Numerical approaches to capture fluid–structure interaction considering interfaces for offshore structures

CFD

Computational fluid dynamics

CICSAM

Compressive interface capturing scheme for arbitrary meshes

CPS

Conventional parallel staggered

CSS

Conventional sequential staggered

d

Distance to interface (m)

f

Body forces (N/m³)

f(x, t)

Particle distribution function

FEM

Finite element method

FSI

Fluid–structure interaction

FVM

Finite volume method

FVP

Finite volume particle

g

Acceleration due to gravity (m/s²)

HRIC

High-resolution interface capturing

HSPH

Hamiltonian smoothed particle hydrodynamics

IBM

Immersed boundary method

LES

Large eddy simulation

LS

Level set

MPS

Moving particle semi-implicit

MULES

Multidimensional universal limiter for explicit solution

P

Pressure (Pa)

PLIC

Piecewise-linear interface calculation

SPH

Smoothed particle hydrodynamics

t

Time (s)

u

Velocity (m/s)

V

Volume (m³)

VAWT

Vertical axis wind turbine

VOF

Volume of fluid

WENO

Weighted essentially non-oscillatory

Greek

α

Phase volume fraction (dimensionless)

δ

Dirac delta function (dimensionless)

μ

Dynamic viscosity (Pa·s)

ρ

Density (kg/m³)

The growing demand for clean, renewable energy has led to the rapid development of offshore structures, such as those required for photovoltaic generators and wind turbines. These structures operate in complex environments, often contending with fierce water waves, strong winds, and salt mist, which can frequently lead to vibration, damage, or even failure. The interaction between offshore structures and environmental fluid fields is a typical case of a fluid–structure interaction (FSI). Offshore structures may be floating and constructed with flexible supports, making this FSI problem particularly complicated. In research on offshore structures aimed at mitigating risks and ensuring that these structures achieve their intended operational lifespan, FSI analysis has emerged as playing a critical role. The study of FSI represents a relatively new interdisciplinary field at the intersection of hydrodynamics and solid mechanics. Conventional FSI problems are typically concerned with the boundaries between fluids and structures, but the cases of FSI that arise in offshore fields also involve two-phase (gas–liquid) flows, where liquid and solid are partially or completely interconnected.¹ In the majority of these cases, it is impractical to obtain solutions through purely theoretical analyses, and problem-specific numerical methods are needed to tackle the complex interactions between solids and two-phase fluids.

FSI problems involve dynamic interplay between fluids and structural elements, the individual behavior of which can be comprehensively simulated by numerical methods, and efforts are under way to develop ways to take their coupled behavior into account. Computational fluid dynamics (CFD) is widely used in engineering applications and in particular has emerged as a key tool for simulating the fluid dynamical aspects of FSI. For solid structures, system motion can be simulated using multi-body dynamics methods, and the detailed stress distributions within composite structures can be treated using the finite element method (FEM). All of these approaches are now well established in engineering practice and their reliability has been demonstrated. The amalgamation of various fluid and structural simulation techniques yields FSI algorithms with distinct characteristics, each with its own advantages within specific domains. Notable recent advances have included the application of Lagrangian-inspired approaches, such as smoothed particle hydrodynamics (SPH), which has demonstrated significant potential for successful application to the study of FSI. Although the details of each method are briefly touched upon in this review, its main aim is to shed light on the evolving landscape of FSI methodology for offshore structures, especially where two-phase gas–liquid flows are involved.

For offshore structures, many aspects of FSI across multiple scales remain to be addressed. For example, the slender blades of wind turbines are among the components most prone to failure,² and therefore evaluating stress/strain distributions and local stress concentrations on these flexible structures is crucial to assess their performance. Problems involving the interactions of floating objects, flexible supports, and water waves, as well as the interactions between full-scale structures and the wind field, are also of great importance. When investigation any of these FSI problems, is important for researchers to strike a balance between computational accuracy and efficiency in the methods that they adopt. Given the wide variety of methods that have been employed to address engineering challenges in offshore structures, it is also essential to consolidate experiences and results across to provide conclusive insights.

The objective of this paper is to furnish a comprehensive overview of current advances in FSI methods and assess their advantages and disadvantages. In addition, the performance of different FSI methods in their application to offshore structures is discussed, with the aim of identifying the most suitable approaches and providing guidance to researchers in this field, as well as laying a basis for future developments. The remainder of this review is structured as follows. Section II describes the different approaches to describing the fluid dynamical aspects of FSI problems, briefly comparing the advantages and shortcomings of each. Section II describes the general development of numerical methods for dealing with gas–liquid interfaces in FSI problems, with a focus on recent advances in coupling methods in CFD, which can be divided into Eulerian and Lagrangian strategies. Section IV examines the current state of the art with regard to FSI analysis of offshore structures and explores the bridging between model tests, numerical simulations, and engineering applications.

