inherent restoring moment when the glider is in trim and heel angle under disturbed ocean current conditions is taken as the evaluation criteria for determining whether the glider is in a stable condition. When the evaluation criteria are satisfied, it is deemed that the glider will return to its stable attitude after certain external flow disturbance.

2.3. Governing Equations

The CFD approach is applied to calculate the fluid mechanical effect on the underwater glider. The Reynolds-averaged Navier–Stokes (RANS) equations are employed. Given the density of the external flow surrounding the glider is a constant, the governing equations are given by

$${\displaystyle \frac{\partial u_i}{\partial x_i}}=0,(i=1,2,3)$$

(3)

$$\rho [{\displaystyle \frac{\partial u_i}{\partial t}}+{\displaystyle \frac{\partial \left(u_i{u}_j\right)}{\partial x_j}}]=-{\displaystyle \frac{\partial P}{\partial x_i}}+\rho g_i+{\displaystyle \frac{\partial }{\partial x_j}}[\rho \mu ({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}})-\rho u_i^{,}{u}_j^{,}],(i,j=1,2,3)$$

(4)

where $u_i$ and $u_j$ are the velocity components of the fluid in the i and j directions, respectively. $x_{i,j}$ are the rectangular coordinates in space, $\rho$ is the density of the fluid in space, P is the mean value of the pressure, $\mu$ is the kinetic viscosity coefficient of the fluid, t is the physical time, $g_i$ is the gravity term, and $\rho u_i^{,}{u}_j^{,}$ is the Reynolds stress term.

To solve the governing equations, two turbulence equations of the standard $k-ϵ$ model are introduced to close the fluid dynamics system, and the equations are given by

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho k\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho k{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}){\displaystyle \frac{\partial k}{\partial x_j}}\right]+G_k+G_b-\rho ϵ-Y_M+S_k$$

(5)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho ϵ\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho ϵu_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[(u+{\displaystyle \frac{u_t}{{\sigma }_{ϵ}}}){\displaystyle \frac{\partial ϵ}{\partial x_j}}\right]+C_{1ϵ}{\displaystyle \frac{ϵ}{k}}(G_k+C_{3e}{G}_b)-C_{2ϵ}\rho {\displaystyle \frac{{ϵ}^2}{k}}+S_{ϵ}$$

(6)

where k and $ϵ$ represent the turbulent kinetic energy and turbulent dissipation rate, ${\mu }_t$ is the turbulent viscosity. $G_k$ and $G_b$ are the turbulent kinetic energy generation of average velocity gradients and the generation of turbulent kinetic energy due to buoyancy, respectively. $Y_M$ represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. $S_k$ and $S_{ϵ}$ represent the source terms. ${\sigma }_k$ and ${\sigma }_{ϵ}$ are the turbulent Prandtl numbers for k and $ϵ$, and $C_{1ϵ}$, $C_{2ϵ}$, and $C_{3ϵ}$ are constants.

Since the $k-ϵ$ model does not accurately describe the boundary layer near the wall, we chose the wall function approach to better capture the boundary layer behavior. $y^{+}$ is chosen to characterize the first layer of the boundary layer mesh and is defined as

$$y^{+}={\displaystyle \frac{\mathsf{\Delta }y\rho {\mu }_{\tau }}{\mu }}={\displaystyle \frac{\mathsf{\Delta }y}{\mathit{v}}}\sqrt{{\displaystyle \frac{{\tau }_{\omega }}{\rho }}}$$

(7)

where $\mathsf{\Delta }y$ is the thickness of the first layer of the grid from the wall, ${\mu }_{\tau }$ is the friction velocity, ${\tau }_{\omega }$ is the shear stress, and $\mu$ and $\mathit{v}$ are the kinetic viscosity coefficient and kinematic viscosity coefficient, respectively. The full $y^{+}$ wall method is a form of wall grid hybrid treatment, which selects coarse and fine grids for calculation based on the value of $y^{+}$. The scalable wall function was selected for the case of high $y^{+}$. For the submerged structures, the $y^{+}$ value needs to be higher than 30, and the standard $k-ϵ$ model with the enhanced wall function or the scalable wall function can obtain acceptable results compared to the experimental values. Therefore, we selected the standard $k-ϵ$ model and the scalable wall function. The $y^{+}$ value was selected as 30, with the growth rate of the boundary layer grid set to 1.5.

