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 中国舰船研究  2018, Vol. 13 Issue (2): 51-59  DOI: 10.3969/j.issn.1673-3185.2018.02.007 0

### 引用本文 [复制中英文]

[复制中文]
YE L Y, WANG C, GUO C Y, et al. Peridynamic model for submarine surfacing through ice[J]. Chinese Journal of Ship Research, 2018, 13(2): 51-59. DOI: 10.3969/j.issn.1673-3185.2018.02.007.
[复制英文]

### 文章历史

Peridynamic model for submarine surfacing through ice
YE Liyu , WANG Chao , GUO Chunyu , CHANG Xin
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: [Objectives] With deepening research on the geographical and climatic environment of the Arctic, the political and military value of submarines in the region has been well recognized. Although the thick ice in the Arctic provides natural protection for submarines, it also poses a risk to submarines during the ice surfacing process. A method for accurately predicting the ice surfacing process and transient ice loads can be the most important issue in the design of submarine shells and choice of ice thickness. [Methods] In this paper, a numerical method for dealing with the submarine and ice contact problem is developed. First, the peridynamic theory on the capture of material fractures is briefly introduced and the feasibility of peridynamics in modeling the ice failure problem is discussed. To reflect the physical reality of submarine-ice interaction, a contact detection algorithm is established to prevent the interpenetration of the submarine surface and ice material particles, and a method for calculating the contact load is introduced. Finally, based on peridynamics and the contact detection algorithm, a numerical program for predicting submarine-ice interaction is compiled. Using the DARPA SUBOFF submarine model, the ice surfacing process of a submarine is simulated. [Results] The results show that this method can vividly capture the ice failure process, which corresponds to observations of the ice surfacing process of American nuclear submarines, and the dynamic ice load can be calculated over time. [Conclusions] This method provides new concepts in the study of submarine-ice interaction, and its results support the optimal shell structure design of arctic submarines.
Key words: submarine    icebreaking surfacing    peridynamics    dynamic behavior    ice load

0 引言

1 近场动力学理论 1.1 基本方程

 $\rho \mathit{\boldsymbol{\ddot u}}\left( {\mathit{\boldsymbol{x}},t} \right) = \int_{{H_x}} {\mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{u}}\left( {\mathit{\boldsymbol{x'}},t} \right) - \mathit{\boldsymbol{u}}\left( {\mathit{\boldsymbol{x}},t} \right),\mathit{\boldsymbol{x'}} - \mathit{\boldsymbol{x}}} \right){\rm{d}}{V_{x'}} + \mathit{\boldsymbol{b}}\left( {\mathit{\boldsymbol{x}},t} \right)}$ (1)

 图 1 物质点x与其临近物质点相互作用 Figure 1 Interaction of a material point x with its neighboring points
 $\left| \mathit{\boldsymbol{\xi }} \right| > \delta \Rightarrow \mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{\eta }},\mathit{\boldsymbol{\xi }}} \right) = 0;\;\;\;\;\forall \mathit{\boldsymbol{\eta }},\mathit{\boldsymbol{\xi }}$ (2)

 $\mathit{\boldsymbol{f}}\left( { - \mathit{\boldsymbol{\eta }}, - \mathit{\boldsymbol{\xi }}} \right) = \mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{\eta }},\mathit{\boldsymbol{\xi }}} \right);\;\;\;\;\forall \mathit{\boldsymbol{\eta }},\mathit{\boldsymbol{\xi }}$ (3)
 $\left( {\mathit{\boldsymbol{\xi }} + \mathit{\boldsymbol{\eta }}} \right) \times \mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{\eta }},\mathit{\boldsymbol{\xi }}} \right) = 0;\;\;\;\;\forall \mathit{\boldsymbol{\eta }},\mathit{\boldsymbol{\xi }}$ (4)
1.2 微观弹脆性(PMB)模型

