Recent Developments in the Nonlinear Hydroelastic Modeling of Sea Ice Interaction with Marine Structures

Abstract

This review provides the recent advancements in nonlinear sea ice modeling for hydroelastic analysis of ice-covered channels and their interaction with floating structures. It surveys theoretical, experimental, and numerical methodologies used to analyze complex coupled sea ice–structure interactions. The paper discusses governing fluid domain solutions, fluid–ice interaction mechanisms, and ice–structure (ship) contact models, alongside experimental techniques and various numerical models. While significant progress has been made, particularly with coupled approaches validated by experimental data, challenges remain in full-scale validation and accurately representing ice properties and dynamic interactions. Findings highlight the increasing importance of understanding sea ice interactions, particularly in the context of climate change, Arctic transportation, and the development of very large floating structures. This review serves as a crucial resource for advancing safe and sustainable Arctic and offshore engineering.

1. Introduction

The increasing feasibility of Arctic shipping and polar resource exploitation is leading to greater maritime and offshore activity in the Arctic regions [1]. In contrast to an open water environment, operations in these areas are subject to both hydrodynamic and ice-induced forces. Therefore, a thorough understanding of sea ice–structure interaction is essential for the effective design and safe operation of polar vessels and offshore platforms [2,3]. The simulation of this interaction is particularly intricate and demanding due to the diverse mechanical characteristics of sea ice and the presence of multimedia and multi-interfaces [4].

The sea ice loads and the relative movement between the structure and an ice feature in the Arctic and polar regions profoundly influence ships and marine structures. Then, the structures experience direct contact loadings from various ice features, including ice floes, ridges, and icebergs. Predicting these ice forces necessitates a comprehensive understanding of sea ice’s mechanical properties alongside detailed knowledge of the intricate contact between ice features and structures [5,6].

In polar regions, when a drifting ice field comes into contact with ships or marine structures, the resulting ice failure can cause hazardous vibrations and high dynamic loads [7]. This phenomenon significantly compromises the ship’s navigation (Figure 1), structural stability, and reliability [8,9,10,11,12,13,14,15,16,17,18].

Ice can fail in various modes, including crushing, bending, buckling, splitting, or under a mixed-mode condition. The manifested failure mode is contingent upon the mechanical properties of sea ice [20], the geometries of the ice feature and structure, collision speed, and the governing boundary conditions [21]. Moreover, when sea ice comes into contact with a structure, it breaks apart into numerous fragments, which can then form a pile that affects the ongoing failure process and eventually clears the active zone. A good example of the process is shown in Figure 2. The movement of a floating ice sheet against an inclined marine structure induces fragmentation from a solid sheet into individual ice blocks, leading to the formation of an ice rubble pile [22]. This accumulation subsequently influences subsequent ice failure events.

The interaction between structures and ice can produce transient hydrodynamic forces, especially prominent during the initial contact and subsequent collision phases and collision forces [23]. Such forces originate from the intricate relationship among the structure’s movement, the inherent properties and actions of the ice, and the enveloping water. Several parameters affect these hydrodynamic forces, including the ice’s velocity, the structure’s geometrical configuration, and the fluid’s characteristics, including its damping capacity.

In addition, the key aspects of managing ice–water coupling involve temperature, pressure, material compatibility [24], ship–ice interaction, glacial processes, collisions between multibody, free surface deformation, fluid–structure interaction, wave drift, fluid cushion, and hydrodynamic damping [25], and influence the trajectory and velocity of ice floes [26].

Especially, ice deformation typically features rafted floes and blocks, broken ice, pressure ridges, shear zones, hummocks, and rubble fields. These deformation features are caused by mechanical forces—including wind stress, ocean currents, and land blockages—which induce spatial variations in the ice’s drift velocity. Observing sea ice deformation through airborne and spaceborne remote sensing holds considerable importance for several reasons [27,28]. The severe hazards posed by ridges and rubble fields to marine operations and offshore structures are widely recognized. Furthermore, accurate data on sea ice deformation is essential for various research areas that investigate the dynamics of the polar climate. The topography of both the ice surface and its bottom significantly influences the forces acting on the ice due to wind and ocean currents. The ice deformation in cold regions has been studied [29], including the spatial and temporal characteristics of sea ice deformation [30].

The increasing recognition of the significance of ice–water coupling has led to the development of more sophisticated hydrodynamic models in recent years. These models dynamically couple ice and water, incorporating field data within a mathematical and numerical framework through various numerical software and methods, and are now extensively applied in engineering [31,32,33,34,35].

It is worth noting that sloping or conical structures are frequently used in Arctic oil and gas operations. This design is favored because it causes ice to fail in bending, significantly reducing horizontal ice loads on the structure. When an ice sheet encounters a conical or sloped structure, the accumulating pressure eventually causes it to fail by bending, resulting in the formation of numerous ice blocks. Following this failure, these blocks are either forced up the sloped structure or they amass as ice rubble in front of it. This problem is treated numerically [36,37,38].

