Source code for felupe.quadrature._gausslegendre

# -*- coding: utf-8 -*-
"""
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This file is part of felupe.

Felupe is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

Felupe is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with Felupe.  If not, see <http://www.gnu.org/licenses/>.

"""

from string import ascii_lowercase

import numpy as np

from . import Scheme


[docs]class GaussLegendre(Scheme): "A n-dimensional Gauss-Legendre quadrature rule." def __init__(self, order: int, dim: int, permute: bool = True): """Arbitrary `order` Gauss-Legendre quadrature rule of `dim` 1, 2 or 3 on the interval [-1, 1] as approximation of ∫ f(x) dx ≈ ∑ f(x_a) w_a (1) with `a` quadrature points `x_a` and corresponding weights `w_a`. """ # integration point weights and coordinates if dim not in [1, 2, 3]: raise ValueError("Wrong dimension.") x, w = np.polynomial.legendre.leggauss(1 + order) points = ( np.stack(np.meshgrid(*([x] * dim), indexing="ij"))[::-1].reshape(dim, -1).T ) idx = list(ascii_lowercase)[:dim] weights = np.einsum(", ".join(idx), *([w] * dim)).ravel() if permute and order == 1 and dim == 2: points = points[[0, 1, 3, 2]] weights = weights[[0, 1, 3, 2]] if permute and order == 1 and dim == 3: points = points[[0, 1, 3, 2, 4, 5, 7, 6]] weights = weights[[0, 1, 3, 2, 4, 5, 7, 6]] if permute and order == 2 and dim == 3: vertices = np.array([0, 2, 8, 6, 18, 20, 26, 24]) edges = np.array([1, 5, 7, 3, 19, 23, 25, 21, 9, 11, 17, 15]) faces = np.array([12, 14, 10, 16, 4, 22]) volume = np.array([13]) permute = np.concatenate((vertices, edges, faces, volume)) points = points[permute] weights = weights[permute] super().__init__(points, weights)
[docs] def inv(self): "Return the inverse quadrature scheme." points = self.points.copy() points[self.points != 0] = 1 / points[self.points != 0] return Scheme(points, self.weights)
[docs]class GaussLegendreBoundary(GaussLegendre): "A n-dimensional Gauss-Legendre quadrature rule on boundaries." def __init__(self, order: int, dim: int, permute: bool = True): """Arbitrary `order` Gauss-Legendre quadrature rule of `dim` 1, 2 or 3 on the interval [-1, 1] as approximation of ∫ f(x) dx ≈ ∑ f(x_a) w_a (1) with `a` quadrature points `x_a` and corresponding weights `w_a`. """ super().__init__(order=order, dim=dim - 1, permute=permute) # reset dimension self.dim = dim if self.dim == 2 or self.dim == 3: # quadrature points projected onto first edge of a quad # or onto first face of a hexahedron self.points = np.hstack((self.points, -np.ones((len(self.points), 1)))) else: raise ValueError("Wrong dimension.")