Numerical Simulation of Earthquake Impacts on Marine Structures: A Comprehensive Review

Abstract

Marine and underwater structures, such as seawalls, piers, breakwaters, and pipelines, are particularly susceptible to seismic events. These events can directly damage the structures or destabilize their supporting soil through phenomena like liquefaction. This review examines advanced numerical modeling approaches, including CFD, FEM, DEM, FVM, and BEM, to assess the impacts of earthquakes on these structures. These methods provide cost-effective and reliable simulations, demonstrating strong alignment with experimental and theoretical data. However, challenges persist in areas such as computational efficiency and algorithmic limitations. Key findings highlight the ability of these models to accurately simulate primary forces during seismic events and secondary effects, such as wave-induced loads. Nonetheless, discrepancies remain, particularly in capturing energy dissipation processes in existing models. Future advancements in computational capabilities and techniques, such as high-resolution DNS for wave–structure interactions and improved near-field seismoacoustic modeling show potential for enhancing simulation accuracy. Furthermore, integrating laboratory and field data into unified frameworks will significantly improve the precision and practicality of these models, offering robust tools for predicting earthquake and wave impacts on marine environments.

1. Introduction

Marine structures are subjected to a variety of dynamic excitations, such as earthquakes, wind forces, traffic loads on bridges, and tsunamis, among others [1]. Over the past decade, several numerical simulations designed specifically for marine structures have emerged, utilizing ocean models of varying complexity. These models often consider the effects of currents, waves, and wind, as well as processes that influence particle interactions during earthquakes, including fragmentation and degradation [2].

Renewable energy, noted for its abundant resources, sustainability, and environmental benefits, has attracted global attention. Earthquakes considerably affect the infrastructure supporting various renewable energy sources in maritime settings. For instance, the seabed composition in Northern Europe is predominantly sandy, while the offshore areas of China feature a mix of sand, silt, and clay—all susceptible to liquefaction. In these regions, earthquakes and liquefaction are crucial factors in ensuring the safety and stability of support structures for offshore wind turbines [3].

The seafloor composition of Japan’s seabed, particularly in regions impacted by the 2011 Tohoku earthquake, exhibits considerable variability influenced by location and the history of land reclamation [4]. In the Tokyo Bay area, the subsurface predominantly consists of fill materials, young alluvium, sandy silt, fine sand, and silty fine sand. These materials are highly susceptible to liquefaction during seismic events due to their low relative density and loose compaction. Additionally, the composition often includes dredged and recycled materials commonly used in land reclamation projects.

Variations in soil properties, such as specific gravity and N-values (standard penetration test counts), play a critical role in determining the extent and severity of liquefaction. However, in some regions, ground improvement measures, such as the installation of sand compaction piles, have significantly enhanced soil stability, effectively reducing liquefaction susceptibility. These interventions underscore the importance of localized soil assessments and mitigation strategies in earthquake-prone coastal areas.

The 2011 Tohoku Earthquake revealed the vulnerability of these structures when a wind turbine, mounted on a monopile foundation, tilted because of seabed liquefaction. Earthquake accelerations can have enduring adverse effects on the operational integrity of marine structures, occasionally leading to failures of wind turbines under extreme conditions. With increasing construction in seismically active zones, a comprehensive evaluation of these risks is imperative. Accurate estimation of earthquake-induced hydrodynamic loads, commonly known as seaquakes, is essential. These loads, resulting from the vertical propagation of seismic waves through compressible water, can considerably increase vibrations [4].

In the past two decades, considerable research has been conducted on the dynamic responses of quay walls. For instance, Sumer et al. [5] investigated the displacements of the breakwater at Eregli Fishery port caused by seismic waves during the 1999 Kocaeli earthquake in Turkey, using field-collected data.

Numerical methods based on finite-element analysis have been developed to evaluate the seismic performance of composite breakwaters and their porous seabed foundations under substantial seismic loads [6]. Furthermore, various studies have broadened the scope beyond earthquake loading to include tunnel behavior under various hydrodynamic pressures, such as blast loading and tunnel excavation. Sharma et al. [7] developed a coupled model to analyze wave interactions with a submerged floating sea tunnel in the presence of a bottom-mounted submerged porous breakwater. The study focused on understanding how the porous breakwater affects wave reflection, transmission, and the forces exerted on the tunnel. Three configurations of breakwater shapes—trapezoidal, circular, and wedge—were evaluated. The analysis employed the Sollitt and Cross model for fluid flow through the porous medium, using an eigenfunction expansion for a uniform seabed and a mild-slope equation for variable shapes. The results showed that the trapezoidal breakwater was most effective in dissipating wave energy, leading to reduced transmission and mitigating wave forces on the tunnel.

In scientific programming, priorities often extend beyond efficiency to consider critical aspects such as correctness, numerical stability, accurate discretization, and flexibility. However, efficiency remains a crucial constraint for current numerical methods and a primary consideration when developing advanced methods. The governing equations of the marine environment and their interactions with relevant structures are highly complex, necessitating approximations for accurate solutions. Although considerations of computational speed cannot override the meticulous design of optimized algorithms, there are inherent limitations to what can be achieved with sophisticated algorithmic strategies, both in terms of maximum speed and the complexity involved in implementing these algorithms. Enhanced computational speed enables more realistic simulations and analyses, offering the opportunity to investigate more intricate phenomena. In marine engineering, several applications continue to be constrained by numerical efficiency [8].

Structural motions are deduced by numerically integrating a simulation model. For bottom-fixed structures, this typically involves using a finite-element model to capture elastic deformations [9]. In contrast, floating structures are often modeled using a rigid multibody approach to efficiently compute global motions. In preliminary analyses, this may be simplified to a single rigid-body model. However, for an accurate assessment of dynamic loads, a more spatially extended model is essential.

This review paper aims to provide insights into the most suitable methods for simulating marine structures in response to seismic activities. We recommend using numerical models based on the boundary-element method (BEM) to analyze seismic responses. The paper is structured as follows: Section 2 outlines the theoretical background and models under review, and it explores the mechanisms of various simulation methods (finite-element method (FEM), computational fluid dynamics (CFD), discrete-element method (DEM), boundary element method (BEM), finite-volume method (FVM), direct numerical simulation (DNS), smoothed-particle hydrodynamics (SPH), lattice Boltzmann method (LBM)) as applied to recent research on marine structures. Then, Section 3 presents the discussion. Finally, Section 4 concludes with the main findings from this review.

2. Numerical Simulation Methods for Marine Structures

The FEM–SPH adaptive method combines the strengths of both FEM and SPH to efficiently perform impact dynamics calculations. Johnson et al. [10] developed and refined this algorithm, presenting a simplified version of the SPH method integrated into a standard Lagrangian code such as EPIC. Building on these insights, they developed an explicit two-dimensional (2D) Lagrangian algorithm that automatically transforms distorted elements into meshless particles during dynamic deformation. This approach proved effective in various scenarios and was later extended to three-dimensional (3D) contexts, resulting in a 3D explicit Lagrangian algorithm capable of converting distorted tetrahedral elements into meshless particles.

Despite their advantages, both FEM and SPH methods face significant challenges [11]. FEM is recognized for its high efficiency and accuracy but struggles with substantial mesh deformation during simulations, potentially leading to significant errors. Meanwhile, SPH handles large mesh deformations adaptively but suffers from tensile instability, complications in applying boundary conditions, and reduced computational efficiency.

The finite-difference method (FDM) solves governing equations by differentiating directly along each coordinate axis, facilitating rapid execution. In contrast, FEM discretizes the domain into elements of a chosen shape and combines them to form the complete system, typically resulting in slower performance compared to FDM. FDM is predominantly used in fluid mechanics and heat transfer, particularly for problems with fixed boundaries, but is less suitable for scenarios involving significant strain or deformation [12].

FEM is highly effective for solving large-deformation problems and offers versatility for complex geometries and material combinations, making it a robust choice for diverse engineering challenges. Similarly, CFD plays a crucial role in marine structure analysis, particularly for hydrodynamic simulations. However, challenges such as hydro-mechanical coupling, surface wave modeling, and extreme wave event simulation pose limitations to CFD’s effectiveness in certain applications [13].

For seismic response analysis, the Boundary Element Method (BEM) is recommended due to its inherent advantage of eliminating the need for artificial domain truncation, a key limitation in FEM and FDM when modeling infinite media. Artificial truncation in dynamic scenarios can lead to numerical inaccuracies caused by wave reflections, which are particularly problematic in seismic and acoustic analyses. While FEM and FDM use techniques like non-reflecting boundaries and infinite elements to address this issue, they are prone to challenges such as numerical wave dispersion, which require additional computational resources and careful tuning to manage effectively.

BEM, on the other hand, operates by reducing the problem dimensionality, focusing computations on the boundaries of the domain rather than its volume. This involves two primary stages [14]:

1.

Solving the boundary integral equation to determine displacements and stresses along the domain’s boundary.

2.

Conducting further computations for all points within the domain using an integral representation formula.

While BEM does face limitations, such as increased computational complexity for non-homogeneous or highly non-linear problems, these shortcomings do not significantly affect its effectiveness in seismic response modeling. The method’s capacity to handle infinite media without artificial truncation and its natural ability to manage boundary conditions make it particularly suitable for scenarios involving seismic wave propagation. Thus, despite its challenges, BEM provides a reliable and efficient approach to addressing seismic responses in marine structures.

