Part 1: Steady conduction1. Mathematical formulation.2. The finite element method.3. Isoparametric elements.Part 2: Transient conduction, non-linearities, convective heat transfer4. Transient conduction.5. Non-linearites.6. Convective heat transfer.Part 3: Coupled problems7. Radiative heat transfer in cavities.8. Fluid-structure interaction in a pipe.9. Metallurgical phase change.10. Thermal and electrical phenomena.

Finite Element Simulation of Heat Transfer

A finite element method is presented to compute the electromagnetic field and hence the temperature rise in complex three-dimensional structures such as power transformers. The temperature dependence of both the heat transfer coefficients and the electric conductivity is taken into account establishing a strong coupling between the thermal and electromagnetic equations

Thermal-electromagnetic coupling in the finite-element simulation of power transformers

The past three decades have witnessed important developments in the theory and practice of model predictive control (MPC). In particular, considerable effort has been devoted to robust MPC theory. There have also been many successful applications. This paper will give a brief overview of existing results and summarise experience gained in two real-world applications. We also present some reflections on issues which, in the authors’ opinion, deserve further attention.

Robust model predictive control: reflections and opportunities

An iterative solution to nonlinear problems of time-periodic eddy currents is performed by directly using the time-harmonic content of the field quantities instead of time-domain techniques employing successive time steps. A linear sinusoidal steady-state field problem is solved to determine the magnetization harmonics at each iteration, with the harmonic values corrected in terms of the actual magnetic induction by applying a fixed-point procedure. To further improve its efficiency, the solution process can be started by retaining a small number of harmonics, with more harmonics subsequently added as needed to achieve the desired accuracy. The proposed method always yields stable results, even when the characteristic B-H is strongly nonlinear, and has a superior computational efficiency with respect to various time-stepping techniques and to the "harmonic balance method."

An Efficient Harmonic Method for Solving Nonlinear Time-Periodic Eddy-Current Problems

For the modeling of induction heating processes, strongly coupled magnetodynamic and thermal problems can be solved together within the same finite element. This is called the direct method. In this case, the electromagnetic quantities are expressed through Fourier series according to the harmonic balance method. In this paper, each harmonic is calculated in the whole space by using the coupling between finite elements and boundary elements. Especially suitable when moving parts are involved and because the mesh of air is unnecessary, it is shown that this coupling is still successful if the direct method is used. At the end, the efficiency of this approach is illustrated with an example.

Coupling between finite elements and boundary elements for the numerical simulation of induction heating processes using a harmonic balance method

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Numerical Simulation Of Induction Heating Processes; Comparison Between Direct Multi-harmonic And Classical Staggered Approaches

Numerical simulation of induction heating operations rests on the modelling of coupled non-linear thermal and magnetodynamic analyses. Strong couplings come from, on one hand, the power dissipated through Joule effect, and on the other hand, electromagnetic properties that depend on temperature. Usually, an indirect or staggered method is used to solve such problems. Several magnetodynamic and thermal simulations are therefore performed alternatively and sequentially. This paper presents a new method solving thermo-metallurgical and magnetodynamic problems together and based on the harmonic balance finite element formulation (HBFEM) of the electromagnetic problem. The temperature and the different harmonics of the Fourier series of the unknown electromagnetic variables are now calculated in a same finite element. An example of the numerical simulation of an induction hardening process is presented. The results obtained with different approaches are compared.

R. Pascal

Papers

Coupling between finite elements and boundary elements for the numerical simulation of induction heating processes using a harmonic balance method

Numerical Simulation Of Induction Heating Processes; Comparison Between Direct Multi-harmonic And Classical Staggered Approaches

A new method for the numerical simulation of induction hardening processes