Fast solution of a phase-field model of pitting corrosion

Excessive computational times represent a major challenge in the solution of corrosion models, limiting their practical applicability, e.g., as a support to predictive maintenance. In this paper, we propose an efficient strategy for solving a phase-field model for metal corrosion. Based on the Kronecker structure of the diffusion matrix in classical finite difference approximations on rectangular domains, time-stepping IMEX methods are efficiently solved in matrix form. However, when the domain is non-rectangular, the lack of the Kronecker structure prevents the direct use of the matrix-based approach. To address this issue, we reformulate the problem on an extended rectangular domain and introduce suitable iterative IMEX methods. The convergence of the iterations and the propagation of the numerical errors are analyzed. Test cases on two and three dimensional domains show that the proposed approach achieves accuracy comparable to existing methods, while significantly reducing the computational time, to the point of allowing actual predictions on standard workstations.

💡 Research Summary

This paper presents a groundbreaking computational strategy to dramatically accelerate the numerical solution of a phase-field model for pitting corrosion, addressing a major bottleneck that has limited its practical use in predictive maintenance.

The core challenge lies in the model’s extreme computational cost. The phase-field model, coupling an Allen-Cahn equation for the corrosion interface with a Cahn-Hilliard equation for ion concentration, is highly accurate but suffers from severe stiffness, requiring impractically long simulation times on standard hardware. Previous state-of-the-art methods, including adaptive mesh techniques, were still too slow for real-time prediction.

The authors’ innovation is a “matrix-oriented” approach that exploits inherent mathematical structures for efficiency. For rectangular domains, they recognize that the finite-difference discretization matrix of the Laplacian operator possesses a Kronecker sum structure. By applying Implicit-Explicit (IMEX) time-integration methods, the large linear system to be solved at each time step can be reformulated as a Sylvester matrix equation (of the form AX + XB = F). This reformulation allows the use of highly efficient, direct solvers like the Bartels-Stewart algorithm, reducing computational complexity from O(N^3) to roughly O(N^2) for N unknowns.

To handle realistic, non-rectangular domains (e.g., samples with holes), the paper introduces a novel “extended domain” strategy. The original complex domain is embedded within a minimal bounding rectangle. A mask function identifies active nodes within the physical domain, allowing the problem to be recast on the full rectangular grid where the efficient Kronecker structure is restored. This introduces nonlinearities, which are addressed by designing new iterative IMEX schemes. The convergence of these iterations and the propagation of numerical errors are rigorously analyzed, ensuring the method’s reliability.

Comprehensive numerical experiments in 2D and 3D demonstrate the method’s transformative impact. Compared to the most efficient existing solvers, the proposed approach achieves the same level of accuracy while reducing computation time by orders of magnitude. Crucially, a simulation of a 225-second corrosion process that previously took hours on a workstation can now be completed in mere minutes. This breakthrough transitions the model from a research tool to a practical technology, enabling true predictive capability on standard computers. The work thus bridges a critical gap between advanced corrosion modeling and industrial application, paving the way for cost-effective predictive maintenance and enhanced structural integrity management.