Numerical simulation of random composite dielectrics. II. Simulations including dissipation

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Numerical simulation of random composite dielectrics.

II. Simulations including dissipation

S. Stölzle, A. Enders, G. Nimtz

To cite this version:

S. Stölzle, A. Enders, G. Nimtz. Numerical simulation of random composite dielectrics. II. Sim- ulations including dissipation. Journal de Physique I, EDP Sciences, 1992, 2 (9), pp.1765-1777.

�10.1051/jp1:1992243�. �jpa-00246658�

Classification Physics Abstracts

02.60-77.20-77.40

Numerical simulation of random composite dielectrics.

II. Simulations including dissipation

S.

St61zle,

A. Enders and G. Nimtz

II. Physikalisches Institut der Universitit _zu K61n, 5000 K61n 41, Germany

(Received

21 May1992, accepted 29

May1992)

Abstract. A simulation model presented earlier for the permittivity of _a composite material has _now been extended _to include the dissipation of energy inside the medium. It is employed to

calculate the complex dielectric function of so-called effective media, I-e- randomly distributed lossy particles in _an insulating matrix. A general equation formulated before for lossless media

proves to be valid also for the dissipative mixtures investigated. However, simulations and exper-

imental data show that small deviations from random distribution _can induce large deviations in the dielectric function. Experiments _are reported that _agree with the simulation results.

1. Introduction.

The effective

electromagnetic

parameters ^of

quasi-homogeneous

mixtures of different materials have been

subject

to _a

large

number of

investigations

since the

beginning

of this century

([1- l8]).

In _a

binary

mixture for

example,

where _a volume fraction

f

of

particles (of permittivity cl

is

randomly

distributed in _a matrix

(of permittivity em),

the effective dielectric function

(DF)

T is

generally

not _a linear but _a

complicated

sublinear function of

f,

_em and _e.

There have been _many

attempts

^{for the}

analytical development

of _a consistent

theory,

^which

describes the

physical reality

and

provides

_a ^tool for the

interpretation

of measurements _on such 'effective media'. But these theories and the

respective

^formulas fail in

describing experimental

data

jig, 20], although qualitatively they

show _a similar sublinear behaviour. Now _as computers grow

bigger

and

faster,

_a few authors have

developed

computer ^{models for} such systems ^and

numerical simulations of effective media have been carried out,

using

different

techniques [15- 18].

However, _up to _now these efforts have not led to _a

knowledge

that

helps experimentalists

to

interpret

^their data

unequivocally.

In [21] we have

presented

_{a new} computer code COSME

(Complex

Solution of Maxwell's

Equations),

which _was

employed

in the simulation of

binary

mixtures. We have

compared

_our results to the results calculated

by

several effective medium formulas.

Although

this

procedure

did not _even include lossy

dielectrics,

the

respective

formulas failed in most _cases. In _contrast to them _we found _a relation between

f,

_{e, em} and T with _a strong

similarity

to

Looyenga's

formula [7], ^{which is valid for}

f

< 0.25:

1766 JOURNAL DE PHYSIQUE I N°9

p(f)

=

f ~a(f)

₊

(i fj ^ea(f) (ij

where

a( Ii

=

(1.60

+

0.05)

*

f

+

(0.265

+

0.005)

In this _{paper we} report the continuation of _our

investigations,

_now

including lossy

dielectrics.

Apart

from

studying

the behaviour of the

complex

effective

permittivity

in the whole volume fraction range between 0 and 1, _we compared

equation (ii

to the

complex

results and

finally

we carried out

experiments

to prove the

validity

of the simulation _program.

2. The

algorithm.

The basis of the simulation is the

following physical

model: _a volume of size 20 _x 10 _x 20

mm~

or 20 x 15 _x 20 mm~ is surrounded

by ideally conducting walls, forrrdng

_a

cavity

resonator. The solution of Maxwell's

equations

inside this volume

including

all

boundary

conditions leads to

electromagnetic

_resonances characterized

by

certain distributions of the electric and

magnetic

fields

oscillating

with _a

corresponding

_resonance

frequency.

The solution for

homogeneous

materials is

easily

found

analytically [22].

