Justin Pounders PhD

Reactors exhibit complex transient behavior. This is primarily because of the multiple time scales that drive different aspects of the physics. Prompt fission neutrons, delayed fission neutrons, heat transfer and fluid dynamics all play an important role, yet the time scales evolving these constituents vary by orders of magnitude.

I am currently investigating methods that will make transient reactor simulations more efficient. This research will ultimately proceed along two fronts. Currently my focus is on developing simple yet robust methods for adaptively selecting time step sizes in neutron transport and diffusion equations. Importantly, these method are implicitly "aware" of the time scales of other physics in the problem. The second important piece of this work is developing efficient schemes for asynchronous communication between different physics solvers. This work will also include nonlinear mechanisms for capturing coupling effects between physics.

The method of characteristics (MOC) is a popular approach to numerically solving radiation transport problems described by a linear Boltzmann equation (e.g. neutron distributions in reactors). The challenging aspect of MOC is computing and storing the characteristic rays along which the solution is ultimately integrated. This problem becomes especially expensive in three dimensions.

In some recent work, I have revisited the method of "short" characteristics as a way of alleviating the overhead of ray tracing and storage. Specifically, one may construct locally high-order spatial solutions by hierarchically refining computational cells (e.g. triangles in 2D). This is a "refinement plus reconstruction" approach akin to some high-order finite volume techniques in computational fluid dynamics. A geometric interpretation of this approach also helps unify the method of short MOC with its more popular counterpart.

A distinct advantage of short MOC is that the problem is more amenable to local decoupling. Once the problem is deconstructed at the local level, technologies such as GP-GPUs may become useful as accelerators.

This line of work emerged from questioning the interpretation of an "eigenvalue" in nonlinear contexts. Specifically, the effective multiplication factor emerged from neutron transport in multiplying media as a way of getting an approximate solution from an otherwise singular equation. In nonlinear (multiphysics) settings, however, the mathematical interpretation of the multiplication factor as an eigenvalue becomes ambiguous at best.

It turns out that expressing some of these problems in the context of variational theory may provide a more rigorous interpretation. In the simple problem of neutron diffusion coupled with heat conduction, for example, one may recast the "nonlinear eigenvalue" simply as a Lagrange multiplier appearing in the minimization of a variational functional. For me, this line of thinking led to a more fundamental question of whether the multiplication factor (eigenvalue) is even necessary as we move to more first-principles-based computational models. Mathematically I believe the answer is "no." Whether it remains a useful artifact, however, should be considered from an engineering/application perspective rather than one of mathematical necessity.