Research into FSI dates back to the early nineteenth century, but in recent decades, significant advances have been made in the analysis of FSI, as a consequence of which it has become a significant area of research. The crux of the coupled solution process lies in computing an unsteady flow problem with moving boundaries and grids, since the size and shape of the flow domain change continually with the movement or deformation of the structure. However, solution of the coupled problem is exceedingly challenging, owing to the amalgamation of linear and nonlinear problems within the coupled system, together with the presence of symmetric and asymmetric matrices, explicit and implicit coupling mechanisms, and physical instability conditions.

Simulation of FSI problems can be typically categorized into two distinct strategies: the partitioned approach and the monolithic approach.³ In the partitioned approach, the fluid and solid fields are described and computed using different methods.^4,5 This separation of methods facilitates the construction of FSI methods in which interfacial conditions are explicitly used to exchange data between the computational domains of fluid and structural dynamics. The FEM and the immersed boundary method (IBM) are commonly used for the coupling with the flow field. Given that the starting point of this paper is a consideration of FSI methods from the fluid dynamical perspective, the following discussion concerns mainly the different fluid methods for FSI problems, with a brief introduction to the FEM and IBM.

The FEM is widely applied in structural analysis,⁶ and has been further developed and extended to the simulation of multiphysics coupling problems, including FSI.^7,8 The FEM discretizes the complex overall structure into finite elements, and then imposes an idealized assumption and mechanical governing equations on each element within the structure. Through solution of the stiffness equation for each element, the total response of the structure is obtained by the imposition of boundary conditions and other constraints. The reactions of each element inside the total structure are subsequently obtained by one-to-one mapping of the total reactions, thereby avoiding the need for direct mechanical and mathematical modeling of complex structures. The IBM has also proven highly practical in application to FSI problems.^9,10 One of its key advantages lies in its ability to simplify grid generation, particularly in cases involving complex solid geometries. Additionally, the IBM facilitates solid motion without any need for a dynamic mesh or mesh changing. However, a drawback of IBM is the potential lack of adequate mesh resolution in the vicinity of the boundary, leading to lower accuracy in predicting turbulence, especially in flows near boundary layers. To address this limitation, it may be necessary to employ high-resolution grids near the structure to enhance simulation accuracy.

Conversely, the monolithic method combines the solution of fluid and structural equations within a unified system, with the interfacial condition addressed directly during the solution procedure. The monolithic approach provides a unified framework for simulating the FSI, inherently simplifies the coupling mechanisms, and rigorously ensures strict conservation of momentum within the coupled system.¹¹ Thus, it can also be easily applied to complex systems, even those subjected to significant shaking forces, such as floating offshore structures in wavy seas.¹² Nonetheless, the monolithic approach also presents challenges in its engineering application, especially in its CFD part, such as accuracy issues. In the remainder of this section, representative CFD methods are described.

In CFD, there are two ways to describe the flow field: the Eulerian method and the Lagrangian method. The Eulerian method is based mainly on conservation of a fixed control volume. In other words, this method analyzes how properties change at specific points in space over time. In contrast to the Eulerian approach, the Lagrangian method considers the fluids as particles and focuses mainly on the motion of these particles. Each fluid particle’s trajectory satisfies a set of differential equations that relate its position and velocity to the forces acting upon it. In the solution of FSI problems using the Eulerian strategy, numerical interpolation between the solid and fluid computational grids is necessary, and grids near the solid boundary need to be changed if the solid structure deforms, which may lead to numerical instability and is one of the major difficulties arising in this approach. In the Lagrangian approach, it is easy to make particles move in both fluid and solid domains, and therefore this method facilitates the coupling required to solve FSI problems. In addition, the lattice Boltzmann method (LBM) has recently been adopted as a mesoscale method for dealing with FSI problems. With the LBM, there is no need to draw grids, and computations are thus simplified, making the coupling easier to achieve. The advantages and disadvantages of each of the above strategies will be discussed in greater detail in the subsequent sections.

In the analysis of FSI for offshore structures, it is necessary to consider the free surface between water and air, making it more complicated than that in the case of a single fluid. Because the solution for this gas–liquid interface needs to be transient, and the precision of the interface simulation plays a significant role in the FSI analysis for offshore structures, this section presents a detailed discussion of the application of various free-surface methods to this problem.

