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Energy-Controllable Time Integration for Elastodynamic Contact

Kevin You, Juntian Zheng, Minchen Li

TL;DR

The paper tackles the challenge of energy-dissipative yet stable time integration for elastodynamic systems with contact. It introduces decoupled alpha-methods, notably A-1 and A-search, which are linearly symplectic and enable adaptive velocity interpolation to target a desired energy $H_{n+1}$ while preserving positional constraints. A-search, in particular, provides flexible control over energy dissipation versus conservation, supports friction, and avoids inversion and interpenetration issues, delivering visually coherent and physically plausible animations at large time steps. Extensive experiments demonstrate that A-search outperforms traditional methods like BDF2 at similar runtimes by maintaining energy in low-frequency modes and reducing manual timestep tuning, with robust performance across a wide range of materials and collision scenarios. Overall, the method offers a practical, energy-controllable tool for high-quality elastodynamic animation in graphics and engineering applications.

Abstract

Dynamic simulation of elastic bodies is a longstanding task in engineering and computer graphics. In graphics, numerical integrators like implicit Euler and BDF2 are preferred due to their stability at large time steps, but they tend to dissipate energy uncontrollably. In contrast, symplectic methods like implicit midpoint can conserve energy but are not unconditionally stable and fail on moderately stiff problems. To address these limitations, we propose a general class of numerical integrators for Hamiltonian problems which are symplectic on linear problems, yet have superior stability on nonlinear problems. With this, we derive a novel energy-controllable time integrator, A-search, a simple modification of implicit Euler that can follow user-specified energy targets, enabling flexible control over energy dissipation or conservation while maintaining stability and physical fidelity. Our method integrates seamlessly with barrier-type energies and allows for inversion-free and penetration-free guarantees, making it well-suited for handling large deformations and complex collisions. Extensive evaluations over a wide range of material parameters and scenes demonstrate that A-search has biases to keep energy in low frequency motion rather than dissipation, and A-search outperforms traditional methods such as BDF2 at similar total running times by maintaining energy and leading to more visually desirable simulations.

Energy-Controllable Time Integration for Elastodynamic Contact

TL;DR

The paper tackles the challenge of energy-dissipative yet stable time integration for elastodynamic systems with contact. It introduces decoupled alpha-methods, notably A-1 and A-search, which are linearly symplectic and enable adaptive velocity interpolation to target a desired energy while preserving positional constraints. A-search, in particular, provides flexible control over energy dissipation versus conservation, supports friction, and avoids inversion and interpenetration issues, delivering visually coherent and physically plausible animations at large time steps. Extensive experiments demonstrate that A-search outperforms traditional methods like BDF2 at similar runtimes by maintaining energy in low-frequency modes and reducing manual timestep tuning, with robust performance across a wide range of materials and collision scenarios. Overall, the method offers a practical, energy-controllable tool for high-quality elastodynamic animation in graphics and engineering applications.

Abstract

Dynamic simulation of elastic bodies is a longstanding task in engineering and computer graphics. In graphics, numerical integrators like implicit Euler and BDF2 are preferred due to their stability at large time steps, but they tend to dissipate energy uncontrollably. In contrast, symplectic methods like implicit midpoint can conserve energy but are not unconditionally stable and fail on moderately stiff problems. To address these limitations, we propose a general class of numerical integrators for Hamiltonian problems which are symplectic on linear problems, yet have superior stability on nonlinear problems. With this, we derive a novel energy-controllable time integrator, A-search, a simple modification of implicit Euler that can follow user-specified energy targets, enabling flexible control over energy dissipation or conservation while maintaining stability and physical fidelity. Our method integrates seamlessly with barrier-type energies and allows for inversion-free and penetration-free guarantees, making it well-suited for handling large deformations and complex collisions. Extensive evaluations over a wide range of material parameters and scenes demonstrate that A-search has biases to keep energy in low frequency motion rather than dissipation, and A-search outperforms traditional methods such as BDF2 at similar total running times by maintaining energy and leading to more visually desirable simulations.
Paper Structure (47 sections, 1 theorem, 59 equations, 19 figures, 2 tables, 1 algorithm)

This paper contains 47 sections, 1 theorem, 59 equations, 19 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Time Integration
  4. Energy Conservation
  5. Adaptive Time Integration
  6. Background
  7. Common Numerical Methods
  8. Symmetric and Symplectic Methods
  9. Stability
  10. Blending
  11. Method
  12. Decoupled Method
  13. Decoupled Symplectic Method
  14. A-1
  15. A-search
  16. ...and 32 more sections

Key Result

theorem 1

The decoupled symplectic method $\Xi_1$ is linearly symplectic for any underlying Runge-Kutta method $\Phi$. Moreover, when $\Phi$ is implicit Euler, A-1 is unconditionally stably symplectic.

Figures (19)

  • Figure 1: Bunny splash. Soft elastic bunny dropped on soft elastic cubes with friction. Static friction holds columns of cubes together when falling. A-search with decay gives dynamical results while respecting static and dynamic frictional contact.
  • Figure 2: Suspended armadillo. Five frames between 3s and 4s. A-search at $h=1/30s$ sees slightly more motion than BDF2 at $h=1/300s$, and much more than BDF2 at $h=1/30s$. Blending has motion only in the foot and artifacts (see video).
  • Figure 3: Vibrating membrane, initialized to $u_{21}$ mode. Left shows frame with largest $u_{21}$ amplitude in the final $0.5s$, right shows frame with most wrinkles. Amplitudes are magnified by 10x for visual clarity. A-search both kept more energy in the $u_{21}$ mode than BDF2, and captured some of wrinkling modes exhibited by the reference.
  • Figure 4: Chain net with Affine Body Dynamics (ABD). A-search handles the complex collisions between chains, while enabling the ball to bounce multiple times. Various frames, top to bottom, then left to right.
  • Figure 5: Bunny stair drop, first five frames showing bounce, then frames in regular $0.2s$ intervals starting from $1s$. A-search at three different energy profiles lead to a natural interpolation between energy conservation and close to implicit Euler dissipation.
  • ...and 14 more figures

Theorems & Definitions (1)

  • theorem 1