References — Papers, Courses, Code & Media

A curated bibliography for the Material Point Method, grouped by type. Every entry has a link, and its annotation says what that work contributes; for where to start, the index names the three sources to read first. If a link rots, search the title — these are all well-indexed works.

1. Peer-reviewed academic papers

Foundational / origin

• Sulsky, D., Chen, Z., Schreyer, H.L. A particle method for history-dependent materials. Computer Methods in Applied Mechanics and Engineering, 1994. — The founding MPM paper. DOI: 10.1016/0045-7825(94)90112-0
• Sulsky, D., Zhou, S.-J., Schreyer, H.L. Application of a particle-in-cell method to solid mechanics. Computer Physics Communications, 1995. DOI: 10.1016/0010-4655(94)00170-7

Graphics landmark & constitutive models

• Simo, J.C. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part I: Continuum formulation. Computer Methods in Applied Mechanics and Engineering 66, 1988. — Primary source for multiplicative finite-strain elastoplasticity, isochoric plastic flow, and the elastic left Cauchy-Green state that a finite-strain J2 (Von Mises) return in MPM carries. DOI: 10.1016/0045-7825(88)90076-X

• Simo, J.C. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part II: Computational aspects. Computer Methods in Applied Mechanics and Engineering 68, 1988. — Primary algorithmic source for the finite-strain radial-return form and its reduction to the classical return-mapping problem. DOI: 10.1016/0045-7825(88)90104-1

• Stomakhin, A., Schroeder, C., Chai, L., Teran, J., Selle, A. A Material Point Method for Snow Simulation. ACM SIGGRAPH 2013. — MPM’s graphics debut; powered Frozen. DOI: 10.1145/2461912.2461948 · Disney Research

• Stomakhin, A., Schroeder, C., Jiang, C., Chai, L., Teran, J., Selle, A. Augmented MPM for phase-change and varied materials (melting/freezing). SIGGRAPH 2014. DOI: 10.1145/2601097.2601176 · PDF

• Klar, G., Gast, T., Pradhana, A., Fu, C., Schroeder, C., Jiang, C., Teran, J. Drucker-Prager Elastoplasticity for Sand Animation. SIGGRAPH 2016. — The engineering-favored sand model. The supplementary technical document is the implementable half: §1 pseudocode, §2 the Drucker-Prager yield-surface derivation, §4 the return-mapping algorithm derived from the plastic flow, and §6 the Hencky-strain derivative lemma. DOI: 10.1145/2897824.2925906 · PDF · supplementary technical document

• Daviet, G., Bertails-Descoubes, F. A Semi-Implicit Material Point Method for the Continuum Simulation of Granular Materials. ACM SIGGRAPH 2016. — The other 2016 sand paper, and the one Klár et al. is usually read without. Same Drucker-Prager yield surface, opposite engineering: the yield criterion is treated as genuinely nonsmooth rather than regularized, volume fraction is a state variable rather than a clamp, and the step is semi-implicit specifically so normal and tangential stress stay coupled through the solve. Read it for the yield-criterion treatment and the varying-volume-fraction rheology. Reference implementation is sand6 (GPL-3.0 — methods only). Same author as the 2026 mixed-MPM entry below, so the two read as one ten-year arc on stiff/implicit MPM. DOI: 10.1145/2897824.2925877 · PDF · annex · project page · code: sand6

• Tampubolon, A.P., Gast, T., Klar, G., Fu, C., Teran, J., Jiang, C., Museth, K. Multi-species simulation of porous sand and water mixtures. SIGGRAPH 2017. — Sand-water coupling on twin grids; and §4.3.4 ships the standard volume-correction fix for the Drucker-Prager apex artifact (see Prerequisites §Drucker-Prager): dispersed sand otherwise re-settles permanently fluffed. Their Figs. 8-9 are the canonical with/without compare. DOI: 10.1145/3072959.3073651 · PDF

• Gao, M., Pradhana, A., Han, X., Guo, Q., Kot, G., Sifakis, E., Jiang, C. Animating Fluid Sediment Mixture in Particle-Laden Flows. ACM SIGGRAPH 2018, TOG 37(4) Article 145. — The sibling to Tampubolon 2017: fluid and sediment two-way coupled by a momentum-exchange force across two MPM grids, with Drucker-Prager sedimentation. Its third contribution — “a density evaluation strategy for converting particles between continuum clumps and discrete debris” — is evaluated from world-space positions rather than integrated velocity gradient: the continuum↔︎grains question treated as a runtime material state instead of a render-time choice. Their Fig. 3 is water-saturated debris flow. Note the method is semi-implicit with a multigrid-preconditioned pressure solve — not a real-time budget. DOI: 10.1145/3197517.3201309 · PDF

• Xiao, X., Wang, H., Yang, X., Desbrun, M., Li, W. Volume-Preserving LBM-MPM Coupling for Air-Water-Sand Mixtures. ACM SIGGRAPH 2026, TOG 45(4). — Lattice-Boltzmann for the weakly-compressible air/water pair, MPM for the sand, inside one continuum formulation. The separable half is the water-retention model — how liquid infiltrates and is held in the granular structure, i.e. the dry-and-frictional → soaked-and-sticky transition curve — a published curve for a transition that Tampubolon 2017 §3.4 leaves implicit. The LBM half is a second solver. CC BY 4.0 — the version of record is free at the ACM DOI; see the header note on ACM 2026 open access. full text (open access) · PDF (Desbrun’s INRIA page) · code

• Ayazi, M., Schroeder, C.A., Shinar, T. Buoyancy-driven Phase Separation in the Material Point Method. ACM SIGGRAPH 2026 Conference Papers. — Density-driven separation of mixed phases inside MPM — the mechanism behind Kot’s 2010 “foamy water” layering (§3), given a published treatment sixteen years later. CC BY 4.0 — free at the ACM DOI, though neither author page carried it, so the DOI is the route. full text (open access)

• Kabore, B.W., Peters, B. Micromechanical model for sintering and damage in viscoelastic porous ice and snow. Part I: Theory. arXiv:1808.04394, 2018. — Not MPM — a micromechanical constitutive theory — but it is the primary source for a curvature-driven evolution law for intergranular neck growth — cohesion as an explicit function of rest time and temperature. Goswami 2024 §2.1 (Surveys & reviews below) carries a survey’s summary of that law rather than the law itself; if a sintering rate curve ever has to be defended or re-derived, derive it from here.* arXiv:1808.04394

Transfer schemes & method refinements

• Jiang, C., Schroeder, C., Selle, A., Teran, J., Stomakhin, A. The Affine Particle-In-Cell Method (APIC). ACM SIGGRAPH 2015. — Stable, angular-momentum-conserving transfers. DOI: 10.1145/2766996 · PDF

