arxiv: 2605.09097 · v1 · submitted 2026-05-09 · 💻 cs.CE · cs.NA· math.NA· physics.comp-ph

Recognition: 2 theorem links

· Lean Theorem

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

Zhaofeng Luo , Minchen Li , Yupeng Jiang

Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:50 UTC · model grok-4.3

classification 💻 cs.CE cs.NAmath.NAphysics.comp-ph

keywords material point methodSchwarz methoddomain decompositionspace-time refinementcomputational mechanicsnonlinear simulationcontact mechanicsadaptive refinement

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The pith

An overlapping Schwarz framework enables local space-time refinement in the material point method while preserving accuracy and reducing computation time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a framework that splits simulations into overlapping coarse and fine subdomains for the material point method. Coupling happens through Schwarz iterations that use mass-weighted transmission between grids and interpolate in time for sub-cycling. This keeps the standard MPM code unchanged in each subdomain. The approach targets problems with localized deformation and contact, delivering accuracy close to full fine-grid runs at a fraction of the cost, as shown by up to 9.15 times speedup in tests. Readers interested in efficient mechanics simulations would see this as a way to handle large nonlinear problems without global high resolution.

Core claim

The central discovery is an overlapping Schwarz space-time refinement method for MPM that decomposes the domain into coarse and fine overlapping regions, applies heterogeneous resolutions, and couples them via MPM-specific Schwarz iterations with mass-weighted spatial transmission and temporal interpolation. This modular approach avoids altering basis functions or enforcing strong interface constraints, instead shifting complexity to interface operators in the alternating procedure. Benchmarks including cantilever beam, Hertzian contact, elastic inclusion, and 3D folding confirm that the method matches reference solutions with good convergence and achieves comparable or lower error than fine

What carries the argument

The overlapping Schwarz alternating procedure combined with mass-weighted spatial transmission and temporal interpolation for sub-cycling between non-matching grids.

Load-bearing premise

The Schwarz iteration with mass-weighted spatial transmission and temporal interpolation converges reliably without introducing artifacts in problems with strong nonlinearity, contact, and large geometric changes.

What would settle it

Running the inclusion or folding benchmark with higher nonlinearity or finer sub-cycling ratios and observing divergence of the Schwarz iterations or non-physical artifacts at the interfaces would falsify the claim of reliable convergence.

Figures

Figures reproduced from arXiv: 2605.09097 by Minchen Li, Yupeng Jiang, Zhaofeng Luo.

**Figure 1.** Figure 1: Illustration of overlapping subdomains. Boundary grid selection Background grid Material boundary Selected boundary grid node Boundary particle Interior particle Shape-function support (of one boundary particle) (a) Boundary grid selection from boundary-particle P2G contributions. ΩS ∂ΩB ΩB ∂ΩS coarse grid fine grid boundary coarse node fine node coincident node (b) Boundary-node ambiguity caused by coinc… view at source ↗

**Figure 2.** Figure 2: Construction of boundary grid nodes for Schwarz coupling. The selected grid nodes form the Dirichlet coupling sets Γ 𝑛 𝛼 ; coincident coarse- and fine-grid boundary nodes are resolved by the mass-based rule in Eq. (24). Naive Interpolation Our Interpolation Interpolation [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison between interpolation-based fine-to-coarse BC computation (left) and our projection-based version (right). Coarse Domain t v n n Fine Domain t n Δt ΔT t n+1/3 Δt t n+2/3 Δt t n+1 v n+1 t n+1 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: Temporal interpolation used for fine-domain sub-cycling. The coarse-domain interface velocities at 𝑡 𝑛 and 𝑡 𝑛+1 are linearly interpolated to provide boundary data at each fine sub-step. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 18 of 27 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: Pipeline overview of our domain-decomposed MPM. Domain Layout Ω𝐿 ∶ [0.05, 0.45], 𝓁 = 0.40 Ω𝑅 ∶ [0.39, 0.90], 𝓁 = 0.51 𝑡 = 0.02 Overlap (𝑥 ∈ [0.39, 0.45]) [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

**Figure 6.** Figure 6: Spatial domain decomposition for the cantilever beam. The structure is dynamically coupled via the intermediate orange overlap zone. The blue domain has width 1.0 and resolution 200 × 200, while the green domain has width 0.5 and resolution 100 × 100. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 19 of 27 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗

**Figure 7.** Figure 7: Master-curve validation for the gravity-driven cantilever beam experiment. Our OS-MPM results agree with the analytical planar-elastica relation between the deformation aspect ratio 𝐻∕𝑊 and the gravito-bending parameter Γ ∗ gb. 0.0 0.2 0.4 0.6 0.8 1.0 x/L −1.0 −0.8 −0.6 −0.4 −0.2 0.0 ¯w Single domain Dual domain [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

**Figure 8.** Figure 8: Normalized displacement comparison for the cantilever beam at Γ ∗ gb = 2.31 × 10−1. The single-domain and dual-domain results nearly overlap. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 20 of 27 [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗

**Figure 9.** Figure 9: Setup of the Hertzian contact benchmark. The fine sub-domain (Ω𝑆 ) resolves the anticipated contact zone near the rigid plane, whereas the coarse sub-domain (Ω𝐵 ) represents the far-field bulk response through an overlapping transition region. 0.35 0.40 0.45 0.50 0.55 0.60 0.65 x (m) 0 2 4 6 8 10 12 Contact pressure (kPa) h=0.015 h=0.01 h=0.0075 h=0.006 h=0.005 Analytical [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

**Figure 10.** Figure 10: Hertzian contact benchmark. Contact pressure distribution along the rigid interface for a sequence of finedomain (Ω𝑆 ) resolutions, with the coarse domain (Ω𝐵 ) held fixed. The numerical profiles converge monotonically toward the analytical Hertz solution as the fine-grid mesh is refined. (a) Coarse Domain Ω𝐵 Matrix 𝑟 = 0.1 (b) Fine Domain Ω𝑆 𝑟 = 0.15 𝑟 = 0.05 Inclusion (c) Composite & Overlap Overlap Zo… view at source ↗

**Figure 11.** Figure 11: Setup for the Elastic Inclusion Problem. Overlap zone is an annular ring bridging Ω𝐵 and Ω𝑆 . Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 21 of 27 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗

**Figure 12.** Figure 12: Stress distribution for the elastic inclusion problem (ℎ = 0.004 m) for stiffness ratio 𝐸in∕𝐸out = 1. Each panel: top row is dual-domain Schwarz MPM; bottom row is single-domain reference. Columns show 𝜎𝑥𝑥, 𝜎𝑦𝑦, and 𝜎𝑥𝑦, respectively. Solid and dashed black circles indicate the coarse-domain (Ω𝐵 ) and fine-domain (Ω𝑆 ) boundaries. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 22 of 27 [PITH_FULL_IMAGE:fi… view at source ↗

**Figure 13.** Figure 13: 𝐸in∕𝐸out = 2. 0.3 0.4 0.5 0.6 0.7 Dual-domain σxx σyy σxy 0.3 0.4 0.5 0.6 0.7 x (m) 0.3 0.4 0.5 0.6 0.7 Single-domain 0.3 0.4 0.5 0.6 0.7 x (m) 0.3 0.4 0.5 0.6 0.7 x (m) −640 −320 0 320 640 Stress (Pa) [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗

**Figure 14.** Figure 14: 𝐸in∕𝐸out = 5. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 23 of 27 [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗

**Figure 15.** Figure 15: 𝜎𝑦𝑦 stress profile along a horizontal cross-section through the inclusion center at each refinement level, for three stiffness ratios (𝐸in∕𝐸out ∈ {1, 2, 5}). Left column: our OS-MPM with dual domains; right column: single-domain MPM. Red dashed line: analytical solution. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 24 of 27 [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗

**Figure 16.** Figure 16: Accuracy and efficiency comparison for the elastic inclusion problem (𝐸in∕𝐸out ∈ {1, 2, 5}). Left column: 𝐿2 stress error vs. fine-domain grid spacing. Right column: 𝐿2 error vs. CPU time. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 25 of 27 [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗

**Figure 17.** Figure 17: Three-dimensional foldable-display-inspired showcase. Top left: von Mises stress distribution at the cross-section 𝑧 = 0.3. Top right: rest configuration and experimental setup. Bottom: solid (left) and particle (right) visualization of the folded configuration. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 26 of 27 [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a modular overlapping Schwarz route to local space-time refinement in MPM that preserves standard discretizations and reports solid cost savings in the benchmarks.