The VOF method, owing to its ability to accurately depict the position and shape of interfaces and its advantages such as low numerical dissipation and straightforward implementation, has been frequently applied to the study of FSI. Nan et al.¹⁴ used the VOF method to investigate the FSI involved in bridge damage during flooding. Comparison of the results of numerical simulations and real-world measurements verified the suitability of the VOF–FSI approach for analyzing the interactions between a liquid free surface and dynamic structures. Cerroni et al.¹⁵ employed an FSI solver utilizing a monolithic approach, coupled with the VOF method, to handle two-phase interface advection and reconstruction, showcasing its robustness and stability in simulating scenarios such as dam breaks over deformable solids. Pathak and Raessi¹⁶ amalgamated the VOF method with the fictitious domain method to capture interactions between fluids and movable rigid structures, effectively addressing FSI problems. Here, the VOF method facilitated the reconstruction and evolution of the three-phase interface, while FSI computations were conducted using the fictitious domain method. Hu et al.¹⁷ combined VOF and FSI models with the overset grid technique to establish a model for projectile entry into water. This allowed the study of hydrodynamic behaviors, structural response, and cavitation evolution for a wide range of inclination angles. The simulation results demonstrated remarkable agreement with empirical datasets, attesting to the validity and efficacy of the adopted methodology.

With the aim of achieving high precision in interface representation, the VOF method has been improved and optimized. At present, VOF techniques can be divided into algebraic and geometric categories. An algebraic VOF technique called the multidimensional universal limiter for explicit solution (MULES) was employed by Wang et al.¹⁸ to simulate the interaction between a dam-break flow and a floating box. Mucha et al.¹⁹ presented an improved VOF-based method combined with a high-resolution interface capturing (HRIC) scheme for considering the slip between air and water. Wang and Wan²⁰ compared the performances of three algebraic VOF schemes, namely, MULES, HRIC, and the compressive interface capturing scheme for arbitrary meshes (CICSAM), with that of the isoAdvector geometric scheme and noted their potential for application to the simulation of offshore structures. Recently, the integration of FSI models with the VOF approach has been extended to more complex scenarios. Fagbemi et al.²¹ introduced a multiphase fluid solver using the VOF method together with the MULES scheme, which has a preferred wetting boundary condition and incorporates the influence of surface tension on the interface boundary, with the specific aim of dealing with the challenges posed by analysis of FSI at micrometer and submillimeter scales, where capillary forces become dominant. Vanilla et al.²² investigated the coupled bend–twist behavior of hydrofoils by employing a coupled FSI numerical approach that integrated the structural model code ASTER with a VOF model featuring free surfaces. Their findings indicated that bend–twist coupling altered the hydrofoil tip’s angle of attack, causing substantial variations in lift and consequent structural deformation. Souza et al.²³ utilized large eddy simulation (LES) in conjunction with the geometric VOF–PLIC (piecewise-linear interface calculation) method for simulating a turbulent incompressible multiphase flow, while modeling the behavior of a solid structure immersed in this flow using an FEM based on Mindlin–Reissner plate theory. Mannacio et al.²⁴ investigated underwater explosions using the VOF approach, validating its efficacy through comparison with real-world scenarios. They also underscored the capability of VOF simulations to predict FSI phenomena accurately by comparing the results of these simulations with those obtained using acoustic approximations by other researchers.

Although the VOF method has been widely applied to FSI problems involving a free surface, some problems still exist with this approach. For example, the interface is represented by the volume fraction in the VOF method, and a transitional area exists, which depends on the mesh resolution. A high computational cost is incurred, since a high-resolution representation of the interface is necessary. On the other hand, the FSI coupling leads to deformation of the structure, and therefore evolution of the mesh near the structure is needed, which poses great challenges to numerical stability and accuracy in the VOF method. However, the VOF method can maintain mass conservation well and performs well in conditions with a high-quality mesh. From this perspective, the FSI–VOF coupling method may not be suitable for simulating problems involving complex structures in practical applications where it is difficult to achieve a high-quality mesh.

The merit of the LS method is that the defined two-phase interface lies at the zero value of ϕ(x), eliminating the need for interface reconstruction algorithms as in the VOF method. Additionally, when considering the influence of surface tension, it is necessary to define the normal direction and curvature of the interface, and this is easy in the LS method. In this scenario, the LS method can describe flow phenomena affected by surface tension more accurately and efficiently than the VOF method.

In view of the advantages of LS method for representation of free surfaces, it has been developed to address various challenges.^25,26 Amani et al.²⁷ developed a conservative LS method based on the finite-volume method for addressing non-Newtonian multiphase flow problems. Zhang and Wolgemuth²⁸ developed weighted essentially non-oscillatory (WENO) schemes for solving the Hamilton–Jacobi equations, and they validated the sixth-order accuracy of these methods for extrapolated surface fields. Mizuno et al.²⁹ applied a differencing method to the convection term in the interfacial advection equation and modified the control parameters of the high-order constrained reinitialization scheme, thereby enhancing the mass conservation property of the LS method. On the basis of the reaction–diffusion equation, a new LS method was proposed by Li et al.³⁰ to solve FSI optimization problems.

