On a regularizing algorithm for calculating current surface density

'41

Short communications

REFERENCES 1, SAMOKHIN, A. B., and TSVETKOV, S. V., On the use of an approximateboundary condition for solving excitation and diffraction problems in inhomogeneous media by the mesh method, Izv, WZOI’.Radiofiz., 20, No. 7,1063-1070,1977. 2. TSVETKOV, S. V., Application of the mesh method for solving diffraction problems in a resonance domain in inhomogeneous media, Dissertation, MFTI, Moscow, 1977. 3. SAMARSKII, A. A.,lntroduction to the theory ofdifference skhem), Nauka, Moscow, 1973.

schemes (Vvedenie v teoriyu raznostnykh

4. SAMARSKII, A. A.. and GULIN, A. V., S?ability of difference schemes (Ustoichivost’ raznostnykh Nauka, Moscow, 1973.

U.S.S.R.

Comput. Maths. Math. Phys. Vol. 22,No. 5. pp. 241-245,

1982.

Printed in Great Britain

skhem),

0041-5553/82SO7.50+.00 01984. Pergamon Press Ltd.

ONAREGULARIZINGALGORITHMFORCALCULATING CURRENTSURFACEDENSlTY* N. N. BONDINA and V. M. MIKHAILOV Kharkov (Received 21 Junua~* 1981)

A REGULARIZING algorithm is described for solving the integro-differential equation for the current surface density in conductors with a plane-meridianal or plane structure of the quasistationary electromagnetic field in the case of a sharp skin effect. The algorithm is based on compensation of the oscillations of the numerical solution resulting from the problem being illposed, and includes an expression for calculating the regularization parameter. Examples are given. The problem of calculating the current surface density 7)on conductors with plane-meridianal or plane structure of the quasi-stationary electromagnetic field when a sharp skin effect is present can be stated as [l]

where A is an integral operator, d* is a parameter, and f is the right-hand side, Equation (1) has to be solved as f + 0 in the light of the initial condition

l.h vjkhisL Mat. mat. Fir., 22,5,1249-1252, US%22:s-P

1982.

h: .2’. Bondina and I: M Mikhailov

242

10‘?(7

10 “k,

2 NoL

b

a

FIG. I. 1 - for kl ; 2 - for k2; 3 - for 7~:4-8 - respectively, ford*=0.25.10--3, 0.3.10-3,0.35.10-3, 0.4.10-J (optima]),

0.44.10-3; 9 - for kl

Node number

FIG. 2. b / d=2, h I d-0.2

FIG. 3. b / R=0.5, b / d=2;

2As t--f 0, Eq. (1) has properties closely similar to an equation

I-

with

of the 1st kind (it transforms

such an equation when t = 0, see [2]). It is well known that, when constructing algorithms for solving ill-posed problems, use is made of supplementary solution

[3]. In the present problem, this information

distribution

cross-section.

This property

into

regularizing

information

about the

will be taken to be the smoothness

of n everywhere with the possible exception

contour of the conductor

d'=0.4~ 10-3,

wjth d&t

of the

of individual points (e.g. nodes) of the

is confirmed

We shall transform (1) into a system of ordinary differential the conductor cross-section contour:

by experimental equations

data [I 1.

for the nodal points of

dq A,,-= -$+f, dt

where Ah is a matrix approximating

the operator A.

We shall solve system (3) in one time 7 step under initial condition (2). Since the matrix Ah is the finite-dimensional analogue of the completely continuous operator A, the inverse matrix Ah-1 is unstable. Hence errors will be introduced into the numerical solution of the problem, first, when

243

Short coninlurlicariom

inverting the ill-posed matrix Ah, and second, when integrating instability

of the numerical

A (see Figs. I-3).

the stiff system 14. 51. The type of

solution depends on the type of quadrature

The Simpson formula leads to an “oscillating”

formula used to approximate

solution; this formula will be used

below. Consider the mechanism

of smoothing

the numerical

solution of (3) by means of the Runge-

Kutta method of 2nd order [6]. We have n = f

(k,+k,) + O(P).

(4)

where k,=TAa,-‘f,

The component

(5)

kl is not smooth, due to the instability

term of (6) the matrix Ah-l their size and polarity.

is multiplied

Let f be a vector with oscillating components. Cortditiorr A. The operation Ah-it

of the matrix Ajl-l.

by kl ; the existing oscillations

In the second

of kl thereby change

We introduce:

changes the amplitude

of oscillation

of all components

of

1 by the same factor. Then, with 5 = f - kl/d*, we can select ad* (which we shall call optimal and denote by dCopt) such that. as a result of addition amplitude

of kl and k2, the oscillations

and opposite signs cancel each other: in view, of the proportional

of kl and kl will intersect at a

the straight lines joining the ordinates of two adjacent components point 9. belonging to the smooth solution.

of the same

change of the amplitude.

The result of the latter is the relation ki-q;=q,

where kZopr is the value of k2 corresponding

-lizoPt,

(7)

t; the smooth solution.

