1,720,984 research outputs found
On the relevance of the scattering anisotropy for a low order transport method
No full textThe present paper is concerned with the extension of the treatment of the scattering anisotropy developed in the AN-SPN (or simply AN) method up to the approximation of the second order. To assess such extension, two 3D examples are considered: the first one is an axial piece of a 17×17 PWR MOX assembly containing the heads of control rods; while the second example is related to a 3D reactor core containing UO2 and MOX homogenized assemblies. It is shown that the passage from the isotropic to the linearly anisotropic approximation of the differential scattering cross section turns out into a remarkable improvement of the results, as regards the multiplication factor as well as the flux distribution. For the kind of problems like those here considered, less important is, on the contrary, the extension to the second order of the anisotropy, although such an upgrade may be included in the calculation with only an almost negligible effort, owing to the irrelevant impact on the computing time
A boundary element–response matrix method for the multigroup criticality problems in the SP3 approximation
No full textThe 3D xyz multigroup BERM-SP3 transport method is here presented. This method, based on the simplified spherical harmonics approximation (SPN), with N = 3, and the assumption of linearly anisotropic scattering, makes use of a Response Matrix (RM) solution procedure, coupled with the Boundary Element Method (BEM), by which the SP3 partial differential equations are reduced to a system of boundary integral equations in terms of partial currents. Numerical problems, all endowed with a complete set of data and the reference results obtained by well assessed transport codes such as the discrete ordinate code TORT and the Monte Carlo code MCNP, illustrate the accuracy and efficiency of the method
Solution of 3D linearly anisotropic scattering, fixed-source multigroup xyz reactor problems by the AN Boundary Element – Response Matrix method
Full text linkThis paper aims at extending the work performed by means of the AN Boundary Element – Response Matrix method in a previous paper, devoted to criticality problems, to non-homogeneous problems (i.e. the problems that involve a fixed external source, as in case of the accelerator driven systems). As in the preceding paper, the interest is given to the 3D, xyz multigroup systems in which linearly anisotropic scattering is allowed. Standard interface conditions have been adopted and no external intervention in order to improve the numerical results, such as some kind of discontinuity factors, has been introduced. The latter choice is motivated by the need of establishing a clean test of the performances of the AN method in itself. Results of 2D and 3D multigroup fixed source problems obtained with the method described in this paper are compared with those obtained by well assessed reference codes such as the MCNP Monte Carlo code and the PARTISN discrete ordinates code. Even in the absence of discontinuity factors and despite of the intrinsically approximate character of the AN method, the accuracy of the numerical results is more than satisfactory. Finally the computational time is in most cases much shorter than that required by the chosen reference code
A Boundary Element - Response Matrix method for 3D neutron diffusion and transport problems
Full text linkAn application of a 3D Boundary Element Method (BEM) coupled with the Response Matrix (RM) technique to solve neutron diffusion and transport equations for multi-region domains is presented. The discussion is here limited to steady state problems, in which the neutrons have a wide energy spectrum, which leads to systems of several diffusion or transport equations. Moreover, the number of regions with different physical constants can be very large. The boundary integral equations concerning each region are solved via a polynomial moment expansion and the multi-fold integrals there involved are reduced to single or double integrals, taking advantage of suitable recurrence formulas. The usual unknowns (the boundary particle density and its normal derivative) are here replaced by the partial currents entering or leaving each cell. The intuitive physical meaning of such quantities facilitates the application of the response matrix technique. Only eigenvalue (criticality) problems will be here considered. As it regards the transport equation, the use of the so called Simplified Spherical Harmonics method allows, through suitable approximations, to cast the problem into a system of differential elliptic equations of the diffusion type, which can still be solved by BEM
A Boundary Element-Response Matrix method for the solution of 3D criticality problems
No full textThe Boundary Element Method (BEM) has been competing for years with the Finite Element Method (FEM) as a solver of partial differential equations of elliptic type. As well known, the origins of BEM can be traced back to I. Fredholm (1900), who first solved, in a general way, the problem of Dirichlet (determine the electric potential inside a domain on the assumption that the limit values of the potential itself, when approaching the domain boundary, is known). Fredholm's solution was based on a transformation of the differential problem into a boundary integral equation in terms of a double-layer of charges distributed on the boundary, thus giving rise to a new research field, characterized by the purpose of obtaining a solution of the boundary value problems without making an explicit reference to what happens in the interior of the domain, at least until the solution procedure is completed.
Part I of the present paper is dedicated to the general theory. In particular, sect. 1 gives an elementary introduction to the so-called BEM "direct approach" (which has often replaced the original method of Fredholm) as applied to the neutron diffusion equation and points out some general features of BEM, as compared with FEM. In the same section the two calculation levels of the method, when considering reactor criticality problems, are specified. Namely, the "cell level", in which the BEM technique is applied to a reactor cell, to yield the outgoing partial currents as a response to the inward currents that are injected into the cell boundary, and the "reactor level", in which the classical Response Matrix method is adopted to connect all the cells of the system and give the final results in terms of the effective multiplication factor and the overall flux distribution.