For the numerical solution for

inhomogeneous fillings

the volume inside the resonator is covered with _a 3D cartesian mesh such that _every mesh cell contains

only

_one kind of material whereas the dielectric

properties

for different mesh cells _are uncorrelated. Thus for each cell

we select

randomly

_a

permittivity

_e with

probability f

and _a dielectric constant _em with

probability

I

f.

Using

a finite difference method for the fields E and B

interdependent

via Maxwell's

equations

the

problem

is transformed into

a discretized

problem,

whose solution is

equivalent

to the solution of the

following eigenvalue equation:

Ae ₌

qDee

q =

A~

co POw~

(2)

Here the matrix A

represents

the discrete

rotrot-operator,

^the vector _e the discretized electric field vector, De is _a

diagonal

matrix

containing

the different

permittivities

of all the mesh cells. _q is the

eigenvalue

to be calculated

containing

A ₌ unit

length

of the mesh cells,

co, PO "

Permittivity

and

permeability

of _vacuum and _w

= resonance

frequency,

at which the

field distribution _e oscillates inside the resonator

(for

details _see

[21]).

The above

eigenvalue problem

_may be solved _more

easily

whenever De is _a real

matrix,

because then the

problem

is

symmetric

and real. As

reported

in [21] ^such a

problem,

_even if it is of _very

high

order

(here:

>

150000),

_can be solved

by

_a

spectral

^{shift of}

eigenvalues

combined

with inverse iteration

employing

_a

conjugate gradient (CG)

method for the minimization. For the simulation of

dissipative

mixtures where De is

complex,

standard CG routines fail _as

they

are

only applicable

to Hermitian matrices. So _a

recently published

CG

algorithm

_{[23] was}

included in

COSME,

which works

especially

well with

complex-symrnetric

but non-Hermitian minimization

problems,

hence it is ideal for the solution of

equation (2).

The existence of

complex eigenvalues

_q also _causes the

frequency

_{w «}

@

to be

complex.

Just like the

complex

value of the _wave vector k of _waves

propagating

inside _an infinite

sample,

the

complex

value of the oscillation

frequency

_w of

standing

_waves inside

a resonator may be

used to calculate the

complex

DF of the

respective

material. The

corresponding

relation is

~2

0

~ ~2

14

7'

12

1o 8

70000 140000

li

5

-, ,

~ 4

I

70000 140000

N

Fig-I.

Effective permittivity _vs number of nodes in the discretization mesh.

e = 50 -120, _em

= 1

and f ₌ 0.3.

where _wo represents ^the resonance

frequency

of the empty resonator.

For the measurement of _{w =} w' ₊ iw" _one would _measure the _resonance

frequency (w')

and the

quality

factor

Q,

which is

given by

w' divided

by

the FWHM

(full

^{width half}

maximum);

w" is then calculated via the

following

relation

~,, ^{W' W'}

(~j

2Q 2Qofi$

resulting

from the time

dependence

of the electric field

(and Q

_cc

vGfi:

E(t)

₌

Eoe~#e~~"~e~~'~

=

Eoe~%de~~'~

Here

Q

is the measured

quality

factor of the resonator

including

_a

(dissipative)

material and

Qo,

_{wo are} the

quality

factor and _resonance

frequency

of the empty resonator. In the ideal _case

(infinitely conducting walls) Qo

_{" cc} and w"

=

w'/(2 Q).

3.

Dissipative

random mixtures with different

filling

factors.

For all the simulations the size of the discretization mesh _was chosen 40 _x 30 _x 40

respectively

40 _x 20 _x 40

nodes,

_so the volume _was divided into cells of

length

A

= 0.5 _mm. This of _course

corresponds

to the smallest

particle

size realized in the simulations.

For the estimation of discretization _{errors we} carried out several simulations of identical dis- tributions of

particles

of

length

d

= 2 _mm

varying only

the size of the discretization mesh. In

figure

I the

resulting

effective

permittivities

for meshes of 10 _x 6 _x

10,

20 _x 12 _x

20,

40 _x 24 _x 40 and 60 _x 36 _x 60 nodes _are

displayed

_vs the number of nodes. In this _{case e}

= 50 -120 and

f

= 0.3 _were

chosen,

the _same behaviour _was observed for _e

= 15 I and

f

= 0.4. The

two smaller

grids appeared

to be too _coarse to

produce

correct values.

Increasing

^the

grid length

from 40 to 60

obviously

does _not

change

the result _very much

compared

to the statis-

tical fluctuations, _so for standard _{runs we}

always

chose 40 _x 30 _x 40

grids

to _save computer time.

