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This paper is devoted to the introduction of a new variant of the extended finite element method (Xfem) for the approximation of elastostatic fracture problems. This variant consists in a reduced basis strategy for the definition of the crack tip enrichment. It is particularly adapted when the asymptotic crack-tip displacement is complex or even un...
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... Sur le même principe, la méthode G-FEM (Generalized Finite Element Method) [Strouboulis et al., 2000] utilise des solutions analytiques, comme des fonctions handbook dans [Strouboulis et al., 2003], des développements asymptotiques [Brancherie et al., 2008] ou des solutions pré-calculées [Chahine et al., 2008, Chahine et al., 2009 en lieu et place des fonctions H et (F j ). ...
Cette thèse vise à proposer des outils innovants pour le calcul de structures aéronautiques évoluant à haute température. En effet, les régimes de fonctionnement des moteurs actuels conduisent à des évolutions élasto-viscoplastiques généralisées dans les pièces métalliques et l’utilisation de modèles simplifiés (élastiques) n’est plus totalement satisfaisante en terme de précision, même en phase de préconception. De même, la géométrie complexe permettant le refroidissement continu des pièces (micro-perforations) doit être prise en compte de manière exacte. Les techniques de calcul standard pour ce genre de problème conduiraient à des simulations lentes et peu flexibles (la moindre modification entraînant une remise en œuvre complète de la chaîne de calcul). Plus précisément, cette thèse étend les méthodes de type global/local non-intrusives au cas de la viscoplasticité généralisée en utilisant deux échelles de temps et d'espace, chacune adaptée aux phénomènes locaux et globaux à capturer. La méthode est ensuite étendue au calcul de nombreux cycles complexes de chargement, par des techniques de saut de cycles. Le schéma de couplage en temps permet alors une adaptation locale du pas de temps par sous-domaine. Des techniques d’accélération de convergence sont proposées, à l’échelle d’un incrément puis à celle de la succession de cycles (sauts de cycles). Ces développements permettent d’obtenir rapidement et précisément une estimation du cycle limite qui alimente un modèle de durée de vie. Le couplage non-intrusif est réalisé dans un script de programmation pilotant un code commercial (dans notre cas le langage Python et Abaqus/Standard). La méthode a été appliquée sur des plateformes de calculs industrielles, en réutilisant directement des maillages et les mises en données issues de modèles intervenant plus tôt dans la chaîne de calcul. Un cas métier, issu d’un bureau d’études de Safran Aircraft Engines, a pu être traité.
... [20,21,22]). More recently, investigations towards the coupling with discontinuous Galerkin methods for multiscale problems [23] or domain-decomposition approaches [24,25,26], spectral element methods [27,28], and extended finite element methods [29,30] have been carried out. ...
In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offline–online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model.
... The solver can be of very different type, depending on the application and the know-how gained in the past. For applications related reduced order models based on a finite element high-fidelity discretization see [71,32,88,81,74,41]), as well as finite volume [34,35,28,52], finite difference method [29,17,51], spectral element method [77,70], extended finite element method [18,64], boundary element method [53,83], isogeometric analysis [53,84] and discontinuous Galerkin methods [43,2,67]. ...
... An alternative is to adopt enrichment functions that are solutions of boundary value problems. This is the basic idea of the GFEM with global-local enrichment functions (GFEM gl ) [40]; the GFEM with mesh-based handbooks [74]; the GFEM with numerically defined harmonic enrichment functions [75]; the method of Menk and Bordas for fracture of bi-material systems [76]; the spider-XFEM [77] and the reduced basis enrichment for the XFEM [78] of Chahine et al. There are other related methods in the literature. ...
This paper presents a novel numerical framework based on the generalized finite element method with global-local enrichments (GFEMgl) for two-scale simulations of propagating fractures in three dimensions. A non-linear cohesive law is adopted to capture objectively the dissipated energy during the process of material degradation without the need of adaptive remeshing at the macro scale or artificial regularization parameters. The cohesive crack is capable of propagating through the interior of finite elements in virtue of the partition of unity (POU) concept provided by the generalized/extended finite element method (GFEM/XFEM), and thus eliminating the need of interfacial surface elements to represent the geometry of discontinuities and the requirement of finite element meshes fitting the cohesive crack surface. The proposed method employs fine-scale solutions of non-linear local boundary value problems extracted from the original global problem in order to not only construct scale-bridging enrichment functions but also to identify damaged states in the global problem, thus enabling accurate global solutions on coarse meshes. This is in contrast with available GFEMgl in which the local solution field contributes only to the kinematic description of global solutions. The robustness, efficiency, and accuracy of this approach are demonstrated by results obtained from representative numerical examples.
