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.
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Acknowledgments
The authors acknowledge the research support of the National Science Foundation through contract grants CMMI-0626481 and DMS-0811025 to the University of California at Davis. We thank John Pask for pointing us to the work of Batcho [47], and for his many suggestions that led to significant improvements in this paper. Helpful comments and suggestions of the anonymous reviewers are also greatly appreciated.
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National Science Foundation, CMMI-0626481, DMS-0811025.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Mousavi, S.E., Sukumar, N. Generalized Duffy transformation for integrating vertex singularities. Comput Mech 45, 127–140 (2010). https://doi.org/10.1007/s00466-009-0424-1
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DOI: https://doi.org/10.1007/s00466-009-0424-1