Research ArticleSimulation of a Vibrant Membrane Using a 2-DimensionalCellular Automaton

I Huerta-Trujillo1 J C Chimal-Eguiacutea1 L Muntildeoz-Salazar23 and J A Martinez-Nuntildeo2

1Modeling and Simulation Laboratory Centro de Investigacion en Computacion Instituto Politecnico Nacional Avenida Juan de DiosBatiz Esquina Miguel Othon de Mendizabal Colonia Nueva Industrial Vallejo 07738 Gustavo A Madero DF Mexico2Postgraduate and Research Section Escuela Superior de Computo Instituto Politecnico Nacional Avenida Juan de Dios BatizEsquina Miguel Othon de Mendizabal Colonia Lindavista 07738 Gustavo A Madero DF Mexico3Instituto Mexicano del Petroleo Eje Central Lazaro Cardenas Norte 152 07730 Gustavo A Madero DF Mexico

Correspondence should be addressed to I Huerta-Trujillo ihuertaipnmx

Received 21 April 2015 Revised 4 June 2015 Accepted 8 June 2015

Academic Editor Tetsuji Tokihiro

Copyright copy 2015 I Huerta-Trujillo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper proposes a 2-dimensional cellular automaton (CA)model and how to derive the model evolution rule to simulate a two-dimensional vibrant membrane The resulting model is compared with the analytical solution of a two-dimensional hyperbolicpartial differential equation (PDE) linear and homogeneous This models a vibrant membrane with specific conditions initial andboundaryThe frequency spectrum is analysed as well as the error between the data produced by the CAmodelThen it is comparedto the data provided by the solution evaluation to the differential equation This shows how the CA obtains a behavior similar tothe PDE Moreover it is possible to simulate nonclassical initial conditions for which there is no exact solution using PDE Veryinteresting information could be obtained from the CA model such as the fundamental frequency

1 Introduction

The infinitesimal calculus and its descendants [1] have beenone of the dominant branches in mathematics since it wasdeveloped by Newton and Leibniz Differential equations arethe main core of calculus and have been the cornerstonefor understanding sciences particularly physics It is oftennecessary to consider more than one variable since it isrequired to change this in function to another one or thetime Therefore to model physical systems the differentialequations have had great success due to more than threecenturies of experience with methods to give the symbolicsolutions Even so few partial differential equations have anexact solution [2]

Moreover the current necessity to experiment with phys-ical systems in order to recognize their behavior make neces-sary to developmodels that simulate the systems and becomeless complex allowing manipulation and approximation asclose as possible to reality In this order of ideas the discretetechniques have had more success when they have been

implemented for simulation purposes An example of this isthe cellular automata (CA) Wolfram [3 4] defined the CA asa mathematical idealization of physical systems whose timeand space are discrete and in which the physical quantitiescan be grouped in a finite set of valuesTheCAare appropriatein physical systems with a highly nonlinear regime suchas chemical or biological systems where there are discretethresholds [5]

For example the two-dimensional wave equation is animportant one since it represents the hyperbolic partial differ-ential equations Although it has many analytical solutionsif the initial or boundary conditions are changed it cannotbe solved There are attempts to simulate the wave equationbehavior using a CA as presented by Chopard and Droz [6]and Kawamura et al [7] they have only been performed ina one-dimensional automaton This due to the complexity ofapplying a suitable rule for the CA evolution

The evolution rule for CA is a fundamental problem thathas no analytical solution [2] In this paper amethodology forclarifying the CA evolution rule that simulates the behavior

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 542507 8 pageshttpdxdoiorg1011552015542507

2 Discrete Dynamics in Nature and Society

of a vibrating membrane is compared with the analyticalsolution of hyperbolic PDE This simulates a proposed fre-quency spectrum observing similarity solutions This showsthat the discretemodeling can be an alternative for systems inwhich the boundary conditions or other circumstances makethe PDE intractable Section 1 offers a brief introductionSection 2 presents the two-dimensional continuous waveequation model as well as the solution to the equation giventhe initial and boundary conditions Section 3 shows themethodology used to make the problem discrete and obtainthe CA evolution rule and the discrete model definitionSection 4 analyses and discusses the results of the continuousmodel against the proposed CA discrete model FinallySection 5 gives the conclusions

2 Vibrating Membrane Classical Model

Themotion equation for amembrane is based on the assump-tion that it is thin and uniform with negligible stiffnessthat is perfectly elastic and without spring vibrating withsmall amplitude movements The equation governing thetransverse membrane vibrations is given by

(12059721199061205971199092 +12059721199061205971199102) = 1

119888212059721199061205971199052 (1)

where 119906(119909 119910 119905) is the membrane deflection and 1198882 = 119879120588where 120588 is themembranemass density [8] and119879 ismembranetension per unit length Equation (1) is called wave equationin two dimensions

Considered as a square membrane (see Figure 1) theboundary conditions are defined as follows

119906 (119909 119910 119905) = 0 for 119909 = 0 119909 = 119887 119910 = 0 119910 = 119887 forall119905 (2)

The initial conditions are defined as

119906 (119909 119910 119905) = 119891 (119909 119910) 120597119906 (119909 119910 119905)

120597119905100381610038161003816100381610038161003816100381610038161003816119905=0 = 119892 (119909 119910) (3)

where 119891(119909 119910) and 119892(119909 119910) are the membrane displacementand initial speed respectively

The initial conditions are defined as

119906 (119909 119910 119905) = 119909119910 (119909 minus 119887) (119910 minus 119887) (4)

120597119906 (119909 119910 119905)120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0 = 0 (5)

Then the repose membrane starts with a spatial distribu-tion that can be seen in Figure 2

x

y

0

b

b

(b b)

Figure 1 Rectangular membrane of 119887 times 119887 length

0

a

0

b

0

c

x

y

z

Figure 2 The membrane system initial and boundary conditions

Differential equation solution (1) with the initial condi-tions and boundary shown in (4) and (5) a length 119887 = 012119898and a constant 119888 = 142829 is

119906 (119909 119910 119905) = 64 (012)41205876

sdot infinsum119898=1119898=odd

infinsum119899=1119899=odd

111989831198993 cos(

120587radic1198982 + 1198992012 142829119905)

sdot sin(119898120587119909012 ) sin(

119899120587119910012)

(6)

The dynamic response of the central membrane point119901 =(1198872 1198872) according to (6) can be observed in Figures 3(a) and3(b)

3 Discrete Membrane Model

This paper considers a synchronous CA [9] where the under-lying topology is a rectangular two-dimensional finite latticeFor this type of CA there are two classic neighborhoods thecase of the CA is considered a Von Neumann neighborhoodtype [10] consisting of a central cell and its four geographicalneighbors to the north south east and west (119899 119904 119890 119908)In this sense the importance of extending the CA to twodimensions lies in that [11] the extension is the emergenceof two-dimensional patterns that have no simple analogue

Discrete Dynamics in Nature and Society 3

times10minus5

15

1

05

0

minus05

minus1

minus15

Am

plitu

de (m

)

0 01 02 03 04 05 06 07 08 09 1

Time (s)

PDE

(a)

times10minus5

1

05

0

minus05

minus1

02 021 022 023 024 025 026 027 028 029 03

Time (s)

PDE

Am

plitu

de (m

)

(b)

Figure 3 PDE graph simulation (a) presents the cell displacement (1198872 1198872) on the 119911-axis (b) zoom-in of the PDE simulation graph

in one dimension and two-dimensional dynamic sometimesallows a direct physical systems comparison

The methodology used for the discretization of the PDEon a 2-dimensional CA is presented

31 Formal Definition of Cellular Automata There are dif-ferent authors [4 9 11 12] that have defined the CA but allcoincide in four elements Taken as a base a CA is as follows

Definition 1 A CA is a 4-tuple CA = (119871 119878 119881Φ) where119871 is a regular lattice 119871 = 119888 isin C119889 for a 119889-dimensionallattice

119878 is a finite set of all possible cell states 119888 isin 119871119881 is a finite set of cells that define the cell neighborhood

Φ 119878119889 rarr 119878 is a transition function applied simultane-ously to the cells that conform the lattice [9]

The objective cell state actualization requires knowing theneighborhood cell states [6 9]

32 AnalyticalModel Discretization Suppose there is amem-branewhich can be thought of as a succession of specificmasspoints connected by springs (see Figure 4) Each point ofthe membrane is attached to the four orthogonal neighborswhere the membrane mass is spread over the attachmentpoints and not on the springs and the membrane boundaryis subject to a fixed surface

Let us call 119889119890(or equilibrium length) the spring length

that joins two masses in an equilibrium stateFor a vibrating membrane system with initial position

conditions described in (4) and starting in repose Eachmembrane junction point is subjected to four forces actingtoward each orthogonal neighbor called 119898

119888 119898119899 119898119904 119898119890

and 119898119908 for the central cell north south east and west

respectively (see Figure 5)

x

y

z

Figure 4 Representation of themembrane as a succession ofmassesand springs each mass joined to its neighbors as north south eastand west

Suppose it is necessary to know the force each neighborapplied over 119898

119888 Following the defined process by Huerta-

Trujillo et al [13] the force that a neighbor exercises over thecentral cell119898

119888is computed (see Figure 6)

Now taking into account Figure 6 gives

997888997888rarrΔ119903119908= 997888rarr119903119908minus997888rarr119903119888 (7)

Knowing that |997888997888rarrΔ119903119908| represents the separation length

between masses it is possible to write this scalar as the sumof the spring length increasing the deformation undergoneby the same spring due to the central mass position changeThus

100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 Δ119903119908 = (119889

119890+Δ119889119890) Δ119903119908 (8)

where Δ119889119890is the spring deformation value therefore it is

necessary to compute this with the purpose of knowing

4 Discrete Dynamics in Nature and Society

x

y

z

mc

mn

ms

me

mw

Figure 5 Representation of the central cell and its four orthogonalneighbors forming a Von Neumann CA neighborhood as well asthe direction in which the neighbor force acts represented by thedirection of the vectors

x

y

z

mcmw

rarr

rc

rarr

Δrw

rarr

rw

Figure 6 Representation of the vectors associated with the centralcell a neighbor and the direction of the force exercised by the springin the neighbor cell direction 997888rarr119903

119888 997888rarr119903119908 and 997888997888rarrΔ119903

119908represent the central

cell vectors the west neighbor and the spring vector that join themrespectively

the force increase from the equilibrium to the analysis pointThus

Δ119889119890Δ119903119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890) Δ119903119908 (9)

Given that both sections of (9) are vectors their compo-nents are represented as follows

Δ119889119890Δ119903119908= (Δ119909

119908 Δ119910119908 Δ119911119908) (10)

and the second vector components

(100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 minus 119889119890) Δ119903119908

= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 minus 119889119890)(

119909119908minus 119909119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 119910119908 minus 119910119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 119911119908 minus 119911119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816) (11)

Equating component to component of (10) and (11) gives

Δ119909119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119909119908minus 119909119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

Δ119910119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119910119908minus 119910119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

Δ119911119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119911119908minus 119911119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

(12)

So the rectangular components Δ119909119908 Δ119910119908 and Δ119911

119908are

obtained corresponding to the displacement increase of theaxis coordinates 119909 119910 and 119911 for the mass 119898

119888for 997888997888rarrΔ119903

119908 The

component values for the vectors 997888997888rarrΔ119903119899 997888997888rarrΔ119903119904 and 997888997888rarrΔ119903

119890are

calculated in the same mannerFollowing the analysis for Hookersquos law given for the119898119888particle there are three forces exercised for 119898

119908in the

direction of 997888997888rarrΔ119903119908due to the vector components 119909 119910 and119911 and similar to each of the particles 119898

119899 119898119904 and 119898

119890in

the directions 997888997888rarrΔ119903119899 997888997888rarrΔ119903119904 and 997888997888rarrΔ119903

119890 respectively Thus the force

applied by the neighbors in the 119909 component is

119865119909= minus 119896119899Δ119909119899minus 119896119904Δ119909119904minus 119896119890Δ119909119890minus 119896119908Δ119909119908 (13)

Considering that the springs that join the masses to themembrane are equal gives 119896

119899= 119896119904= 119896119890= 119896119908= 119896 so this

yields

119865119909= minus 119896 (Δ119909

119899+Δ119909119904+Δ119909119890+Δ119909119908) (14)

In the same manner

119865119910= minus 119896 (Δ119910

119899+Δ119910119904+Δ119910119890+Δ119910119908)

119865119911= minus 119896 (Δ119911

119899+Δ119911119904+Δ119911119890+Δ119911119908) (15)

These are the forces acting over 119898119888 which define that

the acceleration at the time 119898119888is oscillating Since the force

is known and the acceleration stays constant in each timestep using the straight-line motion equation with constantacceleration in terms of the initial speed of 119898

119888for its

rectangular components yields

V119891119909

= V119894119909+ 119886119909119905 (16)

Accordingly

V119891119910= V119894119910+ 119886119910119905 (17a)

V119891119911= V119894119911+ 119886119911119905 (17b)

where 119886 can be computed using second Newtonrsquos law

Discrete Dynamics in Nature and Society 5

The previous equations define the speed of 119898119888after time119905 This permits calculating the new particle position for the

same instant time using the following equations

119909119891119894= 119909119894+ V119894119909119905 + 1

21198861199091199052 (18a)

119910119891119894= 119910119894+ V119894119910119905 + 1

21198861199101199052 (18b)

119911119891119894= 119911119894+ V119894119911119905 + 1

21198861199111199052 (18c)

Speed equations (16) (17a) and (17b) and position (18a)(18b) and (18c) are those used in the evolution function forthe proposed CA

33 Vibrant Membrane Model Using a 2-Dimensional CAUsing the developed analysis defines the CA model for avibrant membrane system fixed at the borders of an area119897 times 119897 as a 4-tuple AC = (119871 119878 119881Φ) where each cell 119898

119888isin 119871

is defined by its mass initial position and speed When themembrane is found in rest state this gives the following

119871 this is a regular 2-dimensional lattice119878

119878 =

0 1 if it is a fixed cell997888997888997888rarr119875119905119898119894119895

vector of position in time 119905119881119905119898119894119895

speed in the time 119905forall119898119894119895isin L

2

(19)

119881 119881 = (119898119899 119898119904 119898119888 119898119890 119898119908)

Φ R3 rarr R3

Φ (a) 997888997888997888rarr119875119905+1119898119888119891

= 997888997888rarr119875119905119898119888119894

+ 997888997888997888rarr119881119894119905119898119888119894119905 + 1

2(sum4

V=1 119886119905V119888)1199052(b) 997888997888997888997888rarr119881119891119905+1

119898119888= 997888997888997888997888rarr119881119894119905119898119888119891

+ sum4V=1

997888rarr119886119905V119888119905where sum4

V=1997888rarr119886119905V119888 is the acceleration that the neighbors of

119898119888exercise over the cell in time 119905 997888997888997888rarr119875119905+1