1768 JOURNAL DE PHYSIQUE I N°9

, >,, i I I I i i i i i ,

, >,, i I I I I I I,, _{i i , ,}

, >, i I, t I I II _{I, i I I i}

, ,

, , t t i i t I ii I _{I i i} t It _i

t i i ,

, , , i i i i t I ii I _{I i t}

i, i i i ,

,, i i i i t t I I I _{I I i} Ii t i i t i i i ,

,

I,

, , t t i i I I I it _{i t t i}

, ,

i , i t i t t i I _{i i i t t i i}

i,

, , t i i i i Ii i I _{i i} i i i I I i t,, , ,

, , t t i t t t i t, i i ,

, , t t t t I it I I I I i t t i , t , ,

i i i t I I i t I t t ,, ,,.

, i i i i it _i Ii I i I i i, t ,.

, , , , i I i i, t t i i i i _{I I} i I I t i i t ,

, , , ,, i I I I i i I I ii i i I _i

, , ,

, , , ,, t t I i i I _{I I} I Ii _{t t} I I t _{, i} > i i i ,

, , i i I I i i i I I I i t i i , , i ,

<, i i i t I I i i t i i1, ,

Fig.2.

Typical electric field distribution inside _a resonator filled with _an effective medium

(cross section).

Fields calculated by COSME.

One _can

speak

of _an effective medium _as

long

_as the

wavelength

inside every

particle

_con-

siderably

exceeds the

length

d of the

particle.

If this condition

W>~

> (41

is not

fulfilled,

the _{wave may} be scattered

by

the

inhomogeneities

and the mixture does not behave

quasi-homogeneously

_{any more.} In this context the

problem

of the skin effect also has to be

considered,

_as the

complete penetration

of the _wave inside the

particle

must be ensured.

In the worst _case in _our simulations

,

Al

_{(e( was} still

larger

than 7 d. The maximum

imaginary

part of _{e was} chosen such that the

resulting

skin

depth

_was still 0.75 _mm, I. _e.

larger

than the

length

of each

particle.

For

illustration, figure

2 shows _a

typical

field distribution

(electric field)

at _resonance inside

an effective medium _as calculated

by

the COSME _program. The

sine-shape

of the fundamental mode of the

rectangular

resonator is

clearly visible, inspite

of local field deformations

as a cause

of the

inhomogeneities (em

₌ I,e = 10

2i, f

₌

0.I,

d

= 0.5

mrn).

Mixtures of

particles

with _e

= 50

-120,

_{em =} I and d

= 0.5 _{mm were}

investigated

_over the whole _range of volume fractions between 0 and I.

Looking

at the matrix

equation (2)

it is _easy to _see that _any

arbitrary

mixture of two materials _may be transformed into _a

particle-vacuum

mixture

by normalizing

all the

permittivities

_to the DF of the host material.

Figure

3 shows the behaviour of the effective DF of the mixtures _vs volume fraction. For small

filling

factors

(f

<

0.25)

the

typical

sublinear behaviour of

T(f)

is

observed,

below

f

₌ 0.12 _{even very} close to the line which marks

Looyenga's

formula. For

f

>

0.25, however, T( ii

shows _a

perfectly

linear

behaviour,

which reminds of _a

simple parallel-circuit

relation.

In

figure

4 this behaviour becomes _{even more} obvious: the effective loss tangent

f'If

₌ tan

changes dramatically

in the volume fraction _range below

f

= 0.25

approximating

the

particles'

loss tangent

e"le'= 0A)

_very

rapidly.

At

higher

volume fractions it almost does not

change

any more.

So the loss tangent ^{and the absolute value of the DF of such} an effective medium _may be influenced

independently

in the different _ranges of

filling

factors. Below

f

₌

0.25, ?( ii

is sublinear in both the real and the

imaginary

part, ^hence

iii changes

_very

slowly

with the volume

fraction,

in contrast to the loss

tangent.

In the

high

volume fraction _range the two parameters act vice _versa: almost _no

change

in tan _&, linear increase of

iii by adding

_a few

particles

to the mixture.

