Robust numerical implementation of non-standard phase-field damage models for failure in solids Jian-Ying Wu
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State Key Laboratory of Subtropical Building Science, South China University of Technology, 510641 Guangzhou, China.
Abstract
Recently several non-standard phase-field models different from the standard one for brittle fracture (Bourdin
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et al., 2000, 2008; Miehe et al., 2010a) have been proposed for the modeling of damage and fracture in solids. On the one hand, linear elastic behavior before the onset of damage and even cohesive cracking induced quasi-brittle
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failure can be considered by such non-standard phase-field models. On the other hand, they present great challenges to the numerical implementation since the damage boundedness is no longer automatically fulfilled. In this work, the numerical implementation of the unified phase-field damage model (Wu, 2017, 2018) is addressed, though it also applies to the standard and other non-standard ones. In particular, an iterative alternate minimization (AM)
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algorithm enhanced with path-following strategies is presented. The phase-field or damage sub-problem is solved by the bound-constrained optimization solver in which the boundedness and irreversibility conditions of the crack phasefield are exactly enforced. Moreover, material softening induced snap-backs typically for localized failure in solids
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can also be effectively dealt with. The equilibrium paths in terms of the fracture surface and the indirect displacement (e.g., the crack mouth opening or sliding displacement) are discussed in the context of the AM algorithm. The AM
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algorithm with the indirect displacement control is advocated, since both prescribed external forces and displacements can be naturally dealt with, which is more versatile than the one with the fracture surface control applicable only for
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prescribed displacements. Representative examples of benchmark tests show that the AM algorithm enhanced with the indirect displacement control is extremely robust even for large increment sizes. The insensitivity of the unified phase-field damage model to the incorporated length scale and mesh size is also confirmed. Keywords:
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Phase-field theory; gradient-damage model; alternate minimization; arc-length method; quasi-brittle failure.
1. Introduction Cracking induced localized failure in solids is usually regarded as prognostics of catastrophic collapse of engineering structures. Therefore, it is of vital importance to predict the occurrence of localized failure and quantify its effects Tel.:
(+86) 20-87112787 Email address:
[email protected] (Jian-Ying Wu)
Preprint submitted to Computer Methods in Applied Mechanics and Engineering
June 6, 2018
on overall structural responses. Ever since the milestone work of Griffith (1920), linear elastic fracture mechanics (LEFM) (Irwin, 1957) and nonlinear one – cohesive zone model (CZM) (Barenblatt, 1959; Dugdale, 1960), have been extensively studied. Almost at the same time, the continuum damage mechanics pioneered by Kachanov (1958) was also established and has been popular since 1980s. Correspondingly to fracture mechanics and continuum damage mechanics, in the computational community discontinuous (or discrete) (Ngo and Scordelis, 1967) and continuous
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(or smeared) (Rashid, 1968) approaches were proposed and eventually evolve into the so-called computational failure
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(fracture/damage) mechanics.
Fracture mechanics mainly deals with behavior of a single (or multiple) predefined crack. However, both the LEFM and CZM themselves are not self-contained. Additional criteria have to be introduced to determine when
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and where a crack nucleates, how much it propagates and in which orientation. Moreover, a branching criterion is also required for dynamic fracture. As the crack has to be represented explicitly, it is indispensable in the numerical implementation to correctly track the crack propagation path. Nevertheless, it is rather cumbersome, if not impossible,
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to accomplish this task, particularly for non-smooth crack surfaces with branching and merging in 3-D cases (Fert´e et al., 2016). Comparatively, continuum damage mechanics focuses on characterizing the stress versus strain softening behavior of cracking solids. The effects of crack propagation are phenomenologically accounted for by the damage variable with an evolution law. It is necessary neither to track the crack path with extra criteria nor to represent the
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crack surface explicitly. However, owing to the lack of a well-defined length scale and to the loss of ellipticity in the governing equation, continuum damage mechanics based local material models cannot be used straightforwardly in the numerical context; otherwise, discretization dependent spurious results would be unavoidable (Baˇzant and Planas,
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1997).
