A novel phase-field monolithic scheme for brittle crack propagation based on the limited-memory BFGS method with adaptive mesh refinement

The phase-field formulation for fracture propagation is widely adopted due to its capability of naturally treating complex crack geometries. The challenges of the phase-field crack simulation include the non-convexity of the underlying energy functional and the expensive computational cost associated with the fine mesh required to resolve the phase-field length-scale around the crack region. We present a novel phase-field monolithic scheme based on the limited-memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, or the L-BFGS method, to address the convergence difficulties usually encountered by a Newton-based approach because of the non-convex energy functional. Comparing with the conventional BFGS method, the L-BFGS monolithic scheme avoids to store the fully dense Hessian approximation matrix. This feature is critical in the context of finite element simulations. To alleviate the expensive computational cost, we integrate the proposed L-BFGS monolithic scheme with an adaptive mesh refinement (AMR) technique. We provide the algorithmic details about the proposed L-BFGS monolithic scheme, especially about how to handle the hanging-node constraints generated during the AMR process as extra linear constraints. Several two-dimensional (2D) and three-dimensional (3D) numerical examples are provided to demonstrate the capabilities of the proposed monolithic scheme, including the accuracy, the robustness, and the computational efficiency regarding the memory consumption and the wall-clock time. Particularly, we emphasize the importance of the appropriately chosen convergence criteria for brute crack propagation. The proposed L-BFGS phase-field monolithic scheme combined with the AMR technique offers an accurate, robust, and efficient approach to model brittle crack propagation in both 2D and 3D problems.