119898119888119891is the final cell

position in space composed of three variables 119901119909 119901119910 and119901

119911 and according to its rectangular coordinates each one is

represented by 32 bits997888997888997888997888rarr119881119891119905+1119898119888

is the final speed in time 119905 + 1composed of three variables V

119909 V119910 and V

119911 and according to

its rectangular coordinates each one is represented by 32 bitsThis implementation allows having real stateswithout contra-dicting the definition of cellular automaton [4] (moreover thetotal states that each cell can take are restricted to 2193 states)

The evolution function Φ is composed of two rules bothapplied simultaneously to all the cells that conform the latticeThis is different from the model proposed by Glabisz [14 15]where the rules are applied in an asymmetric form

The rule (a) defines the cell position at time 119905 + 1 takingthe speed at time 119905 This position is updated to be the new

starting position for 119905+2 and so on Similarly for (b) the finalspeed for time 119905 + 1 is updated with the initial speed for time119905 + 2

An important point in the model definition is the springrestitution coefficient Unlike other models [13] where theconstant restitution is relatively simple to calculate thismodel faces a problem The arrangement of masses andsprings has no serial pattern but shows an array of gridsTo obtain this coefficient it follows the process defined byHuerta-Trujillo et al [16]

34 119896 Restitution Coefficient Calculation To calculate the119896 restitution coefficient value of the springs that bind thecorresponding mass the process begins by assuming that themembrane has a spring constant with a value equal to 119896

119905

Starting with a membrane represented by the proposedCA consisting of 2 times 2 nodes thereupon following thereduction springs connected in series and in parallel obtains

119896 = 119896119905 (20)

Following the process for a CA consisting of 3 times 3 nodesgives

119896 = 3119896119905

2 (21)

carrying on the process the relation for 119899 springs has the form119896 = 119899

2119896119905 (22)

where 119896 is the spring constant 119896119905is the membrane elasticity

constant and 119899 is the number of nodes for a CAof 119899times119899 nodes4 Simulation and Results

CA was implemented applying the object-oriented paradigmusing the C++ language in which each cell is implemented asa Bean object which has as attributes the cell spatial locationthe value of its mass its instantaneous speed or if it is a fixedcell CA space represented by the square lattice is regularlyimplemented as an object composed of cell arrangementsThe CA as such is composed of a grid object and one thatimplements the evolution rule and the definition of the CAneighborhood

The CA dynamic response is given by defining a mem-brane represented by a lattice of 119899times119899 nodes 119899 = 51 based onthe necessary number of cells to simulate a linear system [13]For CA the basis is a membrane 10times 10 cm stretched by 20of its length in the direction of its axes 119909 and 119910 for a lengthequal to that used in the PDE taken as reference node locatedat (1198992 1198992) to match the point (1198872 1198872) of the membraneThe mass of cells is proportional to the density used to solvethe PDE as shown in Section 2 The CA response is shownin Figures 7(a) and 7(b) Qualitatively the CA response isrelevant to the response of the PDE taking the same initialand boundary conditions for both models

The initial CA conditions are the same as those describedfor the PDE model ((4) and (5)) The displacement was

6 Discrete Dynamics in Nature and Society

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

0 01 02 03 04 05 06 07 08 09 1

Time (s)

CA

(a)

times10minus5

15

1

05

0

minus05

minus1

minus15

m

02 021 022 023 024 025 026 027 028 029 03

Time (s)

CA

(b)

Figure 7 Response graph (a) of the simulation made by the CA presenting the cell (1198992 1198992) on the 119911-axis (b) zoom-in of the simulationgraphic

0 01 02 03 04 05 06 07 08 09 1Time (s)

CAPDE

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

(a) Graphic overlay

011 012 013 014 015 016 017 018 019Time (s)

CAPDE

times10minus5

15

1

05

0

minus05

minus1

minus15

m

(b) Zoom graphic overlay

Figure 8 Overlay graphic (a) of the simulation realised by the CA and the PDE shows the displacement of the cell (1198992 1198992) on the 119911-axisin both cases (b) zoom-in graphic overlay

obtained by the PDE membrane ranging 1119904 and took 1 times 104presented samples The displacement that was obtained fromthe CA proposed the same cell (in this case the cell locatedat (1198992 1198992)) at the same time and with the same numberof samples Fifty simulations were performed with the sameaveraged data for comparison against the obtained PDEFigures 8(a) and 8(b) show overlapping graphs obtainingoscillations by CA (in solid line) and the PDE (in dotted line)as well as a zoom-in to the same graphThemean square erroris defined as

MSE = 1119899119899sum119894=1

(Vac119894minusVr119894)2 (23)

where Vac119894is the 119894th value estimated by the CA and Vr

119894is

the reference value taken by the PDE The error between

the measurements generated by the CA and the referencevalues given by the PDE is MSE = 44189 times 10minus11 whichensures that the CA simulates the PDE response

In contrast to other models [7 15] the proposed CAmodel uses 50 less cells in the lattice to get the PDEresponse

In a quantitative form there is a corresponding phasebetween the two models for the same cell Analytically thefundamental frequency is limited by the constants values 119899and119898 The frequency is defined as [17]

119891119899119898

= 1198882radic(119899119887)

2 + (119898119887 )2 (24)

Discrete Dynamics in Nature and Society 7

50 100 150 200 250 3000

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(a) Overlay of frequency spectra of CA and PDE

7570 80 85 90 95 100

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(b) Zoom overlay frequency spectra

Figure 9 Graphic of frequency spectra overlay (a) of the simulation realized by the CA and the PDE (b) zoom-in of the graphic overlay

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus25

minus2

m

0 01 02 03 04 05 06 07 08 09 1

t

CA

(a)

0 200 400 600 800 1000 1200 1400 16000

0002000400060008

0010012001400160018

002

Frequency (Hz)

Am

plitu

de

AC

(b)

Figure 10 Graphic of the simulation of latticersquos central cell for the CAmodel (a) the graphic (b) shows frequency spectra for the central cell

So a fundamental frequency is expected approximately as

11989111 = 841625Hz (25)

Obtaining frequency spectra for both signals and plot-ting them observe that there is a congruence between thefrequency spectra given by the CA and PDE (see Figure 9(a))Figure 9(b) shows the fundamental frequency for both mod-els around 85Hz The fundamental frequency error betweenthe analytical solution and that simulated by the proposedCAis 0995 which is smaller than that found in other models[7 14 18]

Furthermore the robustness of the CA model can betested given a type of irregular initial condition for examplea random initial position for all cells with 0 lt 119911

119888≪ 119887 where119911

119888is the cell position on the 119911-axis The density tension and

boundary conditions are the same as described in Section 2In this case the PDE does not have an analytical solution

since it is not possible to define a function to describethe initial conditions and to be derivable but the pro-posed CA model endures this kind of initial conditions

Simulating the conditions described and plotting the samepoint (1198872 1198872) as in the previous cases the CA modelprovides the central cell response The results can be seenin Figure 10(a) the frequency spectra of this evolution areshown in Figure 10(b) where the fundamental frequency isapproximately 85Hz such as that obtained in the previouscases

5 Conclusions

The 2-dimensional CA model to a vibrant membrane systemshown in this paper was derived from a 1-dimensional CAmodel that simulates a vibrant string [13]The CAmembranemodel is released from the initial conditions so it is notnecessary to redefine it thus it is possible to define the initialconditions and simulate the system behavior immediatelyThis is different to the 2-dimensional wave PDE that issensitive to these conditions Its solution could only befound if the initial conditions are represented by a derivativefunction in all space otherwise the PDE would not have

8 Discrete Dynamics in Nature and Society

an analytical solution The proposed CA model can be set toalmost any initial condition to acquire the response due tothat definition

Themodel expansion from one to two dimensions entailsincreasing the model computational complexity linear in thecase of one dimension to119874(1198992) complexity because CA spacenow has an array of 119899 times 119899 In this sense the parallel model isjustified if119898 is defined as the number of threads that help thedevelopment of CA per unit timeThen the complexity of theCA will be 119874(1198992119898) and if119898 is made large enough such that119898 rarr 119899 then the complexity is reduced to 119874(119899)

On the other hand in our experience it is possible toextend this model to a three-dimensional CA the challengeis to obtain the relationship between the elasticity of thematerial being modeled and the restitution constant for theCA internal springs

Finally it is worth mentioning that in this work thereis not a straight relationship between the CA model and thePDE From this paper it is clear that the results betweenthe PDE and CA are in excellent agreement Moreover thecellular automata could simulate systemswhich are simulatedby PDE under conditions that these equations could notThelatter suggest that perhaps it is possible to find amathematicaltransformation from PDE to CA

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

The authors wish to thank CONACYT COFAA-IPN (Projectno 20151704) and EDI-IPN for the support given for thiswork

References

[1] J Paulos Beyond Numeracy Vintage Series Vintage Books1992

[2] T Toffoli ldquoCellular automata as an alternative to (rather than anapproximation of) differential equations in modeling physicsrdquoPhysica D Nonlinear Phenomena vol 10 no 1-2 pp 117ndash1271984

[3] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[4] S Wolfram A New Kind of Science Wolfram Media Cham-paign Ill USA 2002

[5] S Wolfram ldquoCellular automata as models of complexityrdquoNature vol 311 no 5985 pp 419ndash424 1984

[6] B Chopard and M Droz Cellular Automata Modeling ofPhysical Systems Cambridge University Press New York NYUSA 1998

[7] S Kawamura T Yoshida H Minamoto and Z Hossain ldquoSim-ulation of the nonlinear vibration of a string using the CellularAutomata based on the reflection rulerdquo Applied Acoustics vol67 no 2 pp 93ndash105 2006

[8] M V Walstijn and E Mullan ldquoTime-domain simulation ofrectangular membrane vibrations with 1-d digital waveguidesrdquo

in Proceedings of the 2011 Forum Acusticum pp 449ndash454Aalborg Denmark 2011

[9] J Kari ldquoTheory of cellular automata a surveyrdquo TheoreticalComputer Science vol 334 no 1ndash3 pp 3ndash33 2005

[10] S H White A M del Rey and G R Sanchez ldquoModelingepidemics using cellular automatardquo Applied Mathematics andComputation vol 186 no 1 pp 193ndash202 2007

[11] A Ilachinski Cellular Automata A Discrete Universe WorldScientific 2001

[12] M Kutrib R Vollmar and T Worsch ldquoIntroduction to thespecial issue on cellular automatardquo Parallel Computing vol 23no 11 pp 1567ndash1576 1997

[13] I Huerta-Trujillo J Chimal-Eguıa N Sanchez-Salas and JMartınez-Nuno ldquoModelo de automata celular 1-dimensionalpara una EDP hiperbolicardquo Research in Computing Science vol58 pp 407ndash423 2012

[14] W Glabisz ldquoCellular automata in nonlinear string vibrationrdquoArchives of Civil and Mechanical Engineering vol 10 no 1 pp27ndash41 2010

[15] W Glabisz ldquoCellular automata in nonlinear vibration problemsof two-parameter elastic foundationrdquo Archives of Civil andMechanical Engineering vol 11 no 2 pp 285ndash299 2011

[16] I Huerta-Trujillo E Castillo-Montiel J Chimal-Eguıa NSanchez-Salas and JMartınez-Nuno ldquoAC 2-dimensional comomodelo de una membrana vibranterdquo Research in ComputingScience vol 83 pp 117ndash130 2014

[17] L Kinsler Fundamentals of Acoustics Wiley 2000[18] S Kawamura M Shirashige and T Iwatsubo ldquoSimulation of

the nonlinear vibration of a string using the cellular automationmethodrdquo Applied Acoustics vol 66 no 1 pp 77ndash87 2005

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Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

of a vibrating membrane is compared with the analyticalsolution of hyperbolic PDE This simulates a proposed fre-quency spectrum observing similarity solutions This showsthat the discretemodeling can be an alternative for systems inwhich the boundary conditions or other circumstances makethe PDE intractable Section 1 offers a brief introductionSection 2 presents the two-dimensional continuous waveequation model as well as the solution to the equation giventhe initial and boundary conditions Section 3 shows themethodology used to make the problem discrete and obtainthe CA evolution rule and the discrete model definitionSection 4 analyses and discusses the results of the continuousmodel against the proposed CA discrete model FinallySection 5 gives the conclusions

2 Vibrating Membrane Classical Model

Themotion equation for amembrane is based on the assump-tion that it is thin and uniform with negligible stiffnessthat is perfectly elastic and without spring vibrating withsmall amplitude movements The equation governing thetransverse membrane vibrations is given by

(12059721199061205971199092 +12059721199061205971199102) = 1

119888212059721199061205971199052 (1)

where 119906(119909 119910 119905) is the membrane deflection and 1198882 = 119879120588where 120588 is themembranemass density [8] and119879 ismembranetension per unit length Equation (1) is called wave equationin two dimensions

Considered as a square membrane (see Figure 1) theboundary conditions are defined as follows

119906 (119909 119910 119905) = 0 for 119909 = 0 119909 = 119887 119910 = 0 119910 = 119887 forall119905 (2)

The initial conditions are defined as

119906 (119909 119910 119905) = 119891 (119909 119910) 120597119906 (119909 119910 119905)

120597119905100381610038161003816100381610038161003816100381610038161003816119905=0 = 119892 (119909 119910) (3)

where 119891(119909 119910) and 119892(119909 119910) are the membrane displacementand initial speed respectively

The initial conditions are defined as

119906 (119909 119910 119905) = 119909119910 (119909 minus 119887) (119910 minus 119887) (4)

120597119906 (119909 119910 119905)120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0 = 0 (5)

Then the repose membrane starts with a spatial distribu-tion that can be seen in Figure 2

x

y

0

b

b

(b b)

Figure 1 Rectangular membrane of 119887 times 119887 length

0

a

0

b

0

c

x

y

z

Figure 2 The membrane system initial and boundary conditions

Differential equation solution (1) with the initial condi-tions and boundary shown in (4) and (5) a length 119887 = 012119898and a constant 119888 = 142829 is

119906 (119909 119910 119905) = 64 (012)41205876

sdot infinsum119898=1119898=odd

infinsum119899=1119899=odd

111989831198993 cos(

120587radic1198982 + 1198992012 142829119905)

sdot sin(119898120587119909012 ) sin(

119899120587119910012)

(6)

The dynamic response of the central membrane point119901 =(1198872 1198872) according to (6) can be observed in Figures 3(a) and3(b)

3 Discrete Membrane Model

This paper considers a synchronous CA [9] where the under-lying topology is a rectangular two-dimensional finite latticeFor this type of CA there are two classic neighborhoods thecase of the CA is considered a Von Neumann neighborhoodtype [10] consisting of a central cell and its four geographicalneighbors to the north south east and west (119899 119904 119890 119908)In this sense the importance of extending the CA to twodimensions lies in that [11] the extension is the emergenceof two-dimensional patterns that have no simple analogue

