Dynamic Surrogate Model-Driven Multi-Objective Shape Optimization for Photovoltaic-Powered Underwater Vehicle

Abstract

In this study, a multi-objective shape optimization framework was established for photovoltaic-powered underwater vehicles (PUVs) to systematically investigate multidisciplinary coupled design methodologies. Specifically, a global sensitivity analysis was conducted to identify four critical design parameters with 24 h energy consumption and cabin volume serving as dual optimization objectives. An integrated automated optimization workflow was constructed by incorporating parametric modeling, computational fluid dynamics (CFD) simulations, and dynamic surrogate models. Additionally, a new phased hybrid adaptive lower confidence bound (PHA-LCB) infill criterion was designed under the consideration of error-driven mechanisms, improvement feedback loops, and iterative attenuation factors to develop high-precision dynamic surrogate models. Coupled with the NSGA-II multi-objective genetic algorithm, this framework generated Pareto-optimal front solutions possessing significant engineering value. Furthermore, an optimal design configuration was ultimately determined through multi-criteria decision analysis. Compared to the initial form, it generates an additional 1148.12 Wh of electrical energy within 24 h, with an 22.36% increase in sailing range and a 2.77% improvement in cabin volume capacity. The proposed closed-loop “modeling–simulation–optimization” framework realized multi-objective optimization of PUV shape parameters, providing methodological paradigms and technical foundations for the engineering design of next-generation autonomous underwater vehicles.

1. Introduction

Underwater Unmanned Vehicles (UUV), as self-contained, agile, and technologically sophisticated miniaturized marine platforms capable of substituting human operations in multifarious adverse marine environments, demonstrate pivotal instruments for oceanic exploration and exploitation [1]. As humans progressively explore deep-sea territories, the operational spectrum of UUVs has expanded exponentially, rendering prolonged endurance and extended surveillance capabilities paramount selection criteria for mission-critical applications [2].

The energy constraints inherent in UUV design, arising from the necessity to integrate sensor suites while maintaining compact power sources, constitute a primary technical bottleneck impeding their operational scalability [3]. Regardless of the proliferation of high-energy-density batteries such as Mg-AgCl cells [4], lithium-ion systems [5], and fuel cells [6], these solutions remain insufficient to sustain the round-the-clock, long-duration missions demanded by contemporary oceanographic paradigms. There are two ways to solve this problem; one is to consider how to use external energy sources to power the UUV. UUVs can be powered by establishing a series of underwater charging stations, but this scheme has many disadvantages. First, conventional underwater charging methods typically rely on wet connectors, which are susceptible to corrosion and marine organisms clinging to them during long-term static immersion in seawater, affecting the reliability and safety of charging [7]. Secondly, the underwater charging process places extreme demands on the positioning accuracy of the UUV, especially when docked to a charging station, where any small deviation can cause a charging failure, which increases the difficulty and complexity of charging. Additionally, the complexity of the underwater environment, such as ocean currents, waves, and lighting conditions, can also interfere with the charging process, reducing charging efficiency and success rate [8]. Furthermore, underwater charging equipment is expensive to maintain and overhaul, and the repair process is complex and time-consuming in the event of a failure [9]. The above drawbacks limit the large-scale use of underwater charging stations, while the technology of power generation from environmental energy sources provides a new power supply scheme for UUVs [10]. Zhang [11] and Chen [12], et al., for example, proposed using wave energy for power generation devices but, constrained by the UUV’s internal space, its power generation would be small and would need to involve a three-stage energy conversion device. Its system complex conversion efficiency is low compared to other wave energy-using devices, whereas solar-powered devices, such as SAUV-I and SAUV-II developed by the United States Naval Research Agency [13], with their simple structure and high power generation, are a potential technology [14]. In addition, with advances in photovoltaic panel technology, scholars are currently demonstrating that using solar power to power UUVs near water is a more feasible method, and Rohr et al. have shown that solar panels made using broadband gap semiconductors can achieve efficient power generation within 50 m underwater [15].

From another perspective, the employment of multidisciplinary optimization techniques diminishes energy expenditure during UUV navigation. Among various design considerations, hydrodynamic performance constitutes a focal point for naval architects because an optimized hydrodynamic configuration can effectively mitigate navigational resistance and play a crucial role in enhancing the operational radius of UUVs [16]. With advancements in high-performance computing, Computational Fluid Dynamics (CFD) has been extensively applied to analyze the hydrodynamic characteristics of underwater vehicles. In contrast to conventional tank testing methodologies, CFD methodologies avoid the need for physical prototypes and allow parametric modifications through virtual three-dimensional models to facilitate iterative experiments at substantially reduced costs [17]. By employing ANSYS FLUENT software to generate AUV hull contours based on Myring’s equation, Sener and Aksu [18] investigated the influence of bow configurations on resistance performance and flow characteristics. Their findings reveal significant impacts on both pressure drag and frictional resistance. Qin [19] conducted parametric modeling of a blended-wing-body underwater glider, optimized its lift-to-drag ratio through STAR-CCM+ software, and achieved a 14% improvement in optimal aerodynamic efficiency.

While CFD-based AUV shape optimization offers high precision and cost-effectiveness, its computational demands heavily rely on high-performance computing resources. Concerning complex three-dimensional flow simulations, CFD requires substantial computational resources and time, hindering rapid model iterations [20]. To reinforce optimization efficiency, researchers have integrated CFD with surrogate models for AUV shape optimization, enabling rapid and high-precision iterations. This approach involves the following: (1) sampling representative points within the variable domain by methods such as random, uniform, or optimal Latin hypercube sampling; (2) constructing corresponding shapes and calculating their response values by CFD software to establish a sample library; (3) training surrogate models based on the sample library; and (4) employing optimization algorithms to find optimal AUV shapes that satisfy constraints.

Liu [21] developed surrogate models to optimize AUV resistance, volume, and angle of attack through optimal Latin hypercube sampling and neural networks. The results reveal that shorter midships with longer fore and aft sections reduced resistance at speeds above 1 m/s. Zhang [22] developed an automated optimization platform in Python for BWBUG shape optimization by integrating geometric deformation, CFD solving, surrogate modeling, and optimization strategies, achieving a 20.7% increase in lift-to-drag ratio. Tang [23] proposed a surrogate modeling method through gene expression programming and improved drag prediction accuracy by 13.66% compared to response surface models with the same number of sample points.

The aforementioned studies that utilized single-sampling methods to generate surrogate model sample sets may require large sample sizes for complex optimization problems, resulting in low optimization efficiency. In contrast, dynamic sampling strategies can update sample libraries adaptively, achieve high-precision surrogate models with fewer sample points, and improve optimization efficiency. Luo [24] put forward a dynamic surrogate model based on trust regions and lower confidence bounds to optimize AUV resistance and energy consumption, so as to ensure both local and global optimization capabilities by adaptively adjusting the balance parameter of the confidence bound. To maximize the volume of appendage-equipped AUVs while minimizing resistance, Chen and Feng [25] introduced an adaptive sequential sampling method to update the Kriging model database, lowering sample sizes while maintaining precision.

In summary, dynamic surrogate models, with advantages such as high efficiency, high precision, and low sample requirements for engineering optimization problems, are particularly valuable for computationally expensive optimization tasks. Current research on dynamic surrogate models in the field of UUV optimization remains limited and needs further investigation.

