FARTICLE XODELLING OF SOLIDS, LIQUIDS AND GASES WITH APPLICATION TO A NATURAL SELF-REORG~I~A~~G~

Donald Greeaspan "1

ABSTRACT

A direct computer approa& is developed for model.ling solid, liquid and gas pheno-

mene . The method simulates classical aoLocular mechanics, is computer oriented and

is distinctly different from classical cantinuum modelling. D~verae examples are

described for liquid wave generation, crack development in plates, lunar and binary star

evolution, biological cell sorting, and the inversion of volvox. Also, it is noted

that, by means of particle mode~l~ng, aitl the conservation and symmetry l~s of deter-

ministic physics can be established using only arithmetic.

Eor ciit: first time in the history of mathematical modelling, computers allow the

Simufation of natural phenomena in a fashion which Is more consistent with the classfcal.

theory of matter than is continuum modelling. This type of modellfng is called particle

modeifing.

Particle modelling begins with the assumption that the gross dynamical behavior of

any Liquid, gas, or solid is determined from the dynamicaf behavior of its constituent

stoma and molecules il.]. &applying this principle, we will assume, without loss of

generality, that the fundamental constituents are molecules,) ClassicaLLy, the forces

that act on molecules are of two types, the Long range and the short range. The long

range forces, like gravity, are those which act uniformly on every molecule, Short

range forces are those which act only between any molecule and its immediate neighbors.

This local interaction is of the following general. nature {Id”]. If two malecules are

pushed together they repel each other, if putlad apart they attract each other, and

mutual repulsion is of a greater order of magnitude than is mutual. attraction. Mathe-

matically, this behavior is often formulated as follows. The magnitude F of the farce

between two molecules which are locally r units apart is of the form

where the four parameters C, N, p, q satisfy, typically,

G Y? 0, w ’ 0, q >p27. t21

The major problrm in any simulation of a physical body is that there are too many

component molecules to incorporate into the model.. The classical mathematical approach

is to replace the large, but finite, number of molecules by an infinite set of points.

in so doing, the rich physics of mofecuLar behavior is lost. _- A viable computer alter- p--

native is to replace the large number of molecules by a much smaller number of particles

and then adjust the parameters in formula (1) to COmpenSateS It is this latter

approach which we will follow.

Ths computational procedure can be summarized readily as foU.ows. Any solid,

liquid or gas under study wil.3, be approximated by s finite sot of particles, say, N Of

- -. :. $

Department of &themetics, tTniversity of Texas a~ Arlington, Arlington, Texas 36019 U.S.A.

them. The force F,, on any particle P I

will consist of two parts, a long range com-

ponent and a local component, as described above. The motion of the system will then

be approximated from given initial data by solving the coupled system

Fi = miai, i = l,Z,...,N

numerically by any one of the standard computer procedures available [ll].

The Drop-In-The-Well Problem

Let us consider first a liquid simulation of wide interest [4]. The long range

force is gravity.

Consider a liquid in a cavity, or well, as shown in Figure l(a). The liquid is

approximated by a set of 190 particles, each of unit mass, each shown as an unshaded

circle. Consider, also, a liquid drop which is approximated by a set of 15 particles,

each of mass 2, each represented as a shaded circle. In Figure l(a), an initial con-

figuration is assumed in which the drop has already hit the surface and flattened. The

local force parameters are taken to be H = G = 100, p = 1, q = 3, while the force of

gravity is set at -980mf. The local forces are restricted to radial distances smaller

than 0.25, that is, if the distance between two particles is greater than 0.25, then

the local force between them is taken to be zero.

Figures l(a)-(d) show the resulting dynamical behavior. Figures l(a)-(b) show the

entry and initial dispersion of the drop into the well. Figures l(b)-(c) show clearly

the wave reaction of the well liquid. Figure l(c) shows two developing reactions

within the well, one of which is a backflow over the sinking drop, the second of which

is a wave flow over the right wall. These actions continue in Figure l(d), which shows,

in addition, a distinct decrease in the drop's vertical velocity, which, in the case of

a pollutant, causes a layered effect. Figures Z(a)-(d) show how one can derive addi-

tional information from Figure 1 simply by studying the time deformations of various

liquid columns. The accdrdiantype deformation directly below the drop suggests vortex

circulation, while the column motion to the right of the drap indicates that particles

on the tops of columns tend to flow toward and over the right barrier, while those

below tend to flow back over the submerging drop. Mixing by diffusion is also evident

with time.

Experimental Verification

Since it is important to have experimental verification of a theoretical model

whenever possible, let us show now how to predict and verify the existence of a complex

phenomenon associated with the drop-in-the-well model. This phenomenon is called the

pinching off of a back-drop.

