Time harmonic induction heating problem

Time harmonic induction heating problem (2D)

Hi all,

I’m Damien, a French engineer, currently in my last year of Ph.D. (hoping to defend it in October 2022).

I’m trying to solve a 2d induction heating problem (magnetodynamic) using the time harmonic formulation.

(This is my first post, so I hope to respect the community guidelines)

Problem description

Please find in the picture below the description of my problem :

The domain is composed of the following parts:

  • Steel cylinder (the load)

  • Inductor with 3 copper coils

  • Ambient air

According to the symmetry of the problem, i used the cylindrical coordinate (r, z) in order to simplify the problem.

Also, as the first step, I assume an isotropic medium and therefore omit hysteresis & non-linear magnetic properties (mu(B, H)).


My goal is to compute the temperature evolution over time due to joule heating from eddy current induced in the steel cylinder by the inductor.

My approach is the following:

  1. Solve the time harmonic electromagnetic problem by computing the eddy current distribution in the steel cylinder.

  2. Compute joule heating in the load from the eddy current distribution and set up as a heat source Q for the transient heat equation.

  3. Solve the transient heat equation by computing the temperature evolution in the load.

Issues identification

My trouble is located in the electromagnetic part of the problem.

Some identified tracks :

  • Assignment of material properties in subregions like **electrical conductivity and magnetic permeability

  • 2D time harmonic model formulation

  • Variational formulation implementations of the 2D time harmonic model in cylindrical coordinates.

  • In the computation of magnitudes of interests (eddy current, magnetic flux)

If someone could help me, I will be very grateful ! (Remarks : if someone is interested in the topic, I’m open to a scientific collaboration on the subject. In other words, I would like to publish on the subjects, some helpful contributor will be associate on the paper (with your agreement). Feel free to pm me)

Many thanks in advance for your time and your help ! :grinning:

Best Damien

Please find the code in attachment.
IH_time_harmonic.edp (7.9 KB)

OK, I think this not a difficult problem,

I will try to help you

  1. you to get region numero,
  2. the axisymmetric formulation.
  3. compute the eddy current nd magnetic flux.

Great ! Many thanks in advance !! Your help is welcome !

Premiere remarque, il y a des trous dans les bobines, en non de l’air.

dans la these https://pastel.archives-ouvertes.fr/tel-00443740/document
j’ai trouve la le cas axisymetrique vers page 30

(I allow myself to reply both in English and French to facilitate the understanding of everyone.)

Answer in French

Merci Frédéric pour votre réponse !
À oui en effet ! Très bonne remarque. Dans la machine réelle que j’utilise, il y a une circulation d’eau dans les bobines afin d’éviter la surchauffe de l’inducteur.
Je vais essayer de remplacer les trous dans les bobines par de l’eau ou de l’air.
Merci pour la référence, je vais allez regarder le cas axisymétrique vers la page 30.

(Je me permets de traduire également ma réponse en anglais, si cela intéresse d’autres personnes)

Answer in English

Thanks Frédéric for your answer !
Yes indeed ! That’s a good observation ! In the real device that I used, there is a cooling system inside the coils in order to avoid inductor overheating.
I will try to replace the holes inside the coils with air or water materials.
Thanks for the refs, I will take a look to the axisymmetric case at the page 30.

Although I think that the document that Prof. Hecht linked already provides all necessary derivations I would also like to point towards:

  1. C. Chaboudez, S. Clain, R. Glardon, D. Mari, J. Rappaz, and M. Swierkosz
    “Numerical Modeling in Induction Heating for Axisymmetric Geometries”
    which gives a rather well documented nontrivial example with model parameters and results.
    https://www.esi-group.com/sites/default/files/resource/publication/1334/s_e.inductionlongieee94.pdf might contain some of the missing material properties.

  2. If you progress towards nonlinear magnetic permeability of steel then I found some publications from university of Padova quite helpful e.g.:
    Mattia Spezzapria
    “Multiphysical Finite Element Simulation of Contour Induction Hardening of Gears”
    Chapter 6 “Two dimensional benchmark model” was very instructive.

Keep us up to date on your progress and post example code for further reference ;-).
I think that this topic is very well suited to demonstrate the capabilities of Freefem++.

All the best

Thank you gero for your interest in my topic and your very interesting references !

I will study them carefully.

Of course, I will keep you updated of my progress and post example code for further reference :slight_smile: