Add integration of physical function

[markriegler]

Overview

Integration of physical function

Checklists

• [ ] Documentations are up-to-date.
• [ ] Added example(s)
• [x] Added test(s)

Summary by CodeRabbit

• New Features

  • Enhanced the integration functionality to accept spline objects, improving usability for spline integration.
  • Expanded documentation for the integration function, including detailed parameter descriptions and type safety checks.

• Tests

  • Introduced a new test case to validate the integration of a specific parabolic function over a defined domain, ensuring the accuracy of the integration process.

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Walkthrough

The recent updates enhance the physical_function in splinepy for improved spline integration by introducing a spline parameter for greater clarity and usability. The function's logic is refined to handle various spline types and optimize performance using np.einsum. Additionally, a new test function validates the integration of a parabolic function over a rectangular domain, ensuring the implementation's correctness and robustness.

Changes

Files	Change Summary
splinepy/helpme/integrate.py	Updated physical_function to include a spline parameter, adding comprehensive parameter documentation and error handling. Enhanced internal logic for better integration handling.
tests/helpme/test_integrate.py	Added test_physical_function_integration to validate integration of a parabolic function over a defined rectangular domain, ensuring correctness of the integration logic with B-splines.

Poem

🐰 In a world of splines, oh what a delight,
With functions so clear, they shine ever bright.
Integrate with glee, let the numbers dance,
In the land of math, give new features a chance!
From parabolas bold, to knots that entwine,
Hooray for the changes, all perfectly fine! 🌟

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[clemens-fricke]

I have added a quick test where I integrate a f(x)=1 function over the physical space. Which should be just the volume integral. At first, it looked good but then, I saw that the error was inconsistent between the analytical and numerical integral.

Example implementation

import splinepy
integral_analytical = ((4/3) * np.pi * r**3) * revolve_percentage

def one_function(points):
    return np.ones((1,1))

# quarter sphere
quart_circle = splinepy.helpme.create.sphere(outer_radius=r, angle=(360 * revolve_percentage))

integral = splinepy.helpme.integrate.physical_function(
    quart_circle, one_function
)

I had tested this over multiple revolve percentages and radii. This is the result.

with the error being calculated by np.sqrt((integral[0]-integral_analytical)**2) shown on the z-axis (upwards).

I think the error is too high, but I don't really know what you actually do in the physical_integration. I am also not certain if this is just such a bad approximation or if there is a bug in the implementation.

[markriegler]

Thanks for doing the test.

The issue might have something to do with the splinepy.helpme.create.parametric_view which I used for the integration. I just saw that the function returns a BSpline with degree 1 in every direction, which is different to the degrees of sphere (which has degrees [1 2 2]). I'll have to check

[j042]

Thanks for doing the test.

The issue might have something to do with the splinepy.helpme.create.parametric_view which I used for the integration. I just saw that the function returns a BSpline with degree 1 in every direction, which is different to the degrees of sphere (which has degrees [1 2 2]). I'll have to check

You should be able to pass an argument to get a conforming basis parametric view. But why would you need that?

[clemens-fricke]

Thanks @jzwar. We integrated the fixes from #459 into this PR. The error shown above is still partly present, but it seems to be a numerical error. The relative error is at least e-5, and can be reduced by adding more knots into the spline.

Also thank @markriegler for starting this.