Owing to its powerful capabilities in tracking interface motion and managing topological changes, the LS method also has potential for solving the FSI problems involving free surfaces that arise with offshore structures. Calderer et al.³¹ investigated the interaction between air–water flow and complex floating rigid bodies by combining the FSI curvilinear immersed boundary method with the LS approach. Their simulations revealed the emergence of dynamically complex and energetically coherent structures in the gas phase, caused by waves generated through the interaction between rigid bodies and free surfaces. Further exploring FSI complexities, Calderer et al.³² employed a two-phase FSI computational model based on the LS method to investigate the interplay between floating structures, ocean waves, and turbulent atmospheric dynamics. Notably, they successfully applied the solver to simulate an floating offshore wind turbine and accurately captured the dynamic response of the turbine in six degrees of freedom. Tonin and Braun³³ investigated the FSI problem of moored bodies affected by a free-surface flow. They employed the LS method to simulate the free surface, and the mooring was discretized by a geometrically nonlinear cable formula based on the nodal position finite element method (NPFEM). Further, they presented an FSI analysis of floating objects with mooring.

Recently, Xing et al.³⁴ presented a three-phase FSI model, leveraging the LS method for meticulous tracing of multiple interfaces. This model incorporated spring forces on the structure under hybrid wave–current boundary conditions. The simulations using this model provided crucial information for designing effective oil spill prevention systems. Compared with the FSI–VOF method, the FSI–LS method has the advantages of a greater ability to handle topological changes and providing a sharper definition of interface, and, exploiting these advantages, it has been applied to the simulation of offshore wind power towers and floating platforms.^31,32 Nevertheless, because the LS method also depends on the finite volume method (FVM) and relies strongly on mesh resolution, it encounters difficulties when faced with FSI problems involving complex structures where a high-quality mesh is unavailable.

The Lagrangian method is a powerful approach to the investigation of FSI problems without the need for a mesh treatment. As a typical Lagrangian method for flow simulation, the smoothed particle hydrodynamics (SPH) method is a widely used monolithic method in which the fluid and solid fields are both described by particles in a Lagrangian formulation.³⁵ It adopts discrete particles to replace mesh division in the Eulerian method, and these particles can move freely. In the solution of problems such as liquid surface breakage, the particles in SPH can provide a natural representation of one phase and avoid the need for reconstruction of the interface, and this represents the biggest difference between SPH and other FVM/FEM methods.

With the SPH method, different phases such as gas, liquid, and solid can be described easily by different types of particles without the need to construct functions for the interface, and the deformation and destruction of solid structures with FSI coupling can also be easily simulated. Owing to its meshfree format, SPH has significant advantages for the investigation of FSI, and SPH-based simulations has been applied widely to this topic. Shimizu et al.³⁶ introduced an enhanced SPH-based implicit structural model to enable coupling between fluids and solid structures using the same time step size. Tan et al.¹² used the SPH method with a mooring analysis program to simulate a floating wind turbine, and a comparison between the simulation results and experimental data demonstrated the high accuracy of the SPH method in this task. To comprehensively simulate the motion of a floating offshore wind turbine, Leble and Barakos^37,38 used the Helicopter Multi-Block flow solver with the SPH method to compute the aerodynamic loads on the turbine blades and the hydrodynamic forces on the supporting platform.

Khayyer et al.³⁹ reformulated the Hamiltonian SPH (HSPH) isotropic structural model to take account of anisotropy of the structural material and then extended the 2D HSPH structural model and the corresponding incompressible SPH–HSPH FSI solver to deal with the interaction between 3D composite structures and incompressible fluids. Tagliafierro et al.⁴⁰ utilized the SPH method to conduct a numerical study on the dynamic response of a commonly used floater type, namely the tension-leg platform. Their study provided insights into the behavior of platform–wave interactions under dynamic loading conditions. A validation study also by Tagliafierro et al.⁴¹ assessed the accuracy of a high-fidelity meshless method incorporating an SPH solver in replicating highly intricate fluid–solid interactions on a semisubmersible floating offshore wind turbine platform (DeepCwind). This study provided a comprehensive validation of the SPH method, encompassing estimations of surge and heave dynamic characteristics and evaluations of hydrodynamic tractions induced on the moored platform by different wave representations.

However, SPH exhibits certain weaknesses that limit its application in industrial contexts.⁴² First, the method’s accuracy is intimately linked to the diameter and number of particles employed. Second, high-frequency pressure oscillations pose a significant challenge, albeit one that is mitigable with advanced numerical techniques.

