Approximate full mass matrix methods for the material point method, known as FMPM(k) of order k, can improve the calculation of grid velocities from grid momentum. It can be implemented in any MPM code by inserting a new calculation task whenever grid velocities are needed. The implementation recommended in this paper only needs these calculations once per time step just before when updating particle positions and velocities. FMPM implementation issues arise, however, when its methods are mixed with other MPM feature that rely on lumped mass calculations. Some common lumped-mass MPM features are grid-based, velocity boundary condition, multimaterial contact calculations, crack contact calculations, and imperfect interfaces. This paper first derives a revised FMPM(k) implementation that both simplifies and clarifies the "FMPM Loop" that can be added to MPM codes. Next, that loop is modified to allow FMPM(k) to work well even in simulations that need other MPM features that previously caused conflicts. Two other FMPM(k) issues are apparent loss of stability at very higher order k and inherent computational cost. These issues are discussed in an analysis of temporal stability as a function of order k and in consideration of options to improve efficiency.

翻译：物质点法的近似全质量矩阵方法，称为k阶FMPM(k)，可改进从网格动量计算网格速度的过程。该方法可在任何MPM代码中通过插入新的计算任务（每当需要网格速度时）来实现。本文推荐的实现仅在每个时间步更新粒子位置和速度前执行一次这些计算。然而，当FMPM方法与依赖集中质量计算的其他MPM特性混合使用时，会出现实现问题。一些常见的集中质量MPM特性包括基于网格的速度边界条件、多材料接触计算、裂纹接触计算以及非完美界面。本文首先推导了修订后的FMPM(k)实现方式，简化并明确了可添加到MPM代码中的"FMPM循环"。接着对该循环进行修改，使FMPM(k)即使在需要其他先前引发冲突的MPM特性的模拟中也能良好运行。另外两个FMPM(k)问题涉及高阶k时明显稳定性丧失和固有计算成本。本文通过分析时间稳定性与阶数k的函数关系，并考虑提高效率的选项，对这些问题进行了讨论。