NEUROCOMPUTING

EIXiVIEiR Neurocomputing 13 (19%) 217-230

Application of neural network algorithm to CAD of magnetic systems

Yoshitake Yamzaki a**, Moyuru Ochiai b, Amo Holz ‘, Toshito Hara d

’ Faculty of Computer Science and Systems Engineering, Kyushu Institute of Technology, Ii&a, 8.20, Japan b Department of Electronics, Shohoku College, Atsugi, Kanagawa, 243, Japan

’ Theoretische Physik, Universitiit des Saarlandes, 66123 Saarbriicken, Germany ’ Advantest Corporation, Gyoba, Saitama, 361. Japan

Received 27 March 1995; accepted 11 September 1995

Abstract

Magnetic systems comprise some of the most useful devices in the world, e.g. magnets for magnetic resonance images (MRI), nuclear magnetic resonance (NM@, and linear motors and accelerators. Generally, their constituent magnetic-elements have a strongly nonlinear and finite magnetic permeability, and their magnetic materials are hard and difficult to cut, in addition to being expensive. For determination of the best shape of magnetic systems and the best selection of magnetic materials, it is useful to investigate CAD (computer-aided design) systems based on electromagnetic theory using the material relations (magnetic flux density-magnetization (B-M) curves). Our aims are to construct general CAD systems appropriate for macro- to nano-machines, but in the present paper we focus our attention on the shape design of a simple permanent magnet as an example. Our CAD system is constructed with two parts to determine (1) the self-consistent distribution of magnetization under a certain shape and (2) the best shape which gives the minimum volume. We show that in both cases a neural network algorithm (which we developed) can play a powerful role.

Keywords: Magnetic devices CAD; Neural networks; Nonlinear integral equations; Permanent magnets

* Corresponding author. Email: yamazaki@mse..kyutech.ac.jp

0925-2312/%/$15.00 8 19% Elsevier Science B.V. All rights reserved SSDI 0925-23 12(95)00096-8

218 Y. Yamazaki et al./Neurocomputing 13 (19%) 217-230

1. Introduction

Improving neural network algorithms and applying neurocomputing methods to various fields have been done by researchers, and successful advances have been made. In particular, the neural network algorithm is a very powerful tool for finding optimum values in nonlinear systems, e.g. the Poisson equations and magnetic systems. In computer science and technology, establishing how to design CAD systems in many fields is eagerly desired. Recently, studies on CAD systems for magnetic devices aimed at macro- to nano-machines are required in various fields. Popular examples are superconducting-magnet CAD for medical, biochemical and physical uses, permanent- magnet CAD for similar, various uses, and many, modem magnetic-applications CAD for magnetic floating, magnetic position sensor, brain-wave analysis, and so on. In the near future, more advanced studies, i.e. studies on intelligent-device CAD, using intelligent materials and yielding intelligent mechanisms and functions, will be required, where various kinds of coupling forces (superconducto-magnetic, optomagnetic, piezo- magnetic and thermomagnetic couplings, in addition to elastomagnetic, electromagnetic and thermoelectric couplings) will be appropriately combined and utilized. Particularly, in the field of quantum-semiconductors, LSI, nanomachining and so on, they will be urgently needed. These problems will be solved using neurocomputing with the aid of large scaled modem physics and computer science and technology.

In order to contribute to the fields of large scaled scientific devices (NMR, linear accelerator, . . . ), transport systems (linear motor, linear ship, . . .I, medical systems (MRI, information devices, mechanical devices.. . .) and so on, we have already reported on the fundamental CAD systems for superconducting magnets [l], using the powerful layered neural network algorithm. In the present paper, CAD systems for permanent magnets (available for various uses) is studied and discussed as a simple, fundamental example, which have been designed and constructed as parts of various kinds of machines since ancient times in terms of experimental methods.

