Original articlesA phase field approach to pressurized fractures using discontinuous Galerkin methods
Introduction
In recent years new technologies like enhanced geothermal energy induced a risen interest in numerical simulations of subsurface fracture processes. Thereby special attention must be paid to the formation and evolution of fractures and fracture networks. Taking cracks into account in numerical simulations gives rise to a big challenge. Evolving fractures can form complicated, even changing topologies. Not only can triple points and tips occur, cracks can also join. Classical finite element methods need an exact representation of the crack surface by the mesh to include these cracks into the simulation. In the quasi static case this leads to expensive remeshing at every time step, once the crack propagates. To overcome this difficulty new techniques that avoid remeshing are developed. One possible approach is to enrich the function space locally to take the crack into account. The most well known variant of this approach is the eXtended Finite Element Method (X-FEM) introduced by Moës et al. in [27]; it has many similarities to the Partition of Unity method [2]. The X-FEM method does not require the mesh to resolve the crack, instead new basis functions are introduced in elements cut by a crack. These methods have been successfully applied to crack propagation and for a recent overview we refer to [18], [17]. That method is very sensitive to the choice of enrichment functions, in particular the type of enrichment employed at the tip, as the stresses are singular at the tip. In particular a correct approximation of crack speed and crack direction is challenging [5]. Also the treatment of topological changes, like branching and merging of cracks is not straight forward and requires additional treatment [12].
These difficulties can be avoided by describing the crack implicitly. An alternative approach to simulate crack propagation was introduced by Francfort and Marigo [16]. It uses the idea of an implicit crack description and by this can easily handle topological changes and avoid problems arising from the crack tip modeling. Starting from Griffith’s criterion [20] crack propagation is formulated as an energy minimization problem. The resulting energy strongly resembles the Mumford–Shah Functional from image segmentation [28]. For numerical simulations this functional was regularized by Ambrosio and Tortorelli [1] using the concept of -convergence.
The main idea is to introduce a phase field that regularizes the crack. As the regularization parameter converges to 0, the phase field converges in the sense of -limits to the sharp crack. Thereby one gains an implicit representation of the crack. This does not only avoid the problem of remeshing, but can also handle topological changes implicitly. Neither the forming of triple points, the joining of cracks nor the determination of propagation velocity and direction need any extra attention. The works of Francfort et al. [16] and Bourdin et al. [9] only treat the case, where fractures are driven by linear elasticity. Mikelić et al. [25] then extended their approach by including pressure driven propagation.
This paper is based on the model proposed by Mikelić et al. [25]. Section 2 will introduce the energy formulation, starting with Griffith’s criterion. Afterwards the total energy is derived. Section 3 gives a proof of -convergence in a one-dimensional setting, which is to our knowledge the first proof for the phase field model for fluid filled fractures. In the following Section 4 a discontinuous Galerkin (DG) discretization of the phase field model is derived. In the solver we employ a split approach, similar to [26]. In contrast to them we use a DG formulation for the displacement to take into account displacement jumps along the crack surface. Inequality constraints on the phase field are handled using a non-smooth multigrid solver. Finally in Section 5 we present some numerical experiments followed by a conclusion in Section 6.
Section snippets
Griffith’s criterion
Following the principle of minimal total potential energy the final equilibrium state of a material is broken, if it can be reached by a process only decreasing the total potential energy of the system. According to A. Griffith [20] this energy consists of two parts. First the potential energy of the bulk in some domain , , depending on the displacement and the crack . Second some surface energy that describes the necessary work against cohesive forces between the molecules where
-convergence result
In the following we prove -convergence in the one dimensional setting on the interval , following the proof of Braides [11, Theorem 8.1]. We hope that the proof carries over to two and three dimensions applying slicing properties as in [13], [14], but this is still work in progress.
Let us first recall the definition of -limits [11]. Definition 3.1 A function is said to be the -limit of a sequence if for all holds for every sequence converging to
Discretization
Using the principle of free variations for (5) we derive the Euler–Lagrange equations for our problem.
