Erdogan Madenci, Erkan Oterkus (auth.) - Peridynamic Theory and Its Applicati...
BSc dissertation np
1. Coventry Polytechnic
Computer Simulation of Fractal Patterns
A project submitted for the degree of
Bachelor of Engineering (Hons)
by
M. R. Shaheedullah
Submitted:
April 1989
Department of
Applied Physical Science
2. Abstract
Computer simulations have already been carried out which show that diffusion
limited growth processes produce self-similar patterns. Fractal geometry
defines self-similar patterns as having fractional dimensions and the patterns
are hence termed fractals. Development of known models was carried out in
an attempt to produce a more realistic simulation of diffusion limited growth
processes, with specific reference to electro-deposition. New features
included in the developed models include growth of more than one cluster and
complete coverage of the substrate. Patterns and data were produced from
which fractional dimensions could be calculated. The fractional dimensions
calculated were between one and two which was as expected. Unexpected
results were also observed. For the two developed models, it was found that
the relationship between log of density and log of characteristic length
deviated from its mathematically predicted linearity. It was concluded that this
deviation was the result of impingement with the boundary and possibly
between growing clusters.
3. 1
Contents
1 INTRODUCTION TO FRACTAL GROWTH..........................................................................2
1.1 FRACTAL GEOMETRY.............................................................................................................2
1.2 SELF-SIMILARITY...................................................................................................................3
1.3 THE FRACTAL DIMENSION .....................................................................................................4
1.3.1 Introduction ......................................................................................................................4
1.3.2 Quantitative Consideration of Fractal Dimension ...........................................................6
1.4 AIMS OF THIS PROJECT ..........................................................................................................8
2 EXPERIMENTAL PROCEDURE............................................................................................10
2.1 PARAMETERS .......................................................................................................................10
2.2 PROGRAMMING STRATEGY ..................................................................................................11
3 EXPERIMENTS AND RESULTS ............................................................................................12
3.1 DLA MODEL 1 (SQUARE PERIMETER, ONE SET SEED).........................................................12
3.1.1 Description of Model......................................................................................................12
3.1.2 Description of Results.....................................................................................................13
3.1.3 Screen Prints...................................................................................................................14
3.1.4 Data of No. Centres, Characteristic Length and Time ...................................................15
3.1.5 Plot of ln(ρ) versus ln(L).................................................................................................16
3.1.6 Plot of Fractional Coverage versus Time.......................................................................16
3.1.7 Program..........................................................................................................................17
3.2 DLA MODEL 2 (ONE RANDOM SEED)..................................................................................19
3.2.1 Description of the Model ................................................................................................19
3.2.2 Description of Results.....................................................................................................20
3.2.3 Screen Prints...................................................................................................................21
3.2.4 Data of No. Centres, Characteristic Length and Time ...................................................22
3.2.5 Plot of ln(ρ) versus ln(L).................................................................................................23
3.2.6 ..............................................................................................................................................23
3.2.7 Program..........................................................................................................................24
3.3 DLA MODEL 3 (MORE THAN ONE SEED).............................................................................26
3.3.1 Description of Model......................................................................................................26
3.3.2 Description of Results.....................................................................................................27
3.3.3 Screen prints...................................................................................................................28
3.3.4 Data of No. Centres, Characteristic Length and Time ...................................................29
3.3.5 Plot of ln(ρ) versus ln(L).................................................................................................30
3.3.6 Plot of fractional Coverage versus Time ........................................................................30
3.3.7 Program..........................................................................................................................31
4 DISCUSSION ..............................................................................................................................33
4.1 DISCUSSION OF RESULTS .....................................................................................................33
4.2 DEVELOPMENTS ...................................................................................................................34
5 CONCLUSIONS .........................................................................................................................35
6 ACKNOWLEDGEMENTS........................................................................................................36
7 REFERENCES............................................................................................................................37
4. 2
1 Introduction To Fractal Growth
1.1 Fractal Geometry
Fractal geometry was founded in 1980 by Benoit B. Mandelbrot (1), a
research fellow with IBM. In his words, fractal geometry provides morphology
to the amorphous, form to the formless and shape to the shapeless. In simple
terms it can be said to be the study objects and patterns which have fractional
dimensions, where these objects exhibit self- similarity. Before considering
these concepts in more detail it would be worthwhile to consider the
significance of fractal geometry.
