IMPROVEMENT OF THE TOTAL LAGRANGIAN SPH AND ITS APPLICATION IN IMPACT PROBLEMS
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摘要: 光滑粒子流体动力学(smoothed particle hydrodynamics, SPH)在模拟固体大变形、破碎和裂纹扩展等问题中有天然的优势, 但SPH固有的拉伸不稳定缺陷是SPH在计算固体力学领域进一步应用的一大障碍. 完全拉格朗日SPH (total Lagrangian-SPH, TL-SPH)方法是一种有效的改善拉伸不稳定的措施, 但其仍面临边界区域精度低、界面条件难以施加、损伤裂纹难以模拟等缺陷. 因此, 首先将可达到二阶精度的高阶SPH方法与TL-SPH耦合, 为了节省高阶方法的计算量, 进一步简化粒子选取模式, 提出TL-SFPM (TL-simplified finite particle method)方法; 其次, 将可提高界面精度的DFPM (discontinuous finite particle method)方法与TL-SPH结合, 并提出一种基于黎曼解的界面接触算法, 通过在不同材料粒子间建立黎曼模型求解不同材料间的相互作用, 分别应用于流体−固体接触和固体−固体接触中; 再者, 为了捕捉固体受外载荷后的损伤程度及破坏模式, 提出一种完全拉格朗日框架下的粒子损伤破坏模型; 最后, 通过流−固冲击的带弹性挡板溃坝算例和固−固冲击的子弹撞击靶板算例验证提出的TL-SFPM方法、界面接触算法和损伤破坏模型的合理性和精确性, 进一步扩展TL-SPH方法在计算固体冲击问题中的应用.Abstract: SPH (smoothed particle hydrodynamics) has its natural advantages in dealing with the large deformation of the material, fracture and crack propagation due to the absence of mesh distortion. However, the tensile instability which is an inherent defect encountered in the conventional SPH, is an obstacle for further applying SPH in computational solid mechanics. TL-SPH (total Lagrangian-SPH) is an effective measure to improve the tensile instability, but it still faces some defects. For example, the accuracy may be not enough at the boundary region for the truncated supported domain of the particle. The interface conditions are difficult to be implemented strictly, and the crack propagation cannot be presented under the Total Lagrangian frame. So, first of all, TL-SPH is coupled with the high-order SPH method, which can achieve second-order accuracy. Moreover, the high-order method is simplified by reducing the number of neighbor particles to save the calculational cost, and TL-SFPM (TL-simplified finite particle method) method is proposed with a reasonable neighbor particles selection mode. Secondly, TL-SPH method is combined with the DFPM (discontinuous finite particle method), which can improve the accuracy of the interface. A contact algorithm based on the Riemann solution is proposed by establishing the Riemann model between two particles with different materials. Then the fluid-solid contact algorithm and the solid-solid contact algorithm are introduced, respectively. Moreover, to capture the damage form of the solid under external load, a particle damage model based on the total Lagrangian frame is proposed. Finally, the rationality and accuracy of the proposed TL-SFPM method, the contact algorithm and the damage model are verified by cases of the fluid-solid impact and solid-solid impact, which further extends the application of TL-SPH method in the calculation of solid impact problems. The results of the dam break with an elastic baffle and the bullet impacting target plate also demonstrate the algorithms proposed in this paper has a wide application prospect for simulation of fluid-solid interaction and solid impact problems.
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Key words:
- total Lagrangian SPH /
- tensile instability /
- Riemann contact /
- impact /
- simulation accuracy
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图 1 规则构型下不同区域粒子选取方式
Figure 1. Particles selection modes in different regions under regular configuration
图 2 不同粒子间距下式(7)~式(9)计算三次函数∂f/∂X的均方误差
Figure 2. Mean square error of cubic function ∂f/∂X calculated byEqs. (7)~(9) under different particle spacings
图 3 不规则计算域边界粒子选取方式
Figure 3. Particles selection mode in boundary region under irregular configuration
图 5 界面处TL-SFPM方法中四个粒子的选取方式
Figure 5. Particles selection mode at interface in TL-SFPM method
图 8 不同粒子间距弹性挡板自由端竖直位移变化曲线
Figure 8. Vertical displacement of the free end of elastic baffle with different particle spacings
图 9 弹性挡板和液面位置随时间变化曲线
Figure 9. Curves of the elastic baffle and the water surface positions
图 12 CSPM和TL-SFPM得到钢弹撞击铝合金板同一时刻速度分布云图
Figure 12. Velocity distribution obtained by the traditional SPH and high-order TL-SPH
图 13 TL-SFPM和有限元模拟的不同时刻子弹和板的速度(m/s)云图
Figure 13. Velocity (m/s) distribution at different times obtained by TL-SFPM and FEM
图 14 加入断裂模型后不同冲击速度下的失效粒子分布
Figure 14. Distribution of failure particles at different impact velocities with the fracture model
表 1 弹性挡板材料参数
Table 1. Material parameters of elastic baffle
H/m h/m Z/m s/m ρ/
(kg·m−3)ν E/
MPa0.14 0.079 0.1 0.005 1100 0.4 12 
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表 2 TL-SPH和TL-SFPM模拟时间对比
Table 2. Computational time with TL-SPH and TL-SFPM methods
Method TL-SPH TL-SFPM Computational time/s 9562 9403
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表 3 钢和铝材料参数
Table 3. Material parameters of steel and aluminum
Material ρ/(kg·m−3) Cs/(m·s−1) ν E/GPa A/MPa steel 7800 4596 0.3 200 335 aluminum 2785 5328 0.3 72 265 Material B/MPa n C m Tmelt/K steel 275 0.36 0.022 1.0 1811 aluminum 426 0.34 0.015 1.0 775
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表 4 CSPM和TL-SFPM模拟时间对比
Table 4. Computational time with CSPM and TL-SFPM methods
Method CSPM TL-SFPM Computational time/s 2653 3225
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