Numerical Simulation of an Icebreaker Ramming the Ice Ridge

Abstract

During polar navigation, icebreakers frequently encounter ice ridges, which can significantly reduce navigation efficiency and even pose threats to structural safety. Therefore, studying the ramming of ice ridges by the icebreaker is of great importance. In this study, the ice ridge is decoupled into the consolidated layer and the keel for modeling. The consolidated layer is simplified as layered ice, and an innovative hybrid empirical–numerical method is used to determine the icebreaking loads. For the keel, a failure model is developed using the Mohr–Coulomb criterion in combination with the effective stress principle, accounting for shear failure in porous media and incorporating both cohesion and internal friction angle. The ship is restricted to surge motion only. A comparative analysis with the model test results was conducted to assess the accuracy of the method, with the predicted ice resistance showing deviation of 9.85% in the consolidated ice area and 10.48% in the keel area. Ablation studies were conducted to investigate the effects of different ice ridge shapes, varying retreat distances, and different ship drafts on the performance of ramming the ice ridge. The proposed method can quickly and accurately calculate ice ridge loads and predict their motion responses, providing a suitable tool for on-site rapid navigability assessment and for the design of icebreakers.

1. Introduction

Ice ridges are a common type of sea ice structure encountered by polar ships navigating in ice-covered regions [1]. Under the influence of ocean currents and other external factors, floating ice sheets collide with each other, causing deformation due to compression and shear interactions, resulting in the formation of rugged ice ridges [2,3]. Ice ridges represent extreme conditions and, compared to level ice, impose significantly greater ice loads on ship hulls. The structural forms of ice ridges are diverse, and the internal refrozen ice rubble is highly complex [4]. At present, there is still a lack of accurate constitutive models to describe their behavior, making reliable prediction of ice ridge loads extremely challenging. Accurately predicting the structural and motion responses of ships under ice ridge loading is of great importance for the design and optimization of polar ship structures. “Ramming the ice ridge” refers to the situation where an icebreaker is unlikely to break through the ice ridge in a single attempt. Instead, it needs to retreat a certain distance multiple times, accelerate, and ram again in order to pass through [5]. This study simulates multiple ramming events to investigate the relationship between ice ridge dimensions, operational strategies, and ice-breaking performance. This is beneficial for ensuring the safety and operational efficiency of icebreakers navigating through ice ridges. When encountering ice ridges during actual navigation, key parameters of the ice ridges can be obtained through onboard acoustic equipment or drone inspections. Rapid on-site simulations can then be conducted aboard the vessel to roughly estimate the time required to pass through the ice ridge and the maximum ice loads acting on the ship. This approach helps assess the icebreaker’s current passing performance, ensuring both structural safety and navigation efficiency.

The discrete element method (DEM) simulates objects using numerous particles, which is highly analogous to the composition of ice ridges made up of many discrete ice floes. Therefore, many researchers have employed DEM to numerically simulate the interaction between ships or other offshore structures and ice ridges. Sawamura and Tachibana [6] applied the DEM to simulate rubble ice accumulation and developed both two-dimensional and three-dimensional numerical models of ice ridge–ship interaction. Lau et al. [7] based on the discrete element program DECICE, improved the ship–ice interaction modeling and performed sensitivity analyses on factors affecting ice resistance, such as bow inclination angle, friction coefficient, and penetration depth. Zhu and Ji [8] proposed a discrete element method for ice ridges based on bonded sphere elements and simulated the interaction process between ice ridges and cylindrical vertical structures. However, the modeling techniques and computational efficiency of DEM still need improvement to meet the requirements of on-site real-time assessment of navigability during navigation [9]. The finite element method (FEM) is based on continuum mechanics and employs the “element birth and death” technique to prevent ice cracks from propagating into the interior of elements [10]. This ensures that the numerical simulation can continue even when material failure occurs. Vakulenko and Bolshev [11] constructed an ice ridge material model based on the Mohr–Coulomb criterion and simulated on-site crushing tests using finite element quasi-static analysis. By comparing the simulation results with those from theoretical calculation models, the study demonstrated that finite element methods provide accurate load predictions for ice ridges. Serré [12] studied the constitutive behavior of ice ridges based on the Drucker-Prager yield criterion through experiments and numerical simulations and used ABAQUS 6.11 software to simulate the vertical compression of a truncated cone against an ice ridge keel. However, in current studies using finite element methods to simulate the interaction between structures and ice ridges, the damage and discontinuities of ice ridges are rarely considered. A combined approach of numerical simulation and theoretical formulas can achieve a balance between computational accuracy and efficiency. Zhou, Gao and Li [2] proposed a numerical calculation method for ship-ice ridge interaction based on theoretical formulas, where the ice ridge load consists of consolidated layer load and keel load. According to the inclination angle between the consolidated layer and the ship hull, two failure modes—bending and breaking—are considered. The ice ridge load is calculated by determining the horizontal position of the broken ice relative to the ship and its corresponding motion. Kuuliala et al. [13] combined the ship motion equation with a probabilistic model of ice ridge geometry to propose a numerical method for estimating ice ridge loads, which can be applied to ice route planning and ship maneuverability prediction.

