New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases

Papanicolopulos, Stefanos‐Aldo

doi:10.1016/j.cam.2015.08.001

Cited by 12 publications

(6 citation statements)

References 17 publications

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“…Numerical integration schemes on triangles have been the subject of significant research, with many early contributions in the latter part of the last century . This effort has sustained over the past two decades with greater emphasis on constructing high‐order integration rules on triangles . With an eye on polygonal finite element methods, generalized Gaussian cubature rules have also been constructed for convex and nonconvex polygons .…”

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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“…Variants of this successful strategies provided many near minimal pointsets, even for rather large δ (see e.g. [14,17,30,32] with the references therein).…”

Section: About the Subdivisionmentioning

confidence: 99%

Compressed algebraic cubature over polygons with applications to optical design

Bauman

Sommariva

Vianello

2020

Journal of Computational and Applied Mathematics

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“…In addition, in many applications involving finite elements, it is also desirable that the cubature scheme is symmetric, that is, it is invariant to affine transformation into itself. [4][5][6][7] Symmetric rules are preferred since symmetric distribution of interpolation nodes (e.g., shape functions on a tetrahedron) can be leveraged to deliver fast assembly of stiffness and mass matrices in finite element analysis. More importantly, use of symmetric cubature rules in nonlinear finite element analysis ensures that the simulation results do not depend on the nodal ordering in the element connectivity.…”

Section: Introductionmentioning

confidence: 99%

“…It is desirable that all weights are positive and all nodes are in the interior of the domain, which are collectively referred to as the “PI” criteria. In addition, in many applications involving finite elements, it is also desirable that the cubature scheme is symmetric, that is, it is invariant to affine transformation into itself 4‐7 . Symmetric rules are preferred since symmetric distribution of interpolation nodes (e.g., shape functions on a tetrahedron) can be leveraged to deliver fast assembly of stiffness and mass matrices in finite element analysis.…”

Section: Introductionmentioning

confidence: 99%

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 fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases

Cited by 12 publications

References 17 publications

High‐order cubature rules for tetrahedra

High‐order cubature rules for tetrahedra

Compressed algebraic cubature over polygons with applications to optical design

High‐order symmetric cubature rules for tetrahedra and pyramids

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