Design Optimization and Experiments of Composite Structure Based Pressure Hull for Full-Ocean-Depth Underwater Vehicles

Abstract

This study addresses the limitations of buoyancy factor and compensation capacity in pressure hulls for full-ocean-depth underwater gliders operating in extreme deep-sea conditions. A novel lightweight multifunctional composite structure pressure hull (CSPH) is proposed, utilizing a carbon fiber cylindrical shell as the primary load-bearing structure and silicone oil as the buoyancy compensation medium. A mechanical model of the carbon fiber cylindrical shell under hydrostatic pressure was developed based on three-dimensional elastic mechanics theory. Furthermore, a comprehensive performance evaluation model for the CSPH was created, incorporating both the buoyancy factor (B_f) and buoyancy fluctuation coefficient (f_B). The NSGA-II optimization algorithm was employed to simultaneously minimize B_f and f_B by co-optimizing the carbon fiber ply parameters and the silicone oil volume (V_C). This optimization resulted in a Pareto optimal solution balancing buoyancy and compensation performance. The accuracy of the mechanical model and optimization results was validated through finite element analysis and pressure testing. The results show that, compared to traditional metallic pressure hull designs, the CSPH reduces the buoyancy factor by 48% and enhances buoyancy compensation performance by 2.5 times. The developed CSPH has been successfully deployed on the “Sea-Wing 11000” full-ocean-depth underwater glider, significantly improving its endurance and motion stability for long-term deep-sea observation missions.

1. Introduction

The Hadal Zone, encompassing ocean trench regions beyond 6000 m—with the deepest point reaching approximately 10,987 m—represents a pivotal yet underexplored frontier in marine science [1]. Initiatives such as “HADES” [2], “HADEEP” [3], and “Global Tren D” [4] reflect growing scientific interest, but exploration remains sparse and fragmented [1], largely due to the high cost and limited endurance of existing full-ocean-depth (FOD) platforms. These include human-occupied vehicles (HOVs) [5], remotely operated vehicles (ROVs) [6], and autonomous/remotely operated hybrid vehicles (ARVs) [7]. Underwater gliders, buoyancy-driven autonomous systems offer a promising alternative for sustained observation [8]; yet, their operational depth has been largely confined to about 1000 m [9], restricting their use in hadal zones.

A fundamental constraint in developing hadal gliders is the pressure hull, which must withstand extreme hydrostatic pressure while preserving buoyancy efficiency and payload margin. Traditional metallic hulls (e.g., titanium alloys) become prohibitively heavy at full-ocean depth, necessitating large buoyancy additions that increase volume and degrade energy economy [10]. This has shifted attention to lightweight, high-strength non-metallics. Ceramics were the first employed at full-ocean depth: although excellent in compression, their high brittleness and poor tensile capacity make cutouts, joints, and boundary transitions prone to catastrophic failure; they are also impact-sensitive and require special handling. WHOI’s “Nereus” ARV illustrates these risks: after multiple 10 km-class missions, it was lost in 2014 in the Kermadec Trench due to fracture of ceramic components at depth [11]. Consequently, for cost and safety reasons, WHOI largely abandoned ceramic solutions in subsequent FOD ARV designs. Carbon-fiber-reinforced polymer (CFRP) composites, with high specific strength and stiffness, have emerged as attractive alternatives, as evidenced in systems like “AUTOSUB6000” [12], “Deep Glider” [13], and “SeaWing7000” [14]. However, their application to full-ocean-depth (FOD) vehicles remains unreported—partly because, unlike isotropic ceramics, CFRP exhibit complex, laminate-dependent anisotropy that makes optimization over high-dimensional ply spaces challenging [15].

Current design strategies often rely on finite element analysis (FEA) combined with genetic algorithms to minimize buoyancy factor (Bf). Examples include optimizations of layup patterns under moderate pressures [16], studies of elliptical hulls [17]. To improve design efficiency, prior work has combined hierarchical optimization with surrogate modeling for rib-stiffened structures [18]. In a related effort, Ref. [19] employed finite-element analysis (FEA) and response surface methodology (RSM) to optimize cylindrical CFRP hulls under 20 MPa external pressure. Beyond pressure capacity, the hull must also provide buoyancy compensation—sufficient volumetric compressibility to offset depth-dependent seawater density and reduce energy use. Topology-optimized metallic hulls can supply this in shallow water [20], but at greater depths, the thick walls required for strength/buckling suppress compressibility. Hence, external low-bulk-modulus silicone oil chambers are used; their pressure-induced volume reduction supplies the needed deep-sea-density compensation [21].

Despite these advances, current research exhibits two major limitations:

(1)

Lack of high-fidelity and efficient modeling tools: optimizing thick-walled hulls (30–50 mm, hundreds of plies) for FOD conditions requires evaluating enormous layup combinations. While FEA captures details such as anisotropy and defects, it becomes computationally intractable in high-dimensional design spaces. Surrogate models improve speed but suffer from limited interpretability and extrapolation risks [22]. Researchers have modeled thin-walled cylindrical shells using two-dimensional classical lamination theory (CLT) [23]. However, CLT assumes a sufficiently thin shell and neglects through-thickness stress gradients; although analytically efficient, this simplification leads to biased three-dimensional stress predictions for thick-walled pressure hulls under ultra-high pressure [24].

(2)

Decoupled design of structure and compensation system: existing studies treat structural design and buoyancy compensation as separate steps [21,25]. This approach is more feasible for isotropic hulls, where mechanical response and volumetric compression follow well-characterized laws. For anisotropic CFRP hulls, however, compressibility varies significantly with layup, even for similar collapse strengths. This variability directly affects the volume of silicone oil required for compensation. Since silicone oil is denser than CFRP hulls, its quantity directly impacts net buoyancy. The strong coupling between laminate parameters, structural strength, and compressibility necessitates a co-optimization framework, which has not yet been established.

This study addresses these gaps via the following:

(1)

Deriving a 3D analytical model for thick-walled anisotropic cylinders that efficiently predicts through-thickness stresses/displacements, overcoming CLT limitations and reducing reliance on costly FEA;

(2)

Proposing an integrated co-optimization framework for the CFRP hull and silicone–oil system—equipped with coupled performance metrics, the buoyancy factor (B_f) and buoyancy-fluctuation coefficient (f_B)—to jointly minimize buoyancy (structural) penalty and compensation penalty for FOD gliders;

(3)

Implementing NSGA-II to optimize laminate parameters and oil volume, with validation via pressure tests and Sea-Wing 11000 sea trials, yielding an end-to-end methodology spanning modeling, laboratory calibration, and field deployment.

The paper is structured as follows: Section 2 outlines the overall design and material tests; Section 3 develops the 3D analytical model and performance metrics; Section 4 details the co-optimization process; Section 5 presents experimental and sea trial validation; and Section 6 summarizes conclusions and future work.

2. Overall Structure Design and Core Parameter Testing

2.1. Structure Design

To balance hydrodynamic performance, pressure resistance, and internal space requirements for the full-ocean-depth (FOD) underwater glider, this study adopts a classic torpedo-shaped profile [26]. The primary pressure hull employs a cylindrical as a carbon-fiber-reinforced polymer wound structure. Given that end-cap machining (e.g., drilling and tapping threads for sensor/electronic connections) compromises fiber continuity and structural integrity, titanium alloy end caps are implemented. Further, highly compressible silicone oil is sealed with rubber layers at both hemispherical ends to enable passive deep-sea-density compensation. The compensation mechanism operates as follows:

(i)

During descent: silicone oil volume contraction counteracts seawater density increase under rising pressure and falling temperature.