In summary, both DEM and FEM require long computation times and face difficulties in constructing constitutive models that accurately describe the mechanical properties of ice ridges. In this study, the ice ridge is decoupled into the consolidated layer and the keel. The consolidated layer is similar to level ice in terms of structure and mechanical properties. The failure of the level ice is simulated using a combined empirical–numerical analytical approach. The ridge keel mainly consists of rubble ice blocks, which are regarded in this study as granular materials exhibiting Mohr–Coulomb characteristics. A failure model for the ice ridge keel is established using soil mechanics methods. The ship is constrained to move only in the surge direction during collisions with ice. Conduct simulations of constant-speed collisions with ice ridges to replicate model test scenarios and perform comparative analyses to validate the accuracy of the method. Conduct a multi-factor ablation study to clarify the relationships between these factors and the peak load, the time required to pass, and the number of collisions.

2. Mathematical Methods

2.1. Ship Motion Model

The Earth-fixed coordinate system and the body-fixed coordinate system are used to represent ship motion, both following the right-hand rule. The Earth-fixed coordinate system $O-x_E{y}_E{z}_E$ is attached to the Earth. The body-fixed coordinate system $G-xyz$ is attached to the ship and moves with it, with its origin located at the ship’s center of gravity $G$. The $G_x$ axis points in the forward direction of the ship, while $G_z$ points vertically upward. In this coordinate system, $u\mathrm{a}\mathrm{n}\mathrm{d}v$ represent the velocity components of the ship along the longitudinal and transverse directions, respectively, and $r$ is the yaw angular velocity about the vertical axis. The 3-Dof ship maneuvering equations of motion are expressed as follows

$$\begin{array}{c}\left\{\begin{array}{c}{\mathit{\eta }}^{\prime }=\mathit{J}\mathit{\nu }\\ \left(\mathit{M}+\mathit{A}\right){\mathit{\nu }}^{\prime }+\mathit{B}\mathit{v}+\mathit{C}\mathit{\eta }=\mathit{\tau }\end{array}\right.\end{array}$$

(1)

$$\begin{array}{c}\mathit{J}=\left[\begin{array}{ccc}\mathit{cos}\left(\psi \right)& -\mathit{sin}\left(\psi \right)& 0\\ \mathit{sin}\left(\psi \right)& \mathit{cos}\left(\psi \right)& 0\\ 0& 0& 1\end{array}\right],\mathit{\eta }=\left[\begin{array}{c}x\\ y\\ \psi \end{array}\right],\mathit{\nu }=\left[\begin{array}{c}u\\ v\\ r\end{array}\right],\mathit{\tau }=\left[\begin{array}{c}{\mathit{f}}_b^b\\ {\mathit{m}}_b^b\end{array}\right]\end{array}$$

(2)

Here, $\mathit{J}$ denotes the kinematic transformation matrix from the body-fixed coordinate system to the Earth-fixed coordinate system; $\mathit{\eta }$ is the position vector of the ship; $\mathit{\nu }$ is the velocity vector; and $\mathit{\tau }$ is the external force vector applied to the ship. $\mathit{M}$ is the mass matrix of the hull, $\mathit{A}$ is the added mass matrix of the hull, $\mathit{B}$ is the damping matrix, and $\mathit{C}$ is the restoring force matrix. Since only three degrees of freedom are considered, the damping matrix $\mathit{B}$ due to hydrodynamic resistance and the restoring force matrix $\mathit{C}$ caused by ship motion are set to zero. The hydrodynamic forces acting on the ship in the three degrees of freedom are estimated using another formula. The mass matrix $\mathit{M}$ can be estimated as

$$\begin{array}{c}M=\left[\begin{array}{ccc}\mathsf{\Delta }& 0& 0\\ 0& \mathsf{\Delta }& 0\\ 0& 0& \mathsf{\Delta }{\left(0.25L\right)}^2\end{array}\right]\end{array}$$

(3)

where $\mathsf{\Delta }$ is the displacement and $L$ is the length of the ship. An adaptive step-size Runge–Kutta method is used to solve the equation and obtain the ship’s motion response. In this study, we investigate the ice load experienced by the ship during collisions with ice ridges, the number of collisions, and the time required for passage. Therefore, only the motion of the ship in the surge direction is considered. As a result, the motion equation is effectively reduced to a single degree of freedom.

2.2. Consolidation Layer Failure Model

A two-dimensional Boolean operation between the ship’s waterline and the ice field boundary is used to obtain the overlapping area. After three-dimensional correction, this yields the ship–ice contact area and the resulting compressive force, enabling the prediction of level ice load. This approach has been adopted by multiple researchers and has been validated for its accuracy and efficiency [14,15,16,17]. When a ship navigates through level ice, most of the ice resistance comes from the processes of breaking the ice and displacing the ice floes [18]. Upon contact between the ship and the sea ice, local crushing occurs first, with the crushing force increasing as the contact area expands. When the crushing force exceeds the bearing capacity of the ice, bending failure occurs, causing ice floes to detach from the level ice. The ship continues to move forward, colliding with the broken ice floes. The ice floe initially overturns and may strike the hull, eventually aligning parallel to the hull. Under the combined action of the ship’s thrust and hydrodynamic resistance, the ice floe is pressed beneath the hull, sliding along the hull surface until it detaches and rises to the surface again.