Another interesting aspect of this class of problem is the study of the hydroelastic behavior of floating and/or submerged flexible structures subjected to wave and current actions, with applications to marine structures. In recent years, various analytical models have been developed for 2D, 3D, and oblique wave cases, based on the boundary integral equation method (BIEM), the Boussinesq model, and solution techniques [39,40,41,42,43,44,45,46]. There are numerical models associated with floating rectangular structures under currents and wind [47,48], and circular platforms [49], along with a theoretical floating flexible circular model under current loads using the Timoshenko–Mindlin beam theory [50]. A rectangular structure [51] and one with interconnected platforms [52] were analyzed while investigating the influence of current speed on structural responses and wave quantities. The motion of a large articulated floating platform for marine structures was also analyzed based on an experimental study [53].

As of now, the subject of hydroelasticity associated with sea ice–structure interaction has been well-documented in a range of literature review articles. For instance, ref. [54] provided a comprehensive review of the current state of model-scale ice research. This involves examining the various methods used to create and test model ice, the scaling laws that are applied to relate model-scale results to full-scale scenarios, and the limitations and knowledge gaps in the field. The paper synthesizes findings from a wide range of studies and provides critical insights into the challenges and future directions of model ice research. Mass loading, thin elastic plate, and viscous layer models are among the methodologies reviewed and discussed for different ice types [55], addressing various mathematical frameworks for wave models, including advection, source/sink, and nonlinear interaction terms, as well as wave reflection and transmission between different ice covers, utilizing methods such as Green’s function, matched eigenfunction expansion, and variational methods.

Timco and Weeks [20] offered a comprehensive overview of the engineering characteristics of sea ice. This review covers both first-year and multi-year sea ice. It investigates the physical attributes of the materials, focusing on microstructure, thickness, salinity, porosity, and density, as well as their mechanical behavior, including tensile strength, flexural strength, shear strength, uni-axial and multi-axial compression strength, borehole strength, failure envelope, friction, and elastic properties. Ni et al. [56] review the progress made in addressing the ice–water–structure interaction (IWSI) problem, encompassing common analytical, numerical, and experimental approaches and their primary implementations. An extensive review of recent developments in floating multi-body hydrodynamics and gap resonance was conducted [57], including theoretical, numerical, and experimental research works for numerous applications. Recently, a technical comparative overview of the theoretical and numerical approaches used to evaluate the sensitivity and uncertainty in the hydrodynamic and hydroelastic behavior of floating offshore structures was provided [58] to compare various methodologies linked to the fundamental governing equations, assessing their strengths and weaknesses and exploring their applications in different offshore scenarios. Full-scale measurements of ice loads on offshore structures under various ice conditions (level ice, ridged ice) are reviewed [59], examining load magnitude, duration, and frequency while addressing current standards and their agreement with experimental results. A review of different simulation methods that predict ship efficiency and ice loads for ships navigating continuously in broken ice fields or floating platforms is provided [60,61]. Recently, a comprehensive review examined the ice deformation regarding the characteristics of the flexural gravity waves generated by a moving load acting on a complete ice sheet, an ice sheet with a crack, and an ice sheet with a lead of open water [62].

The above literature confirmed that, to date, no recent review has been presented to the public focusing on the nonlinear analysis of sea ice–structure interactions. Therefore, this paper seeks to present a comprehensive literature review on the theoretical, experimental, and numerical methodologies of the sea ice–structure interactions in Arctic and offshore regions. In Section 2, a comprehensive overview of the ice–structure interaction problem is outlined within various result subsections focusing on ships and offshore floating structures. Further, a comparative discussion of nonlinear methodologies and their applications to modeling for ice–structure and fluid–ice interactions is provided. Section 3 summarizes the review’s concluding remarks and highlights the potential for future advancements in model development.

2. Solutions for Sea Ice–Structure Interaction Problems

Hydroelastic coupling governs the interaction between sea ice and structures. This means the fluid flow and the elastic deformation of the ice/structure must be solved simultaneously for accurate predictions [63]. This is crucial for the application of transport systems in cold regions for operational efficiency and safety. Additionally, it plays a vital role in the effective deployment of air-cushioned vehicles (ACVs) for ice breaking. These diverse methodologies demonstrate the sophisticated mathematical and computational approaches required to understand and predict the complex behavior of hydroelastic systems involving ice and floating structures.

