The simulation of crack initiation and propagation in an elastic material is difficult, as crack paths with complex topologies have to be resolved. Phase-field approach allows to simulate crack behavior by circumventing the need to explicitly model crack paths. However, the underlying mathematical model gives rise to a non-convex constrained minimization problem. In this work, we propose a recursive multilevel trust region (RMTR) method to efficiently solve such a minimization problem. The RMTR method combines the global convergence property of the trust region method and the optimality of the multilevel method. The solution process is accelerated by employing level dependent objective functions, minimization of which provides correction to the original/fine-level problem. In the context of the phase-field fracture approach, it is challenging to design efficient level dependent objective functions as the underlying mathematical model relies on the mesh dependent parameters. We introduce level dependent objective functions that combine fine level description of the crack path with the coarse level discretization. The overall performance and the convergence properties of the proposed RMTR method are investigated by means of several numerical examples in three dimensions.

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Phase-field approaches to fracture are gaining popularity to compute a priori unknown crack paths. In this work the sensitivity of such phase-field approaches with respect to its model specific parameters, that is, the critical length of regularization, the degradation function and the mobility, is investigated. The susceptibility of the computed cracks to the setting of these parameters is studied for problems of linear and finite elasticity. Furthermore, the convergence properties of different solution strategies are analyzed. Monolithic and staggered solution schemes for the solution of the arising nonlinear discrete systems are studied in detail. To conclude, we demonstrate the versatility of the phase-field fracture approach in a real-world problem by comparing different simulations of conchoidal fracture using structured and unstructured meshes. K E Y W O R D Scrack driving force, crack propagation, multilevel methods, phase-field fracture INTRODUCTIONCrack growth is associated with the creation of new internal surfaces of unknown size and evolution. In the literature different possibilities for simulating crack propagation can be found. Besides the cohesive zone model, [1][2][3] the extended finite element methods [4,5] or eigenfracture strategies [6] which allow for modeling the crack as a sharp interface, diffuse interface approaches gain more and more attention. More precisely, the phase-field model is becoming very popular and constitutes a reliable way to predict the crack path. [7][8][9] During the last decades phase-field methods are used to describe a number of topics, that is, thermodynamics, [10] hydraulic fracture, that is, in porous media, [11][12][13][14] ductile materials, [15,16] anisotropy, [17][18][19][20] contact problems, [21] and many more. Most of the phase-field approaches are based on the minimization of the total potential energy that leads to a crack driving force corresponding to Griffith's energy release rate. [22] However, the approach requires a detailed consideration of the fact that only tensile states are responsible for crack growth. One popular way to take this into account is to decompose the strain energy function into a positive and negative part which correspond to the tensile and compressive parts, respectively. [8,9,23] This is not always physically meaningful. Thus, crack driving forces motivated by established fracture criteria have also been introduced. [24] Nevertheless, the phase-field model depends on a series of parameters which influence the results, like the location of crack initialization. In this work we will investigate the influences of these parameters in more detail. In particular, the phase-field This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Non-linear phase field models are increasingly used for the simulation of fracture propagation problems. The numerical simulation of fracture networks of realistic size requires the efficient parallel solution of large coupled non-linear systems. Although in principle efficient iterative multi-level methods for these types of problems are available, they are not widely used in practice due to the complexity of their parallel implementation. Here, we present Utopia, which is an open-source C++ library for parallel non-linear multilevel solution strategies. Utopia provides the advantages of high-level programming interfaces while at the same time a framework to access low-level data-structures without breaking code encapsulation. Complex numerical procedures can be expressed with few lines of code, and evaluated by different implementations, libraries, or computing hardware. In this paper, we investigate the parallel performance of our implementation of the recursive multilevel trust-region (RMTR) method based on the Utopia library. RMTR is a globally convergent multilevel solution strategy designed to solve non-convex constrained minimization problems. In particular, we solve pressure-induced phase-field fracture propagation in large and complex fracture networks. Solving such problems is deemed challenging even for a few fractures, however, here we are considering networks of realistic size with up to 1000 fractures.

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