Calculation of the magnetic system by the solution of inverse problem Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

UDC 537.8

Calculation of the Magnetic System by the Solution of Inverse

Problem

R. V. Polyakova*, I. P. Yudint

* Laboratory of Information Technologies Joint Institute for Nuclear Research Joliot-Curie 6, 141980 Dubna, Moscow region, Russia ^ Laboratory of High Energy Physics Joint Institute for Nuclear Research Joliot-Curie 6, 141980 Dubna, Moscow region, Russia

In this paper we study the problem of searching for the design of the magnetic system for creation of a magnetic field with the required characteristics in the given area. On the basis of analysis of the mathematical model of the magnetic system rather a general approach is proposed to the solving of the inverse problem, which was written by the Fredholm equation:

H(z) = J J(s)G(z, s)ds, z e U, s e s.

s

It was necessary to define the current density distribution function J(s) and the existing winding geometry for creation of a required magnetic field H(z). In the paper a method of solving those by means of regularized iterative processes is proposed. On the base of the concrete magnetic system we perform the numerical study of influence of different factors on the character of the magnetic field being designed.

Key words and phrases: magnet systems, inverse problem, Fredholm equation, regularized iterative processes.

1. Introduction

When designing magnetic system it's necessary to solve the inverse problem, that is, via a given magnetic field to define current parameters or, its geometrical characteristics, or all that simultaneously.

The definition of the beam density distribution in the magnetic system, in which the geometry is known, is a linear inverse problem for the given field.

When the required field must be created with the help of conductors, the value of the current which varies similar to the coordinates of their position providing the current in all the conductors is the same, we come to the solving of the inverse problem [1].

In this paper we consider the construction of a mathematical model of the magnetic system for this kind of the problem and the methods and numerical algorithms for their solution by using the Tikhonov regularization methods. Because a magnetic field is supposed to be given by one of its components (Hx, Hy, Hz) depending on a specific problem, so further H sample will be used for notation.

2. Mathematical Model of the Magnetic System

Let in a region U with the help of the sources of current distributed in the region S a field H should be created with the given characteristics (for example, the whole homogeneous field in the region U). It is known, that the field in any point z of set U is defined by the expression

H(z) = j J(s)G(z,s)ds, z e U, s e 5 (1)

s

Received 31st August, 2011.

where J(s) is a distributed density function of the current in the system, G(z, s) is a Green function, that analytically depends both on the geometry of the source of the magnetic system and on the point z G U.

The inverse problem, namely, a definition over the given density of distribution of current in the magnetic system with the known geometry is a linear inverse problem (model 1).

Then the mathematical problem reduces to the solution of the Fredholm linear integral equation of the first order with unknown function J(s).

If the composition of the magnetic field includes not only variant density of current and arrangement source of current, then we must solve the nonlinear inverse problem (model 2) with unknown J(s) and s G S.

3. Method of Solution of the Inverse Problem (Model 1)

It is known that the problem of solution of the first order Fredholm integral equation (1) is related to the non-correct defined class of the problems, because the large changing in the J(s) solution can correspond the small changing of the input data H(z). To obtain a stable solution of the non-correct defined problem, A. N. Tikhonov developed a regularized algorithm [1].

Here we will use the second order method of a regularization in order to solve the problem.

For this, we construct a smooth parametric functional

Fa[J(s), H(z)] = $[J(s), H(z)] + a Q[J(s)], (2)

where

$[J(s),H(z)] = J[H(z) — J J(s)G(z, s)ds]2dz (3)

u s

is the quadratic deviation of the operator A[z,J(s)] = J J(s)G(z,s)ds of function

H(z), S

Q[J (s)] = J J2(s)ds (4)

s

is the regularizational functional, or a stable one, and a is a numeric parameter of the regularization (a > 0).

Theorem 1. For any function H(z) G L2 and for any a > 0 there exists one and only one 2(n + 1) differencial function J%(s), which realizes the minimum of the smooth functional Fa[J(s),H(z)] of the form (2).

Theorem 2. If H(z) = A[z,J(s)], J(s) G C( n+1), then for any e > 0 and auxiliary values 0 < 72 there exists J) so that, if

1. -\\Hs(z) -*H(z)\\ L2 < 5, where lis(z) G L2;

2. — a = a(S) has the order 52;

S2

33 — * < ^^ < _

then Jgn(s) is a minimum of F%[J£(s), Hs(z)] and

\\J5n(s)(i) —< e, s G S, i =1, 2, ••• ,n with 6<6o(e,^i ,J2,J).