Discrete Dynamics in Nature and Society 3

times10minus5

15

1

05

0

minus05

minus1

minus15

Am

plitu

de (m

)

0 01 02 03 04 05 06 07 08 09 1

Time (s)

PDE

(a)

times10minus5

1

05

0

minus05

minus1

02 021 022 023 024 025 026 027 028 029 03

Time (s)

PDE

Am

plitu

de (m

)

(b)

Figure 3 PDE graph simulation (a) presents the cell displacement (1198872 1198872) on the 119911-axis (b) zoom-in of the PDE simulation graph

in one dimension and two-dimensional dynamic sometimesallows a direct physical systems comparison

The methodology used for the discretization of the PDEon a 2-dimensional CA is presented

31 Formal Definition of Cellular Automata There are dif-ferent authors [4 9 11 12] that have defined the CA but allcoincide in four elements Taken as a base a CA is as follows

Definition 1 A CA is a 4-tuple CA = (119871 119878 119881Φ) where119871 is a regular lattice 119871 = 119888 isin C119889 for a 119889-dimensionallattice

119878 is a finite set of all possible cell states 119888 isin 119871119881 is a finite set of cells that define the cell neighborhood

Φ 119878119889 rarr 119878 is a transition function applied simultane-ously to the cells that conform the lattice [9]

The objective cell state actualization requires knowing theneighborhood cell states [6 9]

32 AnalyticalModel Discretization Suppose there is amem-branewhich can be thought of as a succession of specificmasspoints connected by springs (see Figure 4) Each point ofthe membrane is attached to the four orthogonal neighborswhere the membrane mass is spread over the attachmentpoints and not on the springs and the membrane boundaryis subject to a fixed surface

Let us call 119889119890(or equilibrium length) the spring length

that joins two masses in an equilibrium stateFor a vibrating membrane system with initial position

conditions described in (4) and starting in repose Eachmembrane junction point is subjected to four forces actingtoward each orthogonal neighbor called 119898

119888 119898119899 119898119904 119898119890

and 119898119908 for the central cell north south east and west

respectively (see Figure 5)

x

y

z

Figure 4 Representation of themembrane as a succession ofmassesand springs each mass joined to its neighbors as north south eastand west

Suppose it is necessary to know the force each neighborapplied over 119898

119888 Following the defined process by Huerta-

Trujillo et al [13] the force that a neighbor exercises over thecentral cell119898

119888is computed (see Figure 6)

Now taking into account Figure 6 gives

997888997888rarrΔ119903119908= 997888rarr119903119908minus997888rarr119903119888 (7)

Knowing that |997888997888rarrΔ119903119908| represents the separation length

between masses it is possible to write this scalar as the sumof the spring length increasing the deformation undergoneby the same spring due to the central mass position changeThus

100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 Δ119903119908 = (119889

119890+Δ119889119890) Δ119903119908 (8)

where Δ119889119890is the spring deformation value therefore it is

necessary to compute this with the purpose of knowing

4 Discrete Dynamics in Nature and Society

x

y

z

mc

mn

ms

me

mw

Figure 5 Representation of the central cell and its four orthogonalneighbors forming a Von Neumann CA neighborhood as well asthe direction in which the neighbor force acts represented by thedirection of the vectors

x

y

z

mcmw

rarr

rc

rarr

Δrw

rarr

rw

Figure 6 Representation of the vectors associated with the centralcell a neighbor and the direction of the force exercised by the springin the neighbor cell direction 997888rarr119903

119888 997888rarr119903119908 and 997888997888rarrΔ119903

119908represent the central

cell vectors the west neighbor and the spring vector that join themrespectively

the force increase from the equilibrium to the analysis pointThus

Δ119889119890Δ119903119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890) Δ119903119908 (9)

Given that both sections of (9) are vectors their compo-nents are represented as follows

Δ119889119890Δ119903119908= (Δ119909

119908 Δ119910119908 Δ119911119908) (10)

and the second vector components

(100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 minus 119889119890) Δ119903119908

= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 minus 119889119890)(

119909119908minus 119909119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 119910119908 minus 119910119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 119911119908 minus 119911119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816) (11)

Equating component to component of (10) and (11) gives

Δ119909119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119909119908minus 119909119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

Δ119910119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119910119908minus 119910119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

Δ119911119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119911119908minus 119911119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

(12)

So the rectangular components Δ119909119908 Δ119910119908 and Δ119911

119908are

obtained corresponding to the displacement increase of theaxis coordinates 119909 119910 and 119911 for the mass 119898

119888for 997888997888rarrΔ119903

119908 The

component values for the vectors 997888997888rarrΔ119903119899 997888997888rarrΔ119903119904 and 997888997888rarrΔ119903

119890are

calculated in the same mannerFollowing the analysis for Hookersquos law given for the119898119888particle there are three forces exercised for 119898

119908in the

direction of 997888997888rarrΔ119903119908due to the vector components 119909 119910 and119911 and similar to each of the particles 119898

119899 119898119904 and 119898

119890in

the directions 997888997888rarrΔ119903119899 997888997888rarrΔ119903119904 and 997888997888rarrΔ119903

119890 respectively Thus the force

applied by the neighbors in the 119909 component is

119865119909= minus 119896119899Δ119909119899minus 119896119904Δ119909119904minus 119896119890Δ119909119890minus 119896119908Δ119909119908 (13)

Considering that the springs that join the masses to themembrane are equal gives 119896

119899= 119896119904= 119896119890= 119896119908= 119896 so this

yields

119865119909= minus 119896 (Δ119909

119899+Δ119909119904+Δ119909119890+Δ119909119908) (14)

In the same manner

119865119910= minus 119896 (Δ119910

119899+Δ119910119904+Δ119910119890+Δ119910119908)

119865119911= minus 119896 (Δ119911

119899+Δ119911119904+Δ119911119890+Δ119911119908) (15)

These are the forces acting over 119898119888 which define that

the acceleration at the time 119898119888is oscillating Since the force

is known and the acceleration stays constant in each timestep using the straight-line motion equation with constantacceleration in terms of the initial speed of 119898

119888for its

rectangular components yields

V119891119909

= V119894119909+ 119886119909119905 (16)

Accordingly

V119891119910= V119894119910+ 119886119910119905 (17a)

V119891119911= V119894119911+ 119886119911119905 (17b)

where 119886 can be computed using second Newtonrsquos law

Discrete Dynamics in Nature and Society 5

The previous equations define the speed of 119898119888after time119905 This permits calculating the new particle position for the

same instant time using the following equations

119909119891119894= 119909119894+ V119894119909119905 + 1

21198861199091199052 (18a)

119910119891119894= 119910119894+ V119894119910119905 + 1

21198861199101199052 (18b)

119911119891119894= 119911119894+ V119894119911119905 + 1

21198861199111199052 (18c)

Speed equations (16) (17a) and (17b) and position (18a)(18b) and (18c) are those used in the evolution function forthe proposed CA

33 Vibrant Membrane Model Using a 2-Dimensional CAUsing the developed analysis defines the CA model for avibrant membrane system fixed at the borders of an area119897 times 119897 as a 4-tuple AC = (119871 119878 119881Φ) where each cell 119898

119888isin 119871

is defined by its mass initial position and speed When themembrane is found in rest state this gives the following

119871 this is a regular 2-dimensional lattice119878

119878 =

0 1 if it is a fixed cell997888997888997888rarr119875119905119898119894119895

vector of position in time 119905119881119905119898119894119895

speed in the time 119905forall119898119894119895isin L

2

(19)

119881 119881 = (119898119899 119898119904 119898119888 119898119890 119898119908)

Φ R3 rarr R3

Φ (a) 997888997888997888rarr119875119905+1119898119888119891

= 997888997888rarr119875119905119898119888119894

+ 997888997888997888rarr119881119894119905119898119888119894119905 + 1

2(sum4

V=1 119886119905V119888)1199052(b) 997888997888997888997888rarr119881119891119905+1

119898119888= 997888997888997888997888rarr119881119894119905119898119888119891

+ sum4V=1

997888rarr119886119905V119888119905where sum4

V=1997888rarr119886119905V119888 is the acceleration that the neighbors of

119898119888exercise over the cell in time 119905 997888997888997888rarr119875119905+1

119898119888119891is the final cell

position in space composed of three variables 119901119909 119901119910 and119901

119911 and according to its rectangular coordinates each one is

represented by 32 bits997888997888997888997888rarr119881119891119905+1119898119888

is the final speed in time 119905 + 1composed of three variables V

119909 V119910 and V

119911 and according to

its rectangular coordinates each one is represented by 32 bitsThis implementation allows having real stateswithout contra-dicting the definition of cellular automaton [4] (moreover thetotal states that each cell can take are restricted to 2193 states)

The evolution function Φ is composed of two rules bothapplied simultaneously to all the cells that conform the latticeThis is different from the model proposed by Glabisz [14 15]where the rules are applied in an asymmetric form

The rule (a) defines the cell position at time 119905 + 1 takingthe speed at time 119905 This position is updated to be the new

starting position for 119905+2 and so on Similarly for (b) the finalspeed for time 119905 + 1 is updated with the initial speed for time119905 + 2

An important point in the model definition is the springrestitution coefficient Unlike other models [13] where theconstant restitution is relatively simple to calculate thismodel faces a problem The arrangement of masses andsprings has no serial pattern but shows an array of gridsTo obtain this coefficient it follows the process defined byHuerta-Trujillo et al [16]

34 119896 Restitution Coefficient Calculation To calculate the119896 restitution coefficient value of the springs that bind thecorresponding mass the process begins by assuming that themembrane has a spring constant with a value equal to 119896

119905

Starting with a membrane represented by the proposedCA consisting of 2 times 2 nodes thereupon following thereduction springs connected in series and in parallel obtains

119896 = 119896119905 (20)

Following the process for a CA consisting of 3 times 3 nodesgives

119896 = 3119896119905

2 (21)

carrying on the process the relation for 119899 springs has the form119896 = 119899

2119896119905 (22)

where 119896 is the spring constant 119896119905is the membrane elasticity

constant and 119899 is the number of nodes for a CAof 119899times119899 nodes4 Simulation and Results

CA was implemented applying the object-oriented paradigmusing the C++ language in which each cell is implemented asa Bean object which has as attributes the cell spatial locationthe value of its mass its instantaneous speed or if it is a fixedcell CA space represented by the square lattice is regularlyimplemented as an object composed of cell arrangementsThe CA as such is composed of a grid object and one thatimplements the evolution rule and the definition of the CAneighborhood

The CA dynamic response is given by defining a mem-brane represented by a lattice of 119899times119899 nodes 119899 = 51 based onthe necessary number of cells to simulate a linear system [13]For CA the basis is a membrane 10times 10 cm stretched by 20of its length in the direction of its axes 119909 and 119910 for a lengthequal to that used in the PDE taken as reference node locatedat (1198992 1198992) to match the point (1198872 1198872) of the membraneThe mass of cells is proportional to the density used to solvethe PDE as shown in Section 2 The CA response is shownin Figures 7(a) and 7(b) Qualitatively the CA response isrelevant to the response of the PDE taking the same initialand boundary conditions for both models

The initial CA conditions are the same as those describedfor the PDE model ((4) and (5)) The displacement was

6 Discrete Dynamics in Nature and Society

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

0 01 02 03 04 05 06 07 08 09 1

Time (s)

CA

(a)

times10minus5

15

1

05

0

minus05

minus1

minus15

m

02 021 022 023 024 025 026 027 028 029 03

Time (s)

CA

(b)

Figure 7 Response graph (a) of the simulation made by the CA presenting the cell (1198992 1198992) on the 119911-axis (b) zoom-in of the simulationgraphic

0 01 02 03 04 05 06 07 08 09 1Time (s)

CAPDE

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

(a) Graphic overlay

011 012 013 014 015 016 017 018 019Time (s)

CAPDE

times10minus5

15

1

05

0

minus05

minus1

minus15

m

(b) Zoom graphic overlay

Figure 8 Overlay graphic (a) of the simulation realised by the CA and the PDE shows the displacement of the cell (1198992 1198992) on the 119911-axisin both cases (b) zoom-in graphic overlay

obtained by the PDE membrane ranging 1119904 and took 1 times 104presented samples The displacement that was obtained fromthe CA proposed the same cell (in this case the cell locatedat (1198992 1198992)) at the same time and with the same numberof samples Fifty simulations were performed with the sameaveraged data for comparison against the obtained PDEFigures 8(a) and 8(b) show overlapping graphs obtainingoscillations by CA (in solid line) and the PDE (in dotted line)as well as a zoom-in to the same graphThemean square erroris defined as

MSE = 1119899119899sum119894=1

(Vac119894minusVr119894)2 (23)

where Vac119894is the 119894th value estimated by the CA and Vr

119894is

the reference value taken by the PDE The error between

the measurements generated by the CA and the referencevalues given by the PDE is MSE = 44189 times 10minus11 whichensures that the CA simulates the PDE response

In contrast to other models [7 15] the proposed CAmodel uses 50 less cells in the lattice to get the PDEresponse

In a quantitative form there is a corresponding phasebetween the two models for the same cell Analytically thefundamental frequency is limited by the constants values 119899and119898 The frequency is defined as [17]

119891119899119898

= 1198882radic(119899119887)

2 + (119898119887 )2 (24)

Discrete Dynamics in Nature and Society 7

50 100 150 200 250 3000

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(a) Overlay of frequency spectra of CA and PDE

7570 80 85 90 95 100

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(b) Zoom overlay frequency spectra

Figure 9 Graphic of frequency spectra overlay (a) of the simulation realized by the CA and the PDE (b) zoom-in of the graphic overlay

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus25

minus2

m

0 01 02 03 04 05 06 07 08 09 1

t

CA

(a)

0 200 400 600 800 1000 1200 1400 16000

0002000400060008

0010012001400160018

002

Frequency (Hz)

Am

plitu

de

AC

(b)

Figure 10 Graphic of the simulation of latticersquos central cell for the CAmodel (a) the graphic (b) shows frequency spectra for the central cell

So a fundamental frequency is expected approximately as

11989111 = 841625Hz (25)

Obtaining frequency spectra for both signals and plot-ting them observe that there is a congruence between thefrequency spectra given by the CA and PDE (see Figure 9(a))Figure 9(b) shows the fundamental frequency for both mod-els around 85Hz The fundamental frequency error betweenthe analytical solution and that simulated by the proposedCAis 0995 which is smaller than that found in other models[7 14 18]

Furthermore the robustness of the CA model can betested given a type of irregular initial condition for examplea random initial position for all cells with 0 lt 119911

119888≪ 119887 where119911

119888is the cell position on the 119911-axis The density tension and

boundary conditions are the same as described in Section 2In this case the PDE does not have an analytical solution

since it is not possible to define a function to describethe initial conditions and to be derivable but the pro-posed CA model endures this kind of initial conditions

