INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2009; 80:103–134 Published online 15 July 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2589 Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping Sundararajan Natarajan 1 , St´ ephane Bordas 2, ∗, † and D. Roy Mahapatra 3 1 GE-India Technology Center, Bangalore-560066, India 2 Department of Civil Engineering, University of Glasgow, G12 8LT, Scotland, U.K. 3 Department of Aerospace Engineering, Indian Institute of Science, Bangalore-560012, India SUMMARY 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. Received 7 May 2008; Revised 30 January 2009; Accepted 30 January 2009 KEY WORDS: Schwarz–Christoffel mapping; polygonal finite element; quadrature; integration rule; finite element method; conformal mapping; Wachspress shape functions; numerical integration; discontinuities; natural element method; XFEM 1. INTRODUCTION Mapped elements, such as the well-known isoparametric elements play a vital role in the conven- tional finite element method (FEM). In an effort to overcome the limitations of the FEM, meshfree methods [1–3] were introduced. See [4] for a recent review on meshfree methods. These methods have been recently reviewed and classified in [5]. Although there are many different types of mesh- free methods, it seems consensual that they are in general superior to FEMs for problems where large transformations are of interest and when high-order continuity of the solution is required. However, it also seems clear from the literature that most meshfree methods, especially those relying on a weak form, are more computationally expensive than finite elements [6]. Handling discontinuities in meshfree methods also seems more straightforward as in standard FEM because no meshing or remeshing of the evolving discontinuities is required [7–10]. ∗ Correspondence to: St´ ephane Bordas, Department of Civil Engineering, University of Glasgow, G12 8LT, Scotland, U.K. † E-mail: stephane.bordas@alumni.northwestern.edu Copyright 2009 John Wiley & Sons, Ltd.