1. Introduction
Typical features of sea ice in the polar regions include brash ice, floating ice, layered ice, and rafted ice. Rafted ice is one of the specific ice formations in the polar regions, especially during the initial and final sea ice periods. The dynamic effects of the fractures, extrusion, and accumulation of sea ice cause an increase in sea ice thickness. During navigation, a ship is subjected to non-linear solid ice resistance, which significantly challenges a ship’s safe navigation.
For the navigation safety design of polar ships, researchers have proposed various ship performance prediction methods under sea ice conditions, which can be divided into experimental [
1,
2,
3,
4,
5], analytical [
6,
7,
8,
9,
10], and numerical methods [
11,
12,
13,
14,
15]. Experimental methods include full-scale measurements and model tests. Full-scale ship trials are challenging to replicate and involve high costs, while model tests impose strict requirements on the experimental equipment and methodology. Empirical formula methods involve theoretical analyses of ship–ice interaction processes but often simplify the ship and sea ice models, which has particular limitations for complex sea ice and ship models. In recent years, with the rapid improvement of computer performance, numerical methods have been effectively applied. Numerical methods have significant development potential compared to experimental and empirical analytical methods. Numerical methods applied to ship–ice interactions mainly include the finite element method (FEM), the discrete element method (DEM), and the circumferential crack method (CCM). The FEM has been widely applied in the context of ship–ice interaction problems over the years.
Yu et al. [
16] employed the finite element method to numerically simulate periodic ice loads in the interaction between sea ice and conical structures, and the calculated sea ice bending damage process was similar to that of the results of full-scale measurements. Feng et al. [
17] used the cohesive element method to simulate the interaction between ice and structures and conducted with an analysis of parameter sensitivity. It was found that the structural response was very sensitive to changes in the fracture energy, and the stress–strain curve of the body unit had a significant effect on the simulation. This method was also used by Wang et al. [
18] to simulate the continuous icebreaking process of ships at different heeling angles, and they analyzed the continuous icebreaking process of different ships with different transverse inclinations, with the results showing that the ice resistance capacity of the ship and the extension length of the sea ice crevasse increased with the increase in the ship’s transverse inclination angle. Lee et al. [
19] proposed a method to analyze the ice load in the frequency domain, and the trend of the overall power spectral density with the bow angle was analyzed using different regression methods (linear interpolation, support vector machine, random forest, and deep neural network), and it was found that the deep neural network method performed the best. Shi et al. [
20] proposed an elastic-plastic iceberg material model with temperature gradient effect to study the dynamic collision process between a floating production storage and offloading vessel (FPSO) and an iceberg. The simulation results are compared with the design specification to verify the validity of the iceberg model, and the effects of different iceberg shapes and temperatures on the collision process are analyzed. The results show that the structural damage of a floating production storage and offloading vessel (FPSO) is affected by the structural strength, the iceberg strength, and the localized shape of the iceberg.Based on the interaction process between ships and ice as well as the theory of sea ice fracturing, Lu et al. [
21] proposed an edge-crack theory model. Using the extended finite element method, the mechanism of long crack propagation between parallel ice-breaking channels was studied. The maximum distance between parallel channels without sea ice fracturing was investigated and validated against experimental results.
In the discrete element method (DEM) realm, Hanse et al. [
22] employed a two-dimensional discrete circular-disc viscoelastic model to simulate broken ice and adjusted numerical model calculation parameters according to ice tank experiments. Lau et al. [
23] conducted a series of numerical simulations on the interaction between ice offshore structures and ice ships using the three-dimensional block discrete element model. Liu et al. [
24] calculated the impact of factors such as ship speed, ice thickness, and ship width on the ice resistance of ships using the DEM. Dong et al. [
25] established an ice channel model based on the discrete features of broken ice. Using image segmentation methods to extract ice channel regions and introducing intelligent corner regression networks to accurately delineate ice channel boundaries, this method has shown good accuracy in real ice channel recognition. Xie et al. [
26] simulated the ship–water interaction using a coupled CFD-DEM method and established a discretized propeller model (DPM) and a body force model (BFM). The results indicate that the BFM method can be used effectively for the assessment of the main engine power and hull profile optimization during the ship development and design stages. Regarding the circumferential crack method, Zhou et al. [
27] proposed a method based on the circumferential crack approach to distinguish the forms of sea ice damage according to the ship’s heel angle, and they compared the numerical simulation results with the model test results, which achieved a good consistency. Moreover, Gu et al. [
28] predicted the slewing motion of a polar ship in horizontal ice, considered the effect of hull camber on different damage modes of sea ice, and analyzed the results in comparison with the results of real ruler measurements, and the two results are in good agreement.