We aim here are to construct general CAD systems appropriate from macro- to nano-machines, i.e. how to design shape of simple permanent magnets, choosing one of the best materials, with the aid of the improved neural network algorithm (INNA) [l]. CAD systems, in general, require taking into account classical or quantum effects depending on the system size. Here we consider the classical case and in the near future the quantum case.

The CAD systems are constructed in the following steps: As a preliminary step we make a database with data (magnetic properties, costs, . . . > of magnetic materials, and we specify the user’s requirements (IJR) for the magnets. In the first step of the CAD process one of the best materials are chosen from those available, the best shape of the magnets is computed in the second step, and in the third step the distributions of magnetization {M} and magnetic flux density (B) are computed using our improved neural network algorithm. Next the last three steps are studied and discussed.

For simplicity of consideration we specify simple models with both the z-axis symmetry and the r-4 plane symmetry in Section 2. These magnets consist of single piece, double pieces, - - * , but here we consider up to two piece magnets. Furthermore,

Y. Yamazaki et al./Neurocompnting 13 (19%) 217-230 219

the UR and the fundamental equations (FE) required are summarized from the physical, technical point of view.

In Section 3 the methods for construction of the CAD system is discussed from the neurocomputing point of view.

Results obtained in interesting and important magnetic systems are summarized and discussed in Section 4. It is shown that our CAD system quickly derive the final shapes of magnets satisfying the UR.

In Section 5 concluding remarks are summarized: Our CAD system for the classical case approaches some complete and as our next effort we will proceed to design the CAD systems for the quantum case.

2. Models, user’s requirements and fundamental equations

Our aim here is to consider how to treat fundamental problems arising when we construct CAD systems for permanent magnets. The essentials of modeling the UR and the FE are summarized.

In order to simplify explanation and construction and to shorten computation time, we adopt the following simple model: (1) Shapes of permanent magnets are

(a) symmetric around the z-axis, (b) symmetric with respect to the z = 0 plane, (c) internal radius f,(z) =f,( - z> and external one f2( z) =f2( - z).

(2) Magnets consist of one or two pieces of permanent magnets, which are placed along the z-axis with a gap length g( = 2 gJ

(3) Materials of permanent magnets can be chosen freely (here alnico permanent magnetic materials are used).

Here all quantities of f,, fi, g are nonnegative, magnets with g = 0 (Z 0) are of single-piece (double-pieces) magnet, and shapes of magnets with f,(z) = 0 ( # 0) is rod (pipe)-like. This situation is illustrated in Fig. 1 where the upper (lower) figure illustrates the single piece (double pieces) magnet.

Now let us specify a user’s requirement for the construction of a new magnet. Many requirements are possible, but we choose the following simple requirements: (1) Magnetic-field uniformity is less than or equal to 0.01 within the cube of 1 cm3

whose center is at the origin of the cylindrical coordinates. (2) A set of a half of gap length {g,, g,} is given. (3) The costs required for the magnet-construction are minimum; here minimum volume

of the magnet is required, for simplicity, under a certain chosen material. Even if these are replaced with somewhat more complicated requirements we are able to derive the final solutions. Let us consider the FE of the system and discuss how to solve the FE. The FE are constructed with the following three parts:

(1 I Maxwell equations with shape symmetries (described above) [ 11. As shown in Appendix A, the magnetic system is described by the Maxwell

220 Y. Yamazaki et al./Neurocomputing 13 (1996) 217-230

Fig. 1. Model of permanent magnets with both z-axial and z = 0 plane symmetries. The upper (lower) figure corresponds to single-piece (double-pieces) magnet.