The corresponding Reynolds number (Re) in this study ranges from approximately 1.0 $\times {10}^6$ to 3.5 $\times {10}^6$, depending on the chosen reference length and velocities. The reference lengths for roll and pitch motions are taken as the wingspan of the main foil and the total length of the glider, respectively. The flow velocity also varies accordingly. Across all rotational position conditions, however, the Re falls within the fully turbulent regime, thereby justifying the use of the standard k-$ϵ$ model with scalable wall functions.

2.4. Boundary Conditions

The CFD commercial code STAR-CCM+ is employed to calculate the numerical results of hydrodynamic effects to the glider. To obtain more accurate results, it is necessary to define the size of the computational domain by the physical nature of the research problem. A reasonable external computational domain can increase efficiency and avoid the fluid blocking effect. The blocking ratio (BR) is used to evaluate the setting defects of the flow field in the computational domain, and it is revealed that the blocking effect is insignificant for external computational domains with a BR of less than 6% [22]. The length, width, and height of the computational domain in our research are 10, 25, and 30 times those of the underwater glider, respectively. The computational domain is a standard cuboid, with its six faces classified into three boundary types. The outlet, inlet, and the symmetric boundary are defined as the velocity inlet, pressure outlet, and far-field condition. A detailed computational domain setup is shown in Figure 4.

The structured mesh is employed to generate the mesh for the computational domain, with a basic size of 0.1 m corresponding to the domain size. Figure 5 presents the basic mesh of the external field, and Figure 6 shows the mesh near the surface of the underwater glider applied in our computation.

In Figure 6, it can be seen that the refinement of the mesh is applied to the rear end of the two main foils located at the front of the glider and the rear end of the wake. The refinement affects the contraction pattern of the wake and the distribution of turbulent vortices and therefore impacts the accuracy of the calculation results to some extent. To improve the accuracy of the surface mesh, we refined the computational mesh by controlling the basic size of the external mesh and adjusting the mesh size in the transition region. Then, we selected the initial conditions of the field and analyzed the mesh size for mesh independence. Five types of basic computational mesh numbers were selected in our computation, that is, 1.89 × 10⁵, 9.28 × 10⁵, 12.01 × 10⁵, 19.26 × 10⁵, and 48.11 × 10⁵. The flow velocity at the inlet was set to 1.0 m/s, and the initial flow velocity of the fluid in the domain was equal to the inlet velocity. The hydrodynamic resistance of the underwater glider was monitored and reported during computation.

2.5. Mesh Independence

To verify mesh independence, the stable solution obtained after the convergence of the numerical calculation process was captured based on the mesh size and convergence rate, as presented in

Table 2. Model resistances (N) of different total mesh numbers.

Total Number of Meshes	Resistance
1.89 × 105	231.90
9.28 × 105	118.32
12.01 × 105	115.95
19.26 × 105	115.08
48.11 × 105	114.81

.

Table 2 shows that an increase in the number of meshes affects the results. Specifically, the computed fluid resistance values slightly decrease as the number of meshes increases. The relative error of the fluid resistance in calculation is about 3.65% when the number of meshes is 1 × 10⁶. However, once the mesh number reaches the order of millions, the resistance value stabilizes, and the relative error remains within 1%. To balance computational accuracy and efficiency, we selected a level of mesh number of 1.21 × 10⁶ for subsequent computations. The same mesh refinement conditions were applied to maintain consistency in the computational mesh structure and the spatial arrangement of the computational domain.

2.6. Numerical Method Validation

To verify the accuracy of the numerical model applied in this paper, the current numerical results are compared with both numerical and experimental solutions based on the mesh convergences. Since the SUBOFF model has sufficient numerical solutions and experimental data and has been widely used in engineering as a standard submarine prototype model, we chose it as the reference body to verify our numerical model. The experimental and numerical solutions were selected based on the data in Ref. [23] and Ref. [24], respectively, with the SUBOFF standard model. The selected inlet flow boundary conditions in the domain are 10.00 kn, 11.85 kn, 13.92 kn, 16.00 kn, and 17.79 kn; comparisons among the numerical results in the computation and those in Refs. [23,24] are shown in Figure 7.

