The Computation of Transient 2-D Eddy Current Problem by Domain

Mat/d. Comput. Modelling

Vol. 21, No. 3, pp. 39-51, 1995 Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

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The Computation of Transient 2-D Eddy Current Problem by Domain Decomposition Algorithms I. P. BOGLAEV* AND V. V. SIROTKIN Institute of Microelectronics Technology, Russian Academy of Sciences Moscow District, 142432, Chernogolovka, Russia J. D. LAVERS Department of Electrical and Computer Engineering University of Toronto, Toronto, Ontario, Canada M5S lA4 (Received

and accepted

September

1994)

Abstract-A combination of domain decomposition method and a time discretization for the solution of transient 2-D eddy current problem is considered. A domain decomposition algorithm suitable for parallelization is described and convergence is established. Numerical experiments on a shared memory multiprocessor are presented. Keywords-Transient tion, Parallel computing.

eddy current problem,

Domain

decomposition

method,

Time discretiza-

1. INTRODUCTION The iterative domain decomposition algorithms for the solution of a time harmonic 2-D eddy current problem have been described previously in [1,2]. The Schwarz alternating method [3] and the related computational method from [4] h ave been studied and convergence of these methods has been established. The latter method is suitable for parallel computing and its parallel implementation on a shared memory multiprocessor has demonstrated the effectiveness of this method [2]. In this paper, we describe a combination of domain decomposition method and a time discretization for the solution of transient 2-D eddy current problem. On each time step, this approach reduces a given problem to sequences of problems on appropriate subdomains. As in [1,2], we consider the Schwarz method and the decomposition method from [4] for the solution of a parabolic problem arising in the transient 2-D eddy current problem. This problem can be considered as a singularly perturbed problem (see [5] for details), i.e., the solution exhibits boundary layers. We will use “natural” decomposition: the regions of rapid change of the solution are localized in subdomains. These algorithms are implemented on a 32 processor Kendall Square Research shared memory computer, and results are presented giving the speedups of the algorithms compared to the usual sequential (undecomposed) method. The structure of this paper is as follows. In Section 2, we give the mathematical rnodel describing the transient 2-D eddy current problem. Section 3 develops the iterative algorithms in *The work of this author has been supported by an International Sciences and Engineering Research Council of Canada.

Scientific

Exchange

Award

Typeset 39

of the Natural

by AM-w

I. P. BOGLAEV et al.

40

the case of decomposition of the original domain into two subdomains, and in Section 4, numerical experiments are described. 2.

MATHEMATICAL MODEL OF TRANSIENT EDDY CURRENT PROBLEM

2-D

The mathematical model describing the transient 2-D eddy current problem can be described as a parabolic equation having the following general form (see [6] for details): div(vgradu(P,t))

- (Tv

=

f(P,%

(1)

where P = (cc,y), v and cr are the reluctivity and the conductivity,

respectively,

f

is a given

function. In most cases, the function u(P, t) represents the z-component of the magnetic vector potential A’. For the 2-D problem considered later in this paper, u(P, t) represents the z-component of the magnetic field strength g. In the case of purely sinusoidal regimes with

f =

0, the one-dimensional case of (1) on [0, co]

has the following form:

u”(Z) -

wl7u%(z) =

L = (-1)li2,

0,

u(0) = 1,

where w is the angular frequency (the prime denotes differentiation).

[cos(;)

The solution of this problem

S=[&11’2.

is U(Z)=exp(-:)

u(oo) bounded,

-Asin(;

This expression shows that the decay of the solution is governed by the “penetration depth” 6. In many realistic situations, 6 is a sufficiently small parameter and our problem may be considered as a singularly perturbed problem. The solution has a boundary layer at 5 = 0 of width hs = 4 WV I.

3. DOMAIN

DECOMPOSITION

ALGORITHMS

We assume for simplicity that the domain Rc with eddy currents is a rectangle (0, X) x (0, Y). Consider the following version of (1): YAU - 0 $

= f(P,t),

P E 00,

o = uo(P), tn=n7,

p

E

flo,

7x=1,2 )...)

= f (P, t”),

UYP) = m t”), &a,

7=-.

t %k3x

P E x20,

(3)

Domain Decomposition ASSUMPTION.

Algorithms

41

We assume that the following estimate holds

(4) where u(P, t) and P(P)

are the solutions to (2) and (3), respectively, and constant C is inde-

pendent of T. Now we consider and analyze domain decomposition subdomains.

Introduce the decomposition mains stl, 022 (see Figure 1)

algorithms for the case of two overlapping

of the domain slo into the two overlapping subdo-

X c

a2

Figure 1. Decomposition

3.1.Statement

of the domain Ro.

of Algorithms

We now construct two algorithms. Introduce the two sequences {Vn}, {Wn}, n 2 1, satisfying the following problems

v*v”(p)

_

fl

[VV) - TV)]

= f(p,tn),

PER17

VYP) = de t”),

P E r:,

V”(P)

P E rl;

7-

vAWn(P)

-

c [WV) - Tr

= V”(P),

= f(p

WI

7

p)

1

>

,...)

n,,,,

fi2,

W”(P) = g(P, t”).

P E r;,

W”(P)

P E r2.

= W”(P),

The first algorithm, Al, is the Schwarz alternating algorithm. determined by p E flo\fl,, VYP), u;(P) = Wn(P), P E n2, { n=l,2

p E

u:(p)

= uo(P),

(54

(5b)

Here the function U:(P)

p E fit07

is

(64

I. P. BOGLAEV et al.

42

and the boundary conditions from (5a), (5b) have the forms

T(P) W”(P)

The second algorithm,

= q-l(P),

p

= V”(P),

P E r2.

A2, is constructed

vAZn(p)

E r1,

(6b)

using the following auxiliary problem

PYP) - uJ‘-‘(p)l= f(p

_ cT

p) I

1

P E

e,

7

P(P)

= V”(P), Zn(p)

Z”(P) = W”(P),

P-h,

= dP>

p E 70,

t”),

72=

1,2 )...)

p E Y2, nmaxr

where the subdomain 8 is defined by (see Figure 1)

ec 00,

R1nfi2c8,

-fo==dSlondB,

OlylY},

Yl={f+a:=zB,

ae=y,uy1uy2,

Y2={P:z=zE,

O-cy