(2)

equations which are expressed by (AS). That is, the magnetic flux density {B,, B,} at the cell center (r, z> is described as

B,(r, z) =I$((& M,}) (a=? z). (24

where F,, F, stand for the functions of the magnetization set, expressed with a linear dimensional sum of the first and the second kinds of the complete elliptic integrals. Note that the magnetic flux density distribution {B,, B,) is derived from the magnetization distribution {M,, M,},. Material relations (B-M curves) measured from the magnetic materials 121. Depending on whether the magnetic materials are isotropic or anisotropic, the expressions for the material relations are different. For simplicity we assume that the relation of the absolute values among B and M is derived from the experiment like

M- G(B) (B, M: absolute values) (2.2)

(3)

Y. Yamazaki er al./Neurocomputing 13 (1996) 217-230 221

and that the magnetization components are computed as

M,=MB,/B (a=r, z). (2.3)

Here the function G is determined from the experimental data, and the magnetiza- tion distribution {M,, ML],, is computed from the magnetic flux density distribution

tB,, BL). This means that the new magnetization distribution (M,, M,},, is obtained from (M,, MZ),. i.e. in terms of the iterative computation the final (self-consistent) magnetization distribution is obtained. Volume relation of magnets. The volume is related as

V=2~~[fzz(n) -f:(n)]z(n) az3i2 (F=r/z). (2.4 n

This relation shows that if i> 1 decrease the value of r, otherwise decrease the value of 2, in order to make the value of V minimum.

Finally we consider actually expected shapes of magnets satisfying the UR, on a theoretical basis. They are summarized in Appendix B.

3. Methods of construction for the CAD systems

As mentioned in section 1, the construction of the CAD system is divided into the following three steps in addition to the preliminary step: (p) preliminary step - database on information of magnetic materials. (1) 1 st step - selection of one of the best materials. (2) 2nd step - decision of the best shape of magnets. (3) 3rd step - computation of the self-consistent distributions of magnetizations and

magnetic flux density. In the step p the database is made according to the classification of information on

{magnetic properties (saturation magnetization, coecive force; isotropy or anisotropy; . . . ), and aging properties, - - . , cost, . . .I. The details will be discussed elsewhere.

There are no important problems in the 1st step, because it is enough for the computer to find all plausible materials from the database and to evaluate the minimum cost.

The 2nd step finds the minimum volume of the magnets, according to (2.41, which satisfy the UR. The algorithm is as follows:

Assume shapes (f, , fi), , then find (f,, f,),, with minimum volume satisfying LJR

,!I

(3.1)

modified by means of improved neural network algorithm

until (f, , fJ,, gives minimum volume

222 Y. Yamazaki et al./Neurocomputing I3 (1994) 211-230

The 3rd~step is made so that the computer may iteratively compute and find the self-consistent distribution of magnetization, according to (2.1)-(2.3). The algorithm is as follows:

Assume {M, , ML), , then get {M, , Mz),,

TI

(3.2)

modified in terms of improved neural network algorithm until {M, , M,),, converges

The block diagrams for the last three steps are illustrated in Fig. 2. The first block diagram illustrates the first step. The block ‘determination of shapes’ is illusllated in the second block diagram, which corresponds to the second step. The third block diagram illustrates the block determination of {M,, MZ} related to the third step.

Let us summarize the improved neural network algorithm (reported in [ 1,3] which is applied to the computation in the course of (3.1)-(3.2). This algorithm is a modified Boltzmann machine using a layered neural network and is constructed as follows; (a) layered network where the itb layer corresponds to the ith iteration of computation, (b) data with respect to whole cell points (i.e. all neurons in the ith layer) are replaced

with new data (which correspond to the data in the i + 1st layer) after all computations over whole cell points have been finished.

cc> introduction of error function in each layer.

CEz+F(Eo[-Ei)2 EE{M,, Mz, M9f,9f2}7 I

Boltzmann machines are constructed as

(3.3)

Cd)

(3.4)

(e)

where u, A4 and 5 stand for the weight, physical quantity (like magnetization or shape of magnet) and threshold value, respectively. modified back propagation algorithm which is expressed as follows:

Eir-‘Ei,+aE Ei, [ A /ii* (1 -‘Q/(1 -&)I’

x[(~~~~~+B.x),(~~wM:pr)](+i~-$“) i i

with annealing factor aE, (f) annealing and random search for minimum branch.

Note that our method (e-f) differs from the standard back propagation algorithm in two points; analytically finding the extremum of the system and annealingly and/or randomly searching for the minimum branch of it.

