Numerical analysis of nonlinear multiharmonic eddy current problems

Abstract

This work provides a complete analysis of eddy current problems, ranging from a proof of unique solvability to the analysis of a multiharmonic discretization technique.For proving existence and uniqueness, we use a Schur complement approach in order to combine the structurally different results for conducting and non-conducting regions.For solving the time-dependent problem, we take advantage of the periodicity of the solution. Since the sources usually are alternating current, we propose a truncated Fourier series expansion, i.e. a so-called multiharmonic ansatz, instead of a costly time-stepping scheme. Moreover, we suggest to introduce a regularization parameter for the numerical solution, what ensures unique solvability not only in the factor space of divergence-free functions, but in the whole space H(curl). Finally, we provide estimates for the errors that are due to the truncated Fourier series, the spatial discretization and the regularization parameter.

An 𝐸-based mixed formulation for a time-dependent eddy current problem

“…The techniques used to show that this saddle-point formulation is well posed are similar to the ones given in [7,18]. (Another formulation for a time-dependent eddy current problem in terms of a magnetic vector potential is given in [5].) Mixed finite element schemes have been used extensively for the approximation of evolution problems, mainly in fluid dynamics applications; see, for instance, Johnson and Thomée [16] and Bernardi and Raugel [7].…”

Steady-state, nonlinear analysis of large arrays of electrically actuated micromembranes vibrating in a fluid

“…On the other hand the size of the nonlinear system is multiplied by N h compared to the time-domain approach. (Note that the convergence of the Fourier approximation is generally of order N −1 h [22], but can be much faster for simple excitations [17,23]). …”

hp-Finite element simulation of three-dimensional eddy current problems on multiply connected domains

“…As an alternative, and following [10,65,78], we introduce the primal perturbed problem: Let ε > 0 be a small perturbation parameter, then: Find A ε ∈ H 0 (curl, Ω) such that…”