A Fully Implicit Method for Robust Frictional Contact Handling in Elastic Rods

TL;DR

The paper addresses robust, high-fidelity simulation of frictional contact in slender elastic rods, exemplified by flagella bundling in viscous fluids. It introduces the Fully Implicit Contact Model (IMC), a penalty-based, energy-driven framework integrated with the Discrete Elastic Rods (DER) formulation, featuring a squared, piecewise contact energy and a smooth distance metric to enable aggressive line searches. Friction is handled via a smooth Coulomb-style law with a slipping tolerance, and the entire system is solved with a Newton-based method, yielding fast convergence and stable behavior compared to the state-of-the-art IPC while not guaranteeing strict non-penetration. The approach is demonstrated on flagella bundling scenarios, including frictional effects, with quantitative comparisons showing faster convergence and qualitative visuals illustrating sticking-slipping transitions, underscoring the method’s potential for soft robotics and bio-inspired dynamics in low Reynolds number environments. The work also provides an open-source code release for broader adoption and further development.

Abstract

Accurate frictional contact is critical in simulating the assembly of rod-like structures in the practical world, such as knots, hairs, flagella, and more. Due to their high geometric nonlinearity and elasticity, rod-on-rod contact remains a challenging problem tackled by researchers in both computational mechanics and computer graphics. Typically, frictional contact is regarded as constraints for the equations of motions of a system. Such constraints are often computed independently at every time step in a dynamic simulation, thus slowing down the simulation and possibly introducing numerical convergence issues. This paper proposes a fully implicit penalty-based frictional contact method, Implicit Contact Model (IMC), that efficiently and robustly captures accurate frictional contact responses. We showcase our algorithm's performance in achieving visually realistic results for the challenging and novel contact scenario of flagella bundling in fluid medium, a significant phenomenon in biology that motivates novel engineering applications in soft robotics. In addition to this, we offer a side-by-side comparison with Incremental Potential Contact (IPC), a state-of-the-art contact handling algorithm. We show that IMC possesses comparable performance to IPC while converging at a faster rate.

Figures (5)

• Figure 1: Rendered snapshots of flagella bundling with varying amounts of flagella. Rows contain (a) $M=2$, (b) $M=3$, (c) $M=5$, and (d) $M=10$ flagella. Each column indicates the flagella configuration at the moment of time indicated in the top row.
• Figure 2: (a) Discrete schematic of a flagellar elastic rod. Nodes $\mathbf x_0$, $\mathbf x_1$, and the edge between $\mathbf x_0$ and $\mathbf x_1$ is clamped along the dashed centerline and rotated with an angular velocity $\omega$. The rest of the nodes constitute the helical flagellum which revolves around the centerline. (b) A zoomed in snapshot of two edges showcasing their reference frame, material frame, turning angles, and twist angles. (c) Illustration of two edges approaching contact. The green dots showcase the nodes of the edges while the green dashed lines denote the centerlines of the edges. The red dashed line denotes the vector $\vec{\Delta}$ whose norm is the minimum distance $\Delta$ between the edges. $\vec{\Delta}$ is connected to edges $i$ and $j$ by $\mathbf c_i = \mathbf x_i + \beta_i (\mathbf x_{i+1} - \mathbf x_i)$ and $\mathbf c_j = \mathbf x_j + \beta_j (\mathbf x_{j+1} - \mathbf x_j)$ where $\beta_i, \beta_j \in [0, 1]$. As $\Delta$ approaches the contact threshold $2h$, repulsive forces increase at an exponential rate, thus enforcing non-penetration.
• Figure 3: Plots for the approximation functions in (a) Eq. \ref{['eq:contact_energy']} and (b) Eq. \ref{['eq:gamma']} with varying tolerance values. Note that some of the tolerances displayed are unrealistically large for clarity.
• Figure 4: Rendered snapshots for $M=5$ flagella simulated by (a) IMC and (b) IPC. We can observe that there is great qualitative agreement between both methods at the shown time steps. (c) A top down visualization of boundary conditions applied to the highest nodes (filled in red circles) of each flagella as well as the angular rotation $\omega$ applied to them. The larger hollow red circles represent the rest of the helical flagella. (d) The norm of the average difference in the nodal positions for the flagella simulated by IMC and IPC with respect to time.
• Figure 5: Rendered snapshots for $M=2$ with varying friction coefficients. Each column indicates a moment in time as indicated by the time stamp in the top row. The first row shows the frictionless case $\mu=0$ as a baseline. The second row has $\mu=0.3$ where minor sticking can be observed as the point at where the flagella no longer contact is higher than the frictionless case. Still, $\mu=0.3$ still has plenty of slipping allowing the flagella to not become coiled. As we increase $\mu$ to 0.7 in the third row, we can see the amount of sticking increase, ultimately resulting in the flagella becoming completing coiled.