Papers by Rudnei Dias da Cunha
Progress in Nuclear Energy, 2017
In this work, the formulation of the Analytical Discrete Ordinates (ADO) method for two-dimension... more In this work, the formulation of the Analytical Discrete Ordinates (ADO) method for two-dimensional fixed-source neutron transport problems is extended to problems with anisotropic scattering. The methodology is developed from the discrete ordinates approximation of the two-dimensional transport, whose equations are integrated transversally within homogenized regions of the domain, yielding onedimensional equations for the average angular fluxes. Such one-dimensional equations are solved by the ADO method, after considering approximations for the transverse leakage terms, and their solutions are written in terms of eigenvalues and eigenfunctions. The relevant feature of the ADO method for the isotropic scattering cases, which reduced the order of the eigenvalue problems to half of the number of discrete directions, is also preserved for the present anisotropic cases. Explicit spatial variables expressions are derived for the average angular fluxes in regions of interest defined in the domain. The full domain solution is defined by the coupling of local solutions in different regions, without the use of sweeping procedures. As usual in nodal schemes, auxiliary equations are necessary and, in this case, the unknown region-edge angular fluxes are approximated by constants embedded in the source term. Numerical results for region-average scalar fluxes are obtained and compared with the ones available in the literature to illustrate the feasibility of keeping the computational efficiency already verified in treating problems with isotropic scattering, with the use of the ADO method.
Closed-form solutions for nodal formulations of two dimensional transport problems in heterogeneous media
Annals of Nuclear Energy, 2015
Abstract We present a study of two dimensional, fixed-source neutron transport problems on a hete... more Abstract We present a study of two dimensional, fixed-source neutron transport problems on a heterogeneous medium. It is an extension of a methodology used for solving neutron transport problems on an homogeneous medium, that combines nodal schemes with explicit solutions of the transversal integrated equations via the Analytical Discrete Ordinates method (ADO). We consider closed-form solutions of the integrated discrete ordinates transport equation on a two dimensional cartesian geometry, together with a level symmetric quadrature scheme, on each region of interest, in the domain, possibly characterised by different materials. In this work, each solution in a region is coupled with that of its neighbouring regions to provide the whole solution, without resorting to using iterative schemes. The terms involving unknown angular fluxes that arise using nodal schemes, assumed as constant function approximations, are added to the source term. The model proposed leads to a considerable reduction of the order of the associated eigenvalue problems written as perturbations of diagonal matrices, and the solutions obtained are explicit in terms of the spatial variables. Analytical expressions are also derived for the elementary solutions. The numerical results obtained are shown to be in good agreement with other results available in the literature.
Explicit formulation of a nodal transport method for discrete ordinates calculations in two-dimensional fixed-source problems
Kerntechnik, 2014
In this work a study of two-dimensional fixed-source neutron transport problems, in Cartesian geo... more In this work a study of two-dimensional fixed-source neutron transport problems, in Cartesian geometry, is reported. The approach reduces the complexity of the multidimensional problem using a combination of nodal schemes and the Analytical Discrete Ordinates Method (ADO). The unknown leakage terms on the boundaries that appear from the use of the derivation of the nodal scheme are incorporated to the problem source term, such as to couple the one-dimensional integrated solutions, made explicit in terms of the x and y spatial variables. The formulation leads to a considerable reduction of the order of the associated eigenvalue problems when combined with the usual symmetric quadratures, thereby providing solutions that have a higher degree of computational efficiency. Reflective-type boundary conditions are introduced to represent the domain on a simpler form than that previously considered in connection with the ADO method. Numerical results obtained with the technique are provided...