2.1. FEM Applications in Marine Earthquake Engineering

The Finite Element Method (FEM) is a widely utilized numerical technique in marine earthquake engineering, renowned for its versatility and precision in modeling structural deformations and complex material behaviors under seismic loads. Its ability to handle diverse geometries, non-linear material properties, and multi-physics interactions makes it particularly well-suited for analyzing the dynamic responses of marine structures, such as seawalls, breakwaters, and buried pipelines, during seismic events. FEM is especially effective in capturing the effects of soil–structure interaction, wave propagation, and stress redistribution under earthquake conditions, offering valuable insights into structural performance and vulnerabilities. While FEM requires significant computational resources and careful calibration to mitigate numerical errors such as wave dispersion, its advantages often outweigh these challenges in practical applications. The following evaluation highlights FEM’s strengths in addressing seismic challenges in marine environments, providing a foundation for targeted case studies that demonstrate its application in this field.

The finite-element meshing process for a structure involves selecting nodal points where the final solution is required and assigning finite elements to these points. This task demands both expertise and an in-depth understanding of how structures with specific geometries respond to various external actions, including forces, imposed displacements, and temperature changes. The mesh design must consider factors such as geometry, loading methods, constraints, symmetry, proximity to infinite media, and stress concentrations. Depending on the structure’s geometry, elements may be one-dimensional, 2D, or 3D. Given the limited modeling capabilities of most finite-element analysis software, it is advisable to use specialized computer-aided design (CAD) programs for creating geometry. Meshing replaces the infinite degrees of freedom represented by the geometry with a finite network of elements to simulate the shape accurately. The type and application of loading influence the structure’s specific behavior, necessitating a customized meshing approach. Nodes should be placed at locations where external forces are applied, constraints are present, movements are required, and where there are changes in stiffness, characteristics, or material properties.

The density of the nodal network determines the size of the finite elements. Reducing element size and increasing node count generally improve calculation accuracy. However, when establishing the nodal network, it is crucial to consider an optimal balance between desired accuracy, computational efficiency, and analysis costs. Excessively increasing the number of nodes may not be beneficial, as studies suggest that there is a threshold beyond which additional elements do not enhance the solution and may even degrade it [15].

The ANSYS R14.5 simulation software is widely used across various engineering disciplines, including structural and fluid engineering. Numerical analysis with ANSYS involves creating a CAD geometric model, generating a mesh, applying geometric and mechanical boundary conditions (loads), obtaining the solution, and interpreting the results. The finite element method (FEM) in ANSYS comprises several key steps: meshing the structure into finite elements, selecting shape functions, formulating finite-element equations and their specific matrices, assembling the finite elements while applying boundary conditions, solving the system of equations by calculating loads and displacements, and finally determining the stress and strain states within each finite element. [16].

The study by Ion and Ticu [17] presents a modal analysis of a ship’s hull using a finite-element model tailored to specific study parameters. Given that the vessel is positioned within a continuous medium, no additional boundary conditions are necessary for this free–free type of modal analysis. Typically, the first six vibration modes correspond to rigid-body motions, distinguished by their low frequencies. Although the frequencies for rigid-body motions are theoretically zero, solver inaccuracies can yield non-zero frequency values, sometimes reaching the MHz range. To assess the impact of various stiffener configurations on frequencies and vibration modes, a prior study evaluated three hull construction variations: without stiffeners, with transverse stiffeners, and with both transverse and longitudinal stiffeners. Frequencies and associated modes were obtained for each configuration. Although modes with frequencies above 5 Hz are generally not critical for wave action analysis, sound engineering practices recommend extracting between 100 and 200 modes to account for the diverse range of excitation sources encountered during a ship’s operation [18]. Figure 1 illustrates the mesh preparation and finite element model of the hull with transverse stiffeners, along with the results showing the vibration mode of the transversely stiffened hull at a frequency of 11.209 Hz.

The modal analyses of these structures yield qualitative rather than strictly quantitative data, with visual representations of modal shapes illustrating deformation modes at specific resonance frequencies rather than actual displacements [19]. A key challenge in this research is accurately estimating external forces from ocean waves and the resulting structural stresses on vessels. Addressing this requires a multidisciplinary approach, integrating probability theory, random processes, extreme value statistics, empirical data, naval hydrodynamics, and numerical modeling. Estimating wave characteristics involves more than wave height and frequency; it requires deriving spectral density functions from empirical data to capture a comprehensive view of the sea surface. This approach is essential for assessing vessel behavior in real sea conditions, emphasizing the need to understand wave spectra across different scenarios and regions for accurate structural analysis and extreme value prediction.

A previous study performed static analyses on three structural models of a container port hull: one without stiffeners, one with transverse stiffeners, and one with both transverse and longitudinal stiffeners. These analyses, conducted under various calm water pressure distributions, evaluated the impact of stiffeners and loading methods on hull stress. The application of FEM has introduced advanced approaches for tackling complex structural analysis, enabling more precise resistance calculations and accurate assessment of structural failure criteria. Additionally, FEM supports the iterative optimization of structural dimensions to meet the required standards [20].

2.2. CFD-DEM Simulation of Submarine Landslide

The Discrete Element Method (DEM) is a powerful numerical technique for simulating the behavior of granular materials, making it particularly well-suited for modeling submarine landslides where individual soil particles and their interactions play a critical role. When coupled with Computational Fluid Dynamics (CFD), DEM provides a comprehensive framework for capturing the complex interplay between granular soil dynamics and fluid flow during seismic events. This approach allows for detailed analysis of particle-level mechanisms, such as contact forces and energy dissipation, while also considering the effects of surrounding water pressure and hydrodynamic forces. Despite challenges such as high computational demands and the need for parameter calibration, CFD-DEM simulations have demonstrated remarkable accuracy in reproducing observed landslide behaviors and flow patterns. The following evaluation underscores the effectiveness of DEM in addressing granular soil interactions during submarine landslides, emphasizing its potential to advance our understanding of these processes and improve risk mitigation strategies.

A study simulated a submarine landslide triggered by seismic activity in a region rich in methane hydrates using a coupled computational approach that integrates CFD with DEM [21]. To enhance the accuracy of the simulation for dynamic scenarios, the researchers incorporated dynamic features and considered the Magnus force. They subjected a steep underwater slope containing methane hydrate-bearing sediments, which formed robust inter-layers, to a sinusoidal seismic load. The simulation results indicated a flow-type landslide, leading to a gradual accumulation of debris on a gentler slope. Near the sliding mass, the fluid exhibited an eddy pattern. The findings revealed that the presence of methane hydrate increased sediment strength while reducing its damping properties. At lower methane hydrate saturations (25% and 30%), the combined effects of seismic loading and particle–fluid interactions led to damage in the methane hydrate-rich layer, resulting in settling behind the slope’s crest and upheaval in front of its toe. However, at higher methane hydrate saturations (40% and 50%), the sediment exhibited sufficient strength to resist seismic damage. Increased methane hydrate saturation resulted in reduced sediment damping, which facilitated increased energy transfer from the ground base to the potential sliding mass, initiating the sliding process earlier. The implications of these simulation results for assessing submarine hazards induced by earthquakes were discussed.

Zhang and Luan [22] utilized a continuum–discrete approach known as the CFD–DEM coupling scheme. This technique simulates fluid flow by solving the Navier–Stokes equations within CFD, using coarse-grid local averaging. Concurrently, the motion of individual particles is modeled using Newton’s second law of motion in DEM. The interaction between CFD and DEM is facilitated through the exchange of forces during particle–fluid interactions. The study provided a detailed understanding of this computational scheme. To model submarine landslides triggered by seismic loading in this study, they implemented two significant enhancements to the coupling computation: they expanded the contact model in DEM to include dynamic characteristics and incorporated the Magnus force into fluid–particle interactions. These improvements were briefly outlined in their paper, with more detailed descriptions to be presented in a forthcoming publication. Notably, the DEM simulations followed a 2D approach using disk-shaped particles, while the CFD formulations were primarily 3D; however, the flow velocity in the out-of-plane direction was constrained to zero, effectively treating it as 2D flow. For the purposes of CFD–DEM coupling, the simulation treated these 2D disks as spheres with equivalent diameter and velocity.

A theoretical arrangement of 3D spheres is commonly used to assess porosity and particle volume. This is crucial for calculating interaction forces between particles and the surrounding fluid. These forces are then incorporated into CFD and DEM models to directly impact fluid and particle movement calculations. This approach is necessary since the methane hydrate bond contact model currently exists only in 2D, with a 3D version still in development. While this method may not fully replicate physical reality, it effectively captures general trends in fluid–particle interactions.

In a prior study, to simulate mass flow following slope failure, where finite-element simulations are inadequate, a combination of CFD and DEM (CFD–DEM) was utilized [21]. This integrated approach effectively simulated the initiation, mass transport, and deposition phases of submarine landslides resulting from solid–fluid interactions, as shown in Figure 2 and Figure 3. The study simulated submarine landslides triggered by methane hydrate dissociation, replicating four distinct types of slides—fall, flow, slump–flow, and slump—each varying by the location and extent of methane hydrate dissociation. Given the advantages of the CFD–DEM simulation method in accurately reproducing the entire sliding process, this coupling approach was employed to model seismic loading-induced submarine landslides in methane hydrate-rich regions [23].