read the letter

The main takeaway is that this overlapping Schwarz space-time refinement (OS-MPM) lets you run fine resolution only where needed while using coarser steps elsewhere, all without rewriting the core MPM transfers. The coupling relies on mass-weighted transmission across the overlap plus temporal interpolation for sub-cycling, and the authors position it as more modular than basis-function tweaks or enforced constraints. That combination is the actual new piece for MPM users who already have working codebases. The cantilever beam, Hertz contact, and inclusion tests show convergence to reference solutions, and the inclusion case claims comparable or slightly lower error than a uniform fine grid at the highest resolutions while cutting cost by up to 9.15 times. The 3D folding demo adds a check on generality with large deformation. Those results are the strongest part of the work. The stress-test concern about the inclusion benchmark is worth watching. If the fine subdomain particle density or final Schwarz residual does not exactly match the single-domain reference, the error numbers are not directly comparable. The paper should spell out overlap width, iteration tolerance, and particle-per-cell counts in the fine region to close that gap. For problems with strong nonlinearity or contact, the iteration count and any artifacts from incomplete convergence also need explicit reporting. Overall the construction is coherent and the examples are encouraging, but the accuracy claim rests on those details being clean. This is for people running MPM on mechanics problems with localized features who want to keep their existing solver structure. A reader already working on adaptive or multi-resolution MPM would find the implementation choices and timing numbers useful. I would send it to peer review. The idea is practical and the reported gains are large enough to justify referee time, provided the benchmark setups get a close look.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes an overlapping Schwarz space-time refinement framework (OS-MPM) for the Material Point Method. The domain is decomposed into overlapping coarse and fine subdomains with heterogeneous spatial and temporal resolutions. Coupling is achieved via an MPM-specific Schwarz iteration that employs mass-weighted spatial transmission and temporal interpolation for sub-cycling. The approach preserves standard MPM discretizations within each subdomain and shifts complexity to the interface operators. Numerical examples on a gravity-driven cantilever beam, Hertzian contact, an elastic inclusion problem, and a 3D folding case demonstrate reproduction of reference solutions with good accuracy and convergence; the inclusion benchmark reports comparable or slightly lower error than uniform fine-grid runs while achieving up to 9.15 times computational savings.

Significance. If the accuracy and efficiency claims hold under strictly matched local resolutions, the method offers a modular route to space-time refinement in MPM that avoids alterations to basis functions or strong interface constraints. This would be valuable for problems with localized nonlinearity, contact, and large deformations. The paper supplies concrete numerical demonstrations of convergence behavior and cost reduction, which are positive attributes.

major comments (2)

[Inclusion benchmark] Inclusion benchmark (numerical examples section): The claim of comparable or slightly lower error relative to single-domain fine simulations at the finest resolutions is load-bearing for the efficiency argument. The manuscript must explicitly confirm that particle density per fine cell, overlap width, and final Schwarz residual are identical between the OS-MPM fine subdomain and the reference uniform fine run; otherwise the error comparison is not equivalent.
[Method section] Schwarz iteration description (method section, around the coupling operator): For cases involving strong nonlinearity and contact (Hertzian contact and folding examples), the paper should report iteration counts and residual norms per time step to substantiate that the alternating procedure converges reliably without introducing artifacts or requiring problem-specific tuning.

minor comments (3)

[Figures] Figure captions for the overlap regions (e.g., inclusion and folding figures) could more clearly indicate the overlap width and subdomain boundaries to aid reproducibility.
[Method section] The temporal interpolation operator for sub-cycling is introduced without an explicit statement of its conservation properties or truncation error; a brief derivation or reference would improve clarity.
[Numerical examples] A short table summarizing the Schwarz iteration tolerance, overlap size, and sub-cycling ratio used in each example would help readers assess robustness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment point by point below and will revise the manuscript to incorporate the requested clarifications and data.

read point-by-point responses

Referee: [Inclusion benchmark] Inclusion benchmark (numerical examples section): The claim of comparable or slightly lower error relative to single-domain fine simulations at the finest resolutions is load-bearing for the efficiency argument. The manuscript must explicitly confirm that particle density per fine cell, overlap width, and final Schwarz residual are identical between the OS-MPM fine subdomain and the reference uniform fine run; otherwise the error comparison is not equivalent.

Authors: We confirm that particle density per fine cell, overlap width, and the final Schwarz residual tolerance were set identically between the OS-MPM fine subdomain and the uniform fine-grid reference in the inclusion benchmark. This matching ensures the error comparison is equivalent. We will add an explicit statement in the revised numerical examples section to document these identical settings. revision: yes
Referee: [Method section] Schwarz iteration description (method section, around the coupling operator): For cases involving strong nonlinearity and contact (Hertzian contact and folding examples), the paper should report iteration counts and residual norms per time step to substantiate that the alternating procedure converges reliably without introducing artifacts or requiring problem-specific tuning.