Several scholars have also worked on rafted ice material modeling. Hopkins et al. [
29] utilized the discrete element approach using circular-disk and block models to validate that the relative motion of two flat ice blocks can result in either overlapping or crushing and breaking. The former leads to rafted ice, and the latter is the initial process of ice ridge formation. After observing the natural appearance of rafted ice through experiments, Leppäranta et al. [
30] found that the ice crystals at the contact point between the two layers of level ice in the rafted ice formed a granular structure, and the shear strength of rafted ice was thus lower than that of level ice. Bailey et al. [
31] found that the shear force at the adhesive interface of artificially created rafted ice was approximately 30% lower than that of level ice through experiments. Parmerter et al. [
32] established a numerical sea ice rafting model capable of calculating the bending stress during the ice rafting process. The results showed that the increase in the bending stress of sea ice is proportional to the square of the ice thickness.
Although these methods have been applied to study ship interactions with level ice, ice floes, broken ice, ice ridges, etc., most of the research on rafted ice has focused on its mechanical properties and physical models. There has been relatively less exploration on ship collisions with rafted ice. This paper combines a preset grid method with the circumferential crack icebreaking assumption to establish a numerical model for rafted ice. The model will be used to predict the resistance of ships in the rafted ice region and compare the numerical simulation results with the model test results. On this premise, the effects of different ice thicknesses, ship speeds, and sea ice characteristics on the level of ship ice resistance and rafted ice are studied. This study supports subsequent ship resistance predictions in rafted ice regions more effectively and holds a specific engineering application value.
4. Sensitivity Analysis of Parameters
The numerical model simulating the ship’s motion in rafted ice established in this paper has been well validated through comparisons with previous experiments and numerical results. Additionally, this study focuses on the ice resistance characteristics of ships in the level and rafted ice regions. The model test study from the ice tank of Aalto University in Finland was chosen [
42]. The model ship of MT Uikku had a scale of 1:31.6. The key parameters of both the full-scale and model-scale platforms are presented in
Table 4. The ship model was towed by a trailer at a constant speed on level ice, and various experimental data were obtained by changing the ice thickness and velocity. Three test conditions were selected from this model test for comparisons, which were Case1, 2, and 3, with the parameters of the level ice being listed in
Table 5.
Sequential ice resistance results through Case1, 2, and 3 were analyzed, as shown in
Figure 9. The squares and triangles are the average and maximum ice resistance observed for Case1, 2, and 3, respectively, and the dots and pentagrams are the average and maximum ice resistance for Case1, 2, and 3 in the numerical simulation. In Case1, the numerical simulation’s average value is 598.59 kN, which closely matches the measurement of 560 kN from the ice tank experiment, with an error of approximately 6.8%. In Case2, the numerical simulation’s average value is 759.9 kN, while the measured value in the ice tank experiment is 830 kN, with an error of approximately 8%. In Case3, the average error between the two is 10%. By comparing the maximum ice resistance values under the three test conditions, it can be observed that only in Case2 is there a significant error between the numerical simulation and the ice tank test in the peak ice resistance. The main reason for this error is the inherent variability in ice parameters in different regions of level ice during the model test preparation process. However, the numerical simulation discretizes the ice field using grids coupled with predetermined ice parameters, which, to some extent, affects the peak ice resistance. Based on the above analysis, the numerical results are qualitatively and quantitatively consistent with the experimental data. The numerical method can also predict ice resistance for ships in level ice.
Numerical simulations were conducted under Case1, 2, and 3 to compare the average ice resistance of level and rafted ice of the same ice thicknesses for the three conditions, as shown in
Figure 10. The ice resistance of rafted ice is significantly lower than that of level ice at the same speed, and the difference in ice resistance gradually increases as the ice thickness increases from 0.63 m to 1.03 m. Since the structure of rafted ice is composed of two thin layers of level ice that undergo secondary freezing, its overall strength is lower than that of level ice; therefore, rafted ice is more prone to damage during ship–ice interactions.
