Finite element simulation of eddy current problems using magnetic scalar potentials

Abstract

We propose a new implementation of the finite element approximation of eddy current problems using as principal unknown the magnetic field. In the non-conducting region a scalar magnetic potential is introduced. The method can deal automatically with any topological configuration of the conducting region and, being based on the search of a scalar magnetic potential in the non-conducting region, has the advantage of making use of a reduced number of unknowns. Several numerical tests are presented for illustrating the performance of the proposed method; in particular, the numerical simulation of a new type of transformer of complicated topological shape is shown.

Optimal voltage control of non-stationary eddy current problems

“…We also suppose that the intersection of ∂Ω (1) C and ∂Ω is transversal. The other connected component Ω (2) C is like a hollow cylinder (namely, a torus), and is strictly contained in Ω. One can think of Ω (1) C as an induction coil that envelops the workpiece Ω…”

“…In this situation, one can identify two non-bounding cycles in Ω I : the first one, as in the previous cases, is σ (1) = ∂ + Γ J . The other one is σ (2) , a cycle linking the hollow cylinder Ω (2) C , passing into its hole. Figure 4.…”

“…Let us choose a basis of H µ I . First we fix an orientation on σ (2) ; than one basis field ρ 1 satisfies ∂ + Γ J ρ 1 • τ + J = 1 and σ (2) ρ 1 • τ = 0, whereas the other basis field ρ 2 satisfies ∂ + Γ J ρ 2 • τ + J = 0 and σ (2) ρ 2 • τ = 1. It is readily seen that the voltage system now is given by…”

“…Here, we follow ideas of our former papers [24,25]. Moreover, we slightly modify the model proposed in [5] with a technique, introduced in [1,2], which reduces the computational complexity of the numerical approximation scheme and can be efficiently applied in any geometrical situation.…”

Inverting the discrete curl operator: a novel graph algorithm to find a vector potential of a given vector field

“…The idea is that, instead formulating the vector laplacian by using a vector potential, the scalar potential [6] or the mixed-hybrid [2] formulations could be used instead, which produce linear systems that can be solved in nearly linear time by using algebraic multigrid methods. Second, inverse discrete curl is at the root of efficient algorithms to compute a cohomology basis and source fields for solving magnetostatics and eddy current problems by mimetic or finite element methods [7,8,9,10]. We think h as a discrete magnetic field and i as a discrete current; then (1) expresses the so-called discrete Ampère's law.…”