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FreeForm: Reduced-Order Deformable Simulation from Particle-Based Skinning Eigenmodes
Donglai Xiang, Vismay Modi, Rishit Dagli, Ty Trusty, Gilles Daviet, Anka He Chen, Nicholas Sharp, David I. W. Levin
TL;DR
Reduced-order deformable simulation still faces difficult geometry requirements and costly computation. This paper uses RKPM-based skinning eigenmodes for mesh-free simulation, achieving better accuracy than Simplicits with around 40x faster training.
Problem
Reduced-order elastodynamic simulation remains limited by difficult mesh requirements for arbitrary or Gaussian-splat geometries and the computational cost of high-resolution FEM.
Method
The method parameterizes deformation with RKPM and obtains reduced-order skinning eigenmodes through elastic-energy Hessian eigenanalysis.
Results
Around 40x faster training accompanies consistently better simulation accuracy than Simplicits.
Takeaways & Limitations
With sufficient degrees of freedom, the method can match or surpass the accuracy of full-order MPM and SPH simulations.
Takeaways & Limitations
The reduced smooth basis struggles with high-frequency details, sharp contact, topology changes, and depends on careful kernel, sampling, and particle choices.
Abstract
from arXiv · showhide
We present a novel formulation for mesh-free, reduced-order simulation of deformable hyperelastic objects. Existing work in reduced-order elastodynamic simulation represents the input geometry by either meshes, which can be difficult to obtain due to challenges in scanning and triangulating complex shapes, or by neural fields that require per-shape optimization. We propose to adopt a Reproducing Kernel Particle Method (RKPM) representation, which enables the construction of reduced-order skinning weights by solving a generalized eigensystem on the Hessian matrix of the elastic energy. We demonstrate that this formulation not only leads to a 40x training speedup compared with the per-shape optimization of neural fields, but also achieves lower simulation error when evaluated against the converged results of finite element method. We show our simulation results on a wide variety of objects in different representations including meshes and Gaussian splats, as well as the application of our method in the downstream task of robot simulation.
1. Introduction
Deformable-object simulation is important, but mesh-based FEM and particle methods face representation, computational, and discretization limitations. This work introduces RKPM-based mesh-free reduced-order elastodynamics using skinning eigenmodes and an explicit elastic-energy formulation.
- Motivation and limitations: FEM requires high-quality meshes and many degrees of freedom for accurate high-resolution simulation, limiting its use on arbitrary or Gaussian-splat representations.Volumetric meshing can be difficult for arbitrary shapes and may be undefined for imprecise point-based representations.
- Motivation and limitations: Particle-based MPM and SPH avoid meshes but are sensitive to spatial and temporal discretizations, potentially failing under large strains.These methods have been used for mesh-free simulation of objects represented by 3D Gaussian splats.
- Related work: Existing mesh-free reduced-order simulation requires optimizing a neural field for every input object and can achieve suboptimal simulation accuracy.Simulations must be preceded by per-object optimization in the cited approach.
- Contributions: The proposed method uses RKPM to parameterize a deformation subspace and formulate elastic energy mesh-free, enabling skinning eigenmodes through an explicit representation.The work also derives an easy-to-implement Hessian expression for commonly used Neo-Hookean elastic energy and evaluates efficiency across varied objects.
2. Related Work
Prior elastodynamic simulation commonly uses mesh-based representations, while newer methods support implicit geometries such as NeRFs, Gaussian Splats, and signed-distance functions. Related approaches include neural physics simulation, particle-based methods, and reduced-order models built from compact bases or learned representations.
- Mesh-based elastodynamic simulation traditionally optimizes per-element energies, while recent techniques support implicit representations including NeRFs, Gaussian Splats, and signed-distance functions.
- Neural physics-based simulation: Neural physics simulation spans methods that augment existing algorithms with neural components and methods that replace physics simulation with learned representations.
- Particle-based simulation: Particle-based updated-Lagrangian methods, including Material Point Method and Smoothed Particle Hydrodynamics, simulate Gaussian Splats or NeRFs and support diverse constitutive models.
- Particle-based simulation: Particle-based methods are non-ideal for elastic solids because spatial and temporal discretization sensitivity can cause numerical fracturing under large strains and inexact boundaries.
- Reduced-order simulation: Reduced-order approaches use few degrees of freedom with complex bases or interpolating functions, producing shape-aware motion through modal analysis, exemplars, or learned reduced representations.
3. Methodology
The method represents elastic-object geometry through volume integration, learns RKPM-discretized skinning weights, and then uses them for low-degree-of-freedom elastic simulation. It extends Skinning Eigenmodes to mesh-free objects by selecting orthogonal weights from a Hessian-based approximation of elastic potential.
