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Geometric Contact Potential

Zizhou Huang, Max Paik, Zachary Ferguson, Daniele Panozzo, Denis Zorin

TL;DR

This work introduces a continuum geometric barrier potential for deformable surfaces that satisfies natural, robustness-driven requirements, including locality, differentiability, barrier behavior, and rest-state zero force. By defining interaction sets using local minima and exterior-direction constraints, and by employing an adaptive localization together with mollified directional factors, the authors formulate a discretization that remains barrier-enforcing and differentiable for both smooth and piecewise-smooth surfaces. The resulting framework unifies and improves upon prior barrier and barrier-like approaches, offering discretization-independence, controlled locality, and reduced spurious forces, while enabling efficient discretization and robust large-deformation simulations. The method is validated through extensive 2D/3D experiments, inverse design demonstrations, and comparisons to IPC and repulsive-surface approaches, highlighting improved stability, reduced artifact forces, and favorable convergence properties with respect to mesh refinement.

Abstract

Barrier potentials gained popularity as a means for robust contact handling in physical modeling and for modeling self-avoiding shapes. The key to the success of these approaches is adherence to geometric constraints, i.e., avoiding intersections, which are the cause of most robustness problems in complex deformation simulation with contact. However, existing barrier-potential methods may lead to spurious forces and imperfect satisfaction of the geometric constraints. They may have strong resolution dependence, requiring careful adaptation of the potential parameters to the object discretizations. We present a systematic derivation of a continuum potential defined for smooth and piecewise smooth surfaces, starting from identifying a set of natural requirements for contact potentials, including the barrier property, locality, differentiable dependence on shape, and absence of forces in rest configurations. Our potential is formulated independently of surface discretization and addresses the shortcomings of existing potential-based methods while retaining their advantages. We present a discretization of our potential that is a drop-in replacement for the potential used in the Incremental Potential Contact (IPC) formulation, and compare its behavior to other potential formulations, demonstrating that it has the expected behavior. The presented formulation connects existing barrier approaches, as all recent existing methods can be viewed as a variation of the presented potential, and lays a foundation for developing alternative (e.g., higher-order) versions.

Geometric Contact Potential

TL;DR

This work introduces a continuum geometric barrier potential for deformable surfaces that satisfies natural, robustness-driven requirements, including locality, differentiability, barrier behavior, and rest-state zero force. By defining interaction sets using local minima and exterior-direction constraints, and by employing an adaptive localization together with mollified directional factors, the authors formulate a discretization that remains barrier-enforcing and differentiable for both smooth and piecewise-smooth surfaces. The resulting framework unifies and improves upon prior barrier and barrier-like approaches, offering discretization-independence, controlled locality, and reduced spurious forces, while enabling efficient discretization and robust large-deformation simulations. The method is validated through extensive 2D/3D experiments, inverse design demonstrations, and comparisons to IPC and repulsive-surface approaches, highlighting improved stability, reduced artifact forces, and favorable convergence properties with respect to mesh refinement.

Abstract

Barrier potentials gained popularity as a means for robust contact handling in physical modeling and for modeling self-avoiding shapes. The key to the success of these approaches is adherence to geometric constraints, i.e., avoiding intersections, which are the cause of most robustness problems in complex deformation simulation with contact. However, existing barrier-potential methods may lead to spurious forces and imperfect satisfaction of the geometric constraints. They may have strong resolution dependence, requiring careful adaptation of the potential parameters to the object discretizations. We present a systematic derivation of a continuum potential defined for smooth and piecewise smooth surfaces, starting from identifying a set of natural requirements for contact potentials, including the barrier property, locality, differentiable dependence on shape, and absence of forces in rest configurations. Our potential is formulated independently of surface discretization and addresses the shortcomings of existing potential-based methods while retaining their advantages. We present a discretization of our potential that is a drop-in replacement for the potential used in the Incremental Potential Contact (IPC) formulation, and compare its behavior to other potential formulations, demonstrating that it has the expected behavior. The presented formulation connects existing barrier approaches, as all recent existing methods can be viewed as a variation of the presented potential, and lays a foundation for developing alternative (e.g., higher-order) versions.
Paper Structure (54 sections, 3 theorems, 38 equations, 41 figures, 3 tables, 1 algorithm)

This paper contains 54 sections, 3 theorems, 38 equations, 41 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Barrier potentials
  3. The IPC barrier potential Li2020IPC
  4. Surface/volume double-integral potentials sauer2013computationalKamensky2018Contact
  5. Repulsive shells repulsive2024
  6. Barriers based on the gap functions wriggers2006computational
  7. Related Work
  8. Barrier/interior-point methods
  9. Other approaches
  10. Contact Potential Requirements
  11. Remark
  12. Formulation
  13. Deformable smooth surfaces
  14. Interaction sets
  15. Remark
  16. ...and 39 more sections

Key Result

Proposition 1

The potential eq:potential-deform satisfies req:barrierreq:no-spurious-forcesreq:localizationreq:differentiabilityreq:generality if $f$ is a curvature-continuous surface, with $C(x,f)$ given by def:contact-smooth, distance-to-contact defined as $d_c(x,f) := \min_{y\in C(x,f)} \| f(y) - f(x)\|$, and

Figures (41)

  • Figure 1: We introduce a novel geometric barrier potential satisfying a set of natural properties. None of the other potentials in the literature satisfy all these properties simultaneously, leading to inaccurate results, penetrations, or other undesired artifacts. In this example, we show that the contact potential introduced in Li2020IPC is not zero at the rest pose when the potential extent is larger than the length of an edge, introducing spurious forces (top) and deformation (bottom left). Our potential is zero in the rest pose by construction (bottom right), as this is one of the natural properties of a geometric contact potential.
  • Figure 2: Contact potentials must distinguish points nearing contact (A and B) from nearby non-contacting points (A and C).
  • Figure 3: IPC potential. The extent of the potential is shown in blue. (1) IPC potential is finite, as interactions between edges sharing vertices, and faces with their vertices are dropped. (2) Refinement forces the maximal potential extent $\hat{d}$ to decrease. (3) Spurious forces (red arrows) arise if the surface is compressed horizontally and nearby vertices are closer than the potential extent $\hat{d}$.
  • Figure 4: Surface barrier of Kamensky2018Contact (1) The potential has a self-contact exclusion zone (green), outside the potential extent (blue). For extreme deformations (folding), this allows self-intersections. (2) Extreme compression may push points into the potential zone, leading to spurious forces (red arrows) (3) In quadrature point-to-point discretization, the surface may not remain contact-free, a point may "push through" between other Kamensky2018Contact.
  • Figure 5: (1) The DND gap function is discontinuous when the deformation progresses from the black to the blue curve. (2) The DND gap function can be arbitrarily small for p.w. smooth surfaces as $x_i$ approaches the sharp corner in a continuous setting. (3) The CP gap function changes non-smoothly, as the closest point switches discontinuously (from black to blue).
  • ...and 36 more figures

Theorems & Definitions (8)

  • Definition 1
  • Proposition 1
  • Definition 2: Interaction sets for p.w. smooth surfaces
  • Remark
  • Proposition 2
  • Proposition 3
  • Remark
  • Remark