The interactions between ice and structures that are experienced during the station-keeping operations of vessels and offshore platforms within fragmented ice fields introduce a notable degree of complexity. The manner in which ice failure and the subsequent release of loads transpires is contingent upon the specific geometrical configuration of the local environment and the relative velocity at the points of contact between the ice and the structure [64]. This can manifest through various modes such as local crushing, buckling, bending, shearing, splitting, ridging, rafting, pure displacement, or a combination of these failure mechanisms. Upon detachment from a larger ice floe, the smaller fragments may exhibit acceleration, rotation, collisions, accumulation, submersion, and lateral movement along the hull until they are eventually removed. The hydrodynamic interactions that occur between water, ice, and structures may become pertinent and significant during interactions between broken ice and structures, particularly in scenarios involving substantial ice drifts affecting a stationary structure or high advance velocities of a ship traversing a stationary ice field [65,66].

The ice model is required to incorporate considerations for large deformations and displacements, as well as the phenomena of fracture and fragmentation, and the dynamic motion of the resultant ice fragments. In relation to the structure, the model should encompass both the static and dynamic responses to ice loading; however, the structural response may influence the mechanisms by which the ice experiences failure, thereby calling this phenomenon ice–structure interaction. On the other hand, several works [67,68,69] have highlighted the inadequacy of linear models for describing large-amplitude deflections. This necessitates the use of nonlinear fluid equations and plate theories.

2.1. The Governing Analytical Solution in the Fluid Domain

Here, the basic governing equations of DEM coupled with FEM and CFD are given, as this method is one of the most beneficial case scenarios [6] to solve the sea ice–structure interaction problems. The detailed assessment of the forces and the CFD–DEM coupling process was previously documented [55,70]. Given that the ice-breaking term is less significant for small floe ice compared to level ice, the sea ice model used in the numerical simulations is expected to reflect real-world conditions accurately. In the fluid domain of the coupled CFD–DEM, the continuity equation and the momentum equation must both be satisfied. In the fluid domain part of the coupled CFD–DEM method, the incompressible continuity equation and the momentum equation must be complied with [71]:

$${\displaystyle \frac{\partial \left({\rho }_f{\epsilon }_f\right)}{\partial t}}+\nabla \cdot \left({\rho }_f{\epsilon }_f{\mathit{u}}_f\right)=0$$

(1)

$${\mathit{F}}^{OM}=-\nabla p+\nabla {\mathit{\sigma }}_f+{\rho }_f{\epsilon }_f\mathit{g}-{\displaystyle \frac{\partial \left({\rho }_f{\epsilon }_f{\mathit{u}}_f\right)}{\partial t}}-\nabla \cdot \left({\rho }_f{\epsilon }_f{\mathit{u}}_f{\mathit{u}}_f\right)$$

(2)

$${\mathit{\sigma }}_{\mathit{f}}=\mu \left(\nabla {\mathit{u}}_f+{\left(\nabla {\mathit{u}}_f\right)}^{\prime }\right)-{\displaystyle \frac{2}{3}}\mu \left(\nabla ·{\mathit{u}}_f\right){\delta }_{ij}$$

(3)

Equation (2) is the average volumetric force exerted by the fluid on the solid particle within a given fluid cell, where ${\mathit{\rho }}_f$ is the density, ${\epsilon }_f$ and ${\mathit{u}}_f$ are the volume ratio and velocity of the fluid, respectively. Equation (3) is the fluid viscous stress tensor, where μ is the dynamic viscosity of the fluid, ${\delta }_{ij}$ is the Kronecker delta symbol. These forces include, but are not limited to, fluid drag, fluid pressure, and shear stress as the presence of ice floes influences the flow characteristics around the hull surface during ship–ice interactions [72].

Further, leveraging the fractional volume of fluid (VOF) concept [73], the volume fraction of the water in the cell is used to accurately capture the air–water interface by solving the equation below [74]:

$${\displaystyle \frac{\partial {\epsilon }_{f,1}}{\partial t}}+\nabla \cdot \left({\mathit{u}}_f{\epsilon }_{f,1}\right)=0$$

(4)

where ${\epsilon }_{f,i}$, $i=1$ and 2 refer to water and air, respectively.

The discrete element method (DEM) is employed to simulate the movement of floating ice particles; their movement adheres to the laws of force and moment equilibrium [75]:

$$m{\displaystyle \frac{d{v}_p}{dt}}={\mathit{F}}_g+{\mathit{F}}_f+{\mathit{F}}_c$$

(5)

$$\mathit{I}{\displaystyle \frac{d{\omega }_p}{dt}}={\mathit{M}}_f+{\mathit{M}}_c$$

(6)

where the variables used in Equations (5) and (6) are the same as defined in [33].

2.2. Fluid–Ice Interaction

The fluid force, denoted as ${\mathit{F}}_f$ acting on each ice particle accounts for the interaction between the ice and the surrounding fluid. This force encompasses drag ${\mathit{F}}_d$, pressure gradient ${\mathit{F}}_p$, and lift ${\mathit{F}}_l$ components. These fluid forces are computed for each ice particle based on the properties of the fluid grid. Each ice particle is projected onto the fluid grid, while the surrounding fluid grid contains fluid forces that are interpolated onto the ice particles to contribute to the total force and torque [9].

