All material point method (MPM) codes approximate the full mass matrix with a lumped mass matrix. Because this approach causes dissipation, most MPM simulations rely on so-called FLIP methods to limit dissipation. Recent work to deal with noise caused by those FLIP methods derived the XPIC method (for extended particle in cell method) that filters null space noise from particle velocities using a projection operator. This paper shows that the XPIC projection operator is equivalent to doing grid calculations using an asymptotic expansion of the full mass matrix inverse. From that insight, we derived FMPM($k$) for full mass matrix MPM of order $k$ (where the mass matrix inverse is expanded to $k$ terms). Compared to prior MPM algorithms, FMPM changes the methods used to update particle velocities, positions, stresses, and strains. Several examples show that FMPM is more stable and accurate, has less dissipation than XPIC, and with high enough $k$ has less dissipation than FLIP methods. One challenge in MPM is imposing velocity conditions on the grid and those challenges are amplified in FMPM. This paper describes a new moving-wall approach that improves grid boundary conditions and is beneficial to both FMPM and prior MPM. Finally, the use of FMPM for multimaterial mode MPM, with explicit cracks, and with affine extrapolation options is each discussed.

所有材料点方法（MPM）代码都将总质量矩阵与集总质量矩阵近似。由于这种方法会导致耗散，因此大多数MPM仿真都依赖于所谓的FLIP方法来限制耗散。处理那些由FLIP方法引起的噪声的最新工作派生了XPIC方法（用于单元中的扩展粒子方法），该方法使用投影算子从粒子速度中滤除零空间噪声。本文表明，XPIC投影算子等效于使用全质量矩阵逆的渐近展开进行网格计算。根据这一见解，我们得出了FMPM（$ķ$）对于订单的完整质量矩阵MPM $ķ$ （其中质量矩阵逆扩展为 $ķ$条款）。与以前的MPM算法相比，FMPM更改了用于更新粒子速度，位置，应力和应变的方法。几个示例表明，FMPM更稳定，更准确，耗散性比XPIC小，并且足够高$ķ$比FLIP方法耗散少。MPM的挑战之一是在电网上施加速度条件，而FMPM则加剧了这些挑战。本文介绍了一种新的移动墙方法，该方法可以改善网格边界条件，并且对FMPM和以前的MPM都有利。最后，讨论了将FMPM用于具有明确裂缝和仿射外推选项的多材料模式MPM。