Authors: We agree that reporting Schwarz iteration counts and residual norms per time step would better substantiate reliable convergence in the nonlinear and contact cases. In the revised manuscript we will add this information for the Hertzian contact and 3D folding examples, presented as tables or plots of iterations and residuals versus time step. revision: yes

Circularity Check

0 steps flagged

No circularity in OS-MPM framework construction or claims

full rationale

The paper introduces a new overlapping Schwarz iterative coupling for space-time refinement in MPM, using mass-weighted spatial transmission and temporal interpolation between coarse and fine subdomains while preserving standard MPM discretizations inside each. All load-bearing elements are algorithmic definitions and interface operators, not quantities defined in terms of themselves or fitted parameters renamed as predictions. Benchmark results (e.g., inclusion problem error and 9.15x cost reduction) are presented as empirical simulation outcomes rather than first-principles derivations that reduce to the inputs by construction. No self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the provided text, and the central claim remains an independent numerical validation of the proposed modular coupling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the domain assumption that standard MPM discretizations remain valid inside each subdomain and that the Schwarz iteration converges for the targeted class of problems; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)

domain assumption Standard MPM discretizations can be retained independently inside each overlapping subdomain
The abstract states that the method keeps standard MPM inside each subdomain while moving coupling complexity to the interface operators.

pith-pipeline@v0.9.0 · 5539 in / 1270 out tokens · 61091 ms · 2026-05-12T01:50:00.940893+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

mass-weighted spatial projection Π_S→B (Eq. 26) and temporal interpolation T_m (Eq. 27) inside multiplicative Schwarz iteration for heterogeneous-resolution MPM subdomains
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

variational incremental potential E_h(x̂) minimization (Eq. 12) with Newton line-search for implicit MPM time stepping

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages

[1]

The material-point method for granular materials

Bardenhagen, S., Brackbill, J., Sulsky, D., 2000. The material-point method for granular materials. Computer methods in applied mechanics and engineering 187, 529–541

work page 2000
[2]

Thegeneralizedinterpolationmaterialpointmethod

Bardenhagen,S.G.,Kober,E.M.,2004. Thegeneralizedinterpolationmaterialpointmethod. ComputerModelinginEngineering&Sciences 5, 477

work page 2004
[3]

A novel local grid refinement scheme for material point method

Barzoki, N.K., Sadeghirad, A., 2025. A novel local grid refinement scheme for material point method. Computational Particle Mechanics , 1–30

work page 2025
[4]

Ageneralfluid–sedimentmixturemodelandconstitutivetheoryvalidatedinmanyflowregimes

Baumgarten,A.S.,Kamrin,K.,2019. Ageneralfluid–sedimentmixturemodelandconstitutivetheoryvalidatedinmanyflowregimes. Journal of Fluid Mechanics 861, 721–764. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 15 of 27 Schwarz Space-Time Refinement for MPM

work page 2019
[5]

Formulation and application of a quasi-static material point method

Beuth, L., 2012. Formulation and application of a quasi-static material point method. Ph.D. thesis. Universität Stuttgart. Stuttgart, Germany. URL:http://dx.doi.org/10.18419/opus-419

work page doi:10.18419/opus-419 2012
[6]

The heavy elastica

Bickley, W.G., 1934. The heavy elastica. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 17, 603–622. doi:10.1080/14786443409462419

work page doi:10.1080/14786443409462419 1934
[7]

Unstructured moving least squares material point methods: a stablekernelapproachwithcontinuousgradientreconstructionongeneralunstructuredtessellations

Cao, Y., Zhao, Y., Li, M., Yang, Y., Choo, J., Terzopoulos, D., Jiang, C., 2025. Unstructured moving least squares material point methods: a stablekernelapproachwithcontinuousgradientreconstructionongeneralunstructuredtessellations. ComputationalMechanics75,655–678

work page 2025
[8]

An unconditionally stable total lagrangian material point method for large deformation dynamics

Charlton, T.J., Coombs, C.M., Augarde, C.E., 2017. An unconditionally stable total lagrangian material point method for large deformation dynamics. Computer Methods in Applied Mechanics and Engineering 327, 111–129

work page 2017
[9]

