| Name | triangle-cubature JSON |
| Version |
1.1.0
JSON |
| download |
| home_page | None |
| Summary | cubature rules on triangles |
| upload_time | 2024-08-26 12:53:23 |
| maintainer | None |
| docs_url | None |
| author | None |
| requires_python | >=3.9.18 |
| license | None |
| keywords |
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# 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.
```python
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]
## (Unit) Tests
To run auto tests, you do
```sh
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
Raw data
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"description": "# Triangle Cubature Rules\nThis repo serves as a collection of well-tested triangle cubature rules,\ni.e. numerical integration schemes for integrals of the form\n\n$$\n\\int_K f(x, y) ~\\mathrm{d}x ~\\mathrm{d}y,\n$$\n\nwhere $K \\subset \\mathbb{R}^2$ is a triangle.\nAll cubature rules are based on [1].\n\n## Usage\nUsing the cubature schemes is fairly simple.\n\n```python\nfrom triangle_cubature.cubature_rule import CubatureRuleEnum\nfrom triangle_cubature.integrate import integrate_on_mesh\nfrom triangle_cubature.integrate import integrate_on_triangle\nimport numpy as np\n\n# specifying the mesh\ncoordinates = np.array([\n [0., 0.],\n [1., 0.],\n [1., 1.],\n [0., 1.]\n])\n\nelements = np.array([\n [0, 1, 2],\n [0, 2, 3]\n], dtype=int)\n\n\n# defining the function to be integrated\n# NOTE the function must be able to handle coordinates as array\n# of shape (N, 2)\ndef constant(coordinates: np.ndarray):\n \"\"\"returns 1\"\"\"\n return np.ones(coordinates.shape[0])\n\n\n# integrating over the whole mesh\nintegral_on_mesh = integrate_on_mesh(\n f=constant,\n coordinates=coordinates,\n elements=elements,\n cubature_rule=CubatureRuleEnum.MIDPOINT)\n\n# integrating over a single triangle, e.g.\n# in this case, the \"first\" element of the mesh\nintegral_on_triangle = integrate_on_triangle(\n f=constant,\n triangle=coordinates[elements[0], :],\n cubature_rule=CubatureRuleEnum.MIDPOINT)\n\nprint(f'Integral value on mesh: {integral_on_mesh}')\nprint(f'Integral value on triangle: {integral_on_triangle}')\n\n```\n\n## Available Rules\nThe available cubature rules can be found in `triangle_cubature/cubature_rule.py`.\n\n- `CubatureRuleEnum.MIDPOINT`\n - degree of exactness: 1\n - Ref: [1]\n- `CubatureRuleEnum.LAUFFER_LINEAR`\n - degree of exactness: 1\n - Ref: [1]\n- `CubatureRuleEnum.SMPLX1`\n - degree of exactness: 2\n - Ref: [1]\n\n## (Unit) Tests\nTo run auto tests, you do\n```sh\npython -m unittest discover tests/auto/\n```\n\n> The unit tests use `sympy` to verify the degree of exactness of the\n> implemented cubature rules, i.e. creates random polynomials $p_d$ of the \n> expected degree of exactness $d$ and compares the exact result of\n> $\\int_K p_d(x, y) ~\\mathrm{d}x ~\\mathrm{d}y$ to the value obtained\n> with the cubature rule at hand.\n\n## References\n- [1] Stenger, Frank.\n 'Approximate Calculation of Multiple Integrals (A. H. Stroud)'.\n SIAM Review 15, no. 1 (January 1973): 234-35.\n https://doi.org/10.1137/1015023. p. 306-315\n",
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