PaperlayerView exact revision on arXiv

Source-linked AI summary

Codimensional Incremental Potential Contact

Minchen Li, Danny M. Kaufman, Chenfanfu Jiang

arXiv:2012.04457v3cs.GR

TL;DR

Codimensional contact simulations must remain intersection-free, unify geometric types, and robustly converge across contact conditions. C-IPC extends incremental potential contact with coupled strain limits, thickness handling, and additive CCD, yielding intersection-free simulations with accurate thickness and stable first-impact detection.

  • Problem

    Codimensional contact simulations require intersection-free trajectories, seamless coupling across geometric types, and robust convergence across varying contact conditions.

  • Method

    C-IPC extends incremental potential contact using optimization-based implicit integration, coupled strain limiting, controllable thickness resolution, and additive continuous collision detection.

  • Results

    C-IPC guarantees intersection-free simulation, strict strain-limit satisfaction down to 0.1%, and accurate geometrically meaningful thickness, with stable and accurate first-time-of-impact output.

  • Takeaways & Limitations

    The framework provides unified contact simulation for codimensional and mixed-dimensional structures while maintaining non-intersection throughout all time steps.

  • Takeaways & Limitations

    ACCD can have unbounded worst-case iteration counts when displacement fields remain large while starting distances are small.

Abstract

from arXiv · show

We extend the incremental potential contact (IPC) model for contacting elastodynamics to resolve systems composed of codimensional DOFs in arbitrary combination. This enables a unified, interpenetration-free, robust, and stable simulation framework that couples codimension-0,1,2, and 3 geometries seamlessly with frictional contact. Extending IPC to thin structures poses new challenges in computing strain, modeling thickness and determining collisions. To address these challenges we propose three corresponding contributions. First, we introduce a C2 constitutive barrier model that directly enforces strain limiting as an energy potential while preserving rest state. This provides energetically-consistent strain limiting models (both isotropic and anisotropic) for cloth that enable strict satisfaction of strain-limit inequalities with direct coupling to both elastodynamics and contact via minimization of the incremental potential. Second, to capture the geometric thickness of codimensional domains we extend the IPC model to directly enforce distance offsets. Our treatment imposes a strict guarantee that mid-surfaces (resp. mid-lines) of shells (resp. rods) will not move closer than applied thickness values. This enables us to account for thickness in the contact behavior of codimensional structures and so robustly capture challenging contacting geometries; a number of which, to our knowledge, have not been simulated before. Third, codimensional models, especially with modeled thickness, mandate strict accuracy requirements that pose a severe challenge to all existing continuous collision detection (CCD) methods. To address these limitations we develop a new, efficient, simple-to-implement additive CCD (ACCD) method that applies conservative advancement to iteratively refine a lower bound for deforming primitives, converging to time of impact.

1 INTRODUCTION

C-IPC extends incremental potential contact to mixed codimensional systems by jointly addressing strain limiting, geometric thickness, collision detection, and unified frictional contact. The framework targets intersection-free, convergent simulations whose thickness and strain constraints remain controllable across time steps and contact conditions.

  • Motivation: Thin materials are efficiently modeled with shells, rods, and particles, but reliable simulation must prevent intersections, control strain, model thickness, unify codimensions, and remain robust across time steps.The paper motivates codimensional formulations for cloth, hair, and sand while identifying six requirements for reliable contact dynamics.
  • Framework: C-IPC extends IPC to arbitrary combinations of codimensional degrees of freedom with frictionally coupled volumetric bodies, shells, rods, and particles.The method is presented as a unified framework without distinction or special casing between codimensions.
  • Contributions: A C2 constitutive barrier models strain limiting as an energy potential, preserving rest state and strictly satisfying strain-limit inequalities while coupling elastodynamics and frictional contact.The paper reports verification down to 0.1% and avoids force-splitting artifacts through incremental-potential minimization.
  • Contributions: Offset barriers enforce minimum separations between shell mid-surfaces, rod mid-lines, and particle points, providing controllable finite thickness even for very small modeled scales.This directly addresses the inconsistency of heuristic thickness parameters in prior contact processing.
  • Contributions: ACCD uses conservative advancement to accumulate a lower bound converging to time of impact, providing a simple CCD method for difficult thick codimensional contacts.The authors describe ACCD as easy to implement, with improved performance, robustness, and guarantees compared with prior CCD routines.
  • Results: Across benchmarks, C-IPC maintains intersection-free trajectories, strictly satisfies strain limits, and captures thickness-driven contact behavior while supporting unified frictional simulation.The framework is tested on challenging examples including twisted cloth, hair, garments, and mixed-dimensional contact.

