Triangle Cubature Rules

This repo serves as a collection of well-tested triangle cubature rules, i.e. numerical integration schemes for integrals of the form

$$ \int_K f(x, y) ~\mathrm{d}x ~\mathrm{d}y, $$

where $K \subset \mathbb{R}^2$ is a triangle. All cubature rules are based on [1].

Usage

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}')

Available Rules

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]
• CubatureRuleEnum.DAYTAYLOR
  • degree of exactness: 6
  • Ref: [2]

(Unit) Tests

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.

References

• [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
• [2] D.M. Day and M.A. Taylor 'A new 11 point degree 6 formula for the triangle', PAMM Proc. Appl. Math. Mech. 7 1022501-1022502 (2007).