Comparisons among the numerical results of this model and those in Refs. [23,24] in Figure 7 show that there are certain deviations between the fluid resistance of SUBOFF in this model, large eddy simulation (LES) in Ref. [23], and the physical modeling test in Ref. [24]. The deviation between the result in this research and that in Ref. [23] comes from the differences in their applications of numerical methods and grid scales. The deviation between the result in this research and that in Ref. [24] originates from the deficiency of the numerical method and simplification of the model applied in this research. From the results in Figure 7, it can also be found that with the increase in the flow velocity, the relative error between numerical solutions of this model and the experimental results in Ref. [24] gradually increases. When the inlet flow velocity is 10 kn, their relative deviation is about 3.6%; when the velocity increases to 18 kn, the relative deviation is up to 5.8%. Even though the deviations exist, they are still within the acceptable range in engineering design. In the subsequent optimization process for an underwater glider, the approaches of mesh generation and numerical computation used in this part were still applied based on the fact that the inlet flow velocities were all less than 10 kn, and the deviations were within the engineering acceptable range.

3. Multi-Objective Optimization Method

3.1. Optimization Objectives

The size and layout of the foils, namely, their geometric shape parameters for the underwater glider, were optimized using a surrogate model. Specific parameter information and corresponding ranges are listed in Table 1. The motion stability of the glider in pitch and roll was analyzed by evaluating the moments acting on the glider. The motion stability here refers to the underwater glider’s capability to autonomously return to its original equilibrium state after a disturbance. Typically, fluid disturbances acting on the underwater glider’s main body result in a passive rotation, referred to as the AOA of the hydrofoil. If the glider’s AOA is restored to its original state, the motion stability can be considered satisfactory. To simplify the analysis of disturbance effects, we assume the glider maintains a fixed AOA. Thus, the magnitude of the AOA effectively represents the moment acting upon the glider.

This study primarily focuses on the hydrodynamic response of the underwater glider when a disturbance induces a nonzero AOA. To ensure that the glider can be restored its original stable state, the restoring moment generated by the body must exceed the overturning moment induced by hydrodynamic disturbance. In studies on the motion stability of an underwater glider [7,25], the use of disturbance AOAs in the range of 0°∼40° has been reported as a common practice. And the 20° AOA can be often obtained at the transient rotations when the glider is under the effect of flow disturbances. Accordingly, in the following analysis, the disturbance AOA in both pitch and roll motions was set to 20°. The evaluation criterion for stability is therefore defined as the condition where the overturning moment at a trim and heel angle of 20° remaining smaller than the inherent restoring moment of the glider, ensuring its return to equilibrium.

3.2. Sampling Method and Surrogate Model

The LHS method [26] is a technique for sampling within multidimensional spaces. It ensures a uniform distribution of sampling points across parameter space, thereby providing comprehensive spatial coverage [27]. It is also capable of capturing the interdependencies among parameters [28]. Given these attributes, LHS is extensively applied in fields such as experimental design and parameter optimization. The coverage efficiency of LHS is influenced by the number of sampling strata [29], the randomness in point selection, and the number of strata required to enhance coverage throughout the entire interval. In this research, the LHS method was employed for sampling.

Modeling and analyzing complex systems simultaneously is a challenging task due to their complexity. Therefore, surrogate models are needed to deal with complex systems. The surrogate model adopted in this paper was constructed using Kriging interpolation [30], which is a proxy model for estimating the objective function that considers the spatial correlation among sample points. It effectively approximates nonlinear objective functions and can be expressed by the following equation:

$${\mathit{f}}_K\left(x\right)={\mathit{f}}^T\left(x\right)\beta +Z\left(x\right)$$

(8)

where ${\mathit{f}}^T\left(x\right)$ is the model for simulating the global regression calculation, $\beta$ is the regression coefficient, and $Z\left(x\right)$ is the distribution error, which can be used to calculate the local error. The following equation can express $Z\left(x\right)$:

$$E\left(Z\right(x\left)\right)=0$$

(9)

$$Var\left(Z\left(x\right)\right)={\sigma }^2$$

(10)

$$cov(Z\left(x_i\right),Z\left(x_j\right))={\sigma }^2\left(R(\theta ,x_i,x_j)\right)$$

(11)

where $\sigma$ is the variance, $x_i,x_j$ are any two points in the sample, and $R(\theta ,x_i,x_j)$ is a correlation function with parameter $\theta$, which is often used to represent the spatial correlation of the sample points.