Simulating the conditions described and plotting the samepoint (1198872 1198872) as in the previous cases the CA modelprovides the central cell response The results can be seenin Figure 10(a) the frequency spectra of this evolution areshown in Figure 10(b) where the fundamental frequency isapproximately 85Hz such as that obtained in the previouscases

5 Conclusions

The 2-dimensional CA model to a vibrant membrane systemshown in this paper was derived from a 1-dimensional CAmodel that simulates a vibrant string [13]The CAmembranemodel is released from the initial conditions so it is notnecessary to redefine it thus it is possible to define the initialconditions and simulate the system behavior immediatelyThis is different to the 2-dimensional wave PDE that issensitive to these conditions Its solution could only befound if the initial conditions are represented by a derivativefunction in all space otherwise the PDE would not have

8 Discrete Dynamics in Nature and Society

an analytical solution The proposed CA model can be set toalmost any initial condition to acquire the response due tothat definition

Themodel expansion from one to two dimensions entailsincreasing the model computational complexity linear in thecase of one dimension to119874(1198992) complexity because CA spacenow has an array of 119899 times 119899 In this sense the parallel model isjustified if119898 is defined as the number of threads that help thedevelopment of CA per unit timeThen the complexity of theCA will be 119874(1198992119898) and if119898 is made large enough such that119898 rarr 119899 then the complexity is reduced to 119874(119899)

On the other hand in our experience it is possible toextend this model to a three-dimensional CA the challengeis to obtain the relationship between the elasticity of thematerial being modeled and the restitution constant for theCA internal springs

Finally it is worth mentioning that in this work thereis not a straight relationship between the CA model and thePDE From this paper it is clear that the results betweenthe PDE and CA are in excellent agreement Moreover thecellular automata could simulate systemswhich are simulatedby PDE under conditions that these equations could notThelatter suggest that perhaps it is possible to find amathematicaltransformation from PDE to CA

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

The authors wish to thank CONACYT COFAA-IPN (Projectno 20151704) and EDI-IPN for the support given for thiswork

References

[1] J Paulos Beyond Numeracy Vintage Series Vintage Books1992

[2] T Toffoli ldquoCellular automata as an alternative to (rather than anapproximation of) differential equations in modeling physicsrdquoPhysica D Nonlinear Phenomena vol 10 no 1-2 pp 117ndash1271984

[3] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[4] S Wolfram A New Kind of Science Wolfram Media Cham-paign Ill USA 2002

[5] S Wolfram ldquoCellular automata as models of complexityrdquoNature vol 311 no 5985 pp 419ndash424 1984

[6] B Chopard and M Droz Cellular Automata Modeling ofPhysical Systems Cambridge University Press New York NYUSA 1998

[7] S Kawamura T Yoshida H Minamoto and Z Hossain ldquoSim-ulation of the nonlinear vibration of a string using the CellularAutomata based on the reflection rulerdquo Applied Acoustics vol67 no 2 pp 93ndash105 2006

[8] M V Walstijn and E Mullan ldquoTime-domain simulation ofrectangular membrane vibrations with 1-d digital waveguidesrdquo

in Proceedings of the 2011 Forum Acusticum pp 449ndash454Aalborg Denmark 2011

[9] J Kari ldquoTheory of cellular automata a surveyrdquo TheoreticalComputer Science vol 334 no 1ndash3 pp 3ndash33 2005

[10] S H White A M del Rey and G R Sanchez ldquoModelingepidemics using cellular automatardquo Applied Mathematics andComputation vol 186 no 1 pp 193ndash202 2007

[11] A Ilachinski Cellular Automata A Discrete Universe WorldScientific 2001

[12] M Kutrib R Vollmar and T Worsch ldquoIntroduction to thespecial issue on cellular automatardquo Parallel Computing vol 23no 11 pp 1567ndash1576 1997

[13] I Huerta-Trujillo J Chimal-Eguıa N Sanchez-Salas and JMartınez-Nuno ldquoModelo de automata celular 1-dimensionalpara una EDP hiperbolicardquo Research in Computing Science vol58 pp 407ndash423 2012

[14] W Glabisz ldquoCellular automata in nonlinear string vibrationrdquoArchives of Civil and Mechanical Engineering vol 10 no 1 pp27ndash41 2010

[15] W Glabisz ldquoCellular automata in nonlinear vibration problemsof two-parameter elastic foundationrdquo Archives of Civil andMechanical Engineering vol 11 no 2 pp 285ndash299 2011

[16] I Huerta-Trujillo E Castillo-Montiel J Chimal-Eguıa NSanchez-Salas and JMartınez-Nuno ldquoAC 2-dimensional comomodelo de una membrana vibranterdquo Research in ComputingScience vol 83 pp 117ndash130 2014

[17] L Kinsler Fundamentals of Acoustics Wiley 2000[18] S Kawamura M Shirashige and T Iwatsubo ldquoSimulation of

the nonlinear vibration of a string using the cellular automationmethodrdquo Applied Acoustics vol 66 no 1 pp 77ndash87 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

times10minus5

15

1

05

0

minus05

minus1

minus15

Am

plitu

de (m

)

0 01 02 03 04 05 06 07 08 09 1

Time (s)

PDE

(a)

times10minus5

1

05

0

minus05

minus1

02 021 022 023 024 025 026 027 028 029 03

Time (s)

PDE

Am

plitu

de (m

)

(b)

Figure 3 PDE graph simulation (a) presents the cell displacement (1198872 1198872) on the 119911-axis (b) zoom-in of the PDE simulation graph

in one dimension and two-dimensional dynamic sometimesallows a direct physical systems comparison

The methodology used for the discretization of the PDEon a 2-dimensional CA is presented

31 Formal Definition of Cellular Automata There are dif-ferent authors [4 9 11 12] that have defined the CA but allcoincide in four elements Taken as a base a CA is as follows

Definition 1 A CA is a 4-tuple CA = (119871 119878 119881Φ) where119871 is a regular lattice 119871 = 119888 isin C119889 for a 119889-dimensionallattice

119878 is a finite set of all possible cell states 119888 isin 119871119881 is a finite set of cells that define the cell neighborhood

Φ 119878119889 rarr 119878 is a transition function applied simultane-ously to the cells that conform the lattice [9]

The objective cell state actualization requires knowing theneighborhood cell states [6 9]

32 AnalyticalModel Discretization Suppose there is amem-branewhich can be thought of as a succession of specificmasspoints connected by springs (see Figure 4) Each point ofthe membrane is attached to the four orthogonal neighborswhere the membrane mass is spread over the attachmentpoints and not on the springs and the membrane boundaryis subject to a fixed surface

Let us call 119889119890(or equilibrium length) the spring length

that joins two masses in an equilibrium stateFor a vibrating membrane system with initial position

conditions described in (4) and starting in repose Eachmembrane junction point is subjected to four forces actingtoward each orthogonal neighbor called 119898

119888 119898119899 119898119904 119898119890

and 119898119908 for the central cell north south east and west

respectively (see Figure 5)

x

y

z

Figure 4 Representation of themembrane as a succession ofmassesand springs each mass joined to its neighbors as north south eastand west

Suppose it is necessary to know the force each neighborapplied over 119898

119888 Following the defined process by Huerta-

Trujillo et al [13] the force that a neighbor exercises over thecentral cell119898

119888is computed (see Figure 6)

Now taking into account Figure 6 gives

997888997888rarrΔ119903119908= 997888rarr119903119908minus997888rarr119903119888 (7)

Knowing that |997888997888rarrΔ119903119908| represents the separation length

between masses it is possible to write this scalar as the sumof the spring length increasing the deformation undergoneby the same spring due to the central mass position changeThus

100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 Δ119903119908 = (119889

119890+Δ119889119890) Δ119903119908 (8)

where Δ119889119890is the spring deformation value therefore it is

necessary to compute this with the purpose of knowing

4 Discrete Dynamics in Nature and Society

x

y

z

mc

mn

ms

me

mw

Figure 5 Representation of the central cell and its four orthogonalneighbors forming a Von Neumann CA neighborhood as well asthe direction in which the neighbor force acts represented by thedirection of the vectors

x

y

z

mcmw

rarr

rc

rarr

Δrw

rarr

rw

Figure 6 Representation of the vectors associated with the centralcell a neighbor and the direction of the force exercised by the springin the neighbor cell direction 997888rarr119903

119888 997888rarr119903119908 and 997888997888rarrΔ119903

119908represent the central

cell vectors the west neighbor and the spring vector that join themrespectively

the force increase from the equilibrium to the analysis pointThus

Δ119889119890Δ119903119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890) Δ119903119908 (9)

Given that both sections of (9) are vectors their compo-nents are represented as follows

Δ119889119890Δ119903119908= (Δ119909

119908 Δ119910119908 Δ119911119908) (10)

and the second vector components

(100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 minus 119889119890) Δ119903119908

= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 minus 119889119890)(

119909119908minus 119909119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 119910119908 minus 119910119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 119911119908 minus 119911119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816) (11)

Equating component to component of (10) and (11) gives

Δ119909119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119909119908minus 119909119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

Δ119910119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119910119908minus 119910119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

Δ119911119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119911119908minus 119911119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

(12)

So the rectangular components Δ119909119908 Δ119910119908 and Δ119911

119908are

obtained corresponding to the displacement increase of theaxis coordinates 119909 119910 and 119911 for the mass 119898

119888for 997888997888rarrΔ119903

119908 The

component values for the vectors 997888997888rarrΔ119903119899 997888997888rarrΔ119903119904 and 997888997888rarrΔ119903

119890are

calculated in the same mannerFollowing the analysis for Hookersquos law given for the119898119888particle there are three forces exercised for 119898

119908in the

direction of 997888997888rarrΔ119903119908due to the vector components 119909 119910 and119911 and similar to each of the particles 119898

119899 119898119904 and 119898

119890in

the directions 997888997888rarrΔ119903119899 997888997888rarrΔ119903119904 and 997888997888rarrΔ119903

119890 respectively Thus the force

applied by the neighbors in the 119909 component is

119865119909= minus 119896119899Δ119909119899minus 119896119904Δ119909119904minus 119896119890Δ119909119890minus 119896119908Δ119909119908 (13)

Considering that the springs that join the masses to themembrane are equal gives 119896

119899= 119896119904= 119896119890= 119896119908= 119896 so this

yields

119865119909= minus 119896 (Δ119909

119899+Δ119909119904+Δ119909119890+Δ119909119908) (14)

In the same manner

119865119910= minus 119896 (Δ119910

119899+Δ119910119904+Δ119910119890+Δ119910119908)

119865119911= minus 119896 (Δ119911

119899+Δ119911119904+Δ119911119890+Δ119911119908) (15)

These are the forces acting over 119898119888 which define that

the acceleration at the time 119898119888is oscillating Since the force

is known and the acceleration stays constant in each timestep using the straight-line motion equation with constantacceleration in terms of the initial speed of 119898

119888for its

rectangular components yields

V119891119909

= V119894119909+ 119886119909119905 (16)

Accordingly

V119891119910= V119894119910+ 119886119910119905 (17a)

V119891119911= V119894119911+ 119886119911119905 (17b)

where 119886 can be computed using second Newtonrsquos law

Discrete Dynamics in Nature and Society 5

The previous equations define the speed of 119898119888after time119905 This permits calculating the new particle position for the

same instant time using the following equations

119909119891119894= 119909119894+ V119894119909119905 + 1

21198861199091199052 (18a)

119910119891119894= 119910119894+ V119894119910119905 + 1

21198861199101199052 (18b)

119911119891119894= 119911119894+ V119894119911119905 + 1

21198861199111199052 (18c)

Speed equations (16) (17a) and (17b) and position (18a)(18b) and (18c) are those used in the evolution function forthe proposed CA

33 Vibrant Membrane Model Using a 2-Dimensional CAUsing the developed analysis defines the CA model for avibrant membrane system fixed at the borders of an area119897 times 119897 as a 4-tuple AC = (119871 119878 119881Φ) where each cell 119898

119888isin 119871

is defined by its mass initial position and speed When themembrane is found in rest state this gives the following

119871 this is a regular 2-dimensional lattice119878

119878 =

0 1 if it is a fixed cell997888997888997888rarr119875119905119898119894119895

vector of position in time 119905119881119905119898119894119895

speed in the time 119905forall119898119894119895isin L

2

(19)

119881 119881 = (119898119899 119898119904 119898119888 119898119890 119898119908)

Φ R3 rarr R3

Φ (a) 997888997888997888rarr119875119905+1119898119888119891

= 997888997888rarr119875119905119898119888119894

+ 997888997888997888rarr119881119894119905119898119888119894119905 + 1

2(sum4

V=1 119886119905V119888)1199052(b) 997888997888997888997888rarr119881119891119905+1

119898119888= 997888997888997888997888rarr119881119894119905119898119888119891

+ sum4V=1

997888rarr119886119905V119888119905where sum4

V=1997888rarr119886119905V119888 is the acceleration that the neighbors of

119898119888exercise over the cell in time 119905 997888997888997888rarr119875119905+1

119898119888119891is the final cell

position in space composed of three variables 119901119909 119901119910 and119901

119911 and according to its rectangular coordinates each one is

represented by 32 bits997888997888997888997888rarr119881119891119905+1119898119888

is the final speed in time 119905 + 1composed of three variables V

119909 V119910 and V

119911 and according to

its rectangular coordinates each one is represented by 32 bitsThis implementation allows having real stateswithout contra-dicting the definition of cellular automaton [4] (moreover thetotal states that each cell can take are restricted to 2193 states)

The evolution function Φ is composed of two rules bothapplied simultaneously to all the cells that conform the latticeThis is different from the model proposed by Glabisz [14 15]where the rules are applied in an asymmetric form

The rule (a) defines the cell position at time 119905 + 1 takingthe speed at time 119905 This position is updated to be the new

starting position for 119905+2 and so on Similarly for (b) the finalspeed for time 119905 + 1 is updated with the initial speed for time119905 + 2

An important point in the model definition is the springrestitution coefficient Unlike other models [13] where theconstant restitution is relatively simple to calculate thismodel faces a problem The arrangement of masses andsprings has no serial pattern but shows an array of gridsTo obtain this coefficient it follows the process defined byHuerta-Trujillo et al [16]

34 119896 Restitution Coefficient Calculation To calculate the119896 restitution coefficient value of the springs that bind thecorresponding mass the process begins by assuming that themembrane has a spring constant with a value equal to 119896

119905

Starting with a membrane represented by the proposedCA consisting of 2 times 2 nodes thereupon following thereduction springs connected in series and in parallel obtains

119896 = 119896119905 (20)

Following the process for a CA consisting of 3 times 3 nodesgives

119896 = 3119896119905

2 (21)

carrying on the process the relation for 119899 springs has the form119896 = 119899

2119896119905 (22)

where 119896 is the spring constant 119896119905is the membrane elasticity

constant and 119899 is the number of nodes for a CAof 119899times119899 nodes4 Simulation and Results