Consider a liquid drop, smaller than that shown in Figure l(a), and located cen-

trally over the basin shown in Figure 3(a). (Since column motions will not be emphasized

at present, the particles have been drawn larger than those in Figure 1.) The drop is

allowed to fall freely and the resulting motion in the basin is shown in Figures 3(a)-

(j) at the indicated times [6]. (The motion of each individual particle is displayed

by the attachment of its velocity vector at its center.) ‘In Figures 3(b)-(c) one sees

the drop penetration into the basin. In Figure 3(d) one observes next that the right-

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c r\

a

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0

ono

0000

00

oo

oooo

ooo

~00

0000

00

0000

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0 00

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0~00

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0 ~

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I---

--

(b) t = 0.8

(d) t = 0.15

(f) t - 0.19

(h) t = 0..29

(i) t = 0.39

(j) t = 0.49

Figure 3. Spike and pinch generation

(a)-(j).

32

(k) Experimental verification of a liquid spike and the pinching off of a drop. (Photographed by Lloyd Trefethen, published in ILLUSTRATED EXPERIMENTS IN FLUID MEC~ICS, MIT Press, 1972, and reproduced with per- mission from Educational Development Center, Newton, Mass.)

Figure 3 (continued) Spike and pinch generation.

most particles are developing a w.aoe motion

toward the right wall, the left-most particles

are developing a wave motion toward the left

wall, and the central particles are moving

vertically upward. The central particles

then develop into a vertical spike, or back-

drop, as is seen clearly in Figures 3(e)-(f),

and Figures 3(f)-(g) show a pinching off of

the back-drop. Thereafter, Figure 3(h) shows

the right and left waves colliding with the

walls and Figures 3(i)-(j) show the final

return towards equilibrium.

That such a back-drop does occur is veri-

fiable easily from direct observation by

anyone who will merely use a dropper to drop

water into a filled sink basin. The outward

wave motions are also easily observable in

this way. However, the pinching off of the

top of the back-drop is not so readily

observed. Figure 3(k), however, taken by

means of modern high-speed photographic tech-

niques, does verify that under appropriate

conditions such I pinching off does result.

Cracks and Fractures

Next, let us show how to simulate phenomena related to solids, and, in particular,

let us consider a fundamental problem of interest to engineers, seismologists, and

metallurgists, that is, the site determination of (unwanted) cracks and fractures. To

illustrate the direct application of particle modelling, let us examine how to determine

where hairline cracks will first develop in a metal sheet under stress [B]. For this

purpose, cunslder a metal sheet with a large, centrally located, slanted hole, which is

simulated by the particle arrangement shown in Figure 4. The sheet is now stressed by

slowly stretching its top row upwards and its bottom row downwards. The developing

internal force field is represented by vectors emanating from the centers of the parti-

cles. Figure 5 shows where hairline cracks develop when the elastic limit, that is,

the distance of local force interaction, is first exceeded.

Models with Self-Reorganization

One of the most remarkable properties of various molecular systems is the ability

to self-reorganize. Continuum modelling is especially difficult to apply in this area.

Particle modelling, however, because it is designed to simulate molecular behavior,

applies readily, as the following examples illustrate.

Binary star systems in which two rotating stars are visible separately have long

been of interest to astronomers [12]. For many such visual binaries, judicious combi-

nation of observational data and Kepler's laws allow the determination of the mass of

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Figure 4. Particle set. Figure 5. Hairline crack development due to stress.

each star of the system, which is an exceptional accomplishment, Wowever, interest in

binary systems has become more cogent since application of modern telescopic, spectro-

scopic and computer techniques have revealed that most stars are binary systems, classi-

fied now as visual, spectroscopic, and eclipsing pairs. Let us show, then, how to model

the development of a close binary system. The long range force now will be gravitation.

Consider a relatively circular system of 239 particles, as shown in Figure 6. To

simulate a nonuniformity of density, the particles have been represented by circles of

different radii, where, the larger the circle is, the larger the mass which has been

assigned to that particle. The smallest circles have been left unshaded and will be

called light particles, while all other circles have been shaded and will be called

heavy particles. The entire system is set into counterclockwise motion, with random,

small perturbations incorporated into the velocity components, to simulate a hot,

swirling gas. The resulting motion is described as follows.

As shown in Figure 7, the heavy particles have self-reorganized into four groups,

while the light particles have formed an outer layer of the system. With continued

rotation, the system shows the characteristic distortions of a rotating gas, in addition

to the further self-reorganization of the heavy particles into only two large subgroups.

Figure 8 shows these two concentrations of heavy particles rotating very rapidly about

each other. The angular velocity of each star is sufficiently large that their motion

towards each other is relatively negligible. Rapid rotation, loss of mass in the form

of light particles, and formation of two heavy cores are major characteristics of close

binary systems and are all present in the system shown in Figure 9.

