A high-fidelity material point method for frictional contact problems

TL;DR

This paper presents a penalty-based, Extended B-Spline MPM framework for frictional contact that treats boundary surfaces as deformable, boundary material points and uses EBS interpolation to reduce grid-crossing and numerical integration errors. The approach yields explicit normal and tangential contact forces projected directly onto the Eulerian grid, eliminating premature contact and reducing stress noise at contact interfaces. Benchmarking against analytical solutions and Hertz contact problems, as well as comparisons with OBS and TLMPM variants, shows superior accuracy, energy conservation, and convergence, even on coarse meshes. The method demonstrates robust performance across one- and two-dimensional frictional contact problems and stress-wave scenarios in granular media, indicating strong potential for high-fidelity, large-scale simulations with practical impact in engineering applications.

Abstract

A novel Material Point Method (MPM) is introduced for addressing frictional contact problems. In contrast to the standard multi-velocity field approach, this method employs a penalty method to evaluate contact forces at the discretised boundaries of their respective physical domains. This enhances simulation fidelity by accurately considering the deformability of the contact surface, preventing fictitious gaps between bodies in contact. Additionally, the method utilises the Extended B-Splines (EBSs) domain approximation, providing two key advantages. First, EBSs robustly mitigate grid cell-crossing errors by offering continuous gradients of the basis functions on the interface between adjacent grid cells. Second, numerical integration errors are minimised, even with small physical domains in occupied grid cells. The proposed method's robustness and accuracy are evaluated through benchmarks, including comparisons with analytical solutions, other MPM-based contact algorithms, and experimental observations from the literature. Notably, the method demonstrates effective mitigation of stress errors inherent in contact simulations.

Figures (30)

• Figure 1: \ref{['fig:gov_eqns_frict_cont_probs:continuum_bodies_mpm_approximation:a']} Continuum bodies (discrete fields) into contact and \ref{['fig:gov_eqns_frict_cont_probs:continuum_bodies_mpm_approximation:b']} Material point method approximation.
• Figure 2: Kinematic contact constraints. \ref{['fig:gov_eqns_frict_cont_probs:contact_constraints:a']} normal contact law, \ref{['fig:gov_eqns_frict_cont_probs:contact_constraints:b']} tangential contact law and \ref{['fig:gov_eqns_frict_cont_probs:contact_constraints:c']} Coulomb’s cone for the two-dimensional problem. As a result of employing a penalty function approach and incorporating the penalty parameters $\omega^{nor}$ and $\omega^{tan}$, kinematic constraints are “weakly” imposed, as illustrated by a dashed green line.
• Figure 3: Categorising grid cells and nodes with classified bases on a two-dimensional continuum covered by an Eulerian grid. This is made according to the value of the volume fraction, $\phi_c$, at the boundary grid cells in relation to occupation parameter, $C_c$, chosen for the simulation.
• Figure 4: Derivation of quadratic Extended B-Splines from the Original B-Splines for the case of one-dimensional Eulerian grid. In this example, an occupation parameter at $C_c=0.75$ is utilised. Grid cells with identity, $4$ and $8$ are marked as boundary grid cells, resulting in two degenerated basis functions, associated with grid nodes (control points) $4$ and $10$.
• Figure 5: Boundary tracking process. Step 1: Shared control points indicate potential contact zones. Step 2: Distance of the boundary materials points are less than grid spacing, $\Delta h$. Step 3: The $\textsl{g}^{nor} > 0$ which indicates that the slave material point $p_s$ is not in contact with the master segment, formed by boundary material points $p_{m_1}$ and $p_{m_2}$.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4