In this study, a foldable solar panel integrated into a UUV was designed by combining the two approaches to enhance UUV endurance. It deploys during near-surface charging and retracts during underwater navigation, as shown in Figure 1. The shape of the photovoltaic-powered UUV was optimized following the workflow illustrated in Figure 2. With 24 h energy consumption and cabin volume as optimization objectives, navigation resistance and theoretical solar panel power generation for various UUV shapes at identical speeds were first calculated by STAR-CCM+ 2206 for the establishment of a UUV energy consumption model. The cabin volume was derived from SOLIDWORKS 2021 outputs. These modules were integrated into the ISIGHT 2022 optimization platform, where the CFD calculation module was replaced by a dynamic Kriging model. Additionally, the initial sample points for constructing the surrogate model were generated through an optimized Latin hypercube design and iteratively updated via the proposed Phased Hybrid Adaptive Lower Confidence Bound (PHA-LCB) infill criterion. The main contributions of this research are drawn as follows. (1) A new UUV configuration was proposed, providing a feasible solution for enhancing UUV endurance. (2) The PHA-LCB infill strategy was introduced to significantly improve PUV shape optimization efficiency, guiding the management of similar engineering optimization issues.

2. PUV Numerical Simulation Methodology

2.1. PUV Parametric Modeling

Common AUV hull forms are axisymmetric revolution bodies, with the bow and stern shapes typically governed by specific formulas and the length of the parallel midbody determined by mission-specific requirements. Typical bow and stern configurations include the Myring type [26], Nystrom type, Cox type, and water-drop type [27]. With these hull forms, parametric modification of bow and stern geometries can be achieved by adjusting the generating line equations. Particularly, the Myring-type revolution body has been widely adopted for underwater vehicle geometries because of its advantageous hydrodynamic performance and ease of manufacturing [28], as revealed in platforms such as Remus and MAYA [29]. Consequently, the PUV hull form adopts the Myring-type revolution body, with bilateral 60 mm inward cutouts on the port and starboard sides to accommodate four foldable solar panels. The three-dimensional model and coordinate system employed in this study are illustrated in Figure 3. The governing equations for the Myring-type revolution body, involving the bow section, parallel midbody, and stern section, are expressed as follows [26]:

$$r_1(x)={\displaystyle \frac{1}{2}}d{[1-{({\displaystyle \frac{x-a}{a}})}^2]}^{{\displaystyle \frac{1}{n}}}$$

(1)

$$r_2(x)={\displaystyle \frac{d}{2}}$$

(2)

$$r_3(x)={\displaystyle \frac{1}{2}}d-({\displaystyle \frac{3d}{2c^2}}-{\displaystyle \frac{\tan \theta }{c}}){(x-a-b)}^2+({\displaystyle \frac{d}{c^3}}-{\displaystyle \frac{\tan \theta }{c^2}}){(x-a-b)}^3$$

(3)

The data of PUV model are as follows: a = 500 mm, b = 2000 mm, c = 1000 mm, d = 500 mm, and remain unchanged. n and θ designate shape-controlling parameters for the bow and stern, respectively, with larger values producing more rounded profiles, as depicted in Figure 4.

The primary focus of this work is the optimization of PUV exterior parameters. The definitions and ranges of the design parameters are detailed in

Table 1. PUV configuration design parameter scope.

Parameter Symbol	Parameter Meaning	Initial Value	Parameter Range
n	Bow form index	3	[1, 5]
θ	Stern form index	20	[10, 30]
L	Length of photovoltaic panel	1250	[1000, 1500]
H	Half-height of photovoltaic panel	275	[150, 400]

. The parameter ranges were determined under the consideration of the total area required for solar panels and the necessity to satisfy design constraints for the PUV internal space. Schematic diagrams of each parameter are presented in Figure 5.

2.2. Governing Equations

The hydrodynamic performance of the PUV was obtained by solving the Navier–Stokes equations with STAR-CCM+ 2206. For a three-dimensional, unsteady, continuous, and incompressible fluid, the continuity equation and the momentum equations are as follows:

$${\displaystyle \frac{\partial (\rho \overline{u_i})}{\partial x_i}}=0$$

(4)

$${\displaystyle \frac{\partial (\rho \overline{u_i})}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho \overline{u_i{u}_j}+\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)=-{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+{\displaystyle \frac{\partial {\overline{\tau }}_{ij}}{\partial x_j}}$$

(5)

$\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ describes the Reynolds stress, expressed as

$$-\rho \overline{{u^{\prime }}_i{u^{\prime }}_j}={\mu }_t\left({\displaystyle \frac{\partial \overline{u_i}}{\partial x_i}}+{\displaystyle \frac{\partial \overline{u_j}}{\partial x_i}}\right)-{\displaystyle \frac{2}{3}}\rho {\delta }_{ij}k$$

(6)

A turbulence model should be carefully selected given the differing characteristics exhibited by the Reynolds-Averaged Navier–Stokes (RANS) equations under various flow conditions. Under the consideration of theoretical completeness and engineering applicability, a hybrid wall function was combined with the standard two-layer turbulence model in this study [30]. This model employed a dual-layer solving strategy, enabling the explicit solution of the boundary layer in the near-wall region on the foundation of the standard two-equation model. Notably, such dual-layer models possess the following special attributes. (1) The spatial resolution of the first grid layer near the wall is no longer solely dependent on the post-processed y⁺ parameter distribution. (2) The time-averaged distribution of the wall force coefficient is insensitive to minor variations in the boundary layer grid [31]. Consequently, a separate boundary layer meshing scheme is not required for each shape during the optimization process, contributing to the significantly reinforced optimization efficiency.

The turbulent kinetic energy $k$ and turbulent dissipation rate $\epsilon$ in the standard $k\text{-}\epsilon$ turbulence model are obtained by

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

(7)

$${\displaystyle \frac{\partial }{\partial t}}(\rho \epsilon )+{\displaystyle \frac{\partial }{\partial x_i}}(\rho \epsilon u_i)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]+C_{\epsilon 1}{\displaystyle \frac{\epsilon }{k}}{G}_k-C_{\epsilon 2}\rho {\displaystyle \frac{{\epsilon }^2}{k}}+S_{\epsilon }$$

(8)

The turbulent eddy viscosity in the external region is calculated as follows:

$${\left.{\mu }_t\right|}_{k-\epsilon }=\rho C_{\mu }{\displaystyle \frac{k^2}{\epsilon }}$$

(9)

In the near-wall region influenced by internal viscosity, both turbulent eddy viscosity and dissipation rate are specified in terms of length scales. Specifically, the eddy viscosity relationship is rewritten as

$${\left.{\mu }_t\right|}_{2L}=\rho C_{\mu }{k}^{1/2}{\mathcal{l}}_{\mu }$$

(10)

and the formula for calculating $\epsilon$ is

$$\epsilon ={\displaystyle \frac{k^{3/2}}{{\mathcal{l}}_{\epsilon }}}$$

(11)

where both ${\mathcal{l}}_{\mu }$ and ${\mathcal{l}}_{\epsilon }$ are functions of length scales in exponential form.

Based on Jongen’s recommendation, the following wall-proximity indicator is employed to combine the two-layer formulation with the standard $k\text{-}\epsilon$ model:

$$\lambda ={\displaystyle \frac{1}{2}}\left[1+\tanh \left({\displaystyle \frac{{\mathrm{Re}}_d-{\mathrm{Re}}_y^{*}}{A}}\right)\right]$$

(12)

${\mathrm{Re}}_d=\raisebox{1ex}{$\sqrt{k}d$}\!\left/ \!\raisebox{-1ex}{$\nu $}\right.$ is the Reynolds number based on the distance to the wall.