2 RELATED WORK

Prior work provides codimensional simulation, strain limiting, thickness offsets, and collision detection, but often relies on splitting, heuristic thickness, or numerically fragile CCD. C-IPC combines fully coupled inequality-based strain limiting, offset barriers, and additive conservative-advancement CCD to address these gaps.

  • Codimensional simulation: Codimensional simulation commonly uses implicit or linearly implicit integration with collision filters and penalties, alongside GPU, projection, multigrid, and material-modeling extensions.Related work spans cloth, hair, alternate discretizations, hybrid yarn-shell models, and differentiable simulation.
  • Shells and rods: Prior shell methods can guarantee non-intersection but may require small time steps and simplified friction, whereas C-IPC targets fully implicit unified contact with accurate friction.The comparison is presented as complementary rather than as a direct benchmark result.
  • Strain limiting: Existing strain-limiting methods frequently use equality constraints and timestep splitting, which can produce membrane-locking artifacts and inconsistent material behavior.The paper notes that no prior method is identified as guaranteeing strain-limit enforcement across scenes and time steps.
  • Thickness modeling: Heuristic thickness offsets in prior contact strategies cannot be consistently enforced and may need scene- or speed-dependent adjustment to avoid failures.C-IPC instead uses offset barriers that guarantee requested minimal separation for mid-surfaces, mid-lines, and points.
  • Continuous collision detection: Floating-point root-finding CCD can produce false results under small distances or degenerate configurations, while rational arithmetic may incur unacceptable cost.These numerical issues are especially problematic when finite mid-surface or mid-line separations must coexist with much smaller offset-surface distances.
  • Continuous collision detection: ACCD replaces direct root-finding with conservative advancement and lower bounds on time of impact, achieving robust performance on challenging deformable trajectories and outside C-IPC.The paper reports efficient and accurate success where other methods fail, with similar or better performance when floating-point CCD succeeds.

3 FORMULATION

The formulation generalizes IPC to jointly simulate volumetric bodies, shells, rods, and particles with contact and friction. It combines mixed-dimensional elasticity, incremental-potential time integration, contact barriers, friction, and CCD filtering.

  • Mixed-dimensional formulation: C-IPC targets arbitrary coupling of meshed codimensional models through contact, addressing challenges from thin models and imposed boundary conditions.
  • Elastodynamics with contact: Implicit time stepping minimizes an incremental potential with line search, providing stability and global convergence for elastodynamics.
  • Elastodynamics with contact: IPC augments the incremental potential with contact and friction potentials, enforcing strictly positive primitive-pair distances and friction forces.
  • Elastodynamics with contact: A Newton-type solver uses CCD filtering during every line-search step to maintain intersection-free trajectories.
  • Mixed-dimensional hyperelasticity: Mass and volume for codimensional elements are treated as continuum regions, yielding a unified elasticity potential for volumes, shells, and rods.

4 CONSTITUTIVE STRAIN LIMITING

The paper constructs constitutive strain-limiting barriers that activate near prescribed stretch bounds while preserving the underlying material behavior at rest. The approach supports isotropic and anisotropic cloth models, including data-driven elasticity, and couples strain limiting directly to incremental-potential optimization.

  • Isotropic constitutive strain limiting: A C2 constitutive barrier augments membrane energies with strain limiting while preserving rest-state gradients and Hessians.
  • Isotropic constitutive strain limiting: Isotropic strain limits generally use practical cloth bounds s∈[1.01, 1.1].
  • Isotropic constitutive strain limiting: The barrier has local support, activating only near the strain threshold so the underlying membrane model remains unchanged at lower strains.
  • Isotropic constitutive strain limiting: The strain-limit potential is integrated over cloth volume and optimized within IPC at each time step while line searches prevent barrier violations.
  • Anisotropic constitutive strain limiting: An anisotropic constitutive barrier modifies membrane elasticity while matching the original energy gradient and Hessian at rest.
  • Anisotropic constitutive strain limiting: For data-driven cloth, barriers preserve zero-energy and zero-stress rest behavior, prevent exceeding measured strain bounds, and retain directional anisotropy.
  • Anisotropic constitutive strain limiting: The method avoids physically unsupported extrapolation beyond measured cloth data by imposing stable, controllable strain limits.

