Electromagnetics - Modelling, Simulation, Control and Industrial Applications - Abstract
Valli, Alberto
(joint with Ana Alonso Rodr'iguez, Enrico Bertolazzi, Riccardo Ghiloni) vspace*3mm The aim of this talk is two-fold. First, employing edge finite elements, we construct a discrete approximation of the space of harmonic fields $$ mathbb H_mu(Omega) = bf v in (L^2(Omega))^3 , mathophboxrm curl bf v = bf 0, , ,, mathophboxrm div (mu bf v)=0, , ,, mu bf v cdot bf n=0 hbox on partial Omega $$ (here $Omega$ is a bounded three-dimensional domain with a Lipschitz boundary and $mu$ is a symmetric matrix, uniformly positive definite in $Omega$ and with entries in $L^infty(Omega)$). In particular, we give a simple and efficient computational way for constructing the so-called sl loop fields, i.e., the irrotational vector fields $bf T_0$ that cannot be expressed in $Omega$ as the gradient of any single-valued scalar potential (therefore, there exists a loop in $Omega$ such that the line integral of $bf T_0$ on it is different from 0). These fields are of central importance for numerical electromagnetism in general topological domains. Let us also recall that a maximal set of independent loop fields gives a basis of the first de Rham cohomology group of $Omega$. Second, we furnish a finite element numerical solution to the magnetostatic problem $$ beginarrayll mathophboxrm curl bf H= bf J & hboxin Omega
mathophboxrm div (mu bf H) = 0 & hboxin Omega
mu bf H cdot bf n = 0 & hboxon partial Omega
mu bf H perp mathbb H_mu(Omega) , , endarray $$ where $bf J in (L^2(Omega))^3$ with $mathophboxrm div bf J = 0$ in $Omega$ and $int_(partial Omega)_r bf Jcdot bf n = 0$ for all the connected components $(partial Omega)_r$ of $partial Omega$. In particular, the vector fields satisfying $mathophboxrm curlbf H_e = bf J$ in $Omega$ are often called sl source fields in the electromagnetic literature, and are needed for formulating eddy current problems in terms of a magnetic scalar potential in the insulating region. The proposed method works for general topological configurations and does not need the determination of “cutting” surfaces.