Due to the different formation mechanisms and internal structures of level ice and rafted ice, there are specific differences in their resistance characteristics. As shown in
Figure 11, in Case1, for example, the time history curves of ice resistance for both the level ice and the rafted ice were analyzed. It can be observed that within the first 25 s, the two trends are similar, but the resistance is lower than that of the level ice during the same period. As the icebreaker keeps moving forward, both show an increasing trend in resistance. The ice resistance exhibits periodic fluctuations once the ship enters the stable icebreaking stage. The ice resistance fluctuates with more prominent peaks when interacting with level ice. However, due to the differences in the mechanical properties of the rafted ice layers in the numerical simulation, the ultimate loads on the ice grids are inconsistent. In
Figure 12a, the first rafted ice layer where the ship’s bow is first contacted by the two grid cells has already failed, while in
Figure 12b, the two grid cells of the second rafted ice layer at the same position are in action, and the forces generated by the different layers of ice cause the ship’s ice resistance to fluctuate more significantly. The peaks of the fluctuations are more significant.
4.1. Influence of Ice Thickness
The ice thickness is a crucial component influencing crushed ice. Diverse ice thicknesses of 0.6 m, 0.8 m, 1.0 m, and 1.2 m were selected to simulate the icebreaking of ships in the level and rafted ice regions, and the sailing speed was 1 kn, and the sea ice parameters were referred to in
Table 5 and
Table 6.
Figure 13 compares the mean ice resistance of ships in level and rafted ice regions under varying ice thicknesses. As the ice thickness rises from 0.6 m to 1.0 m, the ship ice resistance of level ice increases from 558.02 kN to 1386.8 kN, while the ship ice resistance of the rafted ice increases from 335.20 kN to 819.14 kN. The ice resistance of ships in both the level and rafted ice regions increases with the growing ice thickness. Under the same ice thickness circumstances, the resistance of ships in the rafted ice region is relatively close to that of the level ice region at 0.6 m ice thickness, but at 1.2 m ice thickness, there is a significant difference between the two. It can be found that the ship ice resistance in level ice is more sensitive to the change in ice thickness compared to the ship ice resistance in the rafted ice area.
Figure 14 illustrates the distribution of ship ice resistance in level and rafted ice conditions under four different ice thicknesses at a certain ship speed. It can be observed that the center of distribution of ship ice resistance in both level and rafted ice increases with the growth of ice thickness. In level ice, the distribution of ship ice resistance gradually transitions from a left-skewed distribution to a right-skewed distribution, indicating that increasing ice thickness increases the probability of encountering peak values in ship ice resistance. In addition, the distribution of ship ice resistance for rafted ice increases with the increase in ice thickness, especially in the conditions of 1 m and 2 m ice thickness, but the distribution of ship ice resistance is lower compared with that of level ice under the same ice thickness conditions. Notably, under different thickness conditions, the distribution of ship ice resistance in rafted ice is more concentrated compared to that of level ice. This suggests that the internal structure of level and rafted ice has distinct influences on the distribution of ice resistance. The structure of rafted ice, being more intricate, results in a more concentrated distribution of ice resistance, potentially leading to increased fatigue effects on the ship’s structure.
4.2. Influence of Ship Speed
This article selects speeds of 2 kn, 3 kn, 4 kn, and 5 kn to simulate ship icebreaking in the level ice and rafted ice regions. The sea ice parameters are shown in
Table 5 and
Table 6, with an ice thickness of 0.76 m.
Figure 15 illustrates the variation in ship ice resistance trends for level and rafted ice at different ship speeds. With an almost linearly increasing relationship, ship speed highly influences the ship’s ice resistance in different ice conditions. A comparison between ship speeds of 2 kn and 5 kn shows that the ship ice resistance of level ice rises by 39.2%, while that of rafted ice increases by 38.4%. It can be observed that the resistance of level ice is more sensitive than that of rafted ice under the influence of ship speed.
