Physics-based Sphere Packing for Lagrangian Mesh Morphing

Jiong Lin; Hod Lipson

cs.GROct 6, 2026

Physics-based Sphere Packing for Lagrangian Mesh Morphing

Authors: Jiong Lin, Hod Lipson

Organizations: Columbia University

Abstract

This paper studies tetrahedral meshes as the body representation for differentiable simulation and computational design. Fixed-connectivity meshes degrade under large morphs, while remeshing from scratch discards node correspondence. We present JamTet, a physics-based sphere-packing framework for volumetric meshing and morphing. We contribute (i) a GPU-parallel mesher combining octree-hierarchical packing with constrained Delaunay tetrahedralization, producing more uniform element volumes than TetGen and fTetWild; (ii) Lagrangian mesh morphing that preserves interior-node identities by re-equilibrating the same spheres within changing shapes and rebuilding the boundary and connectivity, remaining inversion-free where fixed-connectivity and TetSphere meshes invert; and (iii) a differentiable GPU simulator in JAX, with mass-spring edges and a volumetric Neo-Hookean term, integrated with mesh morphing in a design pipeline. In soft-robot morphology design experiments, interior-node gradients improve swimming fitness by 0.73-1.07 over a matched surface-only variant, while voxelized versions of the same designs yield 32-63% lower fitness. These results establish sphere packing as a practical volumetric mesh representation for gradient-based shape optimization. Code and media: under review.

Figures & tables

Panel of Figure 1 — Figure 1: Sphere-to-bunny morph that keeps sphere identities throughout. Top: surface mesh per step; bottom: the spheres, coloured by starting position, settled in each surface and connected into tetrahedra; initial and target meshes cut open, coloured by element quality: regular distorted.

Panel of Figure 2 — Figure 2: Sphere packing mesh in 2D. (a) Mesh generation, left to right: a loose cluster settles into a pile, a boundary force pushes it into the target shape, relaxation gives a tight packing (insets: neighbour distances), and constrained Delaunay triangulation gives the mesh. (b) Morphing: a packed letter “A” morphs into a star, and the balls, coloured by starting position (hue: angle; brightness: radius), keep their identities.

Panel of Figure 3 — Figure 3: Octree packing on the bunny. Top: three levels (each ball split into 8), packing and cut mesh; bottom: packing time and coefficient of variation (CV) of tetrahedron volumes at 64k nodes per shape, octree vs. direct.

Panel of Figure 4 — Figure 4: Sphere-to-bunny morph quality, ${\approx}10$ k spheres. Left: surface per step, shared by all but TetSphere; cut views coloured by AMIPS energy, from blue (3, regular) to deep red (8 or more). Right: mean scaled Jacobian and volume CV per step; crosses: inverted elements present.

Panel of Figure 5 — Figure 5: Morphology design driven by gradients on surface and interior nodes, 1k nodes: starting capsule mid-stroke, five example iterations, and best design mid-stroke; muscle groups red and green; below each design: its interior spheres, coloured by starting position.

Panel of Figure 6 — Figure 6: Fixed-wave and learned-controller swimmers: one design at the same time points. Top: fixed travelling wave, coloured by muscle group; bottom: learned controller, coloured by muscle activation; $\Delta x$ : forward displacement at time $t$ .

Panel of Figure 8 — Figure 8: Mesh and voxel design representations: one swimming design with a smooth surface (left) and its voxel version on the $128^{3}$ design grid (right), with their fitness.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Mesh	Nodes	Tets	Forward (s)	Steps/s	With gradient (s)	Memory (MiB)
Capsule, 1k	1,680	7,954	0.40	60,600	3.56	654 / 1,678
Capsule, 8k	10,768	56,438	0.79	30,200	6.28	656 / 8,852
Bunny, 64k	77,726	425,192	2.27	10,600	–	760 / –

Appendix

Table A3: Simulator speed on one RTX 5090 for a 6 s rollout (24,000 steps); median of five calls after a warm-up, compilation excluded; memory: peak GPU memory, forward / with gradient.

Panel of Figure 9 — Figure A1: Final design of the best seed in every swimming condition of Table 3 , at rest, coloured by muscle group; columns: methods; rows: up–down and left–right swimming; top block 1k, bottom block 8k interior spheres; same scale in all panels.

Panel of Figure 10 — Figure A2: Best 1k up–down swimmers of remesh-fTetWild and ours, rendered as in Figure 7 . Rows 1–2: design-stage wave, coloured by muscle group; rows 3–4: learned controller with $S{=}16$ , coloured by muscle activation; $\Delta x$ : forward displacement at time $t$ ; green dot: start position.

