The Homotopy Analysis Method (HAM) is a widely used analytical approach for solving nonlinear problems, yet its theoretical foundation lacks rigorous justification, and its intrinsic correlation with perturbation theory remains ambiguous, leading to prevalent confusion in the existing literature. This study demonstrates that the fundamental homotopy deformation equation of HAM can be naturally derived from the weak-nonlinearity perturbation theory. We construct a specific analytical expression and optimize the core parameters (the optimal auxiliary linear operator, convergence-control parameter, and auxiliary function) to mitigate the inherent strong nonlinearity of the nonlinear operator. Extending the small parameter εof perturbation theory to the interval [0,1] enables a systematic homotopy deformation process, which connects the linear auxiliary system (at ε=0) with the original nonlinear problem (at ε=1) and confirms HAM as a structured, adaptive generalization of classical perturbation theory. Furthermore, this work provides a rigorous proof that the Homotopy Perturbation Method (HPM) is a special case of HAM: HPM can be directly derived by fixing the optimal auxiliary linear operator as the linear component of the nonlinear system and setting the convergence-control parameter and auxiliary function to specific values, thus making HPM a degenerate form of HAM. This study clarifies the perturbation-theoretic origin of HAM, defines the hierarchical subordination of HPM to HAM, unifies the theoretical framework of homotopy-based nonlinear analytical methods, rectifies common misconceptions in the existing literature, and offers valuable guidance for the rational application, comparative analysis, and further development of such methods.

This study presents an analytical investigation of magnetohydrodynamic (MHD) mixed convection in inclined channels, addressing fundamental gaps in predicting flow reversal phenomena. Novel exact closed-form solutions are developed for the strongly coupled nonlinear equations governing this complex flow, overcoming mathematical challenges from magnetic field effects. The model establishes a comprehensive framework for four distinct flow regimes: stable unidirectional flow (Region I ), localized top-wall reversal (Region II ), bottom-wall reversal (Region III ), and dual-wall reversal (Region IV ). The analysis systematically reveals how critical governing parameters, particularly the magnetic interaction parameter (M), combined with buoyancy intensity, flow inertia, and channel inclination, dynamically reconfigure reversal boundaries and alter flow transition thresholds. Beyond quantitative changes, the analysis demonstrates how variations in these parameters fundamentally transform the topology of reversal domains, with magnetic effects exerting primary control over regime transitions. The results provide new insight into reversal mechanisms, showing how Lorentz forces modify velocity profiles, suppress secondary flows, and reshape thermal distributions. This work introduces a complete analytical mapping of flow reversal behavior in magnetized inclined channels, offering valuable understanding for thermal management system design.

In the paper, a homotopy-based semi-analytical approach has been proposed for investigating the extreme large-deflection nonlinear thermo-mechanical buckling of shallow spherical shells constituted by radially heterogeneous composite materials on non-uniform tri-parameter nonlinear elastic foundation under combined thermo-mechanical loads. The spatially thermal field respectively considering the Dirichlet, Neumann and Robin thermal boundaries at inner and outer surfaces of shells are analytically formulated corresponding to various formulations of resultant thermal moments and forces. The highly coupled and nonlinear governing differential-integral equations for the extremely nonlinear thermo-mechanical buckling model of heterogeneous variable-stiffness shallow spherical shells with arbitrary rotational and translational mechanical boundary constraints have been derived. Nonlinear snap-through thermo-mechanical buckling equilibrium path analysis of spherical shells at largely deformed state with extreme shallow curvatures has been conducted, which indicates the derived shallow coefficient is the vital parameter in description of shallow aggravation. Effects of boundary constraints, the unevenness of initial curvature, the non-uniformity of radially heterogeneous material and foundations, the thermal expansion and conductivity coefficients with thermal amplitudes have been thoroughly investigated, all of which are found to exert a significant influence on the nonlinear thermo-mechanical buckling characteristics of the shell.

