Physics-based Sphere Packing for Lagrangian Mesh Morphing

Motivation

• Fixed connectivity inverts 4.1% of tetrahedra by the end of a sphere-to-bunny morph, while remeshing from scratch assigns new node indices.TetSphere preserves node identity but some of its tetrahedra still invert in the authors’ morph experiments.Source: §1 · pp. 1–2

Background

• The paper also places its approach alongside voxel-based soft-robot design and differentiable particle and mass–spring simulators.The introduction notes that voxel bodies retain one connectivity and particle bodies have no element connectivity.Source: §1 · p. 2

Approach

• JamTet re-settles the same spheres inside each new surface and rebuilds connectivity, preserving every interior node’s identity through morphing.Sphere centres become interior nodes, and constrained Delaunay tetrahedralization connects them to the re-triangulated surface.Source: §1 · p. 2

Contributions

• For soft swimmers, interior-node gradients raise fitness by 0.73–1.07 over a matched surface-only variant in all four swimming conditions.Voxel versions of the designs reach 32–63% lower fitness than the smooth designs.Source: §1 · p. 2

Tetrahedral meshing

• Tetrahedral meshers recover boundaries and control element quality through Delaunay refinement or optimization, while prior particle methods relax packed bubbles or balance forces before triangulation.JamTet adds a two-stage jamming schedule, an octree, GPU execution, and packing persistence across shape changes.Source: §2 · p. 2

Deforming volumetric representations

• Prior deformable representations include particle-based MPM, fixed-grid occupancy, persistent tetrahedralized spheres, and surfaces extracted from deformable background grids.TetSphere is identified as the closest prior representation; level-set methods morph solids by evolving an implicit function, and the paper’s morphs follow such a path.Source: §2 · pp. 2–3

Differentiable simulation and morphology design

• Differentiable simulators have been used for soft-body rollouts and swimmer-shape optimization, while ELDiR evolves robot bodies as it trains controllers.The cited systems include ChainQueen (Hu et al., 2019), DiffTaichi (Hu et al., 2020b), DiffAqua (Ma et al., 2021), and ELDiR (Strgar et al., 2024).Source: §2 · p. 3

3.1 SURFACE-FIRST SPHERE PACKING

• JamTet sets node spacing from domain volume and sphere count, then re-triangulates the boundary, packs interior spheres, and tetrahedralizes the surface and sphere centres.The surface nodes remain fixed during packing, and the spheres together have the volume of the domain while overlapping their neighbours.Source: §3.1 · p. 3
• The two-stage packing uses long-range Gaussian repulsion to fill holes, followed by contact repulsion that cages overlapping spheres into a jammed packing.Centres are projected to remain at least one spacing inside the surface; the first interior layer therefore sits one spacing below it.Source: §3.1 · p. 4

3.2 OCTREE-HIERARCHICAL PACKING

• The octree schedule refines coarse packings by splitting each sphere into eight children and injecting spheres into uncovered regions, reducing repeated neighbour-list work.Each finer level builds a cached neighbour list; the figure depicts three levels and a cut mesh on the bunny.Source: §3.2 · p. 4, Figure 3 · p. 5

3.3 MORPHING WITH PERSISTENT SPHERES

• Morphing carries the same spheres through level-set snapshots, re-settles them inside each re-triangulated shape, and rebuilds connectivity while keeping the sphere count fixed.Snapshots are spaced by resolved source–target symmetric-difference volume and scaled to target volume so sphere spacing remains fixed.Source: §3.3 · p. 4

3.4 DIFFERENTIABLE SIMULATION AND DESIGN

• The simulator combines damped edge springs with a Neo-Hookean volume term, and its gradients update FFD shape parameters before the morph step rebuilds the mesh.The volume term resists collapse and inversion; the morph step maps each updated level set to a valid mesh and passes no gradient.Source: §3.4 · pp. 4–5

4.1 STATIC MESHING

• On the bunny and Spot, our meshes rank first in all four quality measures, tie fTetWild on bunny AMIPS, and rank second in wall time; their Vol CV is 45% of that of TetGen and 25–40% of that of fTetWild.Across all nine shapes, our meshes have the lowest Vol CV on every shape. The octree schedule packs 2.5–6.3× faster than direct packing on the nine shapes and gives a lower Vol CV on every shape.Source: §4.1 · p. 6

4.2 IDENTITY-PRESERVING MORPHING

• Across the sphere-to-bunny morph, JamTet has no inverted tetrahedra and Vol CV 0.124–0.149, while fixed connectivity reaches 4.1% inversion and fTetWild has three times its volume spread.TetSphere inverts after the first step; the table reports mean±sd over morph steps.Source: §4.2 · p. 6, Table 2 · p. 7

