Fourier basis functional expansion tally filter

[tjlaboss]

Description

Part 1: SpatialFourierFilter

Proposing a feature: A new functional expansion tally (FET) filter in the Fourier basis, similar to SpatialLegendreFilter. This is a well-understood, orthonormal orthogonal basis with two attractive properties.

• Sines and cosines are the solution to the Helmholtz equation in a homogeneous slab.
• The basis is periodic. This could lend itself well to problems with periodic boundary conditions and leads to the next part.

Part 2: ArclengthFourierFilter

Or CircumferentialFourierFilter: A new FET filter similar to the SpatialFourierFilter proposed above, but instead of being over a line segment, the functional expansion is over the circumference of a cylinder.

• This essentially tracks the azimuthal dimension of a cylindrical MeshSurfaceFilter for a given radial surface.
• The periodicity of the basis function is essential to ensure continuity at $0 = \theta = 2\pi$.

Status

The Fourier basis is briefly mentioned in Griesheimer's seminal work, but as far as I can tell, has not been implemented before. Assuming this doesn't already exist in a branch somewhere, I intend to make a PR for the full feature in the foreseeable future.

Alternatives

• Could add an argument to use only odd (sin) or even (cos) modes.
• The arclength / circumferential filter could be a new abstract base class that other periodic function bases could inherit from.

Compatibility

The enhancement will not change existing APIs. It will add something new.

[MicahGale]

*orthonormal orthogonal. The norm of a component of a Fourier series is the series period, so in most cases the basis set isn't normalized.

This is probably another issue, but I think there should a system to easily create multi-dimensional separable FETs. Say you want to do a multi-D Legendre as:

$$ \psi(x,y,z) = P_x(x)P_y(y)P_z(z)$$

It would be good to have an easy way to build these basis sets. I could see for a fuel pin having Legendre in $R$, Fourier in $\theta$ and another in $Z$, without resorting to Zernike.