Extended finite element method on polygonal and quadtree meshes

By A. Tabarraei, Natarajan Sukumar. 2008

158 citations recorded by the source corpus.

In this paper, we present mesh-independent modeling of discontinuous flelds on polygonal and quadtree flnite element meshes. This approach falls within the class of extended and generalized flnite element methods, where the partition of unity framework is used to introduce additional (enrichment) functions within the classical displacement-based flnite element approximation. For crack modeling, a discontinuous function and the two-dimensional asymptotic crack-tip flelds are used as enrichment functions. Linearly complete partition of unity approximations are adopted on polygonal (convex and nonconvex elements) and quadtree meshes. Excellent agreement with reference solution results is obtained for mixed-mode stress intensity factors on benchmark crack problems, and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics.

Subfields: Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/e1dcd821d32cc64a5c8bef0ef1679f9b3d807856

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In Computational Mechanics

In Mechanics of Materials

Citations in the corpus

Extended virtual element method for the Laplace problem with singularities and discontinuities (2019, methodology): “Among the methods for fracture on polygonal meshes that are germane to the present study, we mention the X-FEM [4], polygonal finite elements [5] and the scaled boundary finite element method [6].”
Equivalent projectors for virtual element methods (2013, background): “[34]), the Polygonal Finite Element Methods (see for instance [31, 32]) or the Extended Finite Element Methods (see [24] and the references therein).”
Meshless methods: A review and computer implementation aspects (2008, methodology): “Coupling the SFEM with the idea of quad (oct) -tree refinement proposed in [128] or enrichment adaptivity advocated in [119–121] appears promising as suggested by the preliminary results presented in the review paper [109].”
Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method (2010, methodology): “The X-FEM results show excellent agreement with the exact solution and the errors are similar to those obtained by Tabarraei and Sukumar [47].”
Strain smoothing in FEM and XFEM (2010, background): “[3,4,89,61,92,91,88,90] and their extension to partition of unity enrichment in the very recent article by Sukumar’s group [94].”
On three-dimensional modelling of crack growth using partition of unity methods (2010, background): “Special attention may be brought to the combination of quad-tree adaptivity [142] and smoothed (extended) finite element approximations on polygonal domains (SmXFEM).”
Mimetic finite difference method (2014, background): “We mention the seminal paper [268] and the most recent papers [34] [83] [112] [204] [250] [251] [253].”
A multigrid algorithm for the $p$-version of the Virtual Element Method (2017, methodology): “Examples of methods that have been proposed so far include mimetic finite differences [20, 35, 51, 52], polygonal finite element methods [58, 59], discontinuous-Galerkin finite element methods [4, 5, 9, 16, 37, 38, 40] hybridizable and hybrid high hrder methods [42, 43, 45], BEM-based FEM [56] and g”
New perspectives on polygonal and polyhedral finite element methods (2014, background): “To this end, on letting ra = x − xa (ra = ||ra||), we can write [71, 51]”
Polygon scaled boundary finite elements for crack propagation modelling (2012, methodology): “Nodal enrichment has also been recently implemented with polygon-based finite elements by Tabarraei and Sukumar [21] and Zamani and Eslami [22] in polygon finite elements based on barycentric coordinates shape function, Li and Ghosh [23, 24] in the Voronoi cell FEM and by Vu-Bac et al.”
A scaled boundary polygon formulation for elasto-plastic analyses (2014, methodology): “The extension of polygon elements for fracture analysis [6, 7] have also been recently pursued by enriching the basis functions as in the extended finite element method (XFEM) [8].”
Virtual Element Methods for general second order elliptic problems on polygonal meshes (2014, unknown): “We recall for instance [4], [5], [6], [7], [8], [20], [21], [31], [34], [38], [41], [42], [43], [44] [47], [48], [51], [52], [58], [57], [60], [61], [62], [63], [65], [66], [67], [70].”
The Hitchhiker's Guide to the Virtual Element Method (2014, methodology): “On the one hand, VEM can be viewed as an extension of Finite Element Methods to general polygonal and polyhedral elements [11, 21, 29–31, 33–39].”