On the other hand, b>. (6)> k, opl = kl - r.4h1

--$

, opt

Considering

(7) and (8) simultaneously.

we have (9)

While it is not possible to establish in advance that condition does from the result of calculation, d*= Pop*.

i.e. obtaining

A holds, we can conclude that it

the smooth solution of the problem with

N. II;. Bondina and I’. M. Mikhailor

944

b

a

c

FIG. 4

As examples, consider the systems of parallel conductors of circular and rectangular crosssection with forward and reverse current (Fig. 4, a and b) and a one-turn solenoid of rectangular cross-section (Fig. 4, c). In Fig. 1 we show the components of the solution kl and k2 given by (5) and (6) with the optimal value of regularization parameter d*,pt=~.4.i~-J (r/a=O.S). The straight lines joining the ordinates of kl and k2 of nodes 1 and 2 of the coordinate mesh on the conductor contour intersect at one point (Fig. 1, b). In Fig . 2 we show the distribution of r) on the side M-MI of the rectangular bus Use of the rectangles formula leads to a typical deviation from smoothness, i.e. oscillations close to the nodal points of the contour (curves I,?). While smoothing can be achieved by coarsening the mesh step (curve 3), there is then a loss of information about the behaviour of the solution on a substantial piece of the conductor (broken part of curve 3). Curve 4, obtained by means of the present algorithm, provides such information. It must be borne in mind that, when condition A holds, the coefficient of proportionality can have different values on different pieces of the contour (e.g. with different steps of the coordinate mesh on these pieces). Then, d*cpt can be chosen for each piece in accordance with our algorithm. In Fig. 3 we show the distribution of n in the rectangular solenoid. For pieces 1 -MI, N2 - N (Fig. l,c)d*cpt =0.9X10 -3; for piece N1 - N2, deopt = 0.15X 10-Z. On these pieces the numbers of steps are respectively 10, 20 and 10. On the other hand, if the coordinate mesh step is taken to be the same everywhere, then deopt will also have the same value. In the last example, with 10 steps on piece N1 - N2, we have d*,pt = 0.9x 10-3. A regularizing algorithm can be constructed by using a Runge-Kutta method of different order in various modifications [6]. In short, the regularizing algorithm is as follows: 1) we solve system (3) with some value ofd* (this value is taken in the interval (10-5, 10-S)); 2) we calculate &-ikl= 3) we Bnd d*,,,,t from (9);

(k,-k?)/d’;

the last relation is obtained from (6):

Shorr communicarions

245

4) we solve system (3) with d* = deopt, The algorithm results in obtaining a smooth solution whose accuracy is determined by the errors of approximation of the operator A and of the Runge-Kutta method. Translated by D. E. Brown

REFERENCES 1. MIKHAILOV, V. M., Pulsed electromagnetic fields (Impul’snye elektromagnitnye Kharkov, 1979.

polya), Vishch shkola,

2. SHNEERSON, G. A., Calculation of the ac distribution over the surface of a solid of revolution with sharp skin effect, Zh. rekhn. fiz., No. 1, 51-54, 1961. 3. TIKHONOV, A. N., and ARSENIN, V. YA., Methods of solving ill-posed problems (Metody resheniya nekorrektnykh zadachj. Nauka, Moscow, 1979. 4. FADDEEV. D. K., and FADDEEVA. V. N., Computarional methods of linear algebra (Vychislitefnye metody lineinoi algebry), Fizmatgiz, Moscow, 1960. 5. RAKITSKII, YU. V., USTINOV, S. M., and CHERNORUTSKII, I. G., Numerical methodsof solvingstiff systems (Chislennye metody resheniya zhestkikh sistem), Nauka, Moscow, 1979. 6. KRYLOV, V. I., BOBKOV, V. V., and MONASTYRSKII. P. I., Compuational mefhods (Vychislitel’nye metody), Nauka. MOSCOW,1977.

LIS.S.R. Compur. Maths. Math. Phys. Vol. 22. No. 5. pp. 245-249. Printed in Great Britain

1982.

0041-5553/82507.50+.00 01984. Pergamon Press Ltd.

NUMERICAL STUDY OF THE MOTION OF A GAS COMPRESSED BY A SPHERICAL PISTON CONVERGING TO ITS CENTRE* V. P. PARKHOMENKO and S. P. POPOV Moscow (Received 8 December 1978; revised 24 March 1982)

THE PROBLEM of the centrally symmetric flow of gas inside a hollow spherical piston, moving towards its centre according to a power law, is solved numerically for an inviscid thermally nonconducting perfect gas with a ratio of the specific heats x=1.4 . It is shown that, over a wide range of variation of the initial data, as the shock wave approaches the centre, the numerical solution does not yield the solution of the corresponding similarity problem. It is well known that similarity solutions are not only the exact solutions, but are also the asymptotic forms of the solutions of certain classes of non-similarity problems, solvable by ordinary numerical methods. For instance, from a certain instant the explosion of a volume charge is described well by the similarity problem of a strong point explosion, solved in [I]. The nonsimilarity problem on a brief shock also turns into a similarity mode [2,3]. lZh. vychisL Mar. mar. Fiz., 22, 5, 1252-1255,

1982.