• Hu, Y., Fang, Y., Ge, Z., Qu, Z., Zhu, Y., Pradhana, A., Jiang, C. A Moving Least Squares Material Point Method with Displacement Discontinuity and Two-Way Rigid Body Coupling (MLS-MPM). ACM SIGGRAPH 2018. — ~2× faster; eliminates shape-function gradients; adds Compatible Particle-In-Cell (CPIC) cutting & coupling. DOI: 10.1145/3197517.3201293 · PDF · project page

• Gast, T., Schroeder, C., Stomakhin, A., Jiang, C., Teran, J. Optimization Integrator for Large Time Steps (implicit MPM). IEEE Transactions on Visualization and Computer Graphics (TVCG), 2015. DOI: 10.1109/TVCG.2015.2459687 · PDF

• Bardenhagen, S.G., Kober, E.M. The Generalized Interpolation Material Point Method. CMES 5(6):477–496, 2004. — GIMP: shape functions as particle-domain averages of the grid functions — the cell-crossing-noise cure that B-spline MPM inherits by construction. publisher

• Sadeghirad, A., Brannon, R.M., Burghardt, J. A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations. IJNME 86(12):1435–1456, 2011. — CPDI: particle domains tracked through F (semi-side vectors), corner-sampled transfers. DOI: 10.1002/nme.3110

• Hammerquist, C.C., Nairn, J.A. A new method for material point method particle updates that reduces noise and enhances stability. CMAME 318:724–738, 2017. — XPIC(m): removes grid-invisible (“null-space”) velocity noise; $\mathrm{PIC} = \mathrm{XPIC}(1)$; frames FLIP/PIC blends as hidden Δt-dependent dampers. See Theory §5 Transfer filters. DOI: 10.1016/j.cma.2017.01.035 · PDF

• Nairn, J.A., Hammerquist, C.C. Material point method simulations using an approximate full mass matrix inverse. CMAME 377:113667, 2021. — FMPM(k): XPIC reinterpreted as a truncated consistent-mass-matrix inverse, extended to filter the full update (velocity + force) with the cheapest particle update; constraints applied per increment. DOI: 10.1016/j.cma.2021.113667 · PDF

• Fu, C., Guo, Q., Gast, T., Jiang, C., Teran, J. A Polynomial Particle-In-Cell Method (PolyPIC). ACM SIGGRAPH Asia 2017, TOG 36(6). — APIC generalized: each particle’s local velocity field becomes a higher-order polynomial instead of an affine one, and G2P becomes a linear- and angular-momentum-conserving projection onto that reduced basis — less dissipation and better-resolved vorticity at the same grid resolution. The cost is per-particle state, and APIC’s affine C is the cheapest member of the family, so the size of the particle record is what decides whether any higher rung is affordable. DOI: 10.1145/3130800.3130878 · PDF

• Fei, Y., Guo, Q., Wu, R., Huang, L., Gao, M. Revisiting Integration in the Material Point Method: A Scheme for Easier Separation and Less Dissipation (AFLIP / ASFLIP). ACM SIGGRAPH 2021, TOG 40(4). — A different paper from the same first author and year as Principles towards Real-Time Simulation below — do not conflate them. Two changes to the integrator: recover the per-particle momentum the transfer round trip smooths away, and correct particle position instead of advecting by interpolating deformed nodal positions — the latter being what traps particles in a continuum and makes debris refuse to separate. The authors report under 15 lines changed to convert a FLIP integrator. Reference implementation pyasflip (Apache-2.0 — one of the few in this bibliography that is both readable and usable).* DOI: 10.1145/3450626.3459678 · PDF · project page · code

• Sancho, S., Tang, J., Batty, C., Azevedo, V.C. The Impulse Particle-In-Cell Method (IPIC). Eurographics 2024, Computer Graphics Forum 43(2) (DisneyResearch|Studios / ETH Zürich / Waterloo). — APIC extended through the impulse gauge formulation: a coupled advection–stretching during particle advection, aimed at preserving circulation, giving more energetic smoke and liquid with better-retained vortical detail. The authors describe the algorithmic change as simple to implement. Two things to know before reaching for it. First, it is a fluids-vorticity lever — the failure it fixes is lost vortical detail in smoke and energetic liquid, so it buys nothing for a scene that has no vorticity problem. Second, it is an APIC extension from Disney, so it is downstream of US10503845B2 rather than a way around it (Patents) — worth knowing precisely so nobody reaches for it as an escape route. DOI: 10.1111/cgf.15022 · PDF · project page

• Fang, S., Zhu, D., Pan, Z. Enhanced material point method with affine projection stabilizer for efficient hyperelastic simulations. The Visual Computer 41:6547–6560, 2025. — An explicit MLS-MPM stabilizer — an XPBD-style affine projection applied per particle across G2P2G stages — motivated by PB-MPM but keeping the explicit integrator. It is the only located attempt to buy PB-MPM’s timestep tolerance without* adopting its solve. Games §5 already names this work; this is the citation that mention promised. Verified unavailable free — Unpaywall, OpenAlex and Semantic Scholar all report no open-access copy, DBLP lists only the publisher DOI, and no author page was located. The remaining route is an author request, and there is a warm path: coauthor Siyan Zhu is also on DIQ-MPM (Contact & multi-body coupling, below), whose group (Bo Ren, Nankai) posts its own PDFs. This is the only entry here with no free route and no plausible one pending.* DOI: 10.1007/s00371-025-03953-2

• Daviet, G. Mixed Material Point Methods for Stiff Elastoplasticity. ACM SIGGRAPH 2026, TOG 45(4) (NVIDIA). — A mixed (velocity–stress) implicit discretization with compact stencils, aimed squarely at a regime that is still open: stiff elastoviscoplastic materials up to the near-incompressible limit, stepped at CFL rate, on a GPU solver, with two-way rigid coupling. 49M sand particles at ~4 s/frame on one GPU — offline, but the nearest published attack on the stiffness-ratio tax. Note the trade: it keeps rigid bodies and buys stiffness with an implicit solve, where an all-continuum explicit solver dissolves the rigid bodies and pays the CFL tax instead. DOI: 10.1145/3811345 · project page · PDF

• Feng, X., Chen, Y., Yu, C., Su, H., Terzopoulos, D., Yang, Y., Masterjohn, J., Castro, A., Jiang, C. MPM Lite: Linear Kernels and Integration without Particles. ACM SIGGRAPH 2026, TOG 45(4). — Treats the background grid as a voxel hexahedral mesh, resamples particle state onto fixed quadrature points with linear kernels, and then runs force assembly and the whole time integration without touching particles, so solve cost stops scaling with particles-per-cell. Reported 15.9× over implicit MPM and 1.88× over explicit MPM. The strongest published case for the narrow-kernel direction CK-MPM (Real-time / games* below) also takes, and it arrives on the particles-per-cell axis that Steffen et al. 2008 (Accuracy, resolution & dispersion below) treat as an accuracy variable. Note the price: the MLS closed form and the particle-quadrature story both go.* DOI: 10.1145/3811294 · arXiv:2602.07853 · project page

Contact & multi-body coupling

How MPM handles distinct bodies — separate velocity fields per body/material on one grid (the “multi-field” line), per-body background meshes (the “multi-mesh” line), and coupling to rigid bodies. See Theory §5 for the concepts.