(ii)

During ascent: gradual volume expansion compensates for seawater density decrease, preventing thrust decay.

This innovation leverages dynamic conversion between environmental pressure potential energy and molecular potential energy, ensuring real-time constant thrust without active energy input. The composite structure pressure hull (CSPH) overall configuration is shown in Figure 1.

Key design parameters of the CSPH include CFRP cylindrical shell outer diameter D_o, number of winding layers n, length L, stacking sequence [α] = [α₁/α₂/…/α_n]; silicone oil volume Vc. These parameters govern structural strength, buoyancy factor, and compensation performance, constituting core optimization variables. Based on FOD glider requirements, baseline cylindrical dimensions are set as Do = 324 mm, L = 800 mm.

2.2. Parametric Modeling of the CSPH Operational Environment

Seawater pressure, temperature, and density serve as critical inputs for the CSPH optimization, necessitating environmental parameter modeling prior to pressure hull design. According to Ref. [25], the relationships between pressure (P_sea), density (ρ_sea), temperature (T_sea) and depth (H) are expressed as follows:

Pressure (MPa):

$$P_{\mathrm{sea}}\left(H\right)={\displaystyle \frac{\left(1-a_1\right)-\sqrt{(1-a_1{)}^2-4a_{2}H}}{a_2}}$$

(1)

where P_sea(H) is the seawater pressure at depth H, a₁ = 5.95 × 10⁻³ mdb⁻¹, a₂ = 4.42 × 10⁻⁴ mdb⁻².

Density (g/cm³):

$${\rho }_{sea}\left(H\right)={\rho }_0+b_1\left(1-e^{-{\displaystyle \frac{H}{t_1}}}\right)+b_2\left(1-e^{-{\displaystyle \frac{H}{t_2}}}\right)$$

(2)

where ρ_sea(H) is the seawater density at depth H, b₁ = 0.00686, b₂ = 0.32011, t₁ = 269.31, t₂ = 69,196.78, ρ₀ = 1.021 g/cm³.

Temperature (°C):

$$T_{sea}\left(H\right)=18.62e^{-{c_1}^2}+8.049\times 10^2{e}^{-{c_2}^2}+6.448\times 10^5{e}^{-{c_3}^2}$$

(3)

where T_sea(H) is the seawater temperature at depth H, c₁ = (H − 7.318)/156.2, c₂ = (H + 45,360)/8680, c₃ = (H − 333,700)/9.260

As shown in Figure 2, depth-dependent profiles of seawater pressure and density reveal that at H = 11,000 m: ambient pressure reaches 113.5 MPa; and seawater density increases to 1.0748 g/cm³, representing a 5.26% density increase relative to surface conditions. These extreme hydrostatic pressures and pronounced density gradients constitute the primary challenges for the CSPH design.

2.3. Core Parameter Testing for the CSPH

The mechanical properties of carbon fiber reinforced polymer are critically dependent on fiber material selection and manufacturing processes. To ensure reliable marine applications through accurate mechanical assessment of the hull, precise constitutive relationship parameters must be obtained. Unidirectional specimens (Figure 3a) were fabricated using identical manufacturing protocols as the pressure hull. Tested materials include Toray Industries, Inc. (Japan) Carbon Fiber T700. Following ASTM D6641/D6641M [27], unidirectional specimen testing was conducted (Figure 3b), 0° direction stress–strain curves for compression tests (Figure 3c). The resulting mechanical properties are summarized in

Table 1. Mechanical properties of the unidirectional carbon/epoxy laminate.

Elastic Properties	T700	Strength Properties	T700
0° compressive modulus E1 (GPa)	128.89	0° compressive strength XC (MPa)	986.45
90° compressive modulus E2 (GPa)	10.21	0° tensile strength XT (MPa)	2131.55
Shear modulus G12, G13 (GPa)	5.84	90° compressive strength YC (MPa)	160.76
Shear modulus G23 (GPa)	3.83	90° tensile strength YT (MPa)	39.36
Poisson ration μ12, μ13	0.33	Shear strength S12, S13 (MPa)	88.25
Poisson ration μ23	0.32	Shear strength S23 (MPa)	35.93
ρC (g/cm3)	1.6	/	/

.

The mechanical properties were tested at an ambient temperature of 25 °C. Since carbon fiber composites consist of resin and fibers, their mechanical behavior is influenced by temperature, particularly under high temperatures where the resin properties may change. However, according to Equation (3), the operational temperature range of full-ocean-depth gliders is between 2 °C and 35 °C. An existing study, Ref. [28] indicates that within this range, the mechanical properties of the material remain stable without significant variation. Therefore, the measured data are directly adopted in the subsequent design without applying any reduction factors.

This study employs highly compressible silicone oil (ρ_SO = 0.8 g/cm³ @25 °C) as the compensation medium. Its volumetric response exhibits significant temperature-pressure coupling effects. Laboratory simulations (Figure 4) under coupled thermo-pressure conditions were then conducted to obtain depth-dependent volumetric compressibility data k_C(H), These parameters are the critical ones for optimizing CSPH compensation performance. Among them, the seawater temperature-pressure coupling parameters were established based on the seawater pressure Equation (1) and temperature Equation (3): we first constructed the seawater temperature-pressure coupling profiles (Figure 5a).

The experimental setup and testing procedures are shown in Figure 4, with silicone oil’s thermo-pressure compressibility curves presented in Figure 5b.

The parametric model Equation (4) fitted to experimental data demonstrates that at H = 11,000 m: silicone oil achieves 14.7% volumetric compression, substantially exceeding seawater’s 5.26% density increase. This validates its effective density compensation capability for full-ocean-depth applications.

$$k_C\left(H\right)=k_C\left(0\right)+d_1\left(1-e^{-\frac{H}{t_3}}\right)+d_2\left(1-e^{-{\displaystyle \frac{H}{t_4}}}\right)$$

(4)

where d₁ = 0.16956, d₂ = 0.04124, t₃ = 10678.97494, t₄ = 360.23931, k_c(0) = −0.0040329, all dimensionless constants.

In essence, the volume compression of silicone oil increases its density. This increase in density reduces net buoyancy, thereby compensating for the elevated density of seawater at depth. This is the fundamental principle behind using silicone oil for buoyancy compensation.

3. Mechanical Model of Carbon Fiber Cylindrical Shell

The carbon fiber cylindrical shell serves as the primary pressure-bearing component of the CSPH. A precise quantitative analysis of the stress distribution, strain response, and compressive deformation behavior of the carbon fiber cylindrical shell under deep-sea hydrostatic pressure is conducted, thereby providing an essential theoretical foundation for subsequent structural parameter optimization of the CSPH. Therefore, a critical objective of this study is to establish a high-fidelity mechanical model for the carbon fiber cylindrical shell.

