Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme (1912.08620v1)

Abstract: We investigate the potential of quasi-Newton methods in facilitating convergence of monolithic solution schemes for phase field fracture modelling. Several paradigmatic boundary value problems are addressed, spanning the fields of quasi-static fracture, fatigue damage and dynamic cracking. The finite element results obtained reveal the robustness of quasi-Newton monolithic schemes, with convergence readily attained under both stable and unstable cracking conditions. Moreover, since the solution method is unconditionally stable, very significant computational gains are observed relative to the widely used staggered solution schemes. In addition, a new adaptive time increment scheme is presented to further reduces the computational cost while allowing to accurately resolve sudden changes in material behavior, such as unstable crack growth. Computation times can be reduced by several orders of magnitude, with the number of load increments required by the corresponding staggered solution being up to 3000 times higher. Quasi-Newton monolithic solution schemes can be a key enabler for large scale phase field fracture simulations. Implications are particularly relevant for the emerging field of phase field fatigue, as results show that staggered cycle-by-cycle calculations are prohibitive in mid or high cycle fatigue. The finite element codes are available to download from www.empaneda.com/codes.

• The paper introduces a novel monolithic phase field fracture model that employs BFGS quasi-Newton methods to improve convergence and cut computational costs.
• The study demonstrates that the adaptive step scheme accurately handles sudden changes in fracture behavior, drastically reducing the number of required increments.
• Numerical experiments confirm that the quasi-Newton approach outperforms traditional staggered methods by efficiently simulating quasi-static, fatigue, and dynamic fracture scenarios.

Analysis of Quasi-Newton Methods in Phase Field Fracture Modelling

The work presented by Kristensen and Martínez-Pañeda introduces an advanced computational framework for phase field fracture modelling utilizing quasi-Newton methods, specifically focusing on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. This paper addresses the inefficiencies and computational burdens associated with traditional staggered solution schemes in tackling phase field fracture problems.

Phase field fracture methods provide a powerful variational framework to address complex fracture features, treating fracture as an energy minimization problem. The approach has demonstrated its capability in capturing phenomena like crack branching and initiation without remeshing, relevant in applications such as hydrogen embrittlement, battery degradation, and composites.

The primary innovation in this work is the application of quasi-Newton methods to monolithic phase field fracture models, which traditionally suffer from convergence issues due to the non-convex nature of the potential energy functional. The authors overcome these challenges by implementing the BFGS algorithm, known for efficiently handling non-convex minimization problems. Their approach offers significant computational advantages, achieving convergence where conventional Newton methods fail and drastically reducing computational costs compared to staggered schemes.

Notably, the authors implement a new adaptive time-stepping scheme that allows the model to efficiently handle sudden changes in fracture behavior, such as unstable crack growth. This scheme reduces the computational load further, only resorting to small increments when necessary, showing potential computational time reductions by several orders of magnitude.

The paper conducts numerical experiments across various fracture scenarios: quasi-static fracture, phase field fatigue, and dynamic fracture propagation. Key findings include:

• Quasi-static Fracture: The monolithic quasi-Newton framework accurately captures unstable and stable crack growth, with computation times reduced by 100-fold in some cases compared to staggered methods. The approach resolves critical instability points with far fewer increments, which translates to a significant reduction in computational requirements.
• Phase Field Fatigue: Here, quasi-Newton methods prove particularly valuable, enabling cycle-by-cycle calculations that are otherwise prohibitive with staggered solutions. The accuracy and efficiency of fatigue predictions are maintained with as few as four increments per cycle, suggesting the potential expansion of phase field methods into high cycle fatigue analyses, previously inaccessible due to computational constraints.
• Dynamic Fracture: For dynamic crack branching scenarios, the monolithic scheme showcases its robustness in handling highly nonlinear problems, achieving faster convergence and maintaining the ability to simulate complex fracture patterns under dynamic loading conditions.

The implications of these findings extend to both theoretical and practical domains. Theoretically, this approach validates the utility of quasi-Newton methods in non-convex optimization problems within phase field modelling, challenging the traditional dominance of staggered schemes. Practically, it opens potential for application in large-scale fracture simulations, offering an efficient and robust tool for engineering problems requiring detailed fracture analysis, such as infrastructure fatigue and crack propagation in novel materials.

Future directions may explore further optimizations of the method, including its application in large deformation contexts and nearly incompressible materials. Additionally, expanding the framework to address various forms of material degradation could significantly enhance the applicability of phase field models in multi-physics scenarios. Overall, this research represents a significant step toward making large-scale phase field fracture simulations computationally feasible and effective.