Similar to Mikelić et al. [26] we use a split approach to discretize (18), (19). Inside of a fixed-point iteration we solve for the phase field and the displacement in turns. In the following the two discretizations are described in more detail. To ensure sufficient resolution
Numerical examples
To show the capability of the method to cover a wide range of crack propagation problems, it is applied to four different examples. First we will show an example with pure mode I loading and zero pressure. For the chosen configuration the time when propagation starts is given analytically. We will show convergence of our solution to this point with respect to and . Next, two classical benchmark examples are shown and compared to existing literature, namely the single edge notched tension
Conclusion
In this paper the phase field model for fluid filled, pressure driven crack propagation by [25] was adapted to a discontinuous Galerkin setting. Thus continuity requirements along a crack were weakened.
The usage of the phase field regularization in the presence of additional pressure terms in the bulk energy was justified by a proof of the -convergence in the one-dimensional setting. Numerical examples confirm the suitability of our formulation. In two test cases good agreement with an
Acknowledgment
The authors would like to thank Caterina Zeppieri for fruitful discussions regarding the -convergence proof.
References (30)
- et al.
An unfitted discontinuous galerkin method for pore-scale simulations of solute transport
Math. Comput. Simul.
(2011) - et al.
A phase-field description of dynamic brittle fracture
Comput. Methods Appl. Mech. Engrg.
(2012) - et al.
Numerical experiments in revisited brittle fracture
J. Mech. Phys. Solids
(2000) An approximation result for special functions with bounded deformation
J. Math. Pures Appl. (9)
(2004)Addendum to: “An approximation result for special functions with bounded deformation” [J. Math. Pures Appl. (9) 83 (2004), no. 7, 929–954; mr2074682]
J. Math. Pures Appl. (9)
(2005)- et al.
Revisiting brittle fracture as an energy minimization problem
J. Mech. Phys. Solids
(1998) - et al.
A primal–dual active set method and predictor–corrector mesh adaptivity for computing fracture propagation using a phase-field approach
Comput. Methods Appl. Mech. Engrg.
(2015) - et al.
A continuum phase field model for fracture
Eng. Fract. Mech.
(2010) - et al.
A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits
Comput. Methods Appl. Mech. Engrg.
(2010) - et al.
An augmented-Lagrangian method for the phase-field approach for pressurized fractures
Comput. Methods Appl. Mech. Engrg.
(2014)
Approximation of functionals depending on jumps by elliptic functionals via -convergence
Comm. Pure Appl. Math.
The partition of unity method
Internat. J. Numer. Methods Engrg.
A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE
Computing
Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment
Internat. J. Numer. Methods Engrg.
Theory of elasticity and consolidation for a porous anisotropic solid
J. Appl. Phys.
Cited by (12)
Opening-dependent phase field model of hydraulic fracture evolution in porous medium under seepage-stress coupling
2024, Theoretical and Applied Fracture MechanicsA modified combined active-set Newton method for solving phase-field fracture into the monolithic limit
2023, Computer Methods in Applied Mechanics and EngineeringOn the relation of Gamma-convergence parameters for pressure-driven quasi-static phase-field fracture
2022, Examples and CounterexamplesPhase-field implicit material point method with the convected particle domain interpolation for brittle–ductile failure transition in geomaterials involving finite deformation
2022, Computer Methods in Applied Mechanics and EngineeringCitation Excerpt :The second approach can be competent to treat ductile fracture problems in both metals and geomaterials [48,63,64], especially in considering the brittle–ductile failure transition in geomaterials [21,27]. Meanwhile, a great deal of significant numerical methods has been developed to the phase-field fracture modeling, for instance, the finite element method (FEM) [41,44,68], the extended finite elements method (XFEM) [69], the virtual element method (VEM) [70], the discontinuous Galerkin method (DGM) [71], the material point method (MPM) [42,46,72,73] and the combined finite element-finite volume method (FE-FV) [74], etc. Some effective remeshing techniques [75,76] can be also incorporated into these methods to further improve their capacities for capturing the discontinuities in solids.
IPACS: Integrated Phase-Field Advanced Crack Propagation Simulator. An adaptive, parallel, physics-based-discretization phase-field framework for fracture propagation in porous media
2020, Computer Methods in Applied Mechanics and EngineeringA DG/CR discretization for the variational phase-field approach to fracture
2023, Computational Mechanics