The local crushing force between the ship and ice can be obtained by multiplying the effective crushing strength of the ice ${\sigma }_c$ by the contact area $A_c$

$$\begin{array}{c}{F}_{cr}={\sigma }_c{A}_c\end{array}$$

(4)

The planar projected area can be obtained by performing a two-dimensional Boolean union operation between the ship waterline and the boundary of the ice field. Considering the thickness of the level ice and the inclination angle at the ship-ice contact, a three-dimensional correction is applied to obtain the crushing area $A_c$. When the vertical components of the crushing force $F_{cr}$ and the friction force ${\mu }_i{F}_{cr}$ exceed the ice bearing capacity $P_f$, bending failure occurs in the form of a circumferential crack with radius $R$. The bearing capacity of sea ice can be calculated as follows

$$\begin{array}{c}{P}_f=C_f{\left({\displaystyle \frac{\theta }{\pi }}\right)}^2{\sigma }_f{h}_i^2\end{array}$$

(5)

where $C_f$ is an empirical coefficient set to 2.9, ${\sigma }_f$ is the flexural strength of sea ice, $h_i$ is the ice thickness, and $\theta$ the opening angle of the ice wedge. The crack radius can be calculated as

$$\begin{array}{c}R=C_{l}l\left(1.0+C_v{V}_{n,2}^{rel}\right)\end{array}$$

(6)

where $V_{n,2}^{rel}$ is the relative velocity in the normal direction at the ship-ice contact, $C_l$ and $C_v$ are two empirical parameters set to 0.27 and −0.09, respectively, and $l$ is the characteristic length of the ice

$$\begin{array}{c}l={\left[{\displaystyle \frac{E{h}_i^3}{12\left(1-{\nu }^2\right){\rho }_{w}g}}\right]}^{{\displaystyle \frac{1}{4}}}\end{array}$$

(7)

where $E$ is the Young’s modulus of ice, $\nu$ is the Poisson’s ratio, ${\rho }_w$ is the density of seawater, and $g$ is the gravitational acceleration.

The consolidated layer is discretely modeled as a single layer of square ice grids, with each grid having a side length equal to the icebreaking radius $R$ derived from Equation (6). By calculating the overlap area between the hull waterline nodes and the ice grids, the local contact area and corresponding crushing force are determined. When the crushing force exceeds the ice bearing capacity given by Equation (5), the ice grid is considered to fail. Once failure occurs, the affected ice grids near the ship are immediately detached from the consolidated layer and are directly removed in the numerical simulation. A schematic diagram of the discretized layered ice is shown in Figure 1, where white grids represent intact ice, blue grids indicate failed ice, and red grids correspond to the ridge keel area. The submergence resistance caused by the subsequent interaction between the broken ice and the ship is calculated using an empirical formula. Compared to circular, elliptical, or wedge-shaped elements, square grids can fully cover the entire computational domain of the layered ice, which facilitates the implementation of complex ship-ice contact algorithms and more closely resembles actual ice fields. Moreover, the use of square grids simplifies the updating of the ice edge shape after local grid failure and ensures consistency of the ice edge throughout the simulation, effectively improving computational efficiency.

After the ice grids are broken, the resulting ice floes become submerged in water and slide along the hull. This process generates submerged ice resistance. Since the density of ice is lower than that of seawater, the broken ice fragments rely on their buoyancy to adhere to the surface of the hull. The resulting resistance originates directly from the normal force and friction force on the hull surface. In this paper, the formula proposed by Lindqvist [19], which is based on the principle of potential energy and accounts for the influence of velocity, is used to calculate the submerged ice resistance. The component related to submerged resistance is

$$\begin{array}{c}{R}_{sub}=(1+{\displaystyle \frac{9.4V}{\sqrt{gL}}})\left({\rho }_w-{\rho }_i\right)g{h}_{i}B\left[T{\displaystyle \frac{B+T}{B+2T}}+k\right]\end{array}$$

(8)

$$\begin{array}{c}k=\mu \left(0.7L-{\displaystyle \frac{T}{tan\vartheta }}-{\displaystyle \frac{B}{4tan\alpha }}+Tcos\varphi cos\psi \sqrt{{\displaystyle \frac{1}{{\sin }^2\vartheta }}+{\displaystyle \frac{1}{{\tan }^2\alpha }}}\right)\end{array}$$

(9)

where ${\rho }_w$ is the density of seawater; ${\rho }_i$ is the density of sea ice; $L$, $B$, and $T$ are the length, beam, and draft, respectively; $V$ is the ship’s velocity; $\mu$ is the ship-ice friction coefficient; $\vartheta$ is the stem angle, $\alpha$ is the waterline angle.

It should be noted that the Lindqvist formula is used to calculate the submergence resistance experienced by an icebreaker when breaking consolidated layer or level ice. The resistance caused by the interaction between the hull and broken ice generated from the failure of the ice keel is not considered, which is a limitation of this method. However, since the submergence resistance is several orders of magnitude smaller than the icebreaking resistance and the ice-keel breaking resistance, the impact of using this method is relatively small.

2.3. Ice Ridge Keel Failure Model

An ice ridge mainly consists of the ridge sail, the consolidated layer, and the keel, as shown in Figure 2. Here, $h_k$ is the depth of the keel, ${\varphi }_k$ is the keel angle, $h_c$ is the thickness of the consolidated layer, and $h_s$ is the height of the ridge sail. Timco and Burden [1] proposed approximating the cross-sections of the keel and ridge sail as ideal triangular shapes, with the keel height $h_k$ being about 4.4 times that of the ridge sail $h_s$. According to observations by Kankaanpää [20] in the Baltic Sea, the keel height $h_k$ is approximately 6.35 times that of the ridge sail $h_s$. In a state of static equilibrium, the volume of the ridge sail is much smaller than that of the ice ridge keel. Due to its loose and porous nature, the strength of the ridge sail is also lower than that of the consolidated layer; therefore, the influence of the ridge sail is neglected in this study.

