New computationally efficient quadrature formulas for pyramidal elements

Kubatko, Ethan J.; Yeager, Benjamin; Maggi, Ashley L.

doi:10.1016/j.finel.2012.10.012

Cited by 7 publications

(5 citation statements)

References 25 publications

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“…Early contributions in this area have been on low‐order integration rules . Besides tetrahedra in three dimensions (3D), cubature rules have also been proposed for prisms, pyramids, as well as hexahedra . For a comprehensive listing of some of the early studies on cubature rules for domains in two dimensions (2D) and 3D, the interested reader can refer to Cools and Rabinowitz and Cools …”

Section: Introductionmentioning

confidence: 99%

High‐order cubature rules for tetrahedra

Jaśkowiec

Sukumar

2020

Numerical Meth Engineering

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In this paper, we construct new high-order numerical integration schemes for tetrahedra, with positive weights and integration points that are in the interior of the domain. The construction of cubature rules is a challenging problem, which requires the solution of strongly nonlinear algebraic (moment) equations with side conditions given by affine inequality constraints. We present a robust algorithm based on a sequence of three modified Newton procedures to solve the constrained minimization problem. In the literature, numerical integration rules for the tetrahedron are available up to order p = 15. We obtain integration rules for the tetrahedron from p = 2 to p = 20, which are computed using multiprecision arithmetic. For p ≤ 15, our approach provides integration rules that have the same or fewer number of integration points than existing rules; for p = 16 to p = 20, our rules are new. Numerical tests are presented that verify the polynomial-precision of the cubature rules. Convergence studies are performed for the integration of exponential, rational, weakly singular and trigonometric test functions over tetrahedra with flat and curved faces. In all tests, improvements in accuracy is realized as p is increased, though in some cases nonmonotonic convergence is observed. K E Y W O R D Sleast squares, numerical integration, polynomial basis, positive weights, strictly interior integration points, tetrahedral domain 1 2418

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Section: Introductionmentioning

confidence: 99%

High‐order cubature rules for tetrahedra

Jaśkowiec

Sukumar

2020

Numerical Meth Engineering

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“…The traditional objective when constructing quadrature rules is to obtain a rule of strength φ inside of a domain using the fewest number of points. To this end efficient quadrature rules have been derived for a variety of domains: triangles [4][5][6][7][8][9][10][11][12][13], quadrilaterals [11,14,15], tetrahedra [7,9,16,17], prisms [18], pyramids [19], and hexahedra [11,[20][21][22]. For finite element applications it is desirable that (i) points are arranged symmetrically inside of the domain, (ii) all of the points are strictly inside of the domain, and (iii) all of the weights are positive.…”

Section: Introductionmentioning

confidence: 99%

On the identification of symmetric quadrature rules for finite element methods

Witherden

Vincent

2015

Computers & Mathematics with Applications

130

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a b s t r a c tIn this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable of identifying a set of rules with a given strength and a given number of points. We also present polyquad which is an implementation of our methodology. Using polyquad v1.0 we proceed to derive a complete set of symmetric rules on the aforementioned domains. All rules possess purely positive weights and have all points inside the domain. Many of the rules appear to be new, and an improvement over those tabulated in the literature.

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“…19 In this article, we present fully symmetric cubature rules up to order (synonymous with polynomial degree) p = 20 for tetrahedral and pyramidal finite elements. Over the past decade, a few contributions have appeared on the construction of symmetric cubature rules over tetrahedra and pyramids, 4,6,7,9,[20][21][22] with rules that are available for p = 1-10, 14 and p = 1-10 for the tetrahedron and the pyramid, respectively. As noted in Jaśkowiec and Sukumar, 1 constructing high-order 3D cubature rules over finite elements is a challenging problem, which is further compounded herein due to the presence of additional symmetry constraints.…”

Section: Introductionmentioning

confidence: 99%

“…For the tetrahedron and pyramid, such standard tensor‐product constructions are not possible. Use of the Duffy transformation 8 to generate symmetric integration schemes for pyramids has been proposed, 9 but this approach brings additional complications in the numerical integration. The tetrahedral, pyramidal, hexahedral, and prismatic elements are well studied in the literature 10 .…”

Section: Introductionmentioning

confidence: 99%

“…This approach results in a family of symmetric rules up to p = 7 for a tetrahedron. A different approach is utilized in Kubatko et al, 9 where symmetric cubature rules for the pyramid up to p = 8 (PI criteria is met) are obtained by using the Duffy transformation from the hexahedron. In William et al, 6 a sphere close‐packed lattice arrangement of points is utilized in the formulation of symmetric cubature rules on the tetrahedron.…”

Section: Introductionmentioning

confidence: 99%

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High‐order symmetric cubature rules for tetrahedra and pyramids

Jaśkowiec

Sukumar

2020

Numerical Meth Engineering

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In this article, we present an algorithm to construct high-order fully symmetric cubature rules for tetrahedral and pyramidal elements, with positive weights and integration points that are in the interior of the domain. Cubature rules are fully symmetric if they are invariant to affine transformations of the domain. We divide the integration points into symmetry orbits where each orbit contains all the points generated by the permutation stars. These relations are represented by equality constraints. The construction of symmetric cubature rules require the solution of nonlinear polynomial equations with both inequality and equality constraints. For higher orders, we use an algorithm that consists of five sequential phases to produce the cubature rules. In the literature, symmetric numerical integration rules are available for the tetrahedron for orders p = 1-10, 14, and for the pyramid up to p = 10. We have obtained fully symmetric cubature rules for both of these elements up to order p = 20. Numerical tests are presented that verify the polynomial-precision of the cubature rules. Convergence studies are performed for the integration of exponential, weakly singular, and trigonometric test functions over both elements with flat and curved faces. With increase in p, improvements in accuracy is realized, though nonmonotonic convergence is observed.

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New computationally efficient quadrature formulas for pyramidal elements

Cited by 7 publications

References 25 publications

High‐order cubature rules for tetrahedra

High‐order cubature rules for tetrahedra

On the identification of symmetric quadrature rules for finite element methods

High‐order symmetric cubature rules for tetrahedra and pyramids

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