TY - JOUR
T1 - A family of symmetric, optimized quadrature rules for pentatopes
AU - Williams, David M.
AU - Frontin, Cory V.
AU - Miller, Edward A.
AU - Darmofal, David L.
N1 - Funding Information:
The authors would like to thank Dr. Lee Shunn (Research Scientist, Cascade Technologies), Dr. Philip Caplan (Visiting Assistant Professor, Middlebury College), and Dr. Freddie Witherden (Assistant Professor, Texas A&M University) for their involvement in conversations that helped shape this work. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Publisher Copyright:
© 2020 Elsevier Ltd
PY - 2020/9/1
Y1 - 2020/9/1
N2 - This article introduces a new family of quadrature rules for integrating smooth functions on 4-dimensional simplex elements (i.e. pentatopes). These quadrature rules have 1, 5, 15, 35, 70, and 126 points, and are capable of exactly integrating polynomials of degrees 1, 2, 3, 5, 6, and 8, respectively. One main advantage of these rules, is that they have a ‘pentatopic number’ of points, which means that they can exactly interpolate 4-dimensional polynomials of degrees 0 through 5. As a result, the proposed rules can be used for both quadrature and interpolation purposes. Furthermore, these rules are fully symmetric, as they remain invariant under affine transformations (rotations and reflections) of the pentatope back to itself. In addition, these rules are optimal, in the sense that the truncation error associated with each rule has been minimized via a rigorous optimization procedure. Finally, they have positive weights, and all quadrature points reside strictly within the interior of the pentatope.
AB - This article introduces a new family of quadrature rules for integrating smooth functions on 4-dimensional simplex elements (i.e. pentatopes). These quadrature rules have 1, 5, 15, 35, 70, and 126 points, and are capable of exactly integrating polynomials of degrees 1, 2, 3, 5, 6, and 8, respectively. One main advantage of these rules, is that they have a ‘pentatopic number’ of points, which means that they can exactly interpolate 4-dimensional polynomials of degrees 0 through 5. As a result, the proposed rules can be used for both quadrature and interpolation purposes. Furthermore, these rules are fully symmetric, as they remain invariant under affine transformations (rotations and reflections) of the pentatope back to itself. In addition, these rules are optimal, in the sense that the truncation error associated with each rule has been minimized via a rigorous optimization procedure. Finally, they have positive weights, and all quadrature points reside strictly within the interior of the pentatope.
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U2 - 10.1016/j.camwa.2020.07.004
DO - 10.1016/j.camwa.2020.07.004
M3 - Article
AN - SCOPUS:85087976683
SN - 0898-1221
VL - 80
SP - 1405
EP - 1420
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
IS - 5
ER -