An analytical approach for a nodal formulation of a two-dimensional fixed-source neutron transport problem in heterogeneous medium
Kerntechnik, 2015
A nodal formulation of a fixed-source two-dimensional neutron transport problem, in Cartesian geo... more A nodal formulation of a fixed-source two-dimensional neutron transport problem, in Cartesian geometry, defined in a heterogeneous medium, is solved by an analytical approach. Explicit expressions, in terms of the spatial variables, are derived for averaged fluxes in each region in which the domain is subdivided. The procedure is an extension of an analytical discrete ordinates method, the ADO method, for the solution of the two-dimensional homogeneous medium case. The scheme is developed from the discrete ordinates version of the two-dimensional transport equation along with the level symmetric quadrature scheme. As usual for nodal schemes, relations between the averaged fluxes and the unknown angular fluxes at the contours are introduced as auxiliary equations. Numerical results are in agreement with results available in the literature.
Computação paralela : uma breve introdução
Anais do XI International Symposium on Computer Architecture and High Performance Computing (SBAC-PAD 1999), Sep 29, 1999
Jn this paper, we investigate the parallel solution to the rotating internai ftow problems, using... more Jn this paper, we investigate the parallel solution to the rotating internai ftow problems, using the Navier-Stokes equations as proposed in [ST83] and [SPE85]. A Run ge~Kutta time-stepping scheme was applied to the equations and both sequential and message-passing implementations were developed, the latter using MPI , and were tested on an SGI Origin200 distributed, global shared memory parallel computer. The results show that our approach to p arallelize the sequential implementation requires little effort whilst providing good results even for medium-sized problerns, on this particular computer.
Fuzzy Sets and Systems
We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first present... more We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in N.
Journal of Fixed Point Theory and Applications, Sep 27, 2021
We prove that the idempotent Markov operator generated by contractive max plus normalized iterate... more We prove that the idempotent Markov operator generated by contractive max plus normalized iterated function system (IFS) is also a contractive map w.r.t. natural metrics on the space of idempotent measures. This gives alternative proofs of the existence of invariant idempotent measures for such IFSs.
O funcionamento de um reator nuclear esta associado a uma reacao em cadeia sustentavel, ou seja, ... more O funcionamento de um reator nuclear esta associado a uma reacao em cadeia sustentavel, ou seja, o numero de neutrons produzidos deve ser igual ao numero de neutrons que sao absorvidos ou que saem do sistema. [...]
Visualização tridimensional em dinâmica de fluídos computacional
We present algorithms to compute approximations of invariant measures and its attractors for IFS ... more We present algorithms to compute approximations of invariant measures and its attractors for IFS and GIFS, using the deterministic algorithm in a tractable way, with code optimization strategies and use of data structures and search algorithms. The results show that these algorithms allow the use of these (G)IFS in a reasonable running time.
On computing the effective multiplication factor using the ADO method
Annals of Nuclear Energy, 2022
Abstract In this work, the Analytical Discrete Ordinates method, ADO method, is applied to evalua... more Abstract In this work, the Analytical Discrete Ordinates method, ADO method, is applied to evaluate the effective multiplication factor in criticality problems of a nuclear reactor. To address this class of problems, the multigroup ADO formulation in multislab geometry is here derived taking into account real and complex spectrum. Two approaches are presented to find the desired dominant k-eigenvalue. In the first one, a criticality (characteristic) equation is derived from the complete original equation defined in multiplying media and a combined scheme, using regula-falsi and secant root-finding methods, is used to extract a root of that equation. Such scheme is shown to be very efficient and provided high-quality benchmark results, from lower quadrature order approximations than other schemes available in the literature, as found by the application of the ADO formulation also in several other transport problems. As a second proposed approach, the ADO solution is used along with the usual power iteration method in a simpler procedure. This simpler procedure avoids complex eigenvalues, but it is shown to be less accurate due to required approximations in the source approximation procedure.
Communications in Nonlinear Science and Numerical Simulation, 2020
We introduce a discrete version of the Hutchinson-Barnsley theory providing algorithms to approxi... more We introduce a discrete version of the Hutchinson-Barnsley theory providing algorithms to approximate the Hutchinson measure for iterated function systems (IFS) and generalized iterated function systems (GIFS), complementing the discrete version of the deterministic algorithm considered in our previous work.