3.1. Coordinate System Definition and Parametric Design

A cylindrical coordinate system is established for the carbon fiber cylindrical shell, accompanied by parametric design as illustrated in Figure 6. The shell geometry is defined in the (r, θ, z) coordinate system (Figure 6a), where z denotes the axial direction of the pressure hull; r represents the radial direction; θ corresponds to the circumferential direction. The material orientation is described in the material coordinate system (1,2,3) (Figure 6b), with the winding angle α^(k) defined as the orientation of the k-th layer fibers relative to the axial direction z. The laminate stacking sequence is shown in Figure 6c, where the innermost layer is designated as Ply #1; the outermost layer is Ply #n; the inner radius of the k-th ply is r_k; the outer radius of the k-th ply is r_k+1; the inner boundary condition satisfies r₁ = r_i; the outer boundary condition follows r_n+1 = r_i + n*t = r_o; and the constant t denotes that the thickness of a single fiber ply is 0.2 mm.

3.2. Stress and Strain Analysis

According to Ref. [29], a cylindrical shell subjected to uniform hydrostatic pressure exhibits an axisymmetric response. Consequently, the stresses and strains are independent of the circumferential coordinate θ, and the radial displacement u_r and axial displacement u_z are functions solely of z and r, respectively. The resulting stress–strain field comprises the following components:

Circumferential: σ_θ, ε_θ; radial: σ_r, ε_r; axial: σ_z, ε_z; in-plane shear: τ_zθ, γ_zθ.

As depicted in the coordinate system shown in Figure 6a, the pressure hull will develop a three-dimensional displacement field, which is mathematically expressed as follows:

$$u_r=u_r\left(r\right),u_{\theta }=u_{\theta }\left(r,z\right),u_z=u_z\left(z\right)$$

(5)

where u_r, u_θ, u_z represent the radial displacement, circumferential displacement, and axial displacement components of the cylindrical shell, respectively.

According to Ref. [30], the strain–displacement relations for the k-th ply can be expressed as follows:

$$\begin{array}{c}{\epsilon }_r^{\left(k\right)}={\displaystyle \frac{d{u_r}^{\left(k\right)}}{dr}},{\epsilon }_{\theta }^{\left(k\right)}={\displaystyle \frac{{u_r}^{\left(k\right)}}{r}},{\epsilon }_z^{\left(k\right)}={\displaystyle \frac{d{u_z}^{\left(k\right)}}{dz}}={\epsilon }_0\\ {{\gamma }_{zr}}^{\left(k\right)}=0,{{\gamma }_{\theta r}}^{\left(k\right)}=0,{{\gamma }_{z\theta }}^{\left(k\right)}={\displaystyle \frac{d{u_{\theta }}^{\left(k\right)}}{dz}}={\gamma }_{0}r\end{array}$$

(6)

where γ₀ denotes the rate of twist per unit length of the cylindrical shell, while the axial strains remain constant across all layers with a value of ε₀.

Based on the three-dimensional elasticity theory for fibrous composites [31], the off-axis stresses and strains in the k-th ply at an arbitrary winding angle α^(k) can be expressed as follows:

$${\left[\begin{array}{c}{\sigma }_z\\ {\sigma }_{\theta }\\ {\sigma }_r\\ {\tau }_{\theta r}\\ {\tau }_{zr}\\ {\tau }_{z\theta }\end{array}\right]}^{\left(k\right)}={\left[\begin{array}{cccccc}{\overline{C}}_{11}& {\overline{C}}_{12}& {\overline{C}}_{13}& 0& 0& {\overline{C}}_{16}\\ {\overline{C}}_{12}& {\overline{C}}_{22}& {\overline{C}}_{23}& 0& 0& {\overline{C}}_{26}\\ {\overline{C}}_{13}& {\overline{C}}_{23}& {\overline{C}}_{33}& 0& 0& {\overline{C}}_{36}\\ 0& 0& 0& {\overline{C}}_{44}& {\overline{C}}_{45}& 0\\ 0& 0& 0& {\overline{C}}_{45}& {\overline{C}}_{55}& 0\\ {\overline{C}}_{16}& {\overline{C}}_{26}& {\overline{C}}_{36}& 0& 0& {\overline{C}}_{66}\end{array}\right]}^{\left(k\right)}{\left[\begin{array}{c}{\epsilon }_z\\ {\epsilon }_{\theta }\\ {\epsilon }_r\\ {\gamma }_{\theta r}=0\\ {\gamma }_{zr}=0\\ {\gamma }_{z\theta }\end{array}\right]}^{\left(k\right)}$$

(7)

In Equation (7), [C] ^(k) denotes the three-dimensional stiffness matrix of the k-th ply, whose explicit form can be found in Ref. [31].

Based on the infinitesimal element of a single lamina shown in Figure 7a, the mechanical equilibrium relationship illustrated in Figure 7b is established, yielding the following expression:

$$\left({\sigma }_r+d{\sigma }_r\right)\left(r+dr\right)d\theta -{\sigma }_{r}rd\theta -2{\sigma }_{\theta }dr\mathit{sin}{\displaystyle \frac{d\theta }{2}}=0$$

(8)

The infinitesimal element shown in Figure 7b can be regarded as infinitesimal, where dθ approaches an infinitely small value, $\sin {\displaystyle \frac{d\theta }{2}}\approx {\displaystyle \frac{d\theta }{2}}$. Thus, Equation (8) can be simplified to Equation (9).

$${\displaystyle \frac{d{{\sigma }_r}^{\left(k\right)}}{dr}}+{\displaystyle \frac{{{\sigma }_r}^{\left(k\right)}-{{\sigma }_{\theta }}^{\left(k\right)}}{r}}=0$$

(9)

From Equations (6) and (7), it follows that

$${{\sigma }_r}^{\left(k\right)}={{\overline{C}}_{13}}^{\left(k\right)}{\epsilon }_0+{{\overline{C}}_{23}}^{\left(k\right)}{\displaystyle \frac{{u_r}^{\left(k\right)}}{r}}+{{\overline{C}}_{33}}^{\left(k\right)}{\displaystyle \frac{d{u_r}^{\left(k\right)}}{dr}}+{{\overline{C}}_{36}}^k{\gamma }_{0}r$$

(10)

$${{\sigma }_{\theta }}^k={{\overline{C}}_{12}}^k{\epsilon }_0+{{\overline{C}}_{22}}^k{\displaystyle \frac{{u_r}^{\left(k\right)}}{r}}+{{\overline{C}}_{23}}^k{\displaystyle \frac{d{u_r}^{\left(k\right)}}{dr}}+{{\overline{C}}_{26}}^k{\gamma }_{0}r$$

(11)

Combining Equations (9)–(11) yields the ordinary differential equation governing radial displacement u_r^(k):

$${\displaystyle \frac{d^2{u_r}^{\left(k\right)}}{d{r}^2}}+{\displaystyle \frac{d{u_r}^{\left(k\right)}}{rdr}}-{\displaystyle \frac{\frac{{{\overline{C}}^k}_{22}}{{\overline{C}}_{33}}}{r^2}}{u_r}^{\left(k\right)}={\displaystyle \frac{\left({{\overline{C}}^k}_{12}-{{\overline{C}}^k}_{13}\right){\epsilon }_0}{r{{\overline{C}}^k}_{33}}}+{\displaystyle \frac{\left({{\overline{C}}_{26}}^k-2{{\overline{C}}_{36}}^k\right){\gamma }_0}{{{\overline{C}}^k}_{33}}}$$

(12)