Panel of Figure 11 — Figure A3: Top: compliance design of a short cantilever (1:1:4, tip load), uniform section and optimized shape at equal volume, coloured by von Mises stress, 0 5.3 MPa; right: compliance per design step. Bottom: optimized section scale along the beam against the analytical profile; short cantilever: the same design on TetGen and fTetWild meshes.

Structure	Mesh	$C_{0}$ (mJ)	$C_{1}$ (mJ)	$\Delta C$
Cantilever, 1:1:20	ours	210.86	139.10	$-34.0\%$
Cantilever, 1:1:4	ours	1.86	1.28	$-31.1\%$
	TetGen	1.85	1.27	$-31.1\%$
	fTetWild	1.83	1.26	$-31.3\%$
Clamped both ends, 1:1:20	ours	3.68	2.56	$-30.4\%$
I-beam cantilever, 1:1:20	ours	330.39	217.64	$-34.1\%$

Appendix

Table A5: Compliance before (

C_{0}

) and after (

C_{1}

) 60 design steps at fixed volume. Mesh: the volume mesher, at 10.7k–16.7k nodes.

Panel of Figure 13 — Figure A4: Seven shapes at 64k interior spheres. Rows per block: packed spheres; surface mesh; cut view coloured by AMIPS energy: regular distorted.

Explore similar work

HoloTetSphere: Unified TetSphere Mesh Reconstruction for Physical Simulations
Jul 9, 2026YaQiao Dai, Renjiao Yi, Zhirui Gao +3Surface Reconstruction Mesh Generation
National University of Defense Technology, Changsha, China · Institute of AI for Industries, Chinese Academy of Sciences, Nanjing, China
Differentiable Packing of Irregular 3D Objects with Adaptive Container Estimation
Jun 15, 2026Palak Gupta, Shanmuganathan RamanDifferentiable Physics Bounding Box
Indian Institute of Technology Gandhinagar, India
Mesh Field Theory: Port-Hamiltonian Formulation of Mesh-Based Physics
May 1, 2026Satoshi Noguchi, Yoshinobu KawaharaMesh Generation Scalar Field
Center for Mathematical Science and Advanced Technology, JAMSTEC · Center for Advanced Intelligence Project, RIKEN · Graduate School of Information Science and Technology, The University of Osaka.

Jul 9, 2026, cs.GRJ/K move · Enter open · S save

HoloTetSphere: Unified TetSphere Mesh Reconstruction for Physical Simulations

YaQiao Dai, Renjiao Yi, Zhirui Gao, Wei Chen+2

National University of Defense Technology, Changsha, China · Institute of AI for Industries, Chinese Academy of Sciences, Nanjing, China

Surface Reconstruction Mesh Generation Meshes 3D Reconstruction Topology Optimization Physics Simulation

Open paper AskPDF arXiv

Standard pipelines for physics-ready 3D reconstruction rely on a decoupled two-stage paradigm: extracting surface geometry followed by an error-prone tetrahedralization process. While recent Lagrangian methods like TetSphere Splatting attempt to bypass this by directly optimizing volumetric primitives, their homeomorphic constraints prevent topology-adaptive optimization. Consequently, they produce disjoint tetrahedra rather than a single connected mesh, rendering the structures unsuitable for further physical simulations. To address this, we propose a topology-adaptive framework for holistic tetrahedral mesh reconstruction through end-to-end topological and geometric optimization. First, by coupling Gaussian spheres to tetrahedral elements and leveraging edge connections, we estimate a continuous opacity field for differentiable element pruning. Next, jointly minimizing mesh smoothing energy and multi-view Gaussian rendering error drives alternating geometric refinement while preserving topological adaptivity. Consequently, our approach effectively constructs a unified and topologically coherent tetrahedral mesh. Extensive experiments demonstrate that our method outperforms state-of-the-art techniques by achieving superior geometric accuracy and producing coherent, single-connected tetrahedral meshes, thereby effectively bypassing the error-prone conventional tetrahedralization step for reconstructed surface meshes and streamlining downstream physical simulation.

Abstract

Explore similar work

HoloTetSphere: Unified TetSphere Mesh Reconstruction for Physical Simulations

Differentiable Packing of Irregular 3D Objects with Adaptive Container Estimation

Mesh Field Theory: Port-Hamiltonian Formulation of Mesh-Based Physics