Accurate prediction of extreme waves is essential for ensuring the reliability of coastal and offshore infrastructures under complex sea conditions. However, a clear spectral interpretation of the harmonic decomposition employed in directional wave focusing is still lacking. This research develops a fully nonlinear numerical model, validates it against laboratory experiments to investigates the nonlinear evolution and focusing behavior of unidirectional, bidirectional, and multidirectional wave groups. It covers distinct nonlinear phenomena in wave focusing and quantifies the amplitudes and spatial distributions of the components up to the fourth order, providing new insights into the nonlinear mechanisms governing wave focusing. A novel higher-order boundary element method (HOBEM) is developed for directionally focused waves, which employs uniformly distributed inner sources on a three-dimensional cylindrical boundary, coupled with a vector-based control strategy for precise phase alignment and improved computational efficiency. The results demonstrate that unidirectional focused waves exhibit stronger energy transfer from the fundamental frequency to high harmonics than multidirectional waves, with growing differences for steeper waves. For bidirectional focused waves, the strength of the energy transfer to higher harmonics weakens with the increasing directional separation angle. Directional spreading reduces phase coherence, thereby altering the manifestation of nonlinear interactions and influencing the resulting focal behaviour. These findings elucidate the nonlinear characteristics of wave focusing in realistic sea states, offering important theoretical insights into the wave interaction mechanisms under extreme ocean conditions.
Unmanned sailing vessels commonly rely on real-time wind-field measurements as key inputs for multi-level decision-making, including path planning and control. During long-endurance offshore missions, however, wind-sensing systems may be affected by harsh environmental conditions, external mounting constraints, and maintenance difficulties. These issues motivate complementary control strategies that can support autonomous navigation while reducing dependence on real-time wind-measurement channels. In this study, we formulate sail–rudder control under partial observability, where the policy uses onboard observable vessel states and target-relative information without receiving measured wind speed or wind direction. A recurrent SAC-LSTM policy is employed as a wind-input-free control implementation, in which a Long Short-Term Memory (LSTM) module is incorporated into the Soft Actor-Critic (SAC) framework to encode recent observation–action history. This design is used to examine whether temporal context from vessel-motion evolution and previous sail–rudder actions can support continuous sail–rudder decision-making when direct wind variables are excluded from the policy input. In randomized-wind simulations evaluated over five independent training seeds, SAC-LSTM achieved a task success rate of 99.36%, compared with 39.64% for the memoryless wind-input-free SAC baseline and 95.24% for the wind-informed SAC reference. Under the same wind-input-free current-observation setting, SAC-LSTM also reduced the average successful-episode navigation time and rudder-angle variation rate relative to the memoryless SAC baseline. In the present randomized-wind benchmark, SAC-LSTM obtained a higher task success rate than the wind-informed SAC reference without using measured wind speed or wind direction, while showing similar successful-episode navigation time and trajectory length. These results indicate that recurrent observation–action history can provide useful temporal context for wind-input-free sail–rudder control under the tested 4-DOF still-water simulation conditions.
Helical strakes are widely used to suppress vortex-induced vibration (VIV) of marine risers. However, their suppression performance for free-hanging risers remains unexplored. This study experimentally evaluated the effects of strake height, coverage ratio, and spatial arrangement on VIV suppression of a free-hanging riser using helical strakes with a fixed pitch of 17.5D and heights of 0.15D, 0.25D, and 0.35D. The results show that helical strakes cannot effectively reduce the mean bending displacement in the in-line (IL) direction. The mean drag coefficient of the straked risers is generally slightly lower than that of the bare riser and increases with strake height. At high flow velocities, the asymmetric geometry of the helical strakes generates a non-zero mean lift force, and the mean lift coefficient generally increases with strake height, leading to pronounced mean bending displacement and overall deflection in the cross-flow (CF) direction. The CF VIV suppression efficiency increases with strake height, whereas the IL suppression efficiency varies only slightly. The 0.35D strake provides the best suppression efficiency, whereas the 0.25D strake exhibits a noticeable decrease in CF suppression efficiency at high flow velocities, possibly because the increased IL overall deflection increases the axial flow component. Under partial coverage conditions, the uniformly distributed arrangement generally produces smaller mean bending displacements and lower mean lift and drag coefficients while providing better VIV suppression than the upper concentrated arrangement. Notably, the uniformly distributed arrangement with 75% coverage achieves better VIV suppression than full coverage.