The deformation and failure of ice ridge keels under external forces mainly occur in two forms. First, the viscous bonds between ice blocks break due to shear forces. Second, after the failure of these bonds, the behavior of the broken ice depends on the hydrostatic pressure. Under low hydrostatic pressure, the broken ice blocks behave like granular materials with friction, undergoing relative motion and rotation, leading to an increase in volume. Under high hydrostatic pressure, the broken ice blocks are compacted or further fragmented, with smaller pieces filling the voids between larger blocks, resulting in a decrease in volume. As a porous medium, the shear failure mechanism of ice ridges is essentially similar to that of soil. The shear strength of a porous medium follows the Mohr–Coulomb criterion [21,22], typically expressed as:

$$\begin{array}{c}\tau =\sigma tan\varphi +c\end{array}$$

(10)

where $\sigma$ is the total stress, $\varphi$ is the internal friction angle, and $c$ is the cohesion. Porous media undergo deformation under the combined action of external and internal stresses. Its effective stress ${\sigma }^{\prime }$ is equal to the difference between the total stress $\sigma$ and the pore water pressure $u$

$$\begin{array}{c}{\sigma }^{\prime }=\sigma -u\end{array}$$

(11)

Due to the high porosity of the ice keel, during slow loading, the water within the keel’s pores is gradually expelled. Compared to the shear and dilation effects of the broken ice blocks in the keel, the pressure transmitted by the pore fluid flow can be considered negligible. Therefore, in simulations, the total stress is assumed to be approximately equal to the effective stress, and only the internal friction angle $\varphi$ and cohesion $c$ need to be determined to calculate the strength of the ice keel. In engineering practice, the Mohr–Coulomb theory is commonly used to determine these two key parameters and to assess the failure of porous media. The failure of the material is related both to the magnitude of the shear stress on a given plane and the normal stress on that plane. Assuming the element undergoes shear failure, the direction cosines of the failure plane relative to the direction of the normal stress are $(s_1,s_3)$, the area of the failure plane is 1, and the forces acting on the element are shown in Figure 3.

According to the principle of force equilibrium, the normal stress $\sigma$ and shear stress $\tau$ on the shear failure plane can be expressed as

$$\begin{array}{c}\sigma =s_1^2{\sigma }_1+s_3^2{\sigma }_3\end{array}$$

(12)

$$\begin{array}{c}\tau =s_1{s}_3\left({\sigma }_1-{\sigma }_3\right)\end{array}$$

(13)

where ${\sigma }_1$ and ${\sigma }_3$ are the principal normal stresses. Substituting into the effective stress Equation (9), the following expression can be obtained

$$\begin{array}{c}{s}_1{s}_3\left({\sigma }_1-{\sigma }_3\right)=\left(s_1^2{\sigma }_1+s_3^2{\sigma }_3\right)\mathit{tan}\varphi +c\end{array}$$

(14)

As shown in Figure 4, the stress state is analyzed using the Mohr’s stress circle. The Mohr’s circle formed by the maximum and minimum principal normal stresses, ${\sigma }_1$ and ${\sigma }_3$, more easily reaches the limit equilibrium state. According to Mohr’s theory, Equation (13) can be used to derive the limit equilibrium condition

$$\begin{array}{c}{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{{\sigma }_1+{\sigma }_3+2c\mathit{cot}\varphi }}=\mathit{sin}\varphi \end{array}$$

(15)

$$\begin{array}{c}{s}_1=\mathit{cos}\left(45^{°}+{\displaystyle \frac{\varphi }{2}}\right)\end{array}$$

(16)

$$\begin{array}{c}{s}_3=\mathit{cos}\left(45^{°}-{\displaystyle \frac{\varphi }{2}}\right)\end{array}$$

(17)

when the left side of Equation (15) is less than the right side, the porous medium is considered to be intact; otherwise, the porous medium is considered to have failed. By selecting any two or more Mohr’s failure circles at the limit state and solving the corresponding system of equations, the internal friction angle $\varphi$ and cohesion $c$ can be determined.

Once the internal friction angle and cohesion are determined, the strength of the ice ridge can be calculated. The local failure models of ice ridge keels developed by [23] and [24] are adopted, as expressed by the following equations

$$\begin{array}{c}{\mathit{F}}_{\mathit{k}}={\mathit{\mu }}_{\mathit{\varphi }}{\mathit{h}}_{\mathit{e}}{\mathit{D}}_{\mathit{e}}\left({\displaystyle \frac{{\mathit{h}}_{\mathit{e}}{\mathit{\mu }}_{\mathit{\varphi }}{\mathit{\gamma }}_{\mathit{e}}}{\mathit{2}}}+\mathit{2}\mathit{c}\right)\left(\mathit{1}+{\displaystyle \frac{{\mathit{h}}_{\mathit{e}}}{\mathit{6}{\mathit{D}}_{\mathit{e}}}}\right)\end{array}$$

(18)

$$\begin{array}{c}{\mu }_{\varphi }=\mathit{tan}\left(45^o+{\displaystyle \frac{\varphi }{2}}\right)\end{array}$$

(19)

$$\begin{array}{c}{\gamma }_e=\left(1-\eta \right)\left({\rho }_w-{\rho }_i\right)g\end{array}$$

(20)

In the equation, $F_k$ is the ultimate load for local failure of the ice ridge keel, $\varphi$ is the internal friction angle, $c$ is the keel cohesion, ${\mu }_{\varphi }$ is the friction coefficient, ${\gamma }_e$ is the effective buoyancy, $\eta$ is the keel porosity, ${\rho }_w$ is the water density, ${\rho }_i$ is the ice density, $D_e$ is the effective structural width, it varies as a function of the extent of contact, $h_e$ is the effective keel depth.