Impositionofessentialboundaryconditionsinthematerial point method

Cortis,M.,Coombs,W.,Augarde,C.,Brown,M.,Brennan,A.,Robinson,S.,2018. Impositionofessentialboundaryconditionsinthematerial point method. International Journal for Numerical Methods in Engineering 113, 130–152

work page 2018
[10]

Material point method after 25 years: Theory, implementation, and applications

De Vaucorbeil, A., Nguyen, V.P., Sinaie, S., Wu, J.Y., 2020. Material point method after 25 years: Theory, implementation, and applications. Advances in applied mechanics 53, 185–398

work page 2020
[11]

The determination of the elastic field of an ellipsoidal inclusion, and related problems

Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London. Series A 241, 376–396

work page 1957
[12]

Enhancement of the material point method using b-spline basis functions

Gan, Y., Sun, Z., Chen, Z., Zhang, X., Liu, Y., 2018. Enhancement of the material point method using b-spline basis functions. International Journal for numerical methods in engineering 113, 411–431

work page 2018
[13]

An adaptive generalized interpolation material point method for simulating elastoplastic materials

Gao, M., Tampubolon, A.P., Jiang, C., Sifakis, E., 2017. An adaptive generalized interpolation material point method for simulating elastoplastic materials. ACM Transactions on Graphics 36, 1–12

work page 2017
[14]

Implicit time integration for the material point method: quantitative and algorithmic comparisons with the finite element method

Guilkey, J.E., Weiss, J.A., 2003. Implicit time integration for the material point method: quantitative and algorithmic comparisons with the finite element method. International Journal for Numerical Methods in Engineering 57, 1323–1338. doi:10.1002/nme.713

work page doi:10.1002/nme.713 2003
[15]

Barrier-augmentedlagrangianforgpu-basedelastodynamiccontact

Guo,D.,Li,M.,Yang,Y.,Li,S.,Wang,G.,2024. Barrier-augmentedlagrangianforgpu-basedelastodynamiccontact. ACMTransactionson Graphics (TOG) 43, 1–17

work page 2024
[16]

A multi-resolution material point method based on penalty formulation

He, K.Y., Jin, Y.F., Zhou, X.W., Yin, Z.Y., Chen, X., 2025. A multi-resolution material point method based on penalty formulation. International Journal for Numerical and Analytical Methods in Geomechanics 49, 3839–3857

work page 2025
[17]

Explicitphase-fieldmaterialpointmethodwiththeconvectedparticle domaininterpolationforimpact/contactfractureinelastoplasticgeomaterials

Hu,Z.,Zhang,Z.,Zhou,X.,Cui,X.,Ye,H.,Zhang,H.,Zheng,Y.,2023. Explicitphase-fieldmaterialpointmethodwiththeconvectedparticle domaininterpolationforimpact/contactfractureinelastoplasticgeomaterials. ComputerMethodsinAppliedMechanicsandEngineering405, 115851

work page 2023
[18]

Stiffgipc: Advancing gpu ipc for stiff affine-deformable simulation

Huang, K., Lu, X., Lin, H., Komura, T., Li, M., 2025. Stiffgipc: Advancing gpu ipc for stiff affine-deformable simulation. ACM Transactions on Graphics 44, 1–20

work page 2025
[19]

Contact algorithms for the material point method in impact and penetration simulation

Huang, P., Zhang, X., Ma, S., Huang, X., 2011. Contact algorithms for the material point method in impact and penetration simulation. International journal for numerical methods in engineering 85, 498–517

work page 2011
[20]

The affine particle-in-cell method

Jiang, C., Schroeder, C., Selle, A., Teran, J., Stomakhin, A., 2015. The affine particle-in-cell method. ACM Transactions on Graphics 34, 51:1–51:10. doi:10.1145/2766996

work page doi:10.1145/2766996 2015
[21]

The material point method for simulating continuum materials, in: ACM SIGGRAPH 2016 Courses, Association for Computing Machinery, New York, NY, USA

Jiang, C., Schroeder, C., Selle, A., Teran, J., Stomakhin, A., 2016. The material point method for simulating continuum materials, in: ACM SIGGRAPH 2016 Courses, Association for Computing Machinery, New York, NY, USA. pp. 1–52. doi:10.1145/2897826.2927346

work page doi:10.1145/2897826.2927346 2016
[22]

Anangular momentumconserving affine-particle-in-cellmethod

Jiang,C., Schroeder,C., Teran,J., 2017. Anangular momentumconserving affine-particle-in-cellmethod. Journal ofComputationalPhysics 338, 137–164

work page 2017
[23]

