Lagrange multiplier imposition of non-conforming essential boundary conditions in implicit material point method

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.