The study developed by Jiang et al. [21] simulated a submarine landslide by integrating CFD with DEM. A dynamic contact model was implemented within the discrete-element framework to accurately capture the behavior of sediment containing methane hydrates. The simulations also considered minor fluid compressibility within the CFD component. Additionally, the Magnus force was included in the fluid–particle interactions. The study identified that this force considerably influenced particle trajectories within the typical velocity ranges observed during submarine landslides. Sinusoidal seismic loading was applied to trigger failure in the simulated submarine slope, accounting for varying methane hydrate saturations. The results indicated a flow-type sliding pattern, culminating in a gradual accumulation of debris on a gentler slope. Near the sliding mass, the fluid exhibited an eddy pattern: water flowed upward from the slope’s toe to its crest above the surface while moving in the opposite direction below it. The restoration of slope stability was marked by a circular fluid eddy, approximately the height of the slope, moving away from it [24].

2.3. Marine Turbine Hydrodynamics by a Boundary Element Method

The Boundary Element Method (BEM) is a highly efficient numerical approach for solving problems involving infinite or semi-infinite domains, making it particularly suitable for analyzing hydrodynamic interactions in marine turbine systems. Its ability to focus computations on the boundaries of a domain, rather than the entire volume, significantly reduces the computational complexity while maintaining accuracy in capturing wave scattering, flow dynamics, and pressure distributions. BEM is especially effective for modeling the interaction of marine turbines with incident waves, as it naturally accounts for the propagation of hydrodynamic forces without introducing artificial domain truncation, which is a common limitation in other numerical methods like FEM and FVM. Although BEM has limitations in handling non-homogeneous and highly non-linear problems, its application to marine turbine hydrodynamics has demonstrated consistent agreement with theoretical and experimental results, highlighting its potential for improving design and performance analysis in these systems. This evaluation outlines BEM’s strengths and limitations, providing a foundation for the targeted case studies discussed in this section.

This section presents a computational approach for analyzing the hydrodynamics of horizontal-axis marine current turbines using the Integral Boundary Element Method (IBEM), initially developed for marine propellers. Adapted to address the specific flow characteristics of hydrokinetic turbines, this method incorporates semi-analytical trailing wake models and adjustments for viscous flow effects. Previous studies have validated its accuracy by comparing hydrodynamic performance predictions with experimental cases and other numerical models from the literature. This method’s capacity to predict turbine thrust and power across a range of operating conditions has been assessed, factoring in viscous effects from blade flow separation and stall. The results for thrust and power align closely with those from standard blade-element methods in marine turbine design, as illustrated in Figure 3, although accuracy may decrease when blades operate under off-design conditions [25].

Hydrokinetic turbines, developed to harness energy from tidal and ocean currents, are advancing quickly in the field of renewable energy. Large-scale systems often utilize horizontal-axis turbines, either fixed to the seabed or positioned on floating platforms. This rapid progress, in contrast to other ocean energy technologies, is largely due to knowledge transfer from wind energy. While marine turbine blades share design principles with wind turbine blades, they feature a lower aspect ratio, which enhances their durability against the stronger hydrodynamic forces in water. Consequently, blade-element momentum methods—originally formulated for wind turbines—are widely applied in the design and analysis of turbines for tidal and ocean currents [26].

The Tip Speed Ratio (TSR) is a key parameter in marine current turbine performance, defined as the ratio of the blade tip speed to the free-stream water velocity [27]:

$$\mathrm{TSR}={\displaystyle \frac{\Omega R}{V}}$$

(1)

where $\Omega$ is the angular velocity of the turbine, R is the turbine radius, and V is the free-stream water velocity. TSR is critical for understanding the relationship between the rotational speed of the turbine and the flow speed of the water, directly influencing turbine efficiency. It is used to evaluate thrust, torque, and power coefficients, which are essential metrics for optimizing turbine performance. Generally, higher TSR values result in reduced drag and improved efficiency for horizontal-axis turbines, as blade flow remains attached at moderate to high TSRs, enhancing power generation. On the other hand, low TSR values can lead to flow separation and stall, adversely affecting thrust and power output.

When properly calibrated, the blade-element momentum (BEM) method provides a fast and reliable approach for evaluating turbine performance, though certain limitations require careful consideration. The BEM method calculates blade loading based on lift and drag data from 2D airfoil profiles. However, to account for 3D effects such as blade tip interactions and hub dynamics, semi-empirical corrections are necessary to enhance accuracy and capture these complex phenomena effectively. Conversely, marine propeller design often employs boundary-element or panel methods (referred to as IBEM), which assume inviscid flow and consistently model 3D flow around rotors in both steady and unsteady conditions. Although IBEM use in hydrokinetic turbine design is limited, a few studies highlight the challenges that blade-element momentum faces in accurately capturing these turbines’ performance, particularly under high angles of attack where viscosity-induced separation and stall reduce thrust and power. To address this, Baltazar and Falcão de Campos [28] proposed an approach that adjusts the inviscid IBEM predictions for 3D steady flows by modifying the lift force for each blade section based on comparisons of 2D viscous and inviscid lift coefficients. Drag for each blade section was derived from 2D polar curves generated with XFOIL, a viscous-flow simulation tool, using the Kutta–Joukowski law to determine the local incoming velocity at the blade sections. Their approach also incorporated an iterative model to align wake pitch with local flow conditions, though it did not include radial expansion effects.

Figure 3. Marine current turbine. Wake geometry of IBEM model at different operating conditions. From left to right, TSR = 3, 6, 9. The diameter of the turbine was 700 mm [27].

Figure 3. Marine current turbine. Wake geometry of IBEM model at different operating conditions. From left to right, TSR = 3, 6, 9. The diameter of the turbine was 700 mm [27].

A computational method was developed to assess the hydrodynamics of horizontal-axis marine current turbines, using an IBEM designed for inviscid flow along with a tailored trailing wake model for hydrokinetic turbines. To address blade flow separation and stall effects, a viscous-flow correction (VFC) model was added. The VFC applied a semi-empirical adjustment to the IBEM’s inviscid-flow calculations by modifying blade loads based on the lift and drag characteristics of 2D profiles representing the blade sections in 3D flow conditions. Validation of the IBEM–VFC approach was performed by comparing the numerical predictions with experimental data and other computational results from the literature.

The analysis demonstrates that the proposed methodology effectively characterizes turbine performance across varied operational conditions, providing accurate predictions for thrust, torque, and power, especially at medium to high tip-speed ratios where blade flow is mostly attached. It also performs well under lower tip-speed ratios, where blade flow separation and stall contribute to thrust loss and increased drag. This method achieves high accuracy when blade pitch settings match design parameters, although deviations occur under off-design conditions, particularly in predicting thrust and torque. Compared to other computational models, this IBEM-based approach delivers accuracy comparable to commonly used blade-element methods, with the added advantage of a physically consistent 3D flow representation around the turbine across different flow conditions. Unlike blade-element methods—which often require adjustments for tip effects, hub interactions, and blade count—the IBEM approach provides a robust solution for nonuniform flow conditions and turbine cavitation analysis.

Future research could enhance the VFC scheme to align trailing vorticity distributions and induced velocity profiles more accurately with viscosity corrections applied to blade loads. Further validation studies are recommended to evaluate the generalized IBEM–VFC model’s effectiveness in predicting turbine performance at low tip-speed ratios and in off-design operational scenarios [29].

2.4. Analysis of Breaking Wave Forces on a Monopile Structure Using Finite Volume and Finite Difference Numerical Models

The Finite Volume Method (FVM) is a robust numerical approach for solving conservation laws, making it particularly effective in hydrodynamic applications such as wave energy propagation and breaking wave forces on marine structures. Its ability to discretize complex domains and ensure the conservation of mass, momentum, and energy across control volumes provides accurate and reliable solutions for fluid–structure interactions. Paired with the Finite Difference Method (FDM), which excels in efficiently solving partial differential equations, this combination allows for detailed modeling of dynamic wave behaviors and their impact on monopile structures. Despite challenges such as computational intensity and grid refinement requirements, FVM has demonstrated consistent success in simulating realistic wave patterns and force distributions, aligning closely with experimental results. This evaluation highlights the strengths and limitations of these methods, providing a foundation for the subsequent case study on their application to breaking wave forces on monopile structures.

The nonlinear forces from breaking waves present a major challenge in offshore structure design, as the complexities of wave-breaking interactions remain difficult to fully capture. Numerical models are essential tools in studying these interactions, often employing either the Finite Difference Method (FDM) or Finite Volume Method (FVM) to solve the governing equations for wave-breaking phenomena. While both methods have produced reliable results in various studies, their comparative strengths and limitations have not been extensively analyzed.

Numerical models using the Navier–Stokes equations commonly apply FDM or FVM to simulate breaking waves. FDM approximates differential equations through Taylor series expansions on a grid, while FVM uses the integral form of the equations to ensure conservation within finite volumes [30]. The accuracy of each approach depends on the specific discretization used for the governing equations.