The results in
Figure 16 show that the median line of ship ice resistance increases with speed in all cases except for that of the case of a speed of 4 kn in level ice. The variation in ship speed directly influences the size of the grid cells.
Figure 17 illustrates the icebreaking state of vessels in level ice simultaneously under four different ship speeds. It is observed that, under the condition of a 4 kn speed in level ice, continuous crushing occurs between the ship’s side and the grid cells. This leads to a more drastic variation in ship ice resistance, significantly increasing the probability of peak values and causing a more dispersed overall distribution of ice resistance. Therefore, speed not only impacts the peak magnitude of ice resistance but also significantly influences the distribution of ship ice resistance. This indicates that the change in speed may induce alterations in the interaction state between the vessel and ice, consequently affecting the overall characteristics of ice resistance.
4.3. Influence of Bending Strength
In the numerical simulations, bending strength is a crucial parameter directly influencing the load-bearing capacity of each ice grid. To analyze the impact of bending strength on ship ice resistance, especially considering the environmental differences in the growth of rafted ice, which are primarily reflected in the parameter variations of the lower sea ice layer, this paper selected the correction factors of 0.6, 0.7, 0.8, and 0.9 to set the bending strength for both the overall level ice and the lower layer of the rafted ice.
Figure 18 illustrates the linearly increasing trend of ship ice resistance for the level and rafted ice as the correction factor for the bending strength rise from 0.6 to 0.9. The ship ice resistance for level ice increases from 539.15 kN to 701.42 kN, while for rafted ice, it increases from 373.14 kN to 431.87 kN. It is worth noting that the ice resistance generated by the ship in level ice remains higher than that in rafted ice under the influence of bending strength.
As shown in
Figure 19, it can be observed that the median line of the ship ice resistance in level and rafted ice increases with the increasing bending strength, which implies that a higher peak in the ship ice resistance has an effect on the central tendency. Bending strength directly influences the load-bearing capacity of the ice layer, and with an increase in bending strength, the magnitude of ice resistance experienced by the vessel markedly rises. From the analysis of the overall distribution of ship ice resistance, it can be seen that the larger distance between the minima of the median line makes the ship ice resistance in level ice have a right-skewed distribution, while in the rafted ice, the median line and the minima are equally distant from each other, thereby showing a normal distribution. Despite an increase in bending strength, the overall trend changes relatively insignificantly. This suggests that bending strength does not significantly impact the distribution of ice resistance.
4.4. Influence of Crushing Strength
Crushing strength is a crucial parameter in the interaction between ships and ice. In numerical simulations, correction factors of 0.6, 0.7, 0.8, and 0.9 were selected to set the compression strength for level ice and the lower layer of rafted ice. The sailing speed is 0.97 kn, and the ice thickness is 0.76 m.
Figure 20 shows that with the increase of crushing strength, the average and maximum ship ice resistance tend to decline. The icebreaking force between the ship and the ice is affected by the crushing strength of the sea ice. Analysis of individual interactions between the ship and ice grids reveals that enhancing crushing strength shortens the time required for the breaking force to reach the ice’s load-bearing limit. The force remains zero until colliding with the next ice grid, leading to a decreasing trend in the mean ice resistance.
As shown in
Figure 21, it can be observed that the median line of ship ice resistance decreases gradually with the increase in breaking strength in both the level and rafted ice. In the level ice, the median line of ship ice resistance is further away from the end of the minima, which makes the overall left-skewed distribution. In the rafted ice, the median line of ship ice resistance is further away from the extreme value end, and the overall distribution is right-skewed. This is due to the crushing strength directly affecting the icebreaking force between the ship and the ice. The icebreaking force between the ship and the ice increases with the increase in crushing strength. This means that the time of grid cell failure is accelerated under the condition of the same carrying capacity of the grid cells in the level ice. In the process of making contact with the next grid cell, no ice resistance is generated, which leads to a decrease in the trend of the ice resistance of the ship in the level ice. However, the process of collision between the ship and the rafted ice results in the failure of the upper grid cell, which does not mean that the lower grid cell will also fail due to differences in the mechanical properties of the layers of rafted ice. This alternating action leads to a significant difference in the distribution of the overall ice resistance from that in the level ice.