Treatment of near-incompressibility and volumetric locking in higher order material point methods

TL;DR

This paper develops a novel projection-based framework to address near-incompressibility and volumetric locking in higher-order material point methods (MPM). By projecting the dilatational component of the velocity gradient onto a lower-dimensional space—via a two-grid, locally lumped $L^2$ projection—the method yields an improved velocity gradient $oldsymbol{ar{ abla v}}$ that stabilizes the solution without sacrificing the benefits of high-order B-spline backgrounds. The framework extends the Bbar and Fbar concepts to the MPM setting, including finite-strain extensions with a multiplicative split and a compatible reconstruction on material points. Numerical experiments across dynamic elastic and elasto-plastic problems (vibrating bar, Cook’s membrane, elasto-plastic collapse, Taylor bar impact) demonstrate reduced stress oscillations, elimination of locking, and smoother hydrostatic stress fields, with results aligning well with literature. Overall, the proposed approach enables accurate, stable simulations of near-incompressible behavior in higher-order MPM, offering practical impact for large-deformation analyses in solid mechanics.

Abstract

We propose a novel projection method to treat near-incompressibility and volumetric locking in small- and large-deformation elasticity and plasticity within the context of higher order material point methods. The material point method is well known to exhibit volumetric locking due to the presence of large numbers of material points per element that are used to decrease the quadrature error. Although there has been considerable research on the treatment of near-incompressibility in the traditional material point method, the issue has not been studied in depth for higher order material point methods. Using the Bbar and Fbar methods as our point of departure we develop an appropriate projection technique for material point methods that use higher order shape functions for the background discretization. The approach is based on the projection of the dilatational part of the appropriate strain rate measure onto a lower dimensional approximation space, according to the traditional Bbar and Fbar techniques, but tailored to the material point method. The presented numerical examples exhibit reduced stress oscillations and are free of volumetric locking and hourglassing phenomena.

Figures (10)

• Figure 1: Vibrating bar. Convergence of the L$^2$ norm of the displacement error at 0.5 s
• Figure 2: Cook's membrane. The left edge is fixed and vertical traction is applied on the right edge.
• Figure 3: Cook's membrane. Hydrostatic stress at the end of the computation. Discretization M2. Left column from top to bottom: $C^0$-continuous linear shape functions; $C^1$-continuous quadratic B-spline shape functions; $C^2$-continuous cubic B-spline shape functions. Right column from top to bottom: $C^0$-continuous linear shape functions with $\overline{\mathbf{F}}$ projection onto constants; $C^1$-continuous quadratic B-spline shape functions with $\overline{\mathbf{F}}$ projection onto linears; $C^2$-continuous cubic B-spline shape functions with $\overline{\mathbf{F}}$ projection onto quadratics.
• Figure 4: Cook's membrane. Hydrostatic stress at the end of the computation. Discretization M2. Left: $C^1$-continuous quadratic B-spline shape functions with $\overline{\mathbf{F}}$ projection onto constants. Right: $C^1$-continuous quadratic B-spline shape functions with $\overline{\mathbf{F}}$ projection onto linears.
• Figure 5: Cook's membrane. Hydrostatic stress at the end of the computation. $C^1$-continuous quadratic B-spline shape functions with $\overline{\mathbf{F}}$ projection onto linears. From left to right: M1; M2; M3.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6