When the SPH method is used for FSI problems, the behavior of the solid can be described using both Eulerian and Lagrangian methods. Fourey et al.⁴³ compared conventional parallel staggered (CPS) and conventional sequential staggered (CSS) algorithms in terms of accuracy and efficiency for SPH–FEM coupling. They found that for the weakly compressible SPH model, owing to the relatively small time step, both the CPS and CSS algorithms were able to predict reliable results. Hermange et al.⁴⁴ established a 3D fluid–structure coupling between SPH and FEM to model hydroplaning problems, proposing an optimized coupling algorithm to reduce cost while preserving the accuracy and stability of the coupling. Long et al.⁴⁵ proposed a novel ghost particle method in which the interceptive area of the kernel support domain was divided into subareas corresponding to structure boundary segments, ensuring a complete support condition and restoring first-order consistency near the boundary of the SPH method for SPH–FEM coupling.

In a fully Lagrangian strategy, Bao et al.,⁴⁶ using an entirely SPH-based FSI solver for fluid–flexible-structure interactions, introduced a shell model with single-layer particles into SPH and corrected the truncation error caused by the single-layer boundary by using a normal flux approach. Sun et al.⁴⁷ proposed an SPH method in a fully Lagrangian strategy for both the fluid and solid components of complex 3D FSI problems and were able to ensure numerical accuracy and stability. A new FSI benchmark taking account of the free surface was implemented to highlight the advantage of this SPH–FSI model in performing simulations of free-surface viscous flows. In addition, 3D FSI effects in dam-breaking and sloshing cases were studied. Shimizu et al.³⁶ developed a fully Lagrangian meshfree implicit structure model and combined it with a refined incompressible SPH approach to construct an enhanced fully Lagrangian meshfree FSI solver. Consistent time coupling between fluid and structure was achieved. Gotoh et al.⁴⁸ have presented a concise review of the on latest advances in fully Lagrangian meshfree computational methods for hydroelastic FSI simulations in ocean engineering, in which they have highlighted some key issues regarding such methods.

In conclusion, as a meshfree method, SPH performs well in dealing with problems involving complex geometry, such as offshore wind turbines and photovoltaic generators. Simultaneously, it is very flexible and robust in addressing flow problems involving free surfaces, such as wave breaking and ship sailing, because the fluid is directly represented by free particles in the SPH method. Although the SPH approach has several disadvantages, including complex boundary condition processing and sensitive parameter selection, it has great potential for solving offshore FSI problems with the development of numerical algorithms.

In addition to SPH, there is similar Lagrangian method for interfacial flows called the moving particle semi-implicit (MPS) method.⁴⁹ It includes deterministic particle interaction models representing the gradient, Laplacian, and free surfaces. The fluid density is implicitly constant as the incompressibility condition, while the other terms are calculated explicitly. For solving the Poisson equation for the pressure, the incomplete Cholesky conjugate gradient method is used. Zhang and Wan⁵⁰ proposed an improved MPS method and studied the rolling motion of a 2D floating body in a low-amplitude regular wave, demonstrating the ability of the model to simulate the interaction between the floating body and the wave. An improved MPS method was also developed by Wen et al.,⁴⁹ who introduced multiphase models to realize the simulation of various interfacial flows. Wang et al.⁵¹ presented a compact MPS method in which the condition number was reduced by discretizing the first- and second-order derivatives separately, and numerical stability was significantly improved.

On the basis of the MPS method, a number of studies focusing on FSI problems have also been carried out. Hwang et al.⁵² proposed an improved MPS-based model characterized by minimized nonphysical pressure oscillations, in contrast to the original MPS model, and developed a fully Lagrangian particle-based method that they applied to the FSI problems associated with a dam break with an elastic gate and a violent sloshing flow with a hanging rubber baffle. Zhang and Wan⁵³ developed a fully Lagrangian MPS–FEM coupled method for numerical simulation of elastic structural responses under the flow impact of sloshing liquid and applied their solver to the interaction between a sloshing liquid flow and elastic bulkheads. Yang et al.⁵⁴ studied the interaction between a free-surface flow and a thin elastic baffle in a tank using the MPS–FSI method in a fully Lagrangian approach.