Jose et al. [31] conducted a comparative study using two 3D Navier–Stokes solvers, 2PM3D (FDM) and OpenFOAM (FVM), to examine breaking-wave forces on a monopile structure. The study simulated nonbreaking and breaking wave scenarios, comparing results with theoretical predictions and experimental data. Both models closely matched experimental measurements, effectively capturing the interactions between breaking waves and the monopile.

Offshore structures in marine environments face complex nonlinear interactions, such as wave breaking and green water impacts, which can lead to structural damage. Understanding these forces is crucial for designing resilient offshore structures, particularly in shallow waters where breaking wave forces often drive design requirements [32].

The challenge in accurately capturing the interaction between breaking waves and structures lies in the dynamic and complex nature of wave-breaking phenomena. Traditional research has relied on experimental data, but these tests are often restricted to simpler structures and specific conditions. To address these limitations, validated numerical models have become valuable tools for simulating breaking waves and predicting forces. Advances in computational resources and refined numerical codes now make it possible to model wave forces through direct pressure integration over structures, thus reducing the reliance on empirical data. The performance of these models, however, depends significantly on the choice of numerical methods.

Prior studies have validated numerical simulation results using experimental data from Irschik et al. [33]. These simulations employed a numerical wave tank specifically designed to replicate the experimental conditions. The layout of this wave tank, shown in Figure 4, spans a total length of 54 m, excluding the inlet and outlet relaxation zones.

The modeling approaches vary across numerical models. For example, 2PM3D uses a pre-processing application, Geometry 3D, to compute the fractional volumes and areas for fluid flow in the x, y, and z directions, along with normal vectors for each cell in the domain. Figure 5a shows the vertical cylinder and bottom geometry created by Geometry 3D, while Figure 5b displays the cell-edge intersection with the cylinder and the derived parameters for that cell. To reduce wave reflections, numerical dissipation zones extend 2L (where L is the wavelength) at the inlet and outlet boundaries, with the inlet zone mitigating reflections near the wave generation area and the outlet zone preventing them at the boundary. An internal wave generator, positioned at the inlet dissipation zone, produces regular waves. OpenFOAM, using an unstructured mesh and FVM, solves the governing equations flexibly and allows for direct CAD model imports. STL files of the cylinder and slope were generated in AutoCAD and integrated into the domain using the snappyHexMesh tool [34].

Two computational models were employed to simulate breaking waves in a numerical wave tank: 2PM3D, which uses FDM, and OpenFOAM combined with the waves2Foam toolbox, based on FVM. Although both models solve the same governing equations, they handle these equations and the computational geometry differently. To investigate breaking wave effects on a monopile, both models simulated nonbreaking and breaking wave scenarios and were validated against theoretical predictions and experimental data. Results showed strong agreement with experimental and theoretical data, effectively capturing the nonlinear wave characteristics.

However, both models tended to overestimate peak breaking wave forces on the monopile by approximately 7% compared to experimental data, likely due to the incompressible flow models, which did not account for air bubbles in breaking waves. Additionally, the FDM model required more computational cells than the FVM model. Both models calculated secondary load effects accurately but showed a slight delay in comparison to experimental results, possibly due to reduced energy dissipation in the incompressible flow model [35]. Further research is recommended to refine the numerical estimation of secondary load cycles. Figure 6 displays waves2Foam simulation results at four time steps.

2.5. DNS of Wind Turbulence over Breaking Waves

Direct Numerical Simulation (DNS) is a powerful computational technique for resolving fluid dynamics problems at the most detailed level, capturing all relevant turbulence scales without relying on turbulence models. This makes it particularly suited for studying wind turbulence over breaking waves, where the interactions between air and water involve complex, multi-scale processes. DNS provides unprecedented accuracy in simulating these phenomena, enabling insights into energy transfer, turbulence generation, and wave-induced flow structures. However, its high computational demands and limited applicability to large domains are significant challenges. Despite these limitations, DNS has proven effective in replicating realistic turbulence patterns and enhancing the understanding of wave-atmosphere interactions. This evaluation underscores DNS’s strengths and challenges, setting the stage for its application to wind turbulence studies over breaking waves in this section.

Yang et al. [36] studied wind turbulence over breaking waves using direct numerical simulation (DNS) for two-fluid flows, treating air and water as a single system and capturing the interface with a combined level-set and volume-of-fluid method. To obtain turbulence statistics, the study employed ensemble averaging across 100 simulations, focusing on the effects of wave age and steepness on airflow turbulence over breaking waves. Results showed that wave age significantly influences turbulence statistics before wave breaking, with the vertical gradient of mean streamwise velocity transitioning from positive at low wave ages to negative near the wave surface at higher wave ages, indicating a shift from drag- to thrust-dominated conditions due to changing pressure forces. As wave age increases, wave-coherent motions contribute more to momentum and kinetic energy flux (KE-F).

During wave breaking, spilling breakers have minimal impact on the wind field, while plunging breakers create substantial changes in wind turbulence structure near the wave surface. DNS results indicated that during plunging, a high-pressure region ahead of the wavefront accelerates downstream wind and generates a large spanwise vortex, leading to increased Reynolds stress and turbulent kinetic energy (TKE) below the wave crest. KE-F magnitude above the crest rises with plunging waves at small and large wave ages but not at intermediate wave ages. An analysis of KE-F production showed that, at small wave ages, KE-F enhancement is driven by a local maximum in momentum flux, while at large wave ages, it results from a shift in KE-F production due to changes in wave-coherent momentum flux. Intermediate wave ages show no such enhancement, resulting in stable KE-F levels [37].

Wind and ocean wave interactions are essential in determining sea state, influencing the marine atmospheric boundary layer, and impacting upper-ocean dynamics. Strong wind forcing frequently leads to wave breaking, a key factor in air–sea interactions [36]. Breaking waves limit surface wave height while generating currents, vorticity, and turbulence, which in turn, promote the exchange of mass, momentum, and energy between the ocean and atmosphere. Gaining a detailed understanding of wind behavior over breaking waves is crucial for improving models of ocean–atmosphere interactions. Figure 7 shows an example of the computational domain and the coordinate system used in the simulations, where x, y, and z (or $x_1$, $x_2$, and $x_3$) correspond to the streamwise, spanwise, and vertical directions, respectively. The associated velocity components are denoted by u, v, and w (or $u_1$, $u_2$, and $u_3$).

In the aforementioned case, air and water were modeled as a coherent system, where density and viscosity vary according to the fluid phase. The behavior of these fluids is governed by continuity and momentum equations [36]:

$${\displaystyle \frac{\partial u}{\partial x_i}}=0$$

(2)

$$\begin{array}{ccc}\rho \left(\varphi \right)({\displaystyle \frac{\partial u_i}{\partial t}}+u_j{\displaystyle \frac{\partial u_i}{\partial x_j}})=& & \\ -{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial \left(2\mu \left(\varphi \right)S_{ij}\right)}{\partial x_j}}-\rho \left(\varphi \right)g{\delta }_{i3}+\gamma \kappa {\delta }_x\left(\varphi \right)n_i\end{array}$$

(3)

Here, t represents time; $\rho$ and $\mu$ denote the fluid’s density and dynamic viscosity, respectively; $\varphi$ is the level-set function defined as the signed distance from the air–water interface, with positive values in water and negative in air; p represents pressure; g is gravitational acceleration; ${\delta }_{ij}$ is the Kronecker delta; $\gamma$ denotes the surface tension; $\kappa =\nabla ·\left({\displaystyle \frac{\nabla \varphi }{|\nabla \varphi |}}\right)$ at $\varphi =0$ is the curvature of the air–water interface; ${\delta }_s\left(\varphi \right)$ is the Dirac delta function at the interface; $\mathbf{n}={\displaystyle \frac{\nabla \varphi }{|\nabla \varphi |}}$ at $\varphi =0$ is the normal vector at the interface, pointing from water toward air. Periodic boundary conditions are applied in the x and y directions, and a no-slip boundary condition is imposed at the bottom of the water domain. At the top of the air domain, the wind is driven by a constant shear stress ${\tau }_{\mathrm{top}}$ in the streamwise direction.

This DNS study examined three wave ages and two initial wave steepness levels, with 100 simulations per scenario for reliable ensemble averaging. Each simulation maintained consistent initial wind profiles and wave setups, varying only turbulent fluctuations. Results showed that at earlier wave ages, wave-induced motion minimally impacts total momentum flux and kinetic energy, with wind turbulence primarily governing airflow. However, as wave age increases, wave-induced effects become more influential. While spilling waves minimally affect wind turbulence, plunging waves substantially alter turbulence and flow structures near the surface [36].

During wave plunging, the interaction between the overturning jet and air accelerates streamwise airflow, creating a high-pressure region near the jet and facilitating momentum transfer from water to air—distinct from the drag-dominated conditions seen with steep pre-breaking waves. This breaking process enhances wind speed near the wave surface and generates a counterclockwise spanwise vortex, which expands with wave age, intensifying Reynolds shear stress and TKE around it.

At early wave ages, turbulence above the crest temporarily increases during initial plunging, linked to a peak in momentum flux from wave disturbances. With further aging, KE-F above the crest continues to rise during plunging due to a shift in KE-F production from negative to positive, driven by changes in wave-coherent momentum flux. At intermediate wave ages, these processes do not occur, and no transient KE-F growth is observed.

