Journal ArticleDOI
Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping
TLDR
Numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results.Abstract:
This paper presents a new numerical integration technique oil arbitrary polygonal domains. The polygonal domain is mapped conformally to the unit disk using Schwarz-Christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used. This method eliminates the need for a two-level isoparametric mapping Usually required. Moreover, the positivity of the Jacobian is guaranteed. Numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results.read more
Citations
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Journal ArticleDOI
The extended/generalized finite element method: An overview of the method and its applications
TL;DR: An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented in this article, which enables accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements.
Journal ArticleDOI
The Hitchhiker's Guide to the Virtual Element Method
TL;DR: The essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem are presented and its computer implementation is emphasized to enable interested readers to readily implement the method.
Book
hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes
TL;DR: An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analysed.
Journal ArticleDOI
Divergence free virtual elements for the stokes problem on polygonal meshes
TL;DR: This paper develops a new family of Virtual Elements for the Stokes problem on polygonal meshes that can guarantee that the final discrete velocity is pointwise divergence-free, and not only in a relaxed (projected) sense, as it happens for more standard elements.
Journal ArticleDOI
PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes
TL;DR: An efficient Matlab code is presented that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension of linear triangles and bilinear quads and features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation.
References
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Journal ArticleDOI
A finite element method for crack growth without remeshing
TL;DR: In this article, a displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method.
Journal ArticleDOI
Element‐free Galerkin methods
Ted Belytschko,Y. Y. Lu,L. Gu +2 more
TL;DR: In this article, an element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems, where moving least-squares interpolants are used to construct the trial and test functions for the variational principle.
Journal ArticleDOI
Elastic crack growth in finite elements with minimal remeshing
Ted Belytschko,T. Black +1 more
TL;DR: In this article, a minimal remeshing finite element method for crack growth is presented, where Discontinuous enrichment functions are added to the finite element approximation to account for the presence of the crack.
Journal ArticleDOI
The partition of unity finite element method: Basic theory and applications
Jens Markus Melenk,Ivo Babuška +1 more
TL;DR: In this article, the basic ideas and the mathematical foundation of the partition of unity finite element method (PUFEM) are presented and a detailed and illustrative analysis is given for a one-dimensional model problem.
Journal ArticleDOI
Meshless methods: An overview and recent developments
TL;DR: Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined and it is shown that the three methods are in most cases identical except for the important fact that partitions ofunity enable p-adaptivity to be achieved.