Generalized Duffy transformation for integrating vertex singularities

• Published in: Computational mechanics 2010-01, Vol.45 (2-3), p.127-140
• Main Authors: Mousavi, S. E., Sukumar, N.
• Format: Article
• Language: English
• Subjects: Classical and Continuum Physics; Computational Science and Engineering; Corners; Derivatives; Engineering; Estimates; Finite element method; Integrals; Laplace equation; Original Paper; Partitions; Polynomials; Singularities; Theoretical and Applied Mechanics; Three dimensional; Transformations; Transformations (mathematics)
• ISSN: 0178-7675; 1432-0924
• DOI: 10.1007/s00466-009-0424-1

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.