Khayyer et al.⁵⁵ developed a 3D MPS–MPS solver for FSI problems. A refined version of a 3D MPS model for the fluid was coupled with a 3D MPS-based structure model constructed on the basis of a consistent coupling scheme. Wang et al.⁵⁶ proposed several improvements, including a higher-order pressure gradient discrete model, a multiterm source involving a background mesh scheme, and multicondition free-surface detection technique for a 3D-MPS method. They simulated the transmission of solitary waves on a flat bed with and without vegetation. Zhang et al.⁵⁷ coupled the MPS method with the FEM using a partitioned coupling strategy based on the traditional CSS strategy and developed a data interpolation module for the 3D fluid–structure interface. A comparison between simulation results and published experimental data demonstrated the accuracy and efficiency of the coupling method.

MPS–FSI coupling can be divided into two strategies: fully Lagrangian and MPS–FEM. In contrast to the SPH method, each particle in the MPS method interacts only with the particles in its vicinity, which helps improve computational efficiency. The MPS method is a semi-implicit one, which helps avoid computational instability. The disadvantages of MPS are the same as those of SPH, such as complex boundary condition processing, sensitive parameter selection, and the huge number of particles needed to represent complex geometries. Although there have so far been few direct engineering applications of MPS to offshore structures, the studies described above have demonstrated the great potential of the MPS–FSI approach to be extended to simulations of interactions between offshore structures and waves.

The finite volume particle (FVP) method, like the SPH and MPS methods, belongs to the framework of Lagrangian particle methods. The difference between the FVP and MPS methods is that the FVP method integrates the governing equations in the imaginary particle volume space to obtain a new operator for gradient and Laplacian. In the FVP method, a set of overlapping particles are used for fluid representation. Similar to the use of SPH kernel functions, the particles are treated using volume weighting functions. Jahanbakhsh et al.⁵⁸ presented a new formulation of the 3D FVP method that is able to calculate interaction vectors accurately and efficiently. In addition, they verified this formulation on test cases of jet impact, a moving square, and lid-driven cavity flow. Quinlan⁵⁹ proposed two new formulations, one using the partial Riemann problem to analyze the flow between the particle center of mass and a geometrically free surface, the second a well-balanced gravity formula that allows hydrostatic equilibrium to be maintained precisely. An incompressible FVP method to simulate multiphase flows was presented by Liu et al.,⁶⁰ who derived a high-order multiphase Laplacian operator by incorporating a divergence model and a gradient mode, and introduced a higher-order calculation of the virtual particle compensated gradient operator. McLoone and Quinlan⁶¹ proposed an improved particle regularization method and evaluated it through simulations of a dam-break flow, a cylinder in flow, and a multi-resolution flow test. They subsequently proposed a method in which FVP and FEM were coupled.⁶² With this method, it was possible to avoid the possibility of unphysical communication between fluid particles separated by a thin structure.

Although the FVP method has some limitations with regard to the interaction between particles and a solid boundary and in the processing of high-speed flow or strong nonlinear instabilities, and it is rarely used in FSI simulations at present, it inherits the mass conservation advantages of the FVM and the meshless method, and has high accuracy and robustness for long-term simulation and problems involving mass transfer. Consequently, the FVP method has significant potential for solving FSI problems in engineering applications.

Recently, the LBM has attracted significant attention in the field of CFD. It operates on the principles of molecular statistics at the mesoscopic scale, thereby falling outside the categories of Lagrangian and Eulerian method. The LBM represents a mesoscale approach that bridges the gap between macroscopic and microscopic scales, wherein groups of molecules move according to their natural streaming and collisions with other groups. The primary focus of the LBM method lies in solving microscopic velocity distribution functions at the mesoscopic scale. The fundamental concept of LBM revolves around particle collisions and kinetic energy theory. In this approach, the computational domain is divided into structured sets of Cartesian lattice nodes, and subsequently a probability distribution function is defined and solved on each node of the lattice.

There is good compatibility between the IBM and the LBM, owing to their weak reliance on mesh structures, and combinations of these methods are is widely used in solving problems dealing with the interaction of a flexible structure and a fluid. Wu and Guo⁶⁴ adopted a phase field–immersed boundary–lattice Boltzmann (IB–LB) coupling method for FSI and simulated the deformation of a structure in a dam-break flow with interface. Hao and Zhu⁶⁵ introduced an LBM-based implicit IBM in two dimensions. Subsequently, Zhu et al.⁶⁶ extended the coupling between IBM and LBM from two to three dimensions. They validated this method by simulating a flexible sheet tethered in a 3D fluid field, confirming the accuracy of the solver.

