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TEAM Workshop Problem 25: Shape Optimization Of A Die Press
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A large electromagnet that can set up a strong magnetic field is used to orient the magnetic powder in a component. The orientation and strength of the magnetic field should be controlled in order to obtain the required magnetization, in the component that is being magnetized. In this device, the objective is to find the size of the inner die mold and the shape of the outer die mold in order to obtain the desired magnetic field in the cavity shown in the figure. OptiNet will find the radius for the inner mold and the elliptical shape for the outer molds that satisfy these design objectives.
A complete definition of this TEAM (Testing Electromagnetic Analysis Methods) problem can be found on the CompuMag`s website.
METHODS and RESULTS
VARIABLES DEFINITION
This figure shows the parameters that the optimizer will adjust in order to reach the design. The parameters are R1, L2, L3 and L4. R1 is the radius of the inner mold and L2 and L3 are the axes of an ellipse. The parameterization option in MagNet is powerful enough to allow even elliptical shapes to be controlled by parameters. Parameter L4 controls the length of the top piece of the outer mold.
The only constraints in this optimization are the range specified for the values of the parameters R1, L2, L3 and L4.
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OBJECTIVE FUNCTION and VARIABLES FLUCTUATION
The objective of the shape optimization is to obtain a flux density that is radial in the cavity space and whose magnitude along the arc e-f is 0.35 Tesla. The objective function is the mean squared error between the Bx and By values sampled along the arc e-f and what they should be in the case of a radial field that has a magnitude of 0.35 Tesla. Defining this expression in OptiNet is easy.
As the optimization is progressing, OptiNet displays the changes in the goal, variables, objectives, and constraints, all in the form of graphs. In this example, each of the four variables' graphs is updated as OptiNet finds a new design.
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INITIAL FIELD
This figure shows the initial B field between the molds, obtained from a static 2D solver in MagNet. The objective arc is shown here in black; it can be seen that the B field is not at 0.35 Tesla. The displayed field plot was restricted to the values between 0.3 and 0.36 Tesla.
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RESULTS
OptiNet produces a report for each optimization run. In this report, the designs are shown in the order that they are improved. The user can view each design individually. The report also shows the time that it took to arrive at the improved design. The values of all the variables and the optimization function are displayed in this report for each iteration. The values of each parameter can be examined to determine the sensitivity of the design to that particular parameter.
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INITIAL DESIGN RADIAL B FIELD
In the initial design that the user supplied to OptiNet, the field along the contour was far from satisfactory. As shown in this graph, the magnitude of the flux density did not remain at 0.35 Tesla. The integral of the error along the arc sums up to 15.38 Tesla.
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FINAL DESIGN
As can be seen, the objective which is the minimization of the mean squared error has been obtained and the magnitude of the flux density along the contour is 0.35 Tesla, as seen in this graph. Of course, the user can examine previous improved designs. In this case, the error had dropped to 1.70 Tesla, a 9 fold decrease. In OptiNet, it is possible to stop the solution at different points, to examine the design and continue the optimization process, for further improvements.
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