On model-order reduction in neutronic systems via POD-galerkin projection

Abstract

Numerical simulation has become an indispensable tool in a wide range of research areas and industries. In nuclear engineering, such simulations are important to understand the behavior of nuclear reactors under different conditions and, accordingly, to develop optimized designs with established safe operational limits. Estimating quantities of interest like neutron core power, and temperature distributions requires the solution of a set of partial differential equations that model the nuclear reactor physics. These equations can be solved deterministically using a variety of phase-space discretization techniques and, generally, a sufficiently "fine'' phase-space grid is needed to obtain a desired accuracy. However, a brute-force pursuit of accuracy using ever finer grids results in ever larger algebraic systems that can quickly become too expensive to solve. This expense is multiplied for applications that require repeated simulations, such as design optimization and uncertainty quantification. Reduced-Order Models (ROMs) were developed to overcome the high computational cost of numerical simulation for neutronic systems by providing a rapid approximation of the simulated output. The work presented here was primarily exploratory in nature, and the primary goal was to understand the applicability of various model reduction techniques to specific problems of interest in reactor physics. In particular, steady-state and transient neutronic of varying levels of complexity have been analyzed. Efforts began by exploring a 1-D model of a reactor subject to a ramp insertion of reactivity, for which POD-Galerkin projection was tested and selected for more complex problems. These problems included the classic 2-D TWIGL and LRA transient diffusion benchmarks, for which a fine grid solution of the diffusion equations served as the full-order model (FOM) and a source of data for construction of ROMs. To reduce the cost of constructing and applying ROMs for nonlinear problems, the hyper-reduction technique known as Discrete Empirical Interpolation Method (DEIM) and its matrix version were used with POD-Galerkin projection. Although most of the work presented was based on the diffusion equation, a preliminary application to the transport equation was also conducted. A critical difference of the application of the ROM to the transport equation is that the FOM relies on matrix-free operators, which can complicate the use of POD-Galerkin projection. However, the methods developed proved to be applicable to the implementation. Importantly, although the POD-Galerkin method is an intrusive ROM technique, the implemented ROM did not require any major changes to the existing code, making it a trivially intrusive technique. Rather, access only to the system operator, which represents all of the underlying system discretization, was needed. Although many individual cases were considered, the primary conclusions to make are that (1) POD-Galerkin projection of diffusion- or transport-based models yield ROMs that approximate core powers with errors less than the 1% and with computational speedups that range from approximately 3 to 50 depending on the type and numerical fidelity of the underlying FOM and (2) such ROMs can be implemented in the many modern diffusion or transport codes (e.g., the various MOOSE-enabled tools based on finite-element methods¹, or the highly-scalable Denovo discrete-ordinates code²).