These DNS findings offer valuable insights into the complex physics of wind interactions over breaking waves. As computational power improves, high-resolution simulations of increasingly intricate scenarios, such as wind-influenced wave breaking via wave-focusing methods, are possible. Such simulations would require larger computational domains in the streamwise direction, and integrating laboratory and field data could enhance simulation realism [38].

2.6. SPH Simulations of Real Sea Waves Impacting a Large-Scale Structure

The Smoothed Particle Hydrodynamics (SPH) method is a mesh-free computational approach renowned for its ability to accurately simulate free-surface flows and highly deformable interfaces. This makes SPH particularly effective for analyzing complex wave breaking and fluid–structure interactions, such as those encountered in real sea conditions impacting large-scale marine structures. By representing fluids as discrete particles, SPH can naturally handle large deformations and complex boundary geometries, offering a significant advantage over traditional grid-based methods. Despite challenges such as computational cost and sensitivity to particle resolution, SPH has demonstrated strong agreement with experimental data, making it a reliable tool for modeling wave dynamics and assessing structural loads. This section evaluates SPH’s application to large-scale wave impact scenarios, emphasizing its ability to provide detailed insights into free-surface behavior and force distributions on marine structures.

The Pont del Petroli, a decommissioned pier located in Badalona, Spain, holds significant historical and social significance. This structure suffered extensive damage during Storm Gloria which struck southeastern Spain in January 2020 [39]. In preparation for the pier’s reconstruction, a detailed evaluation of the wave-induced forces that led to its structural failure is imperative. To support an upcoming experimental campaign concerning this effort, the authors utilized an advanced CFD code. The simulation used the SPH method, specifically configured to replicate the conditions during Storm Gloria. Considering the substantial computational demands of a full 3D simulation, inlet boundary conditions were implemented to generate waves proximate to the pier structure. The numerical analysis revealed that the forces exerted on the pier during the storm surpassed its designed load capacity, accounting for both its self-weight and accidental loads. The findings suggest that a critical factor in the pier’s failure was the inadequacy of lateral soil resistance. This study represents an innovative application of SPH open-boundary conditions in modeling real-world engineering challenges.

CFD has long been employed in studying fluid–structure interaction (FSI) and wave–structure interaction in both onshore and offshore structural analysis. Initially, potential flow theory dominated, using a velocity potential to satisfy simplified governing equations, typically the Euler equations, under assumptions such as irrotational, inviscid flow, and small displacements. However, for cases with severe wave breaking or extreme wave loads, the Navier–Stokes equations became standard, requiring detailed treatment of viscous effects and the complex air–water interface. These equations are typically discretized on 2D or 3D grids through finite volume (FVM) or finite element methods (FEM). The development of high-performance computing, particularly with CPUs and GPUs, has greatly improved the efficiency and practicality of these approaches, making detailed real-world analyses increasingly feasible.

The water depth and beach profile at the Pont del Petroli site have changed considerably since the pier’s construction. Initially designed for a maximum depth of 12 m at the platform base, both water depth and seabed slope have since shifted. Between 2011 and 2020, The Maritime Engineering Laboratory of the Universitat Politècnica de Catalunya·BarcelonaTech (LIM/UPC) conducted 19 bathymetric surveys, the latest immediately following Storm Gloria. Figure 8 shows these changes, with the original design profile in blue and profiles before and after Storm Gloria in red. The x-axis measures from the landward side, starting at the footbridge’s base, with the platform extending from x = 216 m to x = 240 m. The vertical axis represents the distance from the mean water level. The pier’s rear section near the shore showed minimal slope change, maintaining an average slope of 1:30 to 1:25. However, the front section, from the fourth pile cap to the platform, experienced significant alterations in water depth due to sand accumulation, reducing depth from the original 12 m to nearly 9 m. After Storm Gloria in early 2020, a marked change was observed, with water depth at the platform’s toe reduced to 8 m, 4 m less than initially specified [40].

The open-source software DualSPHysics, based on Smoothed Particle Hydrodynamics (SPH), was used to simulate wave interactions with the Pont del Petroli pier in Badalona, Spain, which sustained extensive damage during Storm Gloria in January 2020. The model aimed to (a) characterize wave loads on the pier in preparation for an experimental campaign in LIM/UPC’s large-scale wave flume, and (b) offer initial insights into the primary failure mechanisms observed during the storm. To approximate the storm conditions, a wave propagation study using the SWAN model was conducted, given the lack of specific local wave data for the storm in Badalona. Visual estimates suggested wave heights of 7–8 m, surpassing the pier platform and footbridge heights. Post-storm bathymetric surveys revealed sand accumulation under the pier, reducing water depth by an average of 1–2 m.

The analysis began with a 2D simulation under various wave conditions and depths, testing wave heights from 6.1 to 9.0 m and wave periods between 9.6 and 12.7 s. Initial water depths at the pier’s toe ranged from 8 to 10 m. Results from the 2D model, capturing load patterns and wave-breaking behaviors, were compared with visual observations from Storm Gloria. Based on these findings, two wave conditions were selected for detailed 3D simulations, with wave heights of 6.5 and 8.0 m and wave periods of 12.0 and 12.7 s, respectively, and a water depth of 8 m at the pier’s toe, reflecting post-Gloria survey data.

The 2D model replicated the layout of LIM/UPC’s wave flume with a wedge-type wavemaker, while the 3D model used an inlet boundary condition to optimize resources by positioning the inlet near the pier and wave-breaking point. Data on free-surface elevation and velocity from the 2D model were applied to the 3D model’s inlet. Horizontal and vertical forces were measured on the pier platform, the initial pile cap, and the first seaward $\pi$-shaped beam of the footbridge. Standard formulas for wave loads on jetties were not directly applicable, as they were developed for jetties on flat seabeds [41].

In this study, the numerical model did not directly simulate the piles; instead, wave forces on the piles were calculated using Morison’s formula, which accounts for slamming loads from breaking waves. The forces measured on the platform and footbridge beam were compared to design loads that included self-weight and accidental loads. Vertical forces generally matched or fell below design specifications, but horizontal forces—especially under extreme conditions (wave height H = 8.0 m, wave period T = 12.7 s)—exceeded expectations.

These findings helped define requirements for load cells and pressure sensors for an experimental campaign at LIM/UPC, focusing on factors such as nominal force/pressure, breaking load, sensitivity, accuracy, and range. SPH simulation snapshots showed that for high waves, breaking began before reaching the platform, causing the plunging wave to transfer significant momentum to the structure. Some energy also dissipated through overtopping, exerting vertical forces. In contrast, smaller waves (H = 6 m, T = 12 s) approached without breaking, resulting in lower platform loads but greater impacts on the pile cap.

The forces on the pile cap were used to calculate the axial load transmitted to the supporting piles. Horizontal forces on the cap were then compared to the soil’s lateral resistance, factoring in embedded pile length and eccentricity. The analysis indicated that horizontal forces surpassed lateral soil resistance, though axial loads remained within acceptable limits.

These SPH simulation findings support the Badalona City Council’s damage report, suggesting that Storm Gloria’s extensive damage resulted from lateral soil resistance being exceeded. Forces on the piles and cap surpassed this resistance, likely leading to the system’s failure and causing the footbridge beam to lose support and fall into the sea [42].

2.7. 3D Modeling of Fluid–Structure Interaction with External Flow Under Earthquake Using Coupled LBM and FEM

The Lattice Boltzmann Method (LBM) is a computational approach known for its efficiency in modeling fluid flows with complex boundary conditions and multi-phase interactions, making it particularly effective for capturing turbulence and wave dynamics in marine environments. When coupled with the Finite Element Method (FEM), which excels at simulating structural deformations and material responses, the combination provides a powerful framework for analyzing fluid–structure interactions under seismic events. This coupled approach enables detailed simulations of external flows interacting with marine structures, accounting for the dynamic effects of both hydrodynamic forces and seismic excitations. While LBM is computationally intensive for large-scale problems, its ability to handle intricate flow phenomena complements FEM’s structural modeling capabilities. This section evaluates the strengths and challenges of the coupled LBM-FEM method, highlighting its application to 3D modeling of fluid–structure interaction in earthquake scenarios.

Kwon and Jo [43] employed a 3D fluid–structure interaction (FSI) approach that combined the lattice Boltzmann method (LBM) for fluid dynamics with FEM for structural behavior. To improve computational efficiency in simulating external flow around embedded pipes, they represented the pipes with 3D beam elements instead of shell elements. They introduced an algorithm that couples these 3D beam elements with lattice Boltzmann grids, enabling precise FSI modeling at the pipes’ outer surfaces. This method was applied to several numerical examples, allowing for an in-depth analysis of FSI characteristics, especially relevant in complex environments like power plants and heat exchangers. Traditionally, structural analysis has used FEM, while flow analysis relied on FVM, leading to the development of coupling methods to integrate FEM and FVM for FSI problems. Additionally, FEM has been applied to multiphysics scenarios, including FSIs, and has been combined with BEM to analyze interactions involving structures and shock waves [44].

