Acceleration of 3D pin-by-pin calculations based on the heterogeneous variational nodal method

The pin-by-pin method with homogenization performed at the pin cell level is of considerable interest in thermal reactor physics calculations. However, significant computational costs are required in pin-by-pin calculations due to the fine spatial meshes. To resolve this issue, we present in this paper the three-dimensional (3D) heterogeneous variational nodal method (VNM), along with dedicated acceleration methods. The combined methods can be utilized to perform pin-by-pin diffusion calculations more efficiently. 3D response matrix (RM) equations are formulated, allowing for incorporation of multiple pin cells into one coarse node without altering the original pin cell configuration. Within the nodes, finite elements in the x-y plane and orthogonal polynomials in the axial direction are employed to describe the piecewise constant heterogeneous geometry. On the nodal interfaces, orthogonal polynomials in the x-y interfaces and piecewise constants in the axial interfaces are adopted to approximate neutron current distributions. The resulting RM equations are solved by the standard Red-Black Gauss-Seidel (RBGS) iteration. Matrix reordering (MR) acceleration and parallelization tailored to the RM formation are incorporated. The coarse nodes acceleration (CNA) is investigated by combining homogenized pin cell nodes into larger heterogeneous nodes. A series of meshing schemes are examined with a small modular reactor core problem. Results show that the implementation of MR and parallelization effectively reduces the RM formation time. Besides, with sufficient radial interface expansion order, CNA is able to reproduce the results obtained with fine node calculations. Furthermore, it is demonstrated that judicious choice of coarse nodes substantially accelerate the RBGS iteration. The combined acceleration schemes achieve favorable accuracy-efficiency trade-off.