This repo serves as a collection of well-tested triangle cubature rules, i.e. numerical integration schemes for integrals of the form
where
Using the cubature schemes is fairly simple.
from triangle_cubature.cubature_rule import CubatureRuleEnum
from triangle_cubature.integrate import integrate_on_mesh
from triangle_cubature.integrate import integrate_on_triangle
import numpy as np
# specifying the mesh
coordinates = np.array([
[0., 0.],
[1., 0.],
[1., 1.],
[0., 1.]
])
elements = np.array([
[0, 1, 2],
[0, 2, 3]
], dtype=int)
# defining the function to be integrated
# NOTE the function must be able to handle coordinates as array
# of shape (N, 2)
def constant(coordinates: np.ndarray):
"""returns 1"""
return np.ones(coordinates.shape[0])
# integrating over the whole mesh
integral_on_mesh = integrate_on_mesh(
f=constant,
coordinates=coordinates,
elements=elements,
cubature_rule=CubatureRuleEnum.MIDPOINT)
# integrating over a single triangle, e.g.
# in this case, the "first" element of the mesh
integral_on_triangle = integrate_on_triangle(
f=constant,
triangle=coordinates[elements[0], :],
cubature_rule=CubatureRuleEnum.MIDPOINT)
print(f'Integral value on mesh: {integral_on_mesh}')
print(f'Integral value on triangle: {integral_on_triangle}')
The available cubature rules can be found in triangle_cubature/cubature_rule.py
.
CubatureRuleEnum.MIDPOINT
- degree of exactness: 1
- Ref: [1]
CubatureRuleEnum.LAUFFER_LINEAR
- degree of exactness: 1
- Ref: [1]
CubatureRuleEnum.SMPLX1
- degree of exactness: 2
- Ref: [1]
To run auto tests, you do
python -m unittest discover tests/auto/
The unit tests use
sympy
to verify the degree of exactness of the implemented cubature rules, i.e. creates random polynomials$p_d$ of the expected degree of exactness$d$ and compares the exact result of$\int_K p_d(x, y) ~\mathrm{d}x ~\mathrm{d}y$ to the value obtained with the cubature rule at hand.
- [1] Stenger, Frank. 'Approximate Calculation of Multiple Integrals (A. H. Stroud)'. SIAM Review 15, no. 1 (January 1973): 234-35. https://doi.org/10.1137/1015023. p. 306-315