The problem of discretizing the kinetic energy variable in nuclear engineering is difficult because, due to resonance effects, the cross sections that characterize neutron and photon interaction with matter can vary by orders of magnitude for small changes in particle energy. (Imagine throwing a dart at a dartboard and having the board increase in size by a factor of 3,000 when you throw the dart at a slightly different speed.) Numerous resonances make brute-force resolution infeasible for full-geometry calculations.
Discretizing highly oscillatory functions – such as the cross sections as functions of particle energy – with relatively few unknowns is a difficult problem. Sub-problems often are used to define a closure, but these carry their own assumptions and approximations.
This talk presents a non-standard finite element method where unknowns have discontiguous support. The support is chosen to minimize variance in solution-like quantities within an element. Machine learning is used to determine this support.
We apply this method to nuclear reactor problems and find first-order convergence in energy even at low unknown counts. We can attain 10^-4 relative error using around 200 energy unknowns in this resonance region.
While the method was developed for nuclear engineering problems, we explore its properties and look forward to a future where machine learning algorithms take on a larger role in mesh generation generally.
