The material point method (MPM) is frequently used to simulate large deformations of nearly incompressible materials such as water, rubber, and undrained porous media. However, MPM solutions to nearly incompressible materials are susceptible to volumetric locking, that is, overly stiff behavior with erroneous strain and stress fields. While several approaches have been devised to mitigate volumetric locking in the MPM, they require significant modifications of the existing MPM machinery, often tailored to certain basis functions or material types. In this work, we propose a locking-mitigation approach featuring an unprecedented combination of simplicity, efficacy, and generality for a family of explicit MPM formulations. The approach combines the assumed deformation gradient ($\bar{\boldsymbol{F}}$) method with a volume-averaging operation built on the standard particle-grid transfer scheme in the MPM. Upon explicit time integration, this combination yields a new and simple algorithm for updating the deformation gradient, preserving all other MPM procedures. The proposed approach is thus easy to implement, low-cost, and compatible with the existing machinery in the MPM. Through various types of nearly incompressible problems in solid and fluid mechanics, we verify that the proposed approach efficiently circumvents volumetric locking in the explicit MPM, regardless of the basis functions and material types.

翻译：物质点方法（MPM）常用于模拟水、橡胶和不排水多孔介质等近不可压缩材料的大变形。然而，MPM在求解近不可压缩材料时易出现体积锁定，即产生过刚行为及错误的应变应力场。尽管已有多种方法被设计用于缓解MPM中的体积锁定，但它们需要对现有MPM框架进行大幅修改，且通常针对特定基函数或材料类型。本研究针对一类显式MPM公式，提出了一种兼具简洁性、有效性和通用性的锁定缓解方法。该方法将假设变形梯度（$\bar{\boldsymbol{F}}$）方法与基于MPM标准粒子-网格传递方案的体积平均操作相结合。在显式时间积分下，这种组合衍生出一种更新变形梯度的全新简单算法，且保留所有其他MPM步骤。因此，所提方法易于实现、成本低廉，且与MPM现有框架兼容。通过固体与流体力学中多种近不可压缩问题的验证，我们证实无论基函数与材料类型如何，该方法均能有效规避显式MPM中的体积锁定。