The lattice Boltzmann method (LBM) is a relatively new approach, compared to established methods like FVM, FEM, and BEM, primarily developed for fluid flow problems since the 1980s. It has demonstrated efficiency in applications involving multiphase, turbulent, and thermal flows, and has also been applied to FSI scenarios, such as flow around rigid structures in artificial heart valve designs. While LBM has been coupled with FDM or FEM to analyze FSIs with flexible structures, most prior work focused on 2D applications, limiting its effectiveness for 3D flow interactions around pipes [45].

LBM evolved from cellular automata principles and incorporated concepts from the Boltzmann equation to overcome cellular automata’s Boolean limitation by using real-valued quantities [46]. This advancement made LBM particularly suited for viscous flow scenarios without accounting for fluid grid deformation. For modeling FSI in flexible pipes, structural FEM typically uses either shell or beam elements. Shell elements capture local deformations and vibrations but require high computational resources, especially for multiple pipes. Beam elements, though less detailed, offer computational efficiency by representing general pipe deformations when local effects are less significant [47,48].

In practical applications, thick-walled pipes generally render local deformations negligible, with overall structural behavior being the primary focus. To address this, a previous study proposed an algorithm linking lattice Boltzmann grid points with 3D beam elements for pipe modeling, assuming linear elasticity with small displacements and disregarding fluid grid deformation [49].

Figure 9 presents a cross-sectional view of a pipe within the surrounding fluid grid points. Since 3D beam elements are represented as lines in finite-element analysis, it is essential to accurately model the pipes’ outer surfaces to enable FSI at these points. For this purpose, the outer surface of each 3D beam element is identified, and the fluid grid points in contact with this surface are marked for interaction. The process for achieving FSI between lattice Boltzmann grids and structural 3D beam elements is described in steps, with each step accompanied by specific algorithms [50].

• Step 1: Identify fluid grid points that interact with the outer surface of the pipe, represented by 3D beam elements.
• Step 2: Organize these fluid grid points so that each group aligns with a specific 3D beam element.
• Step 3: Calculate the representative surface areas for the selected fluid grid points.
• Step 4: For each beam element, compute the resultant pressure force at each fluid grid point, ensuring the force direction is normal to the beam’s surface. This force is then converted into an equivalent nodal force for the beam element, calculated for all associated fluid grid points.
• Step 5: Repeat the force calculations from Step 4 for each beam element.
• Step 6: Conduct a transient analysis of the 3D beam structure using the computed fluid forces, specified boundary conditions, and previous solution state over a defined time increment.
• Step 7: Derive the nodal velocities of the beam structure from the structural analysis. Use shape functions to interpolate these velocities at each fluid grid point in contact with the beam’s surface. Apply these velocities to the fluid grid points through particle-distribution functions, then proceed with LBM fluid flow analysis for the next time step.
• Step 8: Continue iterating through these steps as time advances [51].

Previous research developed a combined LBM and FEM approach to address 3D FSI problems, particularly focusing on external flow around pipes. In this approach, pipes were modeled as 3D beams rather than shells, simplifying fluid interaction at the outer surfaces and enabling efficient finite-element modeling. The fluid was simulated using the D3Q15 lattice in LBM. Sequential solutions of the fluid and structural domains allowed for the exchange of fluid pressure and structural velocity data. To accommodate the different variables used in LBM and FEM at grid points, a method was introduced to accurately decompose the structural velocity into LBM particle-distribution functions [52,53,54].

2.8. Exploring the Role of Artificial Intelligence in Enhancing Numerical Simulations

Various studies have applied artificial intelligence (AI) to enhance the efficiency and accuracy of fluid–structure interaction (FSI) simulations, particularly for tidal turbine blades and hydrofoils. Wang et al. [55] introduced the TurbineNet/FEM framework, which utilizes a neural network to predict hydrodynamic loads based on blade mesh features. This approach significantly reduces computational costs while maintaining high accuracy in structural performance predictions. Similarly, Xu et al. [56] developed the DLFSI model, a deep learning framework that integrates Convolutional Neural Networks (CNN) with Blade Element Momentum theory and the Finite Element Method (FEM). This model facilitates multi-objective optimization of tidal turbine blades, predicting hydrodynamic performance and structural stresses while achieving a computational speedup of 19 times compared to traditional methods. In another contribution, Xu et al. [57] proposed a CNN-FEM framework for FSI analysis of morphing hydrofoils, leveraging CNNs to predict fluid forces and FEM for structural responses. This method achieves equilibrium between fluid and solid domains with over 92% accuracy and computational times up to 100 times faster than conventional software. These AI-driven methodologies highlight the transformative potential of AI in optimizing the design and performance of renewable energy systems.

In the field of offshore structural engineering, Gücüyen et al. [58] employed machine learning to enhance the analysis of jacket-type offshore structures subjected to varying environmental loads. By automating predictions of key structural outputs such as displacement, reaction forces, and stress values, this study demonstrated the effectiveness of models like XGBoost, Random Forest, and Support Vector Regressors. XGBoost was particularly effective for predicting displacement and stress values, while Random Forest excelled in reaction force predictions. These models were trained on datasets derived from Coupled Eulerian-Lagrangian (CEL)-based numerical simulations, bridging the gap between computationally intensive simulations and rapid predictive capabilities. Machine learning proved to be both efficient and reliable, providing accurate predictions and reducing computational costs, thereby serving as a valuable tool for forecasting structural behavior and optimizing design. This study underscores the significant role of machine learning in complementing traditional numerical methods to address complex challenges in offshore structural engineering.

2.9. Advancements in Seismic Amplification and Helical Pile Analysis: Exploring Topographic and Soil Variability Effects

Chen et al. [59] explored the impact of topographic irregularities on seismic site amplification, focusing on input signal frequency, by applying finite element analysis with 3D modeling through SolidWorks, SketchUp, and Abaqus, and incorporating viscoelastic boundaries to simulate seismic wave behavior. Validating their method with a V-shaped valley model, they observed that amplification generally peaked at ridges but noted unexpected amplification in ravines at frequencies above 4 Hz. Their results showed that seismic response intensities correlate with input frequency, particularly near the site’s fundamental period, and that amplification factors often exceeded Eurocode8 recommendations, indicating potential risks in current standards. This study provides a refined approach to evaluating seismic site amplification, especially for complex terrains, and suggests that ignoring such topographic and frequency effects may underestimate seismic risk.

In their study, Cheng et al. [60] aimed to explore the torque–capacity correlation of helical piles in spatially variable clays by using three-dimensional large deformation random finite element analyses with a Monte Carlo framework. The finite element model applied the coupled Eulerian–Lagrangian (CEL) approach to simulate helical pile installation and uplift, incorporating a random field to represent spatial variability in soil strength. The model was verified through comparisons with previous centrifuge tests and numerical studies, showing a close match in predicted installation torques and uplift capacities. The results indicated that spatial variability in soil strength significantly influences the torque–capacity relationship, particularly where interlayer variations (e.g., hard and soft layers) were present. Monte Carlo simulations further demonstrated that torque–capacity correlation coefficients in random soils were generally lower than those in deterministic models. Cheng et al. also proposed a probabilistic framework based on a safety factor to predict allowable uplift capacity, which accounted for varying soil strength distributions. This study underscores the importance of considering soil variability in the design and analysis of helical piles, especially for offshore applications.

3. Discussion

In the field of earthquake engineering, analytical work on civil engineering problems is predominantly conducted through various numerical methods. In this section, these methods are categorized into two tables, summarizing past research on the subject.

The numerical simulation of earthquake impacts on marine structures employs a variety of advanced methods, each suited to capturing different physical phenomena relevant to structural behavior under seismic loads. The finite-element method (FEM) is widely used for structural analysis, offering precise modeling of stress, strain, and deformation within complex geometries. Computational fluid dynamics (CFD) provides insight into fluid–structure interactions, especially when assessing the effect of seismic events on surrounding water. The discrete-element method (DEM) is valuable for simulating granular materials, such as seabeds, and understanding how they respond during seismic disturbances. The finite-volume method (FVM) offers an alternative to CFD, ensuring numerical stability when modeling pressure and velocity fields around structures during earthquakes. Direct numerical simulation (DNS) provides detailed flow field solutions but is computationally intensive, making it suitable for small-scale simulations. Smoothed-particle hydrodynamics (SPH) offers a mesh-free alternative, especially useful for modeling large deformations and free surface flows, such as tsunami effects following seismic events. Lastly, the lattice Boltzmann method (LBM) is gaining traction for its efficiency in simulating fluid flows with complex boundaries, particularly in multiphase systems. The combined use of these methods enables a comprehensive understanding of how marine structures respond to earthquakes, accounting for both structural integrity and fluid–structure interactions.

This review evaluates various numerical methods applied in marine earthquake engineering, each offering unique strengths and addressing specific challenges. A deeper comparative analysis highlights the relative value of each method and their complementary roles in understanding the complex dynamics of marine environments under seismic and hydrodynamic forces.

Finite Element Method (FEM): The Finite Element Method (FEM) is distinguished by its versatility and precision, particularly in modeling structural deformations and soil–structure interactions under seismic loading. Its capability to accommodate non-linear material behavior, multi-physics phenomena, and complex geometries makes it an essential tool for analyzing large-scale marine structures, such as seawalls and buried pipelines. However, FEM’s computational intensity and susceptibility to numerical issues, such as wave dispersion, require meticulous calibration to ensure accurate results. When compared to methods like the Finite Volume Method (FVM) or Discrete Element Method (DEM), FEM excels in capturing detailed stress distributions within solid structures, making it ideal for structural analysis. However, it is less efficient for fluid-dominated problems, where other methods may offer greater computational efficiency and accuracy.

