Thesis Summary:

This thesis analyzes some problems of induced currents in bounded domains in order to solve the problem of three-dimensional electromagnetic forming without any particular symmetry. Particular consideration would be given to the case where current sources are provided in terms of intensities and / or voltages imposed on parts of the domain boundary. Initially we demonstrate the equivalence between two formulations for the problem of currents induced in harmonic regime. The first is a formulation in terms of the magnetic field in the conductor and a magnetic scalar potential in the dielectric. The second is a formulation in terms of the magnetic field in the whole domain and a Lagrange multiplier in the dielectric. The equivalence is displayed at a discrete level. Numerical results are shown highlighting the advantages and disadvantages of each of the formulations. Next, a numerical method is formulated for the formulation in terms of the magnetic field of the transient problem of currents induced with current currents as data. It is shown that the weak formulation has a unique solution that satisfies in a certain sense the starting problem. We propose a space of finite elements for spatial discretization based on edge elements of Nédélec. Then, an implicit Euler scheme is introduced for time discretization. Optimum estimates of error are shown for semi-discrete and totally discrete schemes. In addition, a magnetic scalar potential is introduced into the dielectric to impose the null rotational condition, which leads to significant computational savings. Finally, the method is applied to solve two problems: a test with known analytical solution and an application to electromagnetic conforming. The implementation of the above formulation, magnetic field / magnetic scalar potential, requires the construction of so-called cutting surfaces in the dielectric domain, when it is not simply connected. The construction of these surfaces can be very complex in practice. Thus, the problem of transient induced currents in terms of the electric field primitive is discussed below. In this case the problem is studied with non-local sources in terms of intensities and voltages. In the dielectric it is necessary to introduce a Lagrange multiplier and the analysis leads to the study of a mixed degenerate parabolic problem. Results of solution existence and uniqueness are shown as well as convergence results for the proposed numerical method. The numerical method is validated with an example with known analytical solution; Is also applied to calculate the currents induced in a topology where the introduction of the cutting surface is not trivial. Finally, we address a transient problem of induced currents in a bounded domain where the conducting domain changes over time. In this case, to simplify the analysis, the problem arises with essential boundary conditions and a known volume source. We propose a formulation in terms of the magnetic field for which it is shown that there is a solution. A penalty technique is also proposed to impose the zero rotational constraint on the dielectric. It is shown that this strategy is effective for the fixed conductor problem, both for the continuous problem and for its finite element discretization of Nédélec and an implicit Euler scheme. Likewise, optimal estimates of the error for this numerical scheme are shown uniform with respect to the penalty parameter. Numerical tests are presented confirming the convergence of the proposed penalty method. The numerical method with penalty applies to a case where the driver travels over time. The results show that a convergence order similar to the case of a fixed conductor could be expected.

ISI Publications from the Thesis

Alfredo BERMúDEZ, Bibiana LóPEZ-RODRíGUEZ, Rodolfo RODRíGUEZ, Pilar SALGADO: Numerical solution of a transient three-dimensional eddy current model with moving conductors. International Journal of Numerical Analysis and Modeling, vol. 16, 5, pp. 695-717, (2019).

Alfredo BERMúDEZ, Bibiana LóPEZ-RODRíGUEZ, Rodolfo RODRíGUEZ, Pilar SALGADO: An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 47, 3, pp. 875-902, (2013).

Alfredo BERMúDEZ, Bibiana LóPEZ-RODRíGUEZ, Rodolfo RODRíGUEZ, Pilar SALGADO: Numerical solution of transient eddy current problems with input current intensities as boundary data. IMA Journal of Numerical Analysis, vol. 32, 3, pp. 1001-1029, (2012).

Alfredo BERMúDEZ, Bibiana LóPEZ-RODRíGUEZ, Rodolfo RODRíGUEZ, Pilar SALGADO: Equivalence between two finite element methods for the eddy current problem. Comptes Rendus de l'Académie des Sciences, vol. 348, pp. 769-774, (2010)