The ice ridge keel is modeled as a continuum. As shown in Figure 5, By computing the union of the discretized 3D ship hull and the continuous ice ridge keel, the effective depth and effective structural width are calculated in real time. The half-breadth waterlines at each frame position are extracted from the ship’s 3D model. Since the initially extracted data is not sufficiently detailed, interpolation is applied to obtain smoother and more precise curves, as shown in Figure 6. The refined half-breadth waterline data obtained through interpolation is then layered according to draft depth. Using the design waterline along with its adjacent upper and lower curves, parameters such as the spacing between nodes, flare angle, and waterline angle at each point on the design waterline can be calculated.

2.4. Hydrodynamic and Propulsion Model of the Ship

During the repeated ramming of ice ridges by an icebreaker, the ship can be considered to be in open water during multiple acceleration phases. Therefore, it is necessary to account for the influence of hydrodynamic resistance acting on the vessel. Neglecting the effect of waves, the three-degree-of-freedom hydrodynamic forces and moments acting on the hull are calculated using empirical formulas

$$\begin{array}{c}{X}_H={\displaystyle \frac{0.075}{{2\left({log}_{10}^{R_n}-2\right)}^2}}{\rho }_w{S}_{\omega }u\left|u\right|\end{array}$$

(21)

$$\begin{array}{c}{Y}_H={\displaystyle \frac{1}{2}}{\rho }_w{\int }_L^{\phantom{­}}{C}_D\left(x\right)D\left(x\right)v\left(x\right)\left|v\left(x\right)\right|dx\end{array}$$

(22)

$$\begin{array}{c}{N}_H={\displaystyle \frac{1}{2}}{\rho }_w{\int }_L^{\phantom{­}}{C}_D\left(x\right)D\left(x\right)v\left(x\right)\left|v\left(x\right)\right|xdx\end{array}$$

(23)

where $R_n$ is the Reynolds number, $S_{\omega }$ is the wetted surface area of the hull, $C_D\left(x\right)$ is the drag coefficient at each longitudinal position of the hull—defined as the resistance coefficient of an infinitely long cylinder with a cross-sectional area equivalent to that of the hull at each longitudinal position, and $D\left(x\right)$ is the draft at each longitudinal position. The integration is carried out along the ship length $L$.

The net thrust $T_{net}$ used to overcome ice resistance can be calculated using the formula proposed by Juva and Risk [25]

$$\begin{array}{c}{T}_{net}=T_B\left(1-{\displaystyle \frac{v_1^{rel}}{3V_{ow}}}-{\displaystyle \frac{2}{3}}{\left({\displaystyle \frac{v_1^{rel}}{v_{ow}}}\right)}^2\right)\end{array}$$

(24)

where $T_B$ is the bollard pull, $v_{ow}$ is the maximum open-water speed, and $v_1^{rel}$ is the forward component of the relative velocity between the ship and the ice.

3. Validation

The model tests were conducted in the ice–water basin at Tianjin University. A model ship was towed at a constant speed of 0.3 m/s to collide with an ice ridge, and the ice loads acting on the hull were recorded. The ship parameters at the model scale and the ice parameters are shown in

Table 1. Parameters of the Icebreaker (full scale).

Parameter	Value	Unit
Length (L)	125	m
Breadth (B)	23	m
Depth (T)	8	m
Inclination Angle	20	°
Water angle	34	°

and

Table 2. Parameters of the icefield (model scale).

	Parameter	Value	Unit
Consolidation layer	Ice thickness	0.01	m
	Density	900	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
	Coefficient of friction	0.1	—
	Flexure strength	16.6	kPa
	Crush strength	97.9	kPa
	Elastic modulus	90	MPa
	Poisson ratio	0.3	—
Ice ridge keel	Length	2	m
	Depth	0.133	m
	width	0.54	m
	Angle	50	°
	Porosity	0.3	—
	Internal friction angle	35	°
	Cohesive force	0.1	kPa

, respectively. The scale ratio of the model test is 1:60.

The ice ridge keel is formed by crushing pre-prepared level ice and then refreezing it. Before the test begins, a smooth level ice sheet with a thickness of 5 mm is first prepared. The level ice is then broken into ice fragments approximately 50 mm in size, which are arranged and refrozen in a specific manner to form the ice ridge. Then, freezing was continued until the thickness of the level ice grew to 0.01 m.

The icebreaking process at several typical moments is illustrated in Figure 7. The ice load acting on the ship during the process is shown in Figure 8. Starting from 44.6 s, the ship begins to come into contact with the ice ridge, corresponding to the appearance of load peaks at the same time in Figure 8. The black solid line represents the numerical simulation results, while the red dashed line represents the model test results. The ice loads caused by the ice ridge keel are significantly greater than those in the level ice, causing the load curve to no longer return to zero.

