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MPM: on the dense grid, a world that outgrows the grid's box couples the worlds by one rounding step

Status: open, narrow. Upstream fix proposed (Newton: integer cell origin, cell size exactly the voxel).

What

Implicit MPM's dense grid is one box shared by every world (Newton Grid3D with one environment per world). A particle's cell and weights come from (x - lo) / cell_size, with cell_size = (hi - lo) / res. When the box changes, both round differently, so every world's arithmetic moves by about one float32 ulp.

Alloy keeps a multi-world dense grid's box fixed: the first step's box plus 8 voxels each side (alloy.integration.mpm_worlds). While every particle stays inside it, each world's trajectory is bitwise independent of the others (tests/integration/test_mpm_world_independence.py). When a particle in any world leaves the box, it grows for every world. A warning is logged once, and at that step the worlds couple by about one ulp.

The per-world sparse grids (the default on CUDA) index voxels from the origin and do not have this coupling.

CPU worlds split over worker processes (alloy.build(scene, device="cpu", processes=N)) hold the whole scene's start box in every worker (mpm_worlds.start_box / hold_box, WP8's 9a1bd07da), so a world steps the same in any split. When a particle leaves the held box, only its own worker's box grows. From then on that worker's worlds differ by about one ulp from a run with another split, or with one process.

Number

  • Newton's own dense grid, which follows every world's particles each step: melting world 1 next door moved world 0 by 1.5e-8 m with a fixed iteration count and 3.1e-8 m with each world's solve stopping on its own. That is melt_chocolate's scene, CPU, 2 worlds, 6 periods.
  • With the held box, 0 (bitwise), while no particle leaves it.
  • Three paste worlds in 1, 2 and 3 worker processes: bitwise the same (tests/parallel/test_cpu_worlds.py::test_a_paste_world_steps_the_same_in_any_process). Before the shared start box, the split moved a world by 2.2e-5 m at 2 worlds per process (mayonnaise paste, 0.05 s), and by 3.6 mm at one world per process, which built Newton's sparse single-world grid.

Why

A box that holds still for some worlds while others outgrow it has to depend on them. A box that never moves costs memory with no bound: Newton marks active cells in an array over the whole box every step.

Workaround

  • Use the sparse grid (the default; on CUDA it is per world).
  • On the CPU, where Newton builds no per-world sparse grid, start the material near where it will stay, or set grid_padding so the first box has room for the motion.

Newton sets two material switches from every world's particles when the solver is built: the compliant path (min_young_modulus < inf) and hardening (max_hardening > 0). A world of rigid-plastic particles next to an elastic one moved by 1.9e-5 m in a plain-Newton test. Alloy's material cards always give a finite E, so the compliant path is on in every Alloy MPM scene. Worlds that differ in whether they harden would still couple.

Planned fix

In Newton: a dense grid that locates cells by floor(x / voxel) minus an integer origin, with its cell size exactly the voxel. Alloy's write-up for upstream (an issue text and a repro script, with the per-world stop) is not shipped.