The drag force ${\mathit{F}}_d$ and drag torque ${\mathit{M}}_d$ are defined as:

$${\mathit{F}}_d={\displaystyle \frac{1}{2}}{C}_D{\rho }_f{A}_p\left|{\mathit{v}}_r\right|{\mathit{v}}_r$$

(7)

$${\mathit{M}}_d={\displaystyle \frac{1}{2}}{\rho }_f{\left({\displaystyle \frac{D}{2}}\right)}^5{C}_R\left|{\omega }_r\right|{\omega }_r$$

(8)

where $A_p$ represents the projected area of the particle and $D$ represents the particle diameter when the DEM particle shape is a polyhedron. $v_r$ and ${\omega }_r$ represent the relative velocity and angular velocity between the ice particle and fluid, respectively [76].

The pressure gradient force is calculated using the method described in [33]:

$${\mathit{F}}_p=-V_p\nabla p_s$$

(9)

where $V_p$ represents the volume of ice particles and $\nabla p_s$ represents the gradient of the static pressure in the fluid.

The lift force ${\mathit{F}}_l$, can be decomposed into two components: shear lift force, ${\mathit{F}}_{ls}$, and spin lift force, ${\mathit{F}}_{lr}$ as:

$${\mathit{F}}_l={\mathit{F}}_{lr}+{\mathit{F}}_{ls}$$

(10)

where

$${\mathit{F}}_{lr}=C_{lr}{\displaystyle \frac{\rho \pi }{8}}{D}^2\left|{\mathit{v}}_r\right|{\displaystyle \frac{{\omega }_r\times {\mathit{v}}_r}{\left|{\omega }_r\right|}}$$

$${\mathit{F}}_{ls}=C_{ls}{\displaystyle \frac{\rho \pi }{8}}{D}^3\left({\mathit{v}}_r\times \mathit{\omega }\right)$$

where $\omega$ represents the angular velocity of the fluid, and the coefficients $C_{lr}$ and $C_{ls}$ are determined as in [77].

2.3. Ice–Structure Interaction

The interaction between ice and ice or ship and ice generates contact forces (Fc), which are modeled using a linear spring [70], based on the spring-dashpot principle, which calculates contact forces using a spring for elasticity and a dashpot for viscous damping [78]. The resulting normal force ($F_n$) and tangential force ($F_t$) are defined as:

$${\mathit{F}}_c=F_n\mathit{n}+F_t\mathit{\tau }$$

(11)

Using a similar approach, we can define the contact torque as

$${\mathit{M}}_c=M_n\mathit{n}+M_t\mathit{\tau }$$

(12)

where the subscripts $n$ and $\tau$ indicate the normal and tangential elements, $M_n$, and $M_t$ are normal and tangential torque elements (see [79] for more details).

2.4. Experiments

Ice tanks and controlled ice collision simulations are key forms of experimental modeling that significantly help in understanding and predicting ice–structure interactions. They provide a valuable means to observe complex behaviors in a controlled environment, which is crucial for validating numerical models and gaining a deeper insight into both ice failure mechanisms and the corresponding structural responses [80,81].

Experimental research examined ice–structure interaction using a 20 cm diameter, vertically sided cylindrical pile interacting with an ice sheet (as depicted in Figure 3 and detailed in reference [82]). The study specifically focused on the phenomena of continuous brittle crushing (CBR), frequency lock-in (FLI), and intermittent crushing (IC).

A wave–ice interaction simulation model [83] is introduced to highlight the shortcomings of the conventional ice model, specifically its inadequate stiffness and nonlinear response. A model of ice with virtual equivalent thickness (MIVET) was introduced to improve the representation of sea ice’s elastic stiffness. This was achieved by modifying the thickness and strength of the model ice. The efficiency of the model was tested through physical experiments and compared to the conventional ice model.

Klein et al. [84] discussed the use of transient wave packets in wave–ice interaction experiments. This involves generating wave groups that have a limited duration and specific shape in a controlled laboratory setting. These wave packets are implemented to create simulations of realistic wave conditions for the purpose of analyzing the dynamic response of ice subjected to wave action.

An experimental model test on wave attenuation and dispersion in various ice conditions is conducted in [85], including continuous, fragmented, grease ice, grease pancakes, and wide pancakes (see Figure 4) using a refrigerated wave flume where the wave attenuation and dispersion were measured using ultrasound sensors.

Dolatshah et al. [86] conducted a study on wave depletion and wave-induced ice breakup in a wave–ice flume. This experimental approach involves creating a controlled environment for the wave behavior traveling through ice, enabling precise observation and measurement. The study likely focuses on varying parameters such as wave properties and ice conditions to understand how they affect wave attenuation and ice breakup.