Parameter A determines the width of the wall-proximity indicator, and its definition ensures that the value of $\lambda$ remains within 1% of its far-field value for a given variation in ${\Delta \mathrm{Re}}_y$:

$$A={\displaystyle \frac{\left|{\Delta \mathrm{Re}}_y\right|}{\mathrm{atanh} 0.98}}$$

(13)

Finally, the turbulent viscosity ${\left.{\mu }_t\right|}_{k-\epsilon }$ from the $k\text{-}\epsilon$ model is blended with the two-layer value as follows:

$${\mu }_t=\lambda {\left.{\mu }_t\right|}_{k-\epsilon }+(1-\lambda )\mu {\left({\displaystyle \frac{{\mu }_t}{\mu }}\right)}_{2\mathrm{layer}}$$

(14)

According to the STAR-CCM+ User Manual, the values of the constants for the standard $k\text{-}\epsilon$ two-layer model are as follows:

$C_{\epsilon 1}=1.44$; $C_{\epsilon 2}=1.92$; $C_{\mu }=0.09$; ${\sigma }_k=1.0$; ${\sigma }_{\epsilon }=1.3$; ${\mathrm{Re}}_y^{*}=60$; ${\Delta \mathrm{Re}}_y=10$

A numerical computation method based on a segregated solver was employed to ensure strict satisfaction of the fluid mass conservation and momentum conservation equations. During the computation process, the coupling relationship between the pressure field and the velocity field was achieved through the SIMPLE algorithm, providing stable numerical convergence characteristics for the solution process. The numerical simulation adopted standard seawater at 20 °C as the working medium. In the STAR-CCM+ software, the density and dynamic viscosity coefficient of the medium were precisely configured following the physical properties of seawater. Notably, the settings for other crucial physical parameters involved in the solver are detailed in

Table 2. Solver parameter settings.

Solver Options	Parameter Settings	Solver Options	Parameter Settings
Transient Term	Implicit Time Integration	Segregated Flow Solver	Pressure-Velocity Coupling Algorithm: SIMPLE
Advective Flux	Second-Order Upwind	Reference Velocity U	1.5 m/s
Diffusive Term	Second-Order Gradient	Fluid Density $\rho$	998.21 kg/m3
Implicit Unsteady Solver	Second-Order Time Discretization	Dynamic Viscosity $\mu$	0.00101 Pa·s

.

2.3. Computational Domain and Meshing

STAR-CCM+ for high-fidelity hydrodynamic simulation of the PUV necessitates a meticulous definition of the computational domain. The spatial extent of this domain exhibits direct proportionality to the complexity of flow physics requiring resolution. An insufficiently dimensioned domain may preclude accurate characterization of the vehicle’s intricate hydrodynamic interactions, so as to compromise solution fidelity through the introduction of artificial boundary effects.

In conformance with International Towing Tank Conference (ITTC) recommendations [32], the computational domain was established with specific dimensional and boundary configurations, as illustrated in Figure 6. The forebody and afterbody maintained a two-length-of-vehicle (L_oa) clearance from the inlet boundary and a 4L_oa distance from the outlet, respectively. Vertical and lateral clearances from the vehicle’s centroid measured 2 L_oa in all directions. Boundary conditions were prescribed as uniform velocity inlet at the upstream and lateral surfaces, hydrostatic pressure outlet at the downstream boundary, symmetry planes for the longitudinal extremities, and no-slip walls for the UUV surfaces. Furthermore, both the geometric model and fluid domain were constructed symmetrically about the vehicle’s longitudinal axis to optimize computational efficiency.

In addition to the computational domain, the flow characteristics within the boundary layer significantly influence the hydrodynamic performance of underwater vehicles. The flow details within the boundary layer could be more accurately captured by meshing the boundary layer. A refined boundary layer mesh helps reduce numerical errors and improves the accuracy and reliability of the simulation. In this study, the Reynolds number was 5.89 × 106, which fell within the high Reynolds number range. Under these conditions, the boundary layer structure was divided into two main parts: the outer region and the inner region. The flow characteristics in the outer region were primarily dominated by turbulent effects, while the inner region was further subdivided into three sub-regions (the viscous sublayer, the logarithmic layer, and the buffer layer) based on flow characteristics. Apart from using a two-layer model to accurately solve for the distribution characteristics of the turbulent dissipation rate, a hybrid analytical method was also introduced to couple the velocity and temperature fields within the boundary layer. Particularly, the adopted hybrid wall function was accurately represented regarding the physical field in the buffer layer through the coupling of the viscous sublayer and the logarithmic layer. Optimal grid parameterization ensured faithful reproduction of turbulence evolution adjacent to complex hull geometries and appendage structures. The near-wall first-layer height y+ was determined through a systematic solution of the following governing equations [18]:

$$R{e}_L={\displaystyle \frac{\rho U_{\infty }L\mathrm{oa}}{\mu }}$$

(15)

$$C_f={[2{\log }_{10}(R{e}_L)-0.65]}^{-2.3}$$

(16)

$${\tau }_{\omega }=C_f{\displaystyle \frac{1}{2}}\rho U_{\infty }^2$$

(17)

$$u_{*}=\sqrt{{\displaystyle \frac{{\tau }_{\omega }}{\rho }}}$$

(18)

$$y={\displaystyle \frac{y^{+}\mu }{\rho u_{*}}}$$

(19)

2.4. Grid Convergence Analysis

Grid convergence testing is a crucial step in ensuring the reliability of CFD results. The grid-scale and discretization methods have a direct impact on the accuracy of the numerical solution. A non-converged grid may introduce significant discretization errors. Grid convergence analysis can effectively eliminate uncertainties stemming from grid dependence, laying a reliable numerical foundation for the performance evaluation of the PUV. In this study, convergence analysis was performed on five sets of grids. The drag force calculation results are presented in

Table 3. Grid data and simulation results.

Grid Numbering	Total Number of Grids	ri	Drag Coefficient (e−3)
Grid I	488,902	1.468	4.894
Grid II	858,082	1.217	4.896
Grid III	1,546,680	1.000	4.941
Grid IV	3,361,620	0.772	4.951
Grid V	5,603,474	0.651	4.968

.

According to the ITTC recommendations, the refinement ratio for the base size in each grid set is $\sqrt[4]{2}$ [32]. Thus, grids I, III, and V were selected to calculate the convergence ratio $R_G$ of the simulation results

$$R_G={\displaystyle \frac{{\epsilon }_{35}}{{\epsilon }_{13}}}$$

(20)

where ${\epsilon }_{35}$ represents the error between the results of grid III and grid V, and ${\epsilon }_{13}$ denotes the error between the results of grid I and grid III. Depending on the value of $R_G$, there are four possible scenarios:

• ${0<\mathrm{R}}_{\mathrm{G}}<1$, Monotonic convergence;
• $-{1<\mathrm{R}}_{\mathrm{G}}<0$, Oscillatory convergence;
• ${\mathrm{R}}_{\mathrm{G}}>1$, Monotonic divergence;
• ${\mathrm{R}}_{\mathrm{G}}<-1$, Oscillatory divergence.

After calculation, we get $R_G=0.587$, which falls under the category of monotonic convergence. In this case, two methods can be employed for grid uncertainty quantification according to the 30th ITTC Report [33].