- Overall pipeline: The pipeline first constructs reduced-order skinning bases during training, then uses them for low-degree-of-freedom elastic simulation.The input geometry may use any representation supporting volume integration, such as sampled points.
- Simulation: Orthogonal skinning weights prevent the trivial zero solution and improve the mass matrix’s numerical conditioning for Newton-based time stepping.Simulation time stepping minimizes an incremental potential incorporating inertia, elastic potential, and external-force potential.
- RKPM representation: RKPM discretizes skinning weights with corrected kernels that reproduce polynomial functions up to degree D = 1.The corrected kernels augment radial basis functions with terms enforcing the reproducing condition.
- RKPM representation: The method discretizes skinning-weight fields rather than displacement fields, determining nodal values for each skinning function.This avoids the cost of directly using a large number of RKPM kernels for deformation degrees of freedom.
- Skinning Eigenmode with RKPM: The elastic potential is approximated near rest positions by 1/2 d^T H d, and expressive skinning weights minimize this quadratic form subject to channel orthogonality.The simplified weight-space Hessian Hw = Hxx + Hyy + Hzz prioritizes translation.
4. Evaluation
The evaluation compares the proposed method with mesh-free baselines against FEM on beam, Thingi10K, and Simready experiments. Across these tests, the method improves accuracy over Simplicits, reaches accuracy comparable to full-order methods at sufficient DoFs, and trains around 40x faster.
- Standard beam test: On the standard beam test, the method consistently outperforms Simplicits at the same number of affine transformations and can match or surpass MPM and SPH beyond a DoF threshold.The test includes bending and twisting boundary conditions, with twisting applied up to 720°.
- Thingi10K and Simready datasets: On 20 Thingi10K and 19 Simready shapes, the method is compared with Simplicits using FEM simulations as ground truth across three boundary-condition types.The evaluation reports normalized MSE and maximum displacement error, with shapes tetrahedralized using TetWild.
- Thingi10K and Simready datasets: 40x faster training is achieved than Simplicits while maintaining consistently better simulation accuracy through the eigenanalysis formulation.Training time includes Hessian and mass-matrix computation followed by generalized eigen-decomposition.
- Ablation study on training: The Hessian loss provides better RKPM simulation accuracy than randomly sampled expected elastic-energy loss, while the Hessian-Grid variant is similarly accurate but significantly slower.The comparison uses m = 16 affine transformations for the beam ablation and evaluates training strategies including loss and sampling choices.
- Qualitative results: Qualitative experiments include dropped Gaussian-splat objects, large multi-object scenes, a plinko-machine scenario, and robot-arm interaction with 3DGS objects.The paper presents these scenarios as qualitative demonstrations of the method.
5. Limitation and Future Work
The explicit RKPM discretization has advantages over implicit neural bases but inherits reduced-representation limitations and faces RKPM-specific challenges. Its smooth, globally derived basis struggles with high-frequency wrinkles and large nonlinear effects such as sharp contact.
- Reduced-order representations limit the capture of high-frequency details such as wrinkles.The global basis represents smooth, lowfrequency deformation.
- Smooth bases derived from a linearized formulation make large nonlinear effects, including sharp contact, challenging.
- The explicit RKPM discretization has limitations shared with all reduced representations as well as limitations specific to RKPM.
Supplementary Material · 6. Further Discussion on MPM and SPH
MPM and SPH support diverse material behaviors and topology or phase changes, but their changing interaction stencils make them sensitive to spatial discretization. In contrast, the RKPM total-Lagrangian approach controls sampling in the rest pose and uses fixed shape functions with implicit time stepping, avoiding timestep-size constraints.
- 6. Further Discussion on MPM and SPH: MPM and SPH accommodate diverse material models, including plasticity, topology changes, phase changes, and elastodynamics.
- 6. Further Discussion on MPM and SPH: Changing interaction stencils make MPM and SPH sensitive to spatial discretization.In MPM, particles interact when they share a grid node; in SPH, their kernel supports must overlap.
- 6. Further Discussion on MPM and SPH: As stretching separates particles beyond the critical interaction distance, MPM and SPH can undergo numerical fracture.
- 6. Further Discussion on MPM and SPH: RKPM shares the same sampling criteria but only requires controlling kernel centers and widths in the rest pose.MPM and SPH must instead keep particles well distributed at every timestep.
- 6. Further Discussion on MPM and SPH: Resampling particles over time can address distribution issues, but is impractical when simulating a predefined number of Gaussian splats.
- 6. Further Discussion on MPM and SPH: The RKPM total-Lagrangian approach uses fixed shape functions and implicit time stepping, so it has no constraint on timestep size.Most implicit MPM and SPH variants still advect explicitly and remain subject to the Courant–Friedrichs–Lewy (CFL) condition.