A hybrid material-point spheropolygon-element method for solid and granular material interaction

Jiang, Y., Li, M., Jiang, C., Alonso-Marroquin, F., 2020. A hybrid material-point spheropolygon-element method for solid and granular material interaction. International Journal for Numerical Methods in Engineering 121, 3021–3047

work page 2020
[24]

Erosion and transport of dry soil bed by collisional granular flow: Insights from a combined experimental–numerical investigation

Jiang, Y., Song, P., Choi, C.E., Choo, J., 2023. Erosion and transport of dry soil bed by collisional granular flow: Insights from a combined experimental–numerical investigation. Journal of Geophysical Research: Earth Surface 128, e2023JF007073

work page 2023
[25]

Impact and erosion dynamics of inclusion-enriched dry granular flows on rigid barriers with basal clearance: Numerical insights from hybrid mp-dem simulations

Jiang, Y., Zhang, C., Choi, C.E., 2024. Impact and erosion dynamics of inclusion-enriched dry granular flows on rigid barriers with basal clearance: Numerical insights from hybrid mp-dem simulations. Computers and Geotechnics 176, 106767

work page 2024
[26]

Hybrid continuum–discrete simulation of granular impact dynamics

Jiang, Y., Zhao, Y., Choi, C.E., Choo, J., 2022. Hybrid continuum–discrete simulation of granular impact dynamics. Acta Geotechnica 17, 5597–5612

work page 2022
[27]

Contact Mechanics

Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge

work page 1985
[28]

Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems

Kane, C., Marsden, J.E., Ortiz, M., West, M., 2000. Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems. International Journal for numerical methods in engineering 49, 1295–1325

work page 2000
[29]

Fabricationofpracticaldeformabledisplays:advances and challenges

Kim,D.W.,Kim,S.W.,Lee,G.,Yoon,J.,Kim,S.,Hong,J.H.,Jo,S.C.,Jeong,U.,2023. Fabricationofpracticaldeformabledisplays:advances and challenges. Light: Science & Applications 12, 61

work page 2023
[30]

Extension of b-spline material point method for unstructured triangular grids using powell–sabin splines

de Koster, P., Tielen, R., Wobbes, E., Möller, M., 2021. Extension of b-spline material point method for unstructured triangular grids using powell–sabin splines. Computational particle mechanics 8, 273–288

work page 2021
[31]

Decomposedoptimizationtimeintegratorforlarge-stepelastodynamics

Li,M.,Gao,M.,Langlois,T.,Jiang,C.,Kaufman,D.M.,2019. Decomposedoptimizationtimeintegratorforlarge-stepelastodynamics. ACM Transactions on Graphics (TOG) 38, 1–10

work page 2019
[32]

Energetically consistent inelasticity for optimization time integration

Li, X., Li, M., Jiang, C., 2022. Energetically consistent inelasticity for optimization time integration. ACM Transactions on Graphics (TOG) 41, 1–16

work page 2022
[33]

A mesh-grading material point method and its parallelization for problems with localized extreme deformation

Lian, Y., Yang, P., Zhang, X., Zhang, F., Liu, Y., Huang, P., 2015. A mesh-grading material point method and its parallelization for problems with localized extreme deformation. Computer Methods in Applied Mechanics and Engineering 289, 291–315

work page 2015
[34]

Tied interface grid material point method for problems with localized extreme deformation

Lian, Y., Zhang, X., Zhang, F., Cui, X., 2014. Tied interface grid material point method for problems with localized extreme deformation. International Journal of Impact Engineering 70, 50–61

work page 2014
[35]

Revealing the role of forests in the mobility of geophysical flows

Liang, Z., Choi, C.E., Zhao, Y., Jiang, Y., Choo, J., 2023. Revealing the role of forests in the mobility of geophysical flows. Computers and Geotechnics 155, 105194. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 16 of 27 Schwarz Space-Time Refinement for MPM

work page 2023
[36]

Lions,P.L.,1988. OntheSchwarzalternatingmethod.I,in:Glowinski,R.,Golub,G.H.,Meurant,G.A.,Périaux,J.(Eds.),FirstInternational Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia. pp. 1–42

work page 1988
[37]

On the Schwarz alternating method

Lions, P.L., 1990. On the Schwarz alternating method. III: A variant for nonoverlapping subdomains, in: Chan, T.F., Glowinski, R., Périaux, J., Widlund, O.B. (Eds.), Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia. pp. 202–223

work page 1990
[38]