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Regarding the above facts, some regularized approaches have been advocated during the last two decades, trying to reconcile fracture and damage mechanics (or computationally, discontinuous and continuous approaches). In the
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context of fracture mechanics, a celebrated work was put forward by Francfort and Marigo (1998) who proposed the variational approach to fracture. This approach seeks crack evolution by minimizing the total energy of cracking solids, extending classical Griffith’s theory for brittle fracture (Griffith, 1920). Crack nucleation from an initially sound solid can be considered and arbitrary crack paths can be retrieved without a priori assumption on the crack
topology. Based on this variational approach and motivated by Ambrosio and Tortorelli (1990) regularization of Mumford and Shah (1989) functional for image segmentation, Bourdin et al. (2000, 2008) set forth a numerically
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more amenable formulation – the phase-field model for brittle fracture. An auxiliary variable, say, the crack phase field d.x/ 2 Œ0; 1 with a smooth transition from the intact solid (i.e., d D 0) to the crack surface (i.e., d D 1), as well
as its gradient rd , is introduced to regularize the sharp crack topology by a diffuse damage band with a small but finite length scale. As the length scale parameter approaches to zero, the solution to the original problem is recovered in sense of the
-convergence theorem (Braides, 1998). The above phase-field model was reformulated by Miehe
et al. (2010b) within the framework of thermodynamics. In parallel, similar work was done in the context of continuum damage mechanics with, nevertheless, aiming 2
mainly to suppress the mesh (size and bias) dependence notoriously for local damage models. Among several alternatives, gradient-damage models deserve further comments due to their resemblance to the phase-field model. The gradient-damage model is dated back to Fr´emond and Nedjar (1996) who proposed an extended principle of virtual power in which the damage gradient is incorporated to account for microscopic nonlocal interactions; see also Nedjar (2001). At the same time, Pijaudier-Cabot and Burlion (1996) proposed a similar damage model to analyze
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localization in elastic solids with voids. In particular, besides the displacement field, an extra damage field with an
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evolution law of gradient-type was introduced to characterize the irreversible variation of the volume fraction of material in porous continua. Later on, Lorentz and Andrieux (1999) and Lorentz and Godard (2011) developed a similar gradient-damage model based on an energetic interpretation of the balance equation and of the constitutive relations
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for generalized standard continua. Based on the variational approach to fracture (Francfort and Marigo, 1998), Pham et al. (2011) proposed a gradient-damage model using the optimality and stability conditions. In order to cast the gradient-damage model into the framework of thermodynamics, Polizzotto and Borino (1998) postulated an insula-
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tion condition on the so-called nonlocal energy residual, assuming that no long distance energy is allowed to transfer from the localization band to exterior domains; see Liebe et al. (2001) and Polizzotto (2003) for the details. However, it seems that these gradient-damage models were developed mainly for regularizing their local counterparts rather than for the modeling of fracture. This argument can be seen from the fact that the fracture energy does not enter
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the formulation from the very beginning, but rather, it is later identified heuristically from the 1-D analytical solution. Moreover, the notion of approximating the sharp crack topology by the phase (or damage) field does not appear at all. In both phase-field and gradient-damage models, the governing equations consist of the conventional equilibrium
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equation coupled with an extra evolution law of gradient type for the crack phase-field (or damage). Though it is very
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subtle, the in-between difference lies in their respective interpretations on the incorporated length scale. In phase-field models, the length scale is regarded as a numerical parameter and its values should be taken as small as possible -convergence to the original problem can be guaranteed. However, phase-field models yield numerical
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such that the
results heavily sensitive to the value of the length scale parameter (Miehe et al., 2010b). Consequently, it is generally assumed that, rather than a numerical parameter, the incorporated length scale should be a material property and fixed for a specific problem as in gradient-damage models. If this point of view is accepted, the difference between both models then diminishes. In this work, they are referred to as phase-field damage models. As clarified in Wu (2017, 2018), the phase-field damage model is characterized by two characteristic functions in
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terms of the phase-field variable d , i.e., a geometric crack function ˛.d / approximating the sharp crack surface and an energetic degradation function !.d / defining the free energy potential deteriorated by the diffuse crack (Wu, 2017,
2018). The standard phase-field damage model adopts a quadratic geometric crack function ˛.d / D d 2 . The benefit
is that the admissible range d 2 Œ0; 1 can be intrinsically guaranteed for the resulting exponential distribution of the phase-field (Miehe et al., 2010b). However, this choice virtually regularizes the sharp crack topology into a diffuse one of infinite support. As a consequence, the phase-field evolution equation has to be solved in the whole computational domain to determine the phase-field value at each point (Mo¨es et al., 2011). Moreover, damage initiates at the every 3
beginning and the linear elastic behavior prior to the onset of damage cannot be easily dealt with. Comparatively, in non-standard phase-field damage models the geometric crack function can be quite different. For instance, Pham et al. (2011) proposed using the geometric crack function ˛.d / D d , resulting in a diffuse localization band with parabolic damage distribution and global responses with initially linear stages. The side effect is that the crack phase-field cannot be intrinsically guaranteed in the interval [0, 1] in this case. Consequently, the phase-field boundedness has to
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be explicitly enforced, bringing about extra difficulties to the numerical implementation of non-standard models.