... First, one can cite enrichment methods based on the Partition of Unity Method [66]: the Generalised Finite Element Method [81,27,53] and the eXtended Finite Element Method [67] being the most famous ones. Their principle is to enrich the finite element functional space with specific functions, which can result from asymptotic expansion [16] or pre-computed local finite element problem solution [18,19] for instance. Then, enrichment methods are based on micro-macro models. ...
This paper provides a detailed review of the global/local non-intrusive coupling algorithm. Such method allows to alter a global finite element model, without actually modifying its corresponding numerical operator. We also look into improvements of the initial algorithm (Quasi-Newton and dynamic relaxation), and provide comparisons based on several relevant test cases. Innovative examples and advanced applications of the non-intrusive coupling algorithm are provided, granting a handy framework for both researchers and engineers willing to make use of such process. Finally, a novel nonlinear domain decomposition method is derived from the global/local non-intrusive coupling strategy, without the need to use a parallel code or software. Such method being intended to large scale analysis, we show its scalability. Jointly, an efficient high level Message Passing Interface coupling framework is also proposed, granting an universal and flexible way for easy software coupling. A sample code is also given.
... Both methods show an optimal rate of convergence. The optimal rate of convergence in the L 2 norm for the GFEM/XFEM is also reported in [11,12] for two-dimensional fracture problems. ...
In this paper, we investigate the accuracy and conditioning of the Stable Generalized FEM (SGFEM) and compare it with standard Generalized FEM (GFEM) for a 2-D fracture mechanics problem. The SGFEM involves localized modifications of enrichments used in the GFEM and the conditioning of the stiffness matrix in this method is of the same order as in the FEM. Numerical experiments show that using the SGFEM with only the modified Heaviside functions, which are used as enrichments in the GFEM, to approximate the solution of fracture problems in 2-D, gives inaccurate results. However, the SGFEM using an additional set of enrichment function yields accurate results while not deteriorating the conditioning of the stiffness matrix.Rules for the selection of the optimal set of enrichment nodes based on the definition of enrichment functions used in the SGFEM are also presented. This set leads to optimal convergence rates while keeping the number of degrees of freedom equal to or close to the GFEM. We show that it is necessary to enrich additional nodes when the crack line is located along element edges in 2-D. The selection of these nodes depends on the definition of the enrichment functions at the crack discontinuity.A simple and yet generic implementation strategy for the SGFEM in an existing GFEM/XFEM software is described. The implementation can be used with 2-D and 3-D elements. It leads to an efficient evaluation of SGFEM enrichment functions.
... In order to take into account the effect of orthotropy, orthotropic crack tip enrichments were developed for XFEM and successfully used in static [42][43][44] and dynamic crack propagation problems [45,46] of orthotropic media. Recently, the automatic enrichment technique has also been developed to numerically determine the required enrichment functions for each problem [47][48][49][50][51]. Moreover, new crack tip enrichments have been presented by Esna Ashari and Mohammadi [52] for interlaminar cracks in orthotropic bimaterial domains. ...
In the present study, the extended finite element method (XFEM) has been used for fracture analysis of orthotropic functionally graded materials. Orthotropic crack tip enrichments have been used to reproduce the singular stress field near a crack tip. Moreover, the incompatible interaction integral method has been employed to extract the stress intensity factor components. Accuracy and convergence of the proposed method have been evaluated by numerical examples and quality results have been obtained by far fewer DOFs. Also, crack propagation in isotropic and orthotropic FGMs in the presence of crack tip enrichments has been investigated and various propagation criteria have been compared, and verified, if available, by experimental and numerical data in the literature. Application of XFEM in combination of the maximum circumferential tensile stress criterion for investigation of crack propagation in orthotropic FGM problems is performed for the first time.
... To reproduce the exact fields at the crack tip in orthotropic media, orthotropic enrichment functions were obtained for both static [56][57][58] and dynamic [59,60] cases. Automatic enrichment techniques were proposed to find enrichment functions in arbitrary problems [61][62][63][64]. Furthermore, enrichment functions for interlaminar cracks in orthotropic bimaterials were derived by Esna Ashari and Mohammadi [65]. ...
