Generalized Duffy transformation for integrating vertex singularities

By S. Mousavi, Natarajan Sukumar. 2009

90 citations recorded by the source corpus.

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)/rα, 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 = uβ, 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.

Subfields: Electrical and Electronic Engineering, Mechanics of Materials, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/ee5f660cd716821560d6f8af7375e37b0370d4d2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Electrical and Electronic Engineering

1. Natarajan Sukumar
2. Gianmarco Manzini
3. Elena Benvenuti
4. Andrea Chiozzi
5. S. Mousavi

In Mechanics of Materials

1. S. Mousavi
2. Gianmarco Manzini
3. Elena Benvenuti
4. Natarajan Sukumar
5. Andrea Chiozzi
6. Stéphane P.A. Bordas
7. L. Beirão da Veiga
8. Carlos Armando Duarte

In Numerical Analysis

1. Natarajan Sukumar
2. S. Mousavi
3. Gianmarco Manzini
4. Andrea Chiozzi
5. Elena Benvenuti
6. L. Beirão da Veiga
7. Alessandro Russo
8. Glaucio H. Paulino

Citations in the corpus

• Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method (2010, methodology): “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.”
• Extended virtual element method for the Laplace problem with singularities and discontinuities (2019, methodology): “Comparisons are made with both the standard VEM and standard FEM/X-FEM (from [36]) on quadrilateral and polygonal meshes.”
• Extended finite element method in computational fracture mechanics: a retrospective examination (2015, unknown): “Mousavi and Sukumar (2010a) generalized the Duffy transformation to integrate weakly singular integrands.”