B-spline basis functions to interpolate nodal values (control points) in 1d

[Simon]

Dear all,

my objective is to do an 1d-approximation

F(x) = N_i(x) P_i

where N_i are cubic B-spline basis functions and P_i known nodal values (control points).

To assemble my linear system, I have to compute the first and second derivatives of F at the quadrature points of the triangulation.

Since the number of control points P_i is quite moderate (less than 20), I think it is reasonable to evaluate all basis functions N_i in lieu of implementing an entire finite element class.

I scanned the dealii manual but could only find a CSpline class.

Just to make sure:

B-Spline (basis) functions are not implemented in dealii yet, right?

So I probably will have to rely on an external library.

Can someone make a library recommendation?

I read about tinyspline and splinelib, but have no knowledge regarding their performance, or if they are appropriate for usage in a dealii-program.

As I said, I mainly want to

(i) interpolate control points with B-spline basis functions and

(ii) evaluate first and second derivatives at arbitrary points in the knot interval.

Best

Simon

[Wolfgang Bangerth]

On 2/15/23 06:50, Simon wrote:
>
> my objective is to do an 1d-approximation
> F(x) = N_i(x) P_i
> where N_i are cubic B-spline basis functions and P_i known nodal values
> (control points).
> To assemble my linear system, I have to compute the first and second
> derivatives of F at the quadrature points of the triangulation.
> Since the number of control points P_i is quite moderate (less than 20), I
> think it is reasonable to evaluate all basis functions N_i in lieu of
> implementing an entire finite element class.
>
> I scanned the dealii manual but could only find a CSpline class

> <https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.dealii.org%2Fcurrent%2Fdoxygen%2Fdeal.II%2FclassFunctions_1_1CSpline.html&data=05%7C01%7CWolfgang.Bangerth%40colostate.edu%7C9ac052e12d814289384108db0f5b957c%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C0%7C638120658249146546%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=NqcY%2FpN5QJnuhVYWG3BiLLKxYUZIZNyYDCKKEDq3iYI%3D&reserved=0>.

> Just to make sure:
> B-Spline (basis) functions are not implemented in dealii yet, right?

Simon -- that is correct. I suspect (but don't know) that you also want to use
spline basis functions that span across more than one cell; that is an
uncommon operation for finite element libraries and there is not much support
implemented for these kinds of things -- in particular, there are none of the
spline basis functions one might use for this.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Luca Heltai]

Simon,

If you are willing to do some work, Bernstein basis functions are implemented in deal.II (which are B-spline basis with C0 continuity).

You could exploit BezierExtractor

https://onlinelibrary.wiley.com/doi/full/10.1002/nme.2968

and use the facilities of deal.II to work with cubic B-Splines. This is implemented in deal.II, and available with poor documentation, here:

https://github.com/mathLab/IGA-dealii

You can learn about the data structures by looking at

https://iris.sissa.it/retrieve/dd8a4bf7-09b3-20a0-e053-d805fe0a8cb0/1963_35160_Tezzele_tesi.pdf

Let me know if you need assistance in using the relative classes.

There are two examples that solve a poisson problem and an obstacle problem. We’ll try to make this a gallery example to make it more accessible.

L.

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