Authors: Alfredo Cano Carlos Moreno
Publish Date: 2016/10/26
Volume: 71, Issue: 2, Pages: 571-593
Abstract
The implementations of the eXtended Finite Element Method and the Boundary Element Method need to face the challenge of integrating singular functions Since standard quadrature techniques usually produce inaccurate results a number of specific algorithms have been developed to address this problem We present a general framework for the systematic formulation of the threedimensional case The classical cubic transformation is also considered including an analytical optimization of its parameters for improved practical efficiencyP rtau mr0 r 0r1 In this case P r has two distinct positive roots denoted by tau 2r and tau 3r with tau 2tau 3 Since according to 32 P rtau 00 and P r0=varepsilon it follows from Bolzano’s theorem that 0tau 2tau 0 In other words the root tau 2 is always closer to the imaginary axis than tau 0The term in brackets in 30 is bounded above by 21/3 and thus r 0leqslant 3left fracvarepsilon 2right 2/3 We notice that 36 implies a lower bound for tau 0 namely tau 0geqslant left fracvarepsilon 2right 1/3
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