Addressing integration error for polygonal finite elements through polynomial projections: A patch test connection

Talischi, Cameron; Paulino, Gláucio H.

doi:10.1142/s0218202514400077

Cited by 61 publications

(77 citation statements)

References 48 publications

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“…Regarding the numerical integration of the cut cells, the quadtree-refinement based on a decomposition of the polygonal cell into quadrilaterals is favored and is used if not specified otherwise. The preference of this method is based on the fact that in the wide body of literature it is reported that this approach provides more accurate results [8,34].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Since a standard hextension is performed this behavior is not caused by the geometry approximation error. Instead its source is the inexact numerical integration of the stiffness matrix K and the load vector F. Talischi et al found that all available quadrature schemes are inaccurate even on regular n-gons due to the non-polynomial nature of the basis functions and therefore proposed polynomial projections to circumvent this shortcoming [8,121]. A second reason for this behavior can be attributed to artificial stress singularities that are introduced due to the parabolic mapping.…”

Section: Comparison Of Poly-fcm Results With P-fem Solutionsmentioning

confidence: 99%

“…Not only is the construction of meshes with areas of local refinement more flexible but these element types also simplify mesh adaptation procedures. By deploying polygonal finite elements it is straightforward to perform a local refinement through element splitting and a local coarsening through element aggregation since "hanging nodes" are naturally accommodated [8]. Therefore, the user is further provided with the opportunity to construct conforming quadtree meshes [9][10][11].…”

Section: Introductionmentioning

confidence: 99%

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The finite cell method for polygonal meshes: poly-FCM

Duczek

Gabbert

2016

Comput Mech

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In the current article, we extend the two-dimensional version of the finite cell method (FCM), which has so far only been used for structured quadrilateral meshes, to unstructured polygonal discretizations. Therefore, the adaptive quadtree-based numerical integration technique is reformulated and the notion of generalized barycentric coordinates is introduced. We show that the resulting polygonal (poly-)FCM approach retains the optimal rates of convergence if and only if the geometry of the structure is adequately resolved. The main advantage of the proposed method is that it inherits the ability of polygonal finite elements for local mesh refinement and for the construction of transition elements (e.g. conforming quadtree meshes without hanging nodes). These properties along with the performance of the poly-FCM are illustrated by means of several benchmark problems for both static and dynamic cases.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Comparison Of Poly-fcm Results With P-fem Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The finite cell method for polygonal meshes: poly-FCM

Duczek

Gabbert

2016

Comput Mech

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“…In this section, we briefly discuss aspects of implementation of the proposed formulation. We begin by discussing the issue of numerical quadrature, which is nontrivial for polygonal finite elements (see, for example, [38,30,40]), and proceed to provide explicit expressions for the basis functions for the local spaces V k (E) and their polynomial corrections.…”

Section: Discretization Scheme Based On the Correctionsmentioning

confidence: 99%

A Family of $H(div)$ Finite Element Approximations on Polygonal Meshes

Talischi¹

2015

SIAM J. Sci. Comput.

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In this paper, we present a family of H(div)-compatible finite element spaces on strictly convex n-gons, whose construction makes use of generalized barycentric coordinates. In particular, for integers 0 ≤ k ≤ 2, we define finite element spaces with edge degrees of freedom that include polynomial vector fields of order k and whose vector fields have piecewise kth-order polynomial normal traces along the element boundary. These spaces suffer from the shortcoming that the image of the divergence operator includes nonpolynomial functions and, as such, their direct use in a mixed setting along with polynomial scalar fields may lead to unstable discretizations exhibiting degraded or no convergence. We present a general remedy for restoration of optimal convergence that involves "polynomial corrections" of vector fields and their divergence at the element level. These corrections are consistent with one another and require computation of suitable polynomial projection maps. In addition to the theoretical discussion, the performance of the discretization schemes based on the proposed spaces and the accompanying correction maps is numerically evaluated through their application to H(div) eigenvalue problems, mixed and least-squares approximations of linear and nonlinear porous media flow problems. Introduction.Motivated by the challenges of generating meshes for complex and evolving domains, the development of discretization schemes for solving partial differential equations on arbitrary polygonal and polyhedral meshes has become an active area of research in the past decade. The extension of the finite element method in this direction has been primarily focused on investigation and application of H 1 -conforming spaces involving vertex-based elements [37,36,35,22,39]. This relatively narrow scope can be attributed to the availability and ongoing development of the so-called generalized barycentric coordinates which serve as the underlying basis functions of these finite element spaces. The aim of the present work is to make use of barycentric coordinates to construct H(div)-compatible finite element spaces on convex polygonal partitions. These schemes can be employed for solving variational problems involving the H(div) space including porous media flow or mixed formulations of elasticity and Stokes flow. Since H(curl) in two dimensions is isomorphic to H(div) through a rotation, the present work can also be used to obtain discretizations for problems set in H(curl) such as Maxwells equations.A handful of edge-based finite element formulations are available in the literature for polygons. Kuznetsov and Repin [28] proposed a macroelement formulation using Raviart-Thomas spaces on a simplicial partition of the element. Euler, Schuhmann, and Weiland [19] and more recently Gillette, Rand, and Bajaj [23] have explored

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“…We refer to the recent papers and monographs [19,8,17,9,13,15,28,30,31,33,35,34,36,39,40,24,32,21] as a brief representative sample of the increasing list of technologies that make use of polygonal/polyhedral meshes. We mention here in particular the polygonal finite elements, that generalize finite elements to polygons/polyhedrons by making use of generalized non-polynomial shape functions, and the mimetic discretisation schemes, that combine ideas from the finite difference and finite element methods.…”

Section: Introductionmentioning

confidence: 99%

Divergence free virtual elements for the stokes problem on polygonal meshes

2017

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In the present paper we develop a new family of Virtual Elements for the Stokes problem on polygonal meshes. By a proper choice of the Virtual space of velocities and the associated degrees of freedom, we 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. Moreover, we show that the discrete problem is immediately equivalent to a reduced problem with less degrees of freedom, thus yielding a very efficient scheme. We provide a rigorous error analysis of the method and several numerical tests, including a comparison with a different Virtual Element choice.

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Addressing integration error for polygonal finite elements through polynomial projections: A patch test connection

Cited by 61 publications

References 48 publications

The finite cell method for polygonal meshes: poly-FCM

The finite cell method for polygonal meshes: poly-FCM

A Family of $H(div)$ Finite Element Approximations on Polygonal Meshes

Divergence free virtual elements for the stokes problem on polygonal meshes

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