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The energetic degradation function also plays an important role in phase-field damage models since it characterizes
energy transfers between the phase field and the displacement field. Though various expressions have been proposed in the literature, e.g., Kuhn et al. (2015) and the references therein, the common feature is that almost all of the
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energetic degradation functions are expressed solely in terms of the crack phase field (or damage) and independently of the incorporated length scale. As there is no additional energy dissipation other than crack propagation, the strain energy stored in the bulk releases abruptly and brittle fracture occurs in classical Griffith’s theory (Griffith, 1920).
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In the community of mathematics, Conti et al. (2015) constructed a particular energetic degradation function which is proportional to the incorporated length scale and proved the
-convergence of the resulting phase-field model to
the CZM in the sense of Barenblatt’s theory (Barenblatt, 1959). That is, the fracture energy function is no longer a constant as in Griffith’s theory for brittle fracture, but rather, it depends on the crack opening. However, the numerical
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examples presented later in Focardi and Iurlano (2017) show that the global responses are also heavily sensitive to the length scale parameter.
The numerical implementation of phase-field damage models generally relies on the multi-field finite element
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method in which the nodal unknowns consist of both the displacement and phase field (or damage) degrees of freedom
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(dofs). Two- and three-dimensions can be dealt with straightforwardly in an identical manner. However, a great challenge arises when solving the resulting discrete governing equations. That is, the monolithic algorithm using the
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standard Newton-Raphson iterative scheme performs poorly since the underlying energy functional is not convex with respect to (w.r.t.) the total unknowns (i.e., both the displacement and damage fields). Several attempts, such as the modified Newton scheme (Heister et al., 2015; Wick, 2017), the non-conventional line search method (Gerasimov and De Lorenzis, 2016), the global arc-length method (Singh et al., 2015; May et al., 2016), etc., have been proposed to
enhance robustness of the monolithic algorithm. However, these techniques are not always effective. Comparatively, as the energy functional to be minimized is quadratic and convex w.r.t. the displacement and phase-field separately, the
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discrete governing equations can be solved by fixing the damage and displacement alternately. The resulting alternate minimization (AM) (Bourdin et al., 2000, 2008) or staggered algorithm is very robust. The slow convergence rate can be accelerated by using the over-relaxed strategy and the composite staggered-monolithic algorithm (Farrell and Maurini, 2017) though the numerical implementation is rather cumbersome. Enforcement of the boundedness and irreversibility on the phase-field variable presents another challenge in the numerical implementation of phase-field damage models. Bourdin et al. (2000) considered the later condition only for fully developed cracks, which is not a satisfactory solution for cohesive fracture or quasi-brittle failure. Amor et al. 4
(2009) suggested using the bound-constrained optimization solver included in the Optimization Toolbox of Matlab to solve the damage sub-problem. In order to prevent cracks from healing, Miehe et al. (2010a) proposed using a nondecreasing local history field which represents the maximum free energy potential ever reached. Note that in damage models the similar method has already been a standard strategy in defining the damage threshold since the work of Sim´o and Ju (1987). Owing to its simplicity, this strategy has been widely adopted in phase-field models to deal with
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the crack irreversibility condition. However, as the boundedness d 2 Œ0; 1 cannot be automatically guaranteed, it
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does not apply to non-standard phase-field damage models with a geometric crack function other than the quadratic one.