 $\mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{\eta }},\mathit{\boldsymbol{\xi }}} \right) = \frac{{\mathit{\boldsymbol{\xi }} + \mathit{\boldsymbol{\eta }}}}{{\left| {\mathit{\boldsymbol{\xi }} + \mathit{\boldsymbol{\eta }}} \right|}}\mathit{\boldsymbol{f}}\left( {\left| {\mathit{\boldsymbol{\xi }} + \mathit{\boldsymbol{\eta }}} \right|,\mathit{\boldsymbol{\xi }}} \right);\;\;\;\;\forall \mathit{\boldsymbol{\xi }},\mathit{\boldsymbol{\eta }}$ (5)

 $s = \frac{{\left| {\mathit{\boldsymbol{\xi }} + \mathit{\boldsymbol{\eta }}} \right| - \left| \mathit{\boldsymbol{\xi }} \right|}}{{\left| \mathit{\boldsymbol{\xi }} \right|}} = \frac{{y - \left| \mathit{\boldsymbol{\xi }} \right|}}{{\left| \mathit{\boldsymbol{\xi }} \right|}}$ (6)

 $\mathit{\boldsymbol{f}}\left( {y\left( t \right),\mathit{\boldsymbol{\xi }}} \right) = g\left( {s\left( {t,\mathit{\boldsymbol{\xi }}} \right)} \right)\mu \left( {t,\mathit{\boldsymbol{\xi }}} \right)$ (7)

 $g\left( s \right) = cs;\forall s$ (8)

 $c = \frac{{18\kappa }}{{{\rm{ \mathsf{ π} }}{\delta ^4}}}$ (9)

 $\mu \left( {t,\mathit{\boldsymbol{\xi }}} \right) = \left\{ \begin{array}{l} 1,\;\;s\left( {t',\mathit{\boldsymbol{\xi }}} \right) < {s_0},\;{\rm{for}}\;{\rm{all}}\;\;{\rm{0}} \le t' \le t\\ 0,\;\;{\rm Otherwise} \end{array} \right.$ (10)

 ${s_0} = \sqrt {\frac{{5{G_0}}}{{9\kappa \delta }}}$ (11)

 $\varphi \left( {\mathit{\boldsymbol{x}},t} \right) = \frac{{\int_{{H_x}} {\mathit{\boldsymbol{u}}\left( {\mathit{\boldsymbol{x}},t,\mathit{\boldsymbol{\xi }}} \right){\rm{d}}{V_\xi }} }}{{\int_{{H_x}} {{\rm{d}}{V_\xi }} }}$ (12)

1.3 方程离散化

 $\rho \mathit{\boldsymbol{\ddot u}}_i^m = \sum\limits_j {\mathit{\boldsymbol{f}}\left( {\mathit{\boldsymbol{u}}_j^m - \mathit{\boldsymbol{u}}_i^m,{\mathit{\boldsymbol{x}}_j} - {\mathit{\boldsymbol{x}}_i}} \right){V_j} + \mathit{\boldsymbol{b}}_i^m}$ (13)

 $\mathit{\boldsymbol{\ddot u}}_i^m = \frac{{\mathit{\boldsymbol{u}}_i^{m + 1} - 2\mathit{\boldsymbol{u}}_i^m + \mathit{\boldsymbol{u}}_i^{m - 1}}}{{\Delta {t^2}}}$ (14)

2 潜艇破冰上浮计算方法 2.1 问题描述

 图 2 潜艇破冰上浮实际情况 Figure 2 The actual situation of submarine icebreaking

 图 3 数值计算模型 Figure 3 The numerical model

2.2 模型数值离散化

 图 4 艇体表面网格划分 Figure 4 Mesh division of hull surface
2.3 海冰物质点与潜艇壳体的接触识别

 $\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{A_1}{x_0} + {B_1}{y_0} + {C_1}{z_0} + {D_1} \ge 0}\\ {{\rm{Otherwise}}} \end{array}}&\begin{array}{l} {\rm{Contact}}\\ {\rm{No}}\;{\rm{contact}} \end{array} \end{array}} \right.$ (15)

