DECUHR: an algorithm for automatic integration of singular functions over a hyperrectangular region

Espelid, Terje O.; Genz, Alan

doi:10.1007/bf02142691

Cited by 27 publications

(15 citation statements)

References 7 publications

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“…[12], we use ψ(x) = r 1/3 sin(θ/3) as the enrichment function in the PUFE method. The DECUHR adaptive algorithm [57] (restricted to hyper-rectangular regions) was adopted by Strouboulis et al [12] to compute the enriched contributions in the stiffness matrix, whereas the generalized Duffy transformation is used in this study.…”

Section: L-shaped Domain With a Re-entrant Cornermentioning

confidence: 99%

Generalized Duffy transformation for integrating vertex singularities

Mousavi

Sukumar

2009

Comput Mech

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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.

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Section: L-shaped Domain With a Re-entrant Cornermentioning

confidence: 99%

Generalized Duffy transformation for integrating vertex singularities

Mousavi

Sukumar

2009

Comput Mech

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“…A standard 2 × 2 Gauss integration scheme has been used in the remaining elements. Other procedures are described in References [24][25][26].…”

Section: Finite Element Implementationmentioning

confidence: 99%

A Generalized Finite Element Method for polycrystals with discontinuous grain boundaries

Simone

Duarte

Giessen

2006

Numerical Meth Engineering

165

150

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We present a Generalized Finite Element Method for the analysis of polycrystals with explicit treatment of grain boundaries. Grain boundaries and junctions, understood as loci of possible displacement discontinuity, are inserted into finite elements by exploiting the partition of unity property of finite element shape functions. Consequently, the finite element mesh does not need to conform to the polycrystal topology. The formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach. The proposed methodology can also be used for branched and intersecting cohesive cracks, and comparisons are made to a related approach (Int. J. Numer. Meth. Engng. 2000; 48:1741). Copyright © 2006 John Wiley & Sons, Ltd.

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“…On the other hand, since the box integral is only 2-dimensional, a generalization of the efficient, deterministic method we selected to evaluate the 1-dimensional triangle function is suggestive, and several algorithms have indeed been discussed in the literature [20]. Their application, however, would require analytic knowledge of the location of all singularities.…”

Section: Numerical Evaluationmentioning

confidence: 99%

“…A well suited strategy for these badly-behaved integrands is to divide the integration region into subregions, and apply an iterative, adaptive algorithm. This approach has been used successfully for irregular high-dimensional integrands in combination with Monte Carlo methods [19], but has also been explored for lower-dimensional integrands in connection with integration rules [20].…”

Section: Numerical Evaluationmentioning

confidence: 99%

A numerical evaluation of the scalar hexagon integral in the physical region

2003

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We derive an analytic expression for the scalar one-loop pentagon and hexagon functions which is convenient for subsequent numerical integration. These functions are of relevance in the computation of next-to-leading order radiative corrections to multi-particle cross sections. The hexagon integral is represented in terms of n-dimensional triangle functions and (n+2)-dimensional box functions. If infrared poles are present this representation naturally splits into a finite and a pole part. For a fast numerical integration of the finite part we propose simple one-and two-dimensional integral representations. We set up an iterative numerical integration method to calculate these integrals directly in an efficient way. The method is illustrated by explicit results for pentagon and hexagon functions with some generic physical kinematics.

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DECUHR: an algorithm for automatic integration of singular functions over a hyperrectangular region

Cited by 27 publications

References 7 publications

Generalized Duffy transformation for integrating vertex singularities

Generalized Duffy transformation for integrating vertex singularities

A Generalized Finite Element Method for polycrystals with discontinuous grain boundaries

A numerical evaluation of the scalar hexagon integral in the physical region

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