# Simulation of Sea Ice Fragmentation Based on an Improved Voronoi Diagram Algorithm in an Ice Zone Navigation Simulator

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Physics-Based Cracking Simulation

#### 1.3. Geometry-Based Cracking Simulation

#### 1.4. Scene Management Techniques

#### 1.5. Contributions

- (1)
- The Voronoi algorithm to was used preprocess a sea ice model, and the Voronoi algorithm was improved during processing so that the number of seed points was correlated with the thickness of the sea ice to determine the effects of different thicknesses on the degree of fragmentation.
- (2)
- Not only was the ship–ice interaction process used to generate the channel outline, but the ship’s movement on the sea ice around the channel also generated force conduction. In this study, the effect of conduction cracking on the ice surface near the channel was achieved.
- (3)
- After the sea ice broke up, there was no interaction between ice floes and seawater, i.e., there is no floating effect due to ice floes. After sea ice breaks up, the ice floe should be subject to the buoyancy of seawater, and this study incorporated the floating of the ice floe on the water surface by using a method based on the model grid.
- (4)
- In terms of ice zone scene management, this study improved upon the quadtree scene management technique, used a circular detection area instead of a square detection area, and associated the detection area with the ship’s motion characteristics, leaving a certain amount of buffer for conduction cracking of the ice.

## 2. Methods

#### 2.1. The Voronoi Diagram Algorithm for the Breaking up of Sea Ice Considering Ice Thickness

_{1}, m

_{2}… m

_{n}are generated within the model boundary as seed points for the dissection, denoted as the set C(n).

_{1}, m

_{2}… m

_{x}) in C(n) is denoted as T(x), and the Delaunay tetrahedral mesh is generated with the point-by-point insertion method by inserting the coordinate point m

_{x+1}and calculating the distance between m

_{x+1}and the center of the tetrahedral outer sphere in each T(x), D

_{n}, and the corresponding radius R

_{n}of the outer jointed ball. Following that, the tetrahedral lattices with D

_{n}< R

_{n}are filtered out and denoted as the set ET(x).

_{x+1}is eliminated from the original tetrahedral mesh T(x), and a new tetrahedral mesh NT(x) is generated.

_{x+1}to form a new tetrahedral mesh; this is added to NT(x) and reassigned to T(x).

#### 2.2. Collision Body Setup for Sea Ice Model Sub-Blocks

- (1)
- The Sphere Collider is added to a number of sub-blocks shown in Figure 5a, and the Mesh Collider is added to others.
- (2)
- There is a large redundancy with the sub-block geometry after adding the Sphere Collider enclosure, and there will be a large overlap between the collision bodies of the neighboring sub-blocks, so the radius of the Sphere Collider needs to be adjusted until there is no overlap between any of the Sphere Colliders, as shown in Figure 5c.
- (3)
- The distribution of collision bodies in the final ice model is shown in Figure 5d; the red dashed line indicates where the Mesh Collider was adopted to achieve a higher degree of fitting, while the Sphere Collider was adopted in other parts. This not only reproduces the effect cracking effect when a direct collision occurs, but this also optimizes the realism of conduction cracking.
- (4)
- The collision body information of the processed ice model sub-blocks is stored for calling when the large-area scene is laid out.

#### 2.3. Simulation of the Floating State of the Ice Floes

^{3}[36]; the gravitational acceleration $g$ is set to 9.826 m/s

^{2}at sea level and 70° N [37]; the calculation of ${V}_{\mathrm{s}\mathrm{u}\mathrm{b}}$ makes use of the gridded data intercepted at the surface of the water in the model to derive the volume that should be underwater to satisfy

- (1)
- Two vertices are under the water.
- (2)
- One vertex is under the water.