CA was implemented applying the object-oriented paradigmusing the C++ language in which each cell is implemented asa Bean object which has as attributes the cell spatial locationthe value of its mass its instantaneous speed or if it is a fixedcell CA space represented by the square lattice is regularlyimplemented as an object composed of cell arrangementsThe CA as such is composed of a grid object and one thatimplements the evolution rule and the definition of the CAneighborhood

The CA dynamic response is given by defining a mem-brane represented by a lattice of 119899times119899 nodes 119899 = 51 based onthe necessary number of cells to simulate a linear system [13]For CA the basis is a membrane 10times 10 cm stretched by 20of its length in the direction of its axes 119909 and 119910 for a lengthequal to that used in the PDE taken as reference node locatedat (1198992 1198992) to match the point (1198872 1198872) of the membraneThe mass of cells is proportional to the density used to solvethe PDE as shown in Section 2 The CA response is shownin Figures 7(a) and 7(b) Qualitatively the CA response isrelevant to the response of the PDE taking the same initialand boundary conditions for both models

The initial CA conditions are the same as those describedfor the PDE model ((4) and (5)) The displacement was

6 Discrete Dynamics in Nature and Society

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

0 01 02 03 04 05 06 07 08 09 1

Time (s)

CA

(a)

times10minus5

15

1

05

0

minus05

minus1

minus15

m

02 021 022 023 024 025 026 027 028 029 03

Time (s)

CA

(b)

Figure 7 Response graph (a) of the simulation made by the CA presenting the cell (1198992 1198992) on the 119911-axis (b) zoom-in of the simulationgraphic

0 01 02 03 04 05 06 07 08 09 1Time (s)

CAPDE

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

(a) Graphic overlay

011 012 013 014 015 016 017 018 019Time (s)

CAPDE

times10minus5

15

1

05

0

minus05

minus1

minus15

m

(b) Zoom graphic overlay

Figure 8 Overlay graphic (a) of the simulation realised by the CA and the PDE shows the displacement of the cell (1198992 1198992) on the 119911-axisin both cases (b) zoom-in graphic overlay

obtained by the PDE membrane ranging 1119904 and took 1 times 104presented samples The displacement that was obtained fromthe CA proposed the same cell (in this case the cell locatedat (1198992 1198992)) at the same time and with the same numberof samples Fifty simulations were performed with the sameaveraged data for comparison against the obtained PDEFigures 8(a) and 8(b) show overlapping graphs obtainingoscillations by CA (in solid line) and the PDE (in dotted line)as well as a zoom-in to the same graphThemean square erroris defined as

MSE = 1119899119899sum119894=1

(Vac119894minusVr119894)2 (23)

where Vac119894is the 119894th value estimated by the CA and Vr

119894is

the reference value taken by the PDE The error between

the measurements generated by the CA and the referencevalues given by the PDE is MSE = 44189 times 10minus11 whichensures that the CA simulates the PDE response

In contrast to other models [7 15] the proposed CAmodel uses 50 less cells in the lattice to get the PDEresponse

In a quantitative form there is a corresponding phasebetween the two models for the same cell Analytically thefundamental frequency is limited by the constants values 119899and119898 The frequency is defined as [17]

119891119899119898

= 1198882radic(119899119887)

2 + (119898119887 )2 (24)

Discrete Dynamics in Nature and Society 7

50 100 150 200 250 3000

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(a) Overlay of frequency spectra of CA and PDE

7570 80 85 90 95 100

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(b) Zoom overlay frequency spectra

Figure 9 Graphic of frequency spectra overlay (a) of the simulation realized by the CA and the PDE (b) zoom-in of the graphic overlay

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus25

minus2

m

0 01 02 03 04 05 06 07 08 09 1

t

CA

(a)

0 200 400 600 800 1000 1200 1400 16000

0002000400060008

0010012001400160018

002

Frequency (Hz)

Am

plitu

de

AC

(b)

Figure 10 Graphic of the simulation of latticersquos central cell for the CAmodel (a) the graphic (b) shows frequency spectra for the central cell

So a fundamental frequency is expected approximately as

11989111 = 841625Hz (25)

Obtaining frequency spectra for both signals and plot-ting them observe that there is a congruence between thefrequency spectra given by the CA and PDE (see Figure 9(a))Figure 9(b) shows the fundamental frequency for both mod-els around 85Hz The fundamental frequency error betweenthe analytical solution and that simulated by the proposedCAis 0995 which is smaller than that found in other models[7 14 18]

Furthermore the robustness of the CA model can betested given a type of irregular initial condition for examplea random initial position for all cells with 0 lt 119911

119888≪ 119887 where119911

119888is the cell position on the 119911-axis The density tension and

boundary conditions are the same as described in Section 2In this case the PDE does not have an analytical solution

since it is not possible to define a function to describethe initial conditions and to be derivable but the pro-posed CA model endures this kind of initial conditions

Simulating the conditions described and plotting the samepoint (1198872 1198872) as in the previous cases the CA modelprovides the central cell response The results can be seenin Figure 10(a) the frequency spectra of this evolution areshown in Figure 10(b) where the fundamental frequency isapproximately 85Hz such as that obtained in the previouscases

5 Conclusions

The 2-dimensional CA model to a vibrant membrane systemshown in this paper was derived from a 1-dimensional CAmodel that simulates a vibrant string [13]The CAmembranemodel is released from the initial conditions so it is notnecessary to redefine it thus it is possible to define the initialconditions and simulate the system behavior immediatelyThis is different to the 2-dimensional wave PDE that issensitive to these conditions Its solution could only befound if the initial conditions are represented by a derivativefunction in all space otherwise the PDE would not have

8 Discrete Dynamics in Nature and Society

an analytical solution The proposed CA model can be set toalmost any initial condition to acquire the response due tothat definition

Themodel expansion from one to two dimensions entailsincreasing the model computational complexity linear in thecase of one dimension to119874(1198992) complexity because CA spacenow has an array of 119899 times 119899 In this sense the parallel model isjustified if119898 is defined as the number of threads that help thedevelopment of CA per unit timeThen the complexity of theCA will be 119874(1198992119898) and if119898 is made large enough such that119898 rarr 119899 then the complexity is reduced to 119874(119899)

On the other hand in our experience it is possible toextend this model to a three-dimensional CA the challengeis to obtain the relationship between the elasticity of thematerial being modeled and the restitution constant for theCA internal springs

Finally it is worth mentioning that in this work thereis not a straight relationship between the CA model and thePDE From this paper it is clear that the results betweenthe PDE and CA are in excellent agreement Moreover thecellular automata could simulate systemswhich are simulatedby PDE under conditions that these equations could notThelatter suggest that perhaps it is possible to find amathematicaltransformation from PDE to CA

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

The authors wish to thank CONACYT COFAA-IPN (Projectno 20151704) and EDI-IPN for the support given for thiswork

References

[1] J Paulos Beyond Numeracy Vintage Series Vintage Books1992

[2] T Toffoli ldquoCellular automata as an alternative to (rather than anapproximation of) differential equations in modeling physicsrdquoPhysica D Nonlinear Phenomena vol 10 no 1-2 pp 117ndash1271984

[3] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[4] S Wolfram A New Kind of Science Wolfram Media Cham-paign Ill USA 2002

[5] S Wolfram ldquoCellular automata as models of complexityrdquoNature vol 311 no 5985 pp 419ndash424 1984

[6] B Chopard and M Droz Cellular Automata Modeling ofPhysical Systems Cambridge University Press New York NYUSA 1998

[7] S Kawamura T Yoshida H Minamoto and Z Hossain ldquoSim-ulation of the nonlinear vibration of a string using the CellularAutomata based on the reflection rulerdquo Applied Acoustics vol67 no 2 pp 93ndash105 2006

[8] M V Walstijn and E Mullan ldquoTime-domain simulation ofrectangular membrane vibrations with 1-d digital waveguidesrdquo

in Proceedings of the 2011 Forum Acusticum pp 449ndash454Aalborg Denmark 2011

[9] J Kari ldquoTheory of cellular automata a surveyrdquo TheoreticalComputer Science vol 334 no 1ndash3 pp 3ndash33 2005

[10] S H White A M del Rey and G R Sanchez ldquoModelingepidemics using cellular automatardquo Applied Mathematics andComputation vol 186 no 1 pp 193ndash202 2007

[11] A Ilachinski Cellular Automata A Discrete Universe WorldScientific 2001

[12] M Kutrib R Vollmar and T Worsch ldquoIntroduction to thespecial issue on cellular automatardquo Parallel Computing vol 23no 11 pp 1567ndash1576 1997

[13] I Huerta-Trujillo J Chimal-Eguıa N Sanchez-Salas and JMartınez-Nuno ldquoModelo de automata celular 1-dimensionalpara una EDP hiperbolicardquo Research in Computing Science vol58 pp 407ndash423 2012

[14] W Glabisz ldquoCellular automata in nonlinear string vibrationrdquoArchives of Civil and Mechanical Engineering vol 10 no 1 pp27ndash41 2010

[15] W Glabisz ldquoCellular automata in nonlinear vibration problemsof two-parameter elastic foundationrdquo Archives of Civil andMechanical Engineering vol 11 no 2 pp 285ndash299 2011

[16] I Huerta-Trujillo E Castillo-Montiel J Chimal-Eguıa NSanchez-Salas and JMartınez-Nuno ldquoAC 2-dimensional comomodelo de una membrana vibranterdquo Research in ComputingScience vol 83 pp 117ndash130 2014

[17] L Kinsler Fundamentals of Acoustics Wiley 2000[18] S Kawamura M Shirashige and T Iwatsubo ldquoSimulation of

the nonlinear vibration of a string using the cellular automationmethodrdquo Applied Acoustics vol 66 no 1 pp 77ndash87 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

x

y

z

mc

mn

ms

me

mw

Figure 5 Representation of the central cell and its four orthogonalneighbors forming a Von Neumann CA neighborhood as well asthe direction in which the neighbor force acts represented by thedirection of the vectors

x

y

z

mcmw

rarr

rc

rarr

Δrw

rarr

rw

Figure 6 Representation of the vectors associated with the centralcell a neighbor and the direction of the force exercised by the springin the neighbor cell direction 997888rarr119903

119888 997888rarr119903119908 and 997888997888rarrΔ119903

119908represent the central

cell vectors the west neighbor and the spring vector that join themrespectively

the force increase from the equilibrium to the analysis pointThus

Δ119889119890Δ119903119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890) Δ119903119908 (9)

Given that both sections of (9) are vectors their compo-nents are represented as follows

Δ119889119890Δ119903119908= (Δ119909

119908 Δ119910119908 Δ119911119908) (10)

and the second vector components

(100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 minus 119889119890) Δ119903119908

= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 minus 119889119890)(

119909119908minus 119909119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816 119910119908 minus 119910119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 119911119908 minus 119911119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816) (11)

Equating component to component of (10) and (11) gives

Δ119909119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119909119908minus 119909119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

Δ119910119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119910119908minus 119910119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

Δ119911119908= (100381610038161003816100381610038161003816997888997888rarrΔ119903119908

100381610038161003816100381610038161003816 minus 119889119890)119911119908minus 119911119888100381610038161003816100381610038161003816997888997888rarrΔ119903119908100381610038161003816100381610038161003816

(12)

So the rectangular components Δ119909119908 Δ119910119908 and Δ119911

119908are

obtained corresponding to the displacement increase of theaxis coordinates 119909 119910 and 119911 for the mass 119898

119888for 997888997888rarrΔ119903

119908 The

component values for the vectors 997888997888rarrΔ119903119899 997888997888rarrΔ119903119904 and 997888997888rarrΔ119903

119890are

calculated in the same mannerFollowing the analysis for Hookersquos law given for the119898119888particle there are three forces exercised for 119898

119908in the

direction of 997888997888rarrΔ119903119908due to the vector components 119909 119910 and119911 and similar to each of the particles 119898

119899 119898119904 and 119898

119890in

the directions 997888997888rarrΔ119903119899 997888997888rarrΔ119903119904 and 997888997888rarrΔ119903

119890 respectively Thus the force

applied by the neighbors in the 119909 component is

119865119909= minus 119896119899Δ119909119899minus 119896119904Δ119909119904minus 119896119890Δ119909119890minus 119896119908Δ119909119908 (13)

Considering that the springs that join the masses to themembrane are equal gives 119896

119899= 119896119904= 119896119890= 119896119908= 119896 so this

yields

119865119909= minus 119896 (Δ119909

119899+Δ119909119904+Δ119909119890+Δ119909119908) (14)

In the same manner

119865119910= minus 119896 (Δ119910

119899+Δ119910119904+Δ119910119890+Δ119910119908)

119865119911= minus 119896 (Δ119911

119899+Δ119911119904+Δ119911119890+Δ119911119908) (15)

These are the forces acting over 119898119888 which define that

the acceleration at the time 119898119888is oscillating Since the force

is known and the acceleration stays constant in each timestep using the straight-line motion equation with constantacceleration in terms of the initial speed of 119898

119888for its

rectangular components yields

V119891119909

= V119894119909+ 119886119909119905 (16)

Accordingly

V119891119910= V119894119910+ 119886119910119905 (17a)

V119891119911= V119894119911+ 119886119911119905 (17b)

where 119886 can be computed using second Newtonrsquos law

Discrete Dynamics in Nature and Society 5

The previous equations define the speed of 119898119888after time119905 This permits calculating the new particle position for the

same instant time using the following equations

119909119891119894= 119909119894+ V119894119909119905 + 1

21198861199091199052 (18a)

119910119891119894= 119910119894+ V119894119910119905 + 1

21198861199101199052 (18b)

119911119891119894= 119911119894+ V119894119911119905 + 1

21198861199111199052 (18c)

Speed equations (16) (17a) and (17b) and position (18a)(18b) and (18c) are those used in the evolution function forthe proposed CA

33 Vibrant Membrane Model Using a 2-Dimensional CAUsing the developed analysis defines the CA model for avibrant membrane system fixed at the borders of an area119897 times 119897 as a 4-tuple AC = (119871 119878 119881Φ) where each cell 119898

119888isin 119871

is defined by its mass initial position and speed When themembrane is found in rest state this gives the following

119871 this is a regular 2-dimensional lattice119878

119878 =

0 1 if it is a fixed cell997888997888997888rarr119875119905119898119894119895

vector of position in time 119905119881119905119898119894119895

speed in the time 119905forall119898119894119895isin L

2

(19)

119881 119881 = (119898119899 119898119904 119898119888 119898119890 119898119908)

Φ R3 rarr R3

Φ (a) 997888997888997888rarr119875119905+1119898119888119891

= 997888997888rarr119875119905119898119888119894

+ 997888997888997888rarr119881119894119905119898119888119894119905 + 1

2(sum4

V=1 119886119905V119888)1199052(b) 997888997888997888997888rarr119881119891119905+1