Recently, the author (Wu, 2017) proposed a unified phase-field model with rather generic geometric and energetic
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degradation functions. The classical one for brittle fracture can be regarded as one of its particular examples, and quasi-brittle failure can be dealt with owing to an elegant energetic degradation function. For the latter, the maximum stress and the resulting global responses are independent of the incorporated length scale, as long as the sharp crack
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surface and the non-uniform phase field can be sufficiently resolved. Later on, Wu (2018) proposed an equivalent gradient-damage model using only the standard thermodynamics with internal variables. In particular, the constitutive relation of local damage models still applies, while the damage evolution law emerges naturally from the energetic equivalence between the sharp crack and the geometrically regularized counterpart. The gap between fracture me-
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chanics and continuum damage mechanics (Planas et al., 1993; Mazars and Pijaudier-Cabot, 1996) is largely bridged. In this work, the numerical implementation of the above unified phase-field damage model is addressed as a particular example of non-standard models, though it also applies to the standard one. Specifically, the AM algorithm
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(Bourdin et al., 2000; Amor et al., 2009) is considered and its applicability to non-standard models is verified. The
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boundedness and irreversibility conditions on the damage field are enforced exactly by the bound-constrained optimization solver incorporated in the PETSc toolkit (Balay et al., 2016) to solve the damage sub-problem. In order to
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further improve the robustness in dealing with material softening induced snap-backs, two path-following strategies, i.e., the classical indirect displacement control (de Borst, 1987) and the more recent fracture surface control (Singh et al., 2015), are discussed. Their respective pros and cons in the context of phase-field damage models are identified. The remainder of this paper is structured as follows. Section 2 introduces the unified phase-field damage model
proposed by the author (Wu, 2017, 2018; Feng and Wu, 2018). Both isotropic and hybrid isotropic/anisotropic formu-
lations are considered. Section 3 is devoted to its numerical implementation in the multi-field finite element method.
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The AM algorithm enhanced with path-following strategies in solving the discrete governing equations is discussed in detail. Numerical performances of this solver are investigated in Section 4 regarding the hybrid phase-field model for
cohesive fracture. Several mode-I and mixed-mode benchmark tests of both brittle fracture and quasi-brittle failure are numerically studied. The most relevant conclusions are drawn in Section 5. For the content to be self-contained, the CZM approximated by the unified phase-field damage model is given in Appendix A, and the necessary conditions of the involved model parameters are discussed in Appendix B, closing this paper. Notation. Compact tensor notation is used in the theoretical part of this paper. As general rules, scalars are denoted 5
by italic light-face Greek or Latin letters (e.g. a or ); vectors, second- and fourth-order tensors are signified by italic boldface minuscule, majuscule and blackboard-bold majuscule characters like a, A and A, respectively. The inner products with single and double contractions are denoted by ‘’ and ‘:’, respectively. The Voigt notation of vectors and second-order tensors are denoted by boldface minuscule and majuscule letters like a and A, respectively.
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2. The unified phase-field damage theory
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In this section, the unified phase-field damage theory (Wu, 2017, 2018) is recalled. It extends the phase-field model for brittle fracture (Bourdin et al., 2000, 2008; Miehe et al., 2010b; Pham et al., 2011) to incorporate generic geometric and energetic degradation functions such that quasi-brittle failure can be considered as well.