5 MODELING THICKNESS

The paper extends IPC with finite distance offsets for codimensional thickness and introduces additive CCD for accurately and robustly resolving thin-body contact. The resulting treatment enforces minimum thickness while supporting intersection-free progress in difficult collision queries.

  • Modeling thickness: Standard IPC’s non-intersection constraint is insufficient for codimensional models because thin surfaces require finite separation representing geometric thickness.
  • Thickened boundaries: A larger contact threshold models an elastic responsive layer, while a nonzero core thickness guarantees a minimum thickness under extreme compression.
  • Thickened boundaries: Distance constraints support finite-thickness shells, rods, and particles while reducing to the original IPC constraint for zero-thickness interactions.
  • Thickened boundaries: The thickened barrier diverges at the prescribed thickness and applies nonzero forces only when mid-surfaces are closer than the offset threshold.
  • CCD challenges: Finite thickness creates severe CCD accuracy demands, with standard floating-point CCD returning an erroneous zero time of impact that can halt simulation progress.
  • Additive CCD: ACCD iteratively adds a conservative lower bound using distance evaluations, producing bounded, intersection-free steps toward time of impact without root finding.
  • Additive CCD: ACCD often matches or improves timing when alternatives succeed, but its worst-case iteration count can be unbounded for diverging displacement fields.

6 EVALUATION

The evaluation examines C-IPC’s strain-limiting accuracy, coupling, robustness, and comparison with prior solvers. Across cloth tests, C-IPC enforces prescribed limits while reducing locking artifacts and maintaining coupled contact behavior.

  • Strain limiting: C-IPC restores locking-free cloth behavior under measured strain limits, even with membrane stiffness scaled to 0.01×.For cotton, a 6.08% stretch bound removes membrane locking and unnatural stretching artifacts.
  • Strain limiting: C-IPC satisfies 1% and 0.1% strain limits across tested cloth configurations while avoiding artificial bending stiffness.These limits are below measured limits for standard cloth materials.
  • Limitations: Very tight unilateral strain limits can reactivate locking because many active constraints approach the number of degrees of freedom.The evaluation identifies this as a boundary of unilateral strain limiting rather than a failure to enforce the constraints.
  • Coupled solving: The fully coupled formulation restricts strains to the measured elasticity range, unlike unconstrained cloth stretched well over 10×.The same coupling is evaluated in both contact-free and contact scenarios.
  • Comparison with prior methods: Compared with ARCSim, C-IPC maintains strain-limit satisfaction at larger time steps, whereas ARCSim requires h = 0.001 s and 87 minutes for a 4 s sequence.ARCSim still does not entirely satisfy limits at that smaller step size, and increasing its iteration cap by 10× does not resolve the issue.
  • Comparison with prior methods: ARCSim’s contact tests show worsening strain-limit satisfaction and increasing artifacts as membrane stiffness decreases.With stiff material, limits are reasonably satisfied except for a few time steps, but membrane locking remains.

7 CONCLUSION

C-IPC unifies codimensional and volumetric simulation with controllable thickness, coupled strain limiting, friction, and intersection-free trajectories. The authors also identify infeasible boundary conditions, fully implicit friction convergence, performance, and strain-bound extensions as remaining limitations or future directions.