Soil stabilization is a crucial countermeasure to mitigate marine sediment erosion and scour-related hazards. This study develops a novel method for reinforcing sandy soils in seawater environments via enzymatically induced seawater mineralization, while mitigating bio-generated ammonium contamination through algae cultivation. Unlike traditional bio-cementation methods relying on lab-graded calcium reagents, the proposed approach utilizes naturally occurring calcium and magnesium in seawater to induce carbonate precipitation. Results showed that soybean-extracted urease still exhibited favorable mineralization potential despite a 10% activity inhibition by dissolved salts. A dual-stage cementation treatment was proposed, involving urease fixation followed by continuous injection of seawater-based urea solution. After six cycles of dual-stage treatment, soil samples achieved a maximum unconfined compressive strength of 2.83 MPa. Key factors influencing cementation efficacy during the treatment included urease fixation efficiency, continuous injection rate and ambient temperature. Microscale analysis revealed that soil strength enhancement was mainly attributed to interparticle precipitates of aragonite and magnesium calcite. Moreover, ammonium-rich effluent from the bio-treatment served as a nutrient source for algae growth, reducing ammonium levels by 68.3% within 25 days. This study suggests that enzymatically induced seawater mineralization combined with algae cultivation would provide a sustainable pathway for soil stabilization in marine environments.
Autonomous platforms operating in the oceans require accurate navigation to successfully complete their missions. In this regard, the initial heading estimation accuracy and the time required to achieve it play a critical role. The initial heading is commonly estimated using model-based orientation decomposition methods based on inertial measurements. These methods rely on two assumptions: small platform velocity and known approximate position. Under these assumptions, the inertial quantities in the navigation frame can be determined. However, methods such as dual-vector decomposition and optimization-based attitude decomposition achieve satisfactory heading accuracy only after long alignment times. To address this limitation and enable fast and accurate initial heading estimation, we propose an end-to-end, model-free, neural-assisted framework that uses the same inputs as the model-based approaches. That is, the platform is moored and thus quasi-stationary, satisfying the small velocity condition, while the position is provided by the global navigation satellite system. The proposed approach was trained and validated using a real-world dataset captured by an autonomous surface vehicle. The proposed method demonstrates an average accuracy improvement of 51% and up to an 83% reduction in alignment time. It estimates the initial heading using only inertial measurements as input, producing the heading angle at the end of the alignment period. Thus, the proposed approach enables reduced alignment time and improved accuracy, allowing shorter deployment times and enhanced navigation performance during the mission.
Dredging operations in navigational waterways are essential to maintain channel depth to ensure the safe passage of vessels. Existing shoaling forecasting tools often rely on heuristics or direct modeling of cumulative sedimentation, limiting lead time and predictive reliability. In this work, we propose an end-to-end data-driven forecasting framework that models the rate of change in sedimentation volume, referred to as Channel Infilling Rate (CIR), 30 days ahead, which is then integrated to recover cumulative forecasts. Our approach starts with the calculation of CIR using the Corps Shoaling Analysis Tool, developed by the U.S. Army Corps of Engineers. Following that, we employ variance decomposition-based global sensitivity analysis to identify the most influential upstream streamflow gauges from a large set of candidate gauges, which serve as inputs for CIR prediction. To model the complex relationship between upstream river discharge and the CIR of a region of interest, we explore four data-driven modeling approaches including multivariate linear regression, the long short-term memory (LSTM) networks, a hybrid model architecture that combines Convolutional Neural Network (CNN) with LSTM network (CNN-LSTM), and a Temporal Convolutional Network (TCN) model combined with LSTM network (TCN-LSTM). Finally, the trained models are employed to predict the CIR with uncertainty quantified to provide confidence intervals of the predictions. Unlike previous studies that model aggregate sedimentation volume across entire systems, we perform reach-level modeling to better capture spatial heterogeneity and hydrodynamic variability. Demonstrated on the Houston Ship Channel (HSC) and Southwest Pass (SWP), this framework represents a step toward data-adaptive shoaling forecasts that can support proactive sediment management in complex riverine environments. By providing rolling short-term forecasts with uncertainty bounds, the proposed framework can support risk-aware sediment management and proactive dredging decision-making. The comparison between HSC and SWP further demonstrates that model selection should be adapted to local survey-data availability, with simpler models favored under sparse survey conditions and LSTM-based models favored when dense survey records preserve nonlinear sedimentation dynamics.
