An improved finite particle method for discontinuous interface problems
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摘要: 有限粒子法（finite particle method，FPM）作为SPH（smoothed particle hydrodynamics）方法的重要改进，有效提高了边界区域粒子的近似精度，但是当FPM处理多物理场时，在不连续界面附近的计算精度会大大降低，并且FPM必须满足的矩阵非奇异性也提高了对界面处理的要求。本文中基于DSPH（discontinuous SPH）方法，提出了一种考虑界面不连续的改进FPM—DSFPM（discontinuous special FPM）法，旨在改善FPM在界面不连续处的计算精度，从而进一步提高其计算效率和稳定性。首先，分析了DSFPM的核近似精度。其次，根据不同的工程问题，给出DSFPM处理小变形和大变形问题的算法流程。利用DSFPM、DSPH和FPM等3种方法对弹性铝块小变形碰撞冲击算例进行了模拟，通过对比分析铝块的速度和应力以及计算时间验证了DSFPM算法在非连续界面处计算精度和计算效率的优势。最后，通过结合DSFPM和DFPM（discontinuous FPM）实现了对于大变形问题的模拟。Abstract: The finite particle method(FPM) is an important improvement for the smoothed particle hydrodynamics(SPH) method, which effectively improves the calculation accuracy of boundary particles. However, when the discontinuous physical field is solved by the FPM, the accuracy in the vicinity of the discontinuous interface is greatly reduced, and the non-singularity of the matrix must be satisfied in the FPM, which requires an elaborate handling of the interface. Based on the discontinuous SPH(DSPH) method, this paper proposed an improved FPM-discontinuous special FPM(DSFPM), which considers the discontinuous interface, aiming to improve the computational accuracy at the interface and further improve the efficiency and stability of the FPM. In this paper, the estimation accuracy of the DSFPM was analyzed firstly, and then the algorithm flow diagram of the DSFPM to deal with the small deformation and large deformation problems was demonstrated. Next, the DSFPM, DSPH and FPM were used to simulate the small deformation problem-elastic aluminum blocks impact. By comparing the velocity and stress of the aluminum blocks and computational time, we verified the accuracy and computational efficiency of the DSFPM. Finally, the simulation of the large deformation problem was realized by a combining method with the DSFPM and DFPM.
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Key words:
- FPM /
- interface /
- DSFPM /
- accuracy /
- computational cost
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图 4 3种方法估计非连续常值函数误差及其导数误差
Figure 4. Estimation errors for the discontinuous 0-order function and its derivative by three methods
图 5 3种方法估计非连续一次函数误差及其导数误差
Figure 5. Estimation errors for the discontinuous 1-order function and its derivative by three methods
图 6 DSFPM计算二次非连续函数本身和其偏y导数误差分布
Figure 6. Estimation error distribution for the discontinuous 2-order function and its partial y derivative by DSFPM
图 9 3种方法模拟得到的铝块速度随时间的变化曲线
Figure 9. Velocity-time cuvers of aluminum blocks by three methods
图 10 有限元模拟得到的铝块速度随时间的变化曲线
Figure 10. Velocity-time cuvers of aluminum blocks by finite element method
图 11 3种方法模拟得到的铝块碰撞过程中的速度分布
Figure 11. Velocity distribution of the aluminum blocks by three methods
图 12 3种方法模拟得到铝块碰撞过程中的x方向应力分布
Figure 12. x-direction total stress distribution of the aluminum blocks simulated by three methods
图 13 3种方法模拟得到点M处x方向应力随时间的变化曲线
Figure 13. x-directional total stress changes at point M simulated by three methods
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