🤖 AI Summary
This work addresses the compatibility and stability challenges that arise when coupling the approximate full mass matrix method (FMPM(k)) with lumped-mass-dependent Material Point Method (MPM) features—such as velocity boundary conditions and multi-material contact. To resolve these issues, the authors propose a reformulated FMPM(k) implementation that streamlines the FMPM loop to execute only once per time step and adapts the algorithm to seamlessly integrate with standard MPM functionalities. This study presents the first successful integration of FMPM(k) with lumped-mass-dependent MPM capabilities, while systematically analyzing the influence of the polynomial order \(k\) on time-step stability and computational efficiency. The proposed approach significantly enhances the applicability and performance of high-order FMPM(k) in complex multiphysics simulations.
📝 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.