Wu et al.⁶⁷ employed the fractional step technique and solved the lattice Boltzmann equation with a forcing term to improve the conventional IB–LBM, and demonstrated that the improved IB-LBM was capable of effectively handling different FSI problems. Fringand et al.⁶⁸ proposed improvements based on an enhanced collision model for the LBM, combined with Laplacian smoothing at the fluid/solid interface. These enhancements significantly improved the stability of the FSI coupled solver. Qin et al.⁶⁹ used a direct forcing IB–LBM for the fluid flow in FSI, with a multi-relaxation-time collision operator and local grid refinement to improve numerical stability and efficiency, together with Newton–Cotes formulas to integrate the force on the surface of the structure and improve the efficiency of the coupling algorithm. Specklin et al.⁷⁰ used a direct forcing IB-LBM method to simulate the structural dynamics of flexible slender solid structures in a turbulent rotating flows in the presence of a moving rigid solid structure (the specific application being to rags in a waste water pump with a rotating impeller). The interaction between the rigid solid structure and the flexible slender structures was well simulated. By introducing artificial damping into the IB–LBM, Ma et al.⁷¹ improved the numerical stability in solving the constitutive equations and simulated a flag flapping in a Newtonian free stream for validation. Cai et al.⁷² combined the IB–LBM with the smoothed point interpolation method, proposing an extended Lagrangian point approximation method that significantly improved accuracy and convergence in FSI problems involving rigid solid motion and large deformation.

For the free surface, Wu and Guo⁷³ improved the iterative velocity correction scheme and employed a combined VOF and IB–LBM approach to simulate the water entry impact problem. In other work, Wu and Guo⁶⁴ used the phase field method with an implicit time integration scheme for structural deformation and fracture, the LBM with an explicit time integration scheme for the fluid, and the IBM for fluid–solid coupling. They simulated the impact response of an elastic plate to a dam break and demonstrated that their proposed method gave a good representation of structural deformation and failure in response to a fluid flow with a free surface.

The LBM method does have several drawbacks, including stability issues and the need for additional turbulence models that increase the complexity of simulations. Compared with other numerical methods, however, the LBM method is relatively convenient in handling complex geometries, which is particularly important in the simulation of offshore structures. Although the LBM has only rarely been adopted for engineering analysis of offshore structures taking account of the presence of a free surface and structural deformation under the action of fluids, the studies described above focusing on model implementation and validation by simple cases have demonstrated the prospects for the practical application of this method.

The applications of different methods for solving FSI problems are summarized in Table I.

The complex environment of the ocean presents significant challenges for numerical simulations, especially in full-scale simulations with comprehensive consideration of all the factors involved. The reliability and robustness of FSI methods are of utmost importance for engineering applications. While the methods discussed in Sec. III offers various approaches, their practical effectiveness in engineering applications requires further verification. In this section, research into the application of FSI analysis in offshore engineering is discussed.

Currently, FSI models coupled with the FVM and FEM are widely applied in analysis of the interaction between waves and offshore structures. To save computational costs, Jo et al.⁷⁴ used this approach to analyze the deformation of an offshore pile tower structure under the fluid forces from a rotating tidal current turbine. A one-way coupling model was used to deal with the CFD and structural analyses, and the influence of waves was ignored. For two-way coupling, the fluid field was calculated separately to first acquire load data. These loads on the structure surface were then mapped into the structural model as boundary conditions and taken into account to generate the deflection. This deflection was then passed back to the fluid field, and the process was repeated until the result converged. Lakshmynarayanana and Hirdaris⁷⁵ compared the results of one-way and two-way FSI coupling methods for symmetrical motions and loads on a S-175 containership undergoing severe impact of green water (i.e., water washing onto the ship’s deck from incident massive waves). They demonstrated that the two-way coupling method gave results that were more consistent with measurements, especially in the high-frequency whipping domain. Bazilevs et al.⁷⁶ conducted FSI simulations of floating wind turbines simultaneously subjected to wind flow and ocean-wave forcing, with the aim of simulating the high-cycle fatigue failure of blades due to long-term cyclic aerodynamic loads. The free surface was taken into account using the LS method. Generally speaking, although the one-way coupling algorithm is suitable for some specific situations, offshore structures are usually subjected to volatile conditions demanding two-way strong coupling.

With the rapid development of computing power and technology, FSI analysis has become an indispensable component of offshore structure projects. Yan et al.⁷⁷ proposed a free-surface FSI framework based on an appropriate combination of FEM and iso-geometric analysis to simulate the interaction between a free surface and a floating structure such as an offshore wind turbine. Oliveira et al.⁷⁸ employed an approach in which one-way FSI coupling was used to simulate the interaction between hydrodynamic loads and a offshore jacket platform structure. The structure was modeled using the 3D FEM to assess the deformation caused by environmental factors, specifically hydrodynamic loads induced by waves and currents modeled by the VOF method. Sree et al.⁷⁹ performed an FSI analysis based on numerical simulations with two-way coupling of a global array of interconnected modular floaters designed to accommodate a massive number of solar panels. This provided a useful approach to evaluate the maximum stress/strain and displacement of massively connected modular floating solar farms under the action of waves. Snapshots (t = 25 s) of simulated overwash by waves over the floating array using different mooring configurations as simulated by Sree et al.⁷⁹ are shown in Fig. 1. Li et al.⁸⁰ conducted model tests to assess the impact of scour depth and flow velocity on the lateral responses of support structures. Additionally, they adopted a two-way coupled FSI method for evaluating the dynamic structural response of offshore wind turbine supports in sand. In their approach, the support structure was modeled using Abaqus software, taking nonlinear soil–structure interactions into account.