This paper proposes an enhanced storm surge forecasting approach utilizing novel neural networks, specifically a model based on gated recurrent units (GRUs), and incorporates a distortion loss function known as Distortion Loss Including Shape and Time (DILATE) for improved prediction accuracy and reliability. Historical data from four stations in the hurricane-prone region of the Atlantic Ocean are utilized for both training and testing purposes. Various models are employed to predict storm surges with lead times of 3, 6, and 12 hours, and their performance is evaluated by comparing the predicted values with the observed ones. Furthermore, the influence of various physical factors on storm surge formation is examined. The findings indicate that the GRU-DILATE model is more suitable for storm surge prediction, particularly for longer lead times. This model effectively reduces prediction delays and accurately captures the trends in wave height associated with storm surges. Among the physical factors studied, wind speed and atmospheric pressure emerge as the most important variables influencing storm surges. Additionally, the significance of wind direction is highlighted, as it plays a crucial role in determining whether the sea level rises or falls during the initial stages of a storm surge. It is expected that the proposed model has the potential to enhance response and preparedness measures in hurricane-prone areas, finding applications in coastal planning, infrastructure design, disaster management, and other aspects of ocean engineering. (c) 2026 THE AUTHORS. Published by Elsevier B.V. on behalf of Shanghai Jiao Tong University. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

This study utilizes deep reinforcement learning (DRL) to develop flow control strategies for circular and square cylinders, enhancing energy efficiency and minimizing energy consumption while addressing the limitations of traditional methods.We find that the optimal jet placement for both square and circular cylinders is at the main flow separation point, achieving the best balance between energy efficiency and control effectiveness.For the circular cylinder, positioning the jet at approximately 105{\deg} from the stagnation point requires only 1% of the inlet flow rate and achieves an 8% reduction in drag, with energy consumption one-third of that at other positions. For the square cylinder, placing the jet near the rear corner requires only 2% of the inlet flow rate, achieving a maximum drag reduction of 14.4%, whereas energy consumption near the front corner is 27 times higher, resulting in only 12% drag reduction.In multi-action control, the convergence speed and stability are lower compared to single-action control, but activating multiple jets significantly reduces initial energy consumption and improves energy efficiency. Physically, the interaction of the synthetic jet with the flow generates new vortices that modify the local flow structure, significantly enhancing the cylinder's aerodynamic performance.Our control strategy achieves a superior balance between energy efficiency and control performance compared to previous studies, underscoring its significant potential to advance sustainable and effective flow control.

This paper addresses the challenge of scarcity and discontinuity in spatio-temporal observation data within oceanographic research. The primary goal is to develop a robust model capable of generating missing wave height data and enhancing the understanding of ocean wave behavior. We propose an end-to-end spatio-temporal sequence generation network based on convolutional U-Net, utilizing reanalyzed atmospheric datasets from the European Centre for Medium-Range Weather Forecasts (ECMWF). The model incorporates input features such as wind wave height, wave direction, mean wave period, and wind speed to generate significant wind wave heights in the South China Sea at specified times. Data smoothing was applied at intervals of 12, 24, and 36 h under both moderate and extreme wave conditions, resulting in wave height data at 6-hour intervals. The findings demonstrate that the model effectively captures areas with high wave heights, outperforming traditional interpolation-based methods, especially as input data time intervals increase and wave conditions intensify. This approach offers a novel solution for generating missing spatio-temporal data, providing valuable insights into ocean wave behavior and supporting applications in ocean engineering.

This study investigates active flow control in two-dimensional flows at a Reynolds number of 100 using Deep Reinforcement Learning (DRL). We utilize DRL to develop flow control strategies that enhance energy efficiency and minimize energy consumption, thereby addressing the limitations of traditional methods. We find that the optimal jet placement for both square and circular cylinders is at the main flow separation point, achieving the best balance between energy efficiency and control effectiveness. For the circular cylinder, positioning the jet at approximately 105°from the stagnation point requires only 1% of the inlet flow rate and achieves an 8% reduction in drag, with energy consumption one-third of that at other positions. For the square cylinder, placing the jet near the rear corner requires only 2% of the inlet flow rate, achieving a maximum drag reduction of 14.4%, whereas energy consumption near the front corner is 27 times higher, resulting in only 12% drag reduction. In multi-action control, the convergence speed and stability are lower compared to single-action control, but activating multiple jets significantly reduces initial energy consumption and improves energy efficiency. Physically, the interaction of the synthetic jet with the flow generates new vortices that modify the local flow structure, significantly enhancing the cylinder’s aerodynamic performance. Our control strategy achieves a superior balance between energy efficiency and control performance compared to previous studies, underscoring its significant potential to advance sustainable and effective flow control.