For this theorem we infer that there exists a function J", that is a minimum of

a n

functional in the form (2) which reduces to the solution of the equation (1) J(s The complete demostration of this and other conditions one can find in [1].

When applying the regularizational method, the selection of parameter a is one of the main problems.

The point is that not always for the obtained smooth solution the discrepancy principle is being fulfilled, i.e. the inequality

\\HS - H* II < S, (5)

where 8 is precision of the approximate input data lis, H* is the precise value of the input data, holds.

In practice, for the solution of the non-correct and correct problem it is necessary to find the solution, that satisfies the required precision. In [2] V. A. Morozov suggested, as the main quality criteria to select, the regularization parameter of the deflection principle.

Discrepancy Principle. Set any 0 < S ^ So and any 0 < h ^ h0 with condition X(h, 8, J) = (\\AhJ - AJ\\o + S) x (1 + p(8, h))1/2 < \\AhJ - Hs\\o,

where 0(5, h) is a positive function, such that lim^,h^0 0(8, h) = 0, and Ja is the solution obtained for the minimum of functional Fa(J,H).

Then there exists at least one value of the regulartization parameter a = a(8, h) >

0, so that psh(a(S, h)) = %2(h, S, J), lim^h^o Jfh = Ja and Jfh = J^^.

4. Numerical Algorithm for the Solution of the Problem of

Model 1

In the expression (2), if presenting the integral in the form of sums, we obtain N M M

Fa = Y\Hi(^) - E JM)K%3(z3,s%)]2^z3 + a Y, (6)

j=1 i=1 i=1

where N is a number of points from the set U, M is a number of points from the set S, M < N and Kij = jG(zj ,s)ds.

A Si

Suppose Asi = As = const, Azj = Az = const. The condition of the minimum of the functional Fa is

dFa dFa dFa

rn; = °- in^ = 0^ = 0- <7>

Taking into account (7), we obtain

a pa N N M

= -E HjKijAz + Y Y.JiRijKijAz + aJiAs = 0, I = 1 - M. (8)

=1 =1 =1

In such a way we have a system M linear algebraic equations with the unknowns N of the Ji form:

M N N

Y Ji E KijKijAz + aJi As = Y HjKij Az, I = 1 - M. (9)

=1 =1 =1

Supposing «As = «Az, we obtain

M N N

k13K13 + a'ji = E hjrij , 1 = 1 + M. (10)

i=1 j =1 j =1

Obviously, a' serves the meaning of arbitrary coefficient a. Therefore the system of equations for Ji can be written finally in the form

M N N

Y,Ji T^KijKij + aJi = £HjKij, 1 = 1 + M. (11)

i=l j = l j=l

If the magnetic system is a discrete set of coils, then the field H(z) in any point z £ U is defined in the following way:

M

H(z) = Y,Ji G(z, s)ds, (12)

where M is a number of coils, Ji is the current density in the i-st coil, Asi is the selection of the i-th coil, G(z, s) is a Green function.

Having solved the system of equations (12), we obtain a discrete set Ji, I = 1 ^ M, that is a solution of the problem (1). Similar to that, we define the distribution of the current density in the magnetic system for creation of field H(Zj),j = 1+N, Zj £ U.

5. Particular Case of the Mathematical Model 2

Let in some region S with a disposition M of conductors with the same current

the field H be created.

In the system H (z), z € U we have

M

H(z) = lo ^G(Si, z), (13)

i=1

where G(si, z) is the Green function for the i-th conductor.

Both a current I0 and the coordinates si of the conductors, which would provide the given field H(z), z € U in a best way, should be defined.

The function G(si, z) is usually a nonlinear one concerning the coordinate of the conductors i, therefore the analysed problem is a non-linear inverse problem.

Additional difficulty in the solving of the inverse problem is the restriction in the parameters [3]. However in any particular case one can efficiently find a solution for the given condition of the problem. Let us consider this case.

Let a parameter of the conductors disposition in the region S be only one coordinate, for example x, the region of disposition of the conductors in the axis x being known, x\ < xi < x2.

Then the equation (14) will has the form:

M

H (z) = I0 ^G(xi, z),z € U, x\ <xi < x2. (14)

i=1

We must define I0 , xi to create the field H(z), z € U in the magnetic system. The problem (15) is a non-linear inverse problem.

