The 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 the solution of 3D criticality problems

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*GIUSTI, VALERIO;*

##### 2009

#### Abstract

The 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.