In the model tests, the average ice resistance in the consolidated ice area and the keel area was 43.15 N and 84.41 N, respectively. The ice resistance measured during the experiments was 38.90 N and 75.56 N, resulting in deviation of 9.85% and 10.48%, respectively. The numerical simulation results show good agreement with the model test results in terms of the trend of ice load variation, with similar average and peak load values.

4. Discussion

When encountering a large ice ridge, a ship often needs to perform multiple collisions to break through it. Here, a full-scale simulation case (case 1–3) is presented to illustrate the process of breaking an ice ridge through repeated impacts. The icebreaking process at key moments is shown in Figure 9. The white grids represent the undisturbed consolidated layer, while the red grids in the center indicate the ice ridge keel area. The red grids near the waterline show regions currently in contact with the ship, and the blue grids represent the failed and damaged areas after interacting with the ship.

The time-history curves of displacement, velocity, and ice load experienced by the ship during multiple collisions with ice ridges are shown in Figure 10. At the beginning, the ship is positioned at a certain distance from the ice field and starts accelerating from an initial speed of 2 knots. From 0 to 24 s, the ship sails in open water until it comes into contact with level ice. At this stage, the load consists only of water resistance and partially submerged resistance, both of which increase with the ship’s speed. Between 24 and 56.9 s, the ship enters the level ice field and its speed decreases. At this stage, the load exhibits a periodic rise-and-fall pattern. As the contact area between the ship and the ice increases, the ice load gradually rises. Once it exceeds the bearing capacity of the level ice, the ice fractures, and the ice load drops accordingly. From 56.9 to 68.9 s, the ship collides with the ice ridge, and its speed rapidly decreases to zero. At this point, the ice load exhibits a distinct peak, with the maximum value occurring at this moment. The ship is unable to pass through the ice ridge in a single attempt. Subsequently, the ship is moved backward by 20 m. It should be noted that the backward motion is not simulated dynamically—instead, the ship is instantaneously repositioned 20 m backward. The ship then begins to accelerate for a second collision attempt. During the second collision with the ice ridge, the ship’s speed decreases but does not drop to zero. A distinct load peak also occurs during the second collision. After that, the ship successfully passes through the ice ridge.

4.1. Effect of Different Ice Ridge Shapes on Transit Performance

4.1.1. Varying Consolidated Layer Thicknesses

In this study, ablation studies were conducted under level ice thicknesses of 1.0 m, 1.5 m, and 2.0 m to investigate the corresponding transit performance characteristics. All other parameters were held constant across the test conditions, as shown in

Table 3. Simulation parameters of different consolidated layer thicknesses.

	Parameter	Value	Unit
Consolidation layer.	Ice thickness	1.0/1.5/2.0	m
	Density	900	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
	Coefficient of friction	0.1	—
	Flexure strength	700	kPa
	Crush strength	2800	kPa
	Elastic modulus	5400	MPa
	Poisson ratio	0.3	—
Ice ridge keel	Depth	10	m
	Angle	25	°
	Porosity	0.3	—
	Internal friction angle	60	°
	Cohesive force	32.5	kPa

. The simulation ends after the vessel has completely passed through the ice ridge and proceeded an additional 400 m. The simulation results are shown in

Table 4. Simulation result of different consolidated layer thicknesses.

ID	Ice Thickness (m)	Impact Count	Time Consumed (s)	Maximum Ice Load (kN)
1-1	1.0	1	93.3	45.985
1-2	1.5	1	121.9	61.409
1-3	2.0	2	165.9	55.331

and Figure 11. It can be observed that as the level ice thickness increases, the ship experiences a reduction in speed upon initial contact with the ice ridge. When the level ice thickness reaches 2 m, it exerts significant resistance on the ship, limiting its kinetic energy upon contact with the ice ridge. As a result, the ship is unable to pass through in a single attempt. In addition, the time required to pass through the ice ridge also increases with the level ice thickness. When the level ice thickness is 1.5 m, the maximum peak load occurs during the impact process. A thicker level ice cover results in a reduction in ship speed when impacting ice ridges, thereby weakening the kinetic energy during the collision. In actual navigation, onboard equipment can be used to detect and select areas with thinner level ice for entry, in order to preserve the ship’s kinetic energy as much as possible and reduce the number of collisions required to pass through ice ridges.

4.1.2. Varying Keel Depths

Ablation studies were conducted for keel depths of 10 m, 15 m, and 20 m, with all other parameters kept constant, as shown in

Table 5. Simulation parameters of varying keel heights.

	Parameter	Value	Unit
Consolidation layer	Ice thickness	2.0	m
	Density	900	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
	Coefficient of friction	0.1	—
	Flexure strength	700	kPa
	Crush strength	2800	kPa
	Elastic modulus	5400	MPa
	Poisson ratio	0.3	—
Ice ridge keel	Depth	10/15/20	m
	Angle	25	°
	Porosity	0.3	—
	Internal friction angle	60	°
	Cohesive force	32.5	kPa

. The corresponding ice ridge shapes for each condition are illustrated in Figure 12. When the keel angle remains unchanged, increasing the keel depth effectively enlarges the entire keel. This results in a larger area that the ship must break through in both the vertical and longitudinal directions. The simulation results are shown in

Table 6. Simulation result of varying keel heights.

