The Material Point Method (MPM) is an established and powerful numerical method particularly useful for simulating large-scale, rapid soil deformations. Therefore, it is often used for the numerical investigation of mass movement hazards such as landslides, debris flows, or avalanches. It combines the benefits of both mesh-free and mesh-based continuum-based discretization techniques by discretizing the physical domain with Lagrangian moving particles carrying the history-dependent variables while the governing equations are solved at the Eulerian background grid, which brings many similarities with commonly used finite element methods. However, due to this hybrid nature, the material boundaries do not usually coincide with the nodes of the computational grid, which complicates the imposition of boundary conditions. Furthermore, the position of the boundary may change at each time step and, moreover, may be defined at arbitrary locations within the computational grid that do not necessarily coincide with the body contour, leading to different interactions between the material and the boundary. To cope with these challenges, this paper presents a novel element-wise formulation to weakly impose non-conforming Dirichlet conditions using Lagrange multipliers. The proposed formulation introduces a constant Lagrange multiplier approximation within the constrained elements in combination with a methodology to eliminate superfluous constraints. Therefore, in combination with simple element-wise interpolation functions classically utilized in MPM (and FEM) to approximate the unknown field, a suitable Lagrange multiplier discretization is obtained. In this way, we obtain a robust, efficient, and user-friendly boundary imposition method for immersed methods specified herein for implicit MPM. Furthermore, the extension to frictionless slip conditions is derived. The proposed methodologies are assessed by comparing the numerical results with both analytical and experimental data to demonstrate their accuracy and wide range of applications.

质点法 (MPM) 是一种成熟且强大的数值方法，特别适用于模拟大规模、快速的土壤变形。因此，它经常用于大规模移动灾害（如滑坡、泥石流或雪崩）的数值研究。它结合了无网格和基于网格的连续体离散技术的优点，通过使用携带历史相关变量的拉格朗日运动粒子对物理域进行离散化，同时在欧拉背景网格上求解控制方程，这与常用的有限元方法。然而，由于这种混合性质，材料边界通常不与计算网格的节点重合，这使得边界条件的施加变得复杂。此外，边界的位置可能在每个时间步长发生变化，而且可以定义在计算网格内的任意位置，这些位置不一定与身体轮廓一致，从而导致材料和边界之间的不同相互作用。为了应对这些挑战，本文提出了一种新颖的逐元素公式，使用拉格朗日乘子弱地施加非相容狄利克雷条件。所提出的公式在约束元素内引入了恒定拉格朗日乘数近似，并结合消除多余约束的方法。因此，结合MPM（和FEM）中经典使用的简单逐元素插值函数来近似未知场，获得合适的拉格朗日乘子离散化。通过这种方式，我们获得了一种稳健、高效且用户友好的边界强加方法，用于本文为隐式 MPM 指定的浸没方法。此外，还推导了无摩擦滑动条件的扩展。通过将数值结果与分析和实验数据进行比较来评估所提出的方法，以证明其准确性和广泛的应用范围。