The study of self—similar patterns is a new area. Fractal geometry is a branch
of science for which applications are found by specialists in different fields. It
was not developed for a particular need, the originator being a mathematician
investigating the relationship between chaos and order. Once founded, fractal
geometry was applied to fields as diverse as economics and biology. For
example certain studies of population distribution have found descriptive
graphs to be self—similar. Computer simulations have been written to analyse
development of skin patterns on some wild animals. Fractals occur in nature.
Recognizing that an object is fractal gives one predictive power if the fractal
parameters are known. It is also possible to gain an insight into basic
questions regarding the theory of critical points, relating to scaling powers of
some processes. For example in percolation it was found that scaling powers
of Gibbs Potentials are simply given in terms of fractal dimension.
( , ) ( , )Yh Yt
G L L e LaG h E
L is the characteristic length
Yh and Yt are scaling powers.
Another area in which fractals are used is the analysis of critical paths. The
specification of a cluster structure may be used as describe how oil flows
through a randomly porous material, or how electricity flows through a random
resistor network.
5. 3
1.2 Self-Similarity
It has already been stated that the objects to which fractal geometry can be
applied are commonly those found in nature. Nature provides us with an
almost infinite multitude of shapes and forms, yet we still know little about how
they are created. For example, the shape of any object tends to depend
strongly on the kind of process from which it originated. Synthetic man-made
objects often possess symmetry when observed in detail. Natural objects tend
to seem random when looked at in detail. Yet when inspected by eye they
seem to possess symmetry.
An example of this symmetry is that of tree branches, schematically shown in
figure (1). When looked at from a certain distance the symmetry can be seen.
On magnifying particular regions the symmetry remains, on different scale.
This is self-similarity and is a characteristic of patterns and objects which are
called fractals. Fractal geometry is generally used to describe rough, natural
objects rather than precise man-made objects.
Figure 1 Schematic Diagram of Tree Branches
6. 4
1.3 The Fractal Dimension
1.3.1 Introduction
Generally fractals can be thought of as objects which lie between chaos
(random in nature) and order (precise). They do not exhibit the precise
dimensions of conventional Euclidean geometry. The key difference between
the two geometries is that an object in Euclidean geometry can only have one,
two, or three dimensions whereas in fractal geometry an object can have a
fractional dimension, which is where the term fractal comes from. The fractal
dimension is a way of quantitatively describing a fractal pattern.
The meaning of fractional dimensions can be explained conceptually and by
the use of simple algebra. The first question to answer is why self-similar
object should have a fractional dimension.
Consider a plane. A plane is a two dimensional object ’filled’ with points. Now
consider a frame within which parts are filled with points and parts are unfilled.
If you magnify a specified region of the frame there will come a point when
you will see either a completely filled (two dimensional} or unfilled (zero
dimensions) region. A fractal, self-similar object exhibits different behaviour on
magnification. Regions can be chosen and if magnified, there will never come
a point when it will be either filled or unfilled, it will always be fractionally filled
and is therefore said to have fractional dimension.
Consider the pattern in figure (2), known as the Sierpinski Gasket. It is a
fractal pattern in that it exhibits self- similarity but since it is a mathematical
construct it is also precise. The solid triangles represent that part of the object
which is considered to be ’filled’, white space being unfilled. As drawn there is
a limit to its self—similarity on magnification but as a mathematical construct
its self- similarity is infinite. Any triangle which contains white space (those
with the vertex at the top) is similar to any such within it or of which it is part.
For example triangle ABC is similar to triangle DCE and also triangle FGC.
However much one of these triangles is magnified there will always be some
white space and some filled space hence it has a fractional dimension.
8. 6
1.3.2 Quantitative Consideration of Fractal Dimension
Consider a rectangle of side s and height h. This square has associated with
a characteristic length. This is a quantity related to a relative change in ’space
filled’ whether in two dimensional or three dimensional context.
Note that the term ’mass’ as used here is not physical mass but is used to
describe ’amount of space filled’ whether two dimensional or three
dimensional.
