J.P Hennart

A constructive method for deriving finite elements of nodal type

SummaryIn this paper, we propose an algorithm to derive nodal methods corresponding to various two and three-dimensional nonconforming and mixed finite elements. We show that this algorithm can be used to obtain several classical schemes as well as some more recently developed schemes, and that it leads to a simple proof of unisolvence for these methods. Finally we use our method to obtain a three dimensional nodal scheme of BDM type.

Singularities in the Finite Element Approximation of Two-Dimensional Diffusion Problems

The solution of a two-dimensional elliptic boundary value problem with piecewise smooth external boundaries, interfaces, and diffusion coefficients typical of nuclear reactor structures is known to contain a singular part. The presence of singular functions in the neighborhood of each angular point for a given geometric configuration has important consequences on the convergence orders for approximate solutions of the problem. These consequences are analyzed both theoretically and numerically, in the framework of the finite element method. Some means are described to overcome the damaging effects of the singular points. A thorough numerical study of various reactor configurations extending from liquid-metal fast breeder reactors to pressurized water reactors shows that in the latter case, the use of high-order polynomials is partially unjustified, given the severe limitations on the convergence orders.

A critical discussion of numerical models for multiaquifer systems

Abstract Leaky aquifers constitute complicated hydrological structures, whose inclusion in numerical models of hydrological systems is difficult, because of their three-dimensional nature. Methods for treating such systems can be classified as fully three-dimensional and quasi-three-dimensional. The latter have clear numerical advantages when applicable. In this paper a critical discussion of existing quasi-three-dimensional models is presented.

A generalized nodal finite element formalism for discrete ordinates equations in slab geometry Part III: Numerical results

Abstract In companion papers (hereafter referred to as Part I and Part II), we described a generalized nodal finite element formalism, including practically all existing numerical schemes for solving the discrete ordinates equations in slab geometry. General discrete (or cell-edge) and continuous convergence orders as well as convergence orders for moments of the approximations were proved in several theorems given in Part I and II and they are verified here using a model problem. Finally the methods described in Part I and II are applied to some non-model problems.

Finite Element Formulations of Nodal Schemes for Neutron Diffusion and Transport Problems

Several finite element formulations of nodal schemes for neutron diffusion problems are presented: They include nonconforming primal formulations as well as mixed and mixed-hybrid ones, with exact or approximate evaluation of the matrix coefficients. These different formulations are compared to one another and also related to well-known point- and mesh-centered finite difference schemes. Some numerical results are given in one and two dimensions with different schemes. Applications to two-dimensional neutron transport problems are also proposed through the general approach of transverse integration.

Nodal finite element approximations for the neutron transport equation

Two classes of nodal methods, weakly and strongly discontinuous, are introduced and applied to the numerical solution of the neutron transport equation in two-dimensional Cartesian geometry and discrete ordinates. These methods are then applied for the approximation of the solution of a reference problem well known in the nuclear engineering literature.

A Finite Element Code for Time-Dependent Plasma Diffusion in Arbitrary Toroidal Geometry

A finite element code is described that solves the plasma time-dependent diffusion equations in arbitrary toroidal geometry. For space discretization, linear rectangular elements are used with mapping techniques when the cross section is not circular. The time discretization is achieved by the three-step Crank-Nicolson extrapolation method. Numerical results are exhibited with several linear and nonlinear diffusion models.

A Galerkin method with modified piecewise polynomials for solving a second-order boundary value problem

SummaryThe classical Ritz-Galerkin method is applied to a linear, second-order, self-adjoint boundary value problem. The coefficient functions of the operator exhibit a piecewise smooth behaviour characteristic of some “physical” situations. A trial function is constructed using a modified quintic smooth Hermite space

Nodal diffusion methods for space-time neutron kinetics

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