The Material Point Method (MPM) stands as a continuum-based particle technique designed for addressing large deformation problems. However, the treatment of incompressible materials using MPM remains underexplored. This study focuses on adapting established techniques from the Finite Element Method (FEM) to address incompressibility within MPM for dynamic hyperelastic problems. Firstly, we introduce a mixed displacement-pressure formulation to tackle incompressibility. Secondly, we employ two different stabilization techniques rooted in the Variational Multiscale Method (VMS) to enable the utilization of equivalent low-order spaces for approximating both primary unknowns. The efficacy of these formulations is compared with alternative stabilization techniques and validated across various two- and three-dimensional benchmark problems to assess its accuracy and robustness.

中文翻译：

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使用变分子网格尺度的不可压缩超弹性材料的混合稳定 MPM 公式


物质点法 （MPM） 是一种基于连续体的粒子技术，旨在解决大变形问题。然而，使用 MPM 处理不可压缩材料仍未得到充分探索。本研究的重点是调整有限元方法 （FEM） 的既定技术，以解决动态超弹性问题的 MPM 中的不可压缩性问题。首先，我们引入了一个混合位移-压力公式来解决不可压缩性问题。其次，我们采用了植根于变分多尺度法 （VMS） 的两种不同的稳定技术，以利用等效的低阶空间来近似两个主要的未知数。将这些配方的功效与其他稳定技术进行比较，并在各种二维和三维基准问题中进行验证，以评估其准确性和稳健性。