The application of the theory of boundary integral equations to a multigroup diffusion system in a homogeneous cell is dealt with in detail in sect. 2.
Part II is dedicated to the approximate procedure to numerically solve such boundary integral equations. The moment method has been chosen (sect. 1), since this method, although it implies a very exacting analytic work, does not incur into any difficulty as it regards the boundary singularities, like edges and vertices. The case of a cell with the shape of a prism with a square base, as typical for a square reactor lattice, is treated in sect. 2, where in particular it is shown that the fourfold basic integrals that represent the reciprocal influence of two faces of the prism (via the integral kernel) can be performed very efficiently, without having recourse to more than a few 1D numerical quadratures. Sect. 3 shows how the set of these basic integrals can be translated into a similar set consisting of Legendre-weighted integrals, which are better suited for applying the moment method other than for taking advantage of the symmetry properties of the problem.
The Part III of this report will deal first with the description of the algebraic issues involved in the creation of the nodal response matrix starting from the basic integrals and exploiting the above 4 mentioned symmetry properties to reduce, together with the application of the theory of the circulant matrices, the calculation burden.
Finally, Part IV will provide some details on the iterative procedure settled up in order to solve criticality search problems for systems with cubic cells, focused on the verification of the consistency of the method without care on the execution time. Numerical results will be, in particular, reported as it concerns the IAEA 2D and 3D LWR benchmark problem
A Boundary Element - Response Matrix Method for 3D Neutron Diffusion and Transport Problem
Full text linkAn application of a 3D Boundary Element Method (BEM), coupled with the Response Matrix (RM) technique, to solve the neutron diffusion and transport equations for multi-region domains is presented. The discussion is here limited to steady state problems, in which the neutrons have a wide energy spectrum, which leads to systems of several diffusion or transport equations. Moreover, the number of regions with different physical constants can be very large. The boundary integral equations concerning each region are solved via a polynomial moment expansion and, taking advantage of suitable recurrence formulas, the multi-fold integrals there involved are reduced to single or double integrals. The usual unknowns (the boundary particle density and its normal derivative) are here replaced by the partial currents entering or leaving each computational cell. The intuitive physical meaning of such quantities facilitates the application of the response matrix technique. Only eigenvalue (criticality) problems will be here considered. As it regards the transport equation, the use of the so called Simplified Spherical Harmonics method (SPN) allows, through suitable approximations, to cast the problem into a system of differential elliptic equations of the diffusion type, which can still be solved by BEM
A Boundary Element – Response Matrix method for criticality diffusion problems in xyz geometry
No full textThe Boundary Element-Response Matrix (BERM) method shown in the paper aims to represent an alternative to the Finite Element method in order to solve 3D multigroup diffusion (criticality) problems in xyz geometry. The theory extends the previous work on the diffusion equations in two dimensions and new techniques for the evaluation of the integrals involved in the boundary integral equations, as well as new procedures for solving the resulting linear system, have greatly enhanced the performances of the method. Results show that BERM can achieve an excellent accuracy, still keeping a good computational efficiency
Solution of the one-velocity 2D and 3D source and criticality problems by the boundary element-response matrix (BERM) method in the A2-SP3
No full textSimplified PN and AN methods in neutron transport
No full textThe paper deals with general properties of the odd-order SP N equations. The equivalence between these equations and the so-called AN equations is demonstrated The AN equations have the simple structure of a system of multigroup diffusion equations and are easily transformed into a set of boundary integral equations, which allows to apply a boundary element-response matrix solution technique and obtain a proper setting for the proofs of the boundary and interface conditions which are to be used Some sharper results for the special class of the transport problems in which the total cross section is constant are discussed. Numerical examples conclude the work
Solution of some 2D transport problems by a high order An-SP2n-1 method
No full textA collection of classical 2D transport problems (the escape probability from prisms of various shapes, the current-to-flux ratio of a wedge-shaped reflector, the transport and asymptotic flux as well as the extrapolation length near a corner) are solved by means of the boundary element version of a high order AN method, an equivalent form of the odd order simplified spherical harmonics (SP2N1) method. The use of a high order approximation is motivated by the fact that all the above problems can be made to fulfil the condition of constant total mean free path, which makes AN–SP2N1 to be equivalent, in turn, to the classical odd order spherical harmonics (P2N1) method, so that for these problems AN–SP2N1 shares with the latter method the property that, by increasing the order 2N 1, the error can be made as small as we want. A second purpose of the paper is to show that the boundary element approach can handle such highly singular boundary integrals as those implied by the partial derivatives of the asymptotic flux at the boundary
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