Direct evaluation of singular integrals in boundary element analysis of thick plates Roge ´rio J. Marczak a , Guillermo J. Creus b, * a Department of Mechanical Engineering, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 90050-170, Brazil b Department of Civil Engineering, CEMACOM/UFRGS, Universidade Federal do Rio Grande do Sul, Av. Oswaldo Aranha 99, Porto Alegre, RS 90035-190, Brazil Received 12 December 2001; revised 13 March 2002; accepted 9 April 2002 Abstract This work presents the derivation of the asymptotic expansions for two dimensional elasticity and plate bending problems fundamental solutions, applied to the direct evaluation of BEM singular integrals. Interesting conclusions arise from the resulting analytical expressions, regarding the actual order of singularity of the kernel functions. The expansions were tested for a number of plate bending benchmarks, showing good agreement to analytical solutions for thin and thick plates. The convergence behavior for constant, linear and quadratic elements is analyzed and compared with other integration techniques. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Boundary element method; Singular integrals; Regularization; Thick plate bending; Mindlin plate 1. Introduction The numerical integration of strongly singular kernels plays a key role in the implementation of many integral equation methods, such as the Boundary element method (BEM). In the last two decades, several methodologies have been proposed to perform the task. Nevertheless, only a few of them have generality for use with general fundamental tensors and higher order element shape functions. A comprehensive review can be found in Ref. [24]. The direct method seems to be one of the most general since it imposes no formal restriction on the type of kernel to be integrated and enables the use of standard Gaussian quadrature rules [6–8]. On the other hand, this method requires the knowledge of the analytical asymptotic expansions of the kernels around the singular pole. In spite of the increasing popularity of the method, few publications present the expressions of the asymptotic expansions for common applications like elasticity and plate bending problems. Moreover, it is largely ignored that the analytical derivation of these expansions provides a systematic and didactic insight about the behavior of the singular integrand in the neighborhood of the pole. This work is addressed to the sixth-order plate theories of Mindlin [18] and Reissner [22]. These plate models have been used in BEM context as a substitute for the classical theory because, in addition to the excellent accuracy for both primal and dual variables, they avoid some drawbacks shown by their finite element counterparts, such as poor performance for very thin plates, and the possible generation of non-physical deformation modes. After the original work of van der We ¨een [28,29], many references have reported the application of boundary elements to bending analysis of thick plates, most of them using the Reissner model [12,13,16,33]. Ref. [2] proposed what seems to be one of the first BEM approach genuinely based on the Mindlin model, while Westphal Jr and collaborators [3,30] unified the Mindlin and the Reissner plate models under the same numerical framework. Most of the works in this field have been relying on the use of rigid body movement imposition to indirectly evaluate the singular contributions. Some other approaches suggest the use of analytical treatment based on Taylor expansions of the kernels [21], fully analytic integration for very simple interpolations [34,35] and special quadrature rules for straight elements [19]. Any of these approaches lack generality since their extension to new element shape functions or other types of fundamental solutions require, when feasible, a significant amount of work. The rigid body technique is based on the knowledge of a particular solution. 0955-7997/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0955-7997(02)00029-2 Engineering Analysis with Boundary Elements 26 (2002) 653–665 www.elsevier.com/locate/enganabound * Corresponding author. Tel.: þ 55-513163362; fax: þ 55-512271807. E-mail addresses: creus@ufrgs.br (G.J. Creus), rato@mecanica.ufrgs.br (R.J. Marczak).