We demonstrate the effectiveness of deep reinforcement learning (DRL) in active flow control around a cylinder at Re = 500. A DRL-based algorithm was employed to manipulate synthetic jets for controlling the flow field. The results show that the DRL agent achieves a 49% reduction in drag coefficient and significantly stabilizes lift coefficient fluctuations, showcasing its capability to adaptively optimize control strategies. The vorticity evolution during training reveals a progressive suppression of the Karman vortex street, leading to a fully stabilized wake and smoother flow dynamics. Substantial improvements are observed in the streamwise and cross-stream velocity fields, with the wake transitioning from a chaotic, unsteady state to a stabilized and streamlined structure. Additionally, the pressure field evolves from an asymmetrical, fluctuating distribution to a stabilized, symmetrical profile, reducing drag and ensuring uniform pressure recovery. These findings highlight the potential of DRL-based strategies for robust and energy-efficient flow control.

Microplastic (MP) contamination in the terrestrial environment has received increasing concern in recent years. Due to the richness of surface chemistry in MPs after aging, the complexity of aged MPs with heavy metals often happens and their cotransport in soil could pose a high ecological risk. This study investigated the transport of Pb and Cu in saturated sandy soil coexisting with polystyrene MPs aged by physical coating and chemical oxidation. Iron oxide and extracellular polymeric substances coatings as physical aging processes enabled MPs surface covered with Fe and carboxyl and amide groups, respectively, increasing the MPs surface charge and particle size. Consequently, the inhibited transport of aged MPs and their enhanced sorption capacity resulted in greater accumulation of Pb by11.6 %-31.5 % and Cu by 23.2 %-30.9 % in aged-MPs soil compared to that in the pristine-MPs soil. Chemical aging by ultraviolet irradiation and persulfate oxidation caused surface roughness of MPs with particle size reduction. Correspondently, the MPs transport was enhanced, and co-transport of Pb and Cu in soil increased by up to 27.2 % and 36.2 %, respectively, in comparison with that in the pristine MPs soil. Overall, the transport of Pb and Cu in soil was contrast to physical and chemical aging of plastics in the terrestrial environment which may cause the possibility of the combined pollution of heavy metals with plastics in surface soil and subsurface down to the groundwater, respectively.

This study utilizes reinforcement learning algorithms to develop advanced control strategies for manipulating synthetic jets positioned on a cylinder, aiming to achieve effective flow control. It addresses critical challenges in achieving robust performance across moderate to high Reynolds numbers (Re) and varying jet placement configurations. The results demonstrate that placing synthetic jets near flow separation regions yields optimal control performance. At Re = 500, the multi-objective flow control strategy achieves up to 37.4 % drag reduction and 99 % suppression of lift oscillations, while completely suppressing vortex shedding. At Re = 1000, drag reduction improves to 49.1 %, with equally effective lift suppression and complete vortex shedding suppression. At Re = 2000, the strategy achieves a maximum drag reduction of 58.8 % and suppresses 90 % of lift oscillations, although some residual wake instabilities persist. Additionally, synthetic jets effectively disrupted detached vortices and stabilized the wake, resulting in streamlined flow and suppressed vortex shedding. These findings underscore the pivotal influence of jet placement in optimizing flow control efficacy, while affirming the advanced capability of control strategies to tackle intricate challenges.