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As shown in Figure 1 (left), the reference configuration of a solid ˝ Rndim (ndim D 1; 2; 3) subjected to
distributed body forces b is considered. The solid is kinematically characterized by the displacement field u.x/ and
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(infinitesimal) strain field .x/ WD r sym u.x/, for the symmetric gradient operator r sym ./ with respect to the spatial coordinate x. The external boundary @˝, with the outward unit normal denoted by vector n, is split into two disjointed parts @˝u and @˝ t , i.e., @˝u \ @˝ t D ; and @˝u [ @˝ t D @˝, on which given displacements u .x/ for x 2 @˝u
and tractions t .x/ for x 2 @˝ t are applied, respectively. Accordingly, the admissible space for the displacement is
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defined as usual, i.e.,
n ˇ o n Uu WD uˇu.x/ 2 H 1 .˝/ dim ; u.x/ D u 8x 2 @˝u
(2.1)
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where ŒH 1 .˝/ndim denotes the Sobolev space of vector fields defined in the solid ˝ Rndim . Once a specific failure criterion is fulfilled, a sharp crack set S is initiated in the solid ˝. In the context of phase-
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field damage models, the sharp crack S is smeared over a localization band B ˝ in which the diffuse damage field d.x/ localizes, with the exterior domain ˝nB being intact; see also Figure 1 (right) for illustration. The admissible
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space for the damage is a convex cone imposing the unilateral box constraint, i.e., n ˇ o Ud WD d ˇd.x/ 2 H 1 .˝/; d.x/ 2 Œ0; 1 and dP .x/ 0 8x 2 B
(2.2)
where the second constraint dP 0 accounts for the damage irreversibility, with .P/ being the time derivative. Dirichlet boundary conditions, e.g., d.x/ D 0 for elastic domains and d.x/ D 1 for pre-defined cracks can be considered as
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well.
2.1. Constitutive relations and damage evolution law .; d /, the stress–strain relation reads (Wu, 2017, 2018)
For a cracking solid with the local free energy density WD
@ .; d / @
(2.3a)
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for the stress tensor conjugate to the strain field . The specific expression of the free energy
will be given later.
The associated damage energy release rate (driving force) Y is expressed as Y WD
@ @d
! 0 .d /Yx
.; d / D
@ Yx WD @!
with
(2.3b)
where the effective damage driving force Yx is conjugate to an appropriate energetic (monotonically non-increasing)
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function !.d / W Œ0; 1 ! Œ1; 0 of the damage variable d , with ! 0 .d / 0 being the first derivative.
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In most of the existing phase-field damage models the damage evolution law is usually derived from variational or thermodynamical arguments (Bourdin et al., 2000; Miehe et al., 2010a; Wu, 2017). Alternatively, Wu (2018) proposed using the following postulate of energetic equivalence between the dissipation rates due to evolution of the
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regularized crack
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localization band B and of the original sharp crack S, i.e. Z Z DP D Y dP dV D Gf
.d P / dV Gf APS 0 „ ƒ‚ … B B „ ƒ‚ … sharp crack
for the fracture energy (material property) Gf and the crack surface density functional .d / Z 1p ˇ ˇ2 1 1 with c0 D 4 ˛.ˇ/ dˇ ˛.d / C b ˇrd ˇ
.d / D c0 b 0
(2.4)
(2.5)
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where the geometric crack function ˛.d / 2 Œ0; 1, satisfying the properties ˛.d / D 0 for d D 0 and ˛.d / D 1 for
d D 1 (Wu, 2017), determines the ultimate damage distribution; b is a length scale characterizing the localization bandwidth.
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Applying the divergence theorem to the postulate of energetic equivalence (2.4) and calling for the dissipative
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nature of damage evolution (i.e., dP 0), yield the following damage criterion (Wu, 2018) 8 ˆ dP > 0 2 0, the initial slope k0 WD @=@w
0:5f t2 =Gf and the ultimate crack opening
wc > 0 of a target traction versus separation law .w/; see Appendix A. For the convexity of the energetic degradation
should satisfy (see Appendix B) b
8 lch 0:85lch 3
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function !.d / (and consequently, of the free energy function (2.12) and the modified one (2.15)), the length scale b
(2.20)
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for Griffith’s or Irwin’s characteristic length lch WD E0 Gf =f t2 of the material (400 mm for concrete and 15 mm for
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brittle PMMA). As it is necessary to consider b lch in order to sufficiently resolve the phase-field regularization (2.9), this is an (almost useless) upper bound of the length scale b.