  • Conclusion: C-IPC requires an initially non-interpenetrating geometry; guarantees from a starting tangled state remain future work.The authors note that staging can often create a valid initial garment configuration.
  • Conclusion: C-IPC cannot progress through boundary conditions that impose infeasible intersecting or strain-limit-violating trajectories.Stress tests nevertheless converged when boundary conditions pushed against, but did not violate, contact barriers and strain limits.
  • Conclusion: C-IPC lacks a guarantee of convergence to accurately satisfied fully implicit friction relations, although convergent lagging captures stick-to-slip thresholds.The simulations use one lagged iteration for several complex frictional behaviors.
  • Conclusion: The authors prioritized reliability over performance optimization and identify parallelization and improved IPC solvers as future performance work.They report performance as competitive with state-of-the-art cloth codes when comparable accuracies are available.
  • Conclusion: Future directions include explicit time stepping, differentiable design and training tasks, lower-bound strain limits, and granular-flow applications.The current strain-limiting treatment focuses solely on upper bounds.
  • Conclusion: ACCD is presented as a simple replacement for complex, sensitive CCD modules, with preliminary testing indicating speed-ups outside the C-IPC framework.A comparison with ACM was deferred because licensing concerns prevented access to representative code.
  • Conclusion: C-IPC maintains intersection-free simulation and strict strain-limit satisfaction while accurately capturing geometrically meaningful thickness, independent of elasticity model and time-step size.The strain-limit guarantee was confirmed down to 0.1%.

Supplement to Codimensional Incremental Potential Contact

The supplement identifies the paper, its physical-simulation classification, its focus areas, and its publication format.

  • Supplement metadata: The supplement is authored by Minchen Li, Danny M. Kaufman, and Chenfanfu Jiang.The listed affiliations include UCLA, the University of Pennsylvania, Adobe Research, and Adobe Research respectively.
  • Supplement metadata: The work is classified under Computing methodologies → Physical simulation.
  • Supplement metadata: Its key topics are strain limiting, contact mechanics, mixed-dimensional elastodynamics, and constrained optimization.
  • Supplement metadata: The supplement is a five-page arXiv publication from May 2021.

1 VOLUMETRIC SHELL V.S. DISCRETE SHELL

The paper contrasts volumetric and codimensional shell simulation, showing that volumetric models artificially stiffen bending and cost more, while strain limiting addresses persistent membrane-locking artifacts.

  • Volumetric versus discrete shells: Volumetric thin-shell elements realize bending through shearing deformation, adding a shearing penalty that artificially stiffens bending.
  • Volumetric versus discrete shells: The matched cylinder-twist comparison uses identical boundary conditions, with a 1.5 mm volumetric thickness matching the codimensional contact offset.The volumetric mesh contains 175.2K nodes after extrusion and tetrahedralization.
  • Volumetric versus discrete shells: C-IPC required 1.7 min and 9.5 iterations per time step, versus 3.4 min and 13.1 iterations for volumetric IPC during the first 200 steps.The volumetric model has twice as many nodal degrees of freedom.
  • Membrane locking: Membrane-locking severity depends on both resolution and material because membrane energy introduces an artificial bending penalty.
  • Volumetric versus discrete shells: Volumetric results show smoother, lower-frequency wrinkles and larger gaps between touching layers, indicating artificially stiffened bending relative to codimensional results.
  • Membrane locking: Increasing resolution reduces membrane-locking artifacts for the tested material, becoming nearly negligible at 85K and 246K nodes except near some sharp wrinkle turns.
  • Membrane locking: With 0.01× bending stiffness, membrane-locking artifacts remain obvious even at 246K nodes, whereas the C-IPC strain-limited result removes the visible artifact.

3 ISOTROPIC STRAIN LIMITING DERIVATIVE DERIVATIONS

This section supplies derivative components for isotropic strain limiting, including SVD derivatives, barrier derivatives, efficient computation, and SPD projection of a 6x6 matrix.

  • Derivative components: The derivation requires derivatives of the singular value decomposition.
  • Derivative components: The barrier b_t,i is associated with the i-th singular value of triangle t.
  • Derivative components: The derivative with respect to x is available in the anisotropic strain-limiting implementation details.
  • Derivative components: The section provides derivatives for the barrier and describes an efficient computation step.
  • Derivative components: The symmetric-positive-definite projection processes a 6x6 matrix.

4 ANISOTROPIC STRAIN LIMITING DERIVATIONS

The anisotropic strain-limiting model preserves the rest shape and matches the data-driven model’s local quadratic behavior by using the same stiffnesses. Its constitutive quantities can be computed from the cloth deformation gradient and implemented directly.