• Bardenhagen, S.G., Brackbill, J.U., Sulsky, D. The material-point method for granular materials. Computer Methods in Applied Mechanics and Engineering 187(3–4):529–541, 2000. — Origin of multi-velocity-field grid contact (per-body fields + a common center-of-mass frame); cost linear in body count, no neighbor search. DOI: 10.1016/S0045-7825(99)00338-2 · PDF

• Bardenhagen, S.G., Guilkey, J.E., Roessig, K.M., Brackbill, J.U., Witzel, W.M., Foster, J.C. An Improved Contact Algorithm for the Material Point Method and Application to Stress Propagation in Granular Material. CMES 2(4):509–522, 2001. — The canonical “Bardenhagen contact”: adds the normal-traction separation criterion; the standard separable frictional-contact citation. DOI: 10.3970/cmes.2001.002.509 · PDF

• Hu, W., Chen, Z. A multi-mesh MPM for simulating the meshing process of spur gears. Computers & Structures 81(20):1991–2002, 2003. — First “each body gets its own background mesh” MPM. DOI: 10.1016/S0045-7949(03)00260-8

• Pan, X.F., Xu, A., Zhang, G., Zhu, J., Ma, S., Zhang, X. Three-dimensional multi-mesh material point method for solving collision problems. Communications in Theoretical Physics 49(5):1129–1138, 2008. DOI: 10.1088/0253-6102/49/5/09 · arXiv:0708.3532

• Huang, P., Zhang, X., Ma, S., Huang, X. Contact algorithms for the material point method in impact and penetration simulation. IJNME 85(4):498–517, 2011. — Compares no-slip (single-field) vs frictional separable contact on impact benchmarks. DOI: 10.1002/nme.2981

• Nairn, J.A. Numerical implementation of imperfect interfaces. Computational Materials Science 40(4):525–536, 2007. — Displacement- discontinuity interface springs in MPM, and the contact-area-from-extrapolated- volumes estimate that turns impulse contact laws into traction laws. DOI: 10.1016/j.commatsci.2007.02.010 · PDF

• Nairn, J.A. Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods. CMES 92(3):271–299, 2013. — Multimaterial mode + the average-(volume-)gradient contact normal; imperfect-interface and adhesive laws. publisher · PDF

• Homel, M.A., Herbold, E.B. Field-gradient partitioning for fracture and frictional contact in the material point method. IJNME 109(7):1013–1044, 2017. — Dynamically repartitions material into two velocity fields per node from a damage-field gradient — self-contact through evolving fracture. DOI: 10.1002/nme.5317

• Nairn, J.A., Bardenhagen, S.G., Smith, G.D. Generalized contact and improved frictional heating in the material point method. Computational Particle Mechanics 5(3):285–296, 2018. — Contact laws beyond plain Coulomb — adhesion, velocity-dependent friction — expressed as traction laws via contact area; frictional heating. DOI: 10.1007/s40571-017-0168-1 · PDF

• Hammerquist, C.C., Nairn, J.A. Modeling nanoindentation using the material point method. Journal of Materials Research 33(10):1369–1381, 2018. — Its supplemental material derives the extrapolated-position contact-separation correction (the ~0.8Δx edge-bias constant) — how a position-based contact threshold becomes a computed number instead of a tuned one; see Theory §5 The calibration half. DOI: 10.1557/jmr.2018.75 · PDF · supplemental

• Nairn, J.A., Hammerquist, C.C., Smith, G.D. New material point method contact algorithms for improved accuracy, large-deformation problems, and proper null-space filtering. CMAME 362:112859, 2020. — The modern accuracy revision: logistic-regression normals/separation (superseding gradient-based normals), null-space filtering, 3+-material nodes. DOI: 10.1016/j.cma.2020.112859 · PDF

• Ding, O., Schroeder, C. Penalty Force for Coupling Materials with Coulomb Friction. IEEE TVCG 26(7):2443–2455, 2020. — Penalty-based MPM↔︎rigid coupling. PDF

• Fang, Y., Qu, Z., Li, M., Zhang, X., Zhu, Y., Aanjaneya, M., Jiang, C. IQ-MPM: An Interface Quadrature Material Point Method for Non-sticky Strongly Two-way Coupled Nonlinear Solids and Fluids. SIGGRAPH 2020. — Two-field solid–fluid coupling; frames single-field welding as “sticky numerical artifacts.” The separable part is the interface condition itself — normal velocity continuity only (eq. 6), the solid-side mass-gradient normal (eq. 46), and the deformed-particle interface area (§5.4). The rest — massless “ghost matrix” pressure DOF, implicit elasticity, one monolithic AMG-PCG pressure solve per step at 0.8–12.8 min/frame — is offline machinery, far outside any interactive budget. Reference implementation is ziran2020 (unlicensed). DOI: 10.1145/3386569.3392438 · PDF (lab copy)

• Yue, Y., Smith, B., Chen, P.Y., Chantharayukhonthorn, M., Kamrin, K., Grinspun, E. Hybrid Grains: Adaptive Coupling of Discrete and Continuum Simulations of Granular Media. SIGGRAPH Asia 2018. — Cost-driven partitioning between DEM detail and continuum-MPM tractability — the published statement of the pairwise-vs-grid cost asymmetry. DOI: 10.1145/3272127.3275095

• Zong, Z., Jiang, C., Han, X. A Convex Formulation of Frictional Contact for the Material Point Method and Rigid Bodies. arXiv:2403.13783, 2024; and Yu, C., Du, W., Zong, Z., Castro, A., Jiang, C., Han, X. A Convex Formulation of Material Points and Rigid Bodies with GPU-Accelerated Async-Coupling for Interactive Simulation. arXiv:2503.05046, 2025. — The robotics-grade MPM↔︎rigid line (Drake), pushed to interactive rates. arXiv:2403.13783 · arXiv:2503.05046

• Zhang, K., Cen, R., Zhu, S., Chen, R., Ren, B. DIQ-MPM: Dual Interface Quadrature MPM for Simulating Large Deformation and Fluid-Solid Coupling. IEEE TVCG 32(7):5803–5814, 2026. — The direct successor to IQ-MPM above: implicit total-Lagrangian solid plus Eulerian incompressible fluid, coupled monolithically through a dual interface quadrature that maps interface data between current and reference configurations — which buys strong coupling without overlapping grids. Same caveat as its parent, for the same reason: a global implicit pressure solve per step puts it at offline cost. Read it for the dropped overlapping-grid requirement, not the solver. DOI: 10.1109/TVCG.2026.3674656 · PDF (Bo Ren’s page) · supplemental