#### 2.4. Scenario Management Based on the Improved Quadtree Algorithm

## 3. Experimental Results and Discussion

#### 3.1. Experiments Comparing Different Ice Thicknesses

#### 3.2. Experiments Comparing Different Collision Body Setups

#### 3.3. Floating Effect for Broken Ice Floes

#### 3.4. Improved Quadtree Scene Management

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Comparison of the effects of simulating ice zone scenes: (

**a**) effects achieved by Fan; (

**b**) effects achieved by Sun with the Koch fractal algorithm; (

**c**) effects achieved by Sun with the Voronoi algorithm; (

**d**) effects achieved in this study.

**Figure 2.**Fragmentation pattern [5].

**Figure 7.**Intercept mesh when type = 1: (

**a**) two vertices under the water surface; (

**b**) only one vertex under the water surface.

**Figure 12.**Experimental results of models with different ice thicknesses: (

**a**) $h$ = 0.2 m; (

**b**) $h$ = 0.5 m; (

**c**) $h$ = 1.0 m.

**Figure 13.**Experimental results of the different model preprocessing methods: (

**a**) Mesh Collider; (

**b**) Sphere Collider; (

**c**) Mesh and Sphere Colliders.

**Figure 15.**Visualization of the unit normal vector of ice floes and the floating effect: (

**a**) visualization of the unit normal vector; (

**b**) floating effect.

**Figure 16.**Schematic diagram of the detection radius (The red circles represent the boundary of the detection area.): (

**a**) fragmentation buffer zone; (

**b**) a superimposition of the fragmentation buffer and corner buffer.

**Figure 17.**Schematic diagram of the detection radius: (

**a**) before entering the detection area; (

**b**) after entering the detection area.

**Table 1.**Comparison of the efficacy of ice zone scene simulation based on different geometric methods.

Fractal Algorithm | Conduction Cracking | Floating Effect | Scene Management | |
---|---|---|---|---|

Fan [26] | Voronoi | No | No | Quadtree |

Sun [32] | Improved Koch Curve | No | No | No |

Sun [31] | Voronoi | No | No | Quadtree |

This study | Improved Voronoi | Yes | Yes | Improved Quadtree |

**Table 2.**The input data for a large floating ice floe [5].

Ice thickness | 1 m |

Ice density | 900 kg/m^{3} |

Ice modulus of elasticity | 3 Gpa |

Poisson’s ratio | 0.33 |

Seawater density | 1025 kg/m^{3} |

Acceleration of gravity | 9.81 m/s^{2} |

The radius of the distributed load | 0.5 m |

The uniformly distributed vertical load | 291 kPa |

The flexural strength of the ice | 500 kPa |

**Table 3.**The characteristics of the generated wedges [5].

Number of generated wedges | 3 | 4 | 5 |

The angle of each wedge [deg] | 60 | 45 | 36 |

The distance of the distributed load from the apex [m] | 0.5 | 0.5 | 0.5 |

The maximum load capacity of each wedge [kN] | 113 | 81 | 64 |

The breaking length [m] | 4.5 | 4.5 | 4.5 |

The volume of each broken ice piece [m^{3}] | 11.7 | 8.4 | 6.6 |

The mass of each broken ice piece [tons] | 10.5 | 7.5 | 5.9 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, B.; Ren, H.; Qiu, S.; Yang, X.; Liao, G.; Liang, X.
Simulation of Sea Ice Fragmentation Based on an Improved Voronoi Diagram Algorithm in an Ice Zone Navigation Simulator. *J. Mar. Sci. Eng.* **2023**, *11*, 2047.
https://doi.org/10.3390/jmse11112047

**AMA Style**

Zhang B, Ren H, Qiu S, Yang X, Liao G, Liang X.
Simulation of Sea Ice Fragmentation Based on an Improved Voronoi Diagram Algorithm in an Ice Zone Navigation Simulator. *Journal of Marine Science and Engineering*. 2023; 11(11):2047.
https://doi.org/10.3390/jmse11112047

**Chicago/Turabian Style**

Zhang, Boxiang, Hongxiang Ren, Shaoyang Qiu, Xiao Yang, Gongming Liao, and Xiao Liang.
2023. "Simulation of Sea Ice Fragmentation Based on an Improved Voronoi Diagram Algorithm in an Ice Zone Navigation Simulator" *Journal of Marine Science and Engineering* 11, no. 11: 2047.
https://doi.org/10.3390/jmse11112047