2.4 接触作用力计算

 图 5 物质点的重新分配方案 Figure 5 Relocation of material points inside the ship body
 $d = \frac{{\left| {{A_1}{x_0} + {B_1}{y_0} + {C_1}{z_0} + {D_1}} \right|}}{{\sqrt {A_1^2 + B_1^2 + C_1^2} }}$ (16)

 $\mathit{\boldsymbol{x}}_{\left( k \right)}^{t + \Delta t} = \mathit{\boldsymbol{x}}_{\left( k \right)}^t + {\mathit{\boldsymbol{v}}_0} \cdot \Delta t + d \cdot \mathit{\boldsymbol{n}}$ (17)

 $\mathit{\boldsymbol{\bar v}}_{\left( k \right)}^{t + \Delta t} = \frac{{\mathit{\boldsymbol{\bar u}}_{\left( k \right)}^{t + \Delta t} - \mathit{\boldsymbol{u}}_{\left( k \right)}^t}}{{\Delta t}}$ (18)

 $\mathit{\boldsymbol{F}}_{\left( k \right)}^{t + \Delta t} = - 1 \times {\rho _{\left( k \right)}}\frac{{\mathit{\boldsymbol{\bar v}}_{\left( k \right)}^{t + \Delta t} - \mathit{\boldsymbol{v}}_{\left( k \right)}^{t + \Delta t}}}{{\Delta t}}{V_{\left( k \right)}}$ (19)

 ${\mathit{\boldsymbol{F}}^{t + \Delta t}} = \sum\limits_{k = 1} {\mathit{\boldsymbol{F}}_{\left( k \right)}^{t + \Delta t}\lambda _{\left( k \right)}^{t + \Delta t}}$ (20)

 $\lambda _{\left( k \right)}^{t + \Delta t} = \left\{ \begin{array}{l} 1\;\;\;\;{\rm{Inside}}\;{\rm{impactor}}\\ 0\;\;\;\;{\rm{Outside}}\;{\rm{impactor}} \end{array} \right.$
3 潜艇破冰上浮算例分析 3.1 计算模型及参数设置

 图 6 SUBOFF潜艇模型 Figure 6 The submarine model of SUBOFF
3.2 计算模型的收敛性分析

 图 7 同物质点间距下冰载荷随时间变化的曲线 Figure 7 The time histories of ice load with different spacing of material point

 图 8 不同时间间隔下冰载荷随时间变化曲线 Figure 8 The time histories of ice load with different time step

3.3 潜艇破冰上浮动态特性分析与验证

 图 9 潜艇破冰动态变化过程 Figure 9 Dynamic icebreaking process of submarine

 图 10 美国核潜艇破冰图 Figure 10 The icebreaking picture of U.S. nuclear submarine

 图 11 指挥室围壳上的冰块 Figure 11 The ice piece on submarine conning tower

 图 12 潜艇尾翼冲击海冰结果 Figure 12 The trail fin of the submarine impacting the ice

 图 13 冰—潜艇接触力随时间变化曲线 Figure 13 The time histories of submarine-ice contact forces

 图 14 冰—指挥室围壳冰载荷随时间变化的曲线 Figure 14 The time histories of ice-tower ice load
4 结论

1) 开展了潜艇破冰上浮过程中物质点间隔和时间步长的收敛性分析，计算结果表明，物质点间隔小于Δx = L/500、时间步长小于0.000 948 s时，计算结果可达到收敛。

2) 基于本文建立的数值模型对潜艇破冰上浮过程中海冰的破坏过程进行模拟，很好地再现了海冰的挤压、纵向剪切破坏或弯曲破坏过程，证明了本文计算方法的有效性以及接触识别算法的可靠性。

3) 对潜艇破冰上浮过程冰载荷时程曲线的分析表明，在海冰挤压破坏过程中，冰载荷迅速增大，曲线比较平滑，但海冰弯曲破坏过程一旦发生，冰载荷将出现剧烈的振荡，对艇体结构的安全性不利。

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