The first method, which is based on the Richardson Extrapolation (RE) method, was employed to estimate the numerical error and uncertainty through the solutions from three sets of grids [34]. However, this method is generally only applicable to structured grids. A consistent refinement ratio between different grids should be maintained when unstructured grids are applied. $\Delta {\mathrm{r}}_{i,i+1}$ can be calculated by

$$h_i=\sqrt[3]{{\displaystyle \frac{\mathrm{V}}{N_i}}}$$

(21)

$$\Delta {\mathrm{r}}_{i,i+1}={\displaystyle \frac{h_i}{h_{i+1}}}$$

(22)

In this study, spatial stretching techniques were utilized in conjunction with unstructured grids. Under such computational configurations, the RE method fails to maintain mathematical consistency in error propagation, compromising the credibility of the numerical predictions. Hence, the least-squares-based grid uncertainty quantification methodology proposed by Eça and Hoekstra [35] was adopted to handle the above limitation. This approach necessitates a minimum of four distinct grid configurations through which the exact solution estimate ${\varphi }_0$, discretization error coefficient $\alpha$, and observable accuracy order p are determined by least-squares regression of the governing equation:

$${\varphi }_i={\varphi }_0+\alpha h_i^p$$

(23)

Following different p-values, the error estimator can be selected in the following four forms:

$${\delta }_{RE}=\alpha h_i^p$$

(24)

$${\delta }_1=\alpha h_i$$

(25)

$${\delta }_2=\alpha {h_i}^2$$

(26)

$${\delta }_{12}={\alpha }_1{h}_i+{\alpha }_2{h_i}^2$$

(27)

Ultimately, the calculation formula for uncertainty estimation is determined by the relationship between the standard deviation of the least-squares fitting curve and the data range parameter.

$$\Delta \varphi ={\displaystyle \frac{{\left({\varphi }_i\right)}_{\max }-{\left({\varphi }_i\right)}_{\min }}{n-1}}$$

(28)

When $\sigma <\Delta \varphi$,

$$U({\varphi }_i)=F_S{ϵ}_{\varphi }+\sigma +|{\varphi }_i-{\varphi }_{\mathrm{fit}}|$$

(29)

When $\sigma \ge \Delta \varphi$,

$$U({\varphi }_i)=3{\displaystyle \raisebox{1ex}{$\sigma $}\!\left/ \!\raisebox{-1ex}{$\Delta \varphi $}\right.}\left({ϵ}_{\varphi }+\sigma +|{\varphi }_i-{\varphi }_{\mathrm{fit}}|\right)$$

(30)

The error estimation values ${ϵ}_{\varphi }$ in Equations (29) and (30) are determined by the p-value. When 0.5 < p < 2, ${ϵ}_{\varphi }\cong {\delta }_{RE}$; when p > 2, the one with the smaller $\sigma$ between ${\delta }_1$ and ${\delta }_2$ is used as ${ϵ}_{\varphi }$; and when p < 0.5, the one with the smallest $\sigma$ is chosen among ${\delta }_1$, ${\delta }_2$, and ${\delta }_{12}$. The safety factor $F_s$ in Equations (23)–(30) is selected in accordance with the p-value and the data range parameter $\Delta \varphi$. When 0.5 < p < 2.1 and $\sigma <\Delta \varphi$, $F_s$ = 1.25; otherwise, $F_s$ = 3.

Table 3 lists the grid data and simulation results. Among them, $r_i$ represents the relative grid spacing with respect to Grid III, defined as:

$$r_i=\sqrt[3]{N_3/N_i}$$

(31)

For $\varphi =R$, $\beta =\alpha h_3^p$, and $r_i=h_i/h_3$, the results obtained by Equations (23)–(30) are presented in

Table 4. Grid uncertainty evaluation results.

Parameter Name	Parameter Value	Parameter Name	Parameter Value
${\varphi }_0$	4.99	${ϵ}_{\varphi }({\varphi }_3)$	−0.049
$\beta$	−0.049	$F_s$	1.25
p	1.79	${\varphi }_{\mathrm{fit}}$	4.94
$\sigma$	0.015	$U_{\varphi }({\varphi }_3)$	0.082
$\Delta \varphi$	0.019	$U_{\varphi }(\%{\varphi }_3)$	1.66

. In this case, $U_{\varphi }({\varphi }_3)$ is 1.68% of ${\varphi }_3$, below the recommended 2% of ${\varphi }_3$ [33].

As revealed from the analysis, the mesh generation scheme satisfies the mesh quality requirements and possesses sufficient quality redundancy for shape optimization, as demonstrated in

Table 5. Grid quality check index.

Metric Designation	Operational Envelope	Advisory Limits
Grid Fidelity	[0.115,1]	>1.0 × 10−8
Volumetric Deviation	[0.0976,1]	>1.0 × 10−5
Angular Distortion	[0,60.6]	<85°

.

2.5. Validation of Numerical Model Effectiveness

Since the experimental prototype has not yet been fabricated, the analytical methods presented in this paper, involving mesh generation and boundary condition settings, were applied to the Fully Appended SUBOFF model for the purpose of validating the effectiveness of the results. Since the presence of solar panels is equivalent to an appendage of the vessel, the Fully Appended SUBOFF model is selected. The numerical results were compared with the experimental data provided by the Defense Advanced Research Projects Agency of the United States [36], as depicted in Figure 7. It can be observed that the numerical simulation results exhibited favorable agreement with the experimental data. Therefore, the aforementioned numerical computation scheme can be applied in the subsequent optimization process.

3. Establishment and Validation of the Dynamic Surrogate Model

3.1. Experimental Design

Generally, the more the sample points employed to construct a surrogate model, the higher the accuracy of the model. However, a larger number of sample points imply higher experimental and computational costs. Moreover, an excessive number of sample points can easily lead to overfitting concerning certain engineering problems, resulting in the curtailed accuracy of the model [37]. Therefore, a scientific and effective Design of Experiments (DOE) method should be adopted during the process of constructing a surrogate model to obtain an appropriate number of initial sample points. Various DOE methods are available for obtaining initial sample points, such as full factorial sampling, orthogonal sampling, stratified sampling, random sampling, Latin hypercube sampling, and optimized Latin hypercube sampling. Among them, optimized Latin hypercube sampling can better characterize spatial information with a smaller number of sample points [38]. Hence, it was adopted in this study to collect initial sample points. Referring to the research findings of Schmit et al. [39]., the number of initial sample points in this study was set as (a + 1)(a + 2)/2 + 2a, where a represents the number of influencing factors

A total of twenty-three distinct sample configurations were systematically generated across the predefined parameter domains by the enhanced Latin hypercube sampling technique. The foundational dataset for subsequent analyses was established with rigorously computed corresponding response metrics. Additionally, a sensitivity analysis was conducted based on the Pearson correlation coefficient to characterize the parameter sensitivity relative to the objective function. The results are presented in Figure 8. Among the dimensional parameters, the length (L) and half-height (H) of the solar array, coupled with the bow-shape coefficient (n), demonstrated pronounced dominance in determining the 24 h energy consumption metric (Ne). Conversely, the remaining parameters exhibited comparatively negligible correlations with this performance indicator. Moreover, the cabin volumetric capacity (V) manifested strong dependency on both the bow-shape coefficient (n) and the stern-shape parameter (θ). Hence, the critical design variables H, L, n, and θ were judiciously selected for progressive optimization procedures.