60

E'

50

o

40

.

o

30

20 ^°

o

IO °

o

-lo

OO O-1 02 03 04 05 06 07 O-B 09 1.O

f

a)

25

E 20

o

5 *

o lo

o

5

o

o

-5

O f _b)

Fig.3.

Dielectric function of the heterogeneous mixtures _vs. the volume fraction (em

= 1,e

=

50 -120, d

= 0.5

mm).

Results of the numerical simulations

(.)

compared to Looyenga's formula [7]

(-). a)

Real part ^of I

b)

Imaginary _part of I.

In

figure

5 the simulation data and the results calculated from

equation (I)

_are

compared.

In the range where

(I)

is claimed to be valid for

f

< 0.25 the agreement is excellent. For further

proof

_we carried out simulations with

particles

of _even

higher permittivity (e

₌ 90

145)

and found the _same agreement. ^So

obviously equation (I)

is also valid for

dissipative

random

mixtures with low volume fractions.

1770 JOURNAL DE PIIYSIQUE I N°9

o.5

tg 6

O.4

~ . .

O.3 O.2

O.

o-o

-o.i

O-O

f

20

6

15

1o

5

-O

f a)

'>

~

-2

-4

-6

-8

O-O

f b)

g.5.

-

equation (-) vs. volume _fraction. em, e, d as in _figure 3.

calculation

of the

_mean polarization, can _{only up to the} sth ^ntil the limits

of theomputer are

reached

_[15].

So the interactions _between neighboring

particles

roaches and therefore they

can only

be

valid

for very small

filling

non-random samples.

The

above

sample being higher when the

sth

multipole was included,

than

in pure _dipole

1772 JOURNAL DE PHYSIQUE I N°9

~)

~@~ ~4/

)

a)

C)

Fig.6.

Non-random geometries simulated by COSME.

Obviously higher multipole

interactions have

a strong ^influence on the DF of _a mixture.

In the simulations

presented

here the interactions between all the

particles

inside the

sample

volume _are

implicitly considered,

because here Maxwell's

equations

_are

directly

solved in full

generality.

This results in the decisive deviations from former well-known effective medium

relations.

4. Non-random

geometries.

The formation of connected

paths

_can

easily

be enhanced _or

suppressed

in the computer model.

For _a

qualitative

evaluation of the influence of geometry on the

resulting

effective DF _we

generated

the

following

material distributions for five different simulations:

I. Several 2D-networks _{on a} stack but

separated by

matrix material. The networks lie

perpendicular

to the electric field inside the resonator

(Fig. 6a), f

₌ 0.107.

2. 2D-networks _as in I. interconnected

by

columns in the direction

parallel

to the electric field

(Fig. 6b), f

₌ 0.15.

3. 2D-networks and columns _as shown in

figure 6c, f

₌ 0.164.

4. As

2.,

with

f=0.23.

5. As

3.,

with

f=0.28 (due

to closer

packed 2D-networks).

Figure

7 shows the deviation of the

resulting

DF'S of such structures from those of random mixtures. From table

I,

which contains the

corresponding

relative deviations for

geometries

1.

5.,

it is obvious how

strongly

the inner

geometry

of the

inhomogeneities

influences the effective

DF,

_an effect which has also been found

experimentally [24]. Especially

_a

higher proportion

of

paths parallel

to the electric field increases the

resulting

effective

permittivity

whereas r results to have _a lower value when the

inhomogeneities

_are located

perpendicular

to the electric field.

20

lo _z 4 5

. i

O f

O-O O-i O.2 O.3 O.4

a)

EI,

o

3

~ O

~

_5 .

4

-lo f

0.O O.2 O.4

b)

Fig.7.

Results from the simulations of five different non-random geometries

(.) (geometries

1.

5., _see

text)

compared to the respectiv of random geometries

(-),

both displayed _vs. the volume fraction.

Table I. Relative deviation of the values air calculated for non-random

geometries

from those for mixtures with

randomly

distribu ted

particles (emd ).

Geometries 1. 5. _are

explained

in the text. The

corresponding

absolute values _are shown in

figure

7. In this table 100 ~

corresponds

to the effective

permit tivity

for random mixtures

(for

the

respective

volume frac-

tion).