Fracture & advanced material models

• Wolper, J., Fang, Y., Li, M., Lu, J., Gao, M., Jiang, C. CD-MPM: Continuum Damage Material Point Methods for Dynamic Fracture Animation. SIGGRAPH 2019. DOI: 10.1145/3306346.3322949 · PDF (lab copy)
• Wolper, J., et al. AnisoMPM: Animating Anisotropic Damage Mechanics. SIGGRAPH 2020. DOI: 10.1145/3386569.3392428 · PDF (lab copy)
• Gaume, J., Gast, T., Teran, J., van Herwijnen, A., Jiang, C. Dynamic anticrack propagation in snow. Nature Communications 9:3047, 2018. — MPM slab-avalanche release; snow cohesion/softening in the yield surface. full text (open access) · DOI: 10.1038/s41467-018-05181-w
• O’Brien, J.F., Hodgins, J.K. Graphical Modeling and Animation of Brittle Fracture. SIGGRAPH, 1999. — The seminal physically-based fracture paper, cited for contrast with geometric approaches. DOI: 10.1145/311535.311550
• Müller, M., Chentanez, N., Kim, T.-Y. Real Time Dynamic Fracture with Volumetric Approximate Convex Decompositions. SIGGRAPH, 2013. — Real-time convex-piece fracture — the geometric alternative to a continuum damage model. DOI: 10.1145/2461912.2461934

Fluid-flow validation benchmarks

• Ritter, A. Die Fortpflanzung der Wasserwellen. Zeitschrift des Vereines Deutscher Ingenieure 36(33):947–954, 1892. — Primary source for the parabolic dam-break idealization. Sections 1 and 5 assume a straight horizontal constant-section channel, cross-section-uniform velocity, and negligible friction and cohesion. With $U_0 = \sqrt{g a}$ and $\lambda = U_0 t$, equations 53–54 give the parabolic limiting depth profile; equations 16 and 49–51 give its streamwise velocity, gate depth, and discharge. Sections 6–7 set the scope, and it matters: Ritter says a suddenly withdrawn wall violates the derivation’s acceleration limit, calls the finite-time dam-break curve approximate, and says it approaches the parabola asymptotically. A validation built on it should therefore test compatibility with the named idealization and not call it an exact finite-time physical dam-break solution. primary volume scan (article pp. 947–954)
• Martin, J. C., Moyce, W. J. Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane. Philosophical Transactions of the Royal Society of London A 244(882):312–324, 1952. — Primary source for the dam-break surge-front benchmark. For a column of height $n^{2}a$, it defines surge-front position $Z = z/a$, time $T = n t \sqrt{g/a}$, and speed $U = \mathrm{d}Z/\mathrm{d}T$. Its four plane-rectangular rows report maximum $U$ of 1.62, 1.66, 1.71, and 1.62; the paper’s approximate theory that neglects vertical acceleration gives 2. A benchmark built on it should band around those four published $U_{\max}$ values and not substitute a secondary retelling of the experiment. DOI: 10.1098/rsta.1952.0006 · publisher PDF

Granular-flow validation benchmarks

• Lube, G., Huppert, H.E., Sparks, R.S.J., Hallworth, M.A. Axisymmetric collapses of granular columns. Journal of Fluid Mechanics 508:175–199, 2004. — Primary dry-column benchmark: normalized runout $\zeta = (r_\infty - r_i)/r_i$ follows $1.24a$ below $a = 1.7$ and $1.6\sqrt{a}$ above it for $a = h_i/r_i$; the principal sand has a 30-degree repose angle. The paper also defines $t_\infty$ as the time for the front to cease motion and fits $t_\infty = 3\sqrt{h_i/g}$. Table 1 spans 30–35-degree repose angles across the tested materials — the friction-variation range a benchmark built on it can claim. DOI: 10.1017/S0022112004009036 · publisher · author-uploaded full text
• Lajeunesse, E., Mangeney-Castelnau, A., Vilotte, J.P. Spreading of a granular mass on a horizontal plane. Physics of Fluids 16(7):2371–2381, 2004. — Primary source for the fixed-aspect-ratio shape invariant: normalized deposit profiles at different released masses match when $a$ matches. For $a > 0.74$, $H_f/R_i$ saturates near 0.74; the paper infers $\arctan(0.74) = 36.5$ degrees and observes the internal no-flow boundary at 35–37 degrees. That is an internal yield surface, not the deposit’s free-surface slope. Its runout coefficients are separate from the Lube et al. fit and should not be blended into it. DOI: 10.1063/1.1736611 · author PDF

Accuracy, resolution & dispersion

What a coarse grid does to the physics — quadrature error, dispersion, and under-resolved features. The practical face of these at interactive resolutions: effective stiffness falls below configured stiffness, and features a few cells wide fail before the bulk does.

• Steffen, M., Kirby, R.M., Berzins, M. Analysis and reduction of quadrature errors in the material point method. IJNME 76(6):922–948, 2008. — The foundational MPM error analysis. DOI: 10.1002/nme.2360
• Gan, Y., Sun, Z., Chen, Z., Zhang, X., Liu, Y. Enhancement of the material point method using B-spline basis functions. IJNME 113(3), 2018. — B-spline MPM accuracy/dispersion. DOI: 10.1002/nme.5620
• Homel, M.A., Brannon, R.M., Guilkey, J.E. Controlling the onset of numerical fracture in parallelized implementations of the material point method with convective particle domain interpolation (CPDI) domain scaling. IJNME 107(1), 2016. — “Numerical fracture” — spurious separation of under-resolved material — as a controllable, resolution-set phenomenon. DOI: 10.1002/nme.5151

GPU & performance

• Gao, M., Wang, X., Wu, K., Pradhana, A., Sifakis, E., Yuksel, C., Jiang, C. GPU Optimization of Material Point Methods. ACM SIGGRAPH Asia 2018. — The reference for GPU MPM (particle-to-grid atomics, sparse grid, data layout). Its supplemental technical document — the paper’s own citation key for it is “Gao et al. 2018b” — carries the granular material work, not GPU material: §2 derives the unilateral quartic sand energy density in Hencky-strain space and the Drucker-Prager return mapping corresponding to it (§2.4 yield surface, §2.5 trial/projected strain, §2.6.1 solving the system in 3D). The main paper states only the fit constant $a \approx 6.254$ and defers the derivation there. DOI: 10.1145/3272127.3275044 · PDF · supplemental technical document (mirror)

• Setaluri, R., Aanjaneya, M., Bauer, S., Sifakis, E. SPGrid: A Sparse Paged Grid structure applied to adaptive smoke simulation. SIGGRAPH Asia 2014. — The sparse-grid structure used in GPU MPM. DOI: 10.1145/2661229.2661269 · PDF

