Acceleration of pin-by-pin calculations with the heterogeneous variational nodal method

Due to preferable resolution and accuracy, the pin-by-pin method performing the homogenization over the pin cells is of considerable interest. However, significant computational resources are required due to fine spatial meshes, making the method costly for practical use. As an effort to perform pin-by-pin calculations more efficiently, a three-dimensional (3D) heterogeneous variational nodal method (VNM) is presented. 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 nodal interfaces, orthogonal polynomials in x-y and piecewise constants in axial are adopted to approximate neutron current distributions. The method constructs response matrices (RM) by combining multiple pin cells into one coarse node. Piecewise constant XSS are employed within each coarse node such that the original pin cell configuration is not altered. The resulting RM equations are iteratively solved in within group (WG) iterations by the standard Red-Black Gauss-Seidel (RBGS) algorithm. The matrix re-ordering (MR) acceleration tailored to the RM formation is employed. The coarse nodes acceleration (CNA) is incorporated to accelerate inner iterations, and the coarse mesh finite difference (CMFD) method is utilized to speed up outer iterations. A series of CNA meshing schemes are examined with a 3D pin-by-pin problem. Results show that the implementation of MR effectively reduces the RM formation time. Besides, with sufficient radial interface expansion order, CNA can reproduce the results obtained with fine node calculations. Furthermore, it is demonstrated that CNA substantially accelerates the WG iteration, and the CMFD algorithm is feasible to accelerate outer iterations. The combined acceleration methods yield prominent accuracy-efficiency trade-off than the fine mesh calculation.