Figure 4 - uploaded by Carlos Armando Duarte
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Illustrates the different levels of local elements used to accurately integrate out the local enrichment functions for use in the enriched global problem.
Source publication
This paper presents a two-scale extension of the generalized finite element method (GFEM) which allows for static fracture analyses as well as fatigue crack propagation simulations on fixed, coarse hexa-hedral meshes. The approach is based on the use of specifically-tailored enrichment functions computed on-the-fly through the use of a fine-scale b...
Context in source publication
Context 1
... to that which is presented in great detail in [33], the enriched global shape functions are integrated using the integration points from the local elements. In this instance, further clarification may be beneficial, since multiple levels of local integration rules are required, as shown in Figure 4. In the far left of the figure, the global hexahedral element has been subdivided into tetrahedral elements, and uniformly refined. ...
Citations
... This is because many elements that are not in the high gradient region and do not require refinement will be refined thereby resulting in serious oscillation in the solution [10][11][12]. Thus, using local refinement that selectively refines only the elements within the high gradient region seems to be a better choice [13][14][15][16]. The adaptive finite element method [9,[17][18][19] has the potential to overcome this problem and produce accurate results without refining the mesh of the whole domain. ...
This article considered the traditional finite element method (FEM) and adaptive finite element method (FEM) for the numerical solution of the one-dimensional boundary value problems. We established the preference or the superiority of the h-adaptive FEM to traditional FEM in high gradient problems in terms of accuracy and cost of computation. Numerical examples which confirm the performance and adaptability of the h-adaptive method over the traditional finite element method and the high accuracy of the numerical solution are presented. Detailed error analysis of linear elements was also discussed. In conclusion, h-adaptive FEM is recommended for complex systems with high gradient problems.
... [10]) or are based on so-called enrichment functions, which then often cause conditioning problems, in order to model propagating cracks (see e.g. [24]). The latter is referred to as extended or generalized finite element method. ...
Within this contribution, we present a diffuse interface approach for the simulation of crack nucleation and growth in materials, which incorporates an orientation dependency of the fracture toughness. After outlining the basic motivation for the model from an engineering standpoint, the phase field paradigm for fracture is introduced. Further, a specific phase field model for brittle fracture is reviewed, where we focus on the meaning of the auxiliary parameter differentiating between material phases and the coupling of such a parameter to continuum equations in order to obtain the characteristic self organizing model properties. This specific model, as will be explained, provides the phenomenological and methodical basis for the presented enhancement. The formulation of an appropriate evolution equation in terms of a Ginzburg-Landau type equation will be highlighted and several comments on sharp interface models will be made to present a brief comparison. Following up on the basics we then introduce the formulation of a modified version of the model, which additionally to the handling of cracks in linear elastic materials under quasi static loading is also capable of taking into account the effect of resistance variation with respect to the potential crack extension direction. The strong and also the weak forms of the respective governing equations corresponding to the developed anisotropic phase field model are presented. Utilizing the weak formulation as starting point for the discretization of the two fields (displacement field and the phase field), the computational framework in terms of finite elements is introduced. We finally explain several test cases investigated within simulations and discuss the corresponding numerical results. Besides examples, which are set up to illustrate the general model properties, a comparison with crack paths obtained by experimental investigations will be presented in order to show the potential of the developed phase field model.
... Accordingly, a large number of numerical approaches to fatigue incorporate the phenomenological laws from above. For instance, in [20,29], Paris' law is used to simulate fatigue crack growth with finite elements and generalized finite elements, respectively. The law is also incorporated in a widely used tool for fatigue crack growth simulations, namely NASGRO, [15]. ...
Within this work, we utilize the framework of phase field modeling for fracture in order to handle a very crucial issue in terms of designing technical structures, namely the phenomenon of fatigue crack growth. So far, phase field fracture models were applied to a number of problems in the field of fracture mechanics and were proven to yield reliable results even for complex crack problems. For crack growth due to cyclic fatigue, our basic approach considers an additional energy contribution entering the regularized energy density function accounting for crack driving forces associated with fatigue damage. With other words, the crack surface energy is not solely in competition with the time-dependent elastic strain energy but also with a contribution consisting of accumulated energies, which enables crack extension even for small maximum loads. The load time function applied to a certain structure has an essential effect on its fatigue life. Besides the pure magnitude of a certain load cycle, it is highly decisive at which point of the fatigue life a certain load cycle is applied. Furthermore, the level of the mean load has a significant effect. We show that the model developed within this study is able to predict realistic fatigue crack growth behavior in terms of accurate growth rates and also to account for mean stress effects and different stress ratios. These are important properties that must be treated accurately in order to yield an accurate model for arbitrary load sequences, where various amplitude loading occurs.
