A Splitting Architecture for Exact Reduced Coulomb Friction

Technical Summary: A Splitting Architecture for Exact Reduced Coulomb Friction

Problem Statement
The paper addresses the numerical challenge of simulating frictional contact dynamics for rigid bodies using the exact reduced Coulomb friction law. While the exact law (formulated by De Saxcé and Feng [DSF98]) accurately captures macroscopic behaviors like stable stacking, arch support, and clean stick-slip transitions, it presents significant computational difficulties. Specifically, the law introduces an "augmented velocity" that couples the tangential sliding velocity into the normal direction. This coupling renders the friction law non-associated, meaning the problem cannot be formulated as a single energy minimization or a convex optimization problem.

Existing approaches typically handle this by:

1. Smoothing: Replacing the non-smooth law with a regularized force, introducing a stiffness-accuracy trade-off.
2. Convex Relaxation: Dropping the non-associated coupling term to solve a Cone Complementarity Problem (CCP). This is efficient but produces visible artifacts, such as contacts separating at the cone boundary during sustained sliding.
3. Monolithic Nonsmooth Newton: Solving the full non-smooth system directly. While exact, these methods are difficult to globalize, parallelize, or modularize.

The authors argue that the exact reduced law possesses a natural structural split that has been underutilized in computer graphics solvers: a cone-constrained linear response coupled with a scalar, non-associated term.

Methodology
The authors propose a splitting architecture that treats the exact reduced Coulomb law by decomposing it into two distinct operators:

1. Operator A (Cone-Constrained Linear Response): A strongly convex Second-Order Cone Program (SOCP) representing the linear contact response constrained by the friction cone.
2. Operator B (Scalar Coupling): The non-associated De Saxcé and Feng coupling term, which is cheap to evaluate per contact but globally non-monotone.

The solver employs a Tseng-style Forward-Backward-Forward (FBF) outer iteration scheme:

• Outer Loop: Explicitly evaluates the non-associated coupling $B(\lambda)$ at the current iterate.
• Inner Solve: Solves a strongly convex cone-constrained QP (Operator A) with the coupling term frozen. This inner problem is well-conditioned regardless of the rank of the Delassus operator due to proximal regularization.
• Correction: Applies a lightweight explicit correction to account for the change in the coupling term between the current and intermediate iterates.

Key Contributions

1. Splitting Solver Architecture: The paper presents a solver that adopts the cone-QP/scalar-coupling decomposition identified by Acary et al. [ACLM11] but replaces their Picard outer iteration with a Tseng-style FBF scheme. This structure makes the inner cone solve modular, allowing different convex SOCP solvers (e.g., block Gauss-Seidel, ADMM, interior-point) to be substituted without altering the outer loop.
2. Safeguarded Adaptive Step-Size Rule: Because the De Saxcé coupling is non-monotone, standard convergence theorems for Tseng's method do not directly apply. The authors introduce a step-size control mechanism combining a Lipschitz-based estimate with a local-variation acceptance test and an asymmetric growth policy tuned to the stick-slip boundary.
3. Matrix-Free, Contact-Parallel Implementation: Implemented on top of NVIDIA Warp and the Newton physics engine, the solver avoids forming dense matrices. It utilizes matrix-free operators for the Delassus matrix and parallelizes coupling evaluation and correction over contacts.

Results
The method was evaluated on rigid-body benchmarks designed to stress static support, frictional transmission, and mode transitions:

• Cube on Incline & Backspin Ball: The solver reproduces the exact analytical transition from sliding to sticking and correctly handles the normal force generation from tangential backspin. In contrast, the convex-relaxed MuJoCo solver exhibited spurious normal separation (drifting) and failed to maintain contact.
• Turntable & Painlevé Box: The method correctly captures the transition between capture and ejection under centripetal loading and the slide-to-tumble transition, matching the behavior of the industrial-strength Kamino solver.
• House of Cards & Masonry Arch: In highly coupled static scenarios, the splitting solver maintained structural integrity and local frictional support. MuJoCo failed to maintain the standing configuration (drifting apart), while the proposed method preserved the equilibrium. Notably, the solver successfully balanced a 101-stone arch where Kamino failed under the same setup.
• Performance: On contact-rich benchmarks, the method was competitive with Kamino and an order of magnitude faster than MuJoCo. The inner cone solve scales linearly with contact count and body adjacency.

Significance and Claims
The paper claims that this splitting architecture successfully reproduces the exact reduced Coulomb law without introducing smoothing parameters or convex relaxations. By separating the non-associated coupling from the cone geometry, the method achieves:

• Physical Accuracy: It eliminates the "normal-lifting" artifacts common in convex-relaxed solvers, ensuring that stable stacks and arches remain stable.
• Modularity: The inner solver is decoupled from the outer iteration, allowing hardware-specific or problem-size-specific solvers to be used efficiently.
• Convergence: While the non-monotone nature of the coupling prevents a formal global convergence rate guarantee, the method demonstrates empirical convergence to the single-precision floor on all tested benchmarks.

The authors acknowledge that the explicit treatment of the non-associated coupling can lose contractivity in globally coupled systems (like large arches), requiring more outer iterations. However, they assert that the method remains robust and accurate where other exact solvers struggle, offering a viable path for high-fidelity frictional simulation in computer graphics.