By Ana Alonso Rodriguez, Alberto Valli
This publication offers with the mathematical research and the numerical approximation of time-harmonic eddy present difficulties. it really is self-contained and appropriate for mathematicians and engineers operating within the box, and likewise available for newcomers. reckoning on the alternative of the actual unknowns, those difficulties are formulated in numerous variational methods, with particular awareness to the topology of the computational area. Finite parts of nodal or side kind are used for numerical approximation, and an entire research of convergence is played. a particular characteristic of the booklet is the emphasis given to saddle-point formulations by way of the magnetic and electrical fields. New effects for voltage or present depth excitation difficulties also are offered.
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Additional resources for Eddy Current Approximation of Maxwell Equations: Theory, Algorithms and Applications
Finally, from the Faraday law curl(E−EM ) = −iωμ(H−HM ) we have also obtained H − HM 0,Ω ≤ C|ω|1/2 , which ends the proof. 1 Higher order convergence Under suitable additional assumptions the order of convergence can be improved. The following result can be found in Schmidt et al. , where the eddy current modeling error has been investigated under different points of view. 9. Suppose that div Je = 0 in Ω and that ΩC is simply-connected. 15) for some constant C > 0, independent of ω. Proof.
10) we know that (E−EM ) ∈ Wε (ΩI ; Ω) and that μ−1 curl(E − EM ) · curl z + iω Ω 2 σ(EC − EM C ) · zC = −ω ΩC εEM · z Ω for all z ∈ H(curl; Ω). Taking z = E − EM we have Ω hence μ−1 curl(E − EM ) · curl(E − EM ) + iω = −ω2 Ω εEM · (E − EM ) , ΩC M σ(EC − EM C ) · (EC − EC ) 2 2 νmin curl(E − EM ) 40,Ω + ω2 σmin EC − EM C ≤ ω4 ε2max EM 20,Ω E − EM 20,Ω . 12) therefore we ﬁnd, for a constant C3 > 0 independent of ω, 4 curl(E − EM ) 40,Ω + ω2 EC − EM C 0,ΩC ≤ C3 ω4 EM 20,Ω ( curl(E − EM ) 20,Ω + EC − EM C Using that for each δ > 0 it holds AB ≤ curl(E − EM ) 4 0,Ω 1 A2 2δ + ω2 EC − EM C + δ21 curl(E − EM ) 2 0,ΩC ) .
13)6 some compatibility conditions on the data must be satisﬁed. In fact, one has div(iωμI HI ) = − div curl EI = 0 in ΩI , divτ (EC × nC ) = − divτ (EI × nI ) = − curl EI · nI = iωμI HI · nI iωμI HI · n = − (∂Ω)r curl EI · n = 0 on Γ , ∀ r = 1, . . , p∂Ω . (∂Ω)r This is not enough: for any function vI ∈ H(curl; ΩI ) with curl vI = 0 in ΩI and vI × n = 0 on ∂Ω we have −iω ΩI = μI HI · vI = ΩI curl EI · vI Γ nI × EI · vI = Γ EC × nC · vI . In particular, one can take vI = ρl,I , l = 1, . . 4).
Eddy Current Approximation of Maxwell Equations: Theory, Algorithms and Applications by Ana Alonso Rodriguez, Alberto Valli