Five optimization variables were considered while ensuring sufficient coverage of the sample interval. To construct a surrogate model, 60 sample points were extracted and plugged into the Kriging model.

3.3. Multi-Objective Optimization

The particle swarm algorithm (PSO) [26] is widely used in optimization analysis as an algorithm based on group intelligence. The basic idea of PSO is to find an optimal solution by simulating the information sharing and cooperative behavior of each individual in the group and updating its position while it explores the search space. Its core includes the update rules for particle position and velocity, which gradually make the particle swarm converge to the optimal position through a number of iterations. After continuous updates and improvements [27], PSO, as a simple and efficient optimization method, can be widely applied to optimization problems in continuous space.

For multi-objective optimization problems, multiple conflicting objectives are usually involved. Therefore, a set of optimal solutions in the solution space, determined with the help of a Pareto frontier, is needed, and the selected set of optimal solutions generally has less obvious optimization margins [28]. The MOPSO algorithm is an extension of PSO, designed to solve multi-objective optimization problems. With the characteristics of MOPSO, the optimization performance of each particle in the swarm is evaluated across multiple objectives through a set of Pareto-optimal solutions corresponding to the associated particle.

In this paper, the optimized parameters for enhancing the motion stability of an underwater glider are primarily the dimension and position of the foils. Based on the primary shape of the underwater glider, the optimization process involves sampling in a space using a robust multi-objective approach. Based on the sampled points, a surrogate model determines the primary mapping relationship between the optimized objectives and parameters by delimiting the range of parameter values in the continuous space. Furthermore, the optimization algorithm extracts the global optimal solution.

3.4. CFD-MOPSO Optimization Procedure

In this paper, an optimization framework combing CFD simulations with the surrogate-based MOPSO algorithm was established. The procedure is summarized as follows:

• A set of 60 sample points was generated within the defined parameter ranges using LHS method.
• CFD simulations were performed for each sample points to calculate the roll and pitch moments under the specified disturbance conditions.
• The CFD results were used to construct a Kriging surrogate model, which approximates the relationship between parameters of hydrofoils and moments in the roll and pitch directions.
• The accuracy of the surrogate model was validated, and Sobol sensitivity analysis was conducted to indicate the influence ranking of parameters on motion stability in different directions.
• The surrogate model was coupled with the MOPSO algorithm to search for Pareto-optimized solutions that minimize both roll and pitch moments.
• Selected Pareto solutions were evaluated again by using CFD simulations to confirm their accuracy and effectiveness.

This procedure enables an efficient exploration of the design space while reducing the number of CFD evaluations. The integration of surrogate modeling and MOPSO ensures both computational efficiency and reliable identification of configurations. The detailed process is illustrated in Figure 8.

4. Results and Analysis

4.1. Validation of Surrogate Model Accuracy

To proceed with the optimization process, the accurate surrogate model is required. A total of 60 sample point groups were selected to validate the model’s accuracy. Then the coefficient of determination ($R^2$) was calculated using the following function:

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n(y_i-\widehat{y_i})}{{\sum }_{i=1}^n(y_i-\overline{y})}}$$

(12)

where n is the total number of sample points, $y_i$ is the actual output at position i, $\widehat{y_i}$ is the predicted value at position i by approximating the model, and $\overline{y}$ is the average of the actual output values. The value of $R^2$ is between 0 and 1, and the closer to 1, the better accuracy of the surrogate model. The total number of validation groups was selected to be 12 to validate the accuracy of surrogate model obtained through the Kriging method. The validation results are shown in Figure 9 and Figure 10, where Figure 9 shows the results of test groups in the pitch motion, and Figure 10 shows the results of test groups in the roll motion.