It can be seen that if the characteristic length is 3L
M(3L) = 32 x M(L) - - (2) M(3L) = 38 x M(L) – (4)
A pattern is now emerging.
Comparing equations (1) and (3) we can see that the exponent of the
constant 2 is 2 in the case of a two dimensional object and three in the case
of a three dimensional object.
area(2L) = 2h2s
The ‘mass’ = density x area
M(L) = ρ x hs
M(2L) = ρ x 2s2h
M(2L) = ρ x 22sh
M(2L) = 22 x M(2L) - (1)
vol(2L) = 2h2s2l
The ‘mass’ = density x volume
M(L) = ρ x slh
M(2L) = ρ x 2s2l2h
M(2L) = ρ x 23slh
M(2L) = 23 x M(2L) - (3)
9. 7
We can write that : M(2L) = 2d x M(L)
where d is the dimension of the object
Comparing equations (2) and (4) we can see that where we have increased
the characteristic length by a factor of 2 lthe coefficient of M(L) is 2 and where
we have increased the characteristic length by a factor of three, the coefficient
of M(L) is 3. We write that
M(λL) = λd x M(L) - (5)
where λ is proportionate increase in characteristic length. The actual algebraic
proof of this relationship requires the use of functional mathematics, which is
beyond the scope of this project.
In Euclidean geometry d can only be 1,2 or 3 for any object but we shall see
that in fractal geometry d can be a fraction.
Consider again the Sierpinski gasket figure (2). Let us define triangle DEC as
having a characteristic length of L in which case triangle ABC has a
characteristic length of 2L. The ’mass’ in this case depends on the number of
filled and unfilled triangles. Since M is inversely proportional to the number of
unfilled triangles we can compare M fort he triangles by counting the number
of unfilled triangles. In triangle DEC the number of unfilled triangles is 9. In
triangle ABC the number of unfilled triangles is 27. We can see that the ’mass’
of triangle ABC is 3 x mass of triangle DEC
M(2L) = 3 x M(L)
If we compare this with equation (5)
M(λ L) = λd x M(L)
We can see that λ = 2 and therefore 2d = 3.
Taking logs d x ln(2) = ln3
so d = ln(3) - ln(2)
which is the fractional dimension of the Sierpinski gasket. The general
equation (5) above relates characteristic length to ’mass’. This applies where
a change in L causes a change in the size of the frame of the fractal hence
increasing the ’mass’, with the ’density’ being effectively fixed. However, as
shall be seen later, it is sometimes more convenient to relate L to a change in
density within a fixed frame. L can still be considered to be related to a
change in amount of space filled but now in the context of a fixed frame
whose ’mass’ is being increased.
10. 8
It should be noted that in keeping the size of the frame fixed L can no longer
be considered to be the length of a side of the frame in the case of a square.
In fact we now need a new definition of L related to an increase in density. Let
us consider the case where the fractal increases in mass within the frame.
We know that
’mass’ α density
In the case of a fixed frame with the mass increasing the density can be
considered to be the fractional coverage of the frame therefore
’mass’ α fractional coverage
and M α Ld
so ρ= k x Ld
so ln(ρ) = ln(k) + (d x ln(L))
If a graph of ln(ρ) versus ln(L) is drawn the gradient will be the fractal
dimension.
For certain growth processes assumptions can be made and rules specified
as to how growth occurs. A model of the growth process can therefore be built
and dynamically simulated on a computer. These are termed ’Cellular
Automata’. Such models can lead to interesting patterns and data.
1.4 Aims of This Project
This project concentrates on computer simulations of models created to
represent diffusion limited growth processes. The importance of diffusion
(Brownian motion) in processes such as electro-deposition, colloidal
aggregation and others has long been recognised. Only in recent years has it
been practical to carry out computer simulations of aggregation processes of
aggregation processes using Brownian ’random walk’ trajectories.
A model in contemporary use is the diffusion-limited aggregation model. This
was developed in recognition of the importance of diffusion in the above
growth processes. The first of this type of work was carried by Finegold (2).
however, the simulations were carried out on a small scale and no
quantitative results were reported.