Figures 9 and 10 show a different simulation in which parameter variations resulted

in the final development of a lunar type body. In Figure 10, the hexagons represent

solid particles, while all other particles are in a fluid state. The figure shows

crustal formations, a solid core, and liquid layers. The figure is said to be of lunar

type because its center of mass and its geometric center differ.

Next, let us explore a biological example. Certain biological cells, which origi-

nally were organized by type, will, under appropriate conditions, self-reorganize into

their original structure after having been separated out and mixed. The process is

called cell sorting and is believed to be the result of local forceinteractionsonly [6].

To simulate cell sorting, consider 81 particles of three different adhesion constants,

34

Figure 6. Initial particle set for a hot, rotating gas.

as shown in Figure 11(a). Only local

forces are considered and the particles

are assigned sufficient speeds to assure

a liquid state. Then, Figure 11 shows

the natural self-reorganization into a

layered endoderm, mesoderm, ectoderm

type configuration.

Finally, let us allow G and H

in equation (1) to vary with time,

thereby simulating changing local charge

potentials. In this fashion one can

simulate gross motions like that of the

inversion of volvox [7], a minute

aquatic organism whose flagella are

internal during maturation, but which

inverts upon reaching maturity, This

type of cell rearrangement is shown in

Figure 12 and results merely by a gra-

dual interchange of the roles of G

and H.

Theoretical versus Practical Considerations

Theoretical Newtonian mechanics is characterized by conservation and symmetry laws.

he of the most interesting manifestations of particle modelling is that it can be for-

mulated in such a fashion that the very same conservation and symmetry laws remain valid

I2,51. This has been proved for any number of particles and any choice of parameters

G, 8, p, q in formula (1). It therefore applies to actual molecular configurations of

gases, liquids and solids. In the arithmetic theory, finite sums and differences, which

are constructive, computer compatible concepts, replace integrals and differentials,

respectively. The fact that Newtonian mechanics can be reformulated using only arith-

metic simplifies dramatically the foundational axioms of this classical subject. Never-

theless, practically, we cannot implement such a classical, arithmetic molecular theory

because of the excessive number of molecules required for any dynamical simulation. This

was, of course, the reason why we developed a particle, or quasi-molecular, approach to

modelling in the first place.

It is of interest to observe also that even special relativistic mechanics can be

reformulated using only arithmetic [3], and in such a fashion that computations in the

lab and rocket frames, executed on identical, computers, are related by the Lorentz trans-

formation.

Practically, each possible choice of the parameters C, H, P, and q does consti-

tute a different model. Each such model, when implemented on a computer, constitutes

and experiment In simulation, with the models themselves being related only qualitatively.

Only recently [13] has a direct method been developed for determining these parameters

for a quantitative simulation by an interpolation process from experimental data.

35

FIGURE 7. Binary star evolution - early

Stags.

FIGURE 8, Binary star evolution - late

stage.

FIGURE 9. Lunar evolutlon- FIGURE 10. Lunar evolution- early stage. late stage.

For a final practical consideration, it should be remarked that it would be of great

value to be able to incorporate several million particles into the modelling process.

Development toward such a capability is exactly what is promised by research in the ares

of supercomputers.

36

d)

Figure 11. Biological cell sorting Figure 12. Inversion of Volvox.

REFBRFNCES I 1

[l] R. P. Feynman, X. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1963) Chap. 1, pp. 1-9.

[2] D. Greenspan, pp. 87-93.

[3] D. Greenspan,

[4] D. Greenspan,

[5] D. Greenspan, pp. 20-31.

[6] D. Greenspan,

[7] D. Greenspan,

[8] D. Greenspan,

[9] D. Greenspan,

Discrete Models (Addison-Wesley, Reading, Mass., 1973) Chap. 7,

Int. Journal Theor. Physics, 15, 557 (1976).

Mathematics and Computers in Simulation, 22, 200 (1980).

Arithmetic Applied Mathematics (Pergamon, Oxford, 1980) Chap. 3,

Journal of Math. Biology, I.., 227 (1981).

Syst. Anal. Model. Simul., 1, 5 (1984).

Tech. Rpt. f/208, Math. Dept., Univ. Texas, Arlington, Texas (1984).

Tech. Rpt. 6213, Math. Dept., Univ. Texas, Arlington, Texas (1984).

[lo] J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, N.Y.. 1954) Chap. 1, pp. 22-35.

[ll] R. W. Hackney and J. W. Eastwood, Computer Simulation Using Particles (McGraw-Hill, N. Y., 1981) Chap. 4, pp. 94-119.

[12] Z. Kopal, Dynamics of Close Binary Systems Chap. 1, l-10.

[13] W. R. Reeves and D. Greenspan. Appl. Math.

(D. Reidel, Dordrecht, Holland, 1978)

Modelling, 6, 185 (1982).

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