By analyzing the results in Figure 9 and Figure 10, it is evident that both groups of surrogate model results have achieved high accuracy in fitting. Compared to the roll motion, the fitting accuracy in the pitch motion is slightly better compared with the CFD results. Nevertheless, all $R^2$ values in both pitch and roll exceed 0.95. The $R^2$ values for roll and pitch are 0.966 and 0.996, respectively. The results suggest that the surrogate model is accurately sufficient and that the number of sampling points is sufficient for the subsequent optimization process. It also should be noted that a high $R^2$ value of the surrogate model only reflects the fitting accuracy within the training data, but does not guarantee reliable optimization outcomes. Therefore, in this study, selected Pareto optimization solutions were evaluated again by using CFD simulations to ensure that the optimization results are physically valid.

4.2. Sensitivity Analysis

Global sensitivity analysis is used to assess the sensitivity of the surrogate model to input parameters during the multi-objective optimization process. In the initial stage, variations in output are monitored by adjusting the input parameters of the surrogate model, which helps identify the model’s sensitivity to different input parameters and improves its accuracy. The Sobol method [29], a statistical technique for sensitivity analysis based on the Sobol series, provides feasibility in analyzing the sensitivity of individual parameters in models with global non-linearity.

Additionally, the Sobol method is chosen to analyze the global sensitivity of five types of parameter to the overall research objective. Since the optimization objective is to improve stability of the glider in the roll and pitch, the selected parameters are as follows: the chord length of the main foil, the wingspan of the main foil, the wingspan of the tail foil, the distance from the leading edge of the main body to the main foils, and the distance from the main body to the tail foils. Figure 11 shows the global sensitivity coefficients of the five parameters.

From the configurations in Figure 11, it can be found that the optimization parameters are reasonable. The global sensitivity coefficients of the parameters to the optimization objectives are not concentrated on a specific parameter. When analyzing the two objectives separately, it is evident that WHLength is the primary factor affecting stability in the pitch motion. SLength and CLength are the next most influential factors. Regarding the roll motion, CLength is the most directly influential factor. Unexpectedly, WHLength has a lower effect on stability in the roll motion. Compared to the tail foil, the main foil has a more significant influence on stability in the pitch motion. In addition, it is observed that the dimensions of CLength, SLength, and WHLength have a more significant influence on stability in both roll and pitch motions compared to TWLength and WTLength.

4.3. Optimization Results

In the multi-objective optimization process, we aim to improve the motion stability performance of the underwater glider through adjusting the foil parameters, these parameters include the configurations of shape and position of the foils. The motion stability considered here are roll and pitch of the glider, while the optimized parameters are CLength, SLength, WHLength, TWLength, and WTLength. The improvement of the foil parameters to enhance the motion stability in the roll motion may be accompanied with a deterioration of the stability in the pitch motion. Nevertheless, a Pareto front with non-dominated solution set can solve the multi-objective optimization problem better.

The particle swarm algorithm was implemented to solve the multi-objective optimization problem. A population size of 15,000 particles was employed, and an inertia weight coefficient of 0.4 was set. The maximum iteration of the MOPSO process was set to 100. With these parameters, the distribution of Pareto solution for multi-objective optimization was obtained as illustrated in Figure 12.

From the results in Figure 12, it can be observed that the optimized solutions for the roll and pitch moments are represented by the Pareto frontier distribution of 15,000 particles. In the figure, a confined boundary indicating the minimum values of roll and pitch moments is presented. The minimum optimized solution for the roll moment is 1.75 N·m, and the minimum optimized value for the pitch moment is 54.8 N·m. The two solutions were then used for subsequent study with the corresponding particle points labeled as E and A. The points E and A represent the optimal conditions; that is, the glider will experience minimum moments in roll and pitch motions, respectively, under these conditions. Three additional particle points, labeled B, C, and D, were selected for the sake of follow-up research in this paper. The specific parameter information for all selected points and the original model is listed in

Table 3. Parameter values (m) of the original model and selected optimization points.

Optimization Points	SLength	CLength	TWLength	WHLength	WTLength
original model	1.190	0.210	0.560	0.320	0.260
Model A	1.125	0.121	0.600	0.374	0.251
Model B	1.125	0.100	0.596	0.353	0.266
Model C	1.114	0.100	0.578	0.346	0.272
Model D	1.112	0.100	0.558	0.340	0.275
Model E	1.107	0.100	0.531	0.335	0.275

.

