Parameter unbounded Uzawa and penalty-splitted accelerated algorithms for frictionless contact problems

1. Problem Statement

The paper addresses the numerical solution of frictionless mechanical contact problems in solid mechanics, governed by Hertz-Signorini-Moreau conditions. These problems are inherently non-smooth and nonlinear due to inequality constraints (non-penetration and non-adhesion).

The core challenge lies in the trade-off between existing numerical formulations:

• Lagrange Multiplier (LM) Formulation: Offers high accuracy by satisfying contact conditions exactly but leads to a saddle-point system. Solving this requires handling indefinite matrices, which is computationally expensive and difficult to parallelize for large-scale problems.
• Penalty Method: Transforms the problem into an unconstrained one using a penalty parameter ($k_N$). While it avoids saddle-point systems, it introduces ill-conditioning for large $k_N$ and only provides approximate solutions.
• Iterative Splitting (Uzawa/Operator Splitting): These methods decouple the problem into a "solve step" (standard stiffness system) and an "update step" (dual variable update). While they avoid saddle-point matrices, they suffer from slow convergence and are highly sensitive to parameter choices (augmentation parameter $\rho$ for Uzawa, penalty parameter $k_N$ for penalty methods). Convergence is often guaranteed only within a narrow, theoretically restrictive range of parameters.

2. Methodology

The authors propose a unified iterative framework based on a two-step fixed-point algorithm, enhanced by a specific acceleration strategy.

A. Unified Iterative Framework

The algorithm decomposes the solution process at each iteration $i$:

1. Solve Step: Compute the primal variable (displacements $U^i$) by solving a standard linear system with the stiffness matrix $K$ (which remains constant):
   $$K U^i = F_{ext} - B^\top \lambda^{i-1}$$
   This avoids solving saddle-point or ill-conditioned penalized systems.
2. Update Step: Update the dual variable (contact forces $\lambda^i$) based on the displacement. The update function $G$ depends on the formulation:
   • Uzawa (LM): $G(\lambda, U) = \lambda + \rho(BU - D)$
   • Penalty: $G(\lambda, U) = k_N(BU - D)$
   • A projection step $\Pi_{\mathbb{R}^+}$ (max(0, x)) is applied to enforce non-negativity of contact forces.

B. Acceleration Strategy: Crossed-Secant (CS)

To overcome the slow convergence and parameter sensitivity of standard splitting, the authors integrate the Crossed-Secant (CS) acceleration method (a dynamic relaxation/Aitken-like scheme).

• Mechanism: Unlike standard momentum methods (like Anderson or FISTA) that rely on convex combinations, CS uses a vector secant approach to dynamically adjust the step size.
• Key Innovation: The authors apply the acceleration to the smooth operator $G$ before the projection step ($\Pi_{\mathbb{R}^+} \circ A \circ G$), rather than to the non-smooth composite function $F$. This preserves the intrinsic dynamics of the relaxation scheme.
• Theoretical Insight: The paper demonstrates that CS acts as a dynamic correction to the augmentation/penalty parameter. It effectively renders the iterative scheme parameter-unbounded, allowing convergence even when parameters are far outside the theoretical stability limits (e.g., extremely large penalty parameters or very small augmentation parameters).

C. Comparison with Other Accelerators

The study compares CS against:

• FISTA (with Adaptive Restart): A projected gradient method.
• Anderson Acceleration (AA-1): A secant-based method.
• Results: While FISTA and AA improve convergence within the "in-range" parameter limits, they often fail or diverge when parameters are pushed to extremes. CS remains robust across the entire spectrum.

3. Key Contributions

1. Parameter-Unbounded Convergence: The primary contribution is demonstrating that the Crossed-Secant acceleration allows the Uzawa and penalty-splitted algorithms to converge efficiently regardless of the parameter value. This removes the need for tedious, problem-specific parameter tuning.
2. First Computational Demonstration: This work provides the first evidence that penalty-based splitting strategies can achieve machine-precision accuracy (comparable to Lagrange multipliers) by using extremely large penalty parameters ($k_N \sim 10^{19}$), which were previously considered numerically infeasible due to ill-conditioning.
3. Unified Framework: The paper establishes a generic framework where the "solve" step always involves a standard, constant stiffness matrix, enabling the reuse of matrix factorizations and efficient initialization of iterative solvers.
4. Robustness to Divergence: The CS method is shown to handle oscillatory or divergent fixed-point iterations by actively correcting the step direction, effectively guiding the solution back to the fixed point.

4. Numerical Results

The methods were tested using the Cast3M finite element solver on three types of problems:

A. 3D Academic Benchmark (Hertzian Contact)

• Setup: A sphere indenting a block.
• Findings:
  • Standard Uzawa requires thousands of iterations for small $\rho$ and diverges for large $\rho$.
  • CS-accelerated Uzawa converges in ~130–270 iterations for $\rho \in [10^1, 10^{19}]$, with errors independent of $\rho$.
  • CS-accelerated Penalty achieves machine precision ($10^{-15}$ m gap) for $k_N \ge 10^{16}$, whereas standard penalty methods fail or yield poor accuracy.
  • Convergence order for CS was estimated at 1.41, outperforming FISTA (1.21) and Anderson (1.12).

B. 3D Industrial Case (Pellet-Cladding Interaction)

• Setup: Simulating thermal expansion of nuclear fuel pellets contacting cladding (bamboo-like deformation).
• Findings: The CS-accelerated methods successfully reproduced the complex "bamboo" shape and residual strains. The penalty-splitted approach with large $k_N$ matched the reference saddle-point solution perfectly, validating the ability to use high penalty parameters for industrial-grade accuracy.

C. Multi-Domain and Parallel Performance

• Setup: Increasing the number of contacting spheres (up to 20 domains) and processors.
• Findings:
  • Scalability: The standard saddle-point solver becomes computationally prohibitive as the number of contact domains increases. The proposed splitting methods scale much better.
  • Efficiency: The Uzawa-CS method reduced the total computational time by a factor of ~20 compared to the saddle-point solver. The penalty-CS method achieved a speed-up of 2–3.
  • Parallelism: Since the stiffness matrix $K$ is constant, the factorization is done once. Subsequent iterations are cheap. This makes the method highly suitable for parallel environments, provided the linear solver itself is parallelized.

5. Significance

This paper represents a significant advancement in computational contact mechanics:

• Practicality: It transforms slow, parameter-sensitive splitting algorithms into robust, "plug-and-play" solvers that do not require expert tuning.
• Accuracy vs. Cost: It bridges the gap between the high accuracy of Lagrange multipliers and the computational efficiency of penalty methods, allowing the latter to reach the former's precision without the associated conditioning issues.
• Large-Scale Simulation: By relying on standard stiffness matrices and enabling efficient reuse of factorizations, the proposed framework paves the way for large-scale, parallel simulations of complex multi-contact systems (e.g., in nuclear engineering, aerospace, and manufacturing) that were previously too expensive to solve with high fidelity.

In conclusion, the integration of the Crossed-Secant acceleration strategy effectively removes the "parameter bottleneck" of iterative contact algorithms, offering a unified, robust, and highly efficient solution for frictionless contact problems.