TITLE

Generalized Duffy transformation for integrating vertex singularities

AUTHOR(S)
Mousavi, S. E.; Sukumar, N.
PUB. DATE
January 2010
SOURCE
Computational Mechanics;Jan2010, Vol. 45 Issue 2/3, p127
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
For an integrand with a 1/ r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p( x)/ r α, where p is a trivariate polynomial and α > 0 is the strength of the singularity. We use the map ( u, v, w) → ( x, y, z) : x = u β, y = x v, z = x w, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation ( β = 1) is optimal, whereas if α ≠ 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.
ACCESSION #
45391133

 

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