3.2. Surrogate Model

The surrogate model, also known as an approximate model or metamodel, is a data-driven technique that substitutes complex solution processes. A surrogate model can be utilized to significantly enhance optimization efficiency and lower optimization difficulty [40]. Surrogate models are classified into static and dynamic types based on whether the sample point set changes.

Common approximation techniques include the Kriging model, polynomial response surfaces based on experimental design theory, artificial neural networks, and radial basis function methods. The choice of a surrogate model primarily depends on the nonlinearity and scale of the actual engineering problem [41]. The polynomial response surface method is simple in construction and requires fewer sample points, whereas it is only suitable for engineering issues with low nonlinearity. Although radial basis function methods, neural networks, and Kriging methods can all address problems with higher nonlinearity, neural networks require a large number of sample points and involve complex computations. Compared to other models, the Kriging method provides not only predictions of unknown functions but also prediction variances, guiding the selection of global update points for adaptive surrogate models [42]. Therefore, the Kriging method was used in this study to establish the surrogate model.

3.3. Kriging Model

The Kriging model is a regression analysis method based on stochastic processes, capable of achieving linear unbiased and minimum variance estimation for sampled data. In this study, the Kriging model was constructed with the Design and Analysis of Computer Experiments (DACE) toolbox [43].

The Kriging model consists of a regression part and a non-parametric part, and its formulation is expressed as [44]

$$Y(x)=f(x)+Z(x)$$

(32)

where $f(x)$ denotes a global trend function, representing the mathematical expectation of $Y(x)$, and $Z(x)$ represents a stationary stochastic process with a mean of zero and a variance of ${\sigma }^2$.

$$E[Z(x)]=0$$

(33)

$$Var[Z(x)]={\sigma }^2$$

(34)

$$Cov[Z(x^{(i)}),Z(x^{(j)})]={\sigma }^2[R_{ij}(\theta ,x^{(i)},x^{(j)})]$$

(35)

where $x^{(i)}$ and $x^{(j)}$ indicate any two points in the sample, and $R(\theta ,x^{(i)},x^{(j)})$ stands for a correlation function with hyperparameter $\theta$, expressed as

$$R(\theta ,x^{(i)},x^{(j)})={\displaystyle \prod _l^m{R}_l({\theta }_l,d_l)}$$

(36)

where m represents the dimensionality of x; $d_l={x_l}^{(i)}-{x_l}^{(j)}$.

A multivariate polynomial function is utilized as the regression function, expressed as

$$f(x)={\displaystyle \sum _{i=1}^p{\alpha }_i{b}_i(x)}$$

(37)

where $b_i(x)$ represents the polynomial basis function, and $\alpha =({\alpha }_1,\dots ,{\alpha }_p)$ denotes the corresponding coefficients.

Concerning a given input sample set $S=\{x^{(1)},x^{(2)},\dots ,x^{(n)}\}$ and its corresponding output responses $Y={\{y^{(1)},y^{(2)},\dots ,y^{(n)}\}}^T$, the predicted value at any point $x\in {ℝ}^m$ can be obtained by

$$\widehat{y}(x)=c^{T}Y$$

(38)

Since the Kriging model is an unbiased interpolation, the mean of the error is zero:

$$E[(\widehat{y}(x)-y(x))]=E[(c^{T}1-1)]=0$$

(39)

The prediction variance is

$$\begin{array}{cc}\hfill {\widehat{s}}^2(x)& =E[{(\widehat{y}\left(x\right)-y(x))}^2]\hfill \\ & =E[{(c^{T}Z-Z(x))}^2]\hfill \\ & =E[Z^2(x)+c^{T}Z{Z}^{T}c-2c^{T}ZZ(x)]\hfill \end{array}$$

(40)

3.4. Dynamic Sampling Criteria

The efficacy of optimization outcomes is fundamentally contingent upon the predictive accuracy of the surrogate model when surrogate-based optimization methodologies are employed. In conventional static Kriging frameworks, the sample repository remains invariant throughout the optimization cycle, with its predictive capability inherently reliant on the initial sampling configuration. This static approach progressively compromises representativeness as optimization proceeds, bringing about diminished fidelity in subsequent iterations. With the purpose of mitigating this deficiency, a dynamic surrogate modeling paradigm can be established through adaptive infill sampling strategies that iteratively augment the sample database [45].

The optimization problem under study involves dual objective functions: 24 h energy consumption and cabin volumetric capacity. Notably, the cabin volume objective exhibited measurable linearity with low-order nonlinearity, enabling its surrogate model to be less sensitive to sampling density. Quantitative validation confirms that the cabin volume surrogate achieved a coefficient of determination (R²) value of 0.983. This result reflects exceptional predictive accuracy even with the initial sample set. Furthermore, the deterministic nature of cabin volume eliminated measurement noise, ensuring precise correlation with newly acquired samples from energy consumption evaluations. Consequently, a dynamic surrogate model exclusively for the energy consumption objective was developed in the present study following the PHA-LCB infill criterion.

Regarding the Kriging model, various infill criteria have been proposed, such as Minimum Response Surface (MRS), Maximum Probability of Improvement (MPI), Maximum Expected Improvement (MEI), and Lower Confidence Bound (LCB). JONES D R reported that both MEI and LCB are commonly used global infill criteria [46]. However, MEI is a dense infill criterion and can only find the global optimal solution when there are sufficient sample points. Hence, the improved LCB infill criterion was adopted in this study

The objective function of the LCB infill criterion is

$$f_{LCB}(x)=\widehat{y}(x)-\kappa \widehat{s}(x)$$

(41)

where $\widehat{y}(x)$ represents the predicted value of the Kriging model, $\widehat{s}(x)$ denotes the standard deviation of the prediction point, and $\kappa$ refers to the balance constant.

In the conventional LCB infill criterion, the value of $\kappa$ is set as a fixed value by the designer based on experience, so as to introduce significant uncertainty. Therefore, a Phased Hybrid Adaptive strategy (PHA) was proposed in this study to adaptively adjust parameters by integrating the relative root mean square error (RRMSE) of the model, objective function improvement trends, and iterative process information. Subsequently, the GA-SPSO algorithm [47] was employed to optimize PHA-LCB, generate new sample points, compute their response values, and incorporate them into the sample database. This strategy divides the optimization process into three phases: exploration, balance, and exploitation. The switching conditions between phases and the calculation method for the $\kappa$ value are described as follows.

(1)

Phase Switching Conditions

Introduction of relative root mean square error Threshold:

$$RRMS{E}_{\mathrm{th}}(t)=\min \left(RRMS{E}_{initial},RRMS{E}_{initial}\cdot e^{-\eta t}\right)$$

(42)

The phase division rules are as follows:

Exploration Phase: $RRMS{E}_{\mathrm{curent}}>RRMS{E}_{\mathrm{th}}(t)$, during which global exploration is emphasized.

Balance Phase: $0.25RRMS{E}_{\mathrm{th}}(t)\le RRMS{E}_{\mathrm{curent}}\le RRMS{E}_{\mathrm{th}}(t)$, balancing exploration and exploitation.

Exploitation Phase: $RRMS{E}_{\mathrm{curent}}<0.25RRMS{E}_{\mathrm{th}}(t)$, focusing on local exploitation.