• Wang, X., Qiu, Y., Slattery, S.R., Fang, Y., Li, M., Zhu, S.-C., Zhu, Y., Tang, M., Manocha, D., Jiang, C. A Massively Parallel and Scalable Multi-GPU Material Point Method. SIGGRAPH 2020. — Multi-GPU MPM; G2P2G fused transfers. DOI: 10.1145/3386569.3392442 · PDF (lab copy)

• Blatny, L., Gaume, J. Matter (v1): an open-source MPM solver for granular matter. Geoscientific Model Development 18, 2025. — Primary MPM source for the finite-strain constitutive catalog and its pressure-independent Von Mises law. Use the paper for methods; the companion repository is GPL-3.0 and remains methods-only/never-code under §4. full text (open access)

• Fei, Y., Huang, Y., Gao, M. Principles towards Real-Time Simulation of Material Point Method on Modern GPUs. arXiv:2111.00699, 2021. — Engineering playbook for interactive-rate MPM. arXiv:2111.00699

• Gao, M., Tampubolon, A.P., Jiang, C., Sifakis, E. An adaptive generalized interpolation material point method for simulating elastoplastic materials. SIGGRAPH Asia 2017. — Spatial adaptivity (refinement) — note: adapts resolution, never deactivates resting material. DOI: 10.1145/3130800.3130879 · PDF · technical document

• Fang, Y., Hu, Y., Hu, S.-M., Jiang, C. A Temporally Adaptive Material Point Method with Regional Time Stepping. SCA / Computer Graphics Forum 37(8), 2018. — Closest thing to “sleeping” in published MPM: quiescent regions take larger timesteps (up to ~7.5× speedup) — but material is never frozen, deactivated, or converted to a boundary. DOI: 10.1111/cgf.13524 · PDF (lab copy)

• Zhao, Y., Blatny, L., Feng, X., Juel, M.M., Jiang, C., Gaume, J. Unified sparse framework for large-scale simulations using the material point method. arXiv:2605.28525, 2026. — Sparse-grid construction posed as a general active-node indexing problem, with a scan-based CPU strategy and a hash-based GPU one, reported at one-to-two orders of magnitude less time and memory in strongly sparse cases while reproducing dense MPM exactly. Note the boundary: it skips empty nodes and says nothing about occupied-but-quiescent ones. Its acceptance standard is “identical results as dense MPM” — exact state equivalence rather than visual equivalence. arXiv:2605.28525 · PDF

Real-time / games

• Lewin, C. (EA SEED). A Position Based Material Point Method (PB-MPM). ACM SIGGRAPH 2024 Talks. — Stable at any timestep; built for real-time. Ships liquids + co-rotational elastics; notes its Jacobi-style solve’s slow convergence “manifests as artificial softness” — firmness is the method’s acknowledged weak axis. DOI: 10.1145/3641233.3664323 · SEED publications · code · file-level code study: Implementation §3.3

• Yu, C., Li, X., Lan, L., Yang, Y., Jiang, C. XPBI: Position-Based Dynamics with Smoothing Kernels Handles Continuum Inelasticity. SIGGRAPH Asia 2024. — Real-time continuum inelasticity (plasticine, sand, snow) inside a PBD pipeline. arXiv:2405.11694

• Liu, M., et al. CK-MPM: A Compact-Kernel Material Point Method. SIGGRAPH 2025. — Performance-focused compact kernel. arXiv:2412.10399

• Dharma, D., Manaf, A. Interactive Fluid Simulation Based on Material Point Method for Mobile Devices. ICAICTA 2015 (IEEE). — The only located publication whose subject is MPM on a phone. Small — ~2,000 particles at 60 simulation cycles/second — and the interesting part is the optimization list: grid-size reduction, quadratic B-spline weights, and localized active-node computation — the active-node lever, reached for on a power-limited device. Evidence that the mobile regime is barely charted rather than unattempted. DOI: 10.1109/ICAICTA.2015.7335379 · IEEE Xplore

Surface reconstruction & rendering

The particle-to-pixels half of the subject — what Rendering §1.1 distils. (§5).