A computational method based on the extended finite element method (XFEM) is implemented for fracture analysis of isotropic and orthotropic functionally graded materials (FGMs) under mechanical and steady state thermal loadings. The aim is set to include the thermal effects in loading, governing equations, and the interaction integral for inhomogeneous materials with a complementary study on available crack propagation criteria in orthotropic FGMs under thermal loading conditions. The isotropic and orthotropic crack tip enrichments are applied to reproduce the singular stress field near crack tips. Mixed-mode stress intensity factors are evaluated in isotropic and orthotropic FGMs by means of the interaction integral. In addition, the mesh dependency and number of elements around the crack tip are substantially reduced in comparison with the standard finite element method with the same level of accuracy. Both mode-I and mixed-mode fracture problems with various types of mechanical and thermo-mechanical functionally graded material properties are simulated and discussed to assess the accuracy and efficiency of the proposed numerical method. Good agreements are observed between the predicted results and the reference results available in the literature with far lower degrees of freedom.
... A related method aimed at modeling interactions among multiple static cracks is the multiscale method of Loehnert and Belytschko [33]. Other related methods for two-dimensional static cracks include the spider-XFEM [9] and the reduced basis enrichment for the XFEM [10] of Chahine et al.; the method of Menk and Bordas for fracture of bi-material systems [35]; the harmonic enrichment functions of Mousavi et al. [39] for two-dimensional branched cracks. The outline of the paper is as follows. ...
This paper presents a generalized finite element method (GFEM) for crack growth simulations based on a two-scale decomposition
of the solution—a smooth coarse-scale component and a singular fine-scale component. The smooth component is approximated
by discretizations defined on coarse finite element meshes. The fine-scale component is approximated by the solution of local
problems defined in neighborhoods of cracks. Boundary conditions for the local problems are provided by the available solution
at a crack growth step. The methodology enables accurate modeling of 3-D propagating cracks on meshes with elements that are
orders of magnitude larger than those required by the FEM. The coarse-scale mesh remains unchanged during the simulation.
This, combined with the hierarchical nature of GFEM shape functions, allows the recycling of the factorization of the global
stiffness matrix during a crack growth simulation. Numerical examples demonstrating the approximating properties of the proposed
enrichment functions and the computational performance of the methodology are presented.
KeywordsGeneralized FEM–Extended FEM–Fracture–Crack growth–Fatigue–Multi-scale–Global-local analysis
... A related method aimed at modeling interactions among multiple static cracks is the multiscale method of Loehnert and Belytschko [26]. Other related methods for two-dimensional static cracks include the spider-XFEM [27] and the reduced basis enrichment for the XFEM [28] of Chahine et al.; the method of Menk and Bordas for fracture of bi-material systems [29]; the harmonic enrichment functions of Mousavi et al. [30] for two-dimensional branched cracks. ...
The generalized FEM (GFEM) has been successfully applied to the simulation of dynamic propagating fractures, polycrystalline and fiber-reinforced microstructures, porous materials, etc. A-priori knowledge about the solution of these problems are used in the definition of their GFEM approximation spaces. This leads to more accurate and robust simulations than available finite element methods while relaxing some meshing requirements. This is demonstrated in a simulation of intergranular crack propagation in a brittle polycrystal using simple background meshes. For many classes of problems – like those with material non-linearities or involvingmultiscale phenomena – a-priori knowledge of the solution behavior is limited. In this paper, we present a GFEMbased on the solution of interdependent global (structural) and fine-scale or local problems. The local problems focus on the resolution of fine-scale features of the solution in the vicinity of, e.g., evolving fracture process zones while the global problem addresses the macroscale structural behavior. Fine-scale solutions are accurately solved using an hp-adaptive GFEM and thus the proposed method does not rely on analytical solutions. These solutions are embedded into the global solution space using the partition of unity method. This GFEM enables accurate modeling of problems involving multiple scales of interest using meshes with elements that are orders of magnitude larger than those required by the FEM. Numerical examples illustrating the application of this class of GFEM to high-cycle fatigue crack growth of small cracks and to problems exhibiting localized non-linear material responses are presented.