ID	Keel Depth (m)	Impact Count	Time Consumed (s)	Maximum Ice Load (MN)
2-1	10	2	165.3	55.331
2-2	15	5	228.4	97.244
2-3	20	9	313.4	154.830

and Figure 13. Since the level ice thickness and mechanical properties are identical across all cases, the simulation results before the first impact are the same. With a keel depth of 10 m, the ship passes through the ridge with only 2 impacts. At 15 m, four impacts are required, and at 20 m, the condition is most severe, requiring 9 impacts. The corresponding transit times are 165 s, 227 s, and 312 s, respectively. For every 5 m increase in keel depth, the time required is approximately 1.3 times the original. Since the ice ridges have a fixed angle and are all arranged starting from the leftmost point, the ice loads during the first collision with the ridge are largely aligned in time phase. During the collision process, the peak ice load increases with keel depth and exhibits an approximately linear trend. In actual navigation, field investigation and underwater radar can be used to detect regions with shallower keel depths for impact, which can improve transit efficiency, reduce peak loads, and lower the risk of structural damage.

4.1.3. Varying Angles of Inclination

The simulation parameters for keel inclinations of 20°, 25°, 30°, and 35° kept constant as listed in

Table 7. Simulation parameters of varying angles of inclination.

	Parameter	Value	Unit
Consolidation layer	Ice thickness	2.0	m
	Density	900	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
	Coefficient of friction	0.1	—
	Flexure strength	700	kPa
	Crush strength	2800	kPa
	Elastic modulus	5400	MPa
	Poisson ratio	0.3	—
Ice ridge keel	Depth	15	m
	Angle	20/25/30/35	°
	Porosity	0.3	—
	Internal friction angle	60	°
	Cohesive force	32.5	kPa

. A side view of the ice ridge is shown in Figure 14, where it can be observed that while the keel depth remains unchanged, a smaller inclination angle results in a wider keel. The simulation results are shown in

Table 8. Simulation result of varying angles of inclination.

ID	Angle (m)	Impact Count	Time Consumed (s)	Maximum Ice Load (MN)
3-1	20	7	270.7	100.91
3-2	25	5	228.4	97.244
3-3	30	4	215.4	105.18
3-4	35	3	194.1	107.96

and Figure 15, with all other. This implies that the ship has to impact a longer contact region. With the smallest keel inclination, six impacts were required, and the total collision time was the longest. As the keel inclination decreases, more impacts are needed, and more time is consumed. As the keel angle decreases, the slope of the keel’s side becomes gentler, resulting in a slower rate of load application when the ship makes contact with it. In the ice load curve, this is reflected by a rightward shift in the timing of the peak load as the keel angle decreases. The peak ice load during collisions is around 100 MN, which is attributed to the unchanged keel depth and thus a consistent maximum level of contact between the ship and the ice ridge. It can be observed that the peak ice load is primarily influenced by the keel depth, while changes in keel angle affect the length of the keel region that the ship must pass through, thereby extending the transit time.

4.2. Strategies for Enhancing Transit Efficiency

4.2.1. Different Retreat Distances

All parameters are kept constant, as listed in

Table 9. Simulation parameters of different retreat distance.

	Parameter	Value	Unit
Consolidation layer	Ice thickness	2.0	m
	Density	900	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
	Coefficient of friction	0.1	—
	Flexure strength	700	kPa
	Crush strength	2800	kPa
	Elastic modulus	5400	MPa
	Poisson ratio	0.3	—
Ice ridge keel	Depth	15	m
	Angle	25	°
	Porosity	0.3	—
	Internal friction angle	60	°
	Cohesive force	32.5	kPa

. The simulation results for different retreat distances are shown in

Table 10. Simulation result of different retreat distance.

ID	Retreat Distance (m)	Impact Count	Impact Velocity (kts)	Time Consumed (s)	Second Maximum Ice Load (MN)
4-1	10	5	3.30	226	84.09
4-2	20	5	4.26	228	87.82
4-3	30	4	4.95	222	83.45

and Figure 16. In all conditions, the initial distance between the ship and the ice is the same. The retreat distances considered are 10, 20, and 30 m. A larger retreat distance results in a higher speed during re-impact. Since the initial distance between the ship and the ice ridge is the same in all three scenarios, the velocity and load curves during the first impact are identical. The peak ice load during re-impact shows only minor variation and does not exhibit a clear correlation with retreat distance. The total time required for the ship to pass through is similar for the 10 m and 20 m retreat distances, while the shortest passage time is observed with a 30 m retreat. It should be noted that in actual navigation, reversing a ship is a time-consuming operation, involving pitch adjustment or engine reversal, stopping, and then re-acceleration. This aspect is not considered in the simulation; the simulated time only reflects the cumulative time required for the ship to accelerate and perform the collision. Therefore, if the preparation time required for reversing 30 m is included, the overall time to pass through may be even longer.

4.2.2. Different Drafts

The simulation results for different draft conditions are shown in

Table 11. Simulation result of different drafts.

ID	Draft (m)	Impact Count	Time Consumed (s)	Maximum Ice Load (MN)
5-1	7.5	2	148.1	96.37
5-2	8	5	226.0	97.24
5-3	8.5	8	300.7	93.30

and Figure 17, with all other parameters kept constant as listed in Table 9. The design draft is 8 m, and simulations were conducted by increasing and decreasing the draft by 0.5 m. Considering the effect of ship mass and inertia at different drafts, the ship’s displacement was adjusted accordingly in the simulations. The results show that reducing the draft leads to fewer collisions and a shorter passage time, indicating a significant improvement in ice ridge transit performance. With a draft of 7.5 m, the ship was able to pass through the ice ridge with only two collisions. By reducing the draft, the contact area between the hull and the ice ridge is decreased, thereby lowering the ice load and improving passage performance. In actual navigation, within safe operating regulations, reducing ballast water can enhance the ship’s ability to pass through ice ridges.