• Blinn, J.F. A Generalization of Algebraic Surface Drawing. ACM TOG 1(3), 1982. — The “blobby” implicit surface: a sum of radial kernels thresholded. Every density-splat iso-surface is a descendant, and inherits its per-kernel bumpiness where kernels overlap thinly. DOI: 10.1145/357306.357310
• Müller, M., Charypar, D., Gross, M. Particle-Based Fluid Simulation for Interactive Applications. SCA 2003. — The SPH color field — each particle contributes its volume $m/\rho$, so the interior reads one — thresholded for the surface; the normalization every later method keeps. DOI: 10.5555/846276.846298 · author PDF
• Zhu, Y. & Bridson, R. Animating Sand as a Fluid. SIGGRAPH 2005. — Besides the FLIP sand, the averaged-center signed distance $|\mathbf{x} - \bar{\mathbf{x}}| - \bar{r}$ that removes blob bumps between neighboring particles: Houdini’s “average position” surfacing. Needs a smooth, wide weight. DOI: 10.1145/1186822.1073298 · author PDF
• Adams, B., Pauly, M., Keiser, R., Guibas, L.J. Adaptively Sampled Particle Fluids. SIGGRAPH 2007. — Tracks a per-particle distance to the surface and reconstructs from it; the adaptive-sampling companion to the distance-based surface. DOI: 10.1145/1276377.1276437
• Solenthaler, B., Schläfli, J., Pajarola, R. A unified particle model for fluid–solid interactions. Computer Animation and Virtual Worlds 18(1), 2007. — Fixes the Zhu–Bridson surface’s artifacts in concave regions with a Jacobian-based correction of the averaged center. DOI: 10.1002/cav.162
• van der Laan, W.J., Green, S., Sainz, M. Screen Space Fluid Rendering with Curvature Flow. I3D 2009 (NVIDIA). — Particle depth splat → curvature-flow smoothing of the depth image → shade; thickness by additive splat. The real-time route; its canonical form is patented — Patents row M. DOI: 10.1145/1507149.1507164 · author PDF
• Akinci, G., Cornelis, J., Akinci, N., Teschner, M. Parallel Surface Reconstruction for Particle-Based Fluids. Computer Graphics Forum 31(6), 2012. — Narrow-band, parallel marching cubes over the scalar field; the scalability half of the problem. DOI: 10.1111/j.1467-8659.2012.02096.x
• Yu, J. & Turk, G. Reconstructing Surfaces of Particle-Based Fluids Using Anisotropic Kernels. ACM TOG 32(1), 2013. — Per-particle covariance from a weighted PCA of the neighborhood stretches each kernel along the local particle distribution: flat pools and thin sheets reconstruct flat and thin. The method behind Houdini’s and splashsurf’s defaults. Its position-smoothing pre-pass is mild and SPH-scale; at transfer-kernel scale it becomes mean-shift. DOI: 10.1145/2421636.2421641
• Museth, K. VDB: High-Resolution Sparse Volumes with Dynamic Topology. ACM TOG 32(3), 2013. — The sparse level-set structure production surfacing runs on; with it the particles-to-level-set rasterizer and the dilate/smooth/erode filter chain that closes gaps and de-bumps a particle surface. DOI: 10.1145/2487228.2487235 · code (Apache-2.0)
• Ihmsen, M., Orthmann, J., Solenthaler, B., Kolb, A., Teschner, M. SPH Fluids in Computer Graphics. Eurographics STAR 2014. — The survey; its reconstruction section orders the color-field, distance, anisotropic and screen-space families by what each costs and loses. DOI: 10.2312/egst.20141034
• Bhattacharya, H., Gao, Y., Bargteil, A.W. A Level-set Method for Skinning Animated Particle Data. IEEE TVCG 21(3), 2015. — Constrained smoothing: minimize the surface’s thin-plate energy while staying between the union of particle spheres and its erosion, so corners survive and bumps do not. Frame-independent, parallel, post-process. DOI: 10.1109/TVCG.2014.2362546 · author PDF
• Meng, J., Papas, M., Habel, R., Dachsbacher, C., Marschner, S., Gross, M., Jarosz, W. Multi-Scale Modeling and Rendering of Granular Materials. SIGGRAPH 2015. — Why grains read as grains: explicit per-grain shadowing and inter-reflection near the surface, a scattering medium deeper in. Offline, but it is the physics a real-time sand look imitates. DOI: 10.1145/2766949
• Truong, N. & Yuksel, C. A Narrow-Range Filter for Screen-Space Fluid Rendering. Proc. ACM CGIT 1(1), 2018. — Smooths the particle depth image using only depths within a narrow range of the center pixel, which keeps silhouettes and discontinuities where curvature flow and bilateral filters blur them. The paper specifies the filter completely; its reference code is unlicensed (§4). DOI: 10.1145/3203201 · author PDF
• Oliveira, F., Paiva, A., et al. Narrow-Band Screen-Space Fluid Rendering. Computer Graphics Forum 2022. — The current screen-space state of the art: restricts the work to a narrow band around the visible surface. DOI: 10.1111/cgf.14510
• Hart, J.C. Sphere Tracing: A Geometric Method for the Antialiased Ray Tracing of Implicit Surfaces. The Visual Computer 12, 1996. — The SDF ray-marching algorithm every distance-field renderer uses. DOI: 10.1007/s003710050084
• Quilez, I. Distance functions. iquilezles.org, n.d.. — The working reference for SDF primitives and combinators. article
• Aaltonen, S. GPU-Based Clay Simulation and Ray-Tracing Tech in Claybook. GDC, 2018. — A shipped game built on an SDF representation end to end — simulation and rendering both.
• Tatarchuk, N. Dynamic Parallax Occlusion Mapping with Approximate Soft Shadows. I3D, 2006. — The standard parallax-occlusion-mapping reference. DOI: 10.1145/1111411.1111423
• Policarpo, F., Oliveira, M.M., Comba, J. Real-Time Relief Mapping on Arbitrary Polygonal Surfaces. I3D, 2005. — The ray-marched height-texture predecessor to POM. DOI: 10.1145/1186822.1073292
• Sigg, C., Hadwiger, M. Fast Third-Order Texture Filtering. GPU Gems 2, ch. 20, 2005. — Cubic B-spline filtering built from hardware trilinear fetches — the practical way to sample a density grid smoothly. online

Determinism & reproducibility

Reproducible parallel simulation — the substrate for regression-testing a chaotic sim (and for lockstep networking). Fixed-point accumulation for deterministic GPU scatter long predates MPM use, so cite it properly.

• Le Grand, S., Götz, A.W., Walker, R.C. SPFP: Speed without compromise — A mixed precision model for GPU accelerated molecular dynamics simulations. Computer Physics Communications 184(2):374–380, 2013. — The canonical fixed-point-atomics-for-determinism citation (64-bit fixed-point force accumulation, “ensures that force accumulation is deterministic”). DOI: 10.1016/j.cpc.2012.09.022
• Collange, S., Defour, D., Graillat, S., Iakymchuk, R. Numerical reproducibility for the parallel reduction on multi- and many-core architectures (and the companion reproducible floating-point atomic-addition work, FedCSIS 2015). 2015. — Reproducible accumulation under nondeterministic scheduling. FedCSIS paper · DOI: 10.15439/2015F86
• Ahrens, W., Demmel, J., Nguyen, H.D. Algorithms for Efficient Reproducible Floating Point Summation. ACM TOMS 46(3), Article 22, 2020. — ReproBLAS lineage; bitwise-reproducible summation. DOI: 10.1145/3389360

Adjacent methods (cited in comparisons)

Landmarks from neighboring method families that MPM work is weighed against — see Related Methods for what each method is.

