Discrete Dynamics in Nature and Society

Volume 2015, Article ID 542507, 8 pages

http://dx.doi.org/10.1155/2015/542507

## Simulation of a Vibrant Membrane Using a 2-Dimensional Cellular Automaton

^{1}Modeling and Simulation Laboratory, Centro de Investigación en Computación, Instituto Politécnico Nacional, Avenida Juan de Dios Bátiz, Esquina Miguel Othón de Mendizábal, Colonia Nueva Industrial Vallejo, 07738 Gustavo A. Madero, DF, Mexico^{2}Postgraduate and Research Section, Escuela Superior de Cómputo, Instituto Politécnico Nacional, Avenida Juan de Dios Bátiz, Esquina Miguel Othón de Mendizábal, Colonia Lindavista, 07738 Gustavo A. Madero, DF, Mexico^{3}Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas Norte 152, 07730 Gustavo A. Madero, DF, Mexico

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

Academic Editor: Tetsuji Tokihiro

Copyright © 2015 I. Huerta-Trujillo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 hyperbolic partial differential equation (PDE), linear and homogeneous. This models a vibrant membrane with specific conditions, initial and boundary. The frequency spectrum is analysed as well as the error between the data produced by the CA model. Then it is compared to the data provided by the solution evaluation to the differential equation. This shows how the CA obtains a behavior similar to the PDE. Moreover, it is possible to simulate nonclassical initial conditions for which there is no exact solution using PDE. Very interesting information could be obtained from the CA model such as the fundamental frequency.

#### 1. Introduction

The infinitesimal calculus and its descendants [1] have been one of the dominant branches in mathematics since it was developed by Newton and Leibniz. Differential equations are the main core of calculus and have been the cornerstone for understanding sciences, particularly physics. It is often necessary to consider more than one variable since it is required to change this in function to another one or the time. Therefore, to model physical systems, the differential equations have had great success due to more than three centuries of experience with methods to give the symbolic solutions. Even so, few partial differential equations have an exact solution [2].

Moreover, the current necessity to experiment with physical systems in order to recognize their behavior make necessary to develop models that simulate the systems and become less complex allowing manipulation and approximation as close as possible to reality. In this order of ideas, the discrete techniques have had more success when they have been implemented for simulation purposes. An example of this is the cellular automata (CA). Wolfram [3, 4] defined the CA as a mathematical idealization of physical systems whose time and space are discrete and in which the physical quantities can be grouped in a finite set of values. The CA are appropriate in physical systems with a highly nonlinear regime such as chemical or biological systems, where there are discrete thresholds [5].

For example, the two-dimensional wave equation is an important one since it represents the hyperbolic partial differential equations. Although it has many analytical solutions, if the initial or boundary conditions are changed, it cannot be solved. There are attempts to simulate the wave equation behavior using a CA as presented by Chopard and Droz [6] and Kawamura et al. [7]; they have only been performed in a one-dimensional automaton. This due to the complexity of applying a suitable rule for the CA evolution.

The evolution rule for CA is a fundamental problem that has no analytical solution [2]. In this paper a methodology for clarifying the CA evolution rule that simulates the behavior of a vibrating membrane is compared with the analytical solution of hyperbolic PDE. This simulates a proposed frequency spectrum observing similarity solutions. This shows that the discrete modeling can be an alternative for systems in which the boundary conditions or other circumstances make the PDE intractable. Section 1 offers a brief introduction; Section 2 presents the two-dimensional continuous wave equation model as well as the solution to the equation given the initial and boundary conditions. Section 3 shows the methodology used to make the problem discrete and obtain the CA evolution rule and the discrete model definition. Section 4 analyses and discusses the results of the continuous model against the proposed CA discrete model. Finally, Section 5 gives the conclusions.

#### 2. Vibrating Membrane, Classical Model

The motion equation for a membrane is based on the assumption that it is thin and uniform with negligible stiffness that is perfectly elastic and without spring vibrating with small amplitude movements. The equation governing the transverse membrane vibrations is given bywhere is the membrane deflection and , where is the membrane mass density [8] and is membrane tension per unit length. Equation (1) is called* wave equation in two dimensions*.

Considered as a square membrane (see Figure 1), the boundary conditions are defined as follows: