基于界面重构的三维弹塑性MMALE计算方法

doi:10.19596/j.cnki.1001-246x.8799

摘要/Abstract

摘要：

针对弹塑性流体特点, 通过弹塑性混合网格封闭模型、三维大变形网格上界面重构以及保物理特性弹塑性量重映方法等技术, 给出一种适应弹塑性流体的三维MMALE(multi-material arbitrary Lagragian-Eulerian)算法。首先, 在混合网格封闭模型方面, 针对传统体模量加权模型应用于弹塑性流体导致的压力不平衡问题, 通过引入松弛机制和自适应松弛系数, 实现压力平衡, 同时该封闭模型无须迭代, 避免了在迭代过程可能遇到的不收敛问题, 提升了健壮性; 在界面重构方面, 针对三维多面体网格由于变形及三维特殊性导致的守恒性问题, 通过引入逐级表面三角化算法, 实现了适应三维任意多面体网格上的严格守恒的界面重构。在弹塑性量重映方面, 针对偏应力按分量重映后导致的张量性质破坏, 应变能不守恒的问题, 通过引入对应力张量不变量的重映, 实现重映后应力不破坏屈服准则, 并保持应变能的守恒性。最后, 采用MMALE算法对典型算例进行数值模拟, 验证了其正确性和处理大变形问题的健壮性。

关键词: 弹塑性, 多介质, 界面重构, MMALE

Abstract:

The MMALE (multi-material Arbitrary Lagrangian Eulerian method) is an effective method for the simulation of multi-material flow with large deformation. Traditional MMALE methods are mainly used in pure fluids and focus on two-dimensional problems. In this paper, we propose a three-dimensional MMALE algorithm for elastic-plastic fluids by developing three key technologies: a closure model for elastic-plastic flow, an interface reconstruction method adapted to three-dimensional grids with large deformation, and a physical property preserving remapping method for elastic-plastic quantities. Firstly, to solve the pressure imbalance problem caused by elastic-plastic fluids using traditional bulk modulus weighted closure models, a relaxation mechanism is introduced in our new closure model to achieve the balance of the pressure. At the same time, this model does not require iterations, avoiding the difficulties of non convergence that may be encountered during the iteration process; Secondly, in order to address the conservation issues caused by distorted three-dimensional polyhedral grids, a step-by-step surface triangulation algorithm is introduced to preserve the conservation during the interface reconstruction process. Thirdly, in the aspect of the remapping of elastic-plastic quantities, aiming at the problem that the tensor property is destroyed and the strain energy is not conserved after the deviator stress is remapped component by component, the remapping of invariants of the stress tensor is introduced to preserve the tensor property and maintain the conservation of the strain energy. Finally, the MMALE method is used to simulate several typical examples, and its correctness and robustness are verified.

Key words: elastic-plastic, multiple material, interface reconstruction, MMALE

中图分类号:

郭少冬, 周海兵, 熊俊. 基于界面重构的三维弹塑性MMALE计算方法[J]. 计算物理, 2024, 41(5): 607-618.

Shaodong GUO, Haibing ZHOU, Jun XIONG. A MMALE Method Based on Interface Reconstruction for Three-dimensional Elastic-Plastic Multi-material Flow[J]. Chinese Journal of Computational Physics, 2024, 41(5): 607-618.

图/表 21

图1 弹塑性MMALE算法流程

Fig.1 Flow chart of elastic-plastic MMALE method

图2 逐级表面三角化

Fig.2 Progressive surface triangulation

图3 非凸多面体网格上的界面重构

Fig.3 Interface reconstruction on non convex polyhedral grid

图4 真空黎曼问题初始分布(t=0)

Fig.4 Initial distribution of vacuum Riemann problem(t=0)

图5 t=2.5时，MMALE模拟的密度、压力与解析解

Fig.5 Density and pressure of MMALE simulation and analytical solution with t=2.5

图6 两种混合模型计算的压力随时间变化(a) 体模量加权模型；(b)本文模型

Fig.6 Pressure with time by two mixed closure models (a) bulk modulus weighted model; (b) our model