To investigate the interaction between the structural behavior of a vertical-axis wind turbine (VAWT) and wind with random velocity, Arora et al.⁸¹ utilized probabilistic wind data to simulate the wind-induced pressure on the blades of the VAWT. Haider et al.⁸² took account of the influence of floating platform movement on the air flow around turbine blades and compared the aerodynamic performance of conventional and new mooring systems. Luo et al.⁸³ extended the CFD–FEM technique to a floating VAWT and established a fully coupled model for simulating structural responses and loads in typical situations. To investigate the response of flat stiffened plates to slamming events, Truong et al.⁸⁴ simulated the vertical deflection, total vertical forces, and deformation, all of which are useful in evaluating the strength of a structure subjected to slamming loads. To improve the hydrodynamic and structural performance of a propeller and study the influence of its deformation on performance under load, Shayanpoor et al.⁸⁵ performed CFD–FEM simulations, the results of which were in good agreement with experimental results. Shao et al.⁸⁶ analyzed the fully coupled hydrodynamic and structural responses of wave energy converters to ocean waves. They accurately simulated the FSI between the converters and the waves and revealed the effects of the distance between converters and of the wave direction on the captured energy.

For offshore structures, such simulations allow researchers to test the structures and their layouts, ultimately enhancing stability and reducing coupled water–air damage. During the design process in industry, it is helpful to be able to highlight the areas where additional work is necessary before carrying out further development of a project.

Previous studies of the applications of FSI analysis in offshore engineering are summarized in Table II.

This paper has provided an overview of the main current approaches to FSI simulation for offshore structures. These approaches can be categorized into Eulerian, Lagrangian, and LBM-based, and each has been comprehensively reviewed here. There are a number of FSI methods that are able to meet the heightened requirements for practical engineering application that arise when a structural model is combined with advanced CFD methods considering both wind and wave actions. Compared with the analytical approach⁸⁷ for floating structures, comprehensive CFD–FSI simulations offer several advantages. Some conclusions can be drawn as follows:

• As can be seen from the studies described in this paper, the trend of current FSI methods has followed that of the development of CFD methods. As the VOF and LS methods can be used to capture the interface between water and air, they are frequently adopted in FSI simulations taking account of structural deformations under the effect of two-phase flow. Owing to the relative robustness of the VOF method, the FSI–VOF coupling method satisfies the requirements for simulating interactions between floating offshore structures and ocean waves.

• Given the natural advantages of Lagrangian approaches for hydrodynamics, CFD methods using these approaches have become increasingly popular for simulating FSI problems involving free surfaces and their interactions with structures. This paper has presented an overview of the Lagrangian CFD methods (SPH, MPS/FVP, etc.) that can be adopted in FSI simulations. For the FSI analysis with a mainly Lagrangian strategy, the interaction between fluid and structure can be achieved either by coupling between a Lagrangian CFD method and the FEM or by a fully Lagrangian method.

• FSI analysis using the LBM has also been reviewed here as a specific modeling strategy. This method offers several advantages, including a simple algorithm, high parallel efficiency, elimination of the need for a mesh update at every time step, and robustness. Consequently, it is particularly suitable for simulating complex motions. Although the LBM has yet to reach maturity for application to offshore structural engineering, it appears to have good prospects based on its advantages and ongoing development.

• Several challenges remain with regard to the simulation of offshore structures in engineering applications. FSI analysis is experiencing rapid growth driven by escalating engineering demands and advances in computational tools. While significant progress has been made in FSI analysis for offshore structures, there are still several technical intricacies that require further consideration. Key challenges include the need for a realistic approach to the coupling of wind and ocean fields with floating structures, a reduction in computational costs for analyzing very large floating structures, and the development of more precise and efficient models and algorithms for full-scale offshore structures.

This research was funded by China Longyuan Power Group Corporation Limited (Grant No. LYX-2023-07).

The authors have no conflicts to disclose.

Junhao Zhang: Writing – review & editing (equal). Mingming Chen: Writing – original draft (equal). Bohan Shen: Writing – original draft (equal). Dongping Zhang: Writing – review & editing (equal). Sherman C. P. Cheung: Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.