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Fast solution of a phase-field model of pitting corrosion

Gianluca Frasca-Caccia, Dajana Conte, Beatrice Paternoster

TL;DR

This work tackles the prohibitive computational cost of simulating a two-equation phase-field model of pitting corrosion by exploiting the Kronecker structure of the diffusion operator on rectangular domains and recasting implicit diffusion steps as Sylvester equations solved via a matrix-oriented approach. For nonrectangular geometries with holes, the authors extend the domain to a rectangle using an indicator mechanism and develop iterative, matrix-efficient IMEX schemes with rigorous stability and error analyses. Across 2D and 3D benchmarks, including domains with holes and irregular boundaries, the proposed methods achieve comparable accuracy to existing solvers but reduce wall-clock times by orders of magnitude, enabling practical predictive maintenance workflows on standard workstations. The results demonstrate strong performance of both first- and second-order IMEX schemes, with the second-order iterative variants offering the best overall efficiency, and they establish a solid theoretical foundation for convergence and error control in both Dirichlet and Neumann settings. The methodology shows promise for broader application to other stiff phase-field models and coupled electro-chemo-mechanical problems.

Abstract

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.

Fast solution of a phase-field model of pitting corrosion

TL;DR

This work tackles the prohibitive computational cost of simulating a two-equation phase-field model of pitting corrosion by exploiting the Kronecker structure of the diffusion operator on rectangular domains and recasting implicit diffusion steps as Sylvester equations solved via a matrix-oriented approach. For nonrectangular geometries with holes, the authors extend the domain to a rectangle using an indicator mechanism and develop iterative, matrix-efficient IMEX schemes with rigorous stability and error analyses. Across 2D and 3D benchmarks, including domains with holes and irregular boundaries, the proposed methods achieve comparable accuracy to existing solvers but reduce wall-clock times by orders of magnitude, enabling practical predictive maintenance workflows on standard workstations. The results demonstrate strong performance of both first- and second-order IMEX schemes, with the second-order iterative variants offering the best overall efficiency, and they establish a solid theoretical foundation for convergence and error control in both Dirichlet and Neumann settings. The methodology shows promise for broader application to other stiff phase-field models and coupled electro-chemo-mechanical problems.

Abstract

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.

Paper Structure

This paper contains 30 sections, 20 theorems, 204 equations, 12 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The problem
  3. Matrix-oriented approach on rectangular domains
  4. Space discretization
  5. Time integration
  6. First-order accuracy
  7. Second-order accuracy
  8. Efficient solution of Sylvester problems
  9. Implementation Sketch
  10. The 3D scenario
  11. IMEX methods on domains with holes
  12. Space discretization
  13. Time integration
  14. First-order accuracy
  15. Second-order accuracy
  16. ...and 15 more sections

Key Result

Theorem 1

Any matrix $\mathcal{P}$ in the form matP is invertible.

Figures (12)

  • Figure 1: Stencil configurations for discrete Laplacian. The shaded region belongs to set $\Theta$, the white region to $\widehat{\Omega}$. (a): Stencil with center in $\widehat{\Omega}$ and points in $\Theta$. (b): Stencil with center in $\Theta$ and points in $\widehat{\Omega}$.
  • Figure 2: Pencil electrode test. Configuration of $c$ given by IMEX Euler with $\Delta x=\Delta y=1\mu\text{m},$ and $\Delta t=10^{-3}\text{s}$ at different times.
  • Figure 3: Pencil electrode test. Position of the lower corrosion front in the solution computed by IMEX Euler with $\Delta x=\Delta y=1\mu\text{m},$ and $\Delta t=10^{-3}\text{s}$ at different times.
  • Figure 4: Pencil electrode test. Effect of the spatial grid (left) and of the time step (right) on the computational times of matrix-oriented IMEX methods. We have set $\Delta t = 10^{-3}$s for IMEX Euler and $\Delta t = 10^{-2}$s for IMEX 2SBDF. In all cases, $h=10^{-6}$m and the final time is $T=225$s.
  • Figure 5: Circular pit growth. Configuration of $c$ given by iterative IMEX-E Euler with $\Delta x=\Delta y=1\mu\text{m},$ and $\Delta t=2\cdot 10^{-3}\text{s}$ at different times.
  • ...and 7 more figures

Theorems & Definitions (46)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 36 more