4.3 SOFT ROBOT DESIGN STUDY

• In all four swimming conditions, JamTet exceeds ours-surface by 0.73–1.07 fitness, with disjoint seed ranges; the remeshing baselines finish near ours-surface.The two methods share every step except gradient support, isolating the contribution of interior-node gradients.Source: Table 3 · p. 7, §4.3 · p. 8

4.2 IDENTITY-PRESERVING MORPHING

• Re-packing preserves every interior-node identity through the morph, including at 64,000 spheres, while a fresh packing reaches Vol CV 0.144 against JamTet’s 0.143 but discards identity.The morph uses nine sphere-to-bunny steps; fixed connectivity projects surface nodes and harmonically extends their motion, while fTetWild remeshes each step.Source: §4.2 · pp. 6–7

4.3 SOFT ROBOT DESIGN STUDY

• Each design iteration simulates a capsule with about 1k or 8k interior spheres for 6 s, updates its shape from simulator gradients, and morphs the mesh; a rollout with gradients takes 3.6–6.3 s on one GPU.The study tests up–down and left–right swimming and frictional-ground walking; fitness is forward centre-of-mass displacement averaged over the second half of a rollout.Source: §4.3 · p. 7
• With the largest learned controller, JamTet reaches fitness 4.34, 1.20× its tuned-wave fitness; the method ranking remains unchanged across all six controller test points.The ratio of JamTet to ours-surface falls from 1.84 after design to 1.29 with the largest controller.Source: Figure 7 · p. 8, §4.3 · p. 9
• Interior-node gradients capture how moving material inside the body changes fitness, while surface-node gradients capture outline changes; the drag model gives normal motion six times the resistance of sliding.Both swimmers become plates facing the bending direction, and their bending depends on material distribution through the body.Source: §4.3 · p. 8
• Voxel versions of the swimming designs reach 32–63% lower fitness than the smooth designs at both tested resolutions.The comparison replaces the tetrahedral body with a voxel body using the same gait and objective.Source: Figure 8 · p. 9, §4.3 · p. 9

5 DISCUSSION

• Walking remains inconclusive because ground contact makes its gradient non-smooth; GPU atomics also cause slight variation between repeated gradient evaluations.Other scope qualifications include runs cut short by CDT failures, controller tests with one run per condition, and simulation units with equal node masses.Source: §5 · p. 9
• JamTet’s pipeline requires a GPU, meshes more slowly than TetGen, and thin boundary tetrahedra shorten the stable explicit-integration time step.Only interior nodes persist across morph steps; the surface is re-triangulated to keep its triangles well shaped.Source: §5 · p. 9
• The conclusion presents sphere packing as a practical volumetric mesh representation for gradient-based shape optimization, and reports support for compliance design with linear finite elements.The code, covering meshing, morphing, simulation, and the design loop, is under review with the paper.Source: §5 · p. 9, REPRODUCIBILITY STATEMENT · p. 10

AI USE STATEMENT

• The author used an LLM for language editing, formatting, and literature-review search; scientific ideas, claims, decisions, and cited-literature interpretation remain the researcher’s.Source: AI USE STATEMENT · p. 10

• Table A1 marks free-form bodies as supporting any closed level-set surface, while grid- or particle-based shapes are approximate and primitives or templates are restricted.Source: Table A1 · p. 13
• The pipeline uses a conforming tetrahedral mesh with a smooth surface, built from a closed level-set surface, and simulator gradients with respect to its nodes.Source: A SOFT-ROBOT DESIGN REPRESENTATIONS · p. 13
• Primitive and voxel bodies have blocky surfaces, while particle bodies have no explicit surface.Source: A SOFT-ROBOT DESIGN REPRESENTATIONS · p. 13

• JamTet’s mean AMIPS energy was 3.32–3.47, the lowest or second lowest on every shape.Source: B STATIC MESHING ON NINE SHAPES · p. 13
• JamTet took 35.9–123.4 s per shape, compared with under one second for TetGen and 6.7–1340 s for fTetWild.Source: B STATIC MESHING ON NINE SHAPES · p. 13
• Vol CV was 0.127–0.207 for JamTet, versus 0.293–0.308 for TetGen and 0.310–0.540 for fTetWild across nine shapes.Source: Table A2 · p. 13, B STATIC MESHING ON NINE SHAPES · p. 13
• No mesher produced an inverted element across the nine shapes.Source: Table A2 · p. 13, B STATIC MESHING ON NINE SHAPES · p. 13