... Approaches to simulate crack and rupture initiation and propagations, e.g. Generalized FEM (GFEM) [48], Ex-tended FEM (XFEM) [49,50], or Zencrack [51], usually involve mesh refinement or trial function enrichment. Nevertheless, in the present FPM, as a result of the discontinuous trial and test functions, the algorithm for simulating crack propagation is relatively simple and intuitive, and does not involve either mesh refinement or trial function enrichment. ...
Flexoelectricity refers to a phenomenon which involves a coupling of the mechanical strain gradient and electric polarization. In this study, a meshless Fragile Points Method (FPM), is presented for analyzing flexoelectric effects in dielectric solids. Local, simple, polynomial and discontinuous trial and test functions are generated with the help of a local meshless Differential Quadrature approximation of derivatives. Both primal and mixed FPM are developed, based on two alternate flexoelectric theories, with or without the electric gradient effect and Maxwell stress. In the present primal as well as mixed FPM, only the displacements and electric potential are retained as explicit unknown variables at each internal Fragile Point in the final algebraic equations. Thus the number of unknowns in the final system of algebraic equations is kept to be absolutely minimal. An algorithm for simulating crack initiation and propagation using the present FPM is presented, with classic stress-based criterion as well as a Bonding-Energy-Rate(BER)-based criterion for crack development. The present primal and mixed FPM approaches represent clear advantages as compared to the current methods for computational flexoelectric analyses, using primal as well as mixed Finite Element Methods, Element Free Galerkin (EFG) Methods, Meshless Local Petrov Galerkin (MLPG) Methods, and Isogeometric Analysis (IGA) Methods, because of the following new features: they are simpler Galerkin meshless methods using polynomial trial and test functions; minimal DoFs per Point make it very user-friendly; arbitrary polygonal subdomains make it flexible for modeling complex geometries; the numerical integration of the primal as well as mixed FPM weak forms is trivially simple; and FPM can be easily employed in crack development simulations without remeshing or trial function enhancement.
... Global-local enrichment functions are numerically constructed from the solution of local boundary value problems, using boundary conditions from a global problem defined on a coarse mesh. This numerical strategy is identified here as GFEM gl , and was recently expanded for the interaction and coalescence of multiple crack surfaces [28], fatigue crack simulations [29] and evaluation of stress intensity factors at spot welds [30]. ...
In this paper, the technique of the Stable Generalized Finite Element Method (SGFEM) is applied to the numerically constructed functions of the Generalized Finite Element Method with Global-Local Enrichments (GFEMgl). The application of the resulting approach, named SGFEMgl, is expanded here to 2-D quasi-static crack propagation problems. Crack growth is performed by a two-scale strategy, using local problems generated at each propagation step – whose solutions enrich a single global problem defined on a coarse mesh. Stress Intensity Factors (SIFs) computed along crack growth, strain energy measures, performance in blending elements and the condition number are used to study the accuracy and conditioning of SGFEMgl. The method is compared with the standard GFEMgl. Numerical experiments demonstrate remarkable accuracy of SGFEMgl in linear elastic fracture mechanics problems, considering crack opening modes I and II. Convergence rates analyses also show the superiority of the method, especially with the use of geometrical enrichments.
... scales (e.g., coarse and fine) are generally introduced, and fine and coarse scale problems were solved using various computational methods, e.g., finite element method, generalized/extended finite element method, discrete element method, molecular dynamics, virtual element method, etc. [13][14][15][16][17]. When both fine and coarse scale problems are solved with finite element method, this approach is also called as FE 2 method [18,19]. ...
In this study, a computational framework is proposed to investigate multiscale dynamic fracture phenomena in materials with microstructures. The micro‐ and macro‐scales of a composite material are integrated by introducing an adaptive microstructure representation. Then, the far and local fields are simultaneously computed using the equation of motion, which satisfies the boundary conditions between the two fields. Cohesive surface elements are dynamically inserted where and when needed, and the Park‐Paulino‐Roesler (PPR) cohesive model is employed to approximate nonlinear fracture processes in a local field. A topology‐based data structure (TopS) is utilized to efficiently handle adjacency information during mesh modification events. The efficiency and validity of the proposed computational framework are demonstrated by checking the energy balances and comparing the results of the proposed computation with direct computations. Furthermore, the effects of microstructural properties, such as interfacial bonding strength and unit cell arrangement, on the dynamic fracture behavior are investigated. The computational results demonstrate that local crack patterns depend on the combination of microstructural properties such as unit cell arrangement and interfacial bonding strength; therefore, the microstructure of a material should be carefully considered for dynamic cohesive fracture investigations.
