An Overlapping Schwarz Space-Time Refinement Framework for Material Point Method

TL;DR

The paper introduces an overlapping Schwarz space-time refinement framework for the material point method, enhancing computational efficiency and accuracy in complex deformation problems through localized adaptive resolution.

Contribution

It presents a modular, non-intrusive refinement approach that maintains standard MPM discretizations while enabling efficient local space-time adaptivity.

Findings

Achieves comparable or better accuracy than fine single-domain simulations.

Reduces computational cost by up to 9.15 times.

Demonstrates effectiveness in 3D folding and contact problems.

Abstract

We propose an overlapping Schwarz space-time refinement framework for the material point method (OS-MPM) to improve computational efficiency in problems with strongly localized deformation, contact, and large geometric nonlinearity. The method decomposes the domain into overlapping coarse and fine subdomains with heterogeneous spatial and temporal resolutions, while retaining standard MPM discretizations within each subdomain. Coarse-fine coupling is achieved through an MPM-specific Schwarz iteration combining mass-weighted spatial transmission and temporal interpolation for sub-cycling. In contrast to refinement strategies based on modified basis functions, transition kernels, or strongly enforced interface constraints, the proposed approach preserves the modular structure of standard MPM and shifts the coupling complexity to nonmatching-grid interface operators within the Schwarz alternating procedure. Numerical examples, including a gravity-driven cantilever beam, Hertzian contact, and an elastic inclusion problem, show that the method reproduces analytical or fine-resolution reference solutions with good accuracy and convergence behavior. In the inclusion benchmark, the proposed framework achieves comparable or slightly lower error than single-domain fine simulations at the finest tested resolutions, while reducing computational cost by up to 9.15 times. A three-dimensional folding example further demonstrates the generality of the framework. These results indicate that the proposed method provides an accurate, modular, and efficient route for local space-time refinement in MPM.