Due to the variable-thickness parallelogram plates in ocean structure are characterized by the presence of strong moment singularity at supported obtuse corners, the wide application has turned the nonlinear mechanical analysis into one of the most important engineering concerns. In the paper, a refined geometrically nonlinear bending model of variable-thickness orthotropic parallelogram plates in curved hull under thermo-mechanical loads is proposed, while influences of linearly or quadratically thickened thickness in symmetrical or unsymmetrical profile on mechanical properties are thoroughly investigated. Three kinds of spatially thermal field in parallelogram domain are formulated on account of the coupled interaction of effects between in-plane distribution and thickness variation corresponding to the Dirichlet, Neumann and Robin thermal boundaries. A novel thickness-dependent Airy stress function is introduced overcoming the failure of traditional Airy stress function in equilibrium of in-plane forces, while the highly coupled and variable-coefficient nonlinear governing partial differential equations are firstly derived. The homotopy-based wavelet method is adopted to investigate the nonlinear thermo-elastic bending behaviors, while convergent process is verified and precision of obtained series solutions has been validated in excellent agreement with published results. The significant conclusion can be made that large-deflection nonlinear bending of such plates can be simplified with little discrepancy by omitting terms involving the derivatives of thickness variation in compatibility equation of deformation, which is generalized to the thermo-mechanical bending and greatly simplifies the analyzing procedures.

A brand-new hygro-thermo-mechanical bending model of the inhomogeneous concave and convex composite circular plates having varying thickness undergoing large deformation resting on nonlinear tri-parameter spring and shear elastic layers is proposed. Three-dimensional analytical hygrothermal field of circular plate is demonstrated with non-uniform thermal and moisture diffusion coefficients in axial variation, while thickness and elastic module of material are modeled in quadratically change and sufficiently summarized in normalization. The circled boundaries with arbitrary translational and rotational constraints are parameterized with feasible region of elastic parameters highlighted in rectangular domain by Linear Programming method. Two reduced-order integro-differential governing equations for the composite circular plates under extreme load have been derived, while analytical bending solutions are obtained by an employed homotopy-based analytical scheme with accuracy verified and convergence accelerated by truncation and iteration. Whether thickness or elastic module of composite material is variable, the final outcome on plate structural strength is the discrepancy of bending stiffness with different load capacity, with the former revealing more sensitive than the latter under the condition of same values. Once amplitude and distribution of hygrothermal load are determined, influence on large-deflection bending of circular plates exists at a limited range of nonlinearity, while non-uniform distribution of thermal expansion and moisture concentration aggravates bending effects only within this range.

We propose a novel approach that combines regional ocean data prediction with typhoon trajectory inversion methods to improve the forecasting of typhoon trajectories. This model expands outward from the current path to create a grid within a fixed range of latitudes and longitudes, populated with future ocean environment information to effectively identify the next center point of the trajectory. The model's predictive performance is evaluated using 57 historical typhoons in the North Atlantic for 24- to 72-h lead times through comparative analysis. The results demonstrate a 6.3% reduction in error for 24-h lead-time forecasts and a significant 33.3% reduction for 48-hour lead-time forecasts compared to the popular gated recurrent unit convolutional neural network model, underscoring the enhanced accuracy of the new model, particularly for longer forecasting intervals. In addition, the gradient-weighted class activation mapping technique is employed to examine the relationship between specific inherent features of typhoons, such as wind speed, atmospheric pressure, turning points, and overall movement speed, and the model's prediction error, utilizing statistical relationships to identify conditions under which the model exhibits large or small errors so that we can evaluate its effectiveness in predicting mainstream typhoons while recognizing characteristics that complicate certain forecasts and necessitate increased caution. This provides new insights for artificial intelligence models exploring the physical mechanisms of typhoon path prediction.