• Müller, M., Heidelberger, B., Teschner, M., Gross, M. Meshless Deformations Based on Shape Matching. SIGGRAPH 2005. — Origin of shape-matching “rigidification” of particle sets. DOI: 10.1145/1073204.1073216 · PDF
• Macklin, M., Müller, M., Chentanez, N., Kim, T.-Y. Unified Particle Physics for Real-Time Applications. SIGGRAPH 2014 (NVIDIA FleX). — One particle representation for everything, but rigidity is still ENFORCED (shape-matching constraints) rather than dissolved. DOI: 10.1145/2601097.2601152 · preprint
• Lan, L., Kaufman, D.M., Li, M., Jiang, C., Yang, Y. Affine Body Dynamics: Fast, Stable and Intersection-free Simulation of Stiff Materials. SIGGRAPH 2022. — Replaces rigid bodies with slightly-soft 12-DOF affine bodies to escape rigid-contact machinery inside IPC — the published “soften instead of rigid” move, in another method family (offline; bodies keep a reduced per-body representation). Follow-ups: StiffGIPC (arXiv:2411.06224, SIGGRAPH 2025), M-ABD (arXiv:2603.08079, 2026). arXiv:2201.10022
• Nie, X., et al. An Efficient Sleepy Algorithm for Particle-Based Fluids. Int. J. Computer Games Technology 2014:806095, 2014. — SPH particles “at rest” are skipped, explicitly inspired by rigid-body sleeping — the closest particle-sleeping publication (per-particle skip + wake; no conversion to boundary/terrain). full text (open access) · DOI: 10.1155/2014/806095
• Chentanez, N. & Müller, M. Real-time Simulation of Large Bodies of Water with Small Scale Details. SCA 2010 (NVIDIA). — The height-field water reference, and the technique real-time game water actually uses. Shallow water equations on a staggered height field (explicit, upwind height integration → exact mass conservation) with wet–dry region tracking for shorelines, Perfectly Matched Layer absorbing boundaries for open water, and — the key idea — automatic conversion of height-field water into spray/splash/foam particles wherever the 2.5D representation fails (breaking waves detected by a three-condition test, waterfalls by a terrain-slope threshold, body impacts by swept-triangle tests), reabsorbed on landing. Sub-grid ripples are an advected FFT wave texture faded by the Green strain of its own texture coordinates. Two-way rigid coupling is analytic (prism-volume buoyancy + drag + lift). ~4–18 ms/frame on a GTX 480 at 128²–900×135 grids. Stated limits: volume is not conserved outside the height-integration step, and the stability enhancements are clamps rather than a guarantee. PDF — see Related Methods §1.5.
• Chentanez, N., Müller, M., Kim, T.-Y. Coupling 3D Eulerian, Heightfield and Particle Methods for Interactive Simulation of Large Scale Liquid Phenomena. SCA 2014 / IEEE TVCG 2015. — Settled particles reabsorbed into a cheaper heightfield representation (two-way); the three-representation successor to the 2010 paper above. Successor pattern: Su et al., Real-time Height-field Simulation of Sand and Water Mixtures, SIGGRAPH Asia 2023 — height fields pushed past water into granular material, DOI: 10.1145/3610548.3618159.
• A Digital Twin of River Experiment Infrastructure Based on a 3D Game Engine and Validation of Water Flow with a Real-Scale Experiment. Applied Sciences 15(23):12507, 2025. — A game-engine shallow-water solver (Fluid Flux in Unreal Engine 5, itself an implementation of Chentanez & Müller 2010) used as the fluid model in a peer-reviewed digital twin and validated against a real-scale flow experiment. Useful as evidence of how mature the height-field incumbent is — the bar a continuum solver is measured against for water. full text (open access) · DOI: 10.3390/app152312507
• Xie, T., et al. PhysGaussian: Physics-Integrated 3D Gaussians for Generative Dynamics. CVPR 2024. — Everything-as-MLS-MPM-continuum (elastics, metal, fluids, sand) with no rigid solver — offline/generative visuals, not interactive. arXiv:2311.12198
• Sifakis, E., Barbič, J. FEM Simulation of 3D Deformable Solids. SIGGRAPH Course, 2012. — The standard from-scratch FEM course for graphics — the deformable-solid counterpart to an MPM course. course page
• Hughes, T.J.R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover, 2000. — The standard FEM text, cited for the formulation rather than for graphics practice.
• Gingold, R.A., Monaghan, J.J. Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars. MNRAS 181(3), 1977. — SPH’s origin, published independently of and simultaneously with Lucy 1977. DOI: 10.1093/mnras/181.3.375
• Monaghan, J.J. Smoothed Particle Hydrodynamics. Annu. Rev. Astron. Astrophys. 30, 1992. — The review that carried SPH from astrophysics into everything else. DOI: 10.1146/annurev.aa.30.090192.002551
• Becker, M., Teschner, M. Weakly Compressible SPH for Free Surface Flows. SCA, 2007. — WCSPH — the Tait equation of state that buys a usable timestep without a pressure solve.
• Macklin, M., Müller, M., Chentanez, N. XPBD: Position-Based Simulation of Compliant Constrained Dynamics. MIG, 2016. — Makes PBD stiffness timestep- and iteration-independent, which is what made it usable as physics rather than as an effect. DOI: 10.1145/2994258.2994272 · PDF
• Stam, J. Stable Fluids. SIGGRAPH, 1999. — Unconditionally stable Eulerian fluids by semi-Lagrangian advection — the paper every grid fluid descends from. DOI: 10.1145/311535.311548 · PDF
• Bridson, R. Fluid Simulation for Computer Graphics. CRC Press, 2015. — The standard reference for grid-based fluids, second edition.
• Harlow, F.H., Welch, J.E. Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface. Physics of Fluids 8(12), 1965. — The MAC method: the origin of the staggered grid every fluid solver still uses. DOI: 10.1063/1.1761178
• Kass, M., Miller, G. Rapid, Stable Fluid Dynamics for Computer Graphics. SIGGRAPH, 1990. — First solved water surfaces as a 2D height field — the ancestor of every height-field water system. DOI: 10.1145/97880.97884
• Layton, A.T., van de Panne, M. A Numerically Efficient and Stable Algorithm for Animating Water Waves. The Visual Computer 18, 2002. — Brought the shallow-water equations into graphics. DOI: 10.1007/s003710100131
• O’Brien, J.F., Hodgins, J.K. Dynamic Simulation of Splashing Fluids. Computer Animation, 1995. — Extended the pipe model with splash particles and rigid-body interaction. DOI: 10.1109/CA.1995.393532

Surveys & reviews

• de Vaucorbeil, A., Nguyen, V.P., Sinaie, S., Wu, J.Y. Material Point Method after 25 Years: Theory, Implementation, and Applications. Advances in Applied Mechanics, 2020. — Comprehensive engineering-oriented survey. ScienceDirect
• Jiang, C. PhD thesis: The Material Point Method for the Physics-Based Simulation of Solids and Fluids, UCLA, 2015. PDF
• Ram, D., Gast, T., Jiang, C., Schroeder, C., Stomakhin, A., Teran, J., Kavehpour, P. A Material Point Method for Viscoelastic Fluids, Foams and Sponges. ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 2015. — Oldroyd-B elastic left-Cauchy-Green evolution interpreted as volume-preserving plastic flow without an SVD. PDF · DOI: 10.1145/2786784.2786798
• Zhang, X., Chen, Z., Liu, Y. The Material Point Method: A Continuum-Based Particle Method for Extreme Loading Cases. Academic Press, 2016. ISBN 978-0-12-407716-4. — The book; codifies the multi-mesh / contact material.
• Goswami, P. Snow and Ice Animation Methods in Computer Graphics. Computer Graphics Forum 43, 2024. — Survey of snow/ice simulation, incl. the MPM snow line. DOI: 10.1111/cgf.15059