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Note that the parameter a1 and the resulting energetic degradation function !.d / depend on the length scale b for a constant failure strength f t . As shown in Figure 2, as the length scale b becomes smaller, the energetic degradation function !.d / decreases more rapidly. This property allows defining an equivalent CZM in the 1-D case characterized by the stress versus apparent separation relation .w/ across the localization band B; see Wu (2017) for the details. Consequently, the global responses are insensitive to the length scale b, as long as the sharp crack surface AS is sufficiently approximated by the regularized one Ad . The above fact is in sharp contrast to other phase-field damage
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models which give numerical results heavily dependent on the length scale parameter b.
Remark 2.5 In the context of gradient-damage models (Fr´emond and Nedjar, 1996; Lorentz and Andrieux, 1999),
similar functions were considered in Pham et al. (2011); Lorentz and Godard (2011); Lorentz et al. (2012); Lorentz (2017). However, in all of these references the geometric crack function is adopted as ˛.d / D d , which is a particular
case of the current one (2.18)1 with D 1. More importantly, as clarified in Wu (2017), the geometric crack function
˛.d / D 2d d 2 is optimal for cohesive cracking induced brittle fracture and quasi-brittle failure, since it automatically guarantees a non-decreasing but finite localization bandwidth for almost all those softening laws of practical use; see 10
Appendix A. For instance, the linear softening law, which has been widely adopted, can be considered as well; see the numerical example presented in Section 4.5. Comparatively, this is not the case for other choices, since otherwise the resulting localization band would shrink and the damage irreversibility condition could not be enforced. It is this reason that some strict conditions have to be enforced on the model parameters (Lorentz and Godard, 2011; Lorentz et al., 2012; Lorentz, 2017). Regarding the energetic degradation function, it is Lorentz and Godard (2011) that first d /2 in the standard phase-field damage model
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used a rational function, rather than the quadratic polynomial .1
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(Bourdin et al., 2000, 2008; Miehe et al., 2010a). Motived from this work, the author (Wu, 2017, 2018) proposed the generic expression (2.18)2 which incorporates all those in the aforesaid references as its particular case with p D 2.
As explicitly stated in Wu (2017), the parameter p > 2 results in a positive and infinite ultimate crack opening which
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is necessary for the exponential and hyperbolic softening laws. It is also in the same paper that the involved model parameters (2.19) were given in terms of the standard properties of the material, rather than from data fitting. More
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importantly, almost no extra condition is enforced on the model parameters; see Appendix B for the details. 2.5. Governing equations in weak form
are expressed as 8 ˆ ˆ ˆ ıa > Kuu Kud Ku ˆ = < > = ˆ < > 6 7ˆ 6 du 7 6K Kd d 0 7 ı aN D rd > > ˆ 4 5ˆ ˆ > ˆ > > ˆ ˆ > Ku Kd K :ı; :r ;
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(3.12)
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where the tangent matrices are given by Z Z @ @ N Kuu D BT BdV; Kud D BT NdV; Ku D @ @d ˝ B ! Z Z x du T 0 @Y dd N N T ! 00 Yx C ˛ 00 1 Gf N N C K D N ! BdV; K D N @ c0 b B B @r ; @a
A
Ku D
Kd D
@r ; @Na
K D
@ru D @
Ofext
2b N T N Gf B B dV c0
@r @
(3.13a) (3.13b) (3.13c)
Standard Gauss quadrature rules are used to evaluate the integrals. For the linear constraints (3.8) imposed on the displacement and damage dofs, it follows that 8 9 2 3 8 9 ˆ ˆ > ˆ > LT Kuu L LT Kud LN LT Ku C Kuu aO ˆ ıaf > LT ru > ˆ > ˆ > 6 7< = < = 6 N T du 7 T dd T du T d 6L K L LN K LN 7 ı aN f D LN r LN K aO > > 4 5ˆ ˆ > ˆ ˆ > ˆ : ı > ; ˆ : r > ; Ku L Kd LN K C Ku aO 16
(3.14)
from which the unknowns .af ; aN f ; / can be solved. If the monolithic algorithm worked, the final solution would converge rapidly. However, as the underlying energy functional is non-convex w.r.t. both unknowns .u; d /, the monolithic solver performs poorly for (3.12) and (3.14). A few attempts have been proposed to deal with this issue. For instance, Gerasimov and De Lorenzis (2016) developed a non-conventional line search technique to improve convergence of the monolithic solver; Heister et al. (2015)
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proposed a technique, among other things, to convexify the energy functional, which was further improved in Wick
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(2017) using a modified Newton scheme; Singh et al. (2015) suggested using the fracture surface based arc-length
method and the adaptive time stepping scheme to enhance the robustness. However, these strategies are not always effective and their applicability to non-standard phase-field damage models needs further investigation.