It can be found in the results of Table 3 that the main body’s parameters in the five groups of selected and original models exhibit certain deviations. These deviations provide more clearer characterizations than those of the original ranges as listed in Table 1. For these selected optimized points, the values of SLength (the wingspan of the main foil) are from range of 1.107 m to 1.125 m, while the values of CLength are mainly concentrated in the range between 0.100 m and 0.121 m. It is should noted that the value of CLength in Point A shows a significant difference compared with the other models.

It is can be seen that an increase in CLength is positively related to an increase in rolling moment based on the results by analysis on the data in Table 3 combined with the Pareto frontier optimization as depicted in Figure 12. Similarly, an increase in TWLength is also related to an increase in the rolling moment. Conversely, WHLength displays a different tendency: an increase in the value of WHLength leads to a reduction in the rolling moment. Within the scope of the analyzed parameters, a unidirectional relationship between adjustments to the parameters and resulting changes in the moment’s magnitude are observed, and the relationships exhibit an unidirectional increase or unidirectional decrease.

To represent the dimension differences of the foils determined by the parameters of the selected optimized points and the original model as shown in Table 3, the parameterized models in image forms determined by Models A ∼ E and the original model are illustrated in Figure 13. In Figure 13a–e the images with solid regions depict the parameterized models by Model A to Model E and those with shaded regions the original model.

The data in

Table 4. Rolling and pitching moments (N·m) of original and optimized models.

Models	Rolling Moment	Pitching Moment
Original model	5.63	112.84
Model A	3.72	54.80
Model B	2.49	59.91
Model C	2.01	66.30
Model D	1.82	73.06
Model E	1.75	87.52

presents the rolling and pitching moments for optimized models and original model. From the data in the table, it can be seen that the rolling moment values for all selected optimized points range from 1.75 N·m to 3.72 N·m. In particular, the optimized design Model E achieved a rolling moment of 1.75 N·m, which represents a significant 68.91% decrease compared with the 5.63 N·m of the original model. For the pitching moment, a similar trend is observed: Model A achieved a 51.63% reduction compared to the 112.84 N·m of the original model. It is worth noting that although the optimized improvements in rolling and pitching moments were achieved by different models (Model E and Model A, respectively), an intermediate solution such as Model C provides a more balanced compromise. Specifically, Model C achieved reductions of 64.30% in rolling moment and 41.24% in the pitching moment relative to the original model. In practical terms, such a compromise solution is particularly valuable, as underwater gliders must operate under complex ocean disturbance conditions, where robustness across multiple degrees of freedom is essential for reliable performance.

4.4. Field Analysis

Pressure and streamline distributions around the glider models with different attitudes (trim and heel) and different flow fields were investigated to observe the reason for the difference between rolling and pitching moments on the glider. Figure 14 shows the pressure and streamline distributions around the glider models when the heel angle is set to 20° at a flow velocity of 1.0 m/s.

Under the heeling working condition of the glider, the wingtip of the glider exhibits significant pressure fluctuations in the original model (Figure 14a). As indicated by the red dotted box in Figure 14, a significant pressure variation from negative to positive is observed at the wingtip, suggesting the presence of a flow transition in this region. This flow transition can cause flow instability, leading to vortex shedding. In contrast, a smoother pressure distribution, suggesting a more stable downstream flow, is presented by the optimized models. Additionally, considerable reductions in pressure concentration in the optimized models compared to the original model are shown. The pressure concentration range on the main body of the glider is notably reduced by the optimized models, and weakened pressure intensity regions can be found at the back surface of the glider’s hull. By comparing the differences of the pressure intensity distributions between the original model and the optimized models, it is found that the pressure distribution variation within optimized models is slightly smaller than those in the original model. As shown in Figure 14a–f, the pressure distributions of the optimized models mainly focus on the position of main foil’s wingtip and upper surface of the hull, which are slightly different from those of the original model.

