From: John P. <jwp...@gm...> - 2008-07-03 16:05:41
Attachments:
6th_degree.pdf
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Hi all, For some reason, they don't teach you this stuff in school, or else, I wasn't paying attention that day :-) I found some even-order non tensor-product rules for quadrilateral elements. These rules contain fewer evaluation points than the equivalent-order tensor product rules. Here is the reference: J. W. Wissmann and T. Becker, "Partially symmetric cubature formulas for even degrees of exactness," SIAM J. Numer. Anal. 23 (1986), 676--685. I implemented and tested the degree 4, 6, and 8 rules from the paper. They have 6, 10, and 16 points, respectively, while the equivalent tensor product rules had 9, 16, and 25 points, respectively. I believe this has something to do with the odd-order of the 1D Gauss rules (N points integrate polynomials up to degree 2N-1 exactly) so that 4th and 5th order get lumped together, 6th and 7th, etc. Wissmann's degree 4 and 6 rules are even provably minimal for this element type and order. I also found the arrangement of the points, when plotted, to be quite interesting. A plot of the points in the pair of degree-6 rules (x's for rule 1, o's for rule 2) obtained by Wissmann is attached. -- John |