Y. Yamaz.aki et al./ Neurocomputing 13 (1996) 217-230

CAD system

satisfied & minimum cost

Determination of shapes

Determination of

minimum volume ? Modification of (fl,fzJII

\ I

Yes

Determination of (Yr,Yzl

Distribution iV,.H,l~

Material relation

1 Distribution (X,,UZ)II 1

Convergence of (&,M.

J, Yes

Fig. 2 Block diagrams of CAD system.

223

Using the neurocomputing methods mentioned in this section we designed several important permanent magnet types.

4. Results

We study important simple magnet types, as examples of the CAD study. Generally, whether the gap g is null or not is often one of the requirements based on the user’s requirements. Magnets are classified as follows;

224 Y. Yamazaki etaf./Neurocomputing 13 (1996)217-230

(1) single-piece magnets with or without the hollow (SMH or SM), (2) double-piece magnets with or without the hollow (DMH or DM), (3) multi-piece magnets with or without the hollow (NMH or MM), where class (1) belongs to the gapless case ( g = 0) and the other the case with gap (g Z 0). The classes (SMH, DMI-I) are mainly studied and discussed.

For the hollow we consider the cases of a sphere of radius R and an ellipsoid of radii (R,., R,), which are expected to produce high uniformity fields over wide regions of the hollow as minimum-volume magnets.

For sets of a half gap length (g,, g :), we choose (0.1, 0. L), (0.1, I), (1, 0.1) and (1, 1) in cm unit.

For the number of cell points in the first quadrant we adopt 9 X 9 points, i.e. 9 points in each axial direction. The lengths of the magnet in the first quadrant are denoted by

(H,, HZ). The computation was performed under the UR. In order to satisfy the UR the

computer evaluates the values of (H,, H,), changes the volume of magnet, and finds the shape with the minimum volume.

The results obtained in the cases of the spherical (R = 10) and the ellipsoidal (R, = 10, R, = 15) hollow are illustrated in the left and the right parts of Fig. 3. The magnetic field strength and the field uniformity are indicated with Gauss and l/ 1000 units, respectively.

The features of these results are summarized as follows: (1) The tendencies of these quantities against the gap lengths are reasonable from the

physical point of view. (2) In spite of having expected somewhat more complicated shapes for the magnets, the

computed shapes were simple. The reasons are as follows: (1) The cell number near the boundary of the magnets is too low, i.e. unit cell contribution is large compared with the magnitude of uniformity. Thus the magnetic regions near the boundary have to be divided into fine cells. (2) The computational examples here are too few.

(3) The computing time is very short. (4) The computing accuracy is very high.

Finally, from the CAD-study point of view, we summarize: Our CAD systems were made according to natural laws and we can freely design magnetic machines in completely new forms. We then compare experimentally obtained results to those obtained with our CAD systems to verify the system’s performance.

5. Concluding remarks

The important points of the methods used are summarized as follows: (I) Both parts of the 2nd and 3rd steps in terms of the number of computational

iterations are effectively performed by means of the improved neural network algorithm [ 1,3].

(2) The magnets are divided into cells and each cell is assumed to be uniformly magnetized with magnetization ,at its center. Note that one dimension is reduced and that the surface magnetization current effectively flows.

Y. Yamazaki et al./Neurocompdng 13 (19%) 217-230

Spherical hollow (R=lO;Hr=l5, H,=40)

Ellipsoidal hollow

ml ~

t” (14626. 4. e/1000)

(R,=lO, R,=15:II.=l5. H.=40)

f

R

(lO6OG, 3/1000)

......... 11 . ......... ........

5:::: : : :I 1:: : : : : :I z

l . . l l . . >

T R

(R71G, 3. 6/1000) ......... h ............. -1 .. ....... I

. . . . . . . . . . . . . . Z l

(1346G, 6. 6/1000)

l l . . l l l . * .........

i

........

