The Material Point Method (MPM) is widely used to analyse coupled (solid-water) problems under large deformations/displacements. However, if not addressed carefully, MPM u-p formulations for poro-mechanics can be affected by two major sources of instability. Firstly, inf-sup condition violation can arise when the spaces for the displacement and pressure fields are not chosen correctly, resulting in an unstable pressure field. Secondly, the intrinsic nature of particle-based discretisation makes the MPM an unfitted mesh-based method, which can affect the system's condition number and solvability, particularly when background mesh elements are poorly populated. This work proposes a solution to both problems. The inf-sup condition is avoided using two overlapping meshes, a coarser one for the pressure and a finer one for the displacement. This approach does not require stabilisation of the primary equations since it is stable by design and is particularly valuable for low-order shape functions. As for the system's poor condition number, a face ghost penalisation method is added to both the primary equations, which constitutes a novelty in the context of MPM mixed formulations. This study frequently makes use of the theories of functional analysis or the unfitted Finite Element Method (FEM). Although these theories may not directly apply to the MPM, they provide a robust and logical basis for the research. These rationales are further supported by three numerical examples, which encompass both elastic and elasto-plastic cases and drained and undrained conditions.

翻译：材料点法（MPM）广泛应用于大变形/大位移条件下的耦合（固-水）问题分析。然而，若未谨慎处理，针对孔隙力学的MPM u-p公式化会受到两种主要不稳定来源的影响。其一，当位移场与压力场的空间选择不当时，可能违反inf-sup条件，导致压力场不稳定；其二，基于粒子离散化的固有特性使MPM成为非拟合网格方法，这会影响系统的条件数与可解性，尤其当背景网格单元中粒子分布稀疏时。本研究针对上述两个问题提出解决方案。通过采用两套重叠网格（压力场使用较粗网格，位移场使用较细网格）避免inf-sup条件，该方法因设计内在稳定而无需对主方程进行稳定化处理，对低阶形函数尤具价值。针对系统条件数不佳的问题，在主方程中引入面鬼影惩罚方法，这在MPM混合公式化领域具有创新性。本研究频繁运用泛函分析理论或非拟合有限元法（FEM）的理论框架。尽管这些理论可能无法直接适用于MPM，但为研究提供了坚实且合理的基础。三个数值算例进一步支撑了上述理论论证，涵盖弹性与弹塑性工况、排水与不排水条件。