• A 6 s rollout of the 1k capsule took 0.40 s, 15 times faster than real time; its rest-position gradient took 3.56 s, or 9.0 times the forward cost.Source: Table A3 · p. 13, C SIMULATOR SPEED · pp. 13–14
• The 8k capsule took 0.79 s forward and 6.28 s with its gradient, 7.9 times the forward cost.Source: Table A3 · p. 13, C SIMULATOR SPEED · p. 14
• The 64k bunny forward rollout took 2.27 s, but its gradient was not run because the compiled program was estimated to need more than 82% of the 32 GB GPU memory.Source: Table A3 · p. 13, C SIMULATOR SPEED · p. 14
• Without ground contact, adjoint gradients agreed with central finite differences to 5.5 × 10^-9 relative error; with contact, relative errors rose from 0.19 to 0.97 as the difference step shrank.Source: C SIMULATOR SPEED · p. 14
• Repeated gradient evaluations on one GPU differed by 2.3 × 10^-3 in relative terms because unordered scatter-adds run as atomics.Source: C SIMULATOR SPEED · p. 14

• Swimming used up–down and left–right muscle splits, while walking used the up–down split; the two muscle groups contracted in antiphase at 2 Hz.Source: D ROBOT DESIGNS · p. 14
• The design objective added lateral-drift and body-orientation penalties, and each gradient averaged eight rollouts of perturbed shapes.Source: D ROBOT DESIGNS · p. 14
• With the tuned gait, JamTet reached 1.64× the fitness of its surface-only variant.Source: D ROBOT DESIGNS · p. 14
• The shared controller mapped eight clock signals and each muscle block’s rest position to activation, without reading simulation state.Source: D ROBOT DESIGNS · p. 14
• Figure A2 shows best 1k up–down swimmers under the design-stage wave and a learned S=16 controller, with the floor grid indicating distance travelled.Source: D ROBOT DESIGNS · p. 14

E VOXEL AND SMOOTH SURFACES

• 32–63% lower fitness was measured for voxel bodies than for the smooth designs.The comparison used eight designs from an earlier sweep, distinct from the seeds in Table 3.Source: Table A4 · p. 14, E VOXEL AND SMOOTH SURFACES · p. 15
• The design panels compare methods across up–down and left–right swimming, with 1k- and 8k-sphere designs shown at a common scale.Figure A2 shows design-stage waves by muscle group and learned-controller activation at S=16; Δx denotes forward displacement and the green dot marks the start.Source: Figure A1 · p. 15, E VOXEL AND SMOOTH SURFACES · p. 15, Figure A2 · p. 16
• Each design kept its shape, muscle rule, and fitness measure while its tetrahedral body was replaced by a voxel body at one of two resolutions.One voxel version matched the mesh’s surface-triangle count and used 0.74–0.79× its node count; the other used the design loop’s 128^3 level-set grid.Source: E VOXEL AND SMOOTH SURFACES · p. 15
• Fitness compares stored training fitness for the smooth design with median voxel fitness over three start phases, summarized as mean±sd across designs.Table A4 also reports the change from the smooth design and voxel grid nodes relative to the smooth mesh.Source: Table A4 · p. 14, E VOXEL AND SMOOTH SURFACES · p. 15

F SHAPE DESIGN WITH LINEAR FINITE ELEMENTS

• On short beams, TetGen and fTetWild meshes of the same shape matched the compliance reduction within 0.2 percentage points and had section profiles within 0.001–0.010 RMS of ours.The beam meshes used 8k interior spheres and 11.5k–16.7k nodes; short-beam TetGen and fTetWild meshes had 10.7k–12.9k nodes.Source: F SHAPE DESIGN WITH LINEAR FINITE… · p. 16
• The optimized section scale tapered from 1.48–1.54 at the root to 0.63–0.64 at the tip, with 0.067–0.069 RMS deviation from the uniform-strength profile.For the beam clamped at both ends, deviation from |M(z)|^1/3 was 0.131 because the lower bound s ≥ 0.55 holds the profile above zero where bending moment vanishes.Source: F SHAPE DESIGN WITH LINEAR FINITE… · p. 16
• The design minimizes C = f⊤u for beams under a static 100 N load, using a section scale along the beam axis and rescaling after each step to preserve volume.Figure A3 depicts the short cantilever’s uniform and optimized equal-volume shapes with von Mises stress, compliance over design steps, and section scale against the analytical profile.Source: F SHAPE DESIGN WITH LINEAR FINITE… · pp. 16–17, Figure A3 · p. 17
• 30.4–34.1% compliance reduction was achieved across the four cantilevers after 60 design steps at fixed volume.On the short cantilever, compliance fell from 1.86 to 1.28 mJ, reaching 90% of the drop by step 14.Source: F SHAPE DESIGN WITH LINEAR FINITE… · p. 16, Table A5 · p. 17