........ ....... ....... ....... I_ Z

225

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AR

(603G, 5. 5/l 000)

ml z

R

(055G, 3. l/1000) . . . . . . . . .

‘=, : : : : : : : ....... ‘1 ...... ......

Fig. 3. Shapes of magnets. Magnets with the spherical (ellipsoidal) hollow are illustrated in the left (right) column. The figures starting from the top correspond to the magnets with a half gap length ( g,, g,), (0.1.0. l), (0.1, l), (1.0.1) and (1, 1). The magnetic field strength and the field uniformity are indicated in Gauss and l/lCOO units, respectively.

(3) The integrations over the angular variable 4 in the B,, B, expressions derive one-dimensional sum-expressions over r or z variable, including the first and second kinds of complete elliptic integrals. Note that one dimension is reduced [2,3].

(4) The B-M curves are approximated as a polynomial [2]. (5) Numerical divergences which come from the null denominator do not appear.

The characteristic features of the permanent-magnets CAD system using neurocom- puting are summarized as follows:

226 Y. Yamazaki et al./Neurocompuring 13 (19%) 217-230

(1) The computing time is drastically shortened. (2) The computing accuracy considerably increases. (3) The solutions are found in terms of appropriate annealing procedure. (4) Our method is applicable to any magnetic material. (5) Our method is applicable to systems with macro- to micro-size (i.e. the classical

case). CAD systems of quantum cases. (i.e. nano-size machines) are openquestions. They

are the next target to be attacked in the near future. The problems of magnet shape design have been solved in this paper. These problems

are one of optimizing pattern formations which obey electromagnetic field laws. Neurocomputing techniques have proven useful in this problem. It is considered that studies from these two points of view are required for neurocomputing.

Acknowledgments

It is a pleasure to acknowledge helpful discussions with Professor Jacek M. Zurada.

Appendix A. Maxwell equations in magnetic media with the z-axial and z = 0 plane symmetries

Introducing a vector potential i defined as g = VX L, the Maxwell equations

Vx2=4?TJ/c, v6=0 (A.1)

are rewritten as

vx vx~=4rr\D/c+4?rvX1Ci~4~18,. (A-2)

in the magnetic system. The first (second) term in (A.2) stands for the coil (magnetic) current and Jvr total volume currents. As the expression (A.21 reduces to the Poisson equations

V’K = - 4lrJ,/c. (A-3)

their solutions are expressed in the integral form as

Y. Yamazaki et al./Neurocomputing 13 (1996) 217-230 221

r

Z

Fig. A. Cell structunz of magnet for computation purposes. Magnets are divided into cells whose centers are calculation points. ‘Ihe surface current is assumed to distribute at the side-centers denoted by X , A.

The first (second) term expresses the contributions of the total volume (surface) currents to the vector potential. The normal vector on a surface element ds’ is denoted by S

We take into account the following two facts: (1) Magnetic media are divided into small uniformly magnetized cells, i.e. the first term

in (A.4) vanishes. This situation is illustrated in Fig. A. (2) The z-axial symmetry of the magnets yields A, = A, = 0, i.e. the substitution of the

nonvanishing vector-potential component A, in the expression of B’ leads, e.g. for B,, to

B,(r, z)=~L&~~(&?+~, Jz) +M,(J=- 1, Jz,]

x[n~(-K:cb:E=)+a”(-KX+b~EX)]

+2&[-M,(JR, Jz+ 1) +M,(JR* JZ- I)]

x[,~(-K~+b~E~)+a”(-K+b~E)J

using

a+= (z -z’)[(r’+r)2+(z-z’)*]-1’2/r,

b+= [r’2+r2+(~-z’)2]/[(r-i)2+(~-z’)2]

k:-4r’r/[(r’+r)2+(~-z72], O-=0+ IL~+-Z~ OE[U, b, k]

(A4

(A-6)

228

where

Y. Yamazaki et al./Neeurocompuring 13 (1996) 217-230

AV= A,A,, K( k*) = c”’ dO[l - k* sin* o]-I’*,

dB[l -k* sin* 01”’ (A4

The superscript of X , A, e.g. Ox stands for the variables (8, z’) belonging to the side-center x of the (rectangular) cells and the sum Cx is the summation over all the variables situated at the X -side centers. The cell area AV is expressed as a product of two cell-sides A,, AZ. The first (second) kind of the complete integral is denoted by K (E) in (A.7). The first (second) term in (A.51 is the contribution of the magnetic surface currents in the first (second) quadrant to the magnetic flux density B,.

