Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping

By Sundararajan Natarajan, Stéphane P.A. Bordas, D. Roy Mahapatra. 2009

200 citations recorded by the source corpus.

This paper presents a new numerical integration technique on arbitrary polygonal domains. The polygonal domain is mapped conformally to the unit disk using Schwarz–Christoffel mapping and a midpoint quadrature rule defined on 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. Copyright © 2009 John Wiley & Sons, Ltd.

Subfields: Computational Mechanics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/c1e62be1400c419685f6377583b94e56754167f5

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In Computational Mechanics

In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

Equivalent polynomials for quadrature in Heaviside function enriched elements (2015, methodology): “A class of methods is based on conformal mappings [13, 14].”
Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method (2010, methodology): “[45] use the SchwarzChristoffel conformal mapping from an arbitrary polygon to the unit disk and then apply a midpoint rule to the mapped domain.”
On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM) (2011, methodology): “, [42] and a Schwarz–Christoffel conformal mapping technique was derived in [43, 44] to eliminate the subtriangulation in split and tip elements.”
Polygonal finite elements for topology optimization: A unifying paradigm (2010, unknown): “Alternatively, numerical integration can be carried out using special quadrature rules developed for polygonal domains (see, for example, [30–32]).”
A scaled boundary polygon formulation for elasto-plastic analyses (2014, background): “Compared with the polygon elements based on rational polynomials [1, 2, 3, 4, 11, 12] that require special integration techniques [13, 14], the semi-analytical integration in the present formulation is simpler.”
Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth (2017, background): “This flexibility simplifies local adaptive processes, as the regenerated mesh need not conform to the crack faces [19–21] does not come for free, as (1) numerical integration requires special attention [22–24]; (2) optimal convergence requires enriching a volume independent of the mesh size, thereby”
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].”