Generalized Duffy transformation for integrating vertex singularities

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 = xv, z = xw, 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.

A high-order immersed boundary discontinuous-Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

“…Following [38], we then use a generalized Duffy-trick to integrate each triangle with a proper Gaussrule that respects the order of the singularity depending on the angleˇ˙.…”

Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method

“…This scheme has been used for integration of stiffness matrix entries in the X-FEM where the shape function space is enriched with near-tip functions that have singular derivatives [58]. Mousavi and Sukumar [13] presented a generalized form of the Duffy transformation:…”

“…The Duffy mapping removes the 1/r singularity and the transformed integral is amenable to tensor-product Gauss quadrature. Mousavi and Sukumar [13] have shown that the Duffy transformation is efficient for a 1/r singularity, but is not as efficient for 1/r α singularities when α = 1 (e.g., crack modeling in the X-FEM where α = 1/2 is present). Moreover, for 1 < α < 2 in two dimensions (2 < α < 3 in three dimensions), the Duffy transformation does not remove the singularity.…”

“…In this paper, we present the construction of generalized Gaussian quadrature rules that exactly integrate polynomials on either side of the crack without the need to partition the finite element. Furthermore, we adopt the generalized Duffy transformation introduced by Mousavi and Sukumar [13] to enable the efficient and accurate computation of weakly singular stiffness matrix integrals in fracture computations with the X-FEM.…”

“…Reference [13], a generalized form of the Duffy transformation is introduced in which a parameter β is judiciously selected so that the transformation leads to removal of the weak singularity. Also, in case of a fractional α such as α = 1/2, the generalized Duffy transformation permits dramatic improvements in accuracy.…”