4.3. Effect of Ice Ridge Mechanical Properties on Transit Performance

4.3.1. Varying Internal Friction Angle

The internal friction angle and cohesion determine the strength of an ice ridge keel. Therefore, it is necessary to conduct a trend analysis of these two factors. The simulation input parameters for internal friction angles ranging from 30° to 60° at intervals of 10° are shown in

Table 12. Simulation parameters of varying internal friction angle.

	Parameter	Value	Unit
Consolidation layer	Ice thickness	2.0	m
	Density	900	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
	Coefficient of friction	0.1	—
	Flexure strength	700	kPa
	Crush strength	2800	kPa
	Elastic modulus	5400	MPa
	Poisson ratio	0.3	—
Ice ridge keel	Depth	15	m
	Angle	25	°
	Porosity	0.3	—
	Internal friction angle	30/40/50/60	°
	Cohesive force	32.5	kPa

, and the simulation results are presented in

Table 13. Simulation result of varying internal friction angle.

ID	Internal Friction Angle (m)	Impact Count	Time Consumed (s)	Maximum Ice Load (MN)
6-1	30	1	125	26.96
6-2	40	1	131	31.46
6-3	50	1	142	46.59
6-4	60	1	165	55.32

and Figure 18. It can be observed that as the internal friction angle increases, the peak ice load during keel ramming increases, and the ship experiences greater deceleration during the ramming process, resulting in longer passage times. However, since the ship’s kinetic energy is sufficient, all cases achieve passage in a single ramming attempt.

4.3.2. Varying Cohesive Force

The simulation input parameters for cohesion values ranging from 10 kPa to 40 kPa at intervals of 10 kPa are shown in

Table 14. Simulation parameters of varying cohesive force.

	Parameter	Value	Unit
Consolidation layer	Ice thickness	2.0	m
	Density	900	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
	Coefficient of friction	0.1	—
	Flexure strength	700	kPa
	Crush strength	2800	kPa
	Elastic modulus	5400	MPa
	Poisson ratio	0.3	—
Ice ridge keel	Depth	15	m
	Angle	25	°
	Porosity	0.3	—
	Internal friction angle	60	°
	Cohesive force	10/20/30/40	kPa

, and the simulation results are presented in

Table 15. Simulation result of varying cohesive force.

ID	Cohesive Force (kPa)	Impact Count	Time Consumed (s)	Maximum Ice Load (MN)
7-1	10	1	125	28.38
7-2	20	1	138	46.73
7-3	30	2	160	53.89
7-4	40	2	168	64.64

and Figure 19. It can be observed that both the passage time and the peak ice load exhibit a positive correlation with cohesion. When the cohesion reaches 30 kPa, the ship’s speed drops to zero during the first ramming attempt, requiring a second ramming to pass.

5. Conclusions

This paper proposes a simulation method for multiple ramming events against ice ridges. The main structural components of the ice ridge are decoupled and modeled separately. The consolidated layer is modeled as layered ice. A 2D Boolean operation between the ship’s waterline and the layered ice field is used to compute the overlap area, which is then corrected in 3D to obtain the local contact area for calculating the compressive force. Ice failure is determined based on the ultimate bearing capacity of sea ice. For the keel, a failure model is established based on the Mohr–Coulomb criterion and the effective stress principle, accounting for shear failure in porous media and incorporating both cohesion and internal friction angle. Model-scale validation tests and ablation studies on multiple ramming scenarios were conducted, leading to the following conclusions:

(1)

The accuracy of ice loads during ice ridge ramming was validated by comparison with model test results, with average ice load deviation of 9.85% in the consolidated layer and 10.48% in the ice ridge.

(2)

A thicker consolidated layer reduces the ship’s kinetic energy during ramming, resulting in a greater number of ramming attempts and increased time required to penetrate the ice ridge. Larger internal friction angles and cohesion values both increase the strength of the keel, resulting in higher peak loads and longer passage times.

(3)

Greater keel depth imposes more resistance to the ship, leading to increased time consumption and a higher number of ramming attempts. An increase of 5 m in keel depth results in a time consumption roughly 1.3 times that of the original. Moreover, the peak ice load increases approximately linearly with keel depth, which may cause structural damage or yielding of the hull. Different keel inclinations have a relatively minor effect on the peak load during passage. However, smaller inclination angles result in a longer keel region that the ship must pass through, leading to increased time consumption and a higher number of ramming attempts.

(4)

Increasing the retreat distance allows the ship to accelerate and gain more kinetic energy, resulting in improved performance during ramming. A smaller ship draft reduces the contact area between the ship and the ice ridge, significantly decreasing the number of ramming attempts and the total time required to pass through.

This paper presents a computationally efficient estimation method, suitable for early-stage design or real-time assessment. Some limitations still exist. First, the dynamic process of backing was not simulated. Second, the submergence resistance was estimated using an empirical formula rather than a numerical simulation method. Finally, the ship’s motion response during the ramming of ice ridges was considered in only one forward degree of freedom and needs to be extended to six degrees of freedom. These limitations should be addressed in future work.