Many environmental variables exhibit non-stationary characteristics under global warming, yet such assessments for typhoon-induced multi-hazards remain limited. This study develops an integrated non-stationary joint probability modeling framework that incorporates trend detection, cyclicity analysis, non-stationary marginal extreme value modeling, and multivariate dynamic dependence evaluation. A novel implementation of the Hilbert-Huang transform is proposed to capture the cyclic behavior of constituent variables, potentially caused by various atmospheric circulations, for improved physical realism. Based on the use of ERA5 reanalysis data for China’s offshore regions, the results show that over 40% of sites exhibit increasing trends, and approximately 25% show declining variability and weakening wind-wave dependence. The cyclic effects associated with the El Niño-Southern Oscillation are identified, with a dominant period of about 3.2 years. The generalized additive model-based non-stationary models can better capture nonlinear features compared to linear and stationary models. The variability of time-dependent design loads is mainly governed by non-stationary marginal models rather than by copula-based dependence models. Notably, lifetime exceedance probabilities may be higher under non-stationary conditions compared to those obtained under stationary assumptions. These findings emphasize the importance of non-stationary analysis and modeling of typhoon-induced wind and wave extremes for better-informed offshore structural design and risk adaptation in a changing climate.
In the present work, high resolution wind and wave model data, derived for the needs of metocean climate study for two European areas, are compared with existing in situ measurements and model results from ECHOWAVE coastal database. The data cover two nearshore areas in the Atlantic Ocean around two existing wind farms located in the North Sea (East Anglia One: EAO), and west of Portugal (WindFloat Atlantic: WFA). It is the first time that metocean conditions for these two areas have been produced at such a high spatial (∼1 km for wind and ∼100 m for waves) and temporal (1 hr) resolution for an extent of ten (10) years. For the generation of the model data, two regional numerical models, atmospheric model WRF and wave model SWAN, have been implemented using detailed geographical information acquired from EMODnet (European Marine Observation and Data network) database, and wind and wave boundary conditions from global database ERA5 (ECMWF Re-Analysis, v.5). Both model and measured wind and wave data have been statistically analysed, and compared against each other, using both standard and enhanced error metrics. Comparisons cover the seasonal variability, the directional behaviour, the probability structure, and the duration analysis of the data. Taking into account (a) the differences in the model databases (models used, grid resolution, boundary conditions, quality of geographical information etc.), (b) the various sources of uncertainties in measured data (different temporal extent, distance from the wind farm, missing values, different quality and sampling methods etc.), the presently derived datasets can be considered being in a good agreement with both in situ measurements and ECHOWAVE database.
Unmanned Surface Vehicles (USVs) are increasingly deployed in ocean engineering applications such as hydrographic surveying, environmental monitoring, and maritime security. Achieving accurate path following in realistic ocean environments remains challenging due to nonlinear vessel dynamics, environmental disturbances, and underactuation arising from the absence of direct sway control. This paper proposes an Adaptive Nonlinear Terminal Sliding Mode Controller (ANTSMC) for a three-degree-of-freedom (3-DOF) underactuated USV subject to wind, wave, and current disturbances. The controller integrates a regularised nonlinear terminal sliding surface with a power-rate reaching law and a leakage-regulated adaptive switching gain that adjusts robustness online without a disturbance observer. A benchmark evaluation was conducted across six controllers (LQR, SMC, ASMC, ADRC, FNTSMC, and ANTSMC), four path geometries, and three WMO sea states while accounting for realistic sensor noise and parametric uncertainty. The results show that ANTSMC achieves cross-track accuracy comparable to SMC under calm sea conditions, while demonstrating stronger robustness as environmental disturbances increase. In particular, ANTSMC produces the lowest RMS cross-track error under SS 2 (smooth) and SS 3 (slight) conditions, where the imposed disturbance intensity is higher. Independent validation using the MBZIRC physics-based maritime simulator provides additional evidence of bounded, accurate path-following performance across different path geometries. Moreover, a Monte Carlo study evaluates robustness under expanded parametric and disturbance uncertainties. The results indicate that the proposed observer-free framework provides consistent path-following performance over the tested marine operating conditions.
