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

I Huerta-Trujillo,J A Martinez-Nuño,Laura Muñoz-Salazar ,Juan Carlos Chimal-Eguía

doi:10.1155/2015/542507

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.

Highlights

The infinitesimal calculus and its descendants [1] have been one of the dominant branches in mathematics since it was developed by Newton and Leibniz
The 2-dimensional cellular automata (CA) model to a vibrant membrane system shown in this paper was derived from a 1-dimensional CA model that simulates a vibrant string [13]
The CA membrane model is released from the initial conditions so it is not necessary to redefine it; it is possible to define the initial conditions and simulate the system behavior immediately

Summary

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. 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. The two-dimensional wave equation is an important one since it represents the hyperbolic partial differential equations It has many analytical solutions, if the initial or boundary conditions are changed, it cannot be solved. This simulates a proposed frequency spectrum observing similarity solutions.

Discrete Membrane Model

Simulation and Results

Conclusions

Conflict of Interests