To investigate the reasons for the differences of rolling moments on the glider, the specific pressure distributions along the middle line of the vehicle’s main foil and hull at a heel angle of 20° are provided in Figure 15a–f based on the data of Figure 14a–f. The middle lines are located at the midpoint between the leading and trailing edges of the main foils, extending from the central axis of the hull toward both sides. The horizontal axes in Figure 15a–f represent positions along the middle line, and the vertical axes in the figures represent the pressure values at the corresponding positions of the horizontal axes. The data within the range of −0.1 m to 0.1 m in the horizontal axes corresponds to the position on the hull, while data outside this range represents the position along the main foils. The solid and dashed lines in the figures represent the pressure distribution along the lower and upper surfaces of the main foil and hull, respectively. It can be seen that all models exhibit a similar pressure distribution pattern in these figures: all positive pressure peaks in the figures appear at −0.1 m on the horizontal axes, this position is at the junction point between the foil and the hull, and the negative pressure peaks in the figures are mainly concentrated near the origins of horizontal axes. Compared to the original model, the pressure distribution of the optimized models shows some obvious variations, particularly in the value of the negative pressure peaks. The negative peak value of the lower surface of the original model significantly exceeds that of the optimized model. Furthermore, the distance between the two pressure peaks for the optimized models is shorter than that of the original model. The primary cause for these discrepancies in pressure distribution lies in the alterations to the main foil dimensions and positional configuration. According to the comparisons of the rolling moment values by different models in Table 4, Model E has the smallest rolling moment of all optimized models. In Figure 15f, at the 0.2 m in horizontal axis position, the pressure distribution is smoother and has a smaller value than those of the other models. Collectively, the results shown in Figure 15 suggest that the dimensions and positional configuration of the main foils will directly influence the pressure distribution, which in turn contributes to variations in the moments.

Table 5. Rolling moments (N·m) along the middle line of the hull for selected optimized models.

Models	Rolling Moments of the Upper Surface	Rolling Moments of the Lower Surface
Original model	20.16	−34.35
Model A	15.66	−29.19
Model B	14.64	−28.63
Model C	14.53	−28.10
Model D	13.16	−27.87
Model E	12.98	−27.48

presents the numerical results of the rolling moments along the middle line of the main foils and vehicle’s hull for the original model and five optimized models. The rolling moments are obtained by integrating the surface pressure values along the middle line. The moments values are listed separately for the upper surface and lower surfaces in the table, and all models exhibiting a rolling moment are dominated by the pressure contributions on the lower surfaces. From the results data in Table 5, we can find that compared to the rolling moment of 20.16 N·m in the original model, all rolling moments on the upper surface in the optimized models show a tendency of gradual decrease, and a lowest value of 12.98 N·m of the rolling moment is observed for Model E; this value is 35.6% lower than that of the original model. The gradual tendency of the rolling moments to decrease indicates that the optimized model plays a certain role in suppressing the rolling motion on the glider. We can also find in Table 5 that, compared to the rolling moment of −34.35 N·m in the original model, all the absolute values of the rolling moments on the lower surface are effectively reduced by the optimized models, and a lowest absolute value of −27.48 N·m in rolling moment is found for Model E, whose value is 20.0% lower than that of the original model. The progressive reduction tendency of the rolling moments on the glider caused by the pressure on the upper and lower surface from Model A to Model E is observed and demonstrates that the rolling instability of the glider is effectively mitigated by the optimization process.

Figure 16 provides the images of pressure and streamline distributions of both the original and optimized models when the glider is in a trim angle of 20° at a flow velocity of 2.5 m/s. In the discussion of the trim working condition as shown in Figure 16, it is noticed that the positive pressure regions are concentrated at the front of the main body and the leading edge of the foils, while the negative pressure region is primarily located along the lower portions of the foils and main body. Compared to the results of the original model, noticeably smaller negative pressure regions on the lower part of the main body are exhibited by the optimized model. From the pressure distribution on the upper main body, it can be concluded that differences in foil position and size are among the main reasons for the variations in positive pressure. However, in the optimized models, these variations exhibit a smoother transition compared to the original model, as shown in red areas in Figure 16. The pressure transition regions of the main foil are mainly concentrated at the wingtip. Modifications to the trailing foils can improve the stability of the flow field in the tail region of the main body.