The general solution for the radial displacement in an arbitrary fibrous layer is subsequently derived from Equation (12):

$${u_r}^{\left(k\right)}=D^{\left(k\right)}{r}^{{\beta }^{\left(k\right)}}+E^{\left(k\right)}{r}^{-{\beta }^{\left(k\right)}}+{\alpha }_1^{\left(k\right)}{\epsilon }_{0}r+{\alpha }_2^{\left(k\right)}{\gamma }_0{r}^2$$

(13)

where D^(k), E^(k) represent integration constants for the k-th ply, and β^(k), α₁^(k), α₂^(k) are defined as follows:

$${\beta }^{\left(k\right)}=\sqrt{{\displaystyle \frac{{\overline{C}}_{22}^{\left(k\right)}}{{\overline{C}}_{33}^{\left(k\right)}}}}, {\alpha }_1^{\left(k\right)}={\displaystyle \frac{{\overline{C}}_{12}^{\left(k\right)}-{\overline{C}}_{13}^{\left(k\right)}}{{\overline{C}}_{33}^{\left(k\right)}-{\overline{C}}_{22}^{\left(k\right)}}},{\alpha }_2^{\left(k\right)}={\displaystyle \frac{{\overline{C}}_{26}^{\left(k\right)}-2{\overline{C}}_{36}^{\left(k\right)}}{4{\overline{C}}_{33}^{\left(k\right)}-{\overline{C}}_{22}^{\left(k\right)}}}$$

(14)

3.3. Boundary Conditions and Stress–Strain Solution

For a carbon fiber cylindrical shell wound with n layers, the system yields 2n + 2 unknowns, Coefficients D^(k) and E^(k) (k = 1,2,…,n), ε₀ and γ₀. To solve this indeterminate system, boundary constraints for the cylindrical shell must be established as per Ref. [29].

The mechanical boundary conditions at the inner (r = r_i) and outer (r = r_o) surfaces are

$${\sigma }_r^{\left(n\right)}\left(r_o\right)=-P_d,{\sigma }_r^{\left(1\right)}\left(r_i\right)=0$$

(15)

$${\tau }_{\theta r}^{\left(1\right)}\left(r_o\right)={\tau }_{zr}^{\left(1\right)}\left(r_o\right)=0,{\tau }_{\theta r}^{\left(n\right)}\left(r_i\right)={\tau }_{zr}^{\left(n\right)}\left(r_i\right)=0$$

(16)

In Equations (15) and (16), r_o and r_i represent the outer and inner radii of the cylindrical shell, respectively, where the physical significance is that under external hydrostatic pressure the radial stress at the pressure hull’s inner wall is zero; due to solid mechanics convention (compressive stress being negative), the radial stress at the outer wall σ_r⁽ⁿ⁾ equals -P_d, where P_d is design pressure, and the design pressure P_d is defined as 1.1 times the maximum working pressure P_d = 1.1*P(11,000), yielding P_d = 125 MPa for full-ocean-depth applications, while both inner and outer walls remain shear stress-free is zero.

Assuming no interlayer delamination or slippage occurs in the pressure hull, the continuous deformation boundary conditions between adjacent layers are given by the following expressions with perfect interfacial bonding:

$${\sigma }_r^k\left(r_k\right)={\sigma }_r^{k+1}\left(r_k\right)$$

(17)

$$u_r^k\left(r_k\right)=u_r^{k+1}\left(r_k\right), u_{\theta }^k\left(r_k\right)=u_{\theta }^{k+1}\left(r_k\right)$$

(18)

$${\tau }_{zr}^k\left(r_k\right)={\tau }_{zr}^{k+1}\left(r_k\right), {\tau }_{\theta r}^k\left(r_k\right)={\tau }_{\theta r}^{k+1}\left(r_k\right)$$

(19)

Furthermore, for cylindrical shells under hydrostatic pressure, the axial load resultant and the summation of in-plane shear stresses can be expressed as

$$2\pi \sum _{k=1}^n{\int }_{r_{k-1}}^{r_k}{\sigma }_z^{\left(k\right)}\left(r\right)rdr=-\pi R_o^2{P}_d$$

(20)

$$2\pi \sum _{k=1}^n{\int }_{r_{k-1}}^{r_k}{\tau }_{z\theta }^{\left(k\right)}\left(r\right)r^{2}dr=0$$

(21)

The physical interpretations of Equations (20) and (21) are that the axial force induced by hydrostatic pressure that equals the resultant axial force from all fiber layers, and no rotation occurs in the cylindrical shell under hydrostatic pressure resulting in zero resultant in-plane shear stress; consequently, the 2n + 2 unknown coefficients D^(k) and E^(k)(k = 1,2,…,n), axial strain ε₀, and torsional component γ₀ can be determined from Equation (22), where [Ψ] denotes a (2n + 2) × (2n + 2) matrix.

$$\left\{\begin{array}{c}{D}^{\left(1\right)}\\ ⋮\\ D^{\left(n\right)}\\ E^{\left(1\right)}\\ ⋮\\ E^{\left(n\right)}\\ {\epsilon }_0\\ {\gamma }_0\end{array}\right\}=[\mathit{\psi }{]}^{-1}\left\{\begin{array}{c}0\\ ⋮\\ 0\\ 0\\ ⋮\\ -P_{sea}\\ -{\displaystyle \frac{R_o^2{P}_{sea}}{2}}\\ 0\end{array}\right\}$$

(22)

Once the integration constants D^(k) and E^(k) for each layer, along with the axial strain ε₀ and torsional component γ₀ are determined, the radial displacement u_r^(k) per layer can be obtained from Equation (13), while the strain and stress equations for the cylindrical shell in cylindrical coordinates, specifically radial strain ε_r, circumferential strain ε_θ, and corresponding stresses, are derived from Equations (6) and (7).

$$\left\{\begin{array}{l}{{\epsilon }_r}^{\left(k\right)}={\beta }^{\left(k\right)}{D}^{\left(k\right)}{r}^{{\beta }^{\left(k\right)}-1}-{\beta }^{\left(k\right)}{E}^{\left(k\right)}{r}^{-{\beta }^{\left(k\right)}-1}+{\alpha }_1^{\left(k\right)}{\epsilon }_0+2{\alpha }_2^{\left(k\right)}{\gamma }_{0}r\\ {{\epsilon }_{\theta }}^{\left(k\right)}=D^{\left(k\right)}{r}^{{\beta }^{\left(k\right)}-1}+E^{\left(k\right)}{r}^{-{\beta }^{\left(k\right)}-1}+{\alpha }_1^{\left(k\right)}{\epsilon }_0+{\alpha }_2^{\left(k\right)}{\gamma }_{0}r\end{array}\right.$$

(23)

$$\left\{\begin{array}{c}{\sigma }_z={\overline{C}}_{11}^{\left(k\right)}{{\epsilon }_0}^{\left(k\right)}+{\overline{C}}_{12}^{\left(k\right)}{{\epsilon }_{\theta }}^{\left(k\right)}+{\overline{C}}_{13}^{\left(k\right)}{{\epsilon }_r}^{\left(k\right)}+{\overline{C}}_{16}^{\left(k\right)}{\gamma }_0{r}_{\left(k\right)}\\ {\sigma }_{\theta }={\overline{C}}_{12}^{\left(k\right)}{{\epsilon }_0}^{\left(k\right)}+{\overline{C}}_{22}^{\left(k\right)}{{\epsilon }_{\theta }}^{\left(k\right)}+{\overline{C}}_{23}^{\left(k\right)}{{\epsilon }_r}^{\left(k\right)}+{\overline{C}}_{26}^{\left(k\right)}{\gamma }_0{r}_{\left(k\right)}\\ {\sigma }_r={\overline{C}}_{13}^{\left(k\right)}{{\epsilon }_0}^{\left(k\right)}+{\overline{C}}_{23}^{\left(k\right)}{{\epsilon }_{\theta }}^{\left(k\right)}+{\overline{C}}_{33}^{\left(k\right)}{{\epsilon }_r}^{\left(k\right)}+{\overline{C}}_{36}^{\left(k\right)}{\gamma }_0{r}_{\left(k\right)}\\ {\tau }_{z\theta }={\overline{C}}_{16}^{\left(k\right)}{{\epsilon }_0}^{\left(k\right)}+{\overline{C}}_{26}^{\left(k\right)}{{\epsilon }_{\theta }}^{\left(k\right)}+{\overline{C}}_{36}^{\left(k\right)}{{\epsilon }_r}^{\left(k\right)}+{\overline{C}}_{66}^{\left(k\right)}{\gamma }_0{r}_{\left(k\right)}\end{array}\right.$$

(24)

Since failure criteria for carbon fiber composites must be evaluated in the material coordinate system (1,2,3), the stresses solved in the cylindrical coordinate system (r, θ, z) are transformed via Equation (25) [31], yielding longitudinal stress σ₁, transverse stress σ₂, σ₃, and shear stress τ₁₂ in the material coordinate system.

$${\left[\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\tau }_{23}\\ {\tau }_{13}\\ {\tau }_{12}\end{array}\right]}^{\left(k\right)}={\left[\begin{array}{cccccc}{m}^2& n^2& 0& 0& 0& 2mn\\ n^2& m^2& 0& 0& 0& -2mn\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& m& -n& 0\\ 0& 0& 0& n& m& 0\\ -mn& mn& 0& 0& 0& m^2-n^2\end{array}\right]}^{\left(k\right)}{\left[\begin{array}{c}{\sigma }_z\\ {\sigma }_{\theta }\\ {\sigma }_r\\ {\tau }_{\theta r}\\ {\tau }_{zr}\\ {\tau }_{z\theta }\end{array}\right]}^{\left(k\right)}$$

(25)

where m = cosα^(k), n = sinα^(k), α^(k) is the k-th ply winding angle.

At this stage, the complete solution for stresses and strains in each layer of the carbon fiber cylindrical pressure hull under hydrostatic pressure is obtained through Equations (22)–(25), enabling determination of the critical pressure for hull failure via composite strength criteria; while strength failure dominates structural integrity in extreme deep-sea environments [32], buckling resistance remains essential for safety; therefore, Equation (26) (detailed in Ref. [33] is employed for external pressure stability validation.

$$\left(\left[\mathit{K}\right]+P_{cr}\left[\mathit{L}\right]\right)\left\{\begin{array}{c}{a}_u\\ a_v\\ a_w\\ a_{\phi x}\\ a_{\phi y}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\end{array}\right\}$$

(26)

where [K] and [L] are 5 × 5 matrix, a_u, a_v, a_w, a_φx, a_φy represents the displacement coefficient associated with buckling modes [33], and P_cr denotes the critical buckling load.

3.4. Performance Evaluation Model of the CSPH

The buoyancy factor (B_f), defined as the ratio of the pressure hull weight to the volume of water displaced under normal temperature and pressure [17], is a critical performance indicator for the pressure hull of underwater gliders, directly determining its payload capacity [34]. The CSPH in this paper consists of three parts: carbon fiber shell, silicone oil, and a titanium alloy spherical shell. Predictions from the relevant literature indicate that the buoyancy factor of a carbon fiber cylindrical shell hull is approximately 0.7~0.75 at the design operating depth of 11,000 m [10]; the silicone oil selected in this study has a density of 0.8 g/cm³, thus possessing a buoyancy factor of 0.8. Both of these components exhibit positive buoyancy, while the titanium alloy spherical shell exhibits negative buoyancy [35]. Furthermore, according to the literature [25], the buoyancy factor of a metallic spherical shell depends solely on the material strength and is independent of size, offering limited optimization potential once the material is selected. Therefore, this paper considers the titanium alloy spherical shell as part of the system payload. Consequently, the comprehensive buoyancy factor of the pressure hull is determined by Equation (27).

$$B_f={\displaystyle \frac{M_{hull}}{M_{water}}}={\displaystyle \frac{M_H+M_C}{\left(V_H+V_C\right){\rho }_{sea}\left(0\right)}}$$

(27)

where M_H and M_C represent the masses of the carbon fiber cylindrical shell and the silicone oil, respectively, V_H and V_C denote the displaced water volumes of the cylindrical shell and the silicone oil under normal temperature and pressure, respectively, and ρ_sea(0) is the seawater density at sea level, expressed as follows:

$$\left\{\begin{array}{l}{M}_H={\rho }_{CFRP}\pi (r_o^2-r_i^2)L\\ M_C={\rho }_{so}{V}_C\\ V_H=\pi r_o^{2}L\end{array}\right.$$

(28)

An ideal pressure hull requires its buoyancy B(H) to remain constant at any depth H. However, seawater density exhibits nonlinear variation with depth, the volumetric compression coefficient of the compensation medium changes nonlinearly with depth, while the pressure hull volume deformation with depth H is essentially linear [17]. Consequently, maintaining constant total buoyancy B(H) across all depths is challenging. To evaluate the density compensation effectiveness, we define the buoyancy fluctuation coefficient f_B as the mean square deviation of B(H) relative to its initial value B(0) over the entire operating depth range, where f_B is given by Equation (32).

$$f_B={\displaystyle \frac{1}{H}}{\int }_0^H\left(B\right(H)-B(0){)}^{2}dH$$

(29)

where H = 11,000 m smaller f_B indicates superior buoyancy compensation performance of the CSPH.

To determine the buoyancy B(H) of the CSPH during its operational profile, a parametric model correlating the volume compression variations in hull components with depth H must be established, where the carbon fiber cylindrical shell compression primarily arises from axial and radial deformation components, expressed as follows:

$$\Delta V_H\left(H\right)=\pi R_o^2{\epsilon }_{0}L+\pi L(1-{\epsilon }_0)(R_o^2-(R_o-u_r^{\left(n\right)}{)}^2)$$

(30)

here ε₀ and u_r⁽ⁿ⁾ represent the axial strain and radial displacement of the outermost fibers, respectively, determined for the pressure hull at depth H under ambient pressure P(H) through the mechanical model established in this study.

Simultaneously, the volume variation model of the compensation medium as a function of depth H can be developed according to Equation (4), accounting for its coupled thermal pressure characteristics.

$$\Delta V_C\left(H\right)=k_C\left(H\right)V_C$$

(31)

The real-time buoyancy of the CSPH at varying depths can be derived from Equations (2), (28), (30) and (31), as specified in Equation (32):

$$B\left(H\right)={\rho }_{sea}\left(H\right)\left(V_H+V_C-\Delta V_H\left(H\right)-\Delta V_C\left(H\right)\right)g$$

(32)

where g is gravitational acceleration.

