Discrete-valued design optimization -- induction heating

In this example, the optimal shape design of a multiple coil inductor device uses both continuous-valued and discrete-valued variables. In the ideal design, the inductor coils are configured in such a way that only the top surface of the workpiece reaches a uniform temperature of 1000 degrees Celsius after 25 seconds, while maintaining the efficiency of the system above 75% in order to reduce the input power requirements. OptiNet is used to find this ideal design.

The MagNet and ThermNet simulation systems are used to solve for the coupled electromagnetic-thermal field problem. Simulating the induction heating process requires a transient thermal field solution coupled to several steady-state electromagnetic field solutions.

The Device

The workpiece, shown on the left, exhibits a cylindrical symmetry and is surrounded by six coils made of copper wires. Here, the design variables are the inner radius of each coil and its axial positions; so twelve variables define the shape of the inductor. In this example, the values of the turn radii are discrete in variation, i.e. only a discrete set of values is feasible for them. In addition, continuous-valued design variables (namely, the axial position of turns) are considered at the same time. The workpiece is made of non-ferromagnetic steel; its thermal and electrical conductivities vary with temperature. At t=0, a mmf equal to 1.2 kA/turn (at a frequency f=10 kHz) is supplied to each coil. The workpiece exchanges heat by radiation and convection with the external environment which is maintained at temperature T0=25 °C.


Results

Coupled Electromagnetic-Thermal Simulation

The electromagnetic problem may be considered as steady-state because the associated time constant is negligible with respect to the thermal time constant.

The thermal problem domain is just the workpiece alone. The source of the heat is the losses due to the eddy currents that are distributed throughout the workpiece. The relevant material properties of the workpiece are the thermal conductivity and thermal heat capacity.

The thermal problem is transient and is solved from t=0 to t=25 s. The electromagnetic field solution is performed once at every time step of the thermal transient simulation. The temperature rise is recorded at t=25 s, at a set of points located along the surface of the workpiece.

Optimization

The general goal is to obtain TS = 1000°C at the surface of the end section of the workpiece, 25 s after the application of the electric supply to the inductor. The objective function f, to be minimized, can be cast as the maximum deviation of the temperature values at np points located along the surface of the workpiece from the prescribed temperature TS

f(x) = maxk=1,np|Tk(x) - TS| x ∈ Ω

where x=(x1,...,xnv) is the vector of nv design variables, subject to suitable bounds defining the feasible region Ω; therefore, a min-max optimization problem is to be solved starting from an initial geometry supplied by the designer.

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Variables

The design variables, as well as the steps discretizing their movement, are shown in the illustration to the right.

Here, ri i=1,6 is the radius of i-th coil, z1 is the axial position of coil 1, dk k=1,5 is the axial distance between two adjacent coils starting from coil 1.

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Constraints

As a constraint, the electrical efficiency of the inductor, defined as the ratio between the power transferred to the workpiece and the power supplied to the inductor itself, is prescribed to be not smaller than 0.75.

Optimization Results

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Two shaded plots of the initial setup of the device are shown here on the right. The first plot is the RMS shaded plot of the magnet flux density field with units of Teslas and also presented is the shaded temperature field plot with units of °C.

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Shown here is the final optimized setup of the same shaded plot fields. As can be seen, there is a uniform temperature of 1000°C along the surface of the upper portion of the workpiece.

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This figure shows the temperature distribution along the surface of the workpiece before and after the optimization, respectively.

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When compared to the initial setup it is clear that the variation in temperature along the surface of the end section of the workpiece has been reduced significantly; the graph on the right shows the change in the objective function from the initial to final design. During the optimization, the efficiency of the inductor varied from 0.879 to 0.849; the associated constraint was never violated.

Magnetic mesh: 2230 elements
Degrees of freedom (electromagnetic simulation): 4365
Thermal mesh: 1319 elements
Degrees of freedom (thermal simulation): 2918
Time steps for transient coupled analysis: 26
CPU time [s] for each step: 48
Peak memory usage: 86 MB
Hardware platform: AMD XP Athlon 2800+ 2.08 GHz 2 GB

With a search tolerance equal to 10-4, convergence was reached after 136 iterations.

The case study presented on these pages is based on the following paper:

P. Di Barba1, B. Forghani2, D.A. Lowther3
"Discrete-valued Design Optimization of a Multiple-coil Inductor for Uniform Surface Heating"
COMPEL, vol.24, no.1, 2005, pp.271-280; International Symposium on Heating by Electromagnetic Sources, HES-04, June 23-25, 2004, Padua, Italy, pp.521-528, ISBN 88-86281-92-7

1 University of Pavia, Italy
2 Infolytica Corporation
3 McGill University, Canada