On a class of degenerate abstract parabolic problems and applications to some eddy current models

Pauly, Dirk; Picard, Rainer; Trostorff, Sascha; Waurick, Marcus

doi:10.1016/j.jfa.2020.108847

Cited by 7 publications

(8 citation statements)

References 32 publications

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“…Moreover, the external port is formed by the voltage v and the current i at the conductive contacts, which are, respectively, the input and the output of (1). The variable J int in (26) stands for the current density in Ω induced by the electromagnetic field, whereas E is the electric field intensity. Further, v w is the part of the voltage at the winding which is caused by the resistive effect, and i w is the corresponding current.…”

Section: Proof Since the Expressionmentioning

confidence: 99%

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Passivity, Port-Hamiltonian Formulation and Solution Estimates for a Coupled Magneto-Quasistatic System

Reis¹,

Stykel²

2022

Preprint

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In this paper, we study a quasilinear coupled magneto-quasistatic model from a systems theoretic perspective. First, by taking the injected voltages as input and the associated currents as output, we prove that the magnetoquasistatic system is passive. Moreover, by defining suitable Dirac and resistive structures, we show that it admits a representation as a port-Hamiltonian system. Thereafter, we consider dependence on initial and input data. We show that the current and the magnetic vector potential can be estimated by means of the initial magnetic vector potential and the voltage. We also analyse the free dynamics of the system and study the asymptotic behavior of the solutions for t → ∞.

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Section: Proof Since the Expressionmentioning

confidence: 99%

“…Existence, uniqueness and regularity properties of solutions to (1) have been studied in [5,25,26] for some special cases, while [11] presents well-posedness and regularity results for more general setting. The purpose of this paper is to investigate the dynamic behavior of the coupled MQS model (1) from a systems theoretic point of view.…”

mentioning

confidence: 99%

Passivity, Port-Hamiltonian Formulation and Solution Estimates for a Coupled Magneto-Quasistatic System

Reis¹,

Stykel²

2022

Preprint

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“…The reason being that then Maxwell's equations are merely of parabolic type. We shall refer to [79] and the references therein for an extensive discussion.…”

Section: Proof Ofmentioning

confidence: 99%

Solution Theory for Evolutionary Equations

Seifert

Trostorff

Waurick

2021

Evolutionary Equations

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In this chapter, we shall discuss and present the first major result of the manuscript: Picard’s theorem on the solution theory for evolutionary equations which is the main result of Picard (A structural observation for linear material laws in classical mathematical physics. In Mathematical Methods in the Applied Sciences, vol 32, 2009, pp 1768–1803). In order to stress the applicability of this theorem, we shall deal with applications first and provide a proof of the actual result afterwards. With an initial interest in applications in mind, we start off with the introduction of some operators related to vector calculus.

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“…is connected; 𝜇 ∶ 𝐿2 (Ω) → 𝐿2 (Ω) degenerates as in (6.92); 𝜎 ∶ 𝐿2 (Ω) → 𝐿2 (Ω) is strictly positive definite; moreover, there exists 𝜎 1 , 𝜎 2 > 0such that 0 ≤ 𝜎 1 ≤ 𝜎 ≤ 𝜎 2 ; 𝐇 0 ∈ 𝐇(curl, Ω).by Theorem 4.21 in Ref 51,. there exists 6 a unique weak solution 𝐇 of the 𝐇 problem in (91) belonging to the space 𝐻 𝜌,0 (ℝ; 𝐇 0 (curl, Ω)), where for 𝜌 > 0 we define the weighted 𝐿 2 type space𝐻 𝜌,0 (ℝ; 𝐇 0 (curl, Ω)) = { 𝑓 ∈ 𝐿 2 loc (ℝ; 𝐇 0 (curl, Ω)) ∶ ∫ ℝ ||𝑓(𝑡)|| 2 𝐇(curl,Ω) 𝑒 −2𝜌𝑡 𝑑𝑡 < ∞ } .Let 𝐇 denote this solution; since 𝜎 −1 (𝐯)|∇ × 𝐇|2 ∈ 𝐿 1 (𝑄 𝑇 ) we obtain a weak solution of the parabolic problem just as before.…”

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confidence: 92%

Quantitative analysis of passive intermodulation and surface roughness

Stachura,

Wellander,

Cherkaev

2024

Stud Appl Math

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We explore the relationship between rough surface conductors and the phenomenon of passive intermodulation. The underlying surface is taken to be the boundary of a Lipschitz domain, and a characteristic angle of the domain is used to track boundary dependence on the various fields. To model electro‐thermal passive intermodulation in particular, we consider a specific type of temperature‐dependent conductivity and determine conditions on the conductivity under which one can use fixed point arguments to solve an induction heating and Joule heating problem on a Lipschitz domain. In the latter problem, we also consider a time‐dependent permittivity function . Finally, weak solutions to a magneto‐quasi‐static problem are obtained when the permeability µ is temperature dependent and is allowed to degenerate in a certain way. An interesting effect of the rough surface is the inherently limited Sobolev regularity of the electric field, which can be improved if one assumes additional constraints on the boundary.

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On a class of degenerate abstract parabolic problems and applications to some eddy current models

Cited by 7 publications

References 32 publications

Passivity, Port-Hamiltonian Formulation and Solution Estimates for a Coupled Magneto-Quasistatic System

Passivity, Port-Hamiltonian Formulation and Solution Estimates for a Coupled Magneto-Quasistatic System

Solution Theory for Evolutionary Equations

Quantitative analysis of passive intermodulation and surface roughness

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