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Computational Science, Engineering & Technology Series
ISSN 1759-3158
CSETS: 26
DEVELOPMENTS AND APPLICATIONS IN ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
Chapter 8

Numerical Analysis and Optimisation of Multiphysics Problems Involving Electromagnetic Couplings

F. Bay<sup>1</sup>, D. Cardinaux<sup>2</sup> and R. Naar<sup>1</sup>

1Center for Material Forming (CEMEF), Mines ParisTech – UMR CNRS 7635, Sophia-Antipolis, France
2Transvalor, Mougins, France

Full Bibliographic Reference for this chapter
F. Bay, D. Cardinaux, R. Naar, "Numerical Analysis and Optimisation of Multiphysics Problems Involving Electromagnetic Couplings", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero, (Editors), "Developments and Applications in Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 8, pp 183-199, 2010. doi:10.4203/csets.26.8
Keywords: electromagnetism, multiphysics couplings, numerical analysis, finite elements, optimisation, heat transfer, solid mechanics.

Summary
An increasing number of problems in mechanics and physics involve multiphysics coupled problems. Among these problems, we can often find electromagnetic coupled problems. The computation of coupled multiphysics problems involving electromagnetic fields can be quite consuming in terms of computational time and resources. Moreover, the design stages which may involve both direct modelling and optimisation techniques that are even more demanding in terms of computer power requirements.

Reducing the resources required for solving the electromagnetic problem is one way to enable solving of these problems within reasonable time and memory requirements. It is therefore important to select the most appropriate numerical methods and parameters in order to save computational time and memory requirements for solving the electromagnetic problem.

In the case of induction heating processes, which involve multiphysics couplings with electromagnetism [1,2], many couplings can be modelled but only first order ones should be considered in order to help determine efficient and adapted numerical strategies for solving these problems.

Electromagnetic formulations [3] as well as specific finite elements [4] will be introduced.

If we are considering the case of a paramagnetic material with temperature–independent electric properties, electromagnetic computations need only be carried out once at the beginning of the computations. However, in other cases, it will be more accurate to have a strong coupling procedure between electromagnetic and thermal computations.

A coupling procedure, based on a convergence test over the mean heating power and on tests over the variations of the magnetic parameters which determine the transfer from an electrical to a thermal resolution or inversely from a thermal to an electric one, can be used [5].

Modelling of multiphysics couplings with solid mechanics or metallurgy problems can be carried out using the same coupling procedure [6].

Considering optimisation of problems with electromagnetic couplings implies that one has to deal with direct models involving coupled systems of equations, one of these systems standing for the electromagnetic model.

We shall introduce the optimisation approach for these problems in the case of a coupled electromagnetism-heat transfer problem. The optimisation formalism can be extended and generalised to other kinds of couplings (for example in solid mechanics and metallurgy).

Algorithms and results for first-order [7] and zero-order approaches [8] are presented and discussed.

References
[1]
E.J. Davies, "Conduction and Induction Heating", P. Peregrinus Ltd., London, 1990.
[2]
S. Denis, A. Simon, "Discussion on the role of transformation plasticity in the calculation of quench stresses", Proceedings Int. Conf. Residual Stresses, Garmish, 1-9, 1986.
[3]
M. Chari, A. Konrad, M. Palmo, "Three-dimensional vector potential analysis for machine field problems", IEEE Transactions on Magnetics, 18-2, 436-446, 1982. doi:10.1109/TMAG.1982.1061863
[4]
J.C. Nédélec, "A new family of mixed finite elements in IR3", Numer. Math., 50, 57-81, 1986. doi:10.1007/BF01389668
[5]
F. Bay, V. Labbe, Y. Favennec, J.-L. Chenot, "A numerical model for induction heating processes coupling electromagnetism and thermomechanics", International Journal for Numerical Methods in Engineering, 58, 839-867, 2002. doi:10.1002/nme.796
[6]
D. Cardinaux, F. Bay, "A Three-Dimensional Finite Element method for induction heat treatment computation involving moving parts", 6th International Conference on Electromagnetic Processing of Materials, Dresden, October 2009.
[7]
F. Bay, Y. Favennec, V. Labbe, "Induction Heating Processes Modelling: Optimisation Procedure and Parallel Computing", International Journal of Materials & Product Technology (IJMPT), Special issue "Induction Heating & Hardening", 29(1-4), 52-69, 2006. doi:10.1504/IJMPT.2007.013130
[8]
R. Naar, D. Cardinaux, F. Bay, "Numerical optimisation for induction heat treatment", Heating by Electromagnetic Sources Conference, HES-10, 2010.

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