F. Trevisan

A new set of basis functions for the discrete geometric approach

By exploiting the geometric structure behind Maxwells equations, the so called discrete geometric approach allows to translate the physical laws of electromagnetism into discrete relations, involving circulations and fluxes associated with the geometric elements of a pair of interlocked grids: the primal grid and the dual grid. To form a finite dimensional system of equations, discrete counterparts of the constitutive relations must be introduced in addition. They are referred to as constitutive matrices which must comply with precise properties (symmetry, positive definiteness, consistency) in order to guarantee the stability and consistency of the overall finite dimensional system of equations. The aim of this work is to introduce a general and efficient set of vector functions associated with the edges and faces of a polyhedral primal grids or of a dual grid obtained from the barycentric subdivision of the boundary of the primal grid; these vector functions comply with precise specifications which allow to construct stable and consistent discrete constitutive equations for the discrete geometric approach in the framework of an energetic method.

A methodological analysis of different formulations for solving inverse electromagnetic problems

The authors present a methodological examination of several formulations developed to solve different kinds of inverse electromagnetic problems. Some general numerical formulations of the inverse problems are presented that are defined regardless of the nature of the specific electromagnetic problem. It is shown that the different approaches can be drawn from these general formulations, according to the specific nature of the electromagnetic problems and according to the purpose of the investigations. The most significant aspects of inverse problems, such as existence, uniqueness, and stability of the solution are discussed in detail together with the applicable regularization techniques. >

Symmetric Positive-Definite Constitutive Matrices for Discrete Eddy-Current Problems

We examine the construction of a symmetric positive definite conductance matrix for eddy-current problems, using a discrete approach. We construct a new set of piecewise uniform basis vector functions on both the primal and the dual complex. We define these vector functions for both tetrahedra and prisms

Discrete constitutive equations in A-/spl chi/ geometric eddy-current formulation

Using a geometric formulation for eddy currents, we present a geometric approach to constructing approximations of the discrete magnetic and Ohms constitutive matrices. In the case of Ohms matrix, we also show how to make it symmetric. We compared the impact on the solution of the proposed Ohms matrices, and an iterative technique to obtain a consistent right-hand-side term in the final system is described.

3-D eddy current analysis with the cell method for NDE problems

An algebraic formulation has been developed for the solution of three-dimensional eddy current problems, according to the cell method philosophy. This formulation is presented in the frequency domain and it is based on the circulation of magnetic vector potential along primal edges and on time-integrated electric scalar potential in primal nodes of conducting region. The formulation has been applied to solve a nondestructive evaluation problem. The flaw has been modeled as a zero-thickness defect and the superposition of effects has been applied to compute the impedance variation with respect to the flawless configuration.

Constitutive equations for discrete electromagnetic problems over polyhedral grids

In this paper a novel approach is proposed for constructing discrete counterparts of constitutive equations over polyhedral grids which ensure both consistency and stability of the algebraic equations discretizing an electromagnetic field problem. The idea is to construct discrete constitutive equations preserving the thermodynamic relations for constitutive equations. In this way, consistency and stability of the discrete equations are ensured. At the base, a purely geometric condition between the primal and the dual grids has to be satisfied for a given primal polyhedral grid, by properly choosing the dual grid. Numerical experiments demonstrate that the proposed discrete constitutive equations lead to accurate approximations of the electromagnetic field.

The RFX magnet system

Abstract The reversed field pinch (RFP) magnetic confinement requires both toroidal and poloidal components for the magnetic field induction. As in tokamaks, the former is provided by the toroidal and poloidal components for the the ohmic heating (OH) winding and the equilibrium field (EF) winding. The two induction field components have similar amplitudes, so that the toroidal component required in a RFP is about one order of magnitude lower than that in a tokamak with equal plasma current and aspect ratio. Owing to the local stability properties, the TF coils have to be located as close as possible to the plasma but, at the same time, the magnetic field ripple from TF coils (as well as any kinds of stray fields) must be kept to a minimum. Another peculiarity of the TF winding is that it is required to operate with a time-varying current and at high voltage levels. From these points of view, the design of RFP TF windings presents much less technological problems than a tokamak of similar size. The RFP requires a high toroidal loop voltage during fast current rise and a relatively high toroidal loop voltage during the flat top. The main consequences for the RFX PF magnet systems are as follows: • - large flux and energy to be inductively stored; • - very high voltage across the OH and EF winding terminals, giving rise to substantial insulation problems; • - the very fast rate required for current rise may produce remarkable skin effects within the OH conductors, so that large cross-section conductors have to be avoided; • - the electrodynamic forces acting on the OH winding are large and comparable with those in tokamaks; • - in case of any fault, currents in the coils can rise beyond safety limits at a very high rate, leading to extremely critical conditions for the machine integrity . From a mechanical point of view, both the OH and the EF windings are subject to working conditions similar to those experienced in tokamaks. Thus, RFX windings were manufactured with a similar technology. To detect the TF and PF winding faults a very fast, hard-wired system has been developed, which is able to elaborate signals from specific probes and to decide on necessary protective actions. The present contribution deals with the whole magnet system of RFX, including all TF and PF magnets, mechanical structure, magnetic and electric measurement instrumentation, as well as the fast fault detection system. After a review of the basic concepts representing the theoretical background behind the main choices, all aspects and features concerning the magnet design are presented in detail and deeply discussed. The manufacturing technology is then presented together with the main problems met during manufacture, development and acceptance tests and the methods adopted in order to solve them are explained. On-site assembly procedures, testing and the first integrated RFX operation are finally described.

Automated optimal design techniques for inverse electromagnetic problems

Two methods are considered for the solution of automated digital optimal design problems. The computer procedures based on these methods have been implemented and tested. The first method, based on a deterministic approach, considers a quadratic approximation of the cost function. The second, based on a stochastic approach, is derived from the simulated annealing algorithm. Both methods, implemented as computer codes, have been applied to the solution of a test synthesis problem where the magnetic field is generated by discrete coils. The deterministic method is substantially faster, especially when the calculation of the cost function is time consuming. On the other hand, the stochastic method gives good approximation of the global minimum independently of the initial conditions: as far as CPU time is concerned, the method is more expensive, but can be profitably used when the cost function can be calculated quickly and the number of design variables is large. >

Electromagnetic analysis of superconducting cables and joints in transient regime

The paper deals with the electromagnetic analysis of the superconducting multistrand cables used to manufacture magnets for fusion research. An analysis method is described, in which the cable is represented as an equivalent lumped network, whose parameters are automatically computed starting from the cable geometrical data. The network is then solved in transient regime.

3-D magnetostatic with the finite formulation

The paper presents the solution of a three-dimensional (3-D) magnetostatic problem using the finite formulation of an electromagnetic field. Two different approaches, based on a primal-dual barycentric discretization of the 3-D space, are presented, considering as unknowns either the magnetic fluxes or the circulations of the vector potential. Results on simple reference configurations are reported and discussed.