Engineering topics
Kooreman, Gabriel
Publications and source records attributed to Kooreman, Gabriel.
A Full Multigrid Method for the Self-Adjoint Angular Flux Equation Based on Cellwise Block Jacobi Iteration
Previous work by Densmore et al. has demonstrated that multigrid methods for the self-adjoint angular flux (SAAF) equation with cellwise block Jacobi smoothing can be extremely effective at solving for the angular flux in one dimension. The extension of that work to two dimensions demonstrated that while the smoother works well, the use of a direct coarse-grid approximation can severely limit the rate of convergence for optically thin cells. In that paper, this convergence limit was evaded through the use of a partially Galerkin coarse-grid approximation. However, relying on a Galerkin coarse-grid approximation is not ideal as constructing the coarse-grid discretization in this approach is complicated. The motivation for this summary is to find a method that can work without the need for a Galerkin coarse-grid approximation. The Full Multigrid (FMG) method is well known as a method that can theoretically solve a problem to its discretization accuracy with optimal linear computational complexity, that is in just O(N) operations for N unknowns. Indeed, we develop an FMG method for the 2-D SAAF equation that yields discretization accuracy and does not require the use of a Galerkin coarse-grid approximation. Until recently, aside from proofs about the order of the error for FMG, little work had been done in quantifying when and how FMG can obtain discretization accuracy. However, recent work by Rodrigo et al. has introduced a Fourier analysis technique called worst-case right-hand side analysis used to calculate the FMG accuracy measure. The FMG accuracy measure is the upper bound on the constant of proportionality between the error of the FMG solution and the discretization error. In this summary, we develop and implement an FMG method for the 2-D Cartesian SAAF equations. We then perform worst-case right-hand side analysis and give numerical examples that demonstrate under what circumstances the method can solve the 2-D SAAF equations to discretization accuracy, for optically thin cells, without the need for a Galerkin coarse-grid approximation. (authors)
Error Propagation for Response Matrix Methods in the Many Node Limit
The response matrix method has been a useful solution method for radiation transport problems for decades. The method is desirable for many reasons. It can be extended to both fixed source and eigenvalue problems. In addition, a key aspect of the method is a shift of computational expense to a precomputation phase. Methods and codes employing response matrix or response matrix-like methods have enjoyed formidable computational efficiency while solving large problems. However, current applications of the method rely on large node sizes and few nodes to maximize computational efficiency. As part of a desire to make these methods more amenable to applications such as parallel computing in the future, the number of nodes in problems may increase, and node size will decrease. The effect of this change has not been well studied except for a few cases. In the past, it was determined that a poor approximation of angular dependence of flux on node interfaces could lead to large error propagation for fixed source problems, but a newer study indicates that changing node sizes as well as number of nodes poses little challenge for response methods in the eigenvalue case. This paper presents the unintuitive result that for response matrix methods increasing the spatial resolution of the problem may dramatically decrease accuracy - this effect is demonstrated on a simple analytic 1-D benchmark problem. (authors)
Application of the Hybrid Diffusion-Transport Spatial Homogenization Method to a Pressurized Water Reactor Benchmark Problem - Paper 70
The recently developed Hybrid Diffusion-Transport Spatial Homogenization (DTH) method was previously tested on a benchmark problem typical of a boiling water reactor. In this summary, the DTH method is tested in a 1-D benchmark problem typical of a pressurized water reactor. The method recreates the heterogeneous flux to within 1% average flux error and 34 pcm eigenvalue error in half the time as a heterogeneous transport calculation. (authors)