Where $RRMS{E}_{initial}$ represents the RRMSE of initial model for the test set, $RRMS{E}_{\mathrm{curent}}$ denotes the RRMSE of current model for the test set, $\eta$ denotes the attenuation coefficient, and the RRMSE is calculated using Equation (43).

$$RRMSE=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^M{\left(\frac{f^{(i)}-{\widehat{f}}^{(i)}}{f^{(i)}}\right)}^2}}{M}}}$$

(43)

In the equation, M represents the number of samples in the test set, $f^{(i)}$ denotes the actual value, and ${\widehat{f}}^{(i)}$ denotes the predicted value.

(2)

Phase Weight Allocation

In each phase, the $\kappa$ value is determined by a linear combination of the variance-driven term, the improvement feedback term, and the iteration attenuation term:

$$\kappa (t)=\alpha \cdot {\kappa }_{\sigma }(t)+\beta \cdot {\kappa }_f(t)+\gamma \cdot {\kappa }_t(t)$$

(44)

Error-driven term ${\kappa }_{\sigma }(t)$ reflect the current model uncertainty, defined as

$${\kappa }_{\sigma }(t)={\kappa }_0+\gamma \cdot {\displaystyle \frac{RRMS{E}_{\mathrm{curent}}}{RRMS{E}_{\mathrm{initial}}}}$$

(45)

Improvement feedback term ${\kappa }_f(t)$ is used to perform adjustments based on the magnitude of improvement in the objective function, expressed as

$${\kappa }_f(t)={\kappa }_{prev}\cdot \tanh \left(\Delta f\right)$$

(46)

where $\Delta f$ denotes the change in the objective function, and ${\kappa }_{prev}$ embodies the $\kappa$ value from the previous iteration.

Iteration attenuation term ${\kappa }_t(t)$ attenuates with the number of iterations, mathematically expressed as

$${\kappa }_t(t)={\kappa }_{\mathrm{initial}}\cdot \ln (T_{total}-t)/\ln (T_{total})$$

(47)

where $T_{total}$ represents the total number of iterations, and t denotes the current iteration count.

The weight coefficients for each phase are detailed in

Table 6. Trade-off coefficients for different phases.

Coefficient	$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\gamma }$
Phase
Exploration Phase	0.7	0.2	0.1
Equilibrium Phase	0.4	0.4	0.2
Development Phase	0.1	0.6	0.3

:

3.5. Validation of the Dynamic Surrogate Model

The Six-Hump Camel Function ($f_1(x)$), Shekel Function ($f_2(x)$), and Hartmann 6-Dimensional Function ($f_3(x)$), expressed as follows, were selected for testing to verify the superiority of the PHA-LCB infill criterion. Their detailed information is presented in

Table 7. Test functions information.

Function Index	Value Range	Dimension	Global Optimum
$f_1(x)$	[−3, 3]2	2	−1.0316
$f_2(x)$	[0, 10]4	4	−10.1532
$f_3(x)$	[0, 1]6	6	−3.32237

.

$$f_1(x)=\left(4-2.1x_1^2+{\displaystyle \frac{x_1^4}{3}}\right)x_1^2+x_1{x}_2+(-4+4x_2^2)x_2^2$$

(48)

$$\left\{\begin{array}{c}{f}_2(x)=-{\displaystyle \sum _{i=1}^m{\left({\displaystyle \sum _{j=1}^4{(x_j-C_{ji})}^2}+{\beta }_i\right)}^{-1}}\hfill \\ \beta ={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}{(1,2,2,4,4,6,3,7,5,5)}^T\hfill \\ C=\left(\begin{array}{cccccccccc}4.0& 1.0& 8.0& 6.0& 3.0& 2.0& 5.0& 8.0& 6.0& 7.0\\ 4.0& 1.0& 8.0& 6.0& 7.0& 9.0& 3.0& 1.0& 2.0& 3.6\\ 4.0& 1.0& 8.0& 6.0& 3.0& 2.0& 5.0& 8.0& 6.0& 7.0\\ 4.0& 1.0& 8.0& 6.0& 7.0& 9.0& 3.0& 1.0& 2.0& 3.6\end{array}\right)\hfill \end{array}\right.$$

(49)

$$\left\{\begin{array}{c}{f}_3(x)=-{\displaystyle \sum _{i=1}^4{\alpha }_i}\exp \left(-{\displaystyle \sum _{j=1}^6{A}_{ij}}{(x_j-P_{ij})}^2\right)\hfill \\ \alpha ={(1.0,1.2,3.0,3.2)}^T\hfill \\ A=\left(\begin{array}{cccccc}10& 3& 17& 3.50& 1.7& 8\\ 0.05& 10& 17& 0.1& 8& 14\\ 3& 3.5& 1.7& 10& 17& 8\\ 17& 8& 0.05& 10& 0.1& 14\end{array}\right)\hfill \\ P={10}^{-4}\left(\begin{array}{cccccc}1312& 1696& 5569& 124& 8283& 5886\\ 2329& 4135& 8307& 3736& 1004& 9991\\ 2348& 1451& 3522& 2883& 3047& 6650\\ 4047& 8828& 8732& 5743& 1091& 381\end{array}\right)\hfill \end{array}\right.$$

(50)

Different infill criteria were used to establish dynamic Kriging models to fit and optimize the test functions. Under the same initial sample size and number of iterations, the convergence of different infill criteria during the optimization process is depicted in Figure 9. As revealed by comparing several different infill criteria, the optimization designs based on the EI, PI, LCB and EGRA criteria had slower convergence speeds and were prone to falling into local optimal solutions. However, the PHA-LCB infill criterion proposed in this paper quickly approached the global minimum with fewer iterations, demonstrating preferable global infill capabilities.

4. ISIGHT Integrated Optimization

4.1. Optimization Objectives

This study aimed to optimize the shape of the PUV within the parameter range, so as to minimize its 24 h energy consumption while maximizing the PUV cabin volume. The objective function for 24 h energy consumption is defined as [16].

$$\mathrm{Ne}={\displaystyle \frac{\rho C_x{\mathrm{\Omega }}_T{v}^3{T}_1}{2{\eta }_p}}-P_{0}S{T}_2$$

(51)

where $\rho$ denotes the fluid density; $C_{\mathrm{x}}$ represents the drag coefficient; ${\mathrm{\Omega }}_T$ indicates the wetted surface area of the PUV; ${\eta }_p$ embodies the propeller efficiency, assumed to be 0.9; T1 designates the navigation time within 24 h, which is set at 16 h; P₀ stands for the average power generation per unit area of the solar panel, which is set at 100 W; S refers to the area of the solar panel; and T₂ symbolizes the charging time of the PUV within 24 h, which is set at 8 h.

Another objective function, the cabin volume V, is automatically retrieved by the SOLIDWORKS 2021 module integrated within the ISIGHT 2022 software through accessing the PUV.SLDPRT file.

Based on the above analysis, the optimization model of this study is expressed as

$$\left\{\begin{array}{l}\mathrm{Find} n,\theta ,L,H\\ \max \mathrm{V},\min \mathrm{Ne}\\ s.t.n,\theta ,L,H\in {ℝ}^m\end{array}\right.$$

(52)

4.2. Establishment of the ISIGHT Automatic Optimization Platform

In this study, SOLIDWORKS 2021, STAR-CCM+ 2206, and MATLAB 2019 modules were integrated into ISIGHT to achieve automation of the optimization work with its powerful data coupling function. The optimization flowchart and the constructed ISIGHT integration framework are illustrated in Figure 10 and Figure 11, respectively. The detailed steps of the optimization process are described as follows.