Laboratory experiments were conducted to analyze ice–ice collisions [87], introducing a methodology using robust principal component analysis (RPCA) to identify collision duration, using sensor fusion and Kalman filtering for velocity estimations.

Model tests were conducted in an ice tank to prepare for pack ice conditions [88]. Experiments considered variables such as channel width, broken ice floe size, ice concentration, and ice thickness. Figure 5 depicts a model ship displacing broken ice floes as it navigates through a narrow pack ice channel.

Model-scale glancing impact events between a polar research vessel and giant ice floes were simulated in an ice tank [89]. A key factor in nonlinear transient dynamic analysis is understanding the dynamic characteristics exhibited during ship–ice impacts. Preliminary methodological calibration tests were performed to determine key parameters and processes. Figure 6a depicts a model ship affixed to a carriage, which imparts the necessary velocity for testing, and navigates within the ice floes with specific mass and edge angles (as in Figure 6b).

A model-scale towing experiment to investigate ship resistance in small ice floes, utilized artificial polypropylene ice floes instead of refrigerated model ice [90]. The study focused on the impact of floe size, floe shape, and ice concentration on the resistance of ships. It found that ice resistance increases with floe size and ice concentration. While floe shape has a relatively minor influence on mean resistance compared to other factors, the randomness of floe placement is crucial for accurate simulation (see Figure 7).

2.5. Numerical Models

Huang et al. [91] employed the computational fluid dynamics (CFD) software OpenFOAM to model the hydroelastic response (Figure 8). This approach allows for a detailed simulation of the interaction between the fluid (ocean waves) and the structure (large ice sheet), trapping the elastic response of the ice sheet to wave-induced stresses. OpenFOAM is well-suited to this type of problem as it can handle complex geometries and flow conditions, and it allows for the solution of the coupled fluid–structure interaction equations.

A fully nonlinear numerical FE solver for simulating nonlinear wave processes in the presence of a solid ice sheet is introduced in [92]. The ice sheet is modeled as a linear elastic plate, with its deformation described by the Kirchhoff–Love plate ansatz. To consider the ice sheet’s bending stiffness, the solver integrates a potential flow model with a hydroelastic model.

Based on dimensional analysis and considering the inhomogeneous nature of sea ice [92], a two-layer ice model (Figure 9) is presented to explain wave dissipation within the ice. The described model assumed that wave motion exists in only a fraction of the ice thickness, with the rest of the ice being too viscous to permit wave motion.

A computational fluid–solid dynamic model is used to solve the Navier–Stokes equations for ocean–wave flow and used the Maxwell viscoelastic model for the solid behavior of ice [94], involving a two-way coupling of fluid and solid dynamics solvers. Figure 10 shows results related to waves with an open-water wavelength. The setup with one fixed end is observed to lead to a lower damping rate, compared with the cover having two free ends.

A hybrid approach was employed, utilizing the boundary integral method (BIM) to model the flow under the ice and the finite difference method for the ice plate, where the ice sheet was modeled as a viscoelastic thin plate, while the water was assumed to be of constant depth with potential and linear flow [95]. A computational fully coupled fluid-solid interaction model was adopted to determine the air–water flow using Navier–Stokes equations and refers to the ice as a viscoelastic material [96] while using a nonlinear Winkler–Kelvin–Voigt model to simulate a rigid body on the ice surface.

A 3D computational fluid dynamics (CFD) model is utilized to solve the nonlinear Navier–Stokes equations to simulate wave–ice interaction [97], including wave overwash and scattering, assimilating a volume of fluid (VOF) method to model the air–water interface.

Table 1. Nonlinear methodologies, description, and their applications to modeling for ice–structure and fluid–ice interactions.

Methodologies	Descriptions	Applications	References
Smoothed Particle Hydrodynamics (SPH)	Meshless Lagrangian method that simulates ice mechanics and IWSI problems.	Ice–structure interaction, ice–water interaction, and IWSI.	[15,98,99,100]
Discrete Element Method (DEM)	Particle-based method that simulates ice dynamics and its interaction with structures. Coupled with CFD to consider water effects.	IWSI problems.	[9,66,70,101,102,103,104,105,106,107]
Lattice Boltzmann Method (LBM)	Mesoscopic method for simulating fluid motion.	Simulating interaction between fluid and multi-bodies in IWSI problems.	[108]
Finite Element Method (FEM)	Continuum mechanics problems are tackled using a sophisticated technique. This technique enables the simulation of complex, nonlinear wave processes.	Ice–structure interaction (with simplified models for water effects).	[79,92,109,110]
Boundary Integral Method (BIM)	3D method to compute nonlinear wave interactions with an ice sheet.	The first and second-order hydrodynamic solutions for a floating body interacting with waves.	[67,68,95]
Computational Fluid Dynamics (CFD)	3D model to solve the nonlinear Navier–Stokes equations.	Wave–ice interaction, capturing the elastic deformation of the ice sheet.	[91,97]
Peridynamics (PD)	Meshless method good at solving fracture problems.	Ice mechanics and ice–structure interaction.	[101,111,112]
Experimental Models	In situ tests and model tests.	IWSI problems. kinematics of sea ice.	[80,81,82,83,84,85,86,88,89,90,113]
Winkler–Kelvin–Voigt Model	A 1D model to simulate the counterforce of an ice floe to the rigid body impact.	Arctic sea ice nonlinear simulations.	[96]
Non-smooth Discrete Element Method (NDEM)	Nonlinear method to solve ship–ice interactions.	Ship resistance in broken ice conditions.	[8,31,114]