119898119888= 997888997888997888997888rarr119881119894119905119898119888119891

+ sum4V=1

997888rarr119886119905V119888119905where sum4

V=1997888rarr119886119905V119888 is the acceleration that the neighbors of

119898119888exercise over the cell in time 119905 997888997888997888rarr119875119905+1

119898119888119891is the final cell

position in space composed of three variables 119901119909 119901119910 and119901

119911 and according to its rectangular coordinates each one is

represented by 32 bits997888997888997888997888rarr119881119891119905+1119898119888

is the final speed in time 119905 + 1composed of three variables V

119909 V119910 and V

119911 and according to

its rectangular coordinates each one is represented by 32 bitsThis implementation allows having real stateswithout contra-dicting the definition of cellular automaton [4] (moreover thetotal states that each cell can take are restricted to 2193 states)

The evolution function Φ is composed of two rules bothapplied simultaneously to all the cells that conform the latticeThis is different from the model proposed by Glabisz [14 15]where the rules are applied in an asymmetric form

The rule (a) defines the cell position at time 119905 + 1 takingthe speed at time 119905 This position is updated to be the new

starting position for 119905+2 and so on Similarly for (b) the finalspeed for time 119905 + 1 is updated with the initial speed for time119905 + 2

An important point in the model definition is the springrestitution coefficient Unlike other models [13] where theconstant restitution is relatively simple to calculate thismodel faces a problem The arrangement of masses andsprings has no serial pattern but shows an array of gridsTo obtain this coefficient it follows the process defined byHuerta-Trujillo et al [16]

34 119896 Restitution Coefficient Calculation To calculate the119896 restitution coefficient value of the springs that bind thecorresponding mass the process begins by assuming that themembrane has a spring constant with a value equal to 119896

119905

Starting with a membrane represented by the proposedCA consisting of 2 times 2 nodes thereupon following thereduction springs connected in series and in parallel obtains

119896 = 119896119905 (20)

Following the process for a CA consisting of 3 times 3 nodesgives

119896 = 3119896119905

2 (21)

carrying on the process the relation for 119899 springs has the form119896 = 119899

2119896119905 (22)

where 119896 is the spring constant 119896119905is the membrane elasticity

constant and 119899 is the number of nodes for a CAof 119899times119899 nodes4 Simulation and Results

CA was implemented applying the object-oriented paradigmusing the C++ language in which each cell is implemented asa Bean object which has as attributes the cell spatial locationthe value of its mass its instantaneous speed or if it is a fixedcell CA space represented by the square lattice is regularlyimplemented as an object composed of cell arrangementsThe CA as such is composed of a grid object and one thatimplements the evolution rule and the definition of the CAneighborhood

The CA dynamic response is given by defining a mem-brane represented by a lattice of 119899times119899 nodes 119899 = 51 based onthe necessary number of cells to simulate a linear system [13]For CA the basis is a membrane 10times 10 cm stretched by 20of its length in the direction of its axes 119909 and 119910 for a lengthequal to that used in the PDE taken as reference node locatedat (1198992 1198992) to match the point (1198872 1198872) of the membraneThe mass of cells is proportional to the density used to solvethe PDE as shown in Section 2 The CA response is shownin Figures 7(a) and 7(b) Qualitatively the CA response isrelevant to the response of the PDE taking the same initialand boundary conditions for both models

The initial CA conditions are the same as those describedfor the PDE model ((4) and (5)) The displacement was

6 Discrete Dynamics in Nature and Society

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

0 01 02 03 04 05 06 07 08 09 1

Time (s)

CA

(a)

times10minus5

15

1

05

0

minus05

minus1

minus15

m

02 021 022 023 024 025 026 027 028 029 03

Time (s)

CA

(b)

Figure 7 Response graph (a) of the simulation made by the CA presenting the cell (1198992 1198992) on the 119911-axis (b) zoom-in of the simulationgraphic

0 01 02 03 04 05 06 07 08 09 1Time (s)

CAPDE

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

(a) Graphic overlay

011 012 013 014 015 016 017 018 019Time (s)

CAPDE

times10minus5

15

1

05

0

minus05

minus1

minus15

m

(b) Zoom graphic overlay

Figure 8 Overlay graphic (a) of the simulation realised by the CA and the PDE shows the displacement of the cell (1198992 1198992) on the 119911-axisin both cases (b) zoom-in graphic overlay

obtained by the PDE membrane ranging 1119904 and took 1 times 104presented samples The displacement that was obtained fromthe CA proposed the same cell (in this case the cell locatedat (1198992 1198992)) at the same time and with the same numberof samples Fifty simulations were performed with the sameaveraged data for comparison against the obtained PDEFigures 8(a) and 8(b) show overlapping graphs obtainingoscillations by CA (in solid line) and the PDE (in dotted line)as well as a zoom-in to the same graphThemean square erroris defined as

MSE = 1119899119899sum119894=1

(Vac119894minusVr119894)2 (23)

where Vac119894is the 119894th value estimated by the CA and Vr

119894is

the reference value taken by the PDE The error between

the measurements generated by the CA and the referencevalues given by the PDE is MSE = 44189 times 10minus11 whichensures that the CA simulates the PDE response

In contrast to other models [7 15] the proposed CAmodel uses 50 less cells in the lattice to get the PDEresponse

In a quantitative form there is a corresponding phasebetween the two models for the same cell Analytically thefundamental frequency is limited by the constants values 119899and119898 The frequency is defined as [17]

119891119899119898

= 1198882radic(119899119887)

2 + (119898119887 )2 (24)

Discrete Dynamics in Nature and Society 7

50 100 150 200 250 3000

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(a) Overlay of frequency spectra of CA and PDE

7570 80 85 90 95 100

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(b) Zoom overlay frequency spectra

Figure 9 Graphic of frequency spectra overlay (a) of the simulation realized by the CA and the PDE (b) zoom-in of the graphic overlay

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus25

minus2

m

0 01 02 03 04 05 06 07 08 09 1

t

CA

(a)

0 200 400 600 800 1000 1200 1400 16000

0002000400060008

0010012001400160018

002

Frequency (Hz)

Am

plitu

de

AC

(b)

Figure 10 Graphic of the simulation of latticersquos central cell for the CAmodel (a) the graphic (b) shows frequency spectra for the central cell

So a fundamental frequency is expected approximately as

11989111 = 841625Hz (25)

Obtaining frequency spectra for both signals and plot-ting them observe that there is a congruence between thefrequency spectra given by the CA and PDE (see Figure 9(a))Figure 9(b) shows the fundamental frequency for both mod-els around 85Hz The fundamental frequency error betweenthe analytical solution and that simulated by the proposedCAis 0995 which is smaller than that found in other models[7 14 18]

Furthermore the robustness of the CA model can betested given a type of irregular initial condition for examplea random initial position for all cells with 0 lt 119911

119888≪ 119887 where119911

119888is the cell position on the 119911-axis The density tension and

boundary conditions are the same as described in Section 2In this case the PDE does not have an analytical solution

since it is not possible to define a function to describethe initial conditions and to be derivable but the pro-posed CA model endures this kind of initial conditions

Simulating the conditions described and plotting the samepoint (1198872 1198872) as in the previous cases the CA modelprovides the central cell response The results can be seenin Figure 10(a) the frequency spectra of this evolution areshown in Figure 10(b) where the fundamental frequency isapproximately 85Hz such as that obtained in the previouscases

5 Conclusions

The 2-dimensional CA model to a vibrant membrane systemshown in this paper was derived from a 1-dimensional CAmodel that simulates a vibrant string [13]The CAmembranemodel is released from the initial conditions so it is notnecessary to redefine it thus it is possible to define the initialconditions and simulate the system behavior immediatelyThis is different to the 2-dimensional wave PDE that issensitive to these conditions Its solution could only befound if the initial conditions are represented by a derivativefunction in all space otherwise the PDE would not have

8 Discrete Dynamics in Nature and Society

an analytical solution The proposed CA model can be set toalmost any initial condition to acquire the response due tothat definition

Themodel expansion from one to two dimensions entailsincreasing the model computational complexity linear in thecase of one dimension to119874(1198992) complexity because CA spacenow has an array of 119899 times 119899 In this sense the parallel model isjustified if119898 is defined as the number of threads that help thedevelopment of CA per unit timeThen the complexity of theCA will be 119874(1198992119898) and if119898 is made large enough such that119898 rarr 119899 then the complexity is reduced to 119874(119899)

On the other hand in our experience it is possible toextend this model to a three-dimensional CA the challengeis to obtain the relationship between the elasticity of thematerial being modeled and the restitution constant for theCA internal springs

Finally it is worth mentioning that in this work thereis not a straight relationship between the CA model and thePDE From this paper it is clear that the results betweenthe PDE and CA are in excellent agreement Moreover thecellular automata could simulate systemswhich are simulatedby PDE under conditions that these equations could notThelatter suggest that perhaps it is possible to find amathematicaltransformation from PDE to CA

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

The authors wish to thank CONACYT COFAA-IPN (Projectno 20151704) and EDI-IPN for the support given for thiswork

References

[1] J Paulos Beyond Numeracy Vintage Series Vintage Books1992

[2] T Toffoli ldquoCellular automata as an alternative to (rather than anapproximation of) differential equations in modeling physicsrdquoPhysica D Nonlinear Phenomena vol 10 no 1-2 pp 117ndash1271984

[3] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[4] S Wolfram A New Kind of Science Wolfram Media Cham-paign Ill USA 2002

[5] S Wolfram ldquoCellular automata as models of complexityrdquoNature vol 311 no 5985 pp 419ndash424 1984

[6] B Chopard and M Droz Cellular Automata Modeling ofPhysical Systems Cambridge University Press New York NYUSA 1998

[7] S Kawamura T Yoshida H Minamoto and Z Hossain ldquoSim-ulation of the nonlinear vibration of a string using the CellularAutomata based on the reflection rulerdquo Applied Acoustics vol67 no 2 pp 93ndash105 2006

[8] M V Walstijn and E Mullan ldquoTime-domain simulation ofrectangular membrane vibrations with 1-d digital waveguidesrdquo

in Proceedings of the 2011 Forum Acusticum pp 449ndash454Aalborg Denmark 2011

[9] J Kari ldquoTheory of cellular automata a surveyrdquo TheoreticalComputer Science vol 334 no 1ndash3 pp 3ndash33 2005

[10] S H White A M del Rey and G R Sanchez ldquoModelingepidemics using cellular automatardquo Applied Mathematics andComputation vol 186 no 1 pp 193ndash202 2007

[11] A Ilachinski Cellular Automata A Discrete Universe WorldScientific 2001

[12] M Kutrib R Vollmar and T Worsch ldquoIntroduction to thespecial issue on cellular automatardquo Parallel Computing vol 23no 11 pp 1567ndash1576 1997

[13] I Huerta-Trujillo J Chimal-Eguıa N Sanchez-Salas and JMartınez-Nuno ldquoModelo de automata celular 1-dimensionalpara una EDP hiperbolicardquo Research in Computing Science vol58 pp 407ndash423 2012

[14] W Glabisz ldquoCellular automata in nonlinear string vibrationrdquoArchives of Civil and Mechanical Engineering vol 10 no 1 pp27ndash41 2010

[15] W Glabisz ldquoCellular automata in nonlinear vibration problemsof two-parameter elastic foundationrdquo Archives of Civil andMechanical Engineering vol 11 no 2 pp 285ndash299 2011

[16] I Huerta-Trujillo E Castillo-Montiel J Chimal-Eguıa NSanchez-Salas and JMartınez-Nuno ldquoAC 2-dimensional comomodelo de una membrana vibranterdquo Research in ComputingScience vol 83 pp 117ndash130 2014

[17] L Kinsler Fundamentals of Acoustics Wiley 2000[18] S Kawamura M Shirashige and T Iwatsubo ldquoSimulation of

the nonlinear vibration of a string using the cellular automationmethodrdquo Applied Acoustics vol 66 no 1 pp 77ndash87 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

The previous equations define the speed of 119898119888after time119905 This permits calculating the new particle position for the

same instant time using the following equations

119909119891119894= 119909119894+ V119894119909119905 + 1

21198861199091199052 (18a)

119910119891119894= 119910119894+ V119894119910119905 + 1

21198861199101199052 (18b)

119911119891119894= 119911119894+ V119894119911119905 + 1

21198861199111199052 (18c)

Speed equations (16) (17a) and (17b) and position (18a)(18b) and (18c) are those used in the evolution function forthe proposed CA

33 Vibrant Membrane Model Using a 2-Dimensional CAUsing the developed analysis defines the CA model for avibrant membrane system fixed at the borders of an area119897 times 119897 as a 4-tuple AC = (119871 119878 119881Φ) where each cell 119898

119888isin 119871

is defined by its mass initial position and speed When themembrane is found in rest state this gives the following

119871 this is a regular 2-dimensional lattice119878

119878 =

0 1 if it is a fixed cell997888997888997888rarr119875119905119898119894119895

vector of position in time 119905119881119905119898119894119895

speed in the time 119905forall119898119894119895isin L

2

(19)

119881 119881 = (119898119899 119898119904 119898119888 119898119890 119898119908)

Φ R3 rarr R3

Φ (a) 997888997888997888rarr119875119905+1119898119888119891

= 997888997888rarr119875119905119898119888119894

+ 997888997888997888rarr119881119894119905119898119888119894119905 + 1

2(sum4

V=1 119886119905V119888)1199052(b) 997888997888997888997888rarr119881119891119905+1

119898119888= 997888997888997888997888rarr119881119894119905119898119888119891

+ sum4V=1

997888rarr119886119905V119888119905where sum4

V=1997888rarr119886119905V119888 is the acceleration that the neighbors of

119898119888exercise over the cell in time 119905 997888997888997888rarr119875119905+1

119898119888119891is the final cell

position in space composed of three variables 119901119909 119901119910 and119901

119911 and according to its rectangular coordinates each one is

represented by 32 bits997888997888997888997888rarr119881119891119905+1119898119888

is the final speed in time 119905 + 1composed of three variables V

119909 V119910 and V

119911 and according to

its rectangular coordinates each one is represented by 32 bitsThis implementation allows having real stateswithout contra-dicting the definition of cellular automaton [4] (moreover thetotal states that each cell can take are restricted to 2193 states)

The evolution function Φ is composed of two rules bothapplied simultaneously to all the cells that conform the latticeThis is different from the model proposed by Glabisz [14 15]where the rules are applied in an asymmetric form

The rule (a) defines the cell position at time 119905 + 1 takingthe speed at time 119905 This position is updated to be the new

starting position for 119905+2 and so on Similarly for (b) the finalspeed for time 119905 + 1 is updated with the initial speed for time119905 + 2

An important point in the model definition is the springrestitution coefficient Unlike other models [13] where theconstant restitution is relatively simple to calculate thismodel faces a problem The arrangement of masses andsprings has no serial pattern but shows an array of gridsTo obtain this coefficient it follows the process defined byHuerta-Trujillo et al [16]