Flow control around bluff bodies, such as elliptical cylinders, is crucial in various engineering applications, where drag reduction and vortex shedding suppression are key objectives.This study trains a flow control strategy based on reinforcement learning (RL) to control the flow around an elliptical cylinder between two walls. The theme of this study is to explore whether multi-objective flow control can be achieved for elliptical cylinders with varying aspect ratios (Ar) under low energy consumption conditions. The RL training results indicate that for elliptical cylinders with larger Ar, the control strategy successfully reduces drag, minimizes lift fluctuations, and completely suppresses vortex shedding, all while consuming minimal external energy.However, as the Ar decreases, achieving the desired multi-objective control becomes increasingly difficult, even with substantial external energy consumption.Through physical analysis, we find that the interaction between the blockage ratio (\beta) and Ar limits the effective suppression of vortex shedding, thereby affecting the performance of the control strategy in stabilizing wake dynamics.Further, by reducing \beta, the study demonstrates consistent multi-objective control across all Ar values while maintaining energy efficiency.For extremely slender cylinders, balancing energy consumption with performance remains a challenge, yet vortex shedding is still effectively suppressed.This work underscores the efficacy of RL-driven flow control in achieving stabilization of the flow around slender bluff bodies.

We employ reinforcement learning algorithms to develop control strategies for manipulating synthetic jets on an elliptical cylinder, targeting robust flow control across varying aspect ratios (Ar). Our results show that for high aspect ratios (Ar=1 and 0.75), the learned strategies effectively mitigate vortex shedding, achieving drag reductions of 13.4% and 20.1%, respectively. As the Ar decreases, more complex wake dynamics challenge flow control. For Ar=0.5 and 0.25, drag reductions of 26.88% and 30.37% are achieved, though vortex suppression becomes increasingly difficult. For extremely slender cylinders (Ar=0.1), we enhance control performance by enriching dynamic feature representation and using Kalman filtering for state estimation. Unlike conventional methods, our approach successfully achieves complete vortex shedding suppression, a maximum drag reduction of 32.2%, and eliminates periodic oscillations, stabilizing the flow. These results demonstrate the efficacy of state-enhanced RL in controlling highly slender geometries.

This paper introduces the Haar wavelet homotopy collocation method (HWHCM), a novel approach for numerical approximation of steady and unsteady problems governed by linear or nonlinear ordinary or partial differential equations. The proposed method combines the principles of homotopy analysis and the generalized Haar orthogonal wavelet, which can be considered as an improvement of the wavelet homotopy analysis method (WHAM) for the reduction of calculation complexity and the storage usage. Additionally, the HWHCM method enhances the traditional Haar wavelet collocation method (HWCM) by incorporating homotopy iterative techniques, resulting in improved convergence, stability, and computational accuracy. It offers flexibility in adjusting homotopy parameters and resolution levels, allowing for adaptive balance between accuracy and efficiency tailored to specific problem requirements. The effectiveness and accuracy of the HWHCM are evaluated using rigorous criteria such as relative variance and maximum error. Through its successful application to initial value problems, Poisson equation, Burgers equation, sine-Gordon equation and Schr & ouml;dinger equation, the numerical results support the significant advantages and validity of the HWHCM, confirming its superior accuracy.

This study explores the design of efficient heat and mass transfer systems in sinusoidal channels with corrugated walls, focusing on the influence of wall corrugation and nanofluids. The magnetohydrodynamic behavior of nanofluids is analyzed using the finite element method (FEM) under an angular magnetic field, buoyancy force, and the Buongiorno nanofluid model. Entropy generation analysis is employed to examine the interplay of these factors. A parabolic inlet velocity is considered, with the upper corrugated wall uniformly heated and the lower wall subjected to zero temperature and a prescribed nanoparticle concentration. Effects of magnetic fields, buoyancy forces, Brownian motion, and thermophoresis on heat and mass transfer are systematically studied. The FEM-based solution, utilizing P2P1 shape functions, solves governing equations for velocity, temperature, and nanoparticle concentration profiles. A comprehensive overview of wall corrugation effects exists in the graphical results showing its influence on flow dynamics, thermal behaviour, and species transport behaviour inside the sinusoidal channel. The streamlined shape and its geometry produce modifications in the fluid movement, creating more mixing patterns through recirculating areas and stagnant zones. An improved transport of heat and mass results from the geometric effects, which produce variations in local Nusselt and Sherwood numbers across the domain. The wall undulation mechanisms and nanoparticle movements create localized accumulation patterns and dispersion patterns resulting from thermophoretic effects and Brownian motion. This research establishes the fundamental importance of geometry modifications and nanofluid attributes in enhancing thermal and mass transport capabilities, which provides sound principles for designing improved heating systems, microfluidic devices and other beneficial industrial devices that reduce energy consumption.