(1)

Sample Library Generation

When k = 1(k represents the number of optimization iterations), the DOE module is used to calculate the true response values of 23 sample points obtained by the optimal Latin hypercube; when k ≥ 1, the DOE module is adopted to calculate the true response values of new sample points generated upon the PHA-LCB infill criterion. The SOLIDWORKS module integrated within the ISIGHT framework enables automated generation of parameterized 3D models based on sampled design variables, which are subsequently exported as PUV.SLDPRT files while concurrently interfacing the volumetric data with the cabin volume calculation module. This computational module is specifically designed to evaluate the cabin capacities of geometrically variant models. Concurrently, the STAR-CCM+ integration module initiates the software execution through a *.bat batch command, thereby invoking the GOSTAR.java script to automate sequential operations including 3D model import, adaptive mesh generation, computational fluid dynamics (CFD) simulation, and post-processing data extraction. The energy consumption function is calculated with Formula (51) and the data from the STAR-CCM+ module. After the DOE module completes the calculation, the model parameters and response values are input into the sample library file.

(2)

Surrogate Model Generation

MATLAB is utilized to read the corresponding data in the sample library file, generate surrogate models for the two objective functions, and save them.

(3)

Surrogate Model Optimization

The NSGA-II algorithm is employed to optimize the two surrogate models generated in step 2. The population size is set to 80, the number of generations is 200, the crossover probability is 0.85, the mutation distribution index is 20, and the crossover distribution index is 10.

(4)

Response Value Calculation at the Optimal Solution

A set of optimal solutions is randomly selected from the Pareto front in step 3 to calculate the true response values.

(5)

Generation of New Sample Points

New sample points are generated following the PHA-LCB sampling criterion and passed to the DOE component.

(6)

Termination Condition

The termination condition is determined by the Loop component. The termination condition set in this study are as follows: (1) that the error between the predicted and true values of the energy consumption objective is less than 1%, namely, $\left|(N{\mathrm{e}}_{\text{prediat}}-N{\mathrm{e}}_{\text{actual}})/N{\mathrm{e}}_{\text{actual}}\right|\le 1\mathrm{\%}$, and (2) $RRMS{E}_{\mathrm{curent}}\le 1\%$.

4.3. Surrogate Model Performance Evaluation

According to the above infill criteria, the final sample library was established, and an additional 10 sample points were generated to evaluate the global accuracy of the two surrogate models, as exhibited in Figure 12 and Figure 13. The R² calculation formula is

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left(\widehat{f}(x_i)-f(x_i)\right)}}{{\displaystyle \sum _{i=1}^n\left(\overline{f\left(x_i\right)}-f(x_i)\right)}}},\hspace{1em}\overline{f\left(x_i\right)}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}f(x_i)}$$

(53)

The R² values of both surrogate models are calculated to be greater than 0.95, suggesting that both models have high accuracy and can be used for subsequent optimization analysis.

4.4. Optimization Results

The ultimate goal of this study is to design a photovoltaic-powered UUV with lower 24 h energy consumption and a larger cabin volume. The NSGA-II optimization algorithm was employed to optimize the two conflicting objective functions with the shape parameters of the bow (n), the shape parameters of the stern ($\theta$), the length of the photovoltaic panel (L), and the half-height of the photovoltaic panel (H) as design variables. Finally, a Pareto solution set containing 68 optimal solutions was obtained, as illustrated in Figure 14. Any two points in the optimal solution set were non-dominated, reflecting that improving one objective function to a more desirable level will inevitably compromise the other objective function [48].

Three representative solutions (A, B, and C) from the Pareto front were selected for in-depth comparative analysis, and corresponding parameter values are listed in

Table 8. Parameter values for the three typical solutions.

Typical Solutions	n	θ	L (mm)	H (mm)
A	4.93	29.87	1033.66	221.42
B	4.96	29.97	1488.87	397.42
C	3.73	29.52	1498.73	399.98

. Solution A exhibited the maximum cabin volume, while Solution C demonstrated optimal energy consumption efficiency. Transitioning from Point A to Point B, a marginal reduction of 13.19 cubic decimeters in cabin volume produced a substantial decrease of 1148.12 watt-hour in 24 h energy consumption. In other words, by accepting a 1.41% loss in cabin volume, the energy stored by the PUV over 24 h can be increased by 487.9%; from point B to point C, the cabin volume decreases by 1.13%, while the energy stored by the PUV over 24 h only increases by 9.57%. The optimization trajectory from A to B represents a more favorable design evolution path, where the energy-saving benefit significantly outweighs the minor loss in cabin space.

Figure 15 and Figure 16 illustrate the Pareto optimal solutions demonstrating the evolution of dual-objective functions relative to modifications in PUV’s shape parameters. Then, the variation patterns of the dual objectives—24 h energy consumption and cabin volume capacity—of the PUV in response to design variables were systematically investigated, laying a theoretical foundation for optimizing its morphological configuration. The dashed lines in these visualizations denote the objective function values corresponding to the three critical design points labeled A, B, and C, constructing a multi-dimensional parametric analysis framework for design optimization.

A pronounced observation emerged from the graphical analysis. Specifically, the variation in photovoltaic panel length (L) exhibited significantly greater magnitude compared to other design variables. As photovoltaic panel length increased, 24 h energy consumption underwent a rapid decline, accompanied by a reduction in cabin volume capacity. The reasons for this inverse relationship can be explained as follows. The expansion of the photovoltaic panel’s length led to an increase in the portion cut off by the PUV, contributing to the lessened available cabin capacity space. Among the four design variables examined in this study, photovoltaic panel length exerted the most profound influence on 24 h energy consumption, displaying a contrary trend—a finding congruent with the sensitivity analysis presented in Section 3.1. Concurrently, the bow shape index (n) demonstrated a notable positive correlation with cabin volume capacity since its increment engendered a plumper bow profile that enhanced volumetric efficiency. The impact of this parameter on cabin volume significantly surpassed that of other variables, maintaining a consistent trend with the sensitivity analysis results outlined in Section 3.1.

Paradoxically, the augmentation of bow shape index (n) produced negligible variations in energy consumption. This phenomenon stemmed from the dual-determinant of 24 h energy expenditure: photovoltaic panel area and direct navigation resistance. While increased bow shape index elevated hydrodynamic resistance, photovoltaic panel length served as the dominant factor influencing energy consumption, as confirmed by the preceding analysis. Within the Pareto solution set, design variables exhibited non-uniform variations, resulting in inconspicuous trends between bow shape index and energy consumption function. Similarly, the photovoltaic panel half-height (H) within the Pareto solutions ranges between 190–400 mm, which exceeded hull protrusion thresholds and thus exerted no discernible impact on internal cabin volume. Consequently, the visualization demonstrated an asymptotic relationship between panel half-height and cabin volume. The stern shape index, despite minimal variation, is correlated with divergent energy consumption and cabin volume values. As elucidated in Section 3.1’s sensitivity analysis, the individual influence of this parameter remained comparatively marginal, and the synergistic effects of other variables collectively determined the dual-objective functions. Hence, nearly identical stern shape index values corresponded to varied PUV performance metrics within the Pareto solution set, underscoring the complex interplay of multi-variable optimization.