34 119896 Restitution Coefficient Calculation To calculate the119896 restitution coefficient value of the springs that bind thecorresponding mass the process begins by assuming that themembrane has a spring constant with a value equal to 119896

119905

Starting with a membrane represented by the proposedCA consisting of 2 times 2 nodes thereupon following thereduction springs connected in series and in parallel obtains

119896 = 119896119905 (20)

Following the process for a CA consisting of 3 times 3 nodesgives

119896 = 3119896119905

2 (21)

carrying on the process the relation for 119899 springs has the form119896 = 119899

2119896119905 (22)

where 119896 is the spring constant 119896119905is the membrane elasticity

constant and 119899 is the number of nodes for a CAof 119899times119899 nodes4 Simulation and Results

CA was implemented applying the object-oriented paradigmusing the C++ language in which each cell is implemented asa Bean object which has as attributes the cell spatial locationthe value of its mass its instantaneous speed or if it is a fixedcell CA space represented by the square lattice is regularlyimplemented as an object composed of cell arrangementsThe CA as such is composed of a grid object and one thatimplements the evolution rule and the definition of the CAneighborhood

The CA dynamic response is given by defining a mem-brane represented by a lattice of 119899times119899 nodes 119899 = 51 based onthe necessary number of cells to simulate a linear system [13]For CA the basis is a membrane 10times 10 cm stretched by 20of its length in the direction of its axes 119909 and 119910 for a lengthequal to that used in the PDE taken as reference node locatedat (1198992 1198992) to match the point (1198872 1198872) of the membraneThe mass of cells is proportional to the density used to solvethe PDE as shown in Section 2 The CA response is shownin Figures 7(a) and 7(b) Qualitatively the CA response isrelevant to the response of the PDE taking the same initialand boundary conditions for both models

The initial CA conditions are the same as those describedfor the PDE model ((4) and (5)) The displacement was

6 Discrete Dynamics in Nature and Society

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

0 01 02 03 04 05 06 07 08 09 1

Time (s)

CA

(a)

times10minus5

15

1

05

0

minus05

minus1

minus15

m

02 021 022 023 024 025 026 027 028 029 03

Time (s)

CA

(b)

Figure 7 Response graph (a) of the simulation made by the CA presenting the cell (1198992 1198992) on the 119911-axis (b) zoom-in of the simulationgraphic

0 01 02 03 04 05 06 07 08 09 1Time (s)

CAPDE

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

(a) Graphic overlay

011 012 013 014 015 016 017 018 019Time (s)

CAPDE

times10minus5

15

1

05

0

minus05

minus1

minus15

m

(b) Zoom graphic overlay

Figure 8 Overlay graphic (a) of the simulation realised by the CA and the PDE shows the displacement of the cell (1198992 1198992) on the 119911-axisin both cases (b) zoom-in graphic overlay

obtained by the PDE membrane ranging 1119904 and took 1 times 104presented samples The displacement that was obtained fromthe CA proposed the same cell (in this case the cell locatedat (1198992 1198992)) at the same time and with the same numberof samples Fifty simulations were performed with the sameaveraged data for comparison against the obtained PDEFigures 8(a) and 8(b) show overlapping graphs obtainingoscillations by CA (in solid line) and the PDE (in dotted line)as well as a zoom-in to the same graphThemean square erroris defined as

MSE = 1119899119899sum119894=1

(Vac119894minusVr119894)2 (23)

where Vac119894is the 119894th value estimated by the CA and Vr

119894is

the reference value taken by the PDE The error between

the measurements generated by the CA and the referencevalues given by the PDE is MSE = 44189 times 10minus11 whichensures that the CA simulates the PDE response

In contrast to other models [7 15] the proposed CAmodel uses 50 less cells in the lattice to get the PDEresponse

In a quantitative form there is a corresponding phasebetween the two models for the same cell Analytically thefundamental frequency is limited by the constants values 119899and119898 The frequency is defined as [17]

119891119899119898

= 1198882radic(119899119887)

2 + (119898119887 )2 (24)

Discrete Dynamics in Nature and Society 7

50 100 150 200 250 3000

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(a) Overlay of frequency spectra of CA and PDE

7570 80 85 90 95 100

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(b) Zoom overlay frequency spectra

Figure 9 Graphic of frequency spectra overlay (a) of the simulation realized by the CA and the PDE (b) zoom-in of the graphic overlay

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus25

minus2

m

0 01 02 03 04 05 06 07 08 09 1

t

CA

(a)

0 200 400 600 800 1000 1200 1400 16000

0002000400060008

0010012001400160018

002

Frequency (Hz)

Am

plitu

de

AC

(b)

Figure 10 Graphic of the simulation of latticersquos central cell for the CAmodel (a) the graphic (b) shows frequency spectra for the central cell

So a fundamental frequency is expected approximately as

11989111 = 841625Hz (25)

Obtaining frequency spectra for both signals and plot-ting them observe that there is a congruence between thefrequency spectra given by the CA and PDE (see Figure 9(a))Figure 9(b) shows the fundamental frequency for both mod-els around 85Hz The fundamental frequency error betweenthe analytical solution and that simulated by the proposedCAis 0995 which is smaller than that found in other models[7 14 18]

Furthermore the robustness of the CA model can betested given a type of irregular initial condition for examplea random initial position for all cells with 0 lt 119911

119888≪ 119887 where119911

119888is the cell position on the 119911-axis The density tension and

boundary conditions are the same as described in Section 2In this case the PDE does not have an analytical solution

since it is not possible to define a function to describethe initial conditions and to be derivable but the pro-posed CA model endures this kind of initial conditions

Simulating the conditions described and plotting the samepoint (1198872 1198872) as in the previous cases the CA modelprovides the central cell response The results can be seenin Figure 10(a) the frequency spectra of this evolution areshown in Figure 10(b) where the fundamental frequency isapproximately 85Hz such as that obtained in the previouscases

5 Conclusions

The 2-dimensional CA model to a vibrant membrane systemshown in this paper was derived from a 1-dimensional CAmodel that simulates a vibrant string [13]The CAmembranemodel is released from the initial conditions so it is notnecessary to redefine it thus it is possible to define the initialconditions and simulate the system behavior immediatelyThis is different to the 2-dimensional wave PDE that issensitive to these conditions Its solution could only befound if the initial conditions are represented by a derivativefunction in all space otherwise the PDE would not have

8 Discrete Dynamics in Nature and Society

an analytical solution The proposed CA model can be set toalmost any initial condition to acquire the response due tothat definition

Themodel expansion from one to two dimensions entailsincreasing the model computational complexity linear in thecase of one dimension to119874(1198992) complexity because CA spacenow has an array of 119899 times 119899 In this sense the parallel model isjustified if119898 is defined as the number of threads that help thedevelopment of CA per unit timeThen the complexity of theCA will be 119874(1198992119898) and if119898 is made large enough such that119898 rarr 119899 then the complexity is reduced to 119874(119899)

On the other hand in our experience it is possible toextend this model to a three-dimensional CA the challengeis to obtain the relationship between the elasticity of thematerial being modeled and the restitution constant for theCA internal springs

Finally it is worth mentioning that in this work thereis not a straight relationship between the CA model and thePDE From this paper it is clear that the results betweenthe PDE and CA are in excellent agreement Moreover thecellular automata could simulate systemswhich are simulatedby PDE under conditions that these equations could notThelatter suggest that perhaps it is possible to find amathematicaltransformation from PDE to CA

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

The authors wish to thank CONACYT COFAA-IPN (Projectno 20151704) and EDI-IPN for the support given for thiswork

References

[1] J Paulos Beyond Numeracy Vintage Series Vintage Books1992

[2] T Toffoli ldquoCellular automata as an alternative to (rather than anapproximation of) differential equations in modeling physicsrdquoPhysica D Nonlinear Phenomena vol 10 no 1-2 pp 117ndash1271984

[3] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[4] S Wolfram A New Kind of Science Wolfram Media Cham-paign Ill USA 2002

[5] S Wolfram ldquoCellular automata as models of complexityrdquoNature vol 311 no 5985 pp 419ndash424 1984

[6] B Chopard and M Droz Cellular Automata Modeling ofPhysical Systems Cambridge University Press New York NYUSA 1998

[7] S Kawamura T Yoshida H Minamoto and Z Hossain ldquoSim-ulation of the nonlinear vibration of a string using the CellularAutomata based on the reflection rulerdquo Applied Acoustics vol67 no 2 pp 93ndash105 2006

[8] M V Walstijn and E Mullan ldquoTime-domain simulation ofrectangular membrane vibrations with 1-d digital waveguidesrdquo

in Proceedings of the 2011 Forum Acusticum pp 449ndash454Aalborg Denmark 2011

[9] J Kari ldquoTheory of cellular automata a surveyrdquo TheoreticalComputer Science vol 334 no 1ndash3 pp 3ndash33 2005

[10] S H White A M del Rey and G R Sanchez ldquoModelingepidemics using cellular automatardquo Applied Mathematics andComputation vol 186 no 1 pp 193ndash202 2007

[11] A Ilachinski Cellular Automata A Discrete Universe WorldScientific 2001

[12] M Kutrib R Vollmar and T Worsch ldquoIntroduction to thespecial issue on cellular automatardquo Parallel Computing vol 23no 11 pp 1567ndash1576 1997

[13] I Huerta-Trujillo J Chimal-Eguıa N Sanchez-Salas and JMartınez-Nuno ldquoModelo de automata celular 1-dimensionalpara una EDP hiperbolicardquo Research in Computing Science vol58 pp 407ndash423 2012

[14] W Glabisz ldquoCellular automata in nonlinear string vibrationrdquoArchives of Civil and Mechanical Engineering vol 10 no 1 pp27ndash41 2010

[15] W Glabisz ldquoCellular automata in nonlinear vibration problemsof two-parameter elastic foundationrdquo Archives of Civil andMechanical Engineering vol 11 no 2 pp 285ndash299 2011

[16] I Huerta-Trujillo E Castillo-Montiel J Chimal-Eguıa NSanchez-Salas and JMartınez-Nuno ldquoAC 2-dimensional comomodelo de una membrana vibranterdquo Research in ComputingScience vol 83 pp 117ndash130 2014

[17] L Kinsler Fundamentals of Acoustics Wiley 2000[18] S Kawamura M Shirashige and T Iwatsubo ldquoSimulation of

the nonlinear vibration of a string using the cellular automationmethodrdquo Applied Acoustics vol 66 no 1 pp 77ndash87 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

0 01 02 03 04 05 06 07 08 09 1

Time (s)

CA

(a)

times10minus5

15

1

05

0

minus05

minus1

minus15

m

02 021 022 023 024 025 026 027 028 029 03

Time (s)

CA

(b)

Figure 7 Response graph (a) of the simulation made by the CA presenting the cell (1198992 1198992) on the 119911-axis (b) zoom-in of the simulationgraphic

0 01 02 03 04 05 06 07 08 09 1Time (s)

CAPDE

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus2

m

(a) Graphic overlay

011 012 013 014 015 016 017 018 019Time (s)

CAPDE

times10minus5

15

1

05

0

minus05

minus1

minus15

m

(b) Zoom graphic overlay

Figure 8 Overlay graphic (a) of the simulation realised by the CA and the PDE shows the displacement of the cell (1198992 1198992) on the 119911-axisin both cases (b) zoom-in graphic overlay

obtained by the PDE membrane ranging 1119904 and took 1 times 104presented samples The displacement that was obtained fromthe CA proposed the same cell (in this case the cell locatedat (1198992 1198992)) at the same time and with the same numberof samples Fifty simulations were performed with the sameaveraged data for comparison against the obtained PDEFigures 8(a) and 8(b) show overlapping graphs obtainingoscillations by CA (in solid line) and the PDE (in dotted line)as well as a zoom-in to the same graphThemean square erroris defined as

MSE = 1119899119899sum119894=1

(Vac119894minusVr119894)2 (23)

where Vac119894is the 119894th value estimated by the CA and Vr

119894is

the reference value taken by the PDE The error between

the measurements generated by the CA and the referencevalues given by the PDE is MSE = 44189 times 10minus11 whichensures that the CA simulates the PDE response

In contrast to other models [7 15] the proposed CAmodel uses 50 less cells in the lattice to get the PDEresponse

In a quantitative form there is a corresponding phasebetween the two models for the same cell Analytically thefundamental frequency is limited by the constants values 119899and119898 The frequency is defined as [17]

119891119899119898

= 1198882radic(119899119887)

2 + (119898119887 )2 (24)

Discrete Dynamics in Nature and Society 7

50 100 150 200 250 3000

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(a) Overlay of frequency spectra of CA and PDE

7570 80 85 90 95 100

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(b) Zoom overlay frequency spectra

Figure 9 Graphic of frequency spectra overlay (a) of the simulation realized by the CA and the PDE (b) zoom-in of the graphic overlay

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus25

minus2

m

0 01 02 03 04 05 06 07 08 09 1

t

CA

(a)

0 200 400 600 800 1000 1200 1400 16000

0002000400060008

0010012001400160018

002

Frequency (Hz)

Am

plitu

de

AC

(b)

Figure 10 Graphic of the simulation of latticersquos central cell for the CAmodel (a) the graphic (b) shows frequency spectra for the central cell

So a fundamental frequency is expected approximately as

11989111 = 841625Hz (25)

Obtaining frequency spectra for both signals and plot-ting them observe that there is a congruence between thefrequency spectra given by the CA and PDE (see Figure 9(a))Figure 9(b) shows the fundamental frequency for both mod-els around 85Hz The fundamental frequency error betweenthe analytical solution and that simulated by the proposedCAis 0995 which is smaller than that found in other models[7 14 18]

Furthermore the robustness of the CA model can betested given a type of irregular initial condition for examplea random initial position for all cells with 0 lt 119911

119888≪ 119887 where119911

119888is the cell position on the 119911-axis The density tension and

boundary conditions are the same as described in Section 2In this case the PDE does not have an analytical solution

since it is not possible to define a function to describethe initial conditions and to be derivable but the pro-posed CA model endures this kind of initial conditions

Simulating the conditions described and plotting the samepoint (1198872 1198872) as in the previous cases the CA modelprovides the central cell response The results can be seenin Figure 10(a) the frequency spectra of this evolution areshown in Figure 10(b) where the fundamental frequency isapproximately 85Hz such as that obtained in the previouscases

5 Conclusions

The 2-dimensional CA model to a vibrant membrane systemshown in this paper was derived from a 1-dimensional CAmodel that simulates a vibrant string [13]The CAmembranemodel is released from the initial conditions so it is notnecessary to redefine it thus it is possible to define the initialconditions and simulate the system behavior immediatelyThis is different to the 2-dimensional wave PDE that issensitive to these conditions Its solution could only befound if the initial conditions are represented by a derivativefunction in all space otherwise the PDE would not have