6. Numerical Algorithm of the Solution of the Problem in

Model 2

The solution of the problem (15) was divided in two steps. In the first step, the current density in the twisters is continuously distributed in the range of the given problem. The equation (14) has the form

X2

H(z) = J J(x)G(x,z)dx.

XJ

This problem and the algorithm of its solution was analysed in the points 2, 3. To select the solution Ja(x), (a is the regularization parameter) we calculate the following conditions of the problem:

1. The precision of the calculation of H(z) cannot be worse than the required precision of the magnetic field in the created magnetic system;

2. For all permissible interval [x1,x2], a function Ja(x) must keep the sign;

3. IJa(x)l < Jdop is the permissible current density.

Suppose, that there exists a continuous solution Ja(x), that satisfies all three conditions.

In the second step, we divide the interval [x1,x2] in M subintervals [x\,x\], i = 1 - M.

Then

m X2

H(z) = Y Ja(x)G(x,z)dx, z e U. (15)

i=1

XJ

For each subinterval [x\,x\] satisfies the conditions of the theorems about the mean value (as chosen function Ja(x)), therefore

m X

H(Zj) = E,z3)J Ja(x)dx, j = 1 - N, (16)

i=1

XJ

where N is a number of points in the region U, in which analyses field H,xj is a point in the i-th intervals. The limits x\,x\ chosen such that

Xi X» + !

X 2 X2

y = J Ja(x)dx = Io, (17)

i.e.

X2

10 = J ja(x)dx,

X1

then

M

H (z, ) = Io E G(4 ,Zj ) (18)

i=1

Obviously that for different Zj there exists its point xj, but based on the theorem of the mean values, it is always in the intervals [x\,x\]. This means, that unknown coordinate Xi is also in the i-th interval and it is defined from the condition of minimum

x»+1

1

1

of the functional

N

) = XI

3 = 1

^ 2

J Ja(x)G(x, Zj)dx - I0G(xi, Zj)

m,

(19)

dy(xi) dxi

N

£

3 = 1

Ja(x)G(x, Zj)dx — I0G(xi,Zj)

dG(xj, Zj ) dxi

= 0, x\ < Xi < x\. (20)

Then the solution of the problem is reduced to the solving of M sequential nonlinear equations in the form (20) with one unknown, moreover, the limits of the existence of the solution are known.

Note that we have analyzed the algorithms for creating a magnetic system with infinite thin conductor.

It's easy to demonstrate that for the finite size of the conductor, the algorithms completely persist, but in this case the Green function is under the sign of integration by the section of the conductor.

When the permissible geometrical region of arranging the conductor is defined as the nonlinear one, the density does not involve particular difficulties too and can be described by the similar algorithm.

7. Example of a Numerical Calculation of Real Magnetic

System

Let's consider an example of practically developed application of the algorithm to create non-metallic superconductor (SP) of the bipolar magnet, that was composed by triangular winding of excitement, its geometry is showed in Fig. 1.

Figure 1. The bipolar SP configuration in the plane of the winding; b-one of the possible

real SP configuration

2

2

1

From Fig. 1 it is obvious that the magnetic system was composed by triangular winding and it has a perimeter dimension of the aperture of magnet.

Using the developed numerical algorithm for the nonlinear inverse solution we calculate the mathematical model of the system, with a homogeneous field in which 80% of the aperture represent 10-5 ^ 10-6 for the magnitude of file 4-5 Tl.

The mathematical problem was set in the following mode.

Let in some region U (see Fig. 1a), an halogenous field H(z),z G U be created, using an arrangement M of the conductors of the triangular section in the given limited region S with condition that the current I0 for all conductors is the same. For this

magnetic system

M

H(z) = Io E G(Si,z), si G S,z G U. (21)

i=1

In the Descartes'system of carotenes si = {xi,yi}, z = {x, y} r(a ^ = y - yi + b ln (x - Xi + a)2 + (y - yi + 6)2 +

^(Si,Z)= 2 ln (x - Xi - a)2 + (y - yi + b)2 +

+ y - yi - b in (x - Xi - a)2 + (y - yi - b)2 +

2 (x - Xi + a)2 + (y - yi - b)2

, ,( x - Xi + a x - Xi + a \

+ (x - Xi + a) arctg-r - arctg-—¡- +

v y -yi - b y - yi + b J

s - ^i- a^arctg y-+b - arctg y-- J, (22)

where a is the half-dimension of the tire along x, b is the half-dimension of the tire along y, G(si,z) is the Green function for the triangular tire in the Descartes system of coordinates.

