Implicit frictional dynamics with soft constraints

TL;DR

The paper addresses the challenge of accurately simulating frictional contact in deformable objects by comparing lagged friction models with a fully implicit, smooth friction formulation. It introduces a penalty-based contact scheme with a smooth implicit surface and a volume-change penalty to handle compressible and nearly incompressible media, enabling stable, higher-order time integration and differentiable propagation of derivatives. Key contributions include a direct comparison showing lagged friction underperforms near the stick-slip threshold, a stable high-order integration framework applicable to both lagged and implicit friction, a Stribeck-like friction model, and an adaptive contact-penalty strategy that preserves volume. The work demonstrates improved friction accuracy, robustness to time stepping, and realistic replication of phenomena such as tennis-ball bounce and tire wrinkling, underscoring its practical impact for graphics and elasticity simulations.

Abstract

Dynamics simulation with frictional contacts is important for a wide range of applications, from cloth simulation to object manipulation. Recent methods using smoothed lagged friction forces have enabled robust and differentiable simulation of elastodynamics with friction. However, the resulting frictional behavior can be inaccurate and may not converge to analytic solutions. Here we evaluate the accuracy of lagged friction models in comparison with implicit frictional contact systems. We show that major inaccuracies near the stick-slip threshold in such systems are caused by lagging of friction forces rather than by smoothing the Coulomb friction curve. Furthermore, we demonstrate how systems involving implicit or lagged friction can be correctly used with higher-order time integration and highlight limitations in earlier attempts. We demonstrate how to exploit forward-mode automatic differentiation to simplify and, in some cases, improve the performance of the inexact Newton method. Finally, we show that other complex phenomena can also be simulated effectively while maintaining smoothness of the entire system. We extend our method to exhibit stick-slip frictional behavior and preserve volume on compressible and nearly-incompressible media using soft constraints.

Figures (11)

• Figure 1: An upside-down bowl is lifted using 3 soft pads via friction. The bowl is simulated using the lagged friction model from li20 (top row) and our fully implicit method (Eq. \ref{['eq:be_residual']}, bottom row). The chosen frames between 1 and 150, are frames 35, 40 and 100, at which point the bowl slips out for time steps $h = 0.005$ s, $0.0025$ s, and $0.00125$ s respectively for the lagged method. Using fully implicit integration, the bowl sticks even at the largest time step $h = 0.005$ s as shown. For the pads, $\rho = 1000$ kg/m${}^3$ (density), $E = 600$ KPa (Young's modulus), and $\nu = 0.49$ (Poisson ratio). For the bowl, $\rho = 400$ kg/m${}^3$, $E = 11000$ KPa and $\nu = 0.1$. The friction coefficient is set to $\mu = 0.65$ between the two. See Figure for the plot of the bowl height in all tested configurations.
• Figure 2: Components of the friction model.
• Figure 3: Volume change energy. The energy (negative of work) is plotted for the compressible model in Eq. \ref{['eq:ig_exact_volume_penalty']} (dotted curve), the nearly incompressible model in Eq. \ref{['eq:nif_exact_volume_penalty']} (dashed curve) and the 2nd order approximate model in Eq. \ref{['eq:approx_volume_penalty']} (solid curve). Here $V_0 = 1$ m${}^3$, $\kappa_v = 1$ atm${}^{-1}$, and $P_0 = 1$ atm. The quadratic model approximates both cases, but is ultimately too weak for excessive compression yet too strong during large expansion. Depending on the use case, it may be necessary to model one of Eqs. \ref{['eq:ig_exact_volume_penalty']} or \ref{['eq:nif_exact_volume_penalty']} directly.
• Figure 4: Block slide. Comparisons of analytic stopping conditions of a sliding block to numerical results.
• Figure 5: The height of the bowl in Figure is plotted against frame number as time step $h$ is varied. Here the bowl slips for the lagged friction method, whereas the fully implicit method maintains stable sticking for every time step, hence all plotted lines overlap.