To investigate the reason for the differences of pitching moments on the glider, the specific pressure distributions along the longitudinal middle line of the vehicle’s hull at a trim angle of 20° are provided in Figure 17a–f based on the data of Figure 16a–f. The longitudinal middle lines traverse the complete hull profile from the head to the tail. The horizontal axes in Figure 17a–f represent the position along the longitudinal middle line, and the vertical axes in the figures represent the pressure values at the corresponding positions of the horizontal axes. Zero in the horizontal axis corresponds to the beginning point of the hull’s head. The solid and dashed lines in the figures represent the pressure distribution along the upper and lower surfaces of the hull, respectively. We can observe that all the models exhibit a similar pressure distribution pattern in these figures: all interlaced peaks appear at 1.1 m to 1.2 m on the horizontal axes, and this position corresponds to the location of tail foils. Compared to the original model, the pressure distribution of the optimized models shows obvious differences, particularly in the location of negative pressure peaks. The original model’s pressure peak of the lower surface is located at the position of 0.05 m and 0.61 m on the horizontal axes, while the location of the original model’s pressure peak is concentrated at around 0.65 m on the horizontal axes. Furthermore, the value of pressure peaks of optimized models is smaller than that of the original model. It also can be seen that the pressure distributions of Model A to Model C in Figure 17b–d are similar, which corresponds to a similar pitching moment values for these optimized models.

Table 6. Pitching moments (N·m) along the longitudinal middle line of the hull for selected optimized models.

Models	Pitching Moments of the Upper Surface	Pitching Moments of the Lower Surface
Original model	210.39	−236.50
Model A	191.32	−200.74
Model B	192.24	−201.07
Model C	193.84	−201.31
Model D	195.50	−203.94
Model E	198.43	−204.51

presents the numerical results of the pitching moments along the longitudinal middle line of the vehicle’s hull in the original model and five optimized models. The pitching moments are obtained by integrating the surface pressure values along the middle line. The moments values are listed separately for the upper and lower surfaces in the table. From the results data in Table 6, it can be found that compared to the pitching moment of 210.39 N·m in the original model, all pitching moments on the upper surface of the optimized models show a tendency of gradual decrease from Model E to Model A. Model A has the lowest pitching moment value of 191.32 N·m, which is 9.1% lower than the pitching moment value of the original model. The gradual tendency of the pitching moments to decrease indicates that the optimized process plays a certain role in suppressing pitching moments in the glider model. We can also find in Table 6, compared to the pitching moment of −236.50 N·m in the original model, that all the absolute values of the pitching moments on the lower surface are significantly reduced by the optimized models, with the lowest absolute value of −200.07 N·m in the pitching moment found for Model A, whose value is 15.1% lower than that of the original model. The progressive reduction in the pitching moment tendency on the glider, attributable to pressure differences between the upper and lower surfaces from Model E to Model A, is found in Table 6; this trend indicates that the optimization process effectively mitigates the glider’s pitching instability.

5. Conclusions

The aim in this study was to improve the hydrodynamic performance of an underwater glider in both roll and pitch motions by optimizing its shape, specifically its foil configurations. A multi-objective optimization problem with five parameter sets was studied using a framework that integrates several advanced techniques. The LHS was used to generate 60 sample points, which were then utilized to construct a surrogate model as the Kriging model. Sensitivity analysis was performed on the selected parameters to identify their influence on the optimization objectives. For the optimized model, the parameter WHLength is the primary factor affecting pitch stability, while for the rolling stability, the most influential factor is CLength compared with the other parameters.

The MOPSO method was applied to the surrogate model, a Pareto frontier was generated after 100 iterations with a population size of 15,000, and five optimized solutions were selected through this process. The pitching moments for these optimized solutions range from 54.8 N·m to 87.5 N·m, while the rolling moments range from 1.75 N·m to 3.72 N·m. The optimization achieves maximum improvements of 68.91% in roll stability and 51.63% in pitch stability compared to the original model.

Furthermore, the optimized models exhibit smoother pressure distributions and more stable downstream flows compared with the original design. Specifically, the optimized models show significant reductions in pressure concentration on the glider’s main body and notable decrease in the negative pressure peaks on the lower surface of the hull, which contributes to the decrease in rolling and pitching moments. These improvements confirm that the proposed self-stability control strategy, realized through hydrodynamic optimization of the foil parameters, can effectively enhance the motion stability of underwater gliders under flow disturbance conditions. The established optimization framework couples the hydrodynamic performance of an underwater glider with the parameters affecting glider’s attitude stability into an integrated system, providing researchers with a new method to effectively improve motion stability through parameter optimization. Future research may extend the framework in this study by developing real-time adaptive optimization methods to dynamically adjust hydrofoil configurations in response to various flow conditions.