demonstrates the different recent nonlinear methodologies and a brief discussion on their applications to various problems for the modeling of sea ice interactions with ships and marine structures.

Table 2. Primary objectives in the simulation of sea ice–structure interaction.

Key Parameters	Range/Scenarios	Key Findings	Modeling Approaches	Experimental Validation	References
Ice Thickness	- Thin (e.g., model/MIVET ice) - Moderate - Thick (pack/level ice)	- Resistance increases with thickness. - Thicker ice leads to crushing; thinner causes bending or rafting.	FEM, CFD-DEM, NDEM, SPH	Ice tank tests using varying thicknesses	[83,88,89,90,113,115]
Ship Velocity	- Low (glancing, quasi-static) - Medium (2–5 knots) - High (towing carriage tests)	- Higher speeds increase resistance and hull pressure. - Affects ice breaking and floe dynamics.	CFD-DEM, SPH, NDEM	Towing experiments, impact tests	[89,106,107,114]
Ship Resistance	- Ice concentration: 40–80% - Varied floe size, shape, channel width	- Resistance grows with concentration and floe size. - floe shape has less influence on mean resistance	DEM, NDEM, CFD-DEM	Resistance tests with synthetic/brash ice	[9,12,90,105,114]

gives the key information on the simulation of the sea ice–structure interaction, along with the brief findings, modeling applications, and experimental validations on their required parameters.

The SPH method simulates the complex process of a ship continuously breaking ice, encompassing initial contact, stress buildup, crack initiation and propagation, ice fragmentation, and the movement of broken ice around the hull. Gui et al. [99] also used SPH to simulate the impact of ice on a ship propeller. The propeller was modeled as a rotating rigid body interacting with ice particles. The ice model included brittle failure under impact. The simulation focused on localized ice damage from propeller impact, including crushing, cracking, and fragmentation. SPH is suitable for this high-deformation, fracture-dominated problem. A propulsion machinery system model was used to simulate the dynamic response of the system during the ice–propeller interaction based on different ice classes, number of propeller blades, propeller shaft length, flexible coupling stiffness, and number of cylinders [116]. A novel method was proposed to predict ice propulsion performance through the use of the alternative interaction coefficient system [117] and the critical issues in ship operation in ice; specifically, the scale effect of icebreaker propellers and the potential for an ice interaction coefficient, were addressed.

A coupled SPH–FEM approach is employed in [99]. SPH models the ice (large deformation, fracture), and FEM models the marine structure (elastic deformation). Coupling involves transferring forces and displacements at the ice–structure interface, ensuring momentum and energy conservation. This method is useful when ice undergoes large damage and the structure experiences significant loading. Further, SPH is applied to model ice failure during ship–ice interaction in [100].

A combination of CFD and DEM is likely used in [100]. CFD models fluid flow around the ship and through ice rubble. DEM models ice as discrete particles interacting with each other and the hull via contact forces based on particle properties. CFD–DEM coupling involves the interaction of fluid forces on ice particles and the influence of ice on fluid flow. The model simulates the hydrodynamic interaction between a ship and broken ice, predicting ship resistance and ice particle motion in response to the ship and water flow. To model propeller–ice–water interaction, a coupled CFD–DEM is employed [102]. CFD simulates propeller-induced flow. DEM models ice as discrete particles that interact with each other and with propeller blades via contact forces. The simulation details the physics of the propeller breaking and accelerating ice in water, analyzing propeller forces, ice trajectory and fragmentation, and the resulting flow field which was consistent with the model test results.

To simulate ice load on floating structures in broken ice, a DEM model is employed in [103] to simulate the broken ice as interacting 3D polyhedral elements, using a dilated contact algorithm for efficient interaction detection. The methodology solves the motion of individual ice floes and the structure based on contact forces, providing a detailed prediction of ice loads by capturing the discrete nature of the ice field. Figure 11 shows that at an 80% ice concentration, ice floes are constrained in front of the structure. However, with a 40% concentration, the greater space allows floes to move around the structure.

