Taylor particle-in-cell transfer and kernel correction for material point method
Taylor particle-in-cell transfer and kernel correction for material point method
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DOI：
10.1016/j.cma.2022.115720
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发表时间：
2023-01
影响因子：
7.2
通讯作者：
Keita Nakamura;S. Matsumura;T. Mizutani
中科院分区：
文献类型：
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作者：
Keita Nakamura;S. Matsumura;T. Mizutani
材料点法（MPM）已被广泛用于解决涉及大位移和变形的问题。采用流体隐式粒子（FLIP）传输的标准MPM制剂比粒子在细胞（PIC）传输耗散更少，但也更不稳定。仿射PIC（APIC）转移引入仿射速度，并被开发用于实现稳定的模拟，同时克服PIC中的角动量耗散。本文提出了Taylor-PIC（TPIC）变换，它是一种基于一阶泰勒级数近似的仿射速度和PIC变换相结合的APIC变换。TPIC简单（仅需要粒子速度梯度），并继承了原始APIC的关键优点，例如较少的耗散和稳定性。虽然TPIC不守恒角动量，但与APIC相反，在粒子和网格之间的传输中保持了速度梯度。该速度梯度在其斜对称分量中包含角信息，这使得TPIC能够充分描述角运动，类似于APIC。此外，当明确施加边界条件时，MPM可以在边界附近引起应力振荡，例如，当边界网格速度（或动量）设置为零时。当使用仿射型传输时，这种不稳定性加剧，并且模拟很容易失败。因此，我们提出了一种基于加权最小二乘的边界附近粒子核校正方法。建议的TPIC传输和内核校正验证通过五种类型的模拟，每个使用不同的材料线弹性，冯米塞斯，牛顿流体，和Drucker-Prager。在模拟中，即使对于小的变形，也观察到由于应力振荡而导致的不正确的结果。应用校正核成功地消除了这些寄生振荡，结果与解析解一致。此外，数值结果证实了建议的TPIC转移的准确性和鲁棒性。
The material point method (MPM) has been extensively used to solve problems involving large displacements and deformations. The standard MPM formulation, which adopts fluid-implicit-particle (FLIP) transfer, is less dissipative but also less stable than particle-in-cell (PIC) transfer. The affine PIC (APIC) transfer introduces the affine velocity and was developed to realize stable simulations while overcoming dissipations of the angular momentum in PIC. This paper presents Taylor-PIC (TPIC) transfer, which is a type of APIC transfer that combines the affine velocity based on the first-order Taylor series approximation and PIC transfer. TPIC is simple (only the particle velocity gradient is required) and inherits the key advantages of the original APIC, such as less dissipation and stability. Although TPIC does not conserve angular momentum, in contrast to APIC, the velocity gradient is preserved in the transfer between particles and the grid. This velocity gradient contains the angular information in its skew-symmetric component, which allows TPIC to adequately describe angular motion, similar to APIC.Furthermore, the MPM can cause stress oscillations near boundaries when boundary conditions are explicitly imposed, e.g., when the boundary grid velocity (or momentum) is set to zero. When affine-type transfers are used, this instability is exacerbated, and simulations can easily fail. Therefore, we propose a kernel correction method based on the weighted least squares for particles near boundaries. The proposed TPIC transfer and kernel correction are validated through five types of simulations, each using a different material—linear elastic, von Mises, Newtonian fluid, and Drucker–Prager. In the simulations, incorrect results due to stress oscillations are observed even for small deformations. Applying the corrected kernels successfully removes these spurious oscillations, and the results are consistent with the analytical solutions. Moreover, the numerical results confirm the accuracy and robustness of the proposed TPIC transfer.