7. Implementation Details
The implementation constructs RKPM integration points and kernels, estimates skinning weights through generalized eigenanalysis, and simulates using a CUDA-optimized Kaolin, Warp, and PyTorch stack. Comparisons use matched enhanced implementations, while FEM provides the gold-standard reference and the reduced-order method runs faster than the evaluated alternatives.
- RKPM construction: RKPM integration points are densely sampled on a uniform grid inside the object bounding box, filtered by inside/outside tests when available, or taken directly from downsampled 3DGS points.Points outside watertight meshes are rejected; direct point use avoids out-of-memory errors when necessary.
- Eigenanalysis: The method assembles Hw and M, then uses CUDA double-precision torch.lobpcg generalized eigen-decomposition; the m smallest-eigenvalue eigenvectors provide nodal skinning weights.A global affine constant mode is manually appended, but excluded from reported m values for both methods to ensure fair comparison.
- Simulation: Simulation follows Simplicits and is implemented with Kaolin, Warp, and PyTorch, with CUDA graph kernel launching improving runtime while preserving simulation results.Both methods are tested with the same enhanced implementation for fair runtime comparison.
- Baselines: Simplicits uses a 6-layer, width-64 MLP trained for 10k iterations with Adam at learning rate 10−3, plus elasticity and orthogonality loss weights of 0.1 and 106.At runtime, Simplicits adopts the same simulation implementation as the proposed method.
- Evaluation: FEM supplies the converged gold-standard reference, while the proposed method and Simplicits have similar runtime; FEM is orders of magnitude slower, and MPM and SPH are slower than the reduced-order formulation.The runtime comparison concerns simulating 0.01s in the beam-bending experiment.
8. Proof of Proposition 1
The proof expresses RKPM deformation and its gradient in terms of nodal displacements, then derives the elastic potential and weight-space Hessian. At rest, the Hessian reduces to a weighted integral involving RKPM basis-function gradients.
- Deformation parameterization: The deformation map is Φ(X,d)=X+Σ_k φ_k(X)d_k, with nodal displacements organized as d=[d^x,d^y,d^z]∈R^{K×3}.The three vectors d^x, d^y, and d^z contain nodal displacements in the x, y, and z directions.
- Deformation parameterization: The deformation gradient is F(X,d)=d^T∇φ(X)+I, obtained by differentiating the RKPM deformation map with respect to X.Its entries are assembled from directional derivatives of the basis functions and the displacement components.
- Elastic energy: The total elastic potential is approximated by Monte Carlo quadrature as E_pot(d)≈Σ_i v_i Ψ(F(X_i,d)), using sample points X_i and weights v_i.The strain-energy density is integrated over the domain Ω and evaluated at each sampled deformation gradient.
- Hessian derivation: The weight-space Hessian combines displacement-direction blocks as H_w=H_xx+H_yy+H_zz, each formed from the Hessian of E_pot with respect to nodal displacements around d=0.For each direction, the affine dependence of F on displacement enables a Hessian chain rule through the Jacobian of vec(F_i).
- Hessian derivation: At the rest configuration, the resulting Hessian satisfies (H_w)_ij=∫_Ω(λ(X)+4μ(X))∇φ_i(X)^T∇φ_j(X)dX.The discrete expression is obtained by summing weighted directional basis-gradient outer products and approximates the corresponding domain integral.
9. Additional Evaluation and Analysis
Additional evaluations show that the predicted skinning weights better express FEM deformations than the Simplicits baseline, capture layer-specific behavior in a heterogeneous sphere, and enable visualization of stress concentrations in a twisted beam.
- Basis fitting residuals: The basis-fitting evaluation applies least-squares fitting to FEM results on the standard beam, Thingi10K, and Simready datasets.Residuals are reported in Tables 6 and 7 for reduced-order methods including Simplicits and ours.
- Basis fitting residuals: Our skinning weights better explain FEM simulation results than the Simplicits baseline by isolating basis expressiveness from accumulated dynamic-simulation error.The fitting residual uses analytically solved linear least squares over predicted skinning weights and evaluates average normalized residual across vertices and time steps.
- Heterogeneous sphere: On a heterogeneous stiff-soft-stiff-soft sphere, our method captures distinct deformation behaviors across layers, whereas Simplicits produces only stiff, global deformation.The sphere is dropped onto the ground, requiring a global simulation basis to represent different dynamic patterns across stiffness regions.
- Stress analysis: The twisted-beam evaluation visualizes the three principal Cauchy stress fields, revealing stress distributions and regions of high stress concentration.The principal stresses are the eigenvalues of the symmetric 3 × 3 Cauchy stress tensor in the deformed configuration.