4. Parameter Optimization of the CSPH

The optimal design of the CSPH constitutes a typical multi-objective constrained optimization problem, the core of which lies in simultaneously minimizing the buoyancy factor (B_f) and the buoyancy fluctuation coefficient (f_B) while satisfying constraints on strength and buckling stability. Building upon the mechanical analysis model and performance evaluation model established in Section 3, this study employs the non-dominated Sorting Genetic Algorithm II (NSGA-II) to conduct a co-optimization of the following key design parameters: the ply angle sequence [α] = [α₁/α₂/…/α_n]; the total number of plies, n; and the volume of silicone oil, V_C. Through this multi-objective optimization process, the Pareto optimal solution set for the CSPH is ultimately obtained, addressing the two mutually constraining objectives of B_f and f_B. Figure 8 comprehensively illustrates the entire optimization design workflow for the composite structure hull. The establishment of the non-dominated solution set presents multiple feasible design solutions that satisfy the constraints for the full-ocean-depth underwater glider pressure hull.

4.1. Optimization Model

(1) Design variables

The optimization variables primarily include the total layer count n of the carbon fiber cylindrical shell, the lamination sequence matrix [α] = [α⁽¹⁾/α⁽²⁾/…/α⁽ⁿ⁾], and the volume of silicone oil V_C. Studies [32,33,34,35] demonstrate that for external pressure hulls dominated by compressive strength failure, the four most effective lamination patterns for matrix [α] are cross-ply uniform, [90_p/0_q], [±α_0.5n], [90_p/±α_q], and [0_p/±α_q], where p and q denote layer counts at corresponding winding angles, and α represents the pattern-specific winding angle. Defining the ply ratio coefficient as γ = p/n, the optimization variables are simplified to n, γ, α, and V_C.

(2) Side constraints

Feasible regions must be defined for each design variable to obtain optimal solutions. The literature [10] indicates that the buoyancy factor for carbon fiber cylindrical hulls at 11,000 m design depth ranges from 0.7 to 0.75; process constraints fix single-ply thickness at t = 0.2 mm, yielding a maximum layer count n_max = 220; the winding angle α theoretically varies continuously between 0° and 90°, practical manufacturing constraints limit its values to a discrete set, specifically α ∈ {0°, 5°, 10°, 15°, …, 90°}, since silicone oil’s volumetric compressibility exceeds seawater density variation by over twofold, V_C ≤ V_H. The constraint ranges are as follows:

$$\left\{\begin{array}{l}1\le n\le 220\\ 0\le \gamma \le 1 \\ \alpha \in \{0°, 5°, 10°, 15°, \dots , 90°\}\\ 0\le V_C\le \pi r_o^{2}L\end{array}\right.$$

(33)

(3) Objective function

The optimization objectives are to minimize the CSPH’s buoyancy factor B_f to enhancing buoyancy reserves for battery accommodation in full-ocean-depth gliders. while concurrently minimizing the buoyancy fluctuation coefficient f_B to reduce buoyancy loss during deep-sea operations.

(4) Constraint conditions

In order to ensure the structural stability, the failure and strength constraints imposed on the carbon fiber pressure hull, must be satisfied. Several failure and strength criteria are used as behavior constraints.

The maximum stress criterion [36] serves as the preliminary evaluation method, where failure occurs if any stress component in the principal material direction exceeds the corresponding material strength:

$$F{I}_{Max}=\mathit{max}\{{\displaystyle \frac{\left|{\sigma }_1\right|}{X_c}},{\displaystyle \frac{\left|{\sigma }_2\right|}{Y_c}},{\displaystyle \frac{\left|{\sigma }_3\right|}{Z_c}},{\displaystyle \frac{\left|{\tau }_{12}\right|}{S_{12}}},{\displaystyle \frac{\left|{\tau }_{23}\right|}{S_{23}}},{\displaystyle \frac{\left|{\tau }_{13}\right|}{S_{13}}}\}$$

(34)

Building upon the maximum stress criterion, the Tsai–Hill criterion [37] introduces a more sophisticated approach based on distortion energy theory, accounting for stress component interactions to deliver more realistic failure predictions:

$$F{I}_{Tsai-Hill}={\displaystyle \frac{{{\sigma }_1}^2}{{X_c}^2}}-{\displaystyle \frac{{\sigma }_1{\sigma }_2}{{X_c}^2}}+{\displaystyle \frac{{{\sigma }_2}^2}{{Y_c}^2}}+{\displaystyle \frac{{{\tau }_{12}}^2}{{S_{12}}^2}}\le 1$$

(35)

Representing the most discriminative method in our framework, the Hashin criterion [38] distinguishes failure mechanisms through four discrete modes: fiber tension, fiber compression, matrix tension, and matrix compression:

$$F{I}_{Hashin}=\mathit{max}\{{\displaystyle \frac{\left|{\sigma }_1\right|}{X_c}},{\displaystyle \frac{1}{Y_c}}(({\displaystyle \frac{Y_c}{2S_{23}}}{)}^2-1)({\sigma }_2+{\sigma }_3)+{\displaystyle \frac{1}{4S_{23}^2}}({\sigma }_2+{\sigma }_3{)}^2+{\displaystyle \frac{1}{S_{23}^2}}({\tau }_{23}^2-{\sigma }_2{\sigma }_3)+{\displaystyle \frac{1}{S_{12}^2}}({\tau }_{12}^2+{\tau }_{13}^2)\}\le 1$$

(36)

Given the complex multifactorial nature of composite failure mechanics [38], this study leverages the complementary strengths of these criteria for comprehensive mechanical strength assessment of carbon fiber pressure hull. Consequently, the strength inequality constraints are formulated as

$$\mathrm{g}1: F{I}_{Max}\left(k\right)\le 1,k=1,2,3,\dots n$$

(37)

$$\mathrm{g}2: F{I}_{Tsai-Hill}\left(k\right)\le 1,k=1,2,3,\dots n$$

(38)

$$\mathrm{g}3: F{I}_{Hashin}\left(k\right)\le 1,k=1,2,3,\dots n$$

(39)

Concurrently, the buckling strength factor λ is introduced as a stability constraint, defined as the ratio of critical buckling pressure P_cr to design pressure P_d. To ensure structural integrity, λ must exceed unity (λ > 1).

$$\mathrm{G}4:\lambda ={\displaystyle \frac{P_{cr}}{P_d}}\ge 1$$

(40)

According to the above analysis result, the mathematical optimization model is established as Equation (41).

$$\left\{\begin{array}{c}\begin{array}{c}\mathit{min} B.F\left(n,\gamma ,\alpha ,Vc\right), f_B\left(n,\gamma ,\alpha ,Vc\right)\\ \mathrm{s}.\mathrm{t}. 1\le n\le 220\\ 0\le \gamma \le 1\end{array}\\ \begin{array}{c}\mathsf{\alpha }\in \left\{0°,5°,10°,15°,\dots ,90°\right\}\\ 0\le V_C\le \pi r_o^{2}L\\ FI\left(k\right)\le 1,\lambda \ge 1\end{array}\end{array}\right.$$

(41)

4.2. Optimization Algorithm and Convergence Assessment

The Non-Dominated Sorting Genetic Algorithm II (NSGA-II) was used to solve the optimization model. By incorporating Pareto optimality via a fast non-dominated sorting method [39], NSGA-II is effective for handling multi-objective optimization problems. To ensure reliable convergence, two parameter sets were applied for cross-validation, balancing accuracy and computational cost. As shown in

Table 2. Parameter configuration of NSGA-II.