With the development of maritime inspection, environmental monitoring, and intelligent shipping, visual perception has become a key capability for safe and reliable autonomous navigation. However, real-world waterways involve complex dynamic conditions that degrade the robustness and generalization of existing vision-based perception systems. Although numerous studies have investigated individual perception tasks and algorithmic paradigms, a review from an environmental and application-oriented perspective is still lacking. Our study presents a comprehensive survey of visual perception in complex waterway environments through a unified analytical framework. First, major waterway perception tasks are summarized, and representative challenging conditions are categorized into dynamic water-surface disturbances, complex illumination, adverse weather, and small-target environments. Second, publicly available real-world waterway datasets are systematically organized with emphasis on data diversity, environmental complexity, and task suitability. Third, representative perception methods are comprehensively reviewed under different environmental conditions, focusing on perception bottlenecks, robustness limitations, dataset applicability, and practical deployment constraints. Furthermore, we analyze how environmental complexity affects perception failure mechanisms and discuss the interplay among model design, data quality, and deployment constraints. Finally, emerging directions such as multimodal perception, domain generalization, and robust adverse-condition perception are highlighted. With this review, we aim to provide practical guidance for dataset construction, method design, and system level optimization, thereby supporting safer and more reliable autonomous surface vehicle navigation in complex waterway and maritime engineering applications.
This study presents a modified bond-based peridynamic (PD) model to investigate the fracture behavior of ice covers under regular wave loading. A key improvement lies in the development of an energy-based failure criterion that explicitly links the wave energy input to the critical energy release rate of ice, thereby bridging wave dynamics and fracture mechanics. Wave-induced loads are applied within a one-way coupled hydro-mechanical framework, allowing quantification of energy dissipation during fracture. The model is validated against existing experimental data and systematically applied to examine the effects of wave frequency, amplitude, and ice properties on crack propagation patterns. The results demonstrate that the modified peridynamic approach can effectively capture the fracture initiation and propagation in ice covers under wave action, offering a computationally efficient tool for wave–ice interaction studies.
The tail-slap phenomenon encountered during the high-speed motion of supercavitating vehicles presents a significant challenge to conventional water-rudder control because it may increase hydrodynamic drag and cause thrust loss. To provide an alternative means of attitude control, this study investigates the hydrodynamic response of a supercavitating vehicle to an aft lateral jet and evaluates its attitude-control potential. A three-dimensional numerical framework combining the Volume of Fluid multiphase-flow model, the RNG k–ε turbulence model, and the Schnerr–Sauer cavitation model is employed to examine the interactions among the lateral jet, supercavity morphology, wetted regions, surface-pressure distribution, and vehicle hydrodynamic loads. The effects of freestream velocity are investigated over V=60-110 m/s. The combined effects of angle of attack and lateral-jet intensity are examined at V=100 m/s, with α = 0°-5° and J=0, 0.003, 0.007, 0.013, and 0.029. The results show that the aft lateral jet does not fundamentally alter the overall topology of the supercavity but redistributes the local cavity geometry, wetted regions, and surface pressure, thereby producing pronounced changes in lift and pitching moment. At α = 5°and J=0.029, the drag and lift coefficients increase by 7.7% and 73%, respectively, relative to the no-jet condition, while the pitching-moment coefficient changes from a positive nose-up value to a negative nose-down value. Therefore, the lateral jet causes a certain increase in drag but substantially enhances lift generation and produces a pronounced nose-down static pitching-moment contribution at a large angle of attack. These findings demonstrate the hydrodynamic attitude-control potential of the aft lateral jet and provide a basis for further investigations of supercavitating-vehicle attitude dynamics and control.
The damage mechanisms of thin-walled UHPC shells subjected to underwater contact explosions remain insufficiently understood. This study conducted contact-explosion tests on air-backed UHPC cylindrical shells using 20, 30, and 40 g TNT charges and quantified damage on the blast-facing and rear surfaces through three-dimensional laser scanning. An arbitrary Lagrangian-Eulerian fluid-structure interaction model incorporating strain-rate effects was developed and validated against the experimental results. Parametric analyses were subsequently performed by varying the charge mass, shell thickness, and outer diameter, and the structural responses were normalized using the relative charge scale Πe and relative thickness Πh. The results show that the high dynamic compressive strength of UHPC suppresses localized crushing on the blast-facing surface, whereas tensile-wave reflection at the air-backed rear surface governs spalling. The first 3 ms after detonation contributed 99.19% of the cumulative pressure impulse over 0-30 ms, demonstrating that subsequent bubble evolution had a limited effect on the final damage. Increasing the charge mass promoted through-thickness damage coalescence, whereas increasing the shell thickness suppressed it; near the critical state, an 8.3% increase in charge mass triggered a transition from spalling to punching. Increasing the outer diameter weakened local restraint and enhanced global structural motion. Bending, spalling, and punching failure were distinguished in the Πe-Πh parameter space, providing a dimensionless basis for failure-mode prediction.
