Modeling irregular boundaries using isoparametric elements in the Material Point Method

The Material Point Method (MPM) is a continuum approach for modeling large-deformations in history-dependent materials (Sulsky et al., 1994 and1995). Unlike the classical mesh-based numerical approaches such as the Finite Element Method (FEM), the MPM avoids mesh distortion issues when solving large-deformation problems. The material domain is discretized into a set of Lagrangian material points, which carry information such as their mass, velocity and other history-dependent material variables. The Eulerian background mesh is used purely for solving partial differential equations and the material points can traverse independent of this background mesh. An illustration of the MPM algorithm is shown in Figure 1. Typically in the MPM, rectilinear elements are used to represent a structured mesh. Kinematic boundary constraints are applied on the Eulerian background nodes, independent of the location of the Lagrangian material points. Hence, modeling irregular boundaries such as a natural landslide topography remains a challenge. Figure 2 presents the commonly adopted methods in the MPM to model irregular boundaries in the rectilinear structured mesh. Figure 2a illustrates the application of Dirichlet velocity constraints on the node closest to the boundary. Xu et al. (2018) adopted this to model the 3D runout of Hongshiyan landslide in China. This approach of constraining the boundary nodes often results in a step-wise boundary constraint that does not accurately capture the complex boundaries observed and may cause unrealistic flow constraints near the boundary. Another commonly adopted approach is to represent the irregular boundary surface using a contact algorithm between distinct sets of material points as shown in Figure 2b. Various landslides simulations have utilized this approach such as the Oso landslide simulations by Yerro et al. (2018). This approach is typically used with an unstructured triangular or tetrahedral elements. Figure 2c illustrates a nonconforming implicit boundary condition that can be manipulated independently of the background mesh. In this approach, the Dirichlet boundary conditions are weakly imposed on the finite element space. Cortis et al. (2018) first applied an implicit boundary method in MPM by weakly imposing the Dirichlet boundary conditions over a finite thickness along the boundary. Nevertheless, more research is needed to extend this method for practical applications of MPM. In this study, we explore the suitability of unstructured quadrilateral elements that conform to irregular boundaries as illustrated in Figure 2d. The use of isoparametric representation in the MPM has the following implications: (1) the constraints on boundary surfaces represented using irregular isoparametric elements are often not aligned with the axes of the global coordinate system and have to be defined in the local coordinate system; (2) in the MPM, nodal properties such as the mass, momentum and forces are mapped from the material points based on their location in the natural coordinate system; although mapping from the natural coordinate to the Cartesian coordinate is a linear transformation, the inverse (ξ, ) η (x, ) y mapping is not generally defined for irregular elements. The approach adopted in this study to solve the above two issues are discussed in the following sections. The method is then validated against the analytical solution of a gravity-driven rigid block sliding on an inclined plane and is applied to the simulation of dry sand flow in a controlled experimental inclined flume (Denlinger and Iverson 2001).

In the MPM with a rectilinear Cartesian background mesh, the Dirichlet velocity and frictional constraints are applied on the background nodes, in the directions aligned with the global coordinate axes. However, irregular topologies such as those of a landslide have boundary constraints (velocity vector components) not aligned with the global coordinate axes, and therefore should be applied in the local coordinate axes of each node. This requires a transformation of the nodal velocity and acceleration vectors from the global coordinate system to the local coordinate system through a transformation matrix:

, where is the v v L = T G T transformation matrix, and are the velocity vectors in the global and the local v G v L coordinates, respectively.

The following steps are involved when applying nodal constraints for irregular elements: (1) compute the transformation matrix for each constrained node, (2) transform nodal velocity and acceleration vectors to the local coordinate system using the transformation matrix, (3) apply the Dirichlet velocity and frictional boundary constraints in the local coordinate system, and (4) transform the nodal velocity and acceleration vectors from the local coordinate system back to the global coordinate system.

In contrast to the Gauss integration used in the FEM, the MPM considers the locations of material points as the integration poi

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