2. Courses, tutorials & lecture notes

• Jiang, C., Schroeder, C., Teran, J., Stomakhin, A., Selle, A. The Material Point Method for Simulating Continuum Materials. SIGGRAPH 2016 Course Notes. — The canonical tutorial. Continuum mechanics → hyperelasticity → governing equations → discretization → explicit & implicit integration. PDF · course site: mpm.graphics
• CB-Geo. LearnMPM — concise educational notes on the MPM cycle. geoelements.org/LearnMPM
• Chenfanfu Jiang — curated MPM reading list & homepage. Maintained list of MPM papers, methods (APIC, MLS-MPM, AnisoMPM, Multi-GPU MPM), and code. seas.upenn.edu/~cffjiang/mpm.html · math.ucla.edu/~cffjiang
• Higham, N. Faster SVD via Polar Decomposition — for the singular-value-decomposition (SVD) / polar-decomposition prerequisite. nhigham.com
• IIT Delhi COL865 lecture slides — Plasticity & MPM. cse.iitd.ac.in/~narain/courses/…/21-plasticity-mpm.pdf
• NairnMPM documentation (OSUpdocs) — the most complete practitioner reference for multimaterial-mode MPM contact: per-material-pair selectable contact laws (Coulomb friction, adhesive friction, liquid contact, imperfect interfaces), normals options, detection criteria — a shipping engineering code’s answer to every multi-field design question. Multimaterial MPM · Contact Laws

3. Blogs, articles & media

• nialltl. Notes and examples for the Material Point Method — the most approachable from-scratch MLS-MPM walkthrough (APIC → Neo-Hookean elasticity → real-time fluids). nialltl.neocities.org/articles/mpm_guide

• Wikipedia. Material Point Method — solid overview with history and citations. en.wikipedia.org/wiki/Material_point_method

• Taichi Lang blog & docs — MLS-MPM in Taichi, the 88-line implementation in context. taichi-lang.org

• EA SEED blog/talks — PB-MPM and related real-time simulation research. ea.com/seed · SEED 10-year anniversary

• Grant Kot (@kotsoft) — the most important non-academic figure in real-time MPM, and the root of the hobbyist port tree. No paper describes his engine (his one coauthorship, Gao et al. 2018 in §1, is an academic group’s solver) and almost none of it is open source, so the primary sources are his own pages, demos and repositories:

  • The 2010 Java applet is the ancestor. Its origin chain — Java → Flash → JavaScript → openFrameworks — is recorded verbatim in the header of golanlevin/MPM-Fluid, which Kot then forked back. Also downstream: omgware/fluid-simulator-v2, and nialltl’s MPM guide — the entry point Implementation recommends — which credits him for the altered volume calculations and the softened predictive boundary conditions. Those two tricks are still in use fifteen years later.
  • The arc. Soft bodies and SPH through 2009; the 2D MPM applets and demos in 2010–13; APIC, iOS and multiphase chemistry in 2017–19; a 3D voxel engine on MPM that became Liquid Crystal in 2019–24; and since 2024 a pivot to the browser, to MPM/SPH hybridization, and to chemistry at astrophysical scale.
  • What you can run. Liquid 3D (lc1) — the browser-delivered descendant of the desktop engine, and the one that can be profiled rather than only compared against footage · Tutti Space Program, a browser space simulator whose material system is MPM, backed by a chemistry database of elements, compounds, phase masks and enthalpies · Liquid Crystal Demo 2024, whose configuration disclosure is Games §2. Source repositories are in §4.
  • The non-MPM work, so he is cited correctly. The Liquid Layers demo, his best-starred repository (particle_based_viscoelastic_fluid) and pvfs3d are all Clavet-style particle-based viscoelastic fluid — SPH family, no grid. VoxelMash is a voxel streaming and compression sandbox rather than simulation, despite a URL that grew out of the Liquid Crystal codebase. The lines are converging, though: his 2025 hybrid work moves viscoelastic materials off springs and onto an MPM grid, because springs are expensive.

• Fluid Flux (Imaginary Blend / Krystian Komisarek) — commercial Unreal Engine water system (UE 4.26–5.5+, released April 2022) implementing Chentanez & Müller 2010 directly, and the most concrete public data point on what the height-field incumbent costs and cannot do: ~0.5–0.7 ms simulation on an RTX 2060; 1024×1024 max simulation resolution at ~1 m/pixel (~1 km²); heightfield projection only, so no caves or planets, and domains cannot move or rotate at runtime. Documentation includes per-GPU benchmarks and the full feature list (wet–dry shorelines, waterfalls, advected foam, caustics, wetness, underwater, buoyancy/boats, ocean blending). product page & docs · demo

• Practitioner devlogs — how shipped water and snow are actually built (useful as primary evidence of the incumbent techniques in Related Methods §1.5 and §2.3, and cited in Games §4.1):

  • Micro Voxel Game ocean devlog (MGMishMash, r/proceduralgeneration) — layered simulation in practice: bulk voxel fluid for the water body, waves and tides as oscillating target-height masks, an ocean connectivity flag so rock pools disconnect and retain water as the tide drops, and a separate higher-resolution surface-only ripple pass near the player for wakes.
  • Parallax snow devlog (DimaSerebrennikov, r/IndieDev) — the displacement-plus-parallax occlusion mapping snow pipeline stated by its author (sphere “tracer” primitives per leg, diffusion, normal smoothing, ~140 fps on a laptop RTX 3050), and — more valuable — a large public critique thread that amounts to a catalog of what non-continuum snow cannot do (no compaction into packed paths, no cohesion, no conserved volume, no post-holing, and deep snow reading as sand or flour).

• Erin Catto — solver & determinism articles. Iterative Dynamics with Temporal Coherence (GDC 2005 — the canonical accumulated-impulse warm-starting reference, PDF); Solver2D (2024 survey of rigid-body solver variants, post); Determinism (2024 — Box2D v3’s cross-platform determinism + a golden-hash CI regression test, post).

• Engine determinism documentation — what shipping physics engines promise, and under which static configuration conditions: Rapier enhanced-determinism · Jolt determinism discussion · Box2D FAQ.

• Glenn Fiedler (Gaffer On Games) — Fix Your Timestep! (the fixed-dt accumulator pattern and its “spiral of death,” post); Floating Point Determinism and Deterministic Lockstep (post).

• cebas thinkingParticles — MPM operator docs. Vendor reference for a commercial MPM with per-material constitutive nodes; a concrete parameter catalog for the constitutive models in Theory §6 Material models to explore. Visco · VonMises · Elastic · Linear

• SideFX Houdini — the native MPM solver (Houdini 20.5, July 2024). The single largest change in MPM’s industrial standing since the 2013 snow paper: a GPU-accelerated MPM solver shipped as a first-class DCC feature, written in OpenCL and therefore vendor-agnostic, presented to artists as “the FLIP solver extended to solid mechanics” and shipping presets for snow, mud, soil, concrete, metal, jello, rubber, honey and sand, with multi-material interaction and rigid coupling. It is offline VFX, not real-time — but it moves MPM from “research method a studio implements” to “checkbox in the tool everyone already owns,” and it is the reason a future landscape claim must say shipped game, never “shipped product.” Its parameter surface is also the best available read on which constitutive knobs professionals expect to see, alongside the cebas catalog above. what’s new: MPM · MPM Solver node docs · H20.5 MPM masterclass · example files

4. Open-source implementations

Always open the actual LICENSE file in each repository to confirm current terms for your specific use — license files can be updated.