Discrete Element Method (DEM): Unlike FEM, the Discrete Element Method (DEM) specializes in modeling granular material behavior, making it particularly well-suited for simulating soil particle dynamics in scenarios like submarine landslides. When integrated with Computational Fluid Dynamics (CFD), DEM offers detailed insights into particle-fluid interactions, effectively capturing forces, energy dissipation, and flow patterns with exceptional accuracy. Despite its strengths, DEM’s high computational demands and dependence on precise parameterization can pose challenges for scaling to large domains. Compared to FEM, DEM provides a microscopic perspective, focusing on the behavior of individual particles. This perspective complements FEM’s macroscopic analyses of stress and deformation, making coupled simulations highly effective for addressing complex multiscale problems in marine earthquake engineering.

Boundary Element Method (BEM): The Boundary Element Method (BEM) offers distinct advantages for problems involving infinite or semi-infinite domains, such as wave scattering and hydrodynamic interactions with marine turbines. By focusing computations on boundaries rather than the entire domain, BEM significantly reduces computational complexity, making it an efficient alternative to FEM and FVM for specific applications. However, BEM’s limitations in handling non-homogeneous materials and highly non-linear problems makes it less versatile compared to FEM. Despite these challenges, BEM has demonstrated exceptional effectiveness in modeling wave–structure interactions, particularly in scenarios where domain truncation errors are critical concerns. Its ability to provide precise solutions in such contexts underscores its value as a specialized tool in marine engineering.

Finite Volume Method (FVM): The Finite Volume Method (FVM) excels at solving conservation laws, enabling robust simulations of wave energy propagation and fluid–structure interactions. Its strength lies in ensuring the conservation of mass, momentum, and energy across computational domains, making it a dependable choice for hydrodynamic problems. When integrated with the Finite Difference Method (FDM), FVM offers a comprehensive framework for analyzing complex phenomena, such as breaking wave forces on structures like monopiles. Although computationally intensive, FVM strikes a balance between accuracy and efficiency, particularly for large-scale applications. Compared to Direct Numerical Simulation (DNS), FVM provides less granular detail but is significantly more computationally efficient, making it ideal for scenarios where scalability and practical runtime are critical considerations.

Direct Numerical Simulation (DNS): Direct Numerical Simulation (DNS) offers unmatched accuracy in resolving turbulence across all scales, making it particularly well-suited for studying complex air–sea interactions, such as wind turbulence over breaking waves. However, the method’s extremely high computational demands limit its application to smaller domains or simplified scenarios. Compared to methods like Smoothed Particle Hydrodynamics (SPH) and the Finite Volume Method (FVM), DNS delivers a significantly higher level of detail, capturing fine-scale turbulence dynamics with precision. Despite this advantage, DNS is less practical for large-scale simulations due to its resource-intensive nature, making it more applicable for fundamental research and scenarios requiring detailed turbulence resolution.

Smoothed Particle Hydrodynamics (SPH): Smoothed Particle Hydrodynamics (SPH) offers distinct advantages as a mesh-free method, excelling in modeling free-surface flows and highly deformable interfaces. These capabilities make SPH particularly well-suited for studying wave impacts on large-scale structures and other scenarios involving complex fluid behavior. Although computationally intensive and sensitive to particle resolution, SPH effectively bridges the gap between grid-based methods, such as the Finite Volume Method (FVM), and particle-based methods, like the Discrete Element Method (DEM). By combining the strengths of both approaches, SPH provides high-fidelity insights into fluid–structure interactions, making it a valuable tool for tackling challenging problems in marine and coastal engineering.

Lattice Boltzmann Method (LBM): The Lattice Boltzmann Method (LBM) is highly efficient for modeling complex fluid flows and turbulence, particularly in scenarios involving intricate boundary conditions and multi-phase interactions. When coupled with the Finite Element Method (FEM), LBM provides a robust framework for analyzing fluid–structure interactions under seismic and hydrodynamic loads. This synergy leverages FEM’s strength in structural modeling and LBM’s capability to simulate detailed fluid dynamics, enabling comprehensive and accurate integrated analyses. However, the computational intensity of LBM poses challenges for scalability, particularly in large-scale domains. Despite this limitation, its efficiency and precision in capturing fine-scale fluid behaviors make LBM a valuable tool for addressing specific, high-fidelity applications in marine and coastal engineering.

The selection of a numerical method depends significantly on the specific application, scale, and available computational resources. For structural analyses, the Finite Element Method (FEM) is highly versatile, while the Boundary Element Method (BEM) offers computational efficiency, particularly for problems involving infinite or semi-infinite domains. Particle-based methods, such as the Discrete Element Method (DEM) and Smoothed Particle Hydrodynamics (SPH), excel at modeling granular interactions and free-surface flows, providing detailed insights into soil and fluid dynamics. Meanwhile, the Finite Volume Method (FVM) and Direct Numerical Simulation (DNS) are well-suited for fluid-dominated problems, with FVM balancing accuracy and scalability and DNS delivering unparalleled detail for turbulence studies at the cost of high computational demands. The Lattice Boltzmann Method (LBM) effectively bridges the gap between fluid and structural analyses, particularly when coupled with FEM, enabling robust modeling of fluid–structure interactions. Each method brings unique strengths, making it crucial to align the choice of approach with the problem’s specific requirements and constraints.

Integrating these numerical methods in complementary ways enables researchers and engineers to tackle the multifaceted challenges of marine earthquake engineering, facilitating accurate modeling of both structural and environmental dynamics. Future efforts should focus on developing hybrid approaches that combine the strengths of different methods while leveraging advancements in computational power to enhance their applicability and efficiency in real-world scenarios. Such innovations will enable more robust and scalable solutions for addressing the complexities of marine environments under seismic and hydrodynamic forces.

As previously commented, there are different numerical methods for obtaining approximate solutions to a wide variety of engineering problems. In recent decades, a lot of research has been conducted on the numerical simulation of earthquake effects on marine structures; the main methods identified in this research are presented in

Table 1. Methods for the numerical simulation of earthquake effects on marine structures.

ID	References	Methods	Figures
1	[15,16,17,18,19,47,49]	FEM	Figure 1
2	[21,22]	CFD-DEM	Figure 2
3	[14,28,30]	BEM	Figure 3
4	[32]	FVM	Figure 4, Figure 5 and Figure 6
5	[37]	DNS	Figure 7
6	[40]	SPH	Figure 8
7	[44,53,54,55]	LBM-FEM	Figure 9

.

The classification of numerical methods based on their meshing approach, as shown in

Table 2. Classification of works based on their meshing method.

Mesh Base	Mesh Free	Hybrid
simplified Order customizing [21,61]	Particle Hydrodynamics [41]	Fast liquid Dynamics [62]
correct Orthogonal Decomposition [31]	Fast Multipole Method [63,64]	Particle in Cell Technique [13]
Single value factorization [23,44]	Method of Fundamental Solutions [50]	swirl in Cell Technique [52]
Marker&Cell [37]	bound Pointset Method [65]	Lattice Boltzmann Method [66]

, highlights the diversity of strategies employed for simulating complex systems, particularly in the context of marine structures under dynamic conditions. Mesh-based methods, such as Orthogonal Decomposition and Marker-and-Cell techniques, rely on structured grids to accurately capture geometric details and solve governing equations. These methods are well-suited for high-resolution simulations but can be computationally expensive for large, deformable domains. Mesh-free methods, including Smoothed Particle Hydrodynamics (SPH) and the Fast Multipole Method, eliminate the need for predefined grids, making them highly effective for problems involving large deformations, free surfaces, or complex boundary interactions. Their flexibility enables simulations of scenarios such as wave impacts or sediment displacement during earthquakes. Hybrid methods, such as the Particle-in-Cell and Lattice Boltzmann methods, combine the strengths of both mesh-based and mesh-free approaches. These hybrid techniques offer a balance between accuracy and computational efficiency, making them ideal for multiphase and fluid–structure interaction problems. This classification reflects the ongoing advancements in numerical simulation, emphasizing the importance of selecting the appropriate method based on the specific requirements of the marine structure and seismic scenario being studied.

The dynamics of the marine environment under seismic actions are shaped by a complex interplay of factors, including soil conditions, structural mass distribution, tunnel length, mooring configurations, seismic wave propagation, and tsunami effects. Each of these factors plays a critical role in determining the seismic resilience of marine structures, yet their individual and combined impacts remain insufficiently explored. Soil conditions, for example, are fundamental to foundation stability, especially under liquefaction scenarios, where saturated soils lose strength and behave like a fluid. Structural mass distribution affects inertial forces and resonance behavior during seismic events, with improper distribution potentially amplifying stress and deformation. Similarly, the length of underwater tunnels significantly influences stress distribution and deformation patterns, while mooring configurations determine the stability of floating structures under dynamic loading conditions. Seismic wave characteristics, such as frequency and amplitude, add further complexity to structural responses, particularly when combined with the compounded hydrodynamic forces of tsunamis. These forces create highly dynamic loading conditions that demand a detailed understanding of the interactions between these factors to enhance the seismic resistance and overall resilience of marine structures.