The boundary with an initial opening significantly affects shock wave propagation and bubble dynamics. To investigate the associated loading characteristics, this paper employs an axisymmetric Riemann-SPH method to simulate the shock wave and bubble near a single-layer deformable wall with an initial opening. The numerical model is first validated by simulating a spark-generated bubble experiment. Subsequently, the effects of plate deformation, opening radius, and stand-off distance on shock wave propagation, bubble pulsation, and jet evolution are systematically analyzed. The results show that plate deformation has little influence on shock wave propagation but significantly alters bubble pulsation and jet formation. At the opening center, the pressure history evolves from a double-peak to a single peak with increasing opening radius or stand-off distance. Bubble evolution presents three distinct jet modes: vertical jet, oblique jet tendency, and oblique jet. Small opening radii combined with short stand-off distances promote high-speed oblique jets and toroidal bubble formation, whereas increasing the stand-off distance weakens plate confinement, causing the bubble dynamics to gradually approach those in the free field. The study provides useful insight into shock wave propagation and bubble dynamics near non-intact plates and offers a reference for evaluating underwater explosion loading on damaged structures.
A clear understanding of scour around monopile foundations is increasingly important as offshore wind expands. This study investigated local scour under irregular waves, focusing on the low Keulegan-Carpenter number regime (KC < 6), where scour is typically seen as negligible. Hydraulic model experiments showed that significant scour can still start in this regime, mainly driven by the sediment Froude number (Fr), which increases inertial forces relative to gravitational forces, thereby lowering the threshold for sediment incipient motion. Scour was classified as near field, transitional field, or far field based on where the maximum scour occurred. Deep Near Field scour mainly occurred in the post-shedding regime (KC > 6), with active vortex shedding, while the pre-shedding regime (KC < 6) generally showed Far Field or minor scour due to weaker vortex development. These results show that Fr determines chiefly when and where scour begins in the low KC regime, underscoring the need for new criteria that account for inertial effects. Future research should examine the effect of the Froude number under wave-current conditions to determine whether these findings apply to complex marine environments.
Wave overtopping at harbor breakwaters is an operationally critical hazard, yet visually energetic scenes (e.g., run-up, spray) do not map uniquely onto hydraulically hazardous states. To improve alert reliability, a physics-guided, decision-level fusion framework is proposed. A meta-classifier combines per-frame convolutional neural network (CNN) image inference with synchronized hydrodynamic descriptors, applying the physical state as a soft consistency constraint to re-weight, rather than override, the visual posterior. The framework was evaluated retrospectively on selected keyframes from 13 typhoon events recorded between 2023 and 2025 at an operational harbor breakwater, classifying the hazard state of the current hour across pre- and post-damage structural regimes (293 and 379 held-out event-hours, respectively). Relative to the two single-modality baselines (image-only and hydrodynamic-only), the random-forest (RF) late-fusion preserved severe-event recall at a fixed alert threshold in the image-aligned warning-period evaluation while delivering regime-dependent ranking gains. In that evaluation, the fusion raised average precision (PR-AUC) over the image-only baseline from 0.599 to 0.832 pre-damage and from 0.777 to 0.962 post-damage. In the full-window hourly evaluation, the fusion also improved PR-AUC over the hydrodynamic-only baseline in every regime-by-illumination cell, indicating a consistent contribution from the image score. The single-modality comparison yielding the larger improvement, however, varied with both structural regime and illumination; this is an observed single-site pattern rather than a validated deployment rule. These results provide a single-site field proof of concept for reliability-oriented monitoring under evolving structural conditions; multi-site transferability and uncertainty quantification remain to be established.