图7 两种方法计算的等效塑性应变

Fig.7 Equivalent plastic strain by two methods

图8 两种方法计算的压力

Fig.8 Pressure by two methods

图9 两种混合模型计算的压力(t=2.0) (a)体模量加权模型；(b)本文模型

Fig.9 Pressure by two mixed closure models at t=2.0 (a) bulk modulus weighted model; (b) our model

图10 不同网格尺度计算的应力和等效塑性应变(t=4.0) (a)应力；(b)等效塑性应变

Fig.10 (a) Stress; and (b) equivalent plastic strain with different mesh sizes at t=4.0

图11 Be球壳塑性坍缩问题第一种计算模式

Fig.11 Plastic collapse of Be shell under the first calculation mode

图12 终止时刻的等效塑性应变分布(a)拉氏；(b)MMALE

Fig.12 Equivalent plastic strain at final time (a) Lagrange; (b)MMALE

图13 终止时刻的界面位置(a)拉氏; (b)MMALE

Fig.13 Interface at final time (a) Lagrange; (b)MMALE

图14 Be球壳塑性坍缩问题第二种计算模式

Fig.14 Plastic collapse problem of Be shell under the second calculation mode

图15 终止时刻的等效塑性应变分布(a)拉氏；(b)MMALE

Fig.15 Equivalent plastic strain at final time (a) Lagrange; (b)MMALE

图16 终止时刻的界面位置对比(a)拉氏；(b)MMALE

Fig.16 Interface at final time (a) Lagrange; (b)MMALE

图17 MMALE计算的自由面位置随时间变化

Fig.17 Variation of interface location by the MMALE method with time

图18 能量随时间的变化

Fig.18 Variation of energy by the MMALE method with time

图19 三维R-T不稳定性问题的初始材料界面

Fig.19 The initial material interface of 3D R-T instability problem

图20 三维R-T不稳定性问题模拟的材料界面演化(a)文[26]；(b) 本文模拟

Fig.20 Variation of the interfaces of 3D R-T stability problem (a) Ref. [26]; (b) our method

图21 MMALE模拟的靶丸高速冲击薄靶板过程

Fig.21 Simulation of high-speed impact of the target on the thin target plate by our MMALE method

