Improved Implementation of Approximate Full Mass Matrix Inverse Methods into Material Point Method Simulations
2026-04-08 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and Science
AI summaryⓘ
The authors look at a way to improve how a computer method called the Material Point Method calculates velocities on a grid, using something called FMPM(k). They explain how to add this improvement easily into existing codes and make it work well with other features that usually cause problems when combined with FMPM(k). They also study some challenges like stability issues at very high orders and the extra computing time it requires, suggesting ways to handle these problems.
Material Point Method (MPM)Full Mass MatrixFMPM(k)Grid VelocityLumped MassMultimaterial ContactCrack ContactTemporal StabilityComputational Cost
Authors
John A. Nairn
Abstract
Approximate full mass matrix methods for the material point method, known as FMPM(k) of order k, can improve the calculation of grid velocities from grid momentum. It can be implemented in any MPM code by inserting a new calculation task whenever grid velocities are needed. The implementation recommended in this paper only needs these calculations once per time step just before when updating particle positions and velocities. FMPM implementation issues arise, however, when its methods are mixed with other MPM feature that rely on lumped mass calculations. Some common lumped-mass MPM features are grid-based, velocity boundary condition, multimaterial contact calculations, crack contact calculations, and imperfect interfaces. This paper first derives a revised FMPM(k) implementation that both simplifies and clarifies the "FMPM Loop" that can be added to MPM codes. Next, that loop is modified to allow FMPM(k) to work well even in simulations that need other MPM features that previously caused conflicts. Two other FMPM(k) issues are apparent loss of stability at very higher order k and inherent computational cost. These issues are discussed in an analysis of temporal stability as a function of order k and in consideration of options to improve efficiency.