Population Size	Number of Generation	Crossover Probability	Crossover Index	Mutation Index
12	200	0.9	10	20
40	200	0.9	10	20

, population sizes of 12 and 40 were used, each with 200 iterations; other parameters followed Ref. [40]. The resulting Pareto fronts for the [90_p/±α_q] laminate were generally consistent under both settings (Figure 9), with the larger population providing a denser distribution of solutions. To achieve a balance between computational efficiency and solution accuracy, a configuration with a population size of 12 was selected for subsequent optimization.

4.3. Optimization Results and Discussion

Using the NSGA-II settings in Table 2, we performed a bi-objective co-optimization of four canonical laminate families—[90_p/0_q], [±α_0.5n], [90_p/±α_q] and [0_p/±α_q]—with design variables (n, γ, α, V_C) and strength/buckling constraints enforced. The resulting non-dominated sets are shown in Figure 10. Across families, feasible solutions maintain λ > 1 and failure indices < 1, while exhibiting different trade-offs between the buoyancy factor B_f and the buoyancy–fluctuation coefficient f_B.

Within the aforementioned four Pareto-optimal solution sets, the optimal design parameters and performance metrics for the carbon fiber cylindrical shell employing the four typical stacking sequences are presented in

Table 3. Optimized design parameters and performance metrics for the four typical sequences.

Parameters	[90p/0q]	[±α0.5n]	[90p/±αq]	[0p/±αq]
n	180	180	180	181
γ	0.667	/	0.622	0.249
α	/	55	20	70
Optimal lay up	[90120/060]	[±5590]	[90112/±2034]	[045/±7068]
FImax	0.88	0.84	0.91	0.89
FITsai-Hill	0.99	0.99	0.99	0.99
FIHashin	0.88	0.84	0.91	0.89
Buckling factor (λ)	2.14	1.945	2.18	2.11
ΔVH (L)	1.071	1.083	1.071	1.067

.

Building on the analytical model in Section 3, we solved ply-level 3D stresses and failure indices for each laminate and normalized the radial coordinate as R = (r − r_i)/(r_o − r_i). Figure 11 shows strong through-thickness gradients and multi-axial coupling that are not resolved by conventional 2D/CLT formulations. In [90_p/0_q] laminates, for example, hoop-ply σ₁ peaks at the inner wall and decreases outward, while axial-ply σ₁ exhibits the opposite trend; σ₂ tends to maximize near the outer wall. The radial stress σ₃ rises from ~0 at the inner wall to ~−125 MPa at the outer wall and becomes the same order as in-plane stresses under FOD loads. This alters failure-initiation predictions: Tsai–Hill (interaction-sensitive) tends to trigger at the outer wall, whereas maximum stress/Hashin point to inner-wall fiber-compression initiation. Such differences change which laminates are preferred under identical constraints; hence, resolving σ₃ and its gradients is decision-critical for thick-walled composite shells.

Leveraging detailed resolution of the three-dimensional stress field, this study highlights notable discrepancies in predictions among different failure criteria. For instance, the Tsai–Hill criterion, being sensitive to stress interaction effects, predicts that failure initiates at the outer wall region due to the synergistic contribution of σ₁, σ₂, and σ₃ in that area—even though the value of σ₁ alone is relatively low there. In contrast, the maximum stress and Hashin criteria, based on the high stress concentration of σ₁ captured at the inner wall, determine that fiber compression failure begins from the inner layers. The difference in failure paths (“from outside inward” vs. “from inside outward”) underscores the critical influence of through-thickness stress on failure mode prediction—an effect that can only be accurately captured through 3D analysis. The consistent observation of these stress-failure trends across other stacking sequences ([±α_0.5n], [90_p/±α_q], [0_p/±α_q]) further demonstrates that 3D modeling is universally necessary for evaluating failure in thick-walled composites dominated by stress gradients. These insights indicate that future structural designs should focus on mitigating stress gradients to improve material utilization efficiency and structural safety margins.

Although Table 3 shows similar layer counts n among optima, [90/0]-rich or adjacent-angle routes mitigate strength failure but are challenging to wind at exact 0° [41]. Considering manufacturability (uniform cross-winding) and stability, we adopt [90₁₁₂/±20₃₄], which achieves the highest buckling factor among the four families (λ = 2.18) while keeping failure indices below unity.

On the Pareto set of [90₁₁₂/±20₃₄], we highlight two non-dominated “anchor” solutions (

Table 4. Optimal solutions selected from the Pareto optimal set.

Number	n	γ	α (°)	Vc (L)	Bf	fB
1	180	0.622	20	0	0.632	2.62
2	180	0.622	20	20.85	0.672	0.03

) to expose the benefit of co-designing laminate compressibility with silicone oil volume V_C:

Configuration 1 (min B_f, V_C = 0) attains the lowest B_f = 0.632 but exhibits large depth-dependent buoyancy drift (max increase 24.5 N, Figure 12a). Meeting the ~5 N propulsion-buoyancy threshold at 11,000 m requires >29.5 N surface reserve—an ~80% attenuation over 0–11,000 m—penalizing stability and endurance.

Configuration 2 (min f_B, V_C = 20.85 L) accepts a modest B_f increase to 0.672 (+6.3%) but suppresses fluctuation to f_B = 0.03. The buoyancy profile becomes non-monotonic with a transition near 7040 m; the peak deviation from the surface is only 3.4 N (Figure 12b), an 86% reduction vs. Anchor 1.

Actually, the trade-off between B_f and f_B should not be resolved by “maximizing compensation at all costs.” A larger V_C improves compensation (lower f_B) but adds mass (silicone oil density > CFRP hull), increasing B_f and reducing energy that can be carried. Conversely, minimizing B_f shrinks mass but raises compensation losses, increasing buoyancy-engine work over depth. We therefore propose a system-level selection rule: choose, from the Pareto set, the design that maximizes mission range (or the number of profiles) With the energy model of the sea wing glider [40],

$$\left\{\begin{array}{c}G=2E_{B}h/E_G\mathit{tan}(\gamma )\\ E_G=E_b+E_p+E_c\\ E_B=\left(n_H{V}_H\right(1-B_f)-m_{other})D_B\end{array}\right.$$

(42)

where E_G is the total energy consumption per dive cycle, comprising buoyancy-engine energy E_b, attitude-control energy E_p, and control-sensing energy E_c; h is the maximum dive depth; E_B is the total available energy; γ is the glider angle; and G is the range. n_H denotes the number of CSPHs; m_other is the net mass of all components other than the battery; and D_B is the battery’s energy density.

The endurance of an underwater glider also depends on the energy used by the control-and-sensing payload, E_c = P_cT_G, where P_c is the average power of the control sensor suite and T_G is the dive–climb cycle time. Prior work [40] shows that the Haiyi glider achieves its best range at a pitch of 20° and a cruise speed of 0.2 m/s. Under this operating point, Figure 13 plots the maximum range of a Sea-Wing 11000 versus P_c for two configurations. The results indicate that range is highly sensitive to P_c. When P_c is large (e.g., high-power sensors are carried), the buoyancy system accounts for a smaller fraction of total energy use; optimization should then favor designs with a smaller B_f to free buoyancy margin for more battery energy. Conversely, when P_c is small, the buoyancy system’s share increases, and designs with a smaller f_B (stronger compensation) are preferred to limit depth-induced loss of propulsion buoyancy and reduce pump/valve energy. Therefore, the final choice should be made from the Pareto set by maximizing range under the mission energy budget rather than simply minimizing f_B, as was performed in some earlier studies [10,21].