On the Use of Explicit Formulations in the High Order Discrete Ordinates Solution of the Radiative Transfer Equation in Anisotropic Scattering Media
Procceedings of the 18th Brazilian Congress of Thermal Sciences and Engineering, 2020
Numerical Algorithms, 2020
We provide a new algorithm to generate images of the generalized fuzzy fractal attractors describ... more We provide a new algorithm to generate images of the generalized fuzzy fractal attractors described in [Str17]. We also provide some important results on the approximation of fractal operators to discrete subspaces with application to discrete versions of the deterministic algorithm for fractal image generation in the cases of IFS recovering the classical images from Barnsley et al., Fuzzy IFS from [Vrs92] and GIFS's from [Str16].
Annals of Nuclear Energy, 2017
In this work, recent results on the solution of fixed-source two-dimensional transport problems, ... more In this work, recent results on the solution of fixed-source two-dimensional transport problems, in Cartesian geometry, are reported. Homogeneous and heterogeneous media problems are considered in order to incorporate the idea of arbitrary number of domain division into regions (nodes) when applying the ADO method, which is a method of analytical features, to those problems. The ADO-nodal formulation is developed, for each node, following previous work devoted to heterogeneous media problem. Here, however, the numerical procedure is extended to higher number of domain divisions. Such extension leads, in some cases, to the use of an iterative method for solving the general linear system which defines the arbitrary constants of the general solution. In addition to solve alternative heterogeneous media configurations than reported in previous works, the present approach allows comparisons with results provided by other metodologies generated with refined meshes. Numerical results indicate the ADO solution may achieve a prescribed accuracy using coarser meshes than other schemes.
A Comparative Study of Some iterative Methods to Solve Systems of Linear Equations using Tranputer Networks
A Study on the Parallel, Iterative Solution of Systems of Linear Equations Appearing on Analytical Nodal Schemes for Two-Dimensional Cartesian Geometry Discrete Ordinates Problems
In this work we present our approach to the solution of large linear systems that arise in connec... more In this work we present our approach to the solution of large linear systems that arise in connection with the solution to two-dimensional fixed-source transport problems by the ADO method, based on the use of established iterative methods as Conjugate-Gradients, Generalized Minimum Residuals, Loose Generalized Minimum Residuals and Transpose-free Quasi-Minimal Residuals applied to Jacobi-preconditioned normal equations transformed system. This work was motivated by the use of alternative quadrature schemes to the classical level-symmetric quadrature scheme. These systems are large and sparse and their solution was provided with our own MPI parallel implementations of the iterative methods above. Results obtained show that our approach was successful in solving the systems and the parallel implementations have provided good scalability with respect to the number of processors.
PIM 2.2 The Parallel Iterative Methods package for Systems of Linear Equations User's Guide (Fortran 77 version)
We describe PIM (Parallel Iterative Methods), a collection of Fortran 77 routines to solve system... more We describe PIM (Parallel Iterative Methods), a collection of Fortran 77 routines to solve systems of linear equations on parallel computers using iterative methods. A number of iterative methods for symmetric and nonsymmetric systems are available, including Conjugate-Gradients (CG), Bi-Conjugate-Gradients (Bi-CG), ConjugateGradients squared (CGS), the stabilised version of Bi-Conjugate-Gradients (Bi-CGSTAB), the restarted stabilised version of Bi-Conjugate-Gradients (RBi-CGSTAB), generalised minimal residual (GMRES), generalised conjugate residual (GCR), normal equation solvers (CGNR and CGNE), quasi-minimal residual (QMR), transpose-free quasi-minimal residual (TFQMR) and Chebyshev acceleration. The PIM routines can be used with user-supplied preconditioners, and left-, right- or symmetric-preconditioning are supported. Several stopping criteria can be chosen by the user. In this user's guide we present a brief overview of the iterative methods and algorithms available. The u...
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Papers by Rudnei Dias da Cunha