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

TL;DR

The paper tackles convergence failures in variational phase-field brittle fracture models solved via alternating minimization by introducing an exact line search using a robust bisection strategy to minimize the energy along Newton directions. It proves global convergence for convex subproblems and demonstrates empirical robustness across multiple energy decompositions and irreversibility schemes, including challenging 3D scenarios like the Brazilian test. Through extensive numerical experiments, the approach shows improved reliability and competitive efficiency against common line-search methods, and its practicality is reinforced by public code availability. The work thus offers a minimally invasive, theoretically sound globalization technique that enhances the robustness and efficiency of phase-field fracture simulations in complex, multi-axial loading conditions.

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.

Figures (12)

• Figure 1: Oscillation of $\|\mathbf{R}_{\mathbf{u}}\|$ during Newton iterations for the nucleation test with the volumetric-deviatoric split (a), and behavior of $\|\mathbf{R}_{\mathbf{u}}\|_2 (\lambda)$ and $\phi(\lambda)$ between the previous solution iterate and the full Newton step for selected iterations (b). In (b), the value of $\lambda \in [0,1]$ at which the energy is minimized is highlighted with a vertical dashed line.
• Figure 2: Iterations of the bisection line search algorithm for the illustrative example of Fig. \ref{['fig:toyproblem_linesearch_sampling']} at Newton iteration $k=11$ where $\phi^\prime (0)\phi^\prime (1)<0$ (a), and at Newton iteration $k=12$ where $\phi^\prime (0)\phi^\prime (1)>0$ (b). For each bisection iteration, the iterate $\lambda_l$ is highlighted with a dot, while the bracket $[\lambda_l^{\text{left}}, \lambda_l^{\text{right}}]$ is represented by a segment in the upper plot.
• Figure 3: Staggered paths for various cases from the obstacle course at selected load steps. For each constraint enforcement method, we compare the iterative solution performance using exact line search only for the mechanical problem and for both subproblems.
• Figure 4: Crack-driving contribution of the strain energy density $\psi_{\text{D}}$ with negative values in significant parts of the domain, for (a) the plate with an inclined crack at load step $n=27$ with $\gamma^\star = 5$, and (b) the perforated plate at load step $n=35$ with $\gamma^\star = 5$. Both time steps are prior to crack nucleation. The plots include the isolines with $\psi_{\text{D}}=0$.
• Figure 5: Performance metrics for the nucleation test with the DP-like split (a), the plate with a hole with the star-convex split, $\gamma^\star=5$ (b), the plate with an inclined crack with the star-convex split, $\gamma^\star=1$ (c), and the perforated plate with the spectral split (d).

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Proposition 1
• Proposition 2
• Remark 4