11. 9
The event which contributed most to this area was a discovery by Witten and
Sander (3). This was that a model based on a diffusion-limited growth process
in which particles are added, one at a time, to a growing cluster or aggregate
of particles leads to a fractal, self-similar structure. Because of its relevance to
a wide variety of physical processes including dendritic growth, fluid-fluid
displacement, colloidal aggregation and dielectric breakdown this model has
generated considerable enthusiasm and has been extensively investigated
during the last few years.
It is proposed to develop the Witten-Sander model to be a more realistic
simulation of diffusion limited growth processes. The particular process
chosen to be the basis for the simulation is electro-deposition because it is a
relatively simple two dimensional surface process. The development will be
approached by creating a standard Witten-Sander model on a computer
system, then modifying the system to incorporate changes in the model.
12. 10
2 Experimental Procedure
Original programs were to written in order to simulate the standard Witten-
Sander model and the modifications. From the programming point of view,
once the Witten-Sander model has been simulated modifications would be
relatively simple. The requirements include the simulation of random motion,
time and attachment of centres. These were implemented in the following
manner:
Each point on the screen has four adjacent points and can be considered to
be a square. For a randomly moving centre these four adjacent points are the
possible positions of a randomly moving centre in the next time step. For a
seed, whose nearest neighbours become sites for attachment, the
unoccupied adjacent positions are these sites. This is schematically shown
below
key:
w random walker
possible positions of random walker in next time step
. and unoccupied lattice sites (for simplicity they are not represented in
future schematic diagrams of DLA Models).
Time is implemented simply by incrementing a counter which is initially set to
zero. This counter is implemented only after all random walkers have moved.
The effect of this is that although the computer takes ’real’ time to move all
walkers this is considered to be instantaneous by the simulated time.
2.1 Parameters
There are two parameters that can be varied by changing the program. These
are
(1) the number of centres which nucleate per time step
(2) the size of the matrix, i.e the size of the frame in which the
simulation will take place
13. 11
2.2 Programming Strategy
The major ’strategic’ consideration is the method of checking whether a
moving centre has reached a growing cluster. One approach would be to
check the nearest neighbours of the moving site against the positions of all
fixed centres. If the check proved positive the moving centre would then
undergo the procedure which would make it a fixed centre. The checking of
nearest neighbours against fixed centres would have to be repeated for every
moving centre each time they move. In the later stages of the simulation this
would be a very time consuming process. A possibly much quicker method
would be to move the centre and check whether the new position is a nearest
neighbour of a fixed centre. In this case the checking would be against
unoccupied nearest neighbours of fixed centres rather than fixed centres
themselves. This should be much quicker since the number of unoccupied
nearest neighbours would be much smaller than the number of fixed centres
(except at the very beginning of the program).
14. 12
3 Experiments and Results
3.1 DLA Model 1 (Square Perimeter, One Set Seed)
3.1.1 Description of Model
The model is similar to the standard Witten-Sander DLA model but in this
case the perimeter is a square not a circle. A perimeter is formed of adjacent
sites in the shape of a square. In the middle of this square is a seed.
We start with a square lattice and occupy the centre site with a seed particle.
A particle is, randomly with time, then released into the lattice from the
perimeter of the square. This particle executes a random walk until it reaches
a neighbouring site of the seed particle upon which it ’sticks’ to the seed. The
nearest neighbours of this centre become sites for growth. The model
employs reflective boundary conditions, that is, when a random walker
reaches an edge it is forced to continue its walk within the square. This
process is repeated thousands of times until a large cluster is formed. A
schematic diagram of the process is shown in figure(3).
This model is an example of how totally random motion can give rise to self
similar clusters. This is because the growth rule is non-local. The simulation
ends when any part of the cluster reaches the perimeter since consideration
of interference between the cluster and the perimeter is not included in the
model.
15. 13
Figure(3) Schematic Representation of DLA Model 1
key:
* position from which walker can nucleate
S position of initial seed (fixed at centre of matrix)
nearest unoccupied neighbours of fixed centres (growth sites)
W random walkers
possible positions of walker in next time step
F fixed centres which were once random walkers
3.1.2 Description of Results
The graph of ln(ρ) against ln(L) shows some scatter but it can be seen that
ln(ρ) rises with ln(L) in an approximately linear fashion along the whole curve.
