Unified sparse framework for large-scale simulations using the material point method
Abstract. The material point method (MPM) is a hybrid particle-grid method widely used for large deformation problems with history-dependent behavior, including geophysical mass flows. Standard MPM often relies on a dense background grid, which can be highly inefficient when material occupies a small fraction of the computational domain. Such sparsity is common in many large-scale geophysical mass flow problems. Here, we introduce a unified sparse background-grid framework for large-scale MPM simulation. The framework treats sparse grid construction as a general active-node indexing problem. We develop two architecture-specific implementations to realize the same sparse framework: a scan-based strategy for CPUs and a hash-based strategy for GPUs. Through benchmark problems and a large-scale landslide simulation, we show that the framework provides identical results as standard dense MPM while reducing computational time and memory usage by one to two orders of magnitude in strongly sparse cases.
Status: open (until 30 Nov 2026)
- RC1: 'Comment on egusphere-2026-4500', Anonymous Referee #1, 31 Oct 2026 reply
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- 1
The reviewed manuscript presents a unified sparse background-grid framework for MPM, using scan-based and hash-based implementations for CPUs and GPUs, respectively. The proposed framework aims to reduce computational time and memory usage without modifying the underlying MPM formulation. Through several benchmark problems and a large-scale landslide simulation, the authors demonstrate significant improvements in computational efficiency, particularly for highly sparse problems.
Overall, the manuscript is well written and addresses an important computational challenge in large-scale MPM simulations. I think this will be of interest to the MPM community in general. I have several comments about the implementation details, comparisons with existing approaches, and general applicability of the framework, but otherwise, I am supportive of accepting the manuscript for publication at GMD.
Some comments:
1. Sparse grid allocation has been explored in previous MPM developments, including Setaluri et al. (2014) and Chen et al. (2025). Since the performance improvements are primarily demonstrated against dense MPM, could the authors clarify the advantages of the proposed algorithms over existing sparse approaches? A discussion of differences in indexing, memory allocation, and parallelization would help establish a better understanding of the novelty of this work compared to previous studies.
2. Figs. 3 and 6 show substantial performance improvements over dense MPM. Could the authors clarify whether the dense implementations use particle spatial sorting, comparable parallelization and compiler optimizations, and active-node-only initialization? The elaborated details will help readers understand the fairness of the performance comparisons, i.e., what implementation they are comparing to.
3. Eq. 1: The sparsity ratio in Eq. (1) characterizes the overall fraction of active nodes. However, computational efficiency may also depend on their spatial distribution. For example, material points may not be located around the same location as presented in the numerical tests conducted. How would the proposed approaches perform for multiple disconnected material regions, such as in fragmentation problems? Could the authors discuss whether the distribution and connectivity of active regions influence the computational efficiency?
4. P5, Section 2.1.2: Do the active-node indexing and sparse grid allocation need to be reconstructed at every time step? Could the authors clarify the associated computational overhead and whether information from previous time steps can be reused?
5. P12, Section 3.1: The authors state that the proposed framework is independent of the time integration scheme. Could they briefly discuss how this framework would extend to implicit MPM, particularly regarding global matrix assembly and solving the resulting system? Will the proposed algorithm improve implicit MPM to the same order of efficiency as explicit MPM, or perhaps significantly less because the assembly and matrix-solver parts may dominate the computational cost?
6. P13, Section 3.1: Different MPM interpolation functions have different support sizes, which affect the number of active nodes and shared active nodes per particle. For P2G, for example, this requires an algorithm that can safely handle mapping without race conditions. Could the authors comment on how the computational efficiency of the proposed framework may change when using linear or higher-order B-spline interpolation?