Furthermore, Figure 15 and Figure 16 reveal several insightful correlations that provide invaluable guidance for the morphological design of photovoltaic-powered unmanned underwater vehicles. During the transition from Point A to Point B, both the bow shape index (n) and stern shape index (θ) exhibited negligible variations, maintaining near-constant values throughout this interval. Progressing from Point B to Point C, all parameters remained practically invariant except for the aforementioned index n. These marginally altered geometric descriptors inherently simplified the identification of their optimal configurations, highlighting the methodological superiority of multi-objective optimization frameworks over conventional single-objective approaches. Such frameworks enhanced problem-solving efficacy while expanding the solution space available to designers by generating diverse Pareto-optimal solutions.

Three-dimensional models of the PUV were constructed with parameter combinations corresponding to Points A, B, and C on the Pareto front to validate the optimization outcomes. Subsequent evaluations of the dual objective functions yielded empirical results with a maximum deviation of 6.67% between predicted and actual values, as demonstrated in

Table 9. Comparison of predicted and true values for three candidate points.

Candidate Point	24 h Energy Consumption Predicted (Wh)	24 h Energy Consumption Simulated (Wh)	Error (%)	Volume Predicted (dm3)	Volume True (dm3)	Error (%)
A	235.33	219.63	6.67	936.48	935.70	0.08
B	−912.79	−890.67	2.42	923.29	922.10	0.13
C	−1000.10	−1045.71	4.56	912.85	913.83	0.12
Initial profile	---	274.67	---	---	897.19	---

. This remarkable consistency not only attested to the precision of the optimization process but also reinforced the practical viability of the derived solutions. After these findings were synthesized with the operational requirements of the present study, Point B was ultimately selected as the definitive optimal configuration.

Figure 17 presents a 3D comparative visualization of the initial and optimized hull configurations. The optimized hull exhibits a fuller bow and stern profile, resulting in an increased cabin volume. Concurrently, the optimized design demonstrates elongation in both the length and height of the photovoltaic panels, directly contributing to elevated hydrodynamic resistance. However, the optimized hull achieves a marked enhancement in energy efficiency performance. As detailed in Table 9, despite compensating for additional energy consumption arising from heightened navigation resistance, the optimized configuration attains a net energy gain of 1165.34 Wh compared to the initial design.

As shown in Figure 18, the optimized model demonstrates a primary increase in the pressure drag, which contributes to the overall rise in total drag. Figure 19 and Figure 20 present comparative visualizations of pressure contours and streamline patterns for both the baseline configuration and optimized PUV design at a flow velocity of 1.5 m/s. The graphical analyses suggest that the optimized PUV exhibited pressure distributions highly comparable to the initial model, yet the integration of photovoltaic panels significantly disrupted the hydrodynamic flow regime around the hull. This disruption manifested as pronounced velocity gradients and pressure fluctuations occurring at the leading edge and trailing edge of the panels. Specifically, the impingement of fluid onto the panel’s anterior surface brought about a reduction in flow velocity, generating a localized zone of low-velocity high-pressure stagnation. Downstream from this interface, the flow rapidly transitioned into a high-velocity low-pressure regime, creating substantial pressure differential-induced drag forces. The optimized hull configuration features a more voluminous bow and stern, increasing cabin capacity by 24.91 dm³ over the baseline design. However, this geometric modification also generates elevated pressure concentrations at the bow section, resulting in a corresponding increase in the total drag for the optimized variant.

Nonetheless, this apparent hydrodynamic compromise was strategically offset by the 72.13% expansion in photovoltaic surface area achieved through the optimization process. The enhanced panel dimensions facilitated significantly greater power generation capacity per unit time, and the additional electrical energy produced over a 24 h operational cycle far surpassed the incremental energy expenditure attributable to increased drag. Consequently, the optimized configuration demonstrated a marked reduction in overall 24 h energy consumption compared to the baseline model, while simultaneously demonstrating an increase in cabin volume. In a 24 h day–night cycle, relying solely on solar photovoltaic system power, through rational adjustment of charging times, the optimized PUV sailing range reaches 93.96 km, an increase of 22.36% compared to the initial PUV, as shown in

Table 10. PUV performance comparison before and after optimization.

	Area of Solar Panels (m2)	Charging Time (h)	Navigation Time (h)	Navigation Distance (km)
Initial profile	2.75	8.00	14.22	76.79
Optimized profile	4.72	6.60	17.40	93.96 (+22.36%)

.

5. Conclusions

In this study, a multi-objective optimization was performed to minimize 24 h energy consumption and maximize payload volume for a photovoltaic-powered underwater vehicle. The principal conclusions are systematically drawn as follows.

(1)

A closed-loop optimization pipeline was established by integrating parametric modeling, hydrodynamic analysis, and high-precision dynamic surrogate modeling through the ISIGHT multidisciplinary design optimization platform. Automated geometric iteration was achieved with the Loop component-driven SOLIDWORKS parametric modeling interface. This synchronously triggered batch computational fluid dynamics simulations in STAR-CCM+. The DACE toolbox was employed to construct dynamic surrogate models, enabling intelligent screening and validation of PUV configurations. This integrated workflow lowered operational complexity in shape optimization and provided methodological references for complex appendage design, holding significant engineering value for enhancing energy utilization efficiency in underwater equipment.

(2)

A PHA-LCB infill methodology was proposed. It synergistically integrated global exploration with localized exploitation through a three-tiered adaptive framework. This innovative approach introduced a dynamic equilibrium mechanism that mediated between extensive design space investigation and refined local optimization, wherein the balancing coefficients governing the three operational phases were progressively modulated with real-time surrogate model fidelity metrics and convergence indicators. The methodology employed a hierarchical decision-making architecture where the initial emphasis was on broad spatial reconnaissance to ensure comprehensive domain coverage, followed by iterative refinement cycles for prioritizing regional optimization in proximity to identified promising candidates. Validation against benchmark test functions demonstrated superior precision and convergence rates compared to conventional optimization approaches following EI, PI, LCB and EGRA criteria.

(3)

The dynamic surrogate model constructed by the PHA-LCB method achieved a coefficient of determination exceeding 0.95, satisfying accuracy requirements. Subsequent integration with the NSGA-II multi-objective optimizer yielded a Pareto-optimal solution set within the design variable domain. Rigorous sensitivity analysis specified the interplay between design variables and objective functions while justifying the selection of Point B as the definitive optimal configuration.

(4)

Compared with the initial shape, the final optimized shape increased the bow shape index n and the stern shape index θ by 65.33% and 49.85%, respectively, which increased the cabin volume by 2.77% compared with the initial shape. The length and width of photovoltaic panels increased by 19.12% and 44.52% respectively. This increases the photovoltaic panel area by 72.13% compared to the initial shape, giving it a stronger power generation capacity per unit time, which can generate an additional 1148.12 Wh of electricity within 24 h. By reasonably adjusting the charging time, its cruising range has increased from the initial 76.79 km to 93.96 km, an increase of 22.36%.

Nevertheless, inherent limitations must be acknowledged. The current study relies solely on Kriging model fitting without comparative analysis against other surrogate modeling techniques. While propeller efficiency was assumed constant during energy consumption calculations, real-world performance varies with rotational speed. The average power output of the photovoltaic panel was fixed at 100 W, exceeding potential real-world yields in marine environments. Additionally, the simplified PUV geometry neglected detailed photovoltaic panel features (such as leading-edge fillet radius) and appendage effects (such as rudder and propeller interactions). Thus, further investigation should be conducted to refine the optimization framework’s predictive capabilities and practical applicability.