A high-accuracy framework for phase-field fracture interface reconstructions with application to Stokes fluid-filled fracture surrounded by an elastic medium

TL;DR

The paper tackles the challenge of simulating hydraulically driven cracks embedded in an elastic solid by coupling a variational phase-field fracture model with sharp-interface reconstructions to enable precise interface-tracking. The authors predict the crack path with a phase-field approach and then reconstruct a sharp crack surface using level-set and explicit-mesh strategies, enabling a high-accuracy Eulerian description for subsequent Stokes flow inside the crack and stationary fluid-structure interaction via an ALE framework. They implement a unified algorithm in NGSolve that integrates the phase-field solver, interface reconstruction, mesh remeshing, and a monolithic stationary FSI solver, validating the method on Sneddon benchmarks and extended configurations including two orthogonal cracks. The results show linear convergence of crack-volume and COD with mesh refinement and demonstrate superior geometric and FSI accuracy with explicit-mesh reconstructions compared to unfitted level-set methods, highlighting the framework’s potential for complex multi-crack fracture problems. The work lays a foundation for iterative loops that couple Stokes pressure back into the phase-field update, paving the way for time-dependent, fully coupled simulations.

Abstract

This work considers a Stokes flow in a deformable fracture interacting with a linear elastic medium. To this end, we employ a phase-field model to approximate the crack dynamics. Phase-field methods belong to interface-capturing approaches in which the interface is only given by a smeared zone. For multi-domain problems, the accuracy of the coupling conditions is, however, of utmost importance. Here, interface-tracking methods are preferred, since the interface is resolved on mesh edges up to discretization errors, but it does not depend on the length scale parameter of some smeared zone. The key objective of this work is to construct a robust framework that computes first a crack path via the phase-field method (interface-capturing) and then does an interface-tracking reconstruction. We then discuss several approaches to reconstruct the Eulerian description of the open crack domain. This includes unfitted approaches where a level-set of the crack interface is constructed and an approach where the geometry is re-meshed. Using this reconstructed domain, we can compute the fluid-structure interaction problem between the fluid in the crack and the interacting solid. With the explicit mesh reconstruction of the two domains, we can then use an interface-tracking Arbitrary-Lagrangian-Eulerian (ALE) discretisation approach for the resulting fluid-structure interaction (FSI) problem. Our algorithmic procedure is realised in one final numerical algorithm and one implementation. We substantiate our approach using several numerical examples based on Sneddon's benchmark and corresponding extensions to Stokes fluid-filled regimes.

Figures (9)

• Figure 1: Level set and mesh construction approaches. Left: Explicit Level set construction based on a set of points; Center: Level-Set Transport from the blue to the green iso-surface; Right: Explicit mesh construction with elements inside the crack marked green.
• Figure 2: Example \ref{['sec_test_sneddon']}: Mesh convergence for Sneddon's test. Left: Total crack volume convergence computed from the phase-field integration, rational spline domain area and level-set domain areas respectively. Right: Crack opening displacement computed by integration and point evaluation of the phase-field.
• Figure 3: Example \ref{['sec_test_sneddon']}: Crack opening displacements at a number of points along the crack over the series of meshes considered.
• Figure 4: Example \ref{['sec_test_Stokes']}: Error convergence for the velocity and pressure for the Stokes problem computed on the domains resulting from Sneddon's test using the fitted and unfitted Eulerian domain approximations approaches.
• Figure 5: Example \ref{['sec_test_Stokes']}: Stokes velocity solution magnitude and exact ellipse (red) near the tip of the crack. Top: Mesh level 4, bottom: Mesh level 5. From left to right: Explicit level-set construction, level-set transport, rational spline construction with re-meshing.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3