Structured mesh refinement in generalized interpolation material point (gimp) method for simulation of dynamic problems

Ma, J., Lu, H., Komanduri, R., 2006. Structured mesh refinement in generalized interpolation material point (gimp) method for simulation of dynamic problems. Computer Modeling in Engineering & Sciences 12, 213

work page 2006
[39]

Multiscale simulations using generalized interpolation material point (gimp) method and samrai parallel processing

Ma, J., Lu, H., Wang, B., Roy, S., Hornung, R., Wissink, A., Komanduri, R., 2005. Multiscale simulations using generalized interpolation material point (gimp) method and samrai parallel processing. Computer Modeling in Engineering & Sciences 8, 135

work page 2005
[40]

Comparison study of mpm and sph in modeling hypervelocity impact problems

Ma, S., Zhang, X., Qiu, X., 2009. Comparison study of mpm and sph in modeling hypervelocity impact problems. International journal of impact engineering 36, 272–282

work page 2009
[41]

Afundamentallynewcoupledapproachtocontactmechanicsviathedirichlet-neumann schwarz alternating method

Mota,A.,Koliesnikova,D.,Tezaur,I.,Hoy,J.,2025. Afundamentallynewcoupledapproachtocontactmechanicsviathedirichlet-neumann schwarz alternating method. International Journal for Numerical Methods in Engineering 126, e70039

work page 2025
[42]

The Schwarz alternating method in solid mechanics

Mota, A., Tezaur, I., Alleman, C., 2017. The Schwarz alternating method in solid mechanics. Comput. Methods Appl. Mech. Engrg. 319, 19–51

work page 2017
[43]

The schwarz alternating method for transient solid dynamics

Mota, A., Tezaur, I., Phlipot, G., 2022. The schwarz alternating method for transient solid dynamics. International Journal for Numerical Methods in Engineering 123, 5036–5071

work page 2022
[44]

Iga-mpm:theisogeometricmaterialpointmethod

Moutsanidis,G.,Long,C.C.,Bazilevs,Y.,2020. Iga-mpm:theisogeometricmaterialpointmethod. ComputerMethodsinAppliedMechanics and Engineering 372, 113346

work page 2020
[45]

Micromechanics of Defects in Solids

Mura, T., 2013. Micromechanics of Defects in Solids. Springer Science & Business Media

work page 2013
[46]

Numerical optimization

Nocedal, J., Wright, S.J., 2006. Numerical optimization. Springer

work page 2006
[47]

Aconvectedparticledomaininterpolationtechniquetoextendapplicabilityofthematerial point method for problems involving massive deformations

Sadeghirad,A.,Brannon,R.M.,Burghardt,J.,2011. Aconvectedparticledomaininterpolationtechniquetoextendapplicabilityofthematerial point method for problems involving massive deformations. International Journal for numerical methods in Engineering 86, 1435–1456

work page 2011
[48]

Üeber einen Grenzübergang durch alternirendes Verfahren

Schwarz, H.A., 1870. Üeber einen Grenzübergang durch alternirendes Verfahren. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 15, 272–286

work page
[49]

Bending of a beam or wide strip

Shield, R.T., 1992. Bending of a beam or wide strip. Quarterly Journal of Mechanics and Applied Mathematics 45, 567–573

work page 1992
[50]

Lagrange multiplier imposition of non-conforming essential boundary conditions in implicit material point method

Singer, V., Teschemacher, T., Larese, A., Wüchner, R., Bletzinger, K.U., 2024. Lagrange multiplier imposition of non-conforming essential boundary conditions in implicit material point method. Computational Mechanics 73, 1311–1333

work page 2024
[51]

Evaluationofmaterialpointmethodforuseingeotechnics

Sołowski,W.,Sloan,S.,2015. Evaluationofmaterialpointmethodforuseingeotechnics. InternationalJournalforNumericalandAnalytical Methods in Geomechanics 39, 685–701

work page 2015
[52]

Analysis and reduction of quadrature errors in the material point method (mpm)

Steffen, M., Kirby, R.M., Berzins, M., 2008. Analysis and reduction of quadrature errors in the material point method (mpm). International journal for numerical methods in engineering 76, 922–948

work page 2008
[53]