Hence, all points were fitted to a straight line by using least squares
regression.
The results of the regression were as follows:
Intercept -7.67638
Slope 1.767316
The graph of fractional coverage versus time exhibited parabolic behaviour.
The single screen-print exhibits self-similarity in that the ’fingers’ have
smaller ’fingers’ as branches.
Early stages Later stages
21. 19
3.2 DLA Model 2 (One Random Seed)
3.2.1 Description of the Model
The initial DLA model will now be developed to a stage where progression
can be made from models already presented in fractal literature. This model is
meant to improve on the previous one in that it is to be more practical. In
reality , particles are unlikely to nucleate only from a perimeter. The use of a
perimeter is a purely theoretical consideration. In practice one would expect a
centre to be able to nucleate at any unoccupied site on the lattice. Such a
particle could then execute a random walk as before. The effect of this is that
the random walkers would nucleate randomly with position as well as time.
Also, there is no reason why the seed should be at the centre of the lattice.
The position of the seed is also, therefore, made random.
An important effect of these changes is that the simulation dose not have to
end when the cluster reaches an edge.
figure(4) Schematic Representation of DLA Model 2
key:
S position of initial seed (random)
nearest unoccupied neighbours of fixed centres - (growth sites)
W random walkers
possible positions of walker in next time step
F fixed centres which were once random walkers
Random walkers can nucleate at any unoccupied lattice sites
Early stages Later stages
22. 20
3.2.2 Description of Results
From the data acquired two graphs were plotted. The first was a log-log plot of
density versus characteristic length. Most of the curve is linear, i.e. ln(L) rise
as ln(ρ) rises. The later portion of the graph is not linear. This is not due to an
error in the simulation but is a characteristic of the relationship between the
two quantities at the later stages of the simulation. Since the mathematical
relationship from which the fractal dimension is calculated only holds for a
linear % relationship between ln(L) and ln(ρ), regression is only applied to the
linear portion of the graph. This linearity extends to a characteristic length of
approximately 3.7.
The results from the regression were:
Intercept -5 95484
Slope 1.315398
The fractional coverage were expected to vary in cumulative manner with
respect to time and the graph does in fact exhibit parabolic behaviour.
The screen prints of the patterns expected exhibit the q expected self-
similarity which is most obvious when the number of centres is about 1000. In
the later stages of the simulation when fractional coverage is quite high, self-
similarity is less evident.
28. 26
3.3 DLA Model 3 (More Than One Seed)
3.3.1 Description of Model
On further consideration of the model it was decided that
there is likely to be more than one seed. The number of seeds
would probably be variable but for simplicity it is considered
to be fixed. The number is chosen as five, which is
reasonable for the 64 x 64 lattice used in these simulations.
figure(9) Schematic Representation of DLA Model 3
key:
S positions of initial seeds (randomly positioned)
nearest unoccupied neighbours of fixed centres (growth sites)
W random walkers
possible positions of walker in next time step
F fixed centres which were once random walkers
Random walkers can nucleate at any unoccupied lattice sites.
Early stages Later stages
29. 27
3.3.2 Description of Results
The reasoning behind the regression of only the linear portion of the curve is
similar to that for DLA Model 2, since the shape of the curve is very similar.
This curve is, however, different in that the linear portion exhibits less scatter
than the linear portions of similar graphs for the other two models. The
linearity also ends earlier than that for DLA Model 2, at a characteristic length
of about 2.8.
The results of the regression were:
Intercept -5.531
Slope 1.572
Fractional coverage can be seen to increase in a similar manner to that of
DLA Model 2 but at a greater rate.
The screen-prints show self-similarity of the individual clusters. When they
impinge, self-similarity of the complete structure is observed.
35. 33
4 Discussion
4.1 Discussion Of Results
The most important of the parameters investigated was the relationship
between ln(ρ) and ln(L). The expected result was a linear relationship
between ln(ρ) and 1n(L) which would give a positive gradient. This gradient
would be the fractal dimension. The values of the fractal dimensions were
expected to be between 1 and 2 since the pattern has a dimensionality of two,
i.e. in a two dimensional context.