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3.5. Alternate minimization algorithm
Though it is not convex w.r.t. to both unknowns .u; d /, the underlying energy functional is indeed convex w.r.t.
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u and d separately if the other variable is fixed (Bourdin et al., 2000, 2008). Accordingly, the discrete governing equations (3.5) can be solved by fixing the damage dofs aN or the displacement dofs a alternately, until the final solution converges slowly.
In this work, the resulting alternate minimization (AM) or staggered algorithm is adopted. On the one hand, the
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displacement sub-problem is solved using the standard Newton-Raphson method. On the other hand, the phase field or damage sub-problem, subjected to the boundedness and irreversibility conditions, 0 aNA;n aNA;nC1 1, is dealt
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with by the bound-constrained optimization solver included in the PETSc toolkit (Balay et al., 2016). The phase field ˇ .kC1/ ˇ .k/ ˇ stop criterion (Bourdin et al., 2000), i.e., ˇaN nC1 aN nC1 < for a small positive number D 1:0 10 5 between two consecutive iterations, is adopted in the numerical implementation, though other criteria (Ambati et al., 2015) can
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be considered as well. The resulting algorithm is extremely robust, particularly when it is enhanced with appropriate path-following strategies. Moreover, as now the tangent matrices involved in both sub-problems are symmetric and
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positive-definite, those efficient iterative solvers, e.g., conjugate gradient method, etc., can be used. The above facts are particular useful for large-scale simulations. 3.5.1. Fracture surface control
Singh et al. (2015); May et al. (2016) applied the energy release or dissipation control (Guti´errez, 2004; Verhoosel
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et al., 2009) to the standard phase-field model, resulting in the so-called fracture surface control. In this scheme, the path-following constraint is expressed in terms of the regularized crack surface (2.9) r .Na/ D l
Ad D l C Ad;n
Ad .Na/ D 0
(3.15)
It then follows that u
K
D 0;
Kd D
Z B
1 1 N T 0 N ˛ C 2b BN T rd dV; c0 b 17
K D 0
(3.16)
Note that the above fracture surface based path-following constraint is independent of the unknowns .a; /. As can be seen, neither the governing equation (3.5b) nor the fracture surface based constraint (3.15) depends explicitly on the force vector fext . Accordingly, for the case of prescribed external forces (i.e, Ofext ¤ 0 and aO D 0), Eq. (3.14) becomes 9 2 38 9 8 T uu T ud N T u ˆ T u> > ˆ ˆ > ˆ L K L L K L L K ıa L r > > 6 7ˆ < f> = ˆ < = 6 N T du 7 T dd T d N N N 6L K L L K L 0 7 ı aN f D L r > ˆ > 4 5ˆ ˆ > ˆ > ˆ > ˆ d N > : ; : 0 K L 0 ı r ;
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(3.17)
In this case, it is impossible to solve .a; / while fixing the remaining unknowns aN or solve .Na; / with the nodal unknowns .a; aN ; /, which is, however, not robust enough as we know.
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displacements a fixed. Only the monolithic algorithm discussed in Section 3.4 can be considered to solve all the For prescribed external nodal displacements (i.e., Ofext D 0 and aO ¤ 0), the following AM algorithm is considered:
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(3.14) degenerate to 9 38 9 8 2 N LN T Kdu aO