This study enhances the performance of flow control across various synthetic jet configurations by improving deep reinforcement learning techniques. The training results based on the foundational deep reinforcement learning framework indicate that as the Reynolds number increases, the effectiveness of synthetic jet control becomes increasingly sensitive to the position of the jet. When synthetic jets are positioned near the flow separation region, the control strategy consistently exhibits excellent performance. However, when synthetic jets are located farther from the separation region, the flow control performance diminishes, and the consumption of external energy increases. By enhancing dynamic state features and reshaping the reward function, we significantly improve control performance across various Reynolds numbers and synthetic jet positions. With the optimized framework, we achieve significant drag reduction effects ranging from 8% to 34% within the Reynolds number range of 100–400. The flow control strategy is capable of simultaneously achieving multiple control objectives, including reducing drag, suppressing lift, eliminating vortex shedding, and decreasing energy consumption. These findings highlight the potential of optimizing deep reinforcement learning frameworks to achieve more adaptive flow control strategies for various flow scenarios.

Accurate prediction of ocean wave dynamics remains a critical challenge in coastal engineering and marine energy systems, particularly under extreme sea conditions where conventional models exhibit significant latency and error accumulation. We present a physics-informed hybrid framework, the variational mode decomposition (VMD)-transformer-gated recurrent unit (GRU) model, to address these limitations through synergistic signal decomposition and deep feature learning. By employing the VMD algorithm, nonlinear wave height signals are adaptively decomposed into intrinsic mode functions (IMFs) that isolate distinct frequency–amplitude characteristics, effectively disentangling complex wave interactions into interpretable physical components. These IMFs are subsequently processed through a dual-path architecture: The transformer module captures long-range temporal dependencies via multi-head attention mechanisms, while the GRU network models sequential nonlinearities through gated memory units. Validation across four Atlantic Ocean observation stations demonstrates remarkable forecasting precision, achieving mean absolute error (MAE) reductions of 37% (3 h lead time) and 43% (12 h lead time) compared to conventional long short-term memory/GRU baselines. The model performs particularly well in extreme wave scenarios, significantly reducing phase delays through enhanced trend detection. Quantitative analysis using root mean square error, MAE, and correlation coefficient metrics confirms improved spectral fidelity in reconstructed waveforms, with correlation coefficients exceeding 0.95 across all test cases. This work establishes a new paradigm for wave physics modeling by integrating variational signal processing with attention-based neural operators, offering transformative potential for real-time marine hazard prediction and offshore energy optimization.

This study investigates the heat and mass transport properties of a three-dimensional, laminar, incompressible, magnetized micropolar nanofluid flow between two parallel discs, where the lower disc is stationary and the top disc undergoes motion due to squeezing and an angled magnetic field. The analysis incorporates heat concentration phenomena, Soret and Dufour effects, and Arrhenius activation energy, while applying suction/injection to the lower disc and imposing velocity slip conditions on both discs. Extensive numerical computations are conducted to explore the influence of various physical parameters on the temperature and concentration profiles. Besides, artificial neural network (ANN) and gene expression programming (GEP) regression models are explored to accurately predict the skin-friction coefficient, heat and mass transfer rates at both discs, with the ANN model demonstrating superior accuracy compared to the GEP model. The ANN model demonstrated high correlation coefficients (R values) of 0.999996 and 0.999905 for heat transfer rate at upper and lower disks, respectively, which indicate a strong linear relationship between the predicted values and the actual values. It was also observed that the radial velocity curves decrease, while the microrotation curves exhibit a pattern of increasing and decreasing near the boundaries of the disks due to the enhanced values of the inclination angle. This study also offers a comprehensive analysis of diverse methodologies for determining activation energy and includes a thorough review of prior research on the Arrhenius equation. The findings have broad applicability in obtaining temperature-dependent metrics that are relevant to a range of engineering disciplines.