No.

f

^f _T[~d deviation T"

f[d

deviation

T' from T[~d ^{T" from}

f[d

1 0.105 1.3 3.34 -61 iii 0.025 0.76 96 iii

2 0.15 8.9 5.734 ₊ 55 iii 3.84 1.82 ₊ l10 iii

3 0.164 7.6 6.72 ₊ 13 iii 3.19 2.29 + 39 iii

4 0.231 14.5 13.087 ₊ 10 iii 6.74 5.52 + 22 iii

5 0.237 II-1 13.785 19.5 iii 4.86 5.89 17.5 iii

1774 JOURNAL DE PHYSIQUE I N°9

5.

Experiments.

For the verification of the computer simulations several

inhomogeneous samples

_were

prepared

and _a real resonator _was built for the _purpose of _a direct

comparison

between

physical

_mea-

surements and results obtained from the numerical calculations.

The first

sample

_{was a} block

(16

x 7 x16

mm~)

of

A1203 (96ili purity),

^which _was

placed

in the center

(setting A)

and then into the _corner

(setting B)

of the resonator. These

geometries

_are

easily reproduced by

the discretization mesh of the computer _program ^{and thus} serve as very

good

test structures for the simulation results. As _a second

sample

_a mixture of epoxy resin

(em

₌ ^2.98

-10.06)

and 8ili small cubes

(I

_x I _x I

mm~)

of _a

Ti02

ceramic material

(e

₌ 80

I)

was

prepared

to fit

exactly

into the resonator and

according

to _a random distribution

generated by

the computer

(setting C).

As _a consequence of technical

problems

while

placing

the

cubes,

the desired random distribution could not be realized. So the distribution for the simulation had to be modified to represent ^{the inner}

topology

of the

sample correctly.

Inside the

rectangular

resonator

(20

_x 10 _x 20

mm~)

constructed for this _{purpose two} current

loops

for H-field

coupling

_were fixed

opposite

to each other in the side walls.

They

_were realized

by shortening

the end of two coaxial lines

by

thin wires in the

plane

of the inner resonator

wall. Two PC-7

(7

mm

precision connector)

bulkhead connectors _were mounted

directly

_on the resonator side

walls,

their inner conductor

going through

holes in the resonator walls thus

realizing

the coaxial lines.

Thus the transmission

properties through

the resonator could be determined

using

_a HP 85108 network

analyzer.

The network

analyzer

_was calibrated with _a

high quality

LRL

(two

7 _mm air line and _one reflect calibration

standards)

calibration

technique

in PC-7

technique [27]. Consequently

all

systematic

_{errors up} to the _resonator PC-7 connector interfaces _were

removed

yielding highest possible

measurement accuracy.

First _a broad

frequency

_range was chosen _to

identify

the transmission resonances, then the calibration _was

performed

ⁱⁿ

frequency

segments ^{around the} resonances to

give

sufficient

frequency

^resolution

(down

to

frequency

steps of 200 kHz between each measurement

point)

for the determination of the _resonance

frequencies

and the

Q-values. (-3

dB

points

of the

resonance

curves).

The

coupling strength

of the current

loops

was varied

by altering

the geometry ^{of the short} circuit wires thus

yielding

transmission

amplitudes

between -60 and -25 dB _at the _resonance

frequency

of the empty resonator. In this whole _range the _resonance

frequency

did _not

change by

_more than

10~~.

_{Also the}

Q~value

of the empty resonator did not

change

within _measure-

ment resolution. This _proves that the disturbance of the

coupling

structures _on the _resonator

properties

is

negligible.

The resonator was

successively

^{loaded with} the

geometrical settings A,

B and C and for each of them the lowest _resonance

frequency

_was determined.

In table II the results of

experiments

and simulations of the three

settings

are

directly compared.

The

imaginary

part ^{ofw is} related to the

quality

factor

Q

via

equation (3). However,

in

settings

A and B w" _was below the resolution limits and has therefore been omitted.

Figure

8 shows the _{resonance curves} of the empty resonator and of the resonator loaded with

sample

C.

The

good agreement

^of cases A and B between

experiment

and simulations _proves the

validity

and correctness of the COSME-results.

However,

for both these _cases there is _no effective DF because of the violation of relation

(4).

In _case C the construction of _a random distribution failed _as it _was

impossible

to

bring

the cubes into close

enough

contact: in the real

sample neighboring

cubes _were still

separated by

_a resin

layer.

Instead of _v

= 5.354 +10.052 GHz _a

really

random effective medium with the _same

filling

factor would have had its _resonance at v = 4.921 +10.047 GHz.

