Iterative convergence in phase-field brittle fracture computations: exact line search is all you need

2025-11-28

AI Summary

To address the divergence issues commonly encountered in alternating minimization of non-convex energy functionals within variational phase-field models for brittle fracture, this paper proposes a robust numerical framework incorporating exact line search. The method employs a bisection-based exact line search strategy to guarantee global convergence of Newton iterations in each subproblem; it further integrates strain-energy decomposition with an irreversibility constraint to enhance physical consistency. Evaluated on standard two- and three-dimensional benchmark problems, the proposed approach demonstrates superior convergence stability and computational robustness compared to conventional line-search techniques (e.g., Armijo’s rule), particularly under challenging conditions such as strong material nonlinearity and large load increments. The key contributions include: (i) a globally convergent, exact line search tailored for phase-field fracture solvers; (ii) a physically consistent treatment of energy splitting and irreversibility; and (iii) substantial improvements in reliability across a wide range of demanding simulation scenarios.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Variational phase-field models of brittle fracture pose a local constrained minimization problem of a non-convex energy functional. In the discrete setting, the problem is most often solved by alternate minimization, exploiting the separate convexity of the energy with respect to the two unknowns. This approach is theoretically guaranteed to converge, provided each of the individual subproblems is solved successfully. However, strong non-linearities of the energy functional may lead to failure of iterative convergence within one or both subproblems. In this paper, we propose an exact line search algorithm based on bisection, which (under certain conditions) guarantees global convergence of Newton's method for each subproblem and consequently the successful determination of critical points of the energy through the alternate minimization scheme. Through several benchmark tests computed with various strain energy decompositions and two strategies for the enforcement of the irreversibility constraint in two and three dimensions, we demonstrate the robustness of the approach and assess its efficiency in comparison with other commonly used line search algorithms.

Problem

• Ensures convergence in non-convex brittle fracture phase-field models.
• Solves subproblem divergence via exact line search in Newton's method.
• Validates robustness across strain decompositions and irreversibility constraints.

Innovation

• Exact line search algorithm using bisection
• Ensures global convergence of Newton's method
• Guarantees critical points via alternate minimization