TEAM Workshop Problem No. 25
Optimization of Die Press Model
1. General Description
Fig. 1 shows a model of die press with electromagnet for orientation of
magnetic powder [1]. This is used for producing anisotropic permanent magnet.
The die press and electromagnet are made of steel. The die molds are set to
form the radial flux distribution. The magnetic powder is inserted in the
cavity. The model can be assumed as two-dimensional.
The aim of this problem is to obtain the shape of the die molds by using the
optimization method. The effect of optimization method on number of
iterations, accuracy etc. should also be investigated.
2. Definition of Problem
The B-H curve of the steel shown in Fig. 2 is to be used. The typical
values of B[T] and H[A/m] are shown in Table 1.
2.1. Specified values and unknown variables
The ampere-turns (dc) of each coil are chosen as 4253 AT and 17500 AT
respectively.
(a) Small Ampere-Turns (4253 AT)
x- and y- components Bx and By of flux density
at the points along the line e-f in the cavity are specified
as follows:
where Teta is the angle measured from the x axis.
By the preliminary analysis, it is clarified that the specified flux
distribution can be nearly obtained by assuming the shape of die press by the
combination of straight line, circle and ellipse. The shape of the inner die
mold is assumed as a circle. The inside shape of the outer die mold is
represented by the ellipse and a line parallel to the x- axis as
shown in Fig. 1. Then, the radius R1 of the inner die and the long and
short axis L2 and L3 of the ellipse and the dimension L4 are chosen as
design variables.
The shape g-h of the inner die mold and the inside shape
i-j-k-m of the outer die mold can also be represented by free curves.
(b) Large Ampere-Turns (17500 AT)
Bx and By along the line e-f are specified as follows:
In this case, the shape of the inner die mold and the inside shape of the
outer die mold cannot be represented by a circle and an ellipse like the
case of small ampere-turns. The shapes g-h and i-j-k-m can be
represented by free curves.
2.2. Objective Function
The objective function W is given by
where n is the number of specified points (=10). The subscripts
p and o mean the calculated and specified values respectively.
2.3. Constraints
The constraints of R1, L2, L3 and L4 can be, for example, represented as
follows:
5 < R1 < 9.4
12.6 < L2 < 18
14 < L3 < 45
4 < L4 < 19
3. Items to Compare
3.1. Final Shape (Optimal Shape)
The final shape of die molds which is obtained using the optimization method
should be shown.
3.2. Flux Densities
The amplitude |B| and angle TetaB of flux density vector along the
line e-f (R=11.75) in the cavity are to be compared. |B| and
TetaB of the final shape of die molds which are obtained using the
optimization method should be written in Table 2.
3.3. Solution Form
To compare optimization methods, obtained final shapes, etc., please complete
Table 3. The maximum error Eps_B_max of the amplitude and the maximum
error Eps_Teta_max of the angle of flux density vector are defined as follows:
where the subscripts p and o mean the calculated and specified
values respectively.
4. Measurements
The die molds of initial and final shapes are produced and the flux
distribution in the cavity is measured. The number of turns of each coil is
243. The thickness of the electromagnet and die molds is 100 mm (2-D model).
The x- component of the flux density is measured using a Hall probe
at 0° , and the y- component is measured by a Hall probe at
90° using a goniometer. The comparison of measurement and calculation
is reported in reference [1].
Fig. 1 - Model of die press with electromagnet
Fig. 2 - B-H curve of steel
Table 1 - Data of B-H curve
Table 2 - Flux density in the cavity
(see Fig. 1 and Eqs. (1) or (2))
Table 3 - Solution form
Reference
- N. Takahashi, K. Ebihara, K. Yoshida, T. Najata, K. Ohashi and K.
Miyata: "Investigation of simulated annealing method and its
application to optimal design of die mold for orientation of magnetic
powder", IEEE Trans. on Magnetics, 32, 3, pp. 1210-1213, 1996.