High‐order symmetric cubature rules for tetrahedra and pyramids

Jaśkowiec, Jan; Sukumar, N.

doi:10.1002/nme.6528

High‐order symmetric cubature rules for tetrahedra and pyramids

Journal Article · Thu Oct 15 00:00:00 EDT 2020 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.6528· OSTI ID:1786431

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Faculty of Civil Engineering Cracow University of Technology Cracow Poland
Department of Civil and Environmental Engineering University of California Davis California USA

Abstract

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.

View Accepted Manuscript (Publisher)

Sponsoring Organization:: USDOE

OSTI ID:: 1786431

Journal Information:: International Journal for Numerical Methods in Engineering, Journal Name: International Journal for Numerical Methods in Engineering Journal Issue: 1 Vol. 122; ISSN 0029-5981

Publisher:: Wiley Blackwell (John Wiley & Sons)Copyright Statement

Country of Publication:: United Kingdom

Language:: English

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