Appendix B. Ideal cases of uniform field

It is very helpful for our study to consider the ideal cases of yielding uniform field; (1) uniformly magnetized, parallel infinite planes,

1.

Fig. B. Ideal cases of yielding uniform field: (1) uniformly magnetized, parallel infmite planes; (2) uniformly magnetized, infmite pipe; (3) uniformly magnetized, ellipsoidal magnet; (4) ellipsoidal empty space shaped in

uniformly magnetized infinite medium.

(2) (3)

(4)

Y. Yamazaki et al./Neurocomputing 13 (19%) 217-230

uniformly magnetized, infinite pipe, uniformly magnetized, ellipsoidal magnet, ellipsoidal empty space shaped in uniformly magnetized infinite medium.

229

These cases are illustrated in Fig. B. The completely uniform field is produced in the small empty-space area enclosed with the uniformly magnetized medium in the cases of (l)--(4).

If the uniformity of the field is reduced from the complete uniformity to a certain amount of uniformity, the magnitude of the magnetic medium reduces from the infinite magnitude to some finite amount.

It is expected that shapes of magnets which produce some constant magnetization over the whole volume of the magnets and keep the total volume to a minimum approach ellipsoid-like shapes modified according to the UR, e.g. with gap space near the z = 0 plane and/or around the z-axis.

References

[l] Y. Yamszaki and M. Ochiai, Application of layered neural networks to CAD (computer aided design) for magnetic devices. FSAI 1 ( 1992) 95- 112.

[21 N. Tsuya and Y. Yamazaki, A method for numerical computation of flux density distribution in nonlinear magnetic media, Proc. ht. Conf. on Electronic Relays, Sendai, Japan ( 1963) I%- 199.

I31 Y. Yamazaki, A. Mikami and M. Ochiai, Application of layered neural networks to phase analyzer of laser, FSAI 1 (1992) 81-94.

Yoshitake Yanmzaki received the B.E. and M.E. in electrocommunication engineer- ing from Tohoku University, Sendai in 1962 and 1964. respectively. He recieved the D.S. in statistical physics from University of Tokyo, Tokyo in 1976. He has coworked with the theoretical physics groups in research center Jiilich, Saarland University, Cavendish Laboratory, Universid de Get&e, Universit6 de Paris-Sud and so on since 1977. His research interests are of fundamental researches on neurosystems, mum-science and technology, and phase-transition and critical phe- nomena.

He is now Professor of mechanical system engineering in Kyushu Institute of Technology.

Moyurtt Ochiai recieved the B.S. and M.S. in applied physics, from Waseda University, Tokyo in 1965 and 1967. He recieved also the Ph.D. in theoretical physics from Waseda University in 1970. He has joined the theoretical physics groups at Cemro di Cuhura Sientifica “A. Volta” in Como and Saarland University in Saarbriicken from 1985. His research interests are concerned with fundamental theory of non-equilibrium statistical mechanics, phase transitions, pattern formation, powders and porous materials. He is now Professor of Electronics and Infonnatics at North Shore College of SONY Institute, Atsugi Japan.

230 Y. Yamazaki et al./ Neurocomputing I3 (19%) 217-230

Arno Hdz is professor of theoretical physics at Universitit des Saarlandes (UDS), . SaarbBcken. This main interests conce.m topological defects and their interdisci-

plinary aspects. This leads him to study topological properties of magnetism in nanoscale materials in connection with nanomechanical devices and applications.