G. Palmiotti

Simplified spherical harmonics in the variational nodal method

The multigroup simplified spherical harmonics equations with anisotropic scattering are derived from a variational principle that preserves nodal balance. The resulting equations are discretized using a Ritz procedure with spatial trial functions that are complete polynomials within the nodes and on the interfaces. The resulting equations are case in a response matrix form and incorporated as an option of the variational nodal spherical harmonics code VARIANT. Fixed source and multigroup eigenvalue calculations are performed on benchmark problems. The accuracy and computational efficiency of spherical harmonic and simplified spherical harmonic approximations are compared, and the compensating effects of spatial and angular truncation errors are examined. The results indicate that in most situations, simplified and standard spherical harmonics results of the same order are in close agreement, while the use of simplified spherical harmonics substantially reduces computing costs.

An integral form of the variational nodal method

Abstract An integral form of the variational nodal method is formulated, implemented, and tested. The method combines an integral transport treatment of the even-parity flux within the spatial node with an odd-parity spherical harmonics expansion of the Lagrange multipliers at the node interfaces. The response matrices that result from this formulation are compatible with those in the VARIANT code at Argonne National Laboratory. Spatial discretization within each node allows for accurate treatment of homogeneous or heterogeneous node geometries. The integral method is implemented in Cartesian x-y geometry and applied to three benchmark problems. The method’s accuracy is compared to that of the standard spherical harmonic formulation of the variational nodal method, and the CPU and memory requirements of the two approaches are compared and contrasted. In general, for calculations requiring higher-order angular approximations, the integral method yields solutions with comparable accuracy while requiring substantially less CPU time and memory than the spherical harmonics approach.

A Finite Subelement Generalization of the Variational Nodal Method

Abstract The variational nodal method is generalized by dividing each spatial node into a number of triangular finite elements designated as subelements. The finite subelement trial functions allow for explicit geometry representations within each node, thus eliminating the need for nodal homogenization. The method is implemented within the Argonne National Laboratory code VARIANT and applied to two-dimensional multigroup problems. Eigenvalue and pin-power results are presented for a four-assembly Organization for Economic Cooperation and Development/Nuclear Energy Agency benchmark problem containing enriched UO2 and mixed oxide fuel pins. Our seven-group model combines spherical or simplified spherical harmonic approximations in angle with isoparametric linear or quadratic subelement basis functions, thus eliminating the need for fuel-coolant homogenization. Comparisons with reference seven-group Monte Carlo solutions indicate that in the absence of pin-cell homogenization, high-order angular approximations are required to obtain accurate eigenvalues, while the results are substantially less sensitive to the refinement of the finite subelement grids.

Numerical Optimization of Computing Algorithms of the Variational Nodal Method Based on Transformation of Variables

Abstract Numerical methods based on transformation of variables are developed to improve the computational efficiency of the variational nodal method (VNM). Reordering and orthogonal transformations of the nodal unknowns are found to reduce the coefficient matrices of VNM into block-diagonal forms. These forms make it possible to reduce greatly the number of floating-point operations in matrix manipulations and hence to reduce the computational times. The red-black response matrix acceleration by transformation of interface partial-current variables has been extended to three-dimensional geometries and higher orders of spatial and angular approximations. These combined methods are incorporated within the algorithms currently used in the variational nodal code VARIANT at Argonne National Laboratory. All primary algorithms ranging from the generation of response matrices to the iterative solution method for the response matrix equations are modified to implement the new formulation. The efficiency of the new methods is tested on eigenvalue problems by comparing the computation times of the new and existing methods. Three-dimensional calculations are performed in hexagonal and Cartesian geometry for various spatial and angular approximations. The test results show that very significant gains can be obtained especially for the coupling coefficient calculations in higher angular approximations. More than an order of magnitude reduction of the total computing time is achieved in the best case.

Interface Conditions for Spherical Harmonics Methods

Abstract A set of interface conditions is derived rigorously for the general spherical harmonics solution of the Boltzmann transport equation in three-dimensional Cartesian geometry. The derivation builds upon earlier work of Davidson and Rumyantsev to arrive at sets of interface conditions applicable to both even- and odd-order N spherical harmonics approximations. The exact set of conditions is compared to the approximate set currently employed in the odd-order N variational nodal code VARIANT, and the differences in accuracy and computational effort are summarized. The exact interface conditions are necessary for first-order implementations of spherical harmonics methods.

Red-Black Response Matrix Acceleration by Transformation of Interface Variables

Red-black algorithms for solving response matrix equations in one- and two-dimensional diffusion theory are examined. The definition of the partial currents in terms of the scalar flux and net currents is altered to introduce an acceleration parameter that modifies the values of the response matrix elements while leaving the flux and net current solutions unchanged. The acceleration parameter is selected for response matrices derived analytically for slab geometry and from the variational nodal method for both slab and x-y geometries to minimize the spectral radius of the red-black iteration matrix for homogeneous media. The optimal value is shown to be independent of the mesh spacing in the fine mesh limit and to be a function only of c, the scattering-to-total cross section ratio. The method is then generalized to treat multiregion problems by formulating an approximate expression for the optimum acceleration parameter and demonstrated for a series of benchmark diffusion problems.

Variational nodal transport methods with heterogeneous nodes

The variational nodal transport method is generalized for the treatment of heterogeneous nodes while maintaining nodal balances. Adapting variational methods to heterogeneous nodes requires the ability to integrate over a node with discontinuous cross sections. Integrals are evaluated using composite Gaussian quadrature rules, which permit accurate integration while yielding acceptable computing times. Allowing structure within a nodal solution scheme avoids some of the necessity of cross-section homogenization and more accurately defines the intranodal flux shape. Ideally, any desired heterogeneity can be constructed within the node, but in reality, the finite set of basis functions limits the intranodal complexity that can be modeled. Comparison tests show that the heterogeneous variational nodal method provides accurate results for moderate heterogeneities, even if some improvements are needed for very difficult configurations.