Variations in soil properties—such as type, density, saturation levels, and liquefaction potential—play a crucial role in the stability of marine structures, especially under seismic loading [67]. The mechanical and hydraulic properties of the soil dictate its response to dynamic forces, including wave-induced pressures and earthquake stresses. For instance, loose, saturated soils are highly susceptible to liquefaction, a phenomenon where the soil loses strength and behaves like a fluid, leading to significant foundation instability. Conversely, dense soils, while less prone to liquefaction, can still experience cyclic mobility, where repeated stress cycles cause deformation, potentially compromising structural integrity. Advanced numerical simulations incorporating soil–structure interaction (SSI) models are essential for understanding and predicting these complex behaviors. Methods such as the Finite Volume Method (FVM) and adaptations of Biot’s consolidation equations enable the simulation of coupled dynamics between soil and structures, accounting for pore-pressure changes, shear stresses, and deformation patterns. These models effectively capture intricate phenomena, including pore-pressure accumulation and dissipation, transient soil displacements, and nonlinear stress–strain relationships, under both cyclic and monotonic loading conditions. By leveraging these advanced modeling techniques, engineers can accurately assess risks associated with soil instability and develop strategies to mitigate these challenges. This ensures the safety, resilience, and long-term durability of marine structures in seismically active regions.

The distribution of structural mass plays a critical role in determining the inertial forces acting on marine structures during seismic events, directly influencing their resonance frequencies and overall stability. Non-uniform or improperly distributed mass can amplify specific vibrational modes, resulting in significant stress concentrations and an increased risk of structural failure under dynamic loading conditions, such as earthquakes or wave impacts. Numerical simulations, including the Finite Element Method (FEM) and modal analysis, are invaluable for analyzing these dynamics. These techniques enable engineers to predict how mass distribution affects natural frequencies and deformation patterns in offshore structures, such as jacket platforms and tension-leg platforms when subjected to seismic and hydrodynamic loads [68]. By leveraging these models, engineers can optimize designs to avoid resonance frequencies and ensure that stresses are evenly distributed, thereby enhancing structural stability and safety during extreme events. Research further highlights the effectiveness of vibration control devices, such as tuned mass dampers (TMDs), in mitigating excessive inertial forces. These devices are particularly beneficial for flexible or deepwater structures, where dynamic loads can lead to pronounced vibrations. Incorporating TMDs into the design of marine structures provides an additional layer of resilience, ensuring improved performance and safety under challenging environmental conditions [69].

The relationship between tunnel length and stress distribution during seismic events is a critical factor in determining the structural integrity of underwater tunnels [70]. Numerical simulations, including finite element models (FEM) and coupled fluid–structure interaction (FSI) models, are extensively used to evaluate failure mechanisms in long-span submerged floating tunnels. These studies aim to understand how extended tunnel lengths influence stress and strain distributions under seismic loads, which often result in significant bending moments and torsional forces. For instance, longer tunnel spans are associated with lower natural frequencies, making them more vulnerable to resonant amplification during seismic events. Advanced simulations have also demonstrated how varying anchoring systems and joint configurations can help mitigate these effects, offering practical strategies for improving the seismic resistance of long underwater tunnels. Future research should focus on incorporating soil–structure interaction (SSI) models and accounting for non-linear behaviors to enhance the accuracy of seismic response predictions. By integrating these advanced modeling techniques, engineers can develop more robust designs that ensure the safety and resilience of underwater tunnels in seismically active regions.

Mooring spacing and configurations are crucial in shaping the hydrodynamic response of floating structures during seismic events. Numerical simulations, including those based on Boussinesq equations and panel models, are commonly employed to evaluate the impact of mooring system properties—such as line pretension and fender arrangements—on the stability of moored vessels in harbors. Research has shown that variations in the natural periods of moored ships, which are influenced by mooring line stiffness and spacing, can result in resonance with seismic-induced oscillations. This resonance amplifies ship motions and places significant stress on mooring components, potentially compromising their integrity [71]. To address these challenges, optimizing mooring configurations and spacing has been demonstrated to be an effective mitigation strategy. Time-domain simulations that incorporate seismic excitation, wave-induced forces, and mooring dynamics provide valuable insights into achieving optimal designs. These optimizations enhance the stability and resilience of floating structures in harbors, ensuring better performance under seismic loading conditions.

The characteristics of seismic wave propagation—such as frequency, amplitude, and directionality—play a pivotal role in determining the response of marine structures. Numerical simulations, including Finite Element Method (FEM) models and dynamic stiffness matrix methods, are extensively employed to analyze these effects. These models account for the intricate interactions between soil, water, and structures under obliquely incident seismic waves, enabling accurate simulations of wave behavior and its impact on structural performance. For instance, research has shown that high-frequency seismic waves create localized stress concentrations, while oblique wave angles result in uneven stress distribution across structural elements [72]. Moreover, simulations underscore the importance of soil–water–structure interactions (SWI) and water–structure interactions (WSI) in amplifying or mitigating seismic effects. Factors such as water depth and soil properties significantly influence these interactions, affecting the overall dynamic response of marine structures. These insights are instrumental in guiding the design and optimization of marine structures, ensuring they can withstand complex seismic forces while accounting for the nuanced effects of wave propagation. By incorporating advanced numerical models, engineers can better predict and mitigate the impact of seismic waves, enhancing the safety and resilience of marine infrastructure.

The combined impact of tsunami-induced hydrodynamic forces and seismic effects on marine structures represents a critical area of research. Tsunami waves exert various force components—including hydrostatic, hydrodynamic (drag), buoyant, surge, and debris impact forces—that can severely compromise the integrity of marine and coastal structures. Numerical simulations, utilizing approaches such as finite element models and fluid–structure interaction (FSI) techniques, are essential for evaluating these compounded risks. These models capture the dynamic interactions between advancing tsunami waves and structural elements, allowing for the detailed analysis of critical phenomena such as wave run-up, bore velocities, and debris impacts. Research has identified tsunami velocity and inundation depth as key parameters in determining the magnitude of forces exerted on structures. Notably, broken tsunami waves often produce higher stresses than those predicted by traditional coastal engineering codes, posing significant challenges to structural resilience [73]. Advanced simulations have informed the development of optimized designs, such as breakaway walls to mitigate lateral forces and flexible moorings to reduce stress on floating structures. These findings highlight the necessity of integrating tsunami-specific load scenarios into the design of marine infrastructure. By incorporating advanced modeling techniques and considering the unique dynamics of tsunami events, engineers can enhance the resilience of structures in tsunami-prone regions, ensuring their safety and durability under extreme conditions.

4. Conclusions

This review examined current numerical modeling approaches for assessing earthquake impacts on marine structures, with a focus on the effectiveness of CFD, FEM, DEM, FVM, and BEM methodologies. These models provide cost-effective solutions for analyzing earthquake-induced forces under realistic environmental conditions. However, their application in the naval industry remains limited by computational resource demands and numerical algorithm constraints. The findings highlight that these models often produce results consistent with experimental and theoretical data, underscoring their potential for accurately simulating seismic forces on marine structures.

The review also addressed secondary load effects, such as those occurring during wave breaking. While existing models capture these effects to some extent, discrepancies remain, particularly in the delayed response observed in simulations compared to experimental data. These inconsistencies are likely attributable to the use of incompressible flow models, which inadequately represent energy dissipation after wave breaking. Refinements are needed to improve the modeling of secondary load cycles, which are critical for a comprehensive understanding of the forces acting on marine structures during seismic events.

Future advancements in computational power and modeling techniques promise to enhance the capabilities of numerical simulations for complex marine scenarios. High-resolution DNS studies of wind and wave interactions, for instance, could provide more accurate insights into wave dynamics. Improved earthquake impact modeling should also prioritize near-field seismoacoustic scattering in marine environments, incorporating elements such as free-field calculations, artificial boundary conditions, and a unified fluid–structure interaction (FSI) framework for internal domains. Leveraging parallel computation strategies—such as concentrated-mass explicit finite elements and local-transmission boundary techniques—will enable efficient simulation of large-scale marine scattering problems.

Finally, the integration of laboratory and field data into numerical models will significantly enhance their accuracy and reliability. Such advancements will establish robust tools for predicting the combined impacts of earthquakes and wave dynamics on marine structures, facilitating better design and mitigation strategies.

Future research could focus on integrating emerging technologies, such as artificial intelligence (AI), to enhance and optimize numerical simulation methods for marine structures subjected to seismic actions. AI can be employed to develop surrogate models, accelerate computational processes, and improve the accuracy of multi-physics simulations by capturing complex relationships between seismic, hydrodynamic, and structural parameters. Additionally, advancements in numerical approaches should aim to address complex multi-load conditions, such as the combined effects of seismic forces, wave loading, and soil–structure interactions. Emphasis on developing high-fidelity coupled models that incorporate real-time environmental data and non-linear material behaviors would provide a robust framework for designing resilient marine structures. These research directions not only underscore the potential for innovation in this field but also highlight the relevance of addressing the dynamic challenges in modern marine engineering. By focusing on these areas, future studies can contribute to more reliable and efficient designs that enhance the safety and resilience of marine infrastructure.