... In such cases, a multi-scale analysis can be used under the G/XFEM approach, as proposed by Duarte and Kim (2008). Recent applications of the global-local modelling method to G/XFEM can be found in OHara et al. (2016a), OHara et al. (2016b), Plews and Duarte (2016), Li and Duarte (2018), Gerasimov et al. (2018) and Geelen et al. (2020). Another way of using both fine and coarse-scale meshes is considering a local mesh refinement associated with variable-node elements for the transition from fine to coarse-scale mesh. ...
Purpose
The purpose of this paper is to evaluate some numerical integration strategies used in generalized (G)/extended finite element method (XFEM) to solve linear elastic fracture mechanics problems. A range of parameters are here analyzed, evidencing how the numerical integration error and the computational efficiency are improved when particularities from these examples are properly considered.
Design/methodology/approach
Numerical integration strategies were implemented in an existing computational environment that provides a finite element method and G/XFEM tools. The main parameters of the analysis are considered and the performance using such strategies is compared with standard integration results.
Findings
Known numerical integration strategies suitable for fracture mechanics analysis are studied and implemented. Results from different crack configurations are presented and discussed, highlighting the necessity of alternative integration techniques for problems with singularities and/or discontinuities.
Originality/value
This study presents a variety of fracture mechanics examples solved by G/XFEM in which the use of standard numerical integration with Gauss quadratures results in loss of precision. It is discussed the behaviour of subdivision of elements and mapping of integration points strategies for a range of meshes and cracks geometries, also featuring distorted elements and how they affect strain energy and stress intensity factors evaluation for both strategies.
... In XFEM/GFEM, a level set technique with enrichment functions is used to represent the crack in the domain, hence avoiding the requirement of remeshing during the crack propagation. This method has since been extended for interfacial crack (Sukumar et al., 2004;Pathak et al., 2013a;Kumar et al., 2015b;Hu et al., 2016), fatigue crack growth Singh et al., 2012;Pathak et al., 2015b;Hara et al., 2016a;Pant and Bhattacharya, 2017), elasto-plastic crack growth (Elguedj et al., 2006;Kumar et al., 2014;Kumar et al., 2015c;Kumar et al., 2016), three dimensional crack growth (Areias and Belytschko, 2005;Rabczuk et al., 2010;Pathak et al., 2013b, Pathak et al., 2013c, dynamic crack growth (Zi et al., 2005;Réthoré et al., 2005;Kumar et al., 2015d) fatigue crack growth in functionally graded materials (Singh et al., 2011;Bhattacharya and Sharma, 2014) and interaction of multiple cracks (Hara et al., 2016b). Despite its success in many types of problems, there exist some limitations: (1) it introduces an error during the mapping of discontinuities from the physical space to the natural space (Fries and Belytschko, 2010); (2) the implementation in FEM can be complicated as blending elements are generally required for connecting the enriched elements to standard elements; (3) the numerical solution is sensitive to the numerical integration scheme used for the enriched elements (Rabczuk, 2013); and (4) different enrichment functions are usually required to tackle different material problems. ...
The Floating Node Method (FNM), first developed for modeling the fracture behavior of laminate composites, is here combined with a domain-based interaction integral approach for the generic fracture modeling of quasi-brittle materials from crack nucleation, propagation to final failure. In this framework, FNM is used to represent the kinematics of cracks, crack tips and material interfaces in the mesh. The values of stress intensity factor are obtained from the FNM solution using domain-based interaction integral approach. To demonstrate the accuracy and effectiveness of the proposed method, four benchmark examples of fracture mechanics are considered. Predictions obtained with the current numerical framework compare well against literature/theoretical results.
Within this thesis, a diffuse interface approach for the simulation of crack nucleation and
growth in brittle materials is enhanced to enlarge its scope. Incorporating an additional
energy contribution in the total energy functional of an existing phase field model enables
the handling of fatigue crack growth. In another approach, anisotropic brittle fracture is
studied by a modification of the spatial gradient in the regularized crack energy.
At first, the necessary basics from different fields of mechanics and also for the phase field
method, a method to simulate phase transformations by means of an auxiliary parameter,
are provided. Subsequently, the work gives an outline of the basic motivation for the performed modifications. It is described how the new features are incorporated. Appropriate
evolution equations in terms of Ginzburg-Landau type equations for the new phase field
models are then derived in a detailed way.
The coupled systems of the phase field models are implemented in a nonlinear finite element
formulation, where implicit time integration is employed. The strong and also the weak
forms of the governing equations corresponding to the developed models are presented.
Several test cases are simulated in order to investigate the accuracy of the developed models. These studies illustrate, that the novel phase field frameworks are able to predict
cracking in media that reveal an anisotropic fracture toughness. Furthermore, the fatigue
model accurately predicts important quantities like the crack growth speed. Also phenomena like mean stress or sequence effects are fully covered by the formulation. Simulation
results from both phase field models are compared with experimental findings and reveal
good agreement.