参考文献 29

1	HIRT C W , AMSDEN A A , COOK J L . An arbitrary Lagrangian-Eulerian computing method for all flow speeds[J]. Journal of Computational Physics, 1974, 14 (3): 227- 253. DOI
2	PEERY J S , CARROLL D E . Multi-Material ALE methods in unstructured grids[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 187 (3/4): 591- 619.
3	BENSON D J . Computational methods in Lagrangian and Eulerian hydrocodes[J]. Computer Methods in Applied Mechanics and Engineering, 1992, 99 (2/3): 235- 394.
4	BENSON D J . A mixture theory for contact in multi-material Eulerian formulations[J]. Computer Methods in Applied Mechanics and Engineering, 1997, 140 (1/2): 59- 86.
5	MILLER D S , ZIMMERMAN G B . An algorithm for time evolving volume fractions in mixed zones in Lagrangian hydrodynamics calculations[J]. Russian Journal of Physical Chemistry B, 2009, 3 (1): 117- 121. DOI
6	DESPRES B , LAGOUTIERE F . Numerical resolution of a two-component compressible fluid model with interfaces[J]. Progress in Computational Fluid Dynamics, An International Journal, 2007, 7 (6): 295- 310. DOI
7	田保林, 刘妍, 申卫东, 等. 可压缩多介质流动的整体ALE方法(GALE)[J]. 北京理工大学学报, 2010, 30 (5): 622- 625.
8	SHASHKOV M . Closure models for multimaterial cells in arbitrary Lagrangian-Eulerian hydrocodes[J]. International Journal for Numerical Methods in Fluids, 2008, 56 (8): 1497- 1504. DOI
9	KAMM J R , SHASHKOV M J . A pressure relaxation closure model for one-dimensional, two-material Lagrangian hydrodynamics based on the Riemann problem[J]. Communications in Computational Physics, 2010, 7, 927- 976. DOI
10	ROBINSON A, BRUNNER T, CARROLL S, et al. ALEGRA: An arbitrary Lagrangian-Eulerian multimaterial, multiphysics code[C]//46th AIAA Aerospace Sciences Meeting and Exhibit. Reno: AIAA, 2008: AIAA2008-1235.
11	HIRT C W , NICHOLS B D . Volume of fluid (VOF) method for the dynamics of free boundaries[J]. Journal of Computational Physics, 1981, 39 (1): 201- 225. DOI
12	YOUNGS D. Time-dependent multi-material flow with large fluid distortion[M]//MORTON K W, BAINES M J. Numerical methods for fluid dynamics. New York: Academic Press, 1982: 273-285.
13	RUDMAN M . Volume-tracking methods for interfacial flow calculations[J]. International Journal for Numerical Methods in Fluids, 1997, 24 (7): 671- 691. DOI
14	DYADECHKO V, SHASHKOV M. Moment-of-fluid interface reconstruction[Z]. Los Alamos National Laboratory Report, 2005.
15	DYADECHKOV , SHASHKOV M . Reconstruction of multi-material interfaces from moment data[J]. Journal of Computational Physics, 2008, 227 (11): 5361- 5384. DOI
16	AHN H T , SHASHKOV M . Multi-material interface reconstruction on generalized polyhedral meshes[J]. Journal of Computational Physics, 2007, 226 (2): 2096- 2132. DOI
17	KUCHARIK M , SHASHKOV M , WENDROFF B . An efficient linearity-and-bound-preserving remapping method[J]. Journal of Computational Physics, 2003, 188 (2): 462- 471. DOI
18	MARGOLIN L G , SHASHKOV M . Second-order sign-preserving conservative interpolation (remapping) on general grids[J]. Journal of Computational Physics, 2003, 184 (1): 266- 298. DOI
19	熊俊, 周海兵, 张树道. SALE速度重映算法的改进[J]. 计算物理, 2006, 23 (1): 37- 42.
20	GRANDY J . Conservative remapping and region overlays by intersecting arbitrary polyhedra[J]. Journal of Computational Physics, 1999, 148 (2): 433- 466.
21	KUCHARIK M , SHASHKOV M . Conservative multi-material remap for staggered multi-material arbitrary Lagrangian-Eulerian methods[J]. Journal of Computational Physics, 2014, 258, 268- 304.
22	MARBŒUF A , CLAISSE A , LE TALLEC P . Conservative and entropy controlled remap for multi-material ALE simulations with space-staggered schemes[J]. Journal of Computational Physics, 2019, 390, 66- 92.
23	GALERA S , MAIRE P H , BREIL J . A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface Reconstruction[J]. Journal of Computational Physics, 2010, 229 (16): 5755- 5787.
24	GALERA S , BREIL J , MAIRE P H . A 2D unstructured multi-material cell-centered arbitrary Lagrangian-Eulerian (CCALE) scheme using MOF interface Reconstruction[J]. Computers & Fluids, 2011, 46 (1): 237- 244.
25	JIA Zupeng , LIU Jun , ZHANG Shudao . An effective integration of methods for second-order three-dimensional multi-material ALE method on unstructured hexahedral meshes using MOF interface Reconstruction[J]. Journal of Computational Physics, 2013, 236, 513- 562.
26	CHEN Xiang , ZHANG Xiong , JIA Zupeng . A robust and efficient polyhedron subdivision and intersection algorithm for three-dimensional MMALE remapping[J]. Journal of Computational Physics, 2017, 338, 1- 17.
27	CARAMANA E J , BURTON D E , SHASHKOV M J , et al. The construction of compatible hydrodynamics algorithms utilizing conservation of total energy[J]. Journal of Computational Physics, 1998, 146 (1): 227- 262.
28	CARAMANA E J , ROUSCULP C L , BURTON D E . A compatible, energy and symmetry preserving lagrangian hydrodynamics algorithm in Three-Dimensional cartesian geometry[J]. Journal of Computational Physics, 2000, 157 (1): 89- 119.
29	BARLOW A , HILL R , SHASHKOV M . Constrained optimization framework for interface-aware sub-scale dynamics closure model for multimaterial cells in Lagrangian and arbitrary Lagrangian-Eulerian hydrodynamics[J]. Journal of Computational Physics, 2014, 276, 92- 135.

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