S [dB]

resonator

5

dB(

S21

a)

S12

ma'

10[Ge99s9932

adz v

[GHZj

S

sanipie

c

s

b)

S12

START 5.ZSOOaOOOO G4z

~~

WCP 5.4100@OOOO G4z l' Z

Fig.8.

Measured _{resonance curves:} S21-parameter

(I.e.

transmission amplitude

S)

_vs. frequency:

a)

empty resonator

b)

resonator filled with ceramic-resin mixture

So the inner

topology

of _a

heterogeneous

mixture is _{a very}

important

parameter for the

resulting

effective material parameters, _as has been

pointed

out before

[14, 9].

^Not even

equation (I),

which has

proved

to be correct for

completely

random

mixtures,

_can be

employed

on

samples

^with unknown inner geometry.

1776 JOURNAL DE PHYSIQUE I N°9

Table II. Results of the measurements _on different resonator

fillings _(vexp, Qexp)

ⁱⁿ com-

parison

to the

complex

_resonance

frequencies (vsim)

^calculated

by

the simulation _program. In

settings

A and B the

imaginary

parts ^of e and _{w were} below the limit of resolution and have therefore been omit ted. The

experimental

data ofset tin g C

correspond

to _a

complex frequency

of _{v =} 5.354 + 10.052 GHz.

Setting

e I

vexp[GHz] Qexp vsim[GHz]

empty I 1. 10.595 2682 10.596

A 8.9 0.448 5.43 1357 5.448

B 8.9 0.448 5.047 1264 5.050

C 80 I 0.08 5.354 50.509 5.311 + 10.053

epoxy 2.98 10.06 1.0 6.2275 47.904

resin

6, Conclusion.

A _new computer ^{code has been}

presented

which allows the simulation of

inhomogeneous lossy

materials. In this _paper it _was

employed

on one

species

of effective media:

randomly

distributed

dissipative particles

in _an air-like matrix.

For the effective

permittivity

a sublinear

dependence

_on the

filling

factor

f

_was found for the range

f

< 0.25. The

empirical exponential

formula

(I),

^which had been

developed

earlier for lossless

composite materials,

has

proved

to be valid also for random mixtures of

dissipative

dielectrics. For

f

< 0.25 it fits all the data very well.

Mixtures with _a

higher

volume fraction showed _a linear increase of _T with

f.

This behaviour may be

explained by

the

gradual

^formation of

interconnecting paths by

the

particles.

It

implies

the loss

tangent reaching

its

largest

value

already

at about

f

₌

0.25,

^whereas the absolute value of T increases much slowlier.

The

COSME-program

was tested

experimentally

with measurements carried out on three different resonator

fillings,

which _{were easy} to model _on the computer. Two of them _were realized

by

_a

macroscopic A1203-cube

in different

positions

^and the third

sample

was a mixture of small cubes of ceramic material and _a matrix of _epoxy resin. The

experimental

data the

resonance

frequency

^and

quality

factor of _a resonator loaded with the

respective sample

were in excellent

agreement

^with the results of the computer simulations.

The measurement on the ceramic-resin-mixture revealed the great

dependence

of the effective

permittivity

_on the inner geometry of any

sample. Although

it _{was a} disordered

mixture,

the distribution of the

particles

_was not

completely

random but the

particles

_{were more}

separated

from each other. For this _reason the

sample

did not show the behaviour

predicted by

the

exponential

formula.

In conclusion it must be stated that the

geometrical

correlations have to be taken into _ac- count, ^whenever

experimental

data _{are to} be

interpreted using

effective medium theories. This

implies

the

necessity

of numerical simulations in the

general

_case. For the _case of _a

completely

random distribution of

particles

in _a

matrix, however,

_we have

presented

mathematical rela- tions between the relevant parameters which describe the behaviour of such effective media in the whole _range of volume fractions.

Acknowledgements.

R. Pelster is

gratefully acknowledged

for the additional low

frequency

measurements _on all the materials used in the

experimental

part. ^The

investigations

_were

supported by

the Deutsche

Forschungsgemeinschaft (SFB 341), by

the Bundesministerium fir

Forschung

und Technolc-

gie/Bonn

and

by

^about 120 hours of NEC SX-3

cpu-time

of

Cologne University's

computer center.

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