Reliable characterization of wave energy resources is a prerequisite for the sustainable development of the blue economy, yet it remains challenging due to the spatiotemporal sparsity of satellite altimetry and limited in-situ networks. This study introduces a probabilistic data fusion framework based on Bayesian Neural Fields (BayesNF) that reconstructs continuous, high-resolution significant wave height fields with rigorous uncertainty quantification, using only spatiotemporal coordinates as inputs. The framework is trained on a heterogeneous constellation of nine satellite missions covering the North Tyrrhenian Sea and is assessed through a domain-aware validation strategy that separates satellite-based training from independent SWOT altimetry validation and moored-buoy testing. On the independent buoy record, BayesNF attains a mean absolute error of 0.35 m and a root mean square error (RMSE) of 0.50 m, reducing RMSE by approximately 26% relative to the operational CMEMS L4 product (0.68 m) and 24% relative to a deterministic deep-learning baseline of identical capacity (0.66 m), while approaching the accuracy of the physics-based CMEMS Wave Reanalysis (0.31 m) using no meteorological forcing. Unlike these deterministic products, BayesNF delivers calibrated predictive intervals, with an empirical 95% credible-interval coverage of 95.6%. The framework generates a full-month, high-resolution regional wave field in under a minute (roughly 52 s on a single TPU v6e). Finally, we propagate these stochastic reconstructions into continuous maps of wave energy density, demonstrating how the probabilistic approach captures the non-linear amplification of uncertainty and provides transparent bounds that support preliminary, proxy-based screening of offshore renewable energy potential.
Reliable operation of Autonomous Underwater Vehicles (AUVs) in harsh marine environments depends on effective health monitoring and accurate awareness of their operating condition. However, AUV fault signatures are highly heterogeneous, involving both global statistical shifts and localized high-frequency temporal irregularities, which challenge traditional diagnostic models. To address these challenges and advance failure analysis in maritime systems, this paper proposes a novel hybrid diagnostic framework that combines physically interpretable handcrafted statistical descriptors with learned temporal representations. Using a leakage-free out-of-fold (OOF) stacking strategy, the framework integrates the global statistical characterization provided by a tree-based ensemble with the local temporal pattern learning of a residual 1D CNN. Evaluated on a publicly available experimental AUV benchmark, the proposed method achieves a test accuracy of 99.18% and a Macro-F1 score of 0.9914. To improve interpretability for practical maritime maintenance, feature importance and mean absolute SHAP analyses are applied to the descriptor-based branch. The results indicate that thruster-control variability and vehicle-attitude dynamics provide the largest aggregated physical contributions, supporting the plausibility of the learned diagnostic patterns. These findings demonstrate the potential of the proposed framework as an interpretable decision-support approach for AUV health monitoring and maintenance.
Wave attenuation is essential to mitigate coastal erosion and safeguarding nearby infrastructure. This study presents analytical and computational approaches to model wave evolution over a hybrid coastal protection measure combining vegetation and low-crested breakwaters, based on the Shallow Water Equations (SWEs). In estimating wave transmission coefficient resulting from the interaction between waves, vegetation, and breakwaters, an analytical solution is obtained by applying the separation of variables technique, whereas a staggered finite-volume scheme is developed to conduct numerical simulations. Comparisons of the analytical solutions, numerical simulations, and experimental data demonstrate strong agreement, with relative errors below 0.13% with respect to analytical solutions and below 3.4% compared to experimental measurements, indicating a high degree of accuracy of the proposed model. The results imply that the hybrid system achieves greater wave amplitude reduction than vegetation-only or breakwater-only configurations. Sensitivity analyses reveal that the size and friction factor of both breakwaters and vegetation have comparable effects on wave attenuation, with higher values leading to greater reduction. Notably, adjustments to the friction factor have the most significant impact on wave attenuation compared to the other parameters examined. These findings contribute to enhancing the design of hybrid coastal defense systems and provide valuable insights for sustainable coastal engineering and climate resilience planning.
To visualize the distribution of added resistance in waves over a ship hull, a ship hull integration method is proposed within the framework of linear potential flow theory. The proposed method is based on the near-field method, in which the line integral term is transformed into a surface integral by applying Stokes’ theorem. As a result, all components of the added resistance are expressed exclusively in terms of surface integrals, and it enables direct visualization of their distribution on the hull surface. Since the proposed method requires third-order derivatives of the velocity potential, an approximate calculation method is also proposed based on a shape function. The accuracy and validity of the proposed methods are evaluated through comparisons with the conventional near-field method and available experimental data.