  • At ˜𝐸=0, the element is in its rest shape and all gradients vanish.
  • The second-order derivatives include separate weft and warp terms plus a coupling term, with all other terms equal to zero.
  • At ˜𝐸=0, the model’s constants are recovered, so matching stiffnesses gives a quadratic approximation to the data-driven model.
  • When cloth weft and warp directions align with coordinate axes, ˜𝐸 can be computed from the 3x2 deformation gradient.
  • The constitutive model is implemented using the derived formulation described in the supplement.

5 ROBUST AND EFFICIENT IMPLEMENTATION OF IPC FRICTION

The IPC friction implementation is designed to remain well defined at zero tangential displacement and to maintain robust positive-definiteness handling during friction-force computation.

  • The friction potential f0 is well defined without division by zero.
  • Friction gradients and forces use an algebraically derived f1(||u_k||) that avoids division by zero by construction.
  • The quantity k||u_k|| remains bounded and equals zero when ||u_k||=0.
  • For ||u_k||>0, the relevant matrix is always SPD and needs no SPD projection; otherwise, the 2x2 matrix is projected.

6 FRICTION VALIDATION ON CLOTH

A cloth-on-slope experiment tests whether C-IPC reproduces stick-slip friction behavior under fully implicit force and tangent updates. It matches the critical friction threshold closely.

  • A 0.5m-wide, 8K-node square cloth was simulated for 4s on a 26.565° slope with critical friction coefficient 0.5.Normal forces and tangent operators were iteratively updated to convergence at each time step.
  • 99.8% of the critical coefficient separates the observed behaviors: C-IPC sticks at μ=0.5 and slides at μ=0.49.

7 TERMINATION CRITERIA

The termination criterion uses a velocity-scaled Newton increment and requires sustained convergence across multiple iterations. Default tolerances are fixed across examples, with a tighter setting for card shuffling.

  • The scaled Newton increment p/h has velocity units, enabling convergence monitoring across scenes with different density, dimension, and stiffness scales.
  • A smoother residual-iteration curve is obtained, and termination requires l consecutive iterations satisfying r<ε_d.
  • The default settings are l=3 and ε_d=10^-3 m/s, while card shuffling uses ε_d=5×10^-4 m/s.

8 ROD BENDING MODULUS

The rod formulation links bending stiffness to physical rod properties, so thickness changes automatically affect bending behavior without manually tuning an adjustable stiffness parameter.

  • Rod bending energy: Rod bending energy is integrated directly over the rod length using curvature and rest-edge-length quantities.The discrete-rod formulation expresses bending through vertex curvature binormals weighted by adjacent rest lengths.
  • Rod bending energy: The adjustable bending parameter α controls stiffness in the discrete-rod model but requires manual increases as rod thickness grows.This manual adjustment makes setting up examples inconvenient.
  • Physical bending model: Kirchhoff rod theory defines bending energy using Young’s modulus, rod radius, and curvature.The bending modulus and radius provide physical material and geometric inputs for the rod’s bending response.
  • Physical bending model: The formulation associates α with rod radius, allowing thickness changes under fixed Young’s modulus to automatically modify bending stiffness.This permits examples to be specified from real material thickness and Young’s modulus without tuning α.

9 SHELL ENERGIES

The shell-energy formulation incorporates thickness into membrane and bending behavior, allowing stiffnesses to be set from Young’s modulus, Poisson’s ratio, and measured material settings.

  • Shell-energy parameterization: Shell energies directly include thickness changes while using Young’s modulus and Poisson’s ratio to set stiffnesses.The formulation treats thickness as part of the elasticity parameterization rather than an independently tuned stiffness.
  • Membrane energy: The membrane energy uses Koiter’s shell model with world- and material-space first fundamental forms and thickness-weighted volume.The weighting uses triangle rest area and thickness together with the shell Lamé parameters.
  • Bending energy: The bending energy uses a discrete shell hinge model based on world- and material-space dihedral angles and rest-edge geometry.The formulation includes edge bending stiffness, rest edge length, and an incident-triangle height factor.
  • Bending energy: Bending stiffness is set to half the flexural rigidity, so it adjusts automatically with thickness and permits intuitive material-parameter specification.Young’s modulus and Poisson’s ratio can instead be chosen from measured cloth material parameters.
  • Material settings: For anisotropic cloth, the experiments select the smallest Young’s modulus and Poisson’s ratio across material directions.The settings are drawn from reported cloth-material data.
Loading 2012.04457v3…