Aparticlemethodforhistory-dependentmaterials

Sulsky,D.,Chen,Z.,Schreyer,H.L.,1994. Aparticlemethodforhistory-dependentmaterials. ComputerMethodsinAppliedMechanicsand Engineering 118, 179–196. doi:10.1016/0045-7825(94)90112-0

work page doi:10.1016/0045-7825(94)90112-0 1994
[54]

A local grid refinement scheme for b-spline material point method

Sun, Z., Gan, Y., Huang, Z., Zhou, X., 2020. A local grid refinement scheme for b-spline material point method. International Journal for Numerical Methods in Engineering 121, 2398–2417

work page 2020
[55]

Hierarchical, adaptive, material point method for dynamic energy release rate calculations

Tan, H., Nairn, J.A., 2002. Hierarchical, adaptive, material point method for dynamic energy release rate calculations. Computer Methods in Applied Mechanics and Engineering 191, 2123–2137

work page 2002
[56]

Domain decomposition methods-algorithms and theory

Toselli, A., Widlund, O., 2004. Domain decomposition methods-algorithms and theory. volume 34. Springer Science & Business Media

work page 2004
[57]

Hierarchical optimization time integration for cfl-rate mpm stepping

Wang, X., Li, M., Fang, Y., Zhang, X., Gao, M., Tang, M., Kaufman, D.M., Jiang, C., 2020. Hierarchical optimization time integration for cfl-rate mpm stepping. ACM Transactions on Graphics (TOG) 39, 1–16

work page 2020
[58]

The material point method in large strain engineering problems

Więckowski, Z., 2004. The material point method in large strain engineering problems. Computer methods in applied mechanics and engineering 193, 4417–4438

work page 2004
[59]

Distillationofthematerialpointmethodcellcrossingerrorleadingtoanovelquadrature-based c 0 remedy

Wilson,P.,Wüchner,R.,Fernando,D.,2021. Distillationofthematerialpointmethodcellcrossingerrorleadingtoanovelquadrature-based c 0 remedy. International Journal for Numerical Methods in Engineering 122, 1513–1537

work page 2021
[60]

Simulation of penetration of a foundation element in tresca soil using the generalized interpolation material point method (gimp)

Woo, S.I., Salgado, R., 2018. Simulation of penetration of a foundation element in tresca soil using the generalized interpolation material point method (gimp). Computers and geotechnics 94, 106–117

work page 2018
[61]

Dp-mpm:Domainpartitioningmaterialpointmethodforevolvingmulti-bodythermal–mechanicalcontacts during dynamic fracture and fragmentation

Xiao,M.,Liu,C.,Sun,W.,2021. Dp-mpm:Domainpartitioningmaterialpointmethodforevolvingmulti-bodythermal–mechanicalcontacts during dynamic fracture and fragmentation. Computer Methods in Applied Mechanics and Engineering 385, 114063

work page 2021
[62]

Materialpointmethodenhancedbymodifiedgradientofshapefunction

Zhang,D.Z.,Ma,X.,Giguere,P.T.,2011. Materialpointmethodenhancedbymodifiedgradientofshapefunction. JournalofComputational Physics 230, 6379–6398

work page 2011
[63]

Truncatedhierarchical b-spline material pointmethod for large deformation geotechnical problems

Zhang, K., Shen,S.L., Zhou, A., Balzani,D., 2021. Truncatedhierarchical b-spline material pointmethod for large deformation geotechnical problems. Computers and Geotechnics 134, 104097

work page 2021
[64]

Coupledmaterialpointandlevelsetmethodsforsimulatingsoilsinteractingwithrigidobjectswith complex geometry

Zhao,Y.,Choo,J.,Jiang,Y.,Li,L.,2023. Coupledmaterialpointandlevelsetmethodsforsimulatingsoilsinteractingwithrigidobjectswith complex geometry. Computers and Geotechnics 163, 105708

work page 2023
[65]

Intrinsically flexible displays: key materials and devices

Zhao, Z., Liu, K., Liu, Y., Guo, Y., Liu, Y., 2022. Intrinsically flexible displays: key materials and devices. National Science Review 9, nwac090. Zhaofeng Luo, Minchen Li*, Yupeng Jiang* Page 17 of 27 Schwarz Space-Time Refinement for MPM 𝜕𝜕Ω2⋂Ω1 𝜕𝜕Ω1⋂Ω2 Ω1⋂Ω2 Ω1 Ω2 Figure 1:Illustration of overlapping subdomains. Boundar y grid selection Back gr ound g...

work page 2022