A numerical ice tank based on the adaptation of a lattice Boltzmann (LB)-based free surface flow solver is described for the simulation of complex fluid–ship–ice interactions in [108], optimized for graphics processing unit (GPU) parallelization. The ship is a moving rigid boundary, and ice is a rigid body with defined motion. Interaction is through boundary conditions (no-slip) and contact models. The numerical ice tank efficiently simulates scenarios like ship resistance in level ice or maneuvring around ice, treating ice as rigid. A FE method is used for the modeling [109], aimed at predicting icebreaking vessel resistance in brash ice, which is challenging due to the complex nature of interacting ice floes [118].

A structural model for level ice is presented using FEM in [79], considering fluid–structure interaction (FSI). For better understanding of maneuvrability and structural safety, the model predicts ice loads on a turning ship in level ice, explicitly considering hydrodynamic forces. A 3D coupled thermal mechanical bond-based peridynamics method was used [111] to model and simulate the thermal loading of the thermal de-icing process.. Peridynamics is also used to simulate ice cover fragmentation from complex interactions between ice fragments as the ship moves in [112], and an underwater explosion’s shockwave [115], applied as a pressure pulse. The peridynamic model simulates ice material response, including multiple crack initiation and propagation due to the dynamic loading. The simulation aims to understand ice cover breakup, analyzing crack initiation and propagation under high pressure and explosive loading.

The behavior of a ship in ice channels was studied using model-scale tests, with channels having different widths and ice thicknesses [119]. Numerical simulations of these model test scenarios were carried out utilizing an in-house program designed for ship operation in ice. Subsequent numerical simulations were carried out with other ships to gain general insights into performance in narrow ice channels.

The virtual mass method is employed with CFD and DEM for predicting total resistance in [104]. Optimal virtual mass coefficients were determined through synthetic ice model tests, considering a variety of floe ice distributions and ice conditions.

A combined CFD and DEM approach is used to simulate ship–wave–ice interaction [9]. Two algorithms are introduced to implement computational models with natural ice-floe fields, considering floe size distribution and randomness to investigate the influence of ship speed, ice concentration, thickness, and floe diameter. As depicted in Figure 12 (Fr: is the Froude number), ice resistance varies with various ice concentrations. The amount of resistance caused by ice varies depending on the specific circumstances, which is crucial for calculating power requirements and fuel needs.

Drucker–Prager yield function and fracture energy based on continuum damage mechanics were applied to single-prism ice compression simulations in [120]. Icebreaking patterns were analyzed from cone-ice sheet interaction simulations. The free decay of an icebreaker was numerically analyzed using the developed hydrodynamic plug-in HydroQus. Ice sheet breaking simulations were conducted to analyze the effect of hydrodynamic forces on ice resistance. A series of experiments were performed on hull vibration of a full-scale river icebreaker in [121].

A framework for numerical modeling of ship performance in level ice, including discussions on modeling purposes, hierarchical decomposition of the problem, strategies for accuracy and credibility, and validation methodology, is proposed in [113]. A prototype numerical model was developed based on this framework to simulate ship performance in level ice. Further, numerical simulation results were compared with full-scale measurement data from the ice trial.

A simplified method is proposed to assess ice resistance for large vessels navigating in narrow ice channels behind an icebreaker in [122]. It addresses the lack of mathematical descriptions for ice channel edge breaking by large ships under escort. The approach refines semi-empirical methods commonly employed for estimating ice resistance encountered in both level and broken ice conditions. Ship–ice glancing impacts were analyzed to determine how impact loads changed in space and time. This scenario is a key factor in designing bow structures for polar-class ships [123]. The global spatial migration of ice load along the hull, characterized by a parabola-shaped ice loading trail, was recognized.

A study to predict ice-induced resistance on a container ship in pack ice using a coupled CFD–DEM method was carried out in [105]. A sensitivity analysis of contact model parameters (friction and restitution coefficients) in CFD–DEM was conducted, featuring their importance for simulation accuracy. Their analysis suggested that ice resistance is largely a result of pack ice colliding, accumulating, and extruding at the bow.

An ice-area bulk carrier’s navigation through broken ice fields was simulated using the finite element method (FEM), with the ice modeled as an elastic material, in [110]. The study analyzed the main properties of ice loads, including average and extreme loads, and characteristic frequency, while also examining the effects of sailing speed and ice concentration.

To investigate the influence of channel width on broken-ice resistance, [114] used a numerical model based on the non-smooth discrete element method (NDEM), incorporating a limited impulse method for ice crushing failure. Sensitivity analyses were conducted to investigate the correlation between the channel width effect and factors such as ice concentration, ship velocity, and the thickness of the ice sheet. It was found that wider channels, combined with thicker ice and slower ship speeds, worsen ice resistance by creating more powerful and persistent force chains (see Figure 13).