8 Discrete Dynamics in Nature and Society

an analytical solution The proposed CA model can be set toalmost any initial condition to acquire the response due tothat definition

Themodel expansion from one to two dimensions entailsincreasing the model computational complexity linear in thecase of one dimension to119874(1198992) complexity because CA spacenow has an array of 119899 times 119899 In this sense the parallel model isjustified if119898 is defined as the number of threads that help thedevelopment of CA per unit timeThen the complexity of theCA will be 119874(1198992119898) and if119898 is made large enough such that119898 rarr 119899 then the complexity is reduced to 119874(119899)

On the other hand in our experience it is possible toextend this model to a three-dimensional CA the challengeis to obtain the relationship between the elasticity of thematerial being modeled and the restitution constant for theCA internal springs

Finally it is worth mentioning that in this work thereis not a straight relationship between the CA model and thePDE From this paper it is clear that the results betweenthe PDE and CA are in excellent agreement Moreover thecellular automata could simulate systemswhich are simulatedby PDE under conditions that these equations could notThelatter suggest that perhaps it is possible to find amathematicaltransformation from PDE to CA

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

The authors wish to thank CONACYT COFAA-IPN (Projectno 20151704) and EDI-IPN for the support given for thiswork

References

[1] J Paulos Beyond Numeracy Vintage Series Vintage Books1992

[2] T Toffoli ldquoCellular automata as an alternative to (rather than anapproximation of) differential equations in modeling physicsrdquoPhysica D Nonlinear Phenomena vol 10 no 1-2 pp 117ndash1271984

[3] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[4] S Wolfram A New Kind of Science Wolfram Media Cham-paign Ill USA 2002

[5] S Wolfram ldquoCellular automata as models of complexityrdquoNature vol 311 no 5985 pp 419ndash424 1984

[6] B Chopard and M Droz Cellular Automata Modeling ofPhysical Systems Cambridge University Press New York NYUSA 1998

[7] S Kawamura T Yoshida H Minamoto and Z Hossain ldquoSim-ulation of the nonlinear vibration of a string using the CellularAutomata based on the reflection rulerdquo Applied Acoustics vol67 no 2 pp 93ndash105 2006

[8] M V Walstijn and E Mullan ldquoTime-domain simulation ofrectangular membrane vibrations with 1-d digital waveguidesrdquo

in Proceedings of the 2011 Forum Acusticum pp 449ndash454Aalborg Denmark 2011

[9] J Kari ldquoTheory of cellular automata a surveyrdquo TheoreticalComputer Science vol 334 no 1ndash3 pp 3ndash33 2005

[10] S H White A M del Rey and G R Sanchez ldquoModelingepidemics using cellular automatardquo Applied Mathematics andComputation vol 186 no 1 pp 193ndash202 2007

[11] A Ilachinski Cellular Automata A Discrete Universe WorldScientific 2001

[12] M Kutrib R Vollmar and T Worsch ldquoIntroduction to thespecial issue on cellular automatardquo Parallel Computing vol 23no 11 pp 1567ndash1576 1997

[13] I Huerta-Trujillo J Chimal-Eguıa N Sanchez-Salas and JMartınez-Nuno ldquoModelo de automata celular 1-dimensionalpara una EDP hiperbolicardquo Research in Computing Science vol58 pp 407ndash423 2012

[14] W Glabisz ldquoCellular automata in nonlinear string vibrationrdquoArchives of Civil and Mechanical Engineering vol 10 no 1 pp27ndash41 2010

[15] W Glabisz ldquoCellular automata in nonlinear vibration problemsof two-parameter elastic foundationrdquo Archives of Civil andMechanical Engineering vol 11 no 2 pp 285ndash299 2011

[16] I Huerta-Trujillo E Castillo-Montiel J Chimal-Eguıa NSanchez-Salas and JMartınez-Nuno ldquoAC 2-dimensional comomodelo de una membrana vibranterdquo Research in ComputingScience vol 83 pp 117ndash130 2014

[17] L Kinsler Fundamentals of Acoustics Wiley 2000[18] S Kawamura M Shirashige and T Iwatsubo ldquoSimulation of

the nonlinear vibration of a string using the cellular automationmethodrdquo Applied Acoustics vol 66 no 1 pp 77ndash87 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

50 100 150 200 250 3000

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(a) Overlay of frequency spectra of CA and PDE

7570 80 85 90 95 100

001

002

003

004

005

006

Frequency (Hz)

Am

plitu

de

CAPDE

(b) Zoom overlay frequency spectra

Figure 9 Graphic of frequency spectra overlay (a) of the simulation realized by the CA and the PDE (b) zoom-in of the graphic overlay

times10minus5

15

1

2

05

0

minus05

minus1

minus15

minus25

minus2

m

0 01 02 03 04 05 06 07 08 09 1

t

CA

(a)

0 200 400 600 800 1000 1200 1400 16000

0002000400060008

0010012001400160018

002

Frequency (Hz)

Am

plitu

de

AC

(b)

Figure 10 Graphic of the simulation of latticersquos central cell for the CAmodel (a) the graphic (b) shows frequency spectra for the central cell

So a fundamental frequency is expected approximately as

11989111 = 841625Hz (25)

Obtaining frequency spectra for both signals and plot-ting them observe that there is a congruence between thefrequency spectra given by the CA and PDE (see Figure 9(a))Figure 9(b) shows the fundamental frequency for both mod-els around 85Hz The fundamental frequency error betweenthe analytical solution and that simulated by the proposedCAis 0995 which is smaller than that found in other models[7 14 18]

Furthermore the robustness of the CA model can betested given a type of irregular initial condition for examplea random initial position for all cells with 0 lt 119911

119888≪ 119887 where119911

119888is the cell position on the 119911-axis The density tension and

boundary conditions are the same as described in Section 2In this case the PDE does not have an analytical solution

since it is not possible to define a function to describethe initial conditions and to be derivable but the pro-posed CA model endures this kind of initial conditions

Simulating the conditions described and plotting the samepoint (1198872 1198872) as in the previous cases the CA modelprovides the central cell response The results can be seenin Figure 10(a) the frequency spectra of this evolution areshown in Figure 10(b) where the fundamental frequency isapproximately 85Hz such as that obtained in the previouscases

5 Conclusions

The 2-dimensional CA model to a vibrant membrane systemshown in this paper was derived from a 1-dimensional CAmodel that simulates a vibrant string [13]The CAmembranemodel is released from the initial conditions so it is notnecessary to redefine it thus it is possible to define the initialconditions and simulate the system behavior immediatelyThis is different to the 2-dimensional wave PDE that issensitive to these conditions Its solution could only befound if the initial conditions are represented by a derivativefunction in all space otherwise the PDE would not have

8 Discrete Dynamics in Nature and Society

an analytical solution The proposed CA model can be set toalmost any initial condition to acquire the response due tothat definition

Themodel expansion from one to two dimensions entailsincreasing the model computational complexity linear in thecase of one dimension to119874(1198992) complexity because CA spacenow has an array of 119899 times 119899 In this sense the parallel model isjustified if119898 is defined as the number of threads that help thedevelopment of CA per unit timeThen the complexity of theCA will be 119874(1198992119898) and if119898 is made large enough such that119898 rarr 119899 then the complexity is reduced to 119874(119899)

On the other hand in our experience it is possible toextend this model to a three-dimensional CA the challengeis to obtain the relationship between the elasticity of thematerial being modeled and the restitution constant for theCA internal springs

Finally it is worth mentioning that in this work thereis not a straight relationship between the CA model and thePDE From this paper it is clear that the results betweenthe PDE and CA are in excellent agreement Moreover thecellular automata could simulate systemswhich are simulatedby PDE under conditions that these equations could notThelatter suggest that perhaps it is possible to find amathematicaltransformation from PDE to CA

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

The authors wish to thank CONACYT COFAA-IPN (Projectno 20151704) and EDI-IPN for the support given for thiswork

References

[1] J Paulos Beyond Numeracy Vintage Series Vintage Books1992

[2] T Toffoli ldquoCellular automata as an alternative to (rather than anapproximation of) differential equations in modeling physicsrdquoPhysica D Nonlinear Phenomena vol 10 no 1-2 pp 117ndash1271984

[3] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[4] S Wolfram A New Kind of Science Wolfram Media Cham-paign Ill USA 2002

[5] S Wolfram ldquoCellular automata as models of complexityrdquoNature vol 311 no 5985 pp 419ndash424 1984

[6] B Chopard and M Droz Cellular Automata Modeling ofPhysical Systems Cambridge University Press New York NYUSA 1998

[7] S Kawamura T Yoshida H Minamoto and Z Hossain ldquoSim-ulation of the nonlinear vibration of a string using the CellularAutomata based on the reflection rulerdquo Applied Acoustics vol67 no 2 pp 93ndash105 2006

[8] M V Walstijn and E Mullan ldquoTime-domain simulation ofrectangular membrane vibrations with 1-d digital waveguidesrdquo

in Proceedings of the 2011 Forum Acusticum pp 449ndash454Aalborg Denmark 2011

[9] J Kari ldquoTheory of cellular automata a surveyrdquo TheoreticalComputer Science vol 334 no 1ndash3 pp 3ndash33 2005

[10] S H White A M del Rey and G R Sanchez ldquoModelingepidemics using cellular automatardquo Applied Mathematics andComputation vol 186 no 1 pp 193ndash202 2007

[11] A Ilachinski Cellular Automata A Discrete Universe WorldScientific 2001

[12] M Kutrib R Vollmar and T Worsch ldquoIntroduction to thespecial issue on cellular automatardquo Parallel Computing vol 23no 11 pp 1567ndash1576 1997

[13] I Huerta-Trujillo J Chimal-Eguıa N Sanchez-Salas and JMartınez-Nuno ldquoModelo de automata celular 1-dimensionalpara una EDP hiperbolicardquo Research in Computing Science vol58 pp 407ndash423 2012

[14] W Glabisz ldquoCellular automata in nonlinear string vibrationrdquoArchives of Civil and Mechanical Engineering vol 10 no 1 pp27ndash41 2010

[15] W Glabisz ldquoCellular automata in nonlinear vibration problemsof two-parameter elastic foundationrdquo Archives of Civil andMechanical Engineering vol 11 no 2 pp 285ndash299 2011

[16] I Huerta-Trujillo E Castillo-Montiel J Chimal-Eguıa NSanchez-Salas and JMartınez-Nuno ldquoAC 2-dimensional comomodelo de una membrana vibranterdquo Research in ComputingScience vol 83 pp 117ndash130 2014

[17] L Kinsler Fundamentals of Acoustics Wiley 2000[18] S Kawamura M Shirashige and T Iwatsubo ldquoSimulation of

the nonlinear vibration of a string using the cellular automationmethodrdquo Applied Acoustics vol 66 no 1 pp 77ndash87 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

an analytical solution The proposed CA model can be set toalmost any initial condition to acquire the response due tothat definition

Themodel expansion from one to two dimensions entailsincreasing the model computational complexity linear in thecase of one dimension to119874(1198992) complexity because CA spacenow has an array of 119899 times 119899 In this sense the parallel model isjustified if119898 is defined as the number of threads that help thedevelopment of CA per unit timeThen the complexity of theCA will be 119874(1198992119898) and if119898 is made large enough such that119898 rarr 119899 then the complexity is reduced to 119874(119899)

On the other hand in our experience it is possible toextend this model to a three-dimensional CA the challengeis to obtain the relationship between the elasticity of thematerial being modeled and the restitution constant for theCA internal springs

Finally it is worth mentioning that in this work thereis not a straight relationship between the CA model and thePDE From this paper it is clear that the results betweenthe PDE and CA are in excellent agreement Moreover thecellular automata could simulate systemswhich are simulatedby PDE under conditions that these equations could notThelatter suggest that perhaps it is possible to find amathematicaltransformation from PDE to CA

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

The authors wish to thank CONACYT COFAA-IPN (Projectno 20151704) and EDI-IPN for the support given for thiswork

References

[1] J Paulos Beyond Numeracy Vintage Series Vintage Books1992

[2] T Toffoli ldquoCellular automata as an alternative to (rather than anapproximation of) differential equations in modeling physicsrdquoPhysica D Nonlinear Phenomena vol 10 no 1-2 pp 117ndash1271984

[3] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[4] S Wolfram A New Kind of Science Wolfram Media Cham-paign Ill USA 2002

[5] S Wolfram ldquoCellular automata as models of complexityrdquoNature vol 311 no 5985 pp 419ndash424 1984

[6] B Chopard and M Droz Cellular Automata Modeling ofPhysical Systems Cambridge University Press New York NYUSA 1998

[7] S Kawamura T Yoshida H Minamoto and Z Hossain ldquoSim-ulation of the nonlinear vibration of a string using the CellularAutomata based on the reflection rulerdquo Applied Acoustics vol67 no 2 pp 93ndash105 2006

[8] M V Walstijn and E Mullan ldquoTime-domain simulation ofrectangular membrane vibrations with 1-d digital waveguidesrdquo

in Proceedings of the 2011 Forum Acusticum pp 449ndash454Aalborg Denmark 2011

[9] J Kari ldquoTheory of cellular automata a surveyrdquo TheoreticalComputer Science vol 334 no 1ndash3 pp 3ndash33 2005

[10] S H White A M del Rey and G R Sanchez ldquoModelingepidemics using cellular automatardquo Applied Mathematics andComputation vol 186 no 1 pp 193ndash202 2007

[11] A Ilachinski Cellular Automata A Discrete Universe WorldScientific 2001

[12] M Kutrib R Vollmar and T Worsch ldquoIntroduction to thespecial issue on cellular automatardquo Parallel Computing vol 23no 11 pp 1567ndash1576 1997

[13] I Huerta-Trujillo J Chimal-Eguıa N Sanchez-Salas and JMartınez-Nuno ldquoModelo de automata celular 1-dimensionalpara una EDP hiperbolicardquo Research in Computing Science vol58 pp 407ndash423 2012

[14] W Glabisz ldquoCellular automata in nonlinear string vibrationrdquoArchives of Civil and Mechanical Engineering vol 10 no 1 pp27ndash41 2010

[15] W Glabisz ldquoCellular automata in nonlinear vibration problemsof two-parameter elastic foundationrdquo Archives of Civil andMechanical Engineering vol 11 no 2 pp 285ndash299 2011

[16] I Huerta-Trujillo E Castillo-Montiel J Chimal-Eguıa NSanchez-Salas and JMartınez-Nuno ldquoAC 2-dimensional comomodelo de una membrana vibranterdquo Research in ComputingScience vol 83 pp 117ndash130 2014

[17] L Kinsler Fundamentals of Acoustics Wiley 2000[18] S Kawamura M Shirashige and T Iwatsubo ldquoSimulation of

the nonlinear vibration of a string using the cellular automationmethodrdquo Applied Acoustics vol 66 no 1 pp 77ndash87 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of