From 717b974032153830b0e23ad1a461e3e94553542d Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Fri, 1 Dec 2023 16:27:51 +0000 Subject: [PATCH] Add Xiao-Gimbutas quadrature rules --- FIAT/quadrature_schemes.py | 65 +- FIAT/xg_quad_data.py | 18541 +++++++++++++++++++++++++++++++++++ 2 files changed, 18601 insertions(+), 5 deletions(-) create mode 100644 FIAT/xg_quad_data.py diff --git a/FIAT/quadrature_schemes.py b/FIAT/quadrature_schemes.py index fae0ce855..3c2f930e9 100644 --- a/FIAT/quadrature_schemes.py +++ b/FIAT/quadrature_schemes.py @@ -31,7 +31,7 @@ import numpy # FIAT -from FIAT.reference_element import QUADRILATERAL, HEXAHEDRON, TENSORPRODUCT, TRIANGLE, TETRAHEDRON, UFCTriangle, UFCTetrahedron +from FIAT.reference_element import QUADRILATERAL, HEXAHEDRON, TENSORPRODUCT, TRIANGLE, TETRAHEDRON, UFCTriangle, UFCTetrahedron, default_simplex from FIAT.quadrature import QuadratureRule, make_quadrature, make_tensor_product_quadrature, map_quadrature @@ -323,6 +323,53 @@ def _kmv_lump_scheme(ref_el, degree): return QuadratureRule(T, x, w) +def xg_scheme(ref_el, degree): + """A (nearly) Gaussian simplicial quadrature with very few quadrature nodes, + available for low-to-moderate orders. + + Raises `ValueError` if no quadrature rule for the requested parameters is available. + + See + + H. Xiao and Z. Gimbutas, "A numerical algorithm for the construction of + efficient quadrature rules in two and higher dimensions," Computers & + Mathematics with Applications, vol. 59, no. 2, pp. 663-676, 2010. + http://dx.doi.org/10.1016/j.camwa.2009.10.027 + """ + dim = ref_el.get_spatial_dimension() + if dim == 2: + from FIAT.xg_quad_data import triangle_table as table + elif dim == 3: + from FIAT.xg_quad_data import tetrahedron_table as table + else: + raise ValueError(f"Xiao-Gambutas rule not availale for {dim} dimensions.") + try: + order_table = table[degree] + except KeyError: + raise ValueError(f"Xiao-Gambutas rule not availale for degree {degree}.") + + # Get affine map from the (-1,1)^d triangle to the G-X equilateral triangle + if dim == 2: + A = numpy.array([[1, -1/numpy.sqrt(3)], + [0, 2/numpy.sqrt(3)]]) + b = numpy.array([-1/3, -1/3]) + else: + A = numpy.array([[1, -1/numpy.sqrt(3), -1/numpy.sqrt(6)], + [0, 2/numpy.sqrt(3), -1/numpy.sqrt(6)], + [0, 0, numpy.sqrt(6)/2]]) + b = numpy.array([-1/2, -1/2, -1/2]) + + A = numpy.linalg.inv(A) + Ref1 = default_simplex(dim) + v = numpy.dot(numpy.array(Ref1.vertices) - b[None, :], A.T) + Ref1.vertices = tuple(map(tuple, v)) + + pts_ref = order_table["points"] + wts_ref = order_table["weights"] + pts, wts = map_quadrature(pts_ref, wts_ref, Ref1, ref_el) + return QuadratureRule(ref_el, pts, wts) + + def _triangle_scheme(ref_el, degree): """Return a quadrature scheme on a triangle of specified order. Falls back on canonical rule for higher orders.""" @@ -394,8 +441,12 @@ def _triangle_scheme(ref_el, degree): w[6:12] = 0.082851075618374 w = w/2.0 else: - # Get canonical scheme - return _fiat_scheme(ref_el, degree) + try: + # Get Xiao-Gambutas scheme + return xg_scheme(ref_el, degree) + except ValueError: + # Get canonical scheme + return _fiat_scheme(ref_el, degree) # Return scheme x, w = map_quadrature(x, w, UFCTriangle(), ref_el) @@ -514,8 +565,12 @@ def _tetrahedron_scheme(ref_el, degree): w[12:24] = 0.0482142857142857 w = w/6.0 else: - # Get canonical scheme - return _fiat_scheme(ref_el, degree) + try: + # Get Xiao-Gambutas scheme + return xg_scheme(ref_el, degree) + except ValueError: + # Get canonical scheme + return _fiat_scheme(ref_el, degree) # Return scheme x, w = map_quadrature(x, w, UFCTetrahedron(), ref_el) diff --git a/FIAT/xg_quad_data.py b/FIAT/xg_quad_data.py new file mode 100644 index 000000000..97ec47652 --- /dev/null +++ b/FIAT/xg_quad_data.py @@ -0,0 +1,18541 @@ +# GENERATED, DO NOT EDIT +# Xiao-Gimbutas quadratures +# http://dx.doi.org/10.1016/j.camwa.2009.10.027 +import numpy + + +triangle_table = { # noqa + 1: { + "points": [[-5.55111512312578e-17, 2.22044604925031e-16]], + "weights": [1.73205080756888], + }, + 2: { + "points": [ + [-0.5, -0.288675134594813], + [0.5, -0.288675134594813], + [0.0, 0.577350269189626], + ], + "weights": [0.577350269189626, 0.577350269189626, 0.577350269189626], + }, + 3: { + "points": [ + [-4.16333634234434e-17, -0.390110349270987], + [0.337845472747895, 0.195055174635493], + [-0.337845472747895, 0.195055174635494], + [-0.725271359470687, -0.418735614625928], + [0.725271359470688, -0.418735614625927], + [0.0, 0.837471229251855], + ], + "weights": [ + 0.386908262797819, + 0.386908262797819, + 0.386908262797819, + 0.190442006391806, + 0.190442006391806, + 0.190442006391806, + ], + }, + 4: { + "points": [ + [-4.16333634234434e-17, -0.390110349270987], + [0.337845472747895, 0.195055174635493], + 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