Development and validation of the three-dimensional dynamic code—KIKO3D

Plane waves and response matrices

Recent progress in the development of coarse-mesh nodal methods for the numerical solution of the neutron diffusion and transport equations is reviewed. Compared with earlier nodal simulators, more recent nodal diffusion methods are characterized by the systematic derivation of spatial coupling relationships which are entirely consistent with the multi-group diffusion equation. These relationships most often are derived by developing approximations to the one-dimensional equations obtained by integrating the multidimensional diffusion equation over directions transverse to each coordinate axis. The objective of this research is to develop accurate and efficient spatial homogenization method for coarse mesh analysis of light water reactors. More sophisticated methods for computing approximate equivalence parameters are also introduced and make use of nonlinear iterations between homogenized reactor calculations and local fixed-source calculations to compute equivalence parameters. This special feature induces the need for the study on homogeneous parameters of light water reactor which takes phenomena of different scale and their interaction into account by means of the nodal diffusion method.

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.

Group theoretical considerations originating from Wigner are applied to the few group diffusion equation (DE) and to the linear Boltzmann transport equation (TE). The irreducible subspaces invariant under the symmetries of the TE or DE determine specific incoming current patterns. If these patterns are used in the response matrix formulation, the response matrix will be diagonal. A method for obtaining a solution that belongs to a given irreducible subspace is provided. These considerations are utilized in a coarse-mesh program. It is shown that the solution to a boundary condition problem is determined by the irreducible decomposition of the boundary condition and by the solutions belonging to the irreducible boundary conditions. Finally, the solution of the TE in an infinite lattice is investigated and the results are utilized to solve the TE in a finite lattice.

The relative ease with which high-throughput data can be collected on the same sampling units has raised interest in relating high-dimensional data sets. This problem can be framed as a multivariate regression model with one data set treated as a response matrix and the other as a covariate matrix. The high-dimensionality of both the response and covariate matrices along with the complex correlation structures within each raises statistical challenges. Here, we review a stochastic partitioning method that addresses this problem by fitting a mixture of regression models with variable selection, thereby uncovering in a unified manner group structures and key relationships between the data sets. The problem of variable selection in this context poses added difficulties compared to standard regression settings, as the membership of objects into different components has to be learned simultaneously with the search of component-specific predictors. We emphasize the practical applicability of the method using real data and illustrate features of the associated R package SPaVS.

The asymptotic solution of the neutron transport equation is obtained for large near-critical domains D which possess a cellular, nearly periodic structure. A typical mean free path in D is taken to be of the same order of magnitude as a cell diameter, and these are taken to be small (of order ε) compared to a typical diameter of D. The solution is asymptotic with respect to the small parameter ε. It is a product of two functions, one determined by a detailed cell calculation and the other obtained as the solution of a time dependent diffusion equation. The diffusion equation contains precursor (delayed neutron) densities, equations for which are derived. The coefficients in the diffusion equation, which are determined using the results of the cell calculation, differ from those now used in engineering applications. The initial condition for the diffusion equation is derived, and the problem of determining the boundary condition is discussed.

The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed. In particular, it is proved that when stable alternating direction explicit schemes for solving linear parabolic PDEs are developed to the stochastic case, they retain their unconditional stability properties applying to stochastic advection-diffusion and diffusion SPDEs. Numerically, unconditional stable SADE techniques are significant for approximating the solutions of the proposed SPDEs because they do not impose any restrictions for refining the computational domains. The performance of the proposed methods is tested for stochastic diffusion and advection-diffusion problems, and the accuracy and efficiency of the numerical methods are demonstrated.

In this novel forecasting algorithm to calculate the unit‐impulse response matrix of an unbounded medium to analyse dynamic medium‐structure interaction in the time domain based on the substructure method, a finite‐element region adjacent to the structure–medium interface consisting of two or three rows only is introduced. Its exterior boundary is chosen to be similar to the interior boundary which coincides with the structure–medium interface. At similar boundaries of the unbounded medium the unit‐impulse response matrices are a function of the dimensionless time. The unit‐impulse response matrix at the exterior boundary can be forecasted from that at the interior boundary via this dimensionless time by taking advantage of the time delay of waves propagating between these two boundaries. The forecasting algorithm based solely on the finite‐element method is exact in the finite‐element sense in modelling the unbounded medium.

Our aim is to characterize those matrices that are the response matrix of a semi-positive definite Schr?dinger operator on a circular planar network. Our findings generalize the known results and allow us to consider both nonsingular and non diagonally dominant matrices as response matrices. To this end, we define the Dirichlet-to-Robin map associated with a Schr?dinger operator on general networks, and we prove that it satisfies the alternating property which is essential to characterize the response matrices.

An exact solution of the time dependent diffusion equation for a layered diffusive medium has been obtained by using a method based on the decomposition of the fluence rate in a series of eigenfunctions. Making use of this exact analytical solution inversion procedures have been implemented to retrieve the optical properties of the medium. Time resolved measurements of diffuse reflectance have been generated with Monte Carlo simulations. Inversion procedures have been carried out both for a single time resolved measurement and for multi-distance time resolved measurements. The characteristics of both procedures have been investigated. For the case of a two-layered medium the fitting procedure carried out on a single measurement showed that it is impossible to retrieve correctly all the optical parameters of the layered medium. The inversion procedure correctly retrieves the absorption coefficient of the second layer and the reduced scattering coefficient of the first layer. A significant improvement is obtained by using multi-distance measurements. A particular attention has been devoted to determine the optical properties of muscle with the subcutaneous fat layer.

Nodal Integral method for multi-group neutron diffusion equation in three dimensional cylindrical coordinate system

A finite element method for diffusion theory embracing nodal and difference methods

A response matrix spectral nodal method for energy multigroup X,Y-geometry discrete ordinates problems in non-multiplying media

In an earlier paper, the spherical harmonics method for the solution of the time-dependent transport equation with the Marshak boundary conditions was presented in order to investigate the effect of a strongly anisotropic scattering law on the slab thickness. Here, the previous work is extended to the study of the time-dependent problems in a homogeneous sphere with the same scattering function as used in the previous work. The time-dependent neutron transport equation is solved in the manner used for a critical sphere and all radii for a given time-dependent system are determined by finding the critical radii for the corresponding critical system. The PN calculations of the critical radii were carried out for various combinations of the anisotropy parameters and the fundamental time eigenvalues. Some indications of the accuracy of the method were given for the problem of interest and the variation of the radius with anisotropic scattering was studied. We also obtained numerical values of the critical radii in the range of (1-),1. Finally, some results were discussed and compared with those already obtained by various methods.

Analytical spatial reconstruction scheme for the coarse-mesh solutions generated by the constant spectral nodal method for monoenergetic discrete ordinates transport calculations in X,Y geometry fission–chain reacting systems