Circumventing volumetric locking in explicit material point methods: A simple, efficient, and general approach

TL;DR

Through various types of nearly incompressible problems in solid and fluid mechanics, it is verified that the proposed locking‐mitigation approach efficiently circumvents volumetric locking in the explicit MPM, regardless of the basis functions and material types.

Abstract

The material point method (MPM) is frequently used to simulate large deformations of nearly incompressible materials such as water, rubber, and undrained porous media. However, MPM solutions to nearly incompressible materials are susceptible to volumetric locking, that is, overly stiff behavior with erroneous strain and stress fields. While several approaches have been devised to mitigate volumetric locking in the MPM, they require significant modifications of the existing MPM machinery, often tailored to certain basis functions or material types. In this work, we propose a locking-mitigation approach featuring an unprecedented combination of simplicity, efficacy, and generality for a family of explicit MPM formulations. The approach combines the assumed deformation gradient ($\bar{\boldsymbol{F}}$) method with a volume-averaging operation built on the standard particle-grid transfer scheme in the MPM. Upon explicit time integration, this combination yields a new and simple algorithm for updating the deformation gradient, preserving all other MPM procedures. The proposed approach is thus easy to implement, low-cost, and compatible with the existing machinery in the MPM. Through various types of nearly incompressible problems in solid and fluid mechanics, we verify that the proposed approach efficiently circumvents volumetric locking in the explicit MPM, regardless of the basis functions and material types.

Figures (17)

• Figure 1: MPM update procedure.
• Figure 2: Cook's membrane: problem geometry and boundary conditions.
• Figure 3: Cook's membrane: mean normal stress fields in the standard and $\bar{\bm{F}}$ MPM solutions, obtained with GIMP basis functions.
• Figure 4: Cook's membrane: mean normal stress fields in the standard and $\bar{\bm{F}}$ MPM solutions, obtained with B-splines basis functions.
• Figure 5: Cook's membrane: comparison of our $\bar{\bm{F}}$ GIMP solution with the nonlinear $\bar{\bm{B}}$ GIMP solution in Bisht et al.bisht2021simulating. Both solutions are produced with 5,776 material points.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3