M = Nx + Ny,

Nx is the number of the creep tire of axis x, Ny is the number of the creep tire of axis y, or

k

m = E Ni,

i=1

k is the number of the winding blocks, and Ni is the number of the creeps in the 1-th bloks.

We must define not only I0, but undetermine the block configuration that forms the homogeneus field H(z) for every point z G U with a precision non less than 10-5 - 10-6.

Fig. 2 shows a continuous a continuous distribution and Jy for M = 48 creep and its approximation by continous "blocks" function for each subinterval.

Figure 2. The continuous distribution Ja(s) and its approximation Ja(s) by constant

subinterval function "blocks"

Table 1 contains the numerical calculation for the optimal variant of the magnet; the scheme of this magnet is presented in Fig. 1b.

Table 1

Values of the coordinates of these "blocks" center for x and y, given by the best relative precision of lower harmonics in the homogenous field H

x= =const=3.8 y= =const=3.8

Nkl Nk2 Averaged relative

Ny n2 JVi Vi Vi 2/3 Nx n2 JVi X\ X2 x3 precision of the lower

harmonics C2, C4, ce

10 6

20 5 5 0.509 1.5972 2.8588 9 2 1 1.0017 2.2937 3.1406 0.3 x 10"6 ^0.5 x 10"6

20 12 8 0.8421 2.6494 9 6 2 1 1.6024 2.2468 3.225 0.9 x 10-5 -1- 0.45 x 10"6

CO ¡=

3

o

a

a £

I

"5

3

a

a

o

s

CO

Site

iZ o

o to

13

8. Conclusion

1. In this paper we analyze the method of solving the nonlinear inverse problems which are necessary for the description of the mathematical model of magnetic system of some class.

2. The developed numerical algorithm, based on the method of regularization of the solution of non-correct problems with restrictions in the searched parameters, is reduced to the nonlinear type of problem (15) for the solution of M sequential nonlinear equations with one incognita. It permits to avoid difficulties, related to the solution of the system of the nonlinear equations. This solution is frequently reduced to the inverse problem.

3. To realize the proposed method in computer a numerical algorithm was developed and Fortran programs package was written.

4. Using this complex program, some practical problems [4] were solved, one of those was analyzed as an example in section 7.

References

1. Arsenin V. Y., Tikhonov A. N. Method of the Solution of Non-Correct Problem. — Moscow: Nauka, 1979.

2. Morozov V. A. // Numerical Methods and Programming. — Publication MSU, 1967. — Issue 8. — P. 63.

3. Polyak B. T. Iterationing Methods of Solution of Any Non-Correct Variation Problems // Numerical Methods and Programming. — Publication MSU, 1969. — Issue 12. — Pp. 38-52.

4. Zhidkov E. P., Polyakova R. V., Yudin I. P. A. N. Tikhonov Regularization in a Magnitostatic Problem // 13th conference of MCE. — Dubna, 2006. — Pp. 171177.

УДК 537.8

Расчёт магнитной системы при помощи решения обратной

задачи магнитостатики

Р. В. Полякова*, И. П. Юдин*

* Лаборатория информационных технологий Объединённый институт ядерных исследований ул. Жолио-Кюри, д.6, Дубна, Московская область, 141980, Россия ^ Лаборатория физики высоких энергий Объединённый институт ядерных исследований ул. Жолио-Кюри, д.6, Дубна, Московская область, 141980, Россия

В данной работе решается задача поиска конструкции магнитной системы для создания магнитного поля с требуемыми характеристиками в заданной области. На основе анализа математической модели магнитной системы предлагается достаточно общий подход к решению нелинейной обратной задачи, которая описывается уравнением Фред-гольма:

н(г) = !зв)ав, ^ е и, в е я.

в

Необходимо определить распределение плотности тока 3(в), а также расстановку источников тока для создания поля Н(г). В работе предлагается метод решения этих задач с помощью регуляризованных итерационных процессов. На примере конкретной магнитной системы проводится численное исследование влияния различных факторов на характер создаваемого магнитного поля.

Ключевые слова: магнитные системы, уравнение Фредгольма, метод регуляризации.