5. Finite Element Analysis and Experiment

5.1. Finite Element Analysis

To validate the accuracy of the established mechanical model and computational results for the carbon fiber cylindrical shell, a finite element model (FEM) was constructed using Solid186 layered composite elements within the ANSYS 2022 APDL platform. Stress–strain analysis and buckling analysis were subsequently performed, with the corresponding results illustrated in Figure 14.

Based on finite element analysis results, stress/strain distributions across individual plies in the carbon fiber pressure hull subjected to 125 MPa uniform external pressure were comparatively investigated. The analytical mechanical model (AMM) and numerical model demonstrated excellent agreement (Figure 15), which is specifically manifested in Figure 15.

A comparative error quantification between the FEM and AMM was further conducted.

Table 5. Comparative assessment of maximum stress and strain errors.

Parameter	Layer	AMM Solution	FEM Solution	Error
σ1 (MPa)	20	−738.262	−729.671	1.16%
σ2 (MPa)	4	−84.1909	−85.3908	1.42%
τ12 (MPa)	33	3.42	3.50	2.38%
εθ (με)	4	−6538.56	−6605.07	1%
εz (με)	54	−5350.18	−5950.98	1.24%
εr (με)	18	3092.642	3193.35	3%

details the locations of maximum errors in fiber-direction stress and strain, along with corresponding error values. The global maximum error remained below 3%, demonstrating the predictive capability and computational precision of the proposed mechanical model.

5.2. Hydrostatic Pressure Experiment

To further validate the reliability of the theoretical model and optimized design, a CSPH was fabricated based on the optimized parameters (Figure 16a) and subjected to full-ocean-depth pressure testing. Axial strain (ε_z) and hoop strain (ε_θ) were monitored in real-time using a four-point orthogonal strain rosette configuration (Figure 16c) under quasi-static pressurization (0→125 MPa, 0.05 MPa increments). Experimental data demonstrate excellent agreement with analytical mechanical model predictions (Figure 16d), with maximum deviations of 5.3% in axial strain and 5.8% in hoop strain. The sub-6% error threshold throughout the pressure range confirms the high predictive accuracy of the established 3D elastic mechanical model for stress analysis of thick-walled composite structures.

To assess the ultimate load-bearing capacity of the pressure hull, a stepwise pressurization protocol was implemented for burst pressure testing: incremental loading at 10 MPa intervals with 5 min holding phases until structural failure. Experimental results indicate that catastrophic rupture occurred at 141 Mpa (Figure 17). The actual burst pressure demonstrated a 2.69% deviation from the maximum stress criterion prediction (137.3 MPa,

Table 6. Comparative analysis of multiple failure criteria and burst testing.

Failure Criteria	Failure Pressure	Failure Modes	Error
Tsai–Hill	125	Integrated compressive failure	11.35%
Maximum Stress	137.3	Fiber compression failure	2.69%
Hashin	137.3	Fiber compression failure	2.69%

), validating its practical applicability for strength prediction of thick-walled pressure structures. In contrast, the Tsai–Hill multiaxial failure criterion exhibited substantial conservatism, predicting a critical pressure of 125 MPa This conservatism arises because Tsai–Hill aggregates σ₁, σ₂, τ₁₂ through quadratic interaction terms, penalizing concurrent biaxial/shear components more than component-wise checks, and it does not partition tension–compression modes, both of which are salient for the inner-wall, compression-dominated state of thick shells. In practice, Tsai–Hill is useful as a conservative lower bound, whereas mode-aware criteria (e.g., maximum stress/Hashin) aligned with 3D stress resolution provide test-calibrated sizing. with a relative error of 11.35%. This theoretical discrepancy reveals differential sensitivity to stress interaction effects in composites, providing critical guidance for selecting strength verification methodologies in deep-sea engineering applications.

5.3. Marine Applications and Buoyancy Compensation Performance Validation

From November 2023 to April 2024, the “Sea-Wing 11000” full-ocean-depth glider equipped with the optimized CSPH developed in this study and deployed by the Shenyang Institute of Automation, Chinese Academy of Sciences conducted a 175-day deep-sea trial in the Philippine Trench. The topographic constraints of the operational zone limited the maximum diving depth to 7426 m. As shown in Figure 18d, the hull with configuration 2 demonstrated superior buoyancy compensation performance: its buoyancy variation exhibited close agreement with theoretical predictions in Figure 12b. Specifically, propulsion buoyancy increased post-dive initiation, then decreased beyond 6500 m—showing a 7.6% deviation from the theoretical transition depth (7040 m). Conversely, the hull (configuration 1) suffered >63% velocity attenuation (0–7000 m depth range) due to nonlinear depth-dependent propulsion buoyancy loss. Its buoyancy trend demonstrated broad consistency with the model in Figure 12a. Extended sea trials confirm the optimized hull maintains stable buoyancy compensation under extreme pressure–thermal coupling conditions. This performance validates its engineering viability for full-ocean-depth gliders and submersibles, providing critical technical foundations for 10,000 m-class observation systems.

6. Conclusions

This study proposes a multifunctional Composite Structural Pressure Hull (CSPH) for full-ocean-depth (FOD) underwater gliders, integrating a carbon fiber cylindrical shell with a silicone oil compensator. A three-dimensional analytical model for thick-walled anisotropic cylinders was developed and coupled with an NSGA-II–based co-optimization of laminate parameters and oil volume. Predictions were verified by FEA, hydrostatic tests, and sea trials on the Sea-Wing 11000 platform. The main conclusions are the following:

(1)

The proposed 3D analytical model offers high predictive accuracy (error ≤ 3%) and resolves through-thickness stress gradients and radial coupling effects absent in traditional 2D models. This capability is critical for understanding multi-axial failure modes and accurately assessing the load-bearing capacity of thick composite shells under deep-sea conditions.

(2)

An integrated co-design methodology linking structural and compensation parameters was proposed, forming a closed-loop optimization framework that follows the chain: “ply parameters → mechanical response → compensation volume → buoyancy compensation.” This approach enables concurrent optimization of structural and buoyancy performance from the initial design phase, greatly improving system synergy and design efficiency.

(3)

This study further reveals a trade-off between B_f and f_B within the Pareto-optimal set. Designs with lower B_f enhance payload capacity and battery integration, whereas designs with lower f_B provide stronger density compensation and reduce buoyancy-system energy use. The final choice should be made in light of mission-specific requirements, avoiding an excessive pursuit of compensation maximization.

(4)

Several limitations remain:

Significant stress and failure index gradients within the laminate currently limit full utilization of material strength. Future work will explore variable-stiffness layups along the thickness direction to alleviate stress concentrations, enabling further weight reduction or increased safety margins. Interfacial stress concentration caused by modulus mismatch between carbon fiber and titanium end caps also requires further investigation regarding its effect on long-term structural integrity.