The effect of ice floe shape on ship resistance in low-concentration broken ice conditions was analyzed using a numerical model based on the non-smooth discrete element method (NDEM) in [8]. It highlighted that square-shaped floes lead to the best ice resistance because they make contact more often and over a larger area, while hexagonal and circular floes result in lower resistance.

A procedure for predicting ice-induced hull pressure in ships operating in floe ice fields, combining model tests with CFD–DEM simulations, was introduced in [106]. A novel model test procedure using non-refrigerated synthetic ice and tactile pressure sensors for direct hull pressure measurement was introduced. It was identified that ice-induced pressure and contact duration generally increase with ship speed, especially in higher floe concentrations.

Using direct numerical simulations of the weakly nonlinear Schrödinger equation, [124] systematically investigated the nonlinear wave–ice interaction, demonstrating theoretical nonlinear wave focusing on level ice and identifying key parameters for its observation.

A coupled method of CFD–DEM for ship–ice and ice–ice interactions is used in [107] to simulate a ship navigating in both open-water channels and pack-ice channels, varying channel width, level ice thickness, and ship speed. It was found that channel width has a more pronounced effect on water resistance than the thickness of level ice. For ice resistance, both channel width and level ice thickness significantly influence the interaction, with level ice thickness decreasing and ice resistance decreasing as channel width increases (see Figure 14; $W/B$ refer to the dimensionless channel width).

3. Conclusions

This review highlights the significant advancements in both experimental and numerical methodologies in the field of Arctic and offshore engineering, particularly in accurately predicting the complex and dynamic characteristics of sea ice–structure interactions under various environmental conditions. Emphasizing numerical methods like DEM–CFD and FEM, this review consistently points to the importance of nonlinear coupled hydrodynamic analysis for accurate modeling. Further, experimental model tests, employing both refrigerated and artificial ice, are frequently used to validate the results. Despite the significant progress, several common limitations and challenges still exist in this area as follows.

While many numerical models are validated against model test data, full-scale validation remains challenging and less frequently reported due to the inherent difficulties and costs associated with full-scale ice trials. Many numerical models still rely on simplified ice material properties (e.g., elastic ice models), which may not fully capture the complex, nonlinear, and anisotropic behavior of real ice under various loading conditions.

The dynamic and highly nonlinear nature of ship–ice interaction, particularly concerning ice breaking, clearing, and accumulation (large and small amounts of ice), still presents modeling challenges to avoid catastrophic events that could arise from the navigation point of view. Furthermore, factors such as friction between ice and a ship’s hull, as well as interactions between ice floes, are often oversimplified.

CFD–DEM approaches have also been employed, but the full coupling between fluid dynamics and ice mechanics, especially the detailed modeling of ship-generated waves and their interaction with ice, can be computationally intensive and sometimes simplified. While parametric studies are common, the full range of parameter combinations (e.g., all permutations of ship speed, ice thickness, concentration, and floe geometry) is often too vast for comprehensive investigation within a single study.

It is recommended that the future research directions extend to:

• More extensive and systematic full-scale measurements, which are crucial to rigorously validate numerical models and experimental findings, bridging the gap between theoretical predictions and real-world performance.
• Development and integration of more sophisticated ice constitutive models that accurately represent the complex failure mechanisms (crushing, bending, shearing) and rheological properties of ice under various strain rates and temperatures.
• Further refinement of fully coupled fluid–ice interaction models to better capture the dynamic interplay between the ship, surrounding water, and ice, including the effects of propeller–ice interaction and maneuvering in ice. Further, incorporating uncertainty quantification methods into numerical simulations to account for the inherent variability in ice properties and environmental conditions provides more robust predictions.
• Investigating underwater ice-boundary data and exploring the application of AI and machine learning techniques for real-time prediction of ice resistance, optimization of ship navigation in ice, and analysis of large datasets from full-scale models. On the other hand, developing more standardized methodologies for both model tests and numerical simulations would ensure comparability and reproducibility of results across different research groups.
• Investigating oil spills in iced water in the Arctic region, which can be considered as one of the challenges for marine transport and offshore activities due to the smoothness of the ice sheet, which is non-symmetric between the top of the ice (smooth) and the bottom underwater interface (low smoothness). Also, oil-polluted ice contains pollution along the vertical, and it indicates that ice could be a non-continuous material with vertical lattice.
• Ensuring consistency between the models and experiments in this review; we suggest using satellite observations of ship wakes in ice to validate research findings against real-world data.
• Moving towards multi-physics and multi-scale modeling approaches that can seamlessly integrate different phenomena (e.g., structural response and ice mechanics) at various scales. Translating research findings into practical tools that can assist ship operators in making informed decisions for safe and efficient navigation in ice-covered waters, including route optimization and speed management.

Finally, the present review could be a good reference for sea ice–structure interaction on the nonlinear models problems arising in offshore and Arctic engineering.