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From: John Peterson <jwpeterson@gm...>  20080619 19:11:09

On Thu, Jun 19, 2008 at 1:59 PM, Roy Stogner <roy@...> wrote: > > We might want to split off a QNegativeGauss quadrature class, which > would be nondefault. Some applications are going to be more tolerant > of negative weights than others, and for those applications people > could have the option of choosing quadrature rules with fewer total > points. Yeah, it turns out that Walkington's seventhorder rule with 35 points that I mentioned earlier has negative weights also. This onepointless fifthorder rule may just be an anomaly. The separate class idea is an interesting one, though there would be a lot of empty cases. Another possibility might be a bit in QBase which you could flip to allow computation with potentially more efficient quadrature rules that also just happen to have negative weights. > However, this rule doesn't just have negative weights, it has some > negative barycentric coordinates! That's seriously wrong. Is > "QFuckedUpGauss" too politically incorrect a class name for libMesh > use? As far as I can tell, the barycentric coords of x=(x_1, x_2, ..., x_d) are (x_0, x_1, ..., x_d) where the "zeroth" coordinate is given by x_0 = 1  x_1  x_2  ...  x_d so, I think it's possible for it to be negative. > >> Did you happen to catch the meaning of "class" C_0, C_1, etc.? I'm >> not exactly sure how to get the 87 points out of the tabulated data. > > A "class" is set of points in R^3 s.t. for some symmetry s in G_3 (the > symmetry group of the simplex) the barycentric coordinates > representation of s(x) has the form > lambda(s(x)) = (a0,...,a0,a1,...,a1,a2...) > where a0 appears m0 times, a1 appears m1 times, etc. > > The specific classes here are: > C0: m = [4] > C1: m = [3, 1] > C2: m = [2, 2] > C3: m = [2, 1, 1] > C4: m = [1, 1, 1, 1] > > So I think C0 is the tet center, C1 is along the symmetry lines from > each vertex to the opposite face's centroid, etc. Each C1 point has > four unique permutations, each of which is also a point in the > quadrature rule. Each C2 point has 6 permutations, each C3 point has > 12 permutations. So the 1 + 5 + 1 + 5 lines in that data table become > 1*1 + 5*4 + 1*6 + 5*12 = 87 points. Excellent. Walkington's paper uses similar permutation groups but I just wasn't so sure about this one's notation.  John 
From: Roy Stogner <roy@st...>  20080619 18:25:05

On Thu, 19 Jun 2008, John Peterson wrote: > J. Maeztu and ES de la Maza, "An invariant quadrature rule of degree > 11 for the tetrahedron," C. R. Acad Sci. Paris v. 321 (1995) p. > 12631267. > > If any of our Frenchspeaking users could help out by > finding/translating this paper I'd greatly appreciate it. (Roy, UT > has numbers 14 and 712 of this volume at the PMA "QA 1 C856" but I'm > not sure those cover the page numbers we need...) Vikram found it here, and it's in both French and English. I'll send you a PDF offlist.  Roy 
From: John Peterson <jwpeterson@gm...>  20080619 14:31:07

On Wed, Jun 18, 2008 at 5:18 PM, John Peterson <jwpeterson@...> wrote: > On Wed, Jun 18, 2008 at 1:52 PM, David Knezevic <dave.knez@...> wrote: >> >> hehe, well the way the paper is written doesn't inspire me with confidence >> in their results, so I'd be very interested to hear the results of the >> comparisons. This paper gets even more interesting... Walkington's claimed seventhorder rules have 20 points (d=2) and 35 points (d=3). The only thing we've got at seventhorder for simplices are the "conical product" rules I implemented a long time ago, which have 16 points (d=2) and 64 points (d=3). I'm going to implement and check the 3D version since that appears to be a pretty big win. I don't use seventhorder a lot, but it should be handy for anyone using cubics... While I'm at it: I can't find the following paper online (and I think it might be in French) but it purports to have a degree 11 rule for tets with only 87 points (our current 11thorder rule has 216 pts). J. Maeztu and ES de la Maza, "An invariant quadrature rule of degree 11 for the tetrahedron," C. R. Acad Sci. Paris v. 321 (1995) p. 12631267. If any of our Frenchspeaking users could help out by finding/translating this paper I'd greatly appreciate it. (Roy, UT has numbers 14 and 712 of this volume at the PMA "QA 1 C856" but I'm not sure those cover the page numbers we need...) Thanks,  John 