The log-log plots of density versus characteristic length proved interesting.
The earlier parts of the curve exhibit the expected linearity. Regression of this
part gave the fractal dimension. However the later parts of the curve deviated
significantly from the linear portion. This deviation was not random. The curve
follows a perceivable path, arcing back, indicating that as ln(P} increases ln(L)
decreases. This is behaviour not previously discussed in fractal literature. It
occurs in models 2 and 3, not 1. There may two possible causes.
1) impingement with boundary (edges)
2) impingement between clusters in the case of model 3.
The scans of fractional coverage vs time were as expected.The exponential
increase in the early part of the curve was expected, however, the later parts
of the curve were new results. A possible explanation for this is that once the
cluster reaches the boundary one would expect the rate coverage to decrease
and eventually flatten out. This analysis is borne out by the fact that the scans
for the Witten-Sander model consisted of only the exponential potion. The
uniformity of the curves indicate that the scale of the model is large enough to
provide meaningful results. That is, the number of centres, the size of the
frame and the time were of adequate quantity. The fractal dimensions were as
expected, values between 1 and 2. This helps validate the choice of
characteristic length. Figures deviating from the expected dimensionality
would indicate a misconception in the model with one likely cause being the
characteristic length. The achieved results are significant in that they give
quantitative results for previously unexplored models. The exact significance
would require extensive mathematical analysis including theory of critical
points which is beyond the scope of this project. One important way in which
the developed models are different from the Witten-Sander model is that they
allow complete i coverage of the substrate. This models therefore simulate the
process when it is not diffusion controlled. In the later stages of the simulation
the amount of free substrate is reduced. This means that the extent of random
motion of the walkers before they become fixed is reduced. This decreases
the effect of diffusion in the growth process until the fractional coverage is so
great that only individual free sites are left on the substrate. After this the rate
of growth is completely controlled by the rate of nucleation.
36. 34
4.2 Developments
The models presented could be developed further. Modifications could include
randomizing the number of seeds rather than keeping it fixed. The effect of
impingement may also be investigated. This could concentrate on the
difference between impingement of the cluster on the boundary and
impingement between clusters. This could be done by forcing the seeds to be
close to each other but far from the edges of the frame in one model and
forcing the seeds to be close to the edges but far from each other in another.
Further development could include more detailed consideration of what
happens when two randomly walking particles meet. The possibility that they
stick and move together could be investigated. After this moving cluster
reached a certain size it could become stationary due to its increased mass.
The most important development would be a more detailed analysis of the
non-linear portion of the ln(ρ) ln(L) curve. This would require mathematical
analysis. It may also be possible to calculate fractional dimensions for electro-
deposition by mathematical of the physical quantities and compare the results
with those acquired by computer simulations. This is a technique which has
been used for simulations of other processes such as colloidal aggregation.
37. 35
5 Conclusions
1) The fractal dimension of the standard Witten-Sander DLA model was found
to be 1.8
2) The fractal dimension of DLA Model 2 (one random seed and random
nucleation) was found to be 1.3
3) The fractal dimension of DLA Model 3 (five seeds and random nucleation)
was found to be 1.6
4) The relationship between ln(ρ) and ln(L) for the standard Witten-Sander
model did not deviate significantly from linearity.
5) The relationship between ln(ρ) and ln(L) for DLA Model 2 deviated at a
value of approximately 3.7 for ln(L).
6) The relationship between ln(ρ) and ln(L) for DLA Model 3 deviated at a
value of approximately 2.8 for ln(L).
38. 36
6 Acknowledgements
I would like to thank both my supervisors Dr M.Y. Abyaneh and Dr D. Kirk for
their help and encouragement.
39. 37
7 References
1) Mandelbrot, B.B., "The Fractal Geometry of Nature", W.H.Freeman, San
Francisco (1982)
2) Finegold, L.X.. Biochem. Biophys. Acta 448,393 (1976)
3) Witten, T.A. and Sander, L.M., Phy. rev. lett., 47, 1400 (1981)
4) Stanley, H.E., and Ostrowsky, N., On Growth and Form, Fractal and Non-
Fractal patterns in Physics, NATO ASI, (1985)g
