Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection

By Cameron Talischi, Glaucio H. Paulino. 2013

70 citations recorded by the source corpus.

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations.

Subfields: Atomic and Molecular Physics, and Optics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/b6d43e29697c79742ccb6b844f0f7f18a9600d53

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

Hourglass stabilization and the virtual element method (2015, methodology): “Talischi C, Paulino GH. Addressing integration error for polygonal finite element through polynomial projections: a patch test connection.”
On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes (2013, background): “As discussed in detail in [37], the failure to satisfy patch test (at least asymptotically under mesh reﬁnement) can place a limit on the accuracy that can be achieved in general by the method.”
Some basic formulations of the virtual element method (VEM) for finite deformations (2017, background): “Moreover, especially for nonlinear and three-dimensional problems, efficient yet consistent numerical quadrature rule on general polygons and polyhedrons are difficult to construct and typically contain numerous quadrature points (see [21, 22, 23] for some recent work to alleviate this issue), which”
New perspectives on polygonal and polyhedral finite element methods (